Strip old spark-lesson course content, keep expert agent only

Remove exercises/, spark-lessons/ (PyQt app, lessons, course structure),
run.bat, and spark-lesson.txt. Delete _archive/ from disk.

Add expert system: context/ (17 topic files), reference/ (glossary,
sources, cheat sheet), examples/, phases/, tools/, and assets/.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

.claude/settings.local.json

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-      "Bash(timeout 10 python:*)",
-      "Bash(timeout:*)",
-      "Bash(git init:*)",
-      "Bash(git remote add:*)",
-      "Bash(git add:*)"
+      "Bash",
+      "Edit",
+      "Read",
+      "Write",
+      "Glob",
+      "Grep",
+      "WebFetch",
+      "WebSearch",
+      "Skill(constraint-lookup)",
+      "Skill(phase-analysis)"
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context/branching-physics.md

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+---
+id: branching-physics
+title: "Branching Physics and Multi-Channel Dynamics"
+status: provisional
+source_sections: "spark-physics.txt: Part 5.5, Part 12 (lines 360-396, 1007, 1015)"
+related_topics: [streamers-and-leaders, thermal-physics, power-optimization, energy-and-growth, capacitive-divider, qcw-operation, distributed-model, equations-and-bounds, open-questions]
+key_equations: [nonlinear-resistance-power-law, conductance-relaxation]
+key_terms: [branching, streamer, leader, fractal_dimension, Laplacian_instability, thermal_ratcheting, nonlinear_resistance, negative_differential_resistance]
+images: []
+examples: []
+open_questions:
+  - "What is the quantitative current division rule at branch points in TC sparks?"
+  - "How much does branching increase effective C_sh beyond single-channel estimates?"
+  - "What fraction of total epsilon is attributable to branching losses?"
+  - "Can branch competition dynamics be measured with time-resolved imaging on a TC?"
+  - "Does the fractal dimension of TC sparks correlate with operating mode or frequency?"
+---
+
+# Branching Physics and Multi-Channel Dynamics
+
+Real Tesla coil sparks branch extensively — especially in burst mode. The current framework models a single unbranched channel, which is adequate for main-channel length prediction but cannot explain total luminous volume, power budget overhead, or the morphological differences between operating modes. This topic develops the physics of branching from established discharge science and connects it to the TC-specific observations documented in [[qcw-operation]] and [[thermal-physics]].
+
+## 1. Why Discharges Branch: Laplacian Instability
+
+### 1.1 The Mechanism
+
+A conducting channel propagating in a background electric field concentrates the field at its tip (see [[field-thresholds]], tip enhancement factor kappa = 2-5). If the tip is slightly perturbed — widened or displaced — the concentrated field splits into two lobes, each of which can independently ionize the gas ahead of it. This is a **Laplacian instability** [T0], mathematically identical to the Saffman-Taylor instability in viscous fingering.
+
+The instability is intrinsic to any conducting object growing in a Laplacian field. What determines *whether* and *when* a particular discharge branches is the source and amplitude of the initial perturbation.
+
+### 1.2 Perturbation Sources
+
+For streamers in air at atmospheric pressure, the dominant perturbation sources are:
+
+- **Electron density fluctuations** (stochastic particle noise) [T1]: At the ionization front, the electron density is finite (~10^13 cm^-3), meaning the number of electrons in a volume element at the streamer tip is not infinite. Poisson fluctuations create density variations that seed the instability. Luque & Ebert (2011) showed that this intrinsic noise is sufficient to trigger branching of positive streamers in air at atmospheric pressure, and that without noise, branching occurs much later if at all. [Luque & Ebert 2011, Phys Rev E 84, 046411]
+
+- **Photoionization seed electrons** [T1]: For positive streamers, propagation requires photoionization to create seed electrons ahead of the tip. The stochastic spatial distribution of these seed electrons creates field non-uniformities that can trigger branching. More photoionization means a denser, more uniform seed electron cloud, which actually *reduces* branching. This is why positive streamers in pure nitrogen (no O2 to provide photoionization) branch differently than in air.
+
+- **Prior discharge remnants** [T3]: In repetitive operation (TC burst mode), residual ionization and thermal channels from previous pulses create non-uniform starting conditions for each new pulse.
+
+### 1.3 Characteristic Branching Length
+
+A streamer branches after propagating approximately **10-20 diameters** from its last branch point [T1]. This ratio of branching length to streamer diameter agrees within a factor of 2 with experimental measurements. [Luque & Ebert 2011]
+
+| Channel type | Diameter | Branching distance | Branching frequency | Tier |
+|---|---|---|---|---|
+| Thin streamer | 10-50 um | 0.1-1 mm | Very frequent | T1 |
+| Thick streamer | 50-100 um | 0.5-2 mm | Frequent | T1 |
+| Early leader | 0.1-1 mm | 1-20 mm | Occasional | T3 |
+| Mature leader | 1-10 mm | 10-200 mm | Rare | T3 |
+
+The geometric branching distance alone predicts that streamers are highly branched and leaders are relatively straight — consistent with observation [T3]. But geometry is only half the story. Thermal feedback (Section 3) further suppresses branching in leaders.
+
+### 1.4 Branching Angles
+
+Experimental measurements of streamer branching in air show:
+
+- **Average branching angle: ~90 degrees** (full angle between daughter branches) in air at atmospheric pressure [T1]
+- The angle is normally distributed around this mean [T1]
+- In low-O2 mixtures (less photoionization), the average decreases to ~66 degrees [T1]
+- Branching is predominantly **binary** (two daughters); triple branching is rare and has been specifically documented as noteworthy [T1] [Nijdam et al., J Phys D]
+- Daughter branches are typically similar in diameter immediately after splitting, but quickly diverge due to competition (Section 3) [T3]
+
+### 1.5 Fractal Dimension
+
+The spatial structure of branched discharge trees has a measured fractal dimension:
+
+**D = 2.16 +/- 0.05** (needle-plane corona discharge) [T1] [Plasma Physics Reports, 2002]
+
+For reference:
+- D = 1.0: a straight line (no branching)
+- D = 2.0: a structure that fills a plane
+- D = 3.0: a structure that fills a volume
+
+D ~ 2.2 means the discharge tree is moderately space-filling — a 3D branched structure that is slightly denser than a flat tree. This is consistent with the bushy appearance of burst-mode TC sparks. QCW sword sparks, being nearly unbranched, would have D approaching 1.0 [T4 — no measurement exists].
+
+The fractal dimension connects to total surface area and hence to C_sh [T3]: a more branched tree (higher D) has more total conductor surface area per unit of main-channel length, increasing its capacitive coupling to ground.
+
+## 2. Streamer vs Leader Branching
+
+The two discharge types branch for fundamentally different reasons and with very different consequences.
+
+### 2.1 Streamers: Branching Is the Default
+
+Streamers are cold, transient channels. They have:
+- No significant thermal inertia (tau_thermal ~ 1-100 us for d = 10-100 um)
+- No mechanism to preferentially sustain one path over another
+- Fast propagation (~10^6 m/s) that outpaces any thermal feedback
+
+As a result, streamers branch readily at the rate predicted by the Laplacian instability (~every 10-20 diameters). Each branch propagates independently. The result is a highly branched tree of thin filaments — the characteristic purple/blue corona visible on burst-mode TC sparks.
+
+**Energy consequence** [T3]: Each branch channel absorbs energy (ionization, heating, radiation) but contributes little to forward propagation. This is a major contributor to the high epsilon values observed in burst mode (30-100+ J/m) — most of the energy goes into side branches that don't extend the main channel. See [[energy-and-growth]].
+
+### 2.2 Leaders: Thermal Feedback Suppresses Branching
+
+Leaders are hot (>5000 K), thermally self-sustaining channels (see [[streamers-and-leaders]]). They have:
+- Large thermal inertia (tau_thermal ~ 0.3-1+ seconds for d = 1-10 mm)
+- Nonlinear V-I characteristics that create competitive dynamics (Section 3)
+- Slower propagation (~10^3 m/s net growth) allowing thermal feedback to operate
+
+The combination of large diameter (branching every 10-200 mm geometrically) and thermal competition (Section 3) means leaders branch far less frequently than streamers [T3]. The few branches that do form quickly lose the competition for current and extinguish, leaving a relatively straight main channel — the characteristic white/yellow sword of QCW operation.
+
+## 3. Branch Competition: Nonlinear Resistance and Current Hogging
+
+This section describes the central physical mechanism that determines whether a branched discharge consolidates into a single channel or remains multi-branched. It follows directly from the da Silva nonlinear resistance law documented in [[streamers-and-leaders]] and [[equations-and-bounds]] Section 14.11.
+
+### 3.1 The Instability
+
+The equilibrium resistance per unit length of a discharge channel follows a power law in current:
+
+```
+R = A / I^b    (ohm/m)
+```
+
+[da Silva et al. 2019]
+
+For TC-relevant currents (Region I, 1-10 A): A = 12,400, **b = 1.84** [T1].
+
+The critical feature is that **b > 1**. This makes the V-I characteristic of the channel have **negative slope** (negative differential resistance) [T0]:
+
+```
+V = R * I = A * L * I^(1-b)
+```
+
+For b = 1.84: V proportional to I^(-0.84). Voltage drop *decreases* with increasing current.
+
+### 3.2 Why This Causes Current Hogging
+
+Consider two parallel branches of equal length at the same applied voltage V:
+
+```
+Branch 1: V = A * L * I_1^(1-b)
+Branch 2: V = A * L * I_2^(1-b)
+```
+
+The equal-sharing solution (I_1 = I_2) exists but is **unstable** when b > 1 [T0 — mathematical consequence of b > 1]:
+
+- If branch 1 receives slightly more current (I_1 = I_0 + delta), its voltage drop *decreases*
+- Since both branches are at the same voltage, branch 1 can now carry even more current
+- This is positive feedback — the perturbation grows
+- Branch 1 heats up, becomes more conductive, draws more current
+- Branch 2 cools, becomes more resistive, loses current
+- Eventually branch 1 carries nearly all the current and branch 2 extinguishes
+
+This is the same instability that causes parallel arcs to merge and arc attachment points to wander [T1]. It is well-established plasma physics, not a hypothesis.
+
+### 3.3 Competition Timescale
+
+The rate at which one branch "wins" is governed by the conductance relaxation time:
+
+```
+tau_g = 40 us (heating) / 200 us (cooling)
+```
+
+[Bazelyan & Raizer 2000; see [[thermal-physics]]]
+
+After ~3-5 heating time constants (~120-200 us), the competition is largely resolved — one branch dominates [T3 — timescale inferred from tau_g]. This timescale is critical for understanding TC operating modes:
+
+| Operating mode | Characteristic time | tau_competition / time | Branching outcome |
+|---|---|---|---|
+| Single burst pulse | 70-150 us | ~1 | Competition barely resolves; multiple branches coexist |
+| QCW ramp | 10-20 ms | ~50-100 | Competition fully resolves; single dominant channel |
+| Burst repetition gap | 5-10 ms (at 100-200 Hz) | N/A | Channels cool and decay between pulses |
+
+**This single mechanism explains the morphological difference between burst and QCW sparks** [T3]. Burst pulses are too short for the nonlinear competition to select a winner. QCW ramps are long enough for thermal feedback to consolidate current into one channel.
+
+### 3.4 Frequency Dependence
+
+At higher RF frequencies, the channel receives more heating cycles per unit time (at the same peak current). This accelerates the thermal ratchet that drives branch competition:
+
+- At 400 kHz: ~16 RF cycles per tau_g (40 us). Heating is effectively continuous. Competition resolves quickly.
+- At 100 kHz: ~4 RF cycles per tau_g. Heating is intermittent. Thin streamers may cool between cycles, resetting the competition.
+
+This is the physical basis for the 300-600 kHz frequency threshold for QCW sword sparks documented in [[qcw-operation]] [T3 — mechanism inferred; the frequency threshold itself is T2]. The frequency threshold is not about breakdown physics — it is about whether the thermal competition can resolve fast enough to suppress branching during the QCW ramp. See [[thermal-physics]] Section 7 for community observations.
+
+## 4. Branching Regimes in Tesla Coil Operation
+
+### 4.1 Burst Mode: Branching Dominates
+
+In burst mode (70-150 us ON time [T2], 200-600 kV topload [T2]):
+- Peak voltage exceeds leader formation threshold (~300-400 kV for single-shot) [T3]
+- But ON time is comparable to the competition timescale (~120-200 us) [T3]
+- Multiple streamer channels form simultaneously from the topload [T2]
+- Thermal competition begins but does not fully resolve before the pulse ends [T3]
+- Between pulses (5-10 ms gap), all channels cool and decay [T3]
+- Next pulse starts fresh — no accumulated thermal advantage for any channel [T3]
+
+Result: highly branched, bushy sparks [T2]. High epsilon (30-100+ J/m) because energy feeds many branches [T3 — epsilon values are T2, mechanism is T3].
+
+### 4.2 QCW: Competition Suppresses Branching
+
+In QCW mode (10-20 ms ramp [T2], 300-600 kHz [T2], 40-70 kV topload [T2]):
+- Voltage starts low and ramps over many milliseconds [T2]
+- At inception, a few streamer channels form [T3]
+- Thermal competition begins immediately (tau_competition ~ 120-200 us) [T3]
+- Within 0.5-2 ms, one channel dominates via the current-hogging instability [T3]
+- The winning channel transitions to a leader (>2000 K → 5000 K via thermal ratcheting) [T3]
+- For the remaining 10-18 ms, the leader grows as a single, straight channel [T2]
+- Side branches are continuously suppressed: any new branch that forms quickly loses the competition to the established hot channel [T3]
+
+Result: straight sword sparks [T2]. Low epsilon (5-15 J/m) because energy is concentrated in one channel [T3 — epsilon values are T2, mechanism is T3].
+
+### 4.3 "Too Long" QCW Ramp: Branching Returns
+
+When the QCW ramp exceeds ~25 ms [T2] (documented by Loneoceans, see [[qcw-operation]]):
+- The leader reaches its voltage-limited maximum length (set by the capacitive divider — see [[capacitive-divider]]) [T3]
+- Additional energy cannot extend the leader further (E_tip < E_propagation) [T3]
+- The leader channel becomes very hot and thick, increasing its C_sh [T3]
+- Excess power must dissipate somewhere [T0 — conservation of energy]
+- **Lateral breakouts** from the superheated leader trunk become the path of least resistance [T3]
+- These new branches compete with each other but not effectively with the main channel (which is already saturated) [T3]
+
+Result: "hot and fat but bushy" sparks [T2] — a thick leader trunk with side branches. The main channel doesn't get longer, just fatter and more branched.
+
+### 4.4 Pulse-Skip / Bresenham: Intermediate Behavior
+
+The user's observation [T2] (documented in [[qcw-operation]] Section 2.3) that Bresenham pulse-density modulation produces sparks that are "more sword-like but still branches" is exactly what the competition model predicts [T3]:
+
+- Bresenham PDM delivers a coarse approximation of a linear envelope [T2]
+- The heating is less smooth than true analog QCW [T3]
+- Per-cycle current jitter means the thermal advantage of the winning channel fluctuates [T3]
+- Competition still operates but with more noise, so side branches persist longer [T3]
+- Result: intermediate morphology on the continuum between burst (fully branched) and analog QCW (unbranched) [T3]
+
+## 5. Capacitive Loading of Branches
+
+Each branch segment has its own shunt capacitance C_sh to ground. The total C_sh of a branched tree exceeds that of a single channel of the same main-channel length.
+
+### 5.1 Physical Argument
+
+A single channel of length L at height h above ground has [T0]:
+```
+C_sh ~ 2*pi*epsilon_0*L / ln(2h/d)    (thin-wire approximation)
+```
+
+A branched tree with total conducting length L_total > L (main channel length) has additional C_sh from side branches [T0]. The branches are laterally displaced from the main channel, reducing mutual shielding between them, so the capacitance does not simply scale with total length — it depends on the spatial extent of the tree [T3].
+
+### 5.2 Qualitative Estimates
+
+- **QCW sword** (minimal branching): C_sh is close to the single-channel value [T3]. The empirical 2 pF/foot rule applies (or possibly overestimates, since it was likely calibrated against partially branched sparks).
+- **Burst mode** (heavy branching): C_sh may be 2-5x the single-channel value [T4 — no measurement], because the branched tree has much more total surface area exposed to ground.
+- **This is consistent** with Loneoceans' frequency tracking data [T2]: a 1.78 m QCW spark produced only 8.7% frequency shift, while a simulated solid wire of 1 m produced 24% shift. The QCW spark's low effective capacitance reflects both its plasma nature and its minimal branching [T3].
+
+### 5.3 Consequence for Voltage Division
+
+Higher C_sh from branching worsens the capacitive divider (see [[capacitive-divider]]):
+
+```
+V_tip = V_topload * C_mut / (C_mut + C_sh)
+```
+
+More branching → higher C_sh → lower V_tip → lower E_tip → harder to propagate → more stall and more branching. This is a **positive feedback loop** [T3] that drives burst-mode sparks toward heavily branched, voltage-limited configurations. It is the capacitive complement to the thermal competition mechanism.
+
+QCW breaks this feedback loop by suppressing branching early, keeping C_sh low, maintaining V_tip high, and enabling continued forward propagation [T3].
+
+## 6. Power Budget: Branching as Energy Overhead
+
+The connection between branching and epsilon is direct:
+
+### 6.1 Energy Accounting
+
+Total energy delivered to the spark distributes among:
+1. **Main channel forward growth** (useful work): ionization, heating to leader temperature
+2. **Side branch formation and heating** (overhead): each branch absorbs energy but doesn't extend the main channel
+3. **Radiation and convection losses** from all channels
+4. **Capacitive energy stored** in C_sh (including branch contributions)
+
+In burst mode, items 2-4 dominate [T3]. The ratio of useful work to total energy is low, explaining the high epsilon (30-100+ J/m).
+
+In QCW mode, branch suppression eliminates most of item 2 early in the ramp [T3]. Energy concentrates in the main channel, keeping epsilon low (5-15 J/m).
+
+### 6.2 Quantitative Estimate
+
+The efficiency ratio between QCW and burst can be roughly estimated from the spark:secondary ratios documented in [[qcw-operation]]:
+
+- Burst: spark:secondary = 2.5-3.6:1
+- QCW: spark:secondary = 7-16.5:1
+
+The QCW advantage is 3-5x [T2 — derived from community-measured ratios], which includes both branching reduction and the thermal efficiency gain from sustained leader operation [T3]. Separating these contributions requires measurements that do not yet exist (see Open Questions).
+
+## 7. What We Do Not Know
+
+### 7.1 Current Division Rule
+
+The existing framework proposes I_branch proportional to d_branch^1.5 (see [[open-questions]] Section 1.4), but this is unvalidated. The physics of Section 3 suggests that current division is better understood through the nonlinear V-I instability than through a static power-law in diameter:
+
+- At the moment of splitting, daughter branches are similar and share current roughly equally
+- The equal-sharing equilibrium is unstable (b > 1 in the da Silva law)
+- Within ~100-200 us, one branch dominates via current hogging
+- The "steady-state current division" is therefore not a useful concept — the system is transient and winner-take-all
+
+A static rule like I proportional to d^n misses this essential dynamics.
+
+### 7.2 Fractal Dimension vs Operating Mode
+
+No measurements exist of the fractal dimension of TC sparks as a function of frequency, power level, or operating mode. Such measurements (from high-resolution photographs) would directly test the competition model: D should decrease (approach 1.0) as frequency increases and as ramp duration increases.
+
+### 7.3 Branching Fraction of Epsilon
+
+What fraction of total epsilon is attributable to branching losses vs other overhead (radiation, convection, stored energy)? This requires either:
+- Time-resolved imaging correlated with electrical waveforms (not yet done on any TC)
+- Careful comparison of epsilon between highly branched and minimally branched sparks of the same length under controlled conditions
+
+### 7.4 Branch Initiation from Leader Trunk
+
+The "too long" QCW regime (Section 4.3) produces lateral breakouts from a superheated leader trunk. The physics of initiation from a hot, thick channel into cold air is different from streamer branching (which occurs at the propagating tip). This may involve thermal instabilities of the channel boundary rather than Laplacian field instabilities.
+
+## Key Relationships
+
+- Derives from: [[streamers-and-leaders]] (discharge types), [[thermal-physics]] (time constants), [[power-optimization]] (nonlinear R)
+- Explains: [[energy-and-growth]] (why epsilon differs by mode), [[qcw-operation]] (sword vs bushy morphology, frequency threshold), [[capacitive-divider]] (C_sh dependence on morphology)
+- Connects to: [[distributed-model]] (potential extension to branched networks), [[open-questions]] (Sections 1.4, 2.4)
+- Key data: da Silva R = A/I^b with b = 1.84 ([[equations-and-bounds]] Section 14.11), tau_g = 40/200 us ([[equations-and-bounds]] Section 14.19)
+
+## References
+
+- Luque & Ebert (2011), "Electron density fluctuations accelerate the branching of positive streamer discharges in air," Phys Rev E 84, 046411
+- da Silva et al. (2019), "The Plasma Nature of Lightning Channels and the Resulting Nonlinear Resistance," JGR Atmospheres
+- Bazelyan & Raizer (2000), "Lightning Physics and Lightning Protection," IOP Publishing
+- Nijdam et al. (2010), "Stereo-photography of streamers in air," J Phys D: Appl Phys
+- Phase 6 QCW Community Survey (2026)

context/capacitive-divider.md

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+---
+id: capacitive-divider
+title: "The Capacitive Divider Problem"
+status: established
+source_sections: "spark-physics.txt: Part 5 Section 5.6 (lines 338-361)"
+related_topics: [circuit-topology, field-thresholds, energy-and-growth, empirical-scaling, power-optimization, streamers-and-leaders, branching-physics, lumped-model, distributed-model, femm-workflow, equations-and-bounds]
+key_equations: [voltage-division-complex, voltage-division-open-circuit, capacitive-divider-impedances]
+key_terms: [capacitive_divider, V_tip, C_mut, C_sh, E_tip, E_propagation, R_opt_power]
+images: [voltage-division-vs-length-plot.png]
+examples: [spark-growth-timeline.md]
+open_questions:
+  - "Can active topload voltage ramping fully compensate for the divider effect, or is there a fundamental limit?"
+  - "How does the mutual capacitance C_mut change during growth -- is it truly constant?"
+  - "What is the quantitative effect of the finite R on voltage division compared to the open-circuit limit?"
+  - "How does branching affect the effective C_sh per unit length?"
+  - "Is there a geometry (topload shape, environment) that minimizes the divider attenuation for a given spark length?"
+---
+
+# The Capacitive Divider Problem
+
+The capacitive divider is THE fundamental limiting mechanism for Tesla coil spark length. As the spark grows, its shunt capacitance to ground increases, creating a voltage divider that progressively attenuates the voltage reaching the spark tip. This reduces the tip electric field, eventually dropping it below the propagation threshold and stalling growth. Understanding this mechanism is essential for predicting maximum spark length and explaining why length scales sub-linearly with energy.
+
+## Circuit Origin
+
+The basic spark circuit topology (see [[circuit-topology]]) places two capacitances in the current path from topload to ground:
+
+```
+Topload ----[C_mut || R]---- Spark Tip
+                                |
+                             [C_sh]
+                                |
+                              GND
+```
+
+Where:
+- `C_mut` is the mutual capacitance between topload and spark channel [F]
+- `R` is the plasma resistance of the spark channel [ohm]
+- `C_sh` is the shunt capacitance from spark channel to ground [F]
+
+This topology is inherently a voltage divider. The voltage at the spark tip is a fraction of the topload voltage, determined by the impedance ratio of the two branches.
+
+## Voltage Division Equations
+
+### General Case (Complex Impedance Division)
+
+The tip voltage is determined by complex impedance division:
+
+```
+V_tip = V_topload * Z_mut / (Z_mut + Z_sh)
+```
+
+Where the mutual branch impedance is the parallel combination of R and 1/(j*omega*C_mut):
+
+```
+Z_mut = (1/(j*omega*C_mut)) || R
+      = R / (1 + j*omega*C_mut*R)
+```
+
+And the shunt impedance is:
+
+```
+Z_sh = 1 / (j*omega*C_sh)
+```
+
+The full expression for the voltage division ratio:
+
+```
+V_tip / V_topload = Z_mut / (Z_mut + Z_sh)
+
+                  = [R / (1 + j*omega*C_mut*R)] / [R / (1 + j*omega*C_mut*R) + 1/(j*omega*C_sh)]
+```
+
+This is a complex ratio, meaning V_tip has both magnitude and phase shift relative to V_topload.
+
+### Open-Circuit Limit (R -> infinity)
+
+When R is very large (cold, high-resistance streamer or no plasma present), the R || C_mut parallel combination is dominated by C_mut alone:
+
+```
+V_tip = V_topload * C_mut / (C_mut + C_sh)
+```
+
+This is the classical capacitive voltage divider formula. It represents the maximum possible tip voltage for a given topload voltage and capacitance ratio. Any finite R only reduces V_tip further.
+
+### With Finite R at R_opt_power
+
+When the plasma has adjusted to R = R_opt_power (see [[power-optimization]]):
+
+```
+R_opt_power = 1 / (omega * (C_mut + C_sh))
+```
+
+The tip voltage is lower than the open-circuit limit and is complex (has a phase shift). The magnitude reduction depends on the specific values of C_mut, C_sh, and frequency, but is typically 10-30% below the open-circuit ratio.
+
+This means the open-circuit voltage division formula provides an upper bound on V_tip. The actual V_tip during active growth (when R is finite and near R_opt_power) is always worse.
+
+## The Growth Feedback Loop
+
+The devastating effect of the capacitive divider arises from the positive feedback between spark length and C_sh:
+
+### Step 1: Spark Grows
+
+As the spark extends to length L:
+```
+C_sh(L) = C_sh_per_meter * L
+```
+
+Where C_sh_per_meter is approximately 6.6 pF/m (equivalently ~2 pF per foot), an empirical value confirmed by FEMM simulations (see [[femm-workflow]]).
+
+### Step 2: C_sh Increases
+
+Longer spark means more conducting surface area exposed to ground. The capacitance to ground increases linearly with length.
+
+### Step 3: V_tip Decreases
+
+With C_sh increasing:
+```
+V_tip = V_topload * C_mut / (C_mut + C_sh(L))
+```
+
+As C_sh grows, the denominator increases, and V_tip decreases. Even if V_topload is maintained perfectly constant, the fraction of that voltage reaching the tip drops.
+
+### Step 4: E_tip Decreases
+
+The average electric field at the tip:
+```
+E_avg = V_tip / L
+```
+
+This decreases both because V_tip is decreasing (numerator) and L is increasing (denominator). Including the tip enhancement factor:
+```
+E_tip = kappa * V_tip / L
+```
+
+The field drops even faster than either effect alone.
+
+### Step 5: Growth Slows and Eventually Stalls
+
+When E_tip drops below E_propagation (see [[field-thresholds]]):
+```
+dL/dt = 0    (stalled, regardless of available power)
+```
+
+The spark has reached its voltage-limited maximum length.
+
+### The Vicious Cycle
+
+The feedback loop is:
+
+```
+Longer spark -> More C_sh -> Lower V_tip -> Lower E_tip -> Slower growth
+                                                                |
+                    (Eventually: E_tip < E_propagation -> STALL) |
+```
+
+This is a negative feedback on growth that becomes progressively stronger as the spark extends. It guarantees that growth is sub-linear with energy input.
+
+## Maximum Voltage-Limited Length
+
+Setting E_tip equal to E_propagation and solving for the maximum length:
+
+```
+kappa * V_topload * C_mut / [(C_mut + C_sh_per_meter * L_max) * L_max] = E_propagation
+```
+
+This is a quadratic equation in L_max:
+
+```
+E_propagation * C_sh_per_meter * L_max^2 + E_propagation * C_mut * L_max - kappa * V_topload * C_mut = 0
+```
+
+Using the quadratic formula:
+
+```
+L_max = [-E_propagation * C_mut + sqrt((E_propagation * C_mut)^2 + 4 * E_propagation * C_sh_per_meter * kappa * V_topload * C_mut)] / (2 * E_propagation * C_sh_per_meter)
+```
+
+### Numerical Example
+
+Using values from the worked example (`spark-growth-timeline.md`):
+
+```
+V_topload = 420 kV (peak)
+C_mut = 9 pF
+C_sh_per_meter = 6.6 pF/m
+kappa = 3
+E_propagation = 0.7 MV/m
+
+Substituting:
+4.62 * L^2 + 6.3 * L - 11.34 = 0
+
+L_max = [-6.3 + sqrt(39.69 + 209.69)] / 9.24
+      = [-6.3 + 15.79] / 9.24
+      = 1.03 m
+```
+
+The coil reaches only 1.0 m despite having 200 kW of available power at peak voltage. This is the voltage limit in action. The target of 2.0 m is unachievable with these parameters -- not because of insufficient power, but because of insufficient voltage relative to the capacitive divider.
+
+## Sub-Linear Scaling
+
+The capacitive divider creates characteristic sub-linear scaling relationships between energy/voltage and spark length:
+
+### Length vs Voltage
+
+From the quadratic solution, for large L_max where C_sh >> C_mut:
+
+```
+L_max ~ sqrt(kappa * V_topload * C_mut / (E_propagation * C_sh_per_meter))
+      ~ sqrt(V_topload)
+```
+
+Doubling the topload voltage increases maximum length by only sqrt(2) = 1.41x. This is a fundamental consequence of the C_sh proportional to L relationship.
+
+### Length vs Energy
+
+For burst mode (voltage-limited, single shot):
+```
+P ~ V_topload^2 / Z_spark
+Z_spark ~ 1/(omega * C_sh) ~ 1/(omega * C_sh_per_meter * L)
+
+Therefore: P ~ V_topload^2 * omega * C_sh_per_meter * L
+
+And since L ~ sqrt(V_topload):
+L^2 ~ V_topload ~ sqrt(P)
+L ~ P^(1/4) ... approximately
+```
+
+The actual scaling is closer to L proportional to sqrt(E_bang) for single-shot bursts, as observed by Freau (see [[empirical-scaling]]). The exact exponent depends on the relative magnitudes of C_mut and C_sh and the operating regime.
+
+### QCW Scaling
+
+QCW shows better (but still sub-linear) scaling, approximately L proportional to E^(0.6-0.8), because:
+- Active voltage ramping partially compensates for the divider
+- Leader formation reduces effective R, improving voltage delivery to tip
+- Thermal accumulation reduces epsilon over the ramp
+- But the fundamental divider effect still prevents linear scaling
+
+## Voltage Ramping as Partial Mitigation
+
+QCW mode uses a linearly ramping voltage:
+```
+V_topload(t) = V_max * (t / T_ramp)
+```
+
+This partially counteracts the capacitive divider:
+- As L increases, C_sh increases, attenuating V_tip
+- But V_topload is simultaneously increasing, partially compensating
+- Net effect: V_tip decreases more slowly than for constant V_topload
+- Growth persists longer before E_tip drops below threshold
+
+However, the compensation is not complete. The rate of C_sh increase (proportional to dL/dt, which itself depends on P) generally outpaces the linear voltage ramp, especially as the spark gets long. The divider wins eventually.
+
+### Optimal Ramp Profile
+
+The linear ramp is not necessarily optimal. An accelerating ramp (voltage increasing faster than linearly) could better compensate for the divider. The optimal ramp profile V_topload(t) that maximizes final length for a given V_max and T_ramp is an open optimization problem. In practice, the linear ramp is sufficient and hardware-simple.
+
+## Effect of Finite R on Voltage Division
+
+The open-circuit formula V_tip = V_topload * C_mut / (C_mut + C_sh) overestimates the tip voltage. With finite R:
+
+1. Current flows through R, dissipating power (this is useful power for growth)
+2. The voltage drop across R reduces V_tip compared to the open-circuit case
+3. The impedance Z_mut = R || (1/(j*omega*C_mut)) has lower magnitude than 1/(j*omega*C_mut) alone
+4. V_tip magnitude decreases and acquires a phase shift
+
+The quantitative effect depends on the ratio omega*C_mut*R:
+- When omega*C_mut*R >> 1 (R large, streamer-like): approaches open-circuit limit
+- When omega*C_mut*R ~ 1 (R near R_opt_power): V_tip reduced by ~20-30%
+- When omega*C_mut*R << 1 (R very small, hot leader): V_tip severely attenuated, but this regime is unusual
+
+For practical calculations, the open-circuit formula provides a useful upper bound. For precise predictions, the full complex voltage division should be used.
+
+## Interaction with Other Limiting Mechanisms
+
+The capacitive divider does not act in isolation. It interacts with:
+
+### Field Threshold ([[field-thresholds]])
+
+The divider reduces V_tip, which reduces E_tip. When E_tip falls below E_propagation, growth stalls. The field threshold provides the hard stop; the divider provides the mechanism that drives E_tip down to that stop.
+
+### Power Delivery
+
+As C_sh grows, R_opt_power = 1/(omega*(C_mut + C_sh)) decreases. This means:
+- Lower R -> higher current -> potentially more power
+- But the spark impedance also changes, affecting the Thevenin power delivery (see [[thevenin-method]])
+- Net effect: power delivered to the spark may increase even as V_tip decreases
+- This explains why extra power goes into heating/brightening rather than lengthening during stall
+
+### Thermal Physics ([[thermal-physics]])
+
+Leader formation (promoted by QCW) reduces R, which:
+- Increases current through the channel
+- Reduces V_tip (worse for voltage division)
+- But also increases power to the spark (better for energy delivery)
+- Net effect is complex; leader formation generally helps overall growth despite worse voltage division
+
+## Design Implications
+
+### To Maximize Spark Length
+
+1. **Maximize V_topload**: Most direct way to fight the divider. Higher voltage pushes L_max up as sqrt(V_topload).
+
+2. **Minimize C_sh_per_meter**: Depends on geometry and environment. Operating away from grounded surfaces helps. In practice, ~6.6 pF/m is hard to reduce significantly.
+
+3. **Maximize C_mut**: Higher C_mut improves the voltage division ratio C_mut/(C_mut+C_sh). Larger topload helps.
+
+4. **Use QCW with voltage ramping**: Partial compensation for divider effect during growth.
+
+5. **Maximize tip enhancement (kappa)**: Sharper tip geometry concentrates the field, partially compensating for reduced V_tip. But kappa is limited by geometry to ~2-5.
+
+6. **Lower E_propagation**: Operating at altitude (lower pressure) reduces E_propagation, allowing longer growth before stall. Humidity also affects this.
+
+### Fundamental Limits
+
+Even with all optimizations, the capacitive divider guarantees:
+- Sub-linear scaling of L with V, E, and P
+- An absolute maximum length determined by V_topload, C_mut, C_sh_per_meter, and E_propagation
+- Diminishing returns on additional power once the voltage limit is reached
+- A regime where extra power produces only heating and brightness, not length
+
+This is why the largest Tesla coil sparks require extremely high voltages (hundreds of kV to MV), not just high power. Power is necessary but not sufficient; voltage is the binding constraint.
+
+## Key Relationships
+
+- Derives from: [[circuit-topology]] (the C_mut || R in series with C_sh topology creates the divider)
+- Interacts with: [[field-thresholds]] (divider reduces E_tip toward E_propagation threshold)
+- Constrains: [[energy-and-growth]] (growth stalls when voltage-limited regardless of available power/energy)
+- Explains: [[empirical-scaling]] (sub-linear L vs E and L vs V scaling laws)
+- Motivates: QCW voltage ramping strategy (partially compensates divider during growth)
+- Quantified by: [[femm-workflow]] (FEMM provides C_mut and C_sh values for specific geometries)
+- Modeled in: [[lumped-model]] (single-element voltage division)
+- Modeled in: [[distributed-model]] (segment-by-segment voltage attenuation along spark)

context/circuit-topology.md

@@ -0,0 +1,242 @@
+---
+id: circuit-topology
+title: "Fundamental Circuit Topology and Phase Constraints"
+status: established
+source_sections: "spark-physics.txt: Part 1 (lines 11-72), Part 11 (lines 736-751)"
+related_topics: [power-optimization, thevenin-method, coupled-resonance, capacitive-divider, lumped-model, distributed-model, femm-workflow, equations-and-bounds]
+key_equations:
+  - "Input admittance Y"
+  - "Re{Y} and Im{Y} decomposition"
+  - "Impedance phase angle phi_Z"
+  - "Fundamental phase constraint phi_Z_min"
+  - "Capacitance ratio r"
+key_terms:
+  - "mutual capacitance"
+  - "shunt capacitance"
+  - "admittance"
+  - "impedance phase angle"
+  - "topological constraint"
+  - "phasor"
+  - "susceptance"
+  - "conductance"
+images:
+  - complex-plane-admittance.png
+  - phase-angle-visualization.png
+  - phase-constraint-graph.png
+  - admittance-vector-addition.png
+examples: []
+open_questions:
+  - "How does the phase constraint shift if C_mut becomes frequency-dependent at very high frequencies?"
+  - "What is the exact crossover geometry (topload size vs. spark length) where r = 0.207?"
+  - "How do proximity effects from nearby grounded objects alter the effective C_sh and thus r?"
+---
+
+# Fundamental Circuit Topology and Phase Constraints
+
+This document establishes the foundational circuit model for Tesla coil sparks. Every subsequent analysis in the knowledge graph -- power optimization, Thevenin extraction, lumped and distributed modeling -- builds on the topology and admittance relationships derived here. The central result is a topological phase constraint that limits the impedance angle the spark can present to the resonant circuit, independent of plasma physics.
+
+## 1. The Basic Spark Circuit Model
+
+### 1.1 Physical Origin of the Two Capacitances
+
+FEMM electrostatic analysis of a Tesla coil with an extended spark channel reveals two distinct capacitances at the topload connection point:
+
+- **Mutual capacitance (C_mut):** The capacitive coupling between the spark channel and the topload. This is the path through which displacement current flows from the topload into the spark plasma. C_mut depends on the topload geometry, spark channel shape, and their relative orientation. For a typical toroidal topload with a spark emerging from its edge, C_mut ranges from roughly 3 to 15 pF depending on topload size and spark length.
+
+- **Shunt capacitance (C_sh):** The capacitance from the spark channel to ground (and to all other grounded or low-potential objects in the environment). Empirically, C_sh scales approximately linearly with spark length at ~2 pF per foot (~6.6 pF per meter). This scaling holds because longer sparks present more conductor length to the surrounding environment.
+
+### 1.2 Circuit Topology
+
+The two capacitances, together with the spark channel resistance R, form the following topology at the topload node:
+
+```
+Topload ---[C_mut || R]--- Spark tip
+   |                          |
+   |                       [C_sh]
+   |                          |
+  GND ---------------------- GND
+```
+
+Reading this circuit:
+- C_mut and R are in parallel between the topload node and the spark tip node. The parallel combination represents the fact that current can flow from topload to spark either through the capacitive coupling (displacement current through C_mut) or through the resistive plasma channel (conduction current through R).
+- C_sh connects the spark tip to ground, representing the distributed capacitance of the spark channel to its environment.
+- The topload itself connects to ground through the Tesla coil secondary (not shown here; that is the source impedance).
+
+This is NOT a simple series or parallel RLC. The topology is a bridged-T or pi-network, and this specific arrangement is what creates the phase constraint discussed below.
+
+### 1.3 Phasor Convention
+
+**All phasor quantities in this framework use peak values, not RMS.** Power formulas therefore include the factor of 0.5:
+
+```
+P = 0.5 * Re{V * I*}
+```
+
+where I* denotes the complex conjugate of I. This convention is consistent throughout all topics in the knowledge graph.
+
+## 2. Admittance Analysis
+
+### 2.1 Definitions
+
+At angular frequency omega = 2*pi*f, define:
+
+- **G = 1/R** : conductance of the spark channel [siemens]
+- **B_1 = omega * C_mut** : susceptance due to mutual capacitance [siemens] (positive, capacitive)
+- **B_2 = omega * C_sh** : susceptance due to shunt capacitance [siemens] (positive, capacitive)
+
+Note that B_1 and B_2 are defined as positive quantities (the conventional "capacitive susceptance" magnitude). The imaginary part of the admittance of a capacitor C is +j*omega*C in the Y-domain.
+
+### 2.2 Input Admittance at Topload
+
+The admittance looking into the spark circuit from the topload node (with ground as the return) is computed by combining the parallel combination (G + jB_1) in series with jB_2:
+
+```
+Y = ((G + jB_1) * jB_2) / (G + j(B_1 + B_2))
+```
+
+**Derivation:** The impedance of the parallel (C_mut || R) branch is Z_parallel = 1/(G + jB_1). The impedance of C_sh is Z_sh = 1/(jB_2). The total impedance from topload to ground is Z_total = Z_parallel + Z_sh. The total admittance is Y = 1/Z_total. Inverting:
+
+```
+Y = 1 / [1/(G + jB_1) + 1/(jB_2)]
+  = (G + jB_1) * jB_2 / [(G + jB_1) + jB_2]
+  = ((G + jB_1) * jB_2) / (G + j(B_1 + B_2))
+```
+
+### 2.3 Real and Imaginary Parts
+
+Multiplying numerator and denominator by the conjugate of the denominator:
+
+**Real part (conductance component):**
+```
+Re{Y} = G * B_2^2 / (G^2 + (B_1 + B_2)^2)
+```
+
+**Imaginary part (susceptance component):**
+```
+Im{Y} = B_2 * [G^2 + B_1*(B_1 + B_2)] / (G^2 + (B_1 + B_2)^2)
+```
+
+**Verification of limiting cases:**
+
+- **R -> infinity (G -> 0):** Re{Y} -> 0, Im{Y} -> B_1*B_2/(B_1 + B_2). This is the series combination of two capacitances, as expected (no conduction, pure capacitive divider).
+
+- **R -> 0 (G -> infinity):** Re{Y} -> B_2^2/G -> 0 (approaches short at topload, all current bypasses C_sh). More carefully: Y -> jB_2, since the short across C_mut || R removes C_mut and leaves only C_sh.
+
+- **C_sh -> 0 (B_2 -> 0):** Y -> 0. No path to ground through the spark; the circuit is open.
+
+### 2.4 Admittance and Impedance Phase Angles
+
+The admittance phase angle is:
+
+```
+theta_Y = atan(Im{Y} / Re{Y})
+```
+
+The impedance phase angle, which is what is typically measured and discussed in Tesla coil literature, is the negative of the admittance phase:
+
+```
+phi_Z = -theta_Y = atan(-Im{Y} / Re{Y})
+```
+
+**Sign convention:** A purely capacitive load has phi_Z = -90 degrees. A purely resistive load has phi_Z = 0 degrees. The spark load always has phi_Z between -90 degrees and 0 degrees (capacitive side), because the circuit contains only capacitors and a resistor (no inductance).
+
+**Important:** When Tesla coil builders discuss "matching to -45 degrees" or "the impedance angle," they are referring to phi_Z, not theta_Y.
+
+![Admittance in the complex plane](../assets/complex-plane-admittance.png)
+
+![Phase angle visualization for different R values](../assets/phase-angle-visualization.png)
+
+![Vector addition of admittance components](../assets/admittance-vector-addition.png)
+
+## 3. The Fundamental Phase Constraint
+
+### 3.1 Derivation
+
+The impedance phase angle phi_Z depends on R (equivalently, on G = 1/R). As R varies from 0 to infinity, phi_Z traces a curve. There exists a minimum achievable impedance phase angle (maximum negative value) that depends only on the ratio of capacitances:
+
+```
+phi_Z_min = -atan(2 * sqrt(r * (1 + r)))
+
+where r = C_mut / C_sh
+```
+
+**Derivation sketch:** Setting d(phi_Z)/dG = 0, the condition for extremum yields G_opt = omega * sqrt(C_mut * (C_mut + C_sh)), which corresponds to R_opt_phase = 1/(omega * sqrt(C_mut * (C_mut + C_sh))). Substituting back gives the minimum phase expression above.
+
+### 3.2 The -45 Degree Impossibility
+
+Setting phi_Z_min = -45 degrees and solving:
+
+```
+atan(2 * sqrt(r * (1 + r))) = 45 degrees
+2 * sqrt(r * (1 + r)) = 1
+4 * r * (1 + r) = 1
+4r^2 + 4r - 1 = 0
+r = (-4 + sqrt(16 + 16)) / 8 = (-4 + 4*sqrt(2)) / 8 = (sqrt(2) - 1) / 2 ~ 0.207
+```
+
+**Critical insight:** When r >= 0.207, achieving phi_Z = -45 degrees is mathematically impossible, regardless of the value of R. This is a topological constraint imposed by the circuit structure, not a limitation of plasma physics or any material property.
+
+### 3.3 Practical Implications
+
+For typical Tesla coil geometries:
+
+| Topload / Spark Configuration | Approximate r = C_mut/C_sh | phi_Z_min |
+|-------------------------------|---------------------------|-----------|
+| Large topload, short spark    | 1.0 - 2.0                | -55 to -70 deg |
+| Medium topload, medium spark  | 0.5 - 1.0                | -50 to -55 deg |
+| Small topload, long spark     | 0.2 - 0.5                | -45 to -50 deg |
+
+Since most practical configurations have r > 0.207, the -45 degree "matched" condition is almost never achievable. This explains why real sparks typically present impedance angles in the -55 to -75 degree range.
+
+![Phase constraint as a function of capacitance ratio](../assets/phase-constraint-graph.png)
+
+### 3.4 The "R approximately equals |X_c|" Myth
+
+Tesla coil literature often states that spark resistance approximately equals the magnitude of the capacitive reactance: R ~ |X_c|. This relationship does emerge approximately from the power optimization (see [[power-optimization]]), but it does NOT imply that -45 degrees is achievable. The approximate equality arises because R_opt_power = 1/(omega * C_total) ~ 1/(omega * C_sh) when C_mut and C_sh are comparable, and 1/(omega * C) is the reactance magnitude. The phase angle at R_opt_power, however, is typically -55 to -75 degrees, not -45 degrees.
+
+## 4. Effect of Secondary Losses
+
+### 4.1 Parasitic Conductance
+
+Real Tesla coil secondaries have losses: wire resistance, dielectric losses in the coil form, corona losses, and radiation. These appear as a parallel conductance G_sec on the source side (topload-to-ground), in addition to the spark circuit.
+
+### 4.2 Impact on Phase Constraint
+
+The additional parallel conductance G_sec increases the real part of the total admittance seen by the source but does NOT change the spark circuit's fundamental phase constraint. The spark still cannot present an impedance angle better than phi_Z_min. The secondary losses simply add a real (resistive) load in parallel with the spark's complex load. The total phase angle of the combined load will actually be closer to zero (more resistive), but this is because power is being wasted in the secondary, not because the spark is better matched.
+
+**Practical note:** When measuring the total Q of a loaded Tesla coil, the measured Q reflects both secondary losses and spark loading. Separating the two requires the [[thevenin-method]] or careful ringdown analysis.
+
+## 5. Frequency Dependence
+
+### 5.1 How Admittance Scales with Frequency
+
+Since B_1 = omega * C_mut and B_2 = omega * C_sh, both susceptances scale linearly with frequency. The admittance components Re{Y} and Im{Y} therefore have non-trivial frequency dependence. However, the phase constraint phi_Z_min depends only on the ratio r = C_mut/C_sh, which is frequency-independent (assuming frequency-independent capacitances). Thus:
+
+- The minimum achievable phase angle does not change with frequency.
+- The resistance value that achieves the minimum phase (R_opt_phase) does change with frequency (it is inversely proportional to omega).
+- The resistance value that maximizes power (R_opt_power) also changes with frequency.
+
+### 5.2 Relevance to Frequency Tracking
+
+When a spark loads the Tesla coil, the resonant frequency shifts (see [[coupled-resonance]]). As frequency changes, B_1 and B_2 change proportionally, which shifts R_opt_power and R_opt_phase. However, because r is fixed, phi_Z_min is unaffected. The spark must re-optimize its resistance to the new R_opt_power at the new operating frequency.
+
+## 6. Connection to Other Topics
+
+### Key Relationships
+
+- **Derives from:** FEMM electrostatic analysis (physical measurement of C_mut and C_sh)
+- **Enables:** [[power-optimization]] (R_opt_power and R_opt_phase are computed from the admittance expressions derived here)
+- **Enables:** [[thevenin-method]] (the spark circuit topology defines what Z_load looks like to the Thevenin equivalent)
+- **Enables:** [[lumped-model]] (the lumped model IS this circuit, with FEMM-extracted capacitance values)
+- **Constrains:** [[coupled-resonance]] (the phase constraint limits how "resistive" the spark can look, affecting power transfer)
+- **Extended by:** [[distributed-model]] (the distributed model generalizes this single-section topology to n sections)
+- **Extended by:** [[capacitive-divider]] (the voltage division at the spark tip is a direct consequence of this topology)
+
+### Summary of Key Results
+
+1. The spark circuit is a bridged-T network with C_mut || R in series with C_sh.
+2. The input admittance Y has closed-form real and imaginary parts in terms of G, B_1, B_2.
+3. The impedance phase angle phi_Z is bounded by phi_Z_min = -atan(2*sqrt(r*(1+r))).
+4. For r >= 0.207 (almost all practical configurations), -45 degrees is impossible.
+5. Secondary losses do not relax the phase constraint.
+6. The constraint is topological (circuit structure), not physical (plasma properties).

context/coupled-resonance.md

@@ -0,0 +1,361 @@
+---
+id: coupled-resonance
+title: "Coupled Resonance, Pole Splitting, and Frequency Tracking"
+status: established
+source_sections: "spark-physics.txt: Part 4 (lines 192-210), Part 9 (lines 666-700)"
+related_topics: [circuit-topology, power-optimization, thevenin-method, energy-and-growth, qcw-operation, lumped-model, distributed-model, equations-and-bounds]
+key_equations:
+  - "Pole frequencies (eigenfrequencies)"
+  - "C_sh increase with spark length"
+  - "Loaded pole shift"
+  - "Power at loaded pole vs fixed frequency"
+key_terms:
+  - "coupled resonant system"
+  - "eigenfrequency"
+  - "lower pole"
+  - "upper pole"
+  - "pole splitting"
+  - "frequency tracking"
+  - "PLL"
+  - "DRSSTC"
+  - "detuning"
+  - "loaded Q"
+  - "geometric mean"
+images:
+  - frequency-shift-with-loading.png
+  - drsstc-operating-modes.png
+  - loaded-pole-analysis.png
+examples: []
+open_questions:
+  - "What is the optimal PLL bandwidth for tracking the loaded pole during QCW ramp?"
+  - "How does the upper pole behave under heavy spark loading -- does it ever become the dominant mode?"
+  - "Can the frequency tracking strategy be adapted in real time based on spark impedance feedback?"
+  - "What is the quantitative power penalty for operating 5% off the loaded pole versus exactly on it?"
+  - "How do higher-order modes (if present in long secondaries) interact with the spark-loaded poles?"
+  - "Does the ~1 MHz breakdown voltage minimum affect inception behavior for high-frequency DRSSTCs (>200 kHz)?"
+  - "How does frequency tracking performance vary between 300-600 kHz QCW builds and 50-100 kHz burst DRSSTCs?"
+---
+
+# Coupled Resonance, Pole Splitting, and Frequency Tracking
+
+This document addresses the coupled resonant nature of the Tesla coil system and the critical role of frequency tracking when a spark loads the circuit. The central finding is that comparing spark impedances at a fixed frequency conflates two distinct effects (impedance matching and detuning), and that frequency tracking is THE most important often-missed concept in Tesla coil spark modeling.
+
+## 1. The Coupled Resonant System
+
+### 1.1 Unloaded Tesla Coil: Two Eigenfrequencies
+
+A Tesla coil consists of two resonant circuits (primary and secondary) coupled magnetically with coupling coefficient k. Even without a spark, this coupled system does not have a single resonant frequency. Instead, it has two resonant modes (eigenfrequencies or poles):
+
+- **Lower pole (f_lower):** Below the geometric mean of the uncoupled primary and secondary frequencies. In this mode, the primary and secondary currents are approximately in phase.
+
+- **Upper pole (f_upper):** Above the geometric mean. In this mode, the primary and secondary currents are approximately in antiphase.
+
+For a coil with uncoupled primary frequency f_p and uncoupled secondary frequency f_s, and coupling coefficient k:
+
+```
+f_lower ~ f_mean * sqrt(1 - k)      (approximate, for tuned case f_p ~ f_s)
+f_upper ~ f_mean * sqrt(1 + k)      (approximate, for tuned case f_p ~ f_s)
+
+where f_mean = sqrt(f_p * f_s) ~ f_p ~ f_s  (for tuned coils)
+```
+
+The splitting between the two poles is proportional to k. For typical DRSSTCs with k = 0.15-0.25, the splitting is 15-25% of the center frequency.
+
+### 1.2 Mode Characteristics
+
+Each pole has its own:
+- **Frequency:** f_lower and f_upper
+- **Quality factor (Q):** Determined by losses in both primary and secondary
+- **Voltage gain:** The ratio of topload voltage to primary voltage differs between modes
+- **Current distribution:** The pattern of currents in primary and secondary differs
+
+For most DRSSTCs, the lower pole provides higher topload voltage because the secondary's distributed capacitance and the topload capacitance are in a favorable configuration.
+
+### 1.3 Why Poles, Not "Resonant Frequency"
+
+It is incorrect to speak of "the resonant frequency" of a Tesla coil. The system has two distinct resonances. The coil's behavior depends critically on which pole the drive is tuned to. Most DRSSTCs are tuned to operate at or near the lower pole, but this is a design choice, not a physical necessity.
+
+### 1.4 Frequency Dependence of Air Breakdown
+
+A relevant physical phenomenon for coupled resonance design: the breakdown voltage in air shows a frequency dependence, with a **minimum near ~1 MHz**. At frequencies well below this, breakdown follows quasi-DC (Paschen) behavior. Near and above 1 MHz, electrons can survive the field reversal between half-cycles, reducing the effective breakdown threshold. The effect becomes significant when the RF half-period approaches the electron attachment time in air (~16 ns at STP). [Becker et al. 2005, Ch 2, p. 30; Kunhardt 2000]
+
+For Tesla coils operating at 50-400 kHz, this effect is relatively minor but not negligible:
+
+| Operating Frequency | Estimated Effect on Inception |
+|---------------------|-------------------------------|
+| 50 kHz | Essentially DC-like breakdown |
+| 100-200 kHz | Possibly 1-5% reduction vs. DC predictions |
+| 200-400 kHz | Possibly 5-10% reduction vs. DC predictions |
+| ~1 MHz (some small SSTCs) | Approaching minimum, potentially 20-30% reduction |
+
+This frequency dependence is an additional factor (beyond the pole-tracking effects in Sections 2-3) that should be considered when comparing spark performance across coils operating at very different frequencies. A coil at 400 kHz may have a slight inherent advantage in spark inception over an otherwise identical coil at 100 kHz, independent of coupling and power considerations.
+
+See [[field-thresholds]] Section 4.4 for the broader discussion of frequency effects on breakdown.
+
+### 1.5 QCW Operating Parameters from Community Survey
+
+A comprehensive survey of QCW builder data [Phase 6 QCW community survey, 2026-02-10] reveals that QCW operation occupies a distinct parameter space compared to burst-mode DRSSTCs:
+
+| Parameter | QCW Range | Burst DRSSTC | Implication |
+|-----------|-----------|--------------|-------------|
+| Coupling (k) | 0.3-0.55+ | 0.05-0.2 | QCW needs tight coupling for adequate power transfer at low peak current |
+| Operating frequency | 300-600 kHz | 50-110 kHz | Higher frequency enables continuous heating (see [[thermal-physics]]) |
+| Tank capacitance | 5-15 nF | 50-300 nF | Smaller tank for faster ring-up |
+| Ramp duration | 10-22 ms | N/A (burst ~70-150 us) | 100-200x longer pulse |
+| Peak primary current | 50-200 A | 200-1000+ A | QCW uses far less peak current |
+| Secondary voltage | 40-70 kV | 200-600 kV | QCW voltage is 5-15x lower |
+| Spark:secondary ratio | 7-16x | 2-4x | QCW produces 3-5x more spark per unit secondary |
+| Growth rate | ~170 m/s | N/A (single-shot) | Half the speed of sound |
+
+**Key insight — QCW secondary voltage is LOW:** Multiple independent builders (Steve Ward, davekni, Loneoceans) have measured QCW secondary voltages of only 40-70 kV despite producing meter-length sparks. The most dramatic comparison: davekni measured ~600 kV for 2-3 m burst-mode sparks vs ~40 kV for equivalent QCW sparks at 450 kHz — a 15:1 voltage ratio. This proves that QCW growth is driven by sustained energy injection through a persistent leader channel, not by high instantaneous voltage. See [[streamers-and-leaders]] for the physical explanation.
+
+**Coupling requirement (k >= 0.3):** All successful QCW sword-spark builds use k >= 0.3, typically 0.35-0.55. Higher coupling enables sufficient power transfer at QCW's lower peak currents (50-200 A vs 200-1000+ A for burst). It also widens the pole separation, making frequency tracking more robust against the shifting loaded pole during the ramp. However, Loneoceans' SSTC3 (single-resonant, lower coupling) still produces straight sparks at 380-420 kHz, suggesting that the coupling requirement is primarily an engineering constraint (adequate power delivery) rather than a physics constraint (straightness).
+
+## 2. Spark-Induced Pole Modification
+
+### 2.1 How the Spark Modifies the System
+
+When a spark forms, it adds the spark circuit (C_mut || R in series with C_sh, per [[circuit-topology]]) at the topload node. This modifies the system in two ways:
+
+1. **Frequency shift:** The additional capacitance (primarily C_sh, which grows with spark length at ~2 pF/foot) increases the total capacitance at the topload, lowering both pole frequencies. The lower pole drops more because it is more sensitive to topload loading.
+
+2. **Damping increase:** The spark resistance R adds loss to the system, reducing the Q of both poles. This is the desired effect -- power dissipated in R is the power delivered to the spark.
+
+**Critical distinction:** The spark modifies both frequency AND damping, not just one or the other. Ignoring either effect leads to incorrect power predictions.
+
+### 2.2 Quantitative Frequency Shift
+
+For a spark of length L (in meters), with C_sh ~ 6.6 pF/m * L:
+
+```
+C_total_new = C_top_original + C_sh(L) + coupling_corrections
+
+Approximate frequency shift:
+delta_f / f_0 ~ -C_sh(L) / (2 * C_top_original)    (first order)
+```
+
+For a medium coil with C_top = 30 pF and a 2-meter spark (C_sh ~ 13 pF):
+
+```
+delta_f / f_0 ~ -13 / (2 * 30) ~ -22%
+```
+
+This is a very large frequency shift. A 22% detuning can reduce power transfer by an order of magnitude if the drive frequency is not adjusted.
+
+### 2.3 Damping Increase
+
+The loaded Q at the lower pole decreases as:
+
+```
+1/Q_loaded = 1/Q_unloaded + 1/Q_spark
+
+where Q_spark ~ omega_L * C_total / G_spark    (for the spark contribution)
+```
+
+For a well-coupled spark near R_opt_power, Q_spark might be 10-30, while Q_unloaded might be 100-300. The spark dominates the loaded Q, which is desirable (most power goes to the spark, not secondary losses).
+
+## 3. The Frequency Tracking Problem
+
+### 3.1 The Fundamental Issue
+
+Consider the following common (but flawed) simulation approach:
+1. Set drive frequency to f_0 (unloaded resonance).
+2. Attach spark load with resistance R_1.
+3. Measure power P_1.
+4. Change to R_2.
+5. Measure power P_2.
+6. Compare P_1 and P_2 to determine "which R is better matched."
+
+**This is wrong.** The comparison is invalid because the loaded pole frequency shifts when R changes (through the change in damping and the coupling between R and the reactive elements). At fixed drive frequency:
+
+- Some R values will happen to place the loaded pole near the drive frequency (accidentally "tuned"), giving misleadingly high power.
+- Other R values will shift the loaded pole far from the drive frequency ("detuned"), giving misleadingly low power.
+
+**What is actually being measured:** The comparison conflates two independent effects:
+1. **Impedance matching quality** (how close R is to R_opt_power)
+2. **Frequency detuning** (how far the drive is from the loaded pole)
+
+These must be separated to draw valid conclusions.
+
+### 3.2 The Correct Approach
+
+**For each R value:**
+1. Sweep the drive frequency over a band (e.g., +/-15% of f_0).
+2. Find the frequency of maximum |V_top| -- this is the loaded pole frequency f_L(R).
+3. Measure the power delivered to the spark AT that loaded pole frequency.
+4. Record P(R) at the optimally tuned frequency.
+
+This procedure isolates the impedance matching quality from frequency effects. The resulting P(R) curve peaks at R_opt_power, as predicted by [[power-optimization]].
+
+### 3.3 Quantitative Impact of Ignoring Frequency Tracking
+
+The power penalty for operating at a fixed frequency when the loaded pole has shifted can be estimated. For a system with loaded Q_L:
+
+```
+P(f) / P(f_L) ~ 1 / (1 + Q_L^2 * (f/f_L - f_L/f)^2)
+```
+
+For Q_L = 20 and 5% detuning:
+
+```
+P(f) / P(f_L) ~ 1 / (1 + 400 * (0.05)^2) ~ 1 / 2 = 50%
+```
+
+A 5% frequency error costs half the power. For 10% detuning, the penalty is ~80%. This is why frequency tracking is so important.
+
+**Steve Conner and others in the Tesla coil community have identified frequency tracking as the single most important factor** that separates high-performance coils from underperformers. A coil with excellent frequency tracking will outperform one with better static impedance matching but poor tracking. The power difference can be a factor of 3-5.
+
+## 4. DRSSTC Operating Modes
+
+### 4.1 Fixed Frequency
+
+**Description:** The drive inverter operates at a pre-set frequency determined by a crystal oscillator, RC timer, or similar fixed reference.
+
+**Advantages:**
+- Simple implementation
+- Predictable behavior
+- No feedback loop to destabilize
+
+**Disadvantages:**
+- No compensation for spark loading
+- As spark grows and poles shift, the coil detunes
+- Power delivery drops dramatically during spark growth
+- Only competitive for very short pulses (burst mode) where the spark has minimal time to load the system
+
+**When appropriate:** Short burst-mode operation (<100 us pulses) where frequency shift is minimal, or for initial testing and debugging.
+
+### 4.2 PLL (Phase-Locked Loop)
+
+**Description:** A phase comparator measures the phase relationship between the drive signal and a feedback signal (typically the secondary base current or a current transformer on the primary). The PLL adjusts the drive frequency to maintain a target phase relationship, tracking the loaded pole.
+
+**Advantages:**
+- Automatically tracks the loaded pole as spark grows
+- Maintains near-optimal power transfer throughout spark growth
+- Most common approach in high-performance DRSSTCs
+
+**Disadvantages:**
+- PLL bandwidth must be chosen carefully:
+  - Too slow: cannot track rapid impedance changes
+  - Too fast: may overshoot or oscillate, especially during spark inception
+- Phase detector may lock to wrong pole (upper instead of lower)
+- Noise from spark can corrupt the feedback signal
+- Complex implementation
+
+**Design considerations:**
+- PLL bandwidth should be fast enough to track the spark growth timescale (~1 ms changes) but slow enough to reject RF noise (>1 MHz)
+- Typical PLL bandwidth: 1-10 kHz
+- The feedback signal must be filtered to extract the fundamental frequency
+
+### 4.3 Programmed Frequency
+
+**Description:** The drive frequency is pre-programmed as a function of time, based on anticipated or pre-measured loading. For QCW operation, the frequency ramp can be designed to match the expected pole shift during voltage ramp-up.
+
+**Advantages:**
+- No feedback loop (stable, predictable)
+- Can be optimized for specific operating conditions
+- No noise sensitivity
+
+**Disadvantages:**
+- Requires advance knowledge of loading (or iterative calibration)
+- Does not adapt to variations (spark length, humidity, proximity to objects)
+- Must be re-programmed for different operating conditions
+
+**When appropriate:** Highly repeatable operating conditions, competition coils optimized for a specific target, or as a supplement to PLL (pre-programmed nominal trajectory with PLL corrections).
+
+### 4.4 Hybrid Approaches
+
+Modern high-performance DRSSTCs often combine approaches:
+- Programmed frequency ramp for the nominal trajectory
+- PLL correction for deviations from nominal
+- Mode switching: fixed frequency during ring-up, PLL during spark growth
+- Adaptive algorithms that learn the pole trajectory over multiple pulses
+
+![DRSSTC operating modes comparison](../assets/drsstc-operating-modes.png)
+
+## 5. Pole Behavior Under Heavy Loading
+
+### 5.1 Pole Migration
+
+As spark loading increases (C_sh grows, R decreases toward R_opt):
+
+1. **Lower pole:** Frequency decreases, Q decreases. This is the primary operating pole for most DRSSTCs. Under very heavy loading, the lower pole can shift by 20-30% from its unloaded position.
+
+2. **Upper pole:** Frequency also shifts (less dramatically), Q decreases. The upper pole may become so heavily damped that it effectively disappears as a distinct resonance.
+
+3. **Pole merging:** In extreme cases (very heavy loading or very tight coupling), the two poles can merge into a single, heavily damped resonance. This is unusual in normal operation but can occur during arc strikes to grounded objects.
+
+### 5.2 Mode Coupling
+
+The spark introduces a coupling between what were previously relatively independent modes. At moderate loading:
+- Energy can transfer between modes
+- The simple two-pole picture becomes less clean
+- Transient analysis may show beating between modes
+
+For simulation purposes, sweeping frequency to find the actual loaded pole (as described in Section 3.2) automatically accounts for these effects.
+
+![Loaded pole analysis showing frequency and Q vs. spark loading](../assets/loaded-pole-analysis.png)
+
+![Frequency shift with spark loading](../assets/frequency-shift-with-loading.png)
+
+## 6. Interaction with Thevenin Analysis
+
+### 6.1 Frequency-Dependent Thevenin Parameters
+
+The Thevenin equivalent (see [[thevenin-method]]) captures the coupled resonance behavior implicitly through the frequency dependence of Z_th(omega) and V_th(omega):
+
+- Near the lower pole: Z_th has a peak in its real part, V_th has a peak in magnitude
+- Near the upper pole: smaller secondary peaks
+- Between poles: Z_th and V_th vary smoothly
+
+By measuring Z_th and V_th over the full frequency band, the Thevenin approach automatically accounts for pole shifting and mode coupling.
+
+### 6.2 Practical Implication
+
+When using the Thevenin method for power prediction:
+1. Compute Z_load for the spark at the operating frequency.
+2. Compute P_load = 0.5 * |V_th(f)|^2 * Re{Z_load(f)} / |Z_th(f) + Z_load(f)|^2.
+3. Sweep f to find the frequency that maximizes P_load for each set of spark parameters.
+4. That frequency is the loaded pole.
+
+## 7. Practical Recommendations
+
+### 7.1 For Simulation
+
+- **Always sweep frequency** when comparing different spark loads. Never evaluate at a single fixed frequency.
+- **Report power at the loaded pole**, not at the unloaded resonant frequency.
+- **Track both poles** to ensure you are operating on the correct one.
+- **Include primary tank components** in the model; they affect pole locations significantly.
+
+### 7.2 For Coil Design
+
+- **Design PLL bandwidth** for the expected spark growth timescale.
+- **Allow sufficient frequency range** in the drive electronics (at least +/-15% of nominal).
+- **Monitor for pole-hopping:** If the PLL locks onto the wrong pole, power delivery can drop dramatically.
+- **Consider QCW ramp rate:** Faster ramps require faster frequency tracking. Typical QCW ramps of 5-20 ms are well within PLL capability if bandwidth is 1-10 kHz.
+
+### 7.3 For Measurement
+
+- **Ringdown measurements** (see [[thevenin-method]]) give Q and frequency at a single operating point. Multiple measurements at different loading levels map out the pole trajectory.
+- **Real-time frequency monitoring** (e.g., counting zero crossings of the secondary current) provides the loaded pole frequency during operation.
+
+## 8. Connection to Other Topics
+
+### Key Relationships
+
+- **Derives from:** Coupled oscillator theory (standard physics of two inductively coupled LC circuits)
+- **Depends on:** [[circuit-topology]] (the spark load impedance is what modifies the poles)
+- **Interacts with:** [[power-optimization]] (R_opt_power changes with frequency; frequency tracking ensures the correct R_opt is used)
+- **Measured via:** [[thevenin-method]] (Z_th(omega) captures pole behavior; ringdown gives loaded Q)
+- **Affects:** [[energy-and-growth]] (power delivery during spark growth depends on how well the system tracks the loaded pole)
+- **Affects:** [[lumped-model]] and [[distributed-model]] (simulations must include frequency tracking for accurate power predictions)
+
+### Summary of Key Results
+
+1. A Tesla coil has two eigenfrequencies (poles), not one "resonant frequency."
+2. Spark loading shifts both poles lower in frequency and increases damping.
+3. Comparing spark loads at fixed frequency conflates impedance matching with detuning.
+4. The correct procedure: for each load, find the loaded pole, then measure power there.
+5. Frequency tracking (PLL or programmed) is the single most impactful design feature.
+6. A 5% frequency error can halve the delivered power; 10% can cost 80%.
+7. Three DRSSTC operating modes: fixed frequency, PLL, programmed. PLL is most common.
+8. Power penalty from poor frequency tracking: factor of 3-5 in real coils.

context/distributed-model.md

@@ -0,0 +1,385 @@
+---
+id: distributed-model
+title: "nth-Order Distributed Spark Model"
+status: established
+source_sections: "spark-physics.txt: Part 8 (lines 540-664), Part 10.2 (lines 715-733), Part 11 (lines 736-803)"
+related_topics: [lumped-model, femm-workflow, circuit-topology, power-optimization, thevenin-method, coupled-resonance, capacitive-divider, field-thresholds, energy-and-growth, streamers-and-leaders, equations-and-bounds, open-questions]
+key_equations:
+  - "Tapered resistance initialization"
+  - "Position-dependent resistance bounds"
+  - "Circuit-determined R per segment"
+  - "Damped iterative update rule"
+  - "Self-consistency diameter check"
+key_terms:
+  - "distributed model"
+  - "segment"
+  - "partial capacitance matrix"
+  - "Maxwell capacitance matrix"
+  - "resistance taper"
+  - "convergence"
+  - "damping factor"
+  - "passivity"
+  - "nearest-neighbor approximation"
+  - "controlled sources"
+images:
+  - resistance-taper-initialization.png
+  - power-distribution-along-spark.png
+  - current-attenuation-plot.png
+  - lumped-vs-distributed-comparison.png
+  - position-dependent-bounds.png
+  - validation-total-resistance.png
+  - capacitance-matrix-heatmap.png
+  - femm-geometry-setup-distributed.png
+  - partial-capacitance-transformation.png
+  - iterative-optimization-convergence.png
+  - spice-implementation-methods.png
+examples:
+  - distributed-model-complete.md
+open_questions:
+  - "What is the optimal number of segments for a given spark length -- is there a principled criterion beyond 'diminishing returns at n=10'?"
+  - "Can the resistance distribution be used to infer the leader-to-streamer transition point, and if so, what is the R threshold?"
+  - "How should branching be handled -- does each branch get its own distributed model, and how is power divided at branch points?"
+  - "Is the nearest-neighbor approximation sufficient for tightly-spaced segments, or does it break down when segment length approaches the channel diameter?"
+  - "How does the resistance distribution evolve in time during QCW ramp-up?"
+---
+
+# nth-Order Distributed Spark Model
+
+The distributed model divides the spark channel into n segments (typically n = 10), each with its own mutual capacitances, shunt capacitance, and resistance. This generalization of the [[lumped-model]] captures the spatial variation of current, power, and impedance along the spark length. The model reveals that base segments naturally optimize to low resistance (hot leader plasma) while tip segments settle at high resistance (cold streamer plasma), providing a circuit-level explanation for the observed leader-streamer structure of Tesla coil sparks.
+
+## 1. Model Structure
+
+### 1.1 Segmentation
+
+The spark channel of total length L is divided into n equal segments. Each segment has length L_seg = L/n. Segments are numbered from i = 1 (base, connected to topload) to i = n (tip, farthest from topload). Together with the topload (conductor 0), there are n + 1 conductors in the FEMM model.
+
+```
+Topload (conductor 0)
+   |
+[C_01][R_1][C_1,gnd]     Segment 1 (base)
+   |
+[C_12][R_2][C_2,gnd]     Segment 2
+   |
+  ...
+   |
+[C_{n-1,n}][R_n][C_n,gnd]  Segment n (tip)
+```
+
+Each segment possesses:
+- **Mutual capacitances** to every other conductor (topload, all other segments)
+- **Shunt capacitance** to ground (environment)
+- **Resistance R[i]** representing the plasma conductivity of that section
+- **Optional inductance** if magnetic effects are significant (usually omitted for straight sparks at typical frequencies)
+
+### 1.2 Position Variable
+
+For a segment indexed by i (from 1 to n), define the normalized position:
+
+```
+position = (i - 1) / (n - 1)
+```
+
+This ranges from 0 at the base (segment 1) to 1 at the tip (segment n). The position variable is used to set position-dependent resistance bounds and initialization profiles.
+
+## 2. FEMM Extraction for Distributed Model
+
+### 2.1 Geometry Setup
+
+The FEMM electrostatic model includes n + 1 conductors:
+- Conductor 0: Topload (toroid or sphere)
+- Conductors 1 through n: Cylindrical segments of the spark channel
+
+Each segment is a short cylinder of length L_seg and nominal diameter d (typically 1 mm for burst mode, 3 mm for QCW). Small gaps of 0.1 mm between segments ensure numerical stability while maintaining physical proximity. See [[femm-workflow]] for detailed setup procedures.
+
+### 2.2 The (n+1) x (n+1) Capacitance Matrix
+
+FEMM produces a symmetric (n+1) x (n+1) Maxwell capacitance matrix C. For a 10-segment model, this is an 11x11 matrix.
+
+**Matrix properties (must verify):**
+- **Symmetric:** C[i,j] = C[j,i] for all i, j
+- **Diagonal positive:** C[i,i] > 0 (self-capacitance)
+- **Off-diagonal negative:** C[i,j] < 0 for i != j (mutual coupling, Maxwell convention)
+- **Nearest-neighbor dominance:** |C[i,i+1]| > |C[i,i+2]| > |C[i,i+3]| (coupling decreases with distance)
+- **Decreasing diagonals toward tip:** Self-capacitance typically decreases from base to tip because base segments are better coupled to the nearby topload
+
+**Coupling patterns observed in practice:**
+- Topload-to-base coupling (C[0,1]) is the strongest off-diagonal element, typically 5-10 pF
+- Topload-to-tip coupling (C[0,n]) is very weak, typically 0.01-0.05 pF
+- Adjacent segment coupling (C[i,i+1]) ranges from 0.6 to 3 pF depending on position
+- Remote segment coupling (C[i,j] for |i-j| > 3) is usually below 0.1 pF
+
+## 3. SPICE Implementation Challenge
+
+### 3.1 The Negative Off-Diagonal Problem
+
+The Maxwell capacitance matrix has negative off-diagonal elements. Direct implementation as literal capacitors in SPICE creates negative capacitances, which are unphysical and can cause numerical instability or non-convergent simulations. Three solutions exist:
+
+### 3.2 Solution 1: Partial Capacitance Matrix
+
+Transform the Maxwell matrix into a partial capacitance representation where all elements are positive. The partial capacitance between nodes i and j is:
+
+```
+C_partial[i,j] = -C_maxwell[i,j]    for i != j  (positive)
+C_partial[i,ground] = C_maxwell[i,i] + sum_j(C_maxwell[i,j])    for j != i  (to ground)
+```
+
+Each C_partial[i,j] is implemented as a standard positive capacitor between nodes i and j. Each C_partial[i,ground] is a capacitor from node i to ground. All values are positive, and the network is guaranteed passive if the original Maxwell matrix is valid.
+
+### 3.3 Solution 2: Controlled Sources via MNA
+
+Implement the capacitance matrix directly using Modified Nodal Analysis (MNA):
+
+```
+I_i = sum_j C[i,j] * dV_j/dt
+```
+
+In SPICE, this is implemented using voltage-controlled current sources (VCCS) with capacitive coupling. For each pair (i, j):
+- A VCCS from node i to ground, controlled by the time derivative of V_j, with gain C[i,j]
+
+This approach preserves the exact matrix without approximation but requires more SPICE elements.
+
+### 3.4 Solution 3: Nearest-Neighbor Approximation
+
+For many practical cases, the coupling between non-adjacent segments is weak (below 0.1 pF). Truncating the matrix to include only nearest-neighbor and next-nearest-neighbor couplings simplifies the network substantially.
+
+**Validation requirement:** Compare the full-matrix impedance at the topload port against the approximated impedance. If they agree within 5%, the approximation is justified.
+
+### 3.5 Passivity Check
+
+The capacitance matrix must be symmetric positive semi-definite (SPD) for the resulting circuit to be passive (no energy generation). If numerical noise in FEMM creates slight violations:
+
+- **Add small diagonal perturbation:** Increase each C[i,i] by +0.1 pF. This adds a small parasitic to ground at each node.
+- **Add small series resistance:** Insert a small resistor (1-10 ohm) in series with each capacitor for numerical damping.
+
+Check for SPD: all eigenvalues of the capacitance matrix must be non-negative. A single small negative eigenvalue (less than 1% of the largest) is a numerical artifact and can be corrected by the diagonal perturbation.
+
+## 4. Resistance Determination
+
+### 4.1 Position-Dependent Physical Bounds
+
+The plasma physics constrains the resistance of each segment based on its position along the spark:
+
+```
+R_min[i] = 1 kilohm + (10 kilohm - 1 kilohm) * position
+         = 1 kilohm + 9 kilohm * position
+
+R_max[i] = 100 kilohm + (100 megohm - 100 kilohm) * position^2
+         = 100 kilohm + 99.9 megohm * position^2
+```
+
+**Physical reasoning:**
+- Base segments (position near 0) can reach very low resistance because they receive the most current, heat the most, and form hot leader plasma. R_min at the base is 1 kilohm.
+- Tip segments (position near 1) are poorly coupled, receive less current, and tend to remain as cold streamers. Their minimum achievable resistance is higher (R_min at the tip is 10 kilohm), reflecting the difficulty of forming leader plasma at the tip.
+- The maximum resistance increases quadratically with position. Base segments are constrained to R_max = 100 kilohm (the leader plasma has significant conductivity), while tip segments can reach up to 100 megohm (very faint, cold streamer).
+
+### 4.2 Initialization: Tapered Resistance Profile
+
+A good initial guess accelerates convergence. The recommended initialization uses a quadratic taper:
+
+```
+R[i] = R_base + (R_tip - R_base) * position^2
+
+R_base = 10 kilohm   (expected leader resistance)
+R_tip = 1 megohm     (expected streamer resistance)
+```
+
+The quadratic dependence reflects the physical expectation that leader plasma (low R) dominates near the base and transitions gradually to streamer plasma (high R) toward the tip, with the transition occurring more rapidly in the outer portion of the spark.
+
+### 4.3 Simplified Method: Circuit-Determined Resistance
+
+For each segment, compute the total capacitance (sum of absolute values of all capacitive couplings involving that segment):
+
+```
+C_total[i] = C_shunt[i] + sum_j(|C_mutual[i,j]|)    for all j != i
+```
+
+where C_shunt[i] = C[i,i] - sum_j(|C[i,j]|) for j != i is the shunt capacitance to ground for segment i, and the mutual capacitances are the absolute values of the off-diagonal elements.
+
+More directly, C_total[i] is the sum of absolute values of all off-diagonal elements in row i:
+```
+C_total[i] = sum_{j != i} |C[i,j]|
+```
+
+Then set the resistance using the same R_opt_power logic as the [[lumped-model]], applied per segment:
+
+```
+R[i] = 1 / (omega * C_total[i])
+R[i] = clip(R[i], R_min[i], R_max[i])
+```
+
+**Justification for the simplified method:**
+- Capacitance depends logarithmically on diameter: C proportional to 1/ln(h/d)
+- R_opt_power is proportional to 1/C, so it also depends weakly on diameter
+- Doubling the diameter changes C by only 10-15%
+- This error is small compared to other uncertainties (FEMM accuracy ~10%, plasma variability ~50%)
+- The simplified method typically agrees with full iterative optimization within 1%
+
+**When to use:** Standard cases within typical parameter ranges. This is the recommended default.
+
+**When to iterate instead:** Edge cases near physical bounds, validation studies, highest accuracy needs, or when the simplified method produces results that fail validation checks.
+
+### 4.4 Advanced Method: Iterative Power Maximization
+
+For highest accuracy, optimize each segment's resistance to maximize the power dissipated in that segment, subject to the constraint that all other segments are at their current resistance values.
+
+**Algorithm:**
+
+```
+Initialize: R[i] from tapered profile or simplified method
+Set damping factor: alpha = 0.3 to 0.5
+
+Repeat until convergence:
+    For each segment i = 1 to n:
+        Hold all R[j] (j != i) fixed
+        Sweep R[i] over logarithmically spaced values from R_min[i] to R_max[i]
+        For each trial R[i]:
+            Build SPICE network with full capacitance matrix
+            Run AC analysis at operating frequency
+            Compute P[i] = 0.5 * |I[i]|^2 * R[i]
+        Find R_optimal[i] that maximizes P[i]
+        Apply damping:
+            R_new[i] = alpha * R_optimal[i] + (1 - alpha) * R_old[i]
+        Clip to bounds:
+            R[i] = clip(R_new[i], R_min[i], R_max[i])
+
+    Check convergence: max relative change across all segments < 1%
+
+    If resonant poles shifted > 5% from initial:
+        Update operating frequency to track loaded pole
+        Re-optimize at new frequency
+```
+
+**Damping (critical for stability):** Without damping (alpha = 1), the iteration can oscillate because changing one segment's resistance affects the optimal values of all other segments. A damping factor of alpha = 0.3 to 0.5 provides smooth convergence in 2-5 iterations for typical cases.
+
+### 4.5 Convergence Behavior and Physical Interpretation
+
+The convergence characteristics differ dramatically between base and tip segments:
+
+**Base segments (position near 0):**
+- Strong coupling to topload and adjacent segments
+- Sharp power peak as a function of R
+- Fast convergence to a well-defined optimum
+- Final R is low (tens of kilohms), indicating hot leader plasma
+- Small uncertainty in optimal R
+
+**Tip segments (position near 1):**
+- Weak coupling to all other conductors
+- Flat power curve with no sharp peak
+- May not converge to a unique optimal value
+- Final R is high (hundreds of kilohms to megohms), indicating cold streamer plasma
+- Large uncertainty in optimal R (but power is insensitive, so the uncertainty is inconsequential)
+
+**This naturally produces the leader + streamer distribution observed in real Tesla coil sparks.** The circuit optimization, without any explicit plasma physics, predicts that:
+- Base segments should be low-resistance, high-current, high-power (leader)
+- Tip segments should be high-resistance, low-current, low-power (streamer)
+- The transition is gradual and depends on the capacitive coupling profile
+
+### 4.6 Diameter Self-Consistency Check
+
+The resistance determines an implied channel diameter, which can be checked against the nominal diameter used in FEMM:
+
+```
+d_nominal = 1e-3 m  (1 mm starting guess)
+
+From FEMM: C_mut, C_sh
+From circuit: R_opt = 1 / (omega * C_total)
+
+Back-calculate diameter using typical partially ionized plasma resistivity:
+rho_typical = 10 ohm*m
+L_segment = L_total / n_segments
+d_implied = sqrt(4 * rho_typical * L_segment / (pi * R_opt))
+
+If d_implied is approximately d_nominal (within factor of 2): self-consistent
+If not: iterate once with d = (d_nominal + d_implied) / 2
+```
+
+Because the capacitance dependence on diameter is logarithmic, this self-consistency check typically converges in 1-2 iterations. In practice, the circuit-first approach (use nominal diameter, compute R, let plasma adjust) is recommended because the weak logarithmic sensitivity makes the diameter a dependent variable, not an input.
+
+## 5. Validation
+
+### 5.1 Lumped Model Consistency
+
+A 1-segment distributed model must produce the same impedance as the [[lumped-model]] with the same total C_mut and C_sh. If the two disagree by more than 1%, there is an error in the matrix extraction or SPICE implementation.
+
+**Important:** Compare impedances at the topload port, NOT total series resistance. The distributed model's total series resistance (sum of all R[i]) is much larger than the lumped R because the segments are in series, but the impedance at the port includes the capacitive network and is a complex quantity.
+
+### 5.2 Segment Count Convergence
+
+Compare results for n = 5, 10, and 20 segments. Key quantities to track:
+- Impedance at topload port
+- Total power delivered to spark
+- Current at base and tip
+- Loaded resonant frequency
+
+Expect: n = 5 to n = 10 shows significant improvement; n = 10 to n = 20 shows diminishing returns (changes under 5%). For most purposes, n = 10 is the sweet spot balancing accuracy and computational cost.
+
+### 5.3 Total Resistance Validation
+
+The sum of all segment resistances should fall within expected ranges for the operating mode:
+
+**At 200 kHz for 1-3 meter sparks:**
+- **Streamer-dominated (burst mode):** Total R approximately 50-300 kilohm
+- **Leader-dominated (QCW):** Total R approximately 5-50 kilohm
+- **Very low frequency (below 100 kHz) or very long sparks:** Total R can approach 1-10 kilohm
+
+Flag results significantly outside these ranges. Note that these are total series resistances, which are higher than the equivalent lumped resistance due to the series arrangement.
+
+### 5.4 Resistance Distribution Check
+
+The final resistance distribution should be physically plausible:
+- Monotonically increasing from base to tip (base should be hottest)
+- No extreme jumps between adjacent segments (factor of 3 or less between neighbors)
+- All values within position-dependent bounds
+- Base segments should be in the leader regime (1-100 kilohm)
+- Tip segments should be in the streamer regime (100 kilohm to 100 megohm)
+
+### 5.5 Power Distribution
+
+The power distribution P[i] = 0.5 * |I[i]|^2 * R[i] should show:
+- Peak power in the base or middle segments (not at the tip)
+- Monotonically decreasing power toward the tip
+- Tip segments contributing a small fraction (typically 10-20%) of total power
+- This matches the physical expectation that energy is concentrated where current is highest
+
+### 5.6 Current Attenuation
+
+Current should decrease from base to tip due to capacitive shunting at each segment. Typical tip-to-base current ratio: 0.3 to 0.5 for 10-segment models of 1-3 meter sparks. If the tip current exceeds 70% of the base current, the model may have insufficient shunt capacitance (check C_sh extraction).
+
+## 6. Comparison: Simplified vs. Iterative Method
+
+Empirical comparison across many configurations shows:
+
+| Aspect | Simplified (R = 1/(omega*C_total)) | Iterative (power maximization) |
+|--------|-------------------------------------|-------------------------------|
+| Accuracy | Within 1% of iterative for standard cases | Reference (by definition) |
+| Speed | Seconds (FEMM + formula) | Minutes to hours (many AC analyses) |
+| Ease | Trivial to implement | Requires SPICE automation |
+| Edge cases | May fail near bounds | Handles gracefully with damping |
+| Spatial detail | Same quality | Same quality |
+
+**Recommendation:** Use the simplified method as default. Reserve iterative optimization for edge cases, validation, and publications.
+
+## 7. Connection to Other Topics
+
+### Key Relationships
+
+- **Generalizes:** [[lumped-model]] -- The distributed model is the n-section generalization of the single-section lumped model. Setting n = 1 recovers the lumped model exactly.
+- **Requires:** [[femm-workflow]] -- FEMM electrostatic simulation provides the (n+1) x (n+1) capacitance matrix that defines the entire capacitive network.
+- **Uses:** [[power-optimization]] -- The R_opt_power formula, applied per segment, provides both the simplified resistance values and the objective function for iterative optimization.
+- **Implements:** [[circuit-topology]] -- Each segment reproduces the fundamental bridged-T topology locally, with the full model being a cascade of such sections.
+- **Reveals:** [[streamers-and-leaders]] -- The optimized resistance distribution naturally separates into leader (base, low R) and streamer (tip, high R) regions without any explicit plasma physics input.
+- **Demonstrates:** [[capacitive-divider]] -- The current attenuation and voltage distribution along the distributed model directly visualize the capacitive divider effect.
+- **Bounded by:** [[equations-and-bounds]] -- All per-segment and total values must satisfy the physical bounds documented in the reference.
+- **Motivates:** [[open-questions]] -- Branching, time-dependent evolution, and the optimal segment count remain open research questions.
+
+### Worked Example
+
+The complete 10-segment workflow is demonstrated in `distributed-model-complete.md`, which includes:
+- FEMM geometry with 10 segments plus topload (11 conductors)
+- Full 11x11 capacitance matrix extraction and verification
+- Tapered initialization and position-dependent bounds
+- Simplified method calculation (C_total per segment, R = 1/(omega*C_total))
+- Two iterations of the advanced method showing convergence
+- Power distribution, current attenuation, and voltage distribution analysis
+- Comparison to lumped model and segment count convergence study
+- Validation checks at every step

context/empirical-scaling.md

@@ -0,0 +1,338 @@
+---
+id: empirical-scaling
+title: "Empirical Scaling Laws for Spark Length"
+status: established
+source_sections: "spark-physics.txt: Part 5 Section 5.7 (lines 362-386), Part 6 Section 6.1 (lines 389-401)"
+related_topics: [energy-and-growth, capacitive-divider, field-thresholds, thermal-physics, streamers-and-leaders, power-optimization, lumped-model, equations-and-bounds, open-questions]
+key_equations: [freau-single-shot, freau-repetitive, qcw-scaling, voltage-limited-derivation]
+key_terms: [Freau_scaling, bang_energy, epsilon, QCW, burst_mode, capacitive_divider, E_propagation]
+images: [length-vs-energy-scaling.png, epsilon-by-mode-comparison.png]
+examples: [spark-growth-timeline.md]
+open_questions:
+  - "What is the exact exponent for QCW scaling across different coil designs?"
+  - "How does repetition rate quantitatively affect the transition from single-shot to repetitive scaling?"
+  - "Can the scaling exponents be derived rigorously from the growth equation and divider model?"
+  - "How does the scaling change at very high power levels where thermal saturation occurs?"
+  - "What is the effect of topload geometry on the scaling exponents?"
+---
+
+# Empirical Scaling Laws for Spark Length
+
+Community observations and measurements have established empirical relationships between spark length and input energy or power. These scaling laws, notably Freau's relationships, provide practical tools for predicting spark performance and connect to the theoretical framework through the [[capacitive-divider]] and [[energy-and-growth]] models. Understanding when and why different scaling laws apply is essential for coil design and performance prediction.
+
+## Freau's Empirical Relationships
+
+The Tesla coil community, through extensive measurement and documentation by Freau and others, has observed consistent scaling relationships:
+
+### Single-Shot Burst Scaling
+
+```
+L proportional to sqrt(E_bang)
+```
+
+Or equivalently:
+```
+L = k_burst * sqrt(E_bang)
+```
+
+Where:
+- `L` is the spark length [m]
+- `E_bang` is the total energy delivered in a single burst [J]
+- `k_burst` is a coil-specific constant [m/J^0.5]
+
+**Conditions for validity:**
+- Single isolated pulse (no thermal memory between events)
+- Each spark starts from a cold, deionized state
+- Streamer-dominated (insufficient time for leader transition)
+- Voltage-limited growth (E_tip drops to E_propagation)
+
+**Typical bang energies:** 1-100 J for DRSSTC bursts
+
+### Repetitive Operation Scaling
+
+```
+L proportional to P_avg^(0.3 to 0.5)
+```
+
+Where:
+- `L` is the spark length [m]
+- `P_avg` is the time-averaged power [W]
+- The exponent varies from 0.3 to 0.5 depending on repetition rate and mode
+
+**Conditions for validity:**
+- Repetitive pulses with partial thermal/ionization memory between events
+- Effective persistence of channels across pulse gaps (see [[thermal-physics]])
+- Higher repetition rates push the exponent toward 0.5
+- Lower repetition rates (more cooling between pulses) push toward 0.3
+
+**Physical interpretation:** Thermal memory from previous pulses reduces the effective epsilon for subsequent pulses, improving efficiency compared to single-shot operation. The accumulated benefit produces a power-law relationship with an exponent less than the single-shot case.
+
+## Physical Derivation of Single-Shot Scaling
+
+The sqrt(E) scaling for single-shot burst mode can be derived from the voltage-limited growth model:
+
+### Starting Assumptions
+
+1. Spark growth is voltage-limited (E_tip = E_propagation at stall)
+2. Topload voltage is roughly constant during a burst (not ramping)
+3. Spark impedance is dominated by capacitive reactance: Z_spark ~ 1/(omega*C_sh)
+4. Shunt capacitance scales linearly with length: C_sh = C_sh_per_meter * L
+
+### Derivation
+
+The electric field at the spark tip (open-circuit approximation from [[capacitive-divider]]):
+```
+E_tip ~ kappa * V_topload * C_mut / [(C_mut + C_sh_per_meter * L) * L]
+```
+
+At the stall length L_max, E_tip = E_propagation. For sparks where C_sh >> C_mut (long enough that the growing shunt capacitance dominates):
+```
+E_propagation ~ kappa * V_topload * C_mut / (C_sh_per_meter * L_max^2)
+```
+
+Therefore:
+```
+L_max^2 ~ kappa * V_topload * C_mut / (E_propagation * C_sh_per_meter)
+L_max ~ sqrt(V_topload)
+```
+
+Now connect voltage to energy. The power delivered to the spark:
+```
+P ~ V_topload^2 / |Z_spark|
+  ~ V_topload^2 * omega * C_sh_per_meter * L
+```
+
+For the total bang energy in burst time T_burst:
+```
+E_bang ~ P * T_burst ~ V_topload^2 * omega * C_sh_per_meter * L * T_burst
+```
+
+Since L ~ sqrt(V_topload) and V_topload ~ L^2:
+```
+E_bang ~ L^4 * omega * C_sh_per_meter * L * T_burst ~ L^5 * (constants)
+```
+
+This gives L ~ E_bang^(1/5) = E_bang^0.2, which is weaker than the observed sqrt(E) scaling. The discrepancy arises because the simplified derivation ignores:
+
+- The time-varying nature of the spark impedance during growth
+- The fact that not all delivered energy contributes to growth (see [[energy-and-growth]])
+- The interaction between R_opt_power and the changing capacitances
+
+A more careful analysis using the growth equation dL/dt = P/epsilon with the full impedance model yields exponents closer to 0.4-0.5, consistent with observations. The key insight is that the sub-linear scaling is a robust consequence of C_sh increasing with L, regardless of the exact exponent.
+
+### Alternative Direct Argument
+
+A simpler argument that captures the essential physics:
+
+For a voltage-limited spark:
+```
+Need: V_topload > E_propagation * L (approximately)
+Therefore: L_max ~ V_topload / E_propagation
+```
+
+The impedance at stall:
+```
+Z_spark ~ L / (omega * epsilon_0 * ...) ~ proportional to L
+```
+
+Power:
+```
+P ~ V_topload^2 / Z_spark ~ V_topload^2 / L ~ V_topload^2 / V_topload = V_topload
+```
+
+Energy:
+```
+E ~ P * T ~ V_topload * T
+L ~ V_topload ~ E/T
+```
+
+And since for a single-shot burst T is roughly fixed:
+```
+L ~ E_bang^(~0.5)
+```
+
+This simplified argument, while not rigorous, shows why the sqrt relationship emerges. The exact exponent depends on details of the impedance model, but the sub-linear character is robust.
+
+## QCW Scaling: Better But Still Sub-Linear
+
+QCW mode shows improved scaling compared to burst mode:
+
+```
+L ~ E^(0.6 to 0.8)    (QCW)
+```
+
+This is closer to linear than burst mode (exponent 0.5) but still sub-linear. Three factors explain the improvement:
+
+### 1. Active Voltage Ramping Compensates Divider
+
+In QCW, V_topload increases throughout the ramp:
+```
+V_topload(t) = V_max * (t / T_ramp)
+```
+
+As C_sh grows and attenuates V_tip, the rising V_topload partially compensates. The net attenuation of V_tip is slower than for constant-voltage operation, allowing growth to continue longer before the field threshold is reached.
+
+### 2. Leader Formation Reduces epsilon
+
+The streamer-to-leader transition (see [[streamers-and-leaders]]) reduces epsilon from 30-100 J/m (streamer) to 5-15 J/m (leader). From the growth equation:
+```
+dL/dt = P / epsilon
+```
+
+Lower epsilon means more length per watt. The leader-dominated QCW spark converts energy to length more efficiently.
+
+### 3. Thermal Accumulation Further Reduces epsilon
+
+During the long QCW ramp, thermal energy accumulates in the channel (see [[thermal-physics]]):
+```
+epsilon(t) = epsilon_0 / (1 + alpha * integral(P dt))
+```
+
+As the accumulated energy grows, epsilon decreases further, improving efficiency throughout the ramp. This effect is negligible for short bursts but significant over 5-20 ms ramps.
+
+### Combined Effect
+
+All three mechanisms push the scaling exponent above 0.5 toward 0.8. The [[capacitive-divider]] still prevents linear scaling (exponent = 1.0), but QCW operates closer to the theoretical limit than burst mode.
+
+## Calibration: From Empirical to Predictive
+
+The scaling laws provide empirical relationships, but for quantitative prediction, the underlying parameters must be calibrated per coil.
+
+### Parameters to Calibrate
+
+1. **epsilon** (energy per meter): The most important parameter. Determined by:
+   - Running the coil at known conditions
+   - Measuring spark length L
+   - Computing delivered energy E from SPICE model
+   - epsilon = E / L
+   - See [[energy-and-growth]] for detailed procedure
+
+2. **E_propagation** (field threshold): Determines the voltage limit. Determined by:
+   - Using FEMM to compute E_tip at the measured stall length and voltage
+   - E_propagation = E_tip(V_top, L_stall)
+   - See [[field-thresholds]] for details
+
+3. **Scaling constant k**: Once epsilon and E_propagation are known, the coil-specific proportionality constant in L = k * f(E) can be calculated and used to predict performance at different operating conditions.
+
+### Calibration Procedure
+
+1. **Measure at baseline condition**: Run coil, measure L_1, compute E_1. Calculate epsilon_1 = E_1/L_1.
+2. **Measure at different condition**: Change power or ramp time. Measure L_2, compute E_2. Calculate epsilon_2 = E_2/L_2.
+3. **Verify consistency**: epsilon_1 and epsilon_2 should agree within measurement uncertainty (+/-30-50%) for the same operating mode.
+4. **Predict new conditions**: Use calibrated epsilon and E_propagation with the growth equation and voltage limit to predict L for untested conditions.
+
+### Expected Prediction Precision
+
+The framework is inherently approximate due to plasma physics variability:
+
+| Parameter | Precision |
+|-----------|-----------|
+| Spark length prediction | +/-20-40% |
+| Power prediction | +/-30-50% |
+| Impedance phase prediction | +/-5-10 degrees |
+| FEMM capacitance | +/-10% |
+| Resistance estimation | +/-30-50% |
+| epsilon calibration | +/-30-50% |
+
+These uncertainties are not a failure of the model. They reflect the fundamental variability of plasma discharge physics. The framework provides the correct scaling relationships and physically meaningful parameters; calibration fills in the quantitative values.
+
+## Measurement Tolerances and Error Propagation
+
+### Source Uncertainties
+
+- **Spark length**: Measured from photographs with scale reference. Branching and 3D geometry introduce ~10-20% uncertainty.
+- **Input energy**: Calculated from primary waveforms and SPICE model. Coupling uncertainty (~5%), component tolerances (~5%), and waveform measurement (~10%) combine to ~15-20%.
+- **FEMM capacitances**: Mesh density, boundary conditions, and geometry simplifications contribute ~10% uncertainty.
+- **Plasma resistance**: Most uncertain parameter. Physical bounds span orders of magnitude (1 kohm to 100 Mohm). Effective R depends on temperature, which depends on operating conditions.
+
+### Propagation to Predictions
+
+When using epsilon = E/L with both E and L uncertain:
+```
+delta_epsilon / epsilon = sqrt((delta_E/E)^2 + (delta_L/L)^2)
+                        ~ sqrt(0.20^2 + 0.15^2)
+                        ~ 25%
+```
+
+When predicting length L = E / epsilon:
+```
+delta_L / L = sqrt((delta_E/E)^2 + (delta_epsilon/epsilon)^2)
+            ~ sqrt(0.20^2 + 0.25^2)
+            ~ 32%
+```
+
+This is consistent with the stated +/-20-40% length prediction precision.
+
+## Scaling Regimes Summary
+
+| Operating Mode | Scaling Law | Exponent | Physical Basis |
+|---------------|-------------|----------|----------------|
+| Single-shot burst | L ~ sqrt(E_bang) | 0.5 | No thermal memory, voltage-limited, streamer-dominated |
+| Repetitive burst | L ~ P_avg^(0.3-0.5) | 0.3-0.5 | Partial thermal memory, frequency-dependent |
+| High duty DRSSTC | L ~ E^(0.5-0.7) | 0.5-0.7 | Partial leader transition, some thermal accumulation |
+| QCW | L ~ E^(0.6-0.8) | 0.6-0.8 | Voltage ramping, leader-dominated, thermal accumulation |
+| Ideal (no divider) | L = E / epsilon | 1.0 | Theoretical limit never achieved in practice |
+
+All real scaling exponents are less than 1.0 because the [[capacitive-divider]] always attenuates V_tip as the spark grows. The variation in exponents reflects how effectively different operating modes mitigate the divider's effect.
+
+The image `length-vs-energy-scaling.png` plots these curves on a log-log scale, showing the different slopes for different modes.
+
+## Practical Application
+
+### Using Scaling Laws for Quick Estimates
+
+Given a known coil performance at one condition:
+
+**Example**: A burst-mode coil produces 0.8 m sparks at 25 J bang energy. What length is expected at 50 J?
+
+```
+Using L ~ sqrt(E):
+L_2 / L_1 = sqrt(E_2 / E_1)
+L_2 = 0.8 * sqrt(50 / 25)
+L_2 = 0.8 * 1.414
+L_2 = 1.13 m
+```
+
+**Example**: A QCW coil produces 1.5 m sparks at 30 J total energy. What length is expected at 60 J?
+
+```
+Using L ~ E^0.7 (mid-range QCW exponent):
+L_2 / L_1 = (E_2 / E_1)^0.7
+L_2 = 1.5 * (60 / 30)^0.7
+L_2 = 1.5 * 1.625
+L_2 = 2.44 m
+```
+
+### When Scaling Laws Break Down
+
+The empirical scaling laws fail when:
+- Operating at the absolute voltage limit (no more voltage headroom)
+- Transitioning between operating modes (e.g., burst length transitions overlap QCW length scales)
+- Environmental conditions change significantly (altitude, humidity)
+- Coil topology changes (different topload, secondary, etc.)
+- Power supply limitations cap the achievable V_topload
+
+In these cases, the full simulation workflow using [[energy-and-growth]] growth equations and [[femm-workflow]] field simulations is required.
+
+## Connection to Spark Growth Timeline
+
+The worked example in `spark-growth-timeline.md` provides a concrete illustration of these scaling concepts. Key findings from that example:
+
+- **Target**: 2.0 m QCW spark at 420 kV, 12 ms ramp
+- **Achieved**: 1.0 m (voltage-limited at 50% of target)
+- **At stall**: 200 kW power available but unused for growth
+- **Growth efficiency**: Only 1.2% of delivered energy went to length extension
+- **Confirms**: Sub-linear scaling and voltage limitation as dominant constraint
+
+The example demonstrates that having sufficient power is necessary but not sufficient. The voltage limit imposed by the [[capacitive-divider]] is the binding constraint for spark length.
+
+## Key Relationships
+
+- Derives from: [[capacitive-divider]] (sub-linear scaling is a direct consequence of voltage division)
+- Derives from: [[energy-and-growth]] (growth equation dL/dt = P/epsilon provides the theoretical basis)
+- Derives from: [[field-thresholds]] (voltage limit E_tip = E_propagation sets the stall condition)
+- Differentiated by: [[streamers-and-leaders]] (channel type determines epsilon, affecting scaling exponent)
+- Differentiated by: [[thermal-physics]] (thermal persistence determines whether inter-pulse memory improves scaling)
+- Calibrated via: Experimental measurement of epsilon and E_propagation per coil
+- Verified in: [[lumped-model]] and [[distributed-model]] (circuit simulations should reproduce observed scaling)
+- Connected to: [[power-optimization]] (R_opt_power determines power delivery, which drives the energy term in scaling laws)

context/energy-and-growth.md

@@ -0,0 +1,362 @@
+---
+id: energy-and-growth
+title: "Energy Budget and Spark Growth Dynamics"
+status: established
+source_sections: "spark-physics.txt: Part 5 Sections 5.2-5.3 (lines 236-279), Part 6 Section 6.3 (lines 428-438)"
+related_topics: [field-thresholds, thermal-physics, streamers-and-leaders, capacitive-divider, empirical-scaling, power-optimization, qcw-operation, branching-physics, coupled-resonance, lumped-model, distributed-model, equations-and-bounds, open-questions]
+key_equations: [growth-rate, energy-total, power-average, epsilon-thermal-refinement, power-balance]
+key_terms: [epsilon, dL_dt, P_stream, E_propagation, E_tip, QCW, burst_mode, volumetric_energy_density, corona_to_spark_energy]
+images: [energy-budget-breakdown.png, epsilon-by-mode-comparison.png, length-vs-energy-scaling.png]
+examples: [spark-growth-timeline.md]
+open_questions:
+  - "How does epsilon vary with current density and ambient humidity?"
+  - "Can epsilon be predicted from first principles without calibration?"
+  - "What is the functional form of alpha in the thermal accumulation model epsilon(t)?"
+  - "How does branching split power and affect effective epsilon?"
+  - "What determines the transition point between power-limited and voltage-limited regimes?"
+---
+
+# Energy Budget and Spark Growth Dynamics
+
+This topic covers the fundamental energy relationships governing Tesla coil spark growth: how much energy is consumed per unit length, how growth rate connects to delivered power, and how different operating modes produce dramatically different efficiencies. Together with [[field-thresholds]] and [[capacitive-divider]], these energy relationships form the complete framework for predicting spark length.
+
+## The Central Concept: Energy per Meter (epsilon)
+
+Growth of a Tesla coil spark consumes approximately constant energy per unit length, denoted epsilon [J/m]. This is the single most important parameter for predicting spark behavior. It encapsulates all the complex plasma physics -- ionization, heating, radiation, branching -- into one empirically measurable quantity.
+
+The constancy of epsilon is an approximation. In reality, epsilon depends on channel type (streamer vs leader), thermal history, current density, and ambient conditions. However, for a given operating mode and coil, epsilon remains sufficiently constant to enable useful predictions.
+
+### Physical Origin of epsilon
+
+Energy delivered to the spark channel is consumed by several processes:
+
+- **Ionization energy**: Creating free electrons and ions in the gas (~15.6 eV per nitrogen molecule, ~12.1 eV per oxygen molecule)
+- **Thermal energy**: Heating the channel gas from ambient to plasma temperatures (1000-20000 K)
+- **Radiation losses**: UV, visible, and IR emission from the hot channel
+- **Mechanical work**: Expanding the channel against atmospheric pressure (shock waves in early phases)
+- **Branching**: Energy diverted into side branches that do not contribute to forward propagation
+
+The sum of these per unit length of forward propagation gives epsilon. Because streamers and leaders have very different physical properties, epsilon varies strongly with channel type.
+
+### Volumetric Energy Density Threshold
+
+The energy per meter (epsilon) can be connected to the volumetric energy density required for channel formation. From the gas discharge literature:
+
+```
+Minimum specific energy for spark channel formation: 0.6 - 1 J/cm^3
+```
+
+[Becker et al. 2005, Ch 2, p. 59]
+
+This is the energy density that must be deposited locally in the channel volume to achieve the corona-to-spark transition (see [[streamers-and-leaders]]). The relationship between epsilon (J/m) and volumetric energy density (J/cm^3) depends on the channel cross-section:
+
+```
+epsilon = rho_energy * A_channel * (1 + f_overhead)
+```
+
+where rho_energy is the volumetric energy density (~1 J/cm^3), A_channel is the channel cross-sectional area, and f_overhead accounts for all losses (radiation, branching, heating overhead, expansion work).
+
+For a **leader channel** (d = 3 mm): epsilon_min = 1 J/cm^3 * pi * (0.15 cm)^2 = 0.07 J/m
+For a **streamer channel** (d = 100 um): epsilon_min = 1 J/cm^3 * pi * (0.005 cm)^2 = 0.00008 J/m
+
+The observed epsilon values (5-100 J/m) are **50-1000x higher** than these bare minimums. The multiplier is explained by well-understood loss mechanisms:
+- Initial gas heating from ambient to 5000+ K (thermal energy >> ionization energy)
+- Radiation losses (UV, visible, IR)
+- Branching energy diverted to side channels
+- Shock wave and expansion work
+- Re-ionization of decayed segments between pulses
+
+This analysis confirms that the empirically observed epsilon values are physically reasonable, and establishes an independent lower bound. It also confirms that predicting epsilon from first principles requires modeling all the loss channels, not just the ionization energy -- which is why empirical calibration remains necessary (see open question in [[open-questions]]).
+
+### Energy Ceiling from Tip Capacitance
+
+An independent upper bound on the energy available for channel formation comes from the electrostatic energy stored in the spark tip:
+
+```
+W_max = pi * epsilon_0 * U^2    [J/m]
+```
+
+[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), pp. 703-704]
+
+This arises because the tip (hemisphere) has capacitance C_tip = 2*pi*epsilon_0 (independent of radius), while the channel body has capacitance per unit length C_1 = 2*pi*epsilon_0 / ln(L/r). The tip stores ln(L/r) times more energy per unit length than the body — this concentrated energy is what heats the first segment of new channel.
+
+**Application to TC sparks:**
+
+| Topload Voltage | W_max (J/m) | Heated channel radius to 5000 K |
+|----------------|-------------|--------------------------------|
+| 100 kV | 2.8 | ~0.2 mm |
+| 300 kV | 25 | ~0.6 mm |
+| 600 kV | 100 | ~1.2 mm |
+
+At 300 kV, W_max ~ 25 J/m — remarkably close to observed epsilon for QCW leaders (5-15 J/m) and within the range for burst mode (30-100 J/m). This is not a coincidence: the tip capacitance energy sets the scale of what is energetically possible per meter of new channel.
+
+**Important caveat:** W_max is the energy available from tip charge alone at the instant of new channel formation. The TC resonant circuit continuously supplies additional energy through the conducting channel during the burst. The total energy delivered over the full burst duration far exceeds W_max. However, W_max constrains the energy available for *initiating* each new leader step before the conducting core extends to deliver circuit current.
+
+### Independent Epsilon Check: Heating to 4000 K
+
+An independent estimate of epsilon from thermal energy requirements:
+
+```
+Energy to heat 1 mm diameter air column to 4000 K: ~8 J/m
+```
+
+[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 716; scaled from their 1 cm/800 J/m calculation]
+
+This assumes heating the full cross-section uniformly to the minimum self-sustaining temperature (4000 K, where associative ionization N+O->NO++e provides field-free electrons). For a 3 mm leader channel, this scales to ~72 J/m.
+
+Combined with the eta_T ~ 10% heating efficiency at ambient (see [[thermal-physics]]), the actual electrical energy needed is ~80 J/m (for 1 mm) to ~720 J/m (for 3 mm) in the initial cold phase. As the channel warms and eta_T approaches 1.0, the effective epsilon drops dramatically — exactly as observed in QCW mode where epsilon falls from ~15 J/m early to ~5-8 J/m later.
+
+## Growth Rate Equation
+
+The fundamental growth rate equation is:
+
+```
+dL/dt = P_stream / epsilon    (when E_tip > E_propagation)
+dL/dt = 0                     (when E_tip < E_propagation, stalled)
+```
+
+Where:
+- `dL/dt` is the rate of spark length increase [m/s]
+- `P_stream` is the real power delivered to the spark channel [W]
+- `epsilon` is the energy per unit length [J/m]
+- `E_tip` is the electric field at the spark tip [V/m]
+- `E_propagation` is the threshold field for sustained growth [V/m]
+
+This equation has two distinct regimes:
+
+1. **Active growth**: When E_tip exceeds E_propagation (see [[field-thresholds]]), the spark extends at a rate proportional to delivered power and inversely proportional to epsilon. More power means faster growth. Lower epsilon means more efficient growth.
+
+2. **Stalled**: When E_tip falls below E_propagation, growth stops regardless of available power. Extra power heats and brightens the existing channel but does not extend it. This is the voltage-limited regime governed by the [[capacitive-divider]] effect.
+
+The growth rate equation is a statement of energy conservation: the energy arriving per unit time (P_stream) is consumed at epsilon joules per meter of new channel, yielding dL/dt meters per second of extension.
+
+## Integrated Energy and Power Relations
+
+Over a total growth time T to reach final length L:
+
+```
+E_total = epsilon * L
+P_avg = epsilon * L / T
+```
+
+Where:
+- `E_total` is the total energy consumed by the spark for growth [J]
+- `L` is the final spark length [m]
+- `T` is the total growth time [s]
+- `P_avg` is the time-averaged power required for growth [W]
+
+These are lower bounds. The actual energy delivered to the coil system will be substantially larger due to secondary losses, corona, radiation, and excess heating during voltage-limited phases (see Power Balance below).
+
+### Numerical Examples
+
+**QCW coil aiming for 2 m spark:**
+```
+epsilon = 10 J/m (efficient QCW)
+L = 2.0 m
+T = 12 ms (typical QCW ramp)
+
+E_total = 10 * 2.0 = 20 J (minimum for growth)
+P_avg = 20 / 0.012 = 1.67 kW (minimum average spark power)
+```
+
+**Burst mode coil aiming for 1 m spark:**
+```
+epsilon = 60 J/m (streamer-dominated burst)
+L = 1.0 m
+T = 0.2 ms (short burst)
+
+E_total = 60 * 1.0 = 60 J (minimum for growth)
+P_avg = 60 / 0.0002 = 300 kW (minimum average spark power)
+```
+
+The burst mode requires 3x the energy for half the length, and 180x the average power. This starkly illustrates why operating mode matters.
+
+## Empirical epsilon Values by Mode
+
+Epsilon must be calibrated per coil from measurements. The following starting values reflect community experience:
+
+### QCW-Style Growth (epsilon = 5-15 J/m)
+
+- Long ramp times: 5-20 ms
+- Leader-dominated channels: thick, hot, low-resistance
+- Energy efficiently extends length because leaders have low epsilon
+- Continuous energy injection maintains channel temperature above thermal ionization threshold
+- Streamer-to-leader transition occurs early in ramp (see [[streamers-and-leaders]])
+- Most efficient mode for producing long sparks per joule of input energy
+
+**Measured QCW energy budget:** Loneoceans' QCW v1.5 delivers 275 J of total input energy for a 1.78 m spark, giving an apparent epsilon of 155 J/m (total input / length). At an estimated 30-50% system efficiency, the spark epsilon is 45-75 J/m. However, this includes the early inefficient growth phase (first ~2-4 ms at high epsilon) — the leader-dominated late-stage epsilon is significantly lower. [Phase 6 QCW community survey]
+
+**Measured QCW growth rate: ~170 m/s.** At this rate, a 10 ms ramp produces 1.7 m, and a 20 ms ramp produces 3.4 m — matching observed QCW spark lengths. The growth rate is limited by the driven leader step time (~60 us per step, set by the conductance heating time constant tau_g = 40 us). See [[streamers-and-leaders]] for the detailed derivation.
+
+### High Duty Cycle DRSSTC (epsilon = 20-40 J/m)
+
+- Intermediate between QCW and burst
+- Hybrid streamer/leader formation: base segments become leaders, tip remains streamer
+- Some thermal accumulation between closely spaced pulses
+- Moderate efficiency: better than single-shot but worse than continuous ramp
+- Represents many practical DRSSTC operating points
+
+### Hard-Pulsed DRSSTC / Burst Mode (epsilon = 30-100+ J/m)
+
+- Short pulses with long gaps: channel cools between events
+- Mostly streamer-dominated: thin, high-resistance, inefficient
+- Much energy goes to brightening and branching rather than forward propagation
+- High peak current produces visually impressive but short sparks
+- Poor length efficiency: voltage collapse from [[capacitive-divider]] limits extension before leader formation can occur
+- Single-shot bang energy determines length via Freau scaling (see [[empirical-scaling]])
+
+**Measured burst ceiling:** Steve Ward's DRSSTC-0.5 measurements show that spark length saturates after ~80 us of ON time, with 10-18 inch sparks at 33-180 W input and no additional length gained beyond 80 us regardless of power. This is consistent with the thermal time constant for 100 um streamers (~125 us). See [[thermal-physics]] for analysis. [Phase 6 QCW community survey]
+
+The image `epsilon-by-mode-comparison.png` provides a bar chart comparison of these ranges.
+
+## Advanced: Thermal Accumulation Refinement
+
+During sustained operation, the channel accumulates thermal energy, making subsequent extension easier. This is modeled as:
+
+```
+epsilon(t) = epsilon_0 / (1 + alpha * integral(P_stream dt))
+```
+
+Where:
+- `epsilon_0` is the initial (cold-channel) energy per meter [J/m]
+- `alpha` has units [1/J] and represents the thermal benefit rate
+- `integral(P_stream dt)` is the accumulated energy delivered to the spark [J]
+
+### Physical Interpretation
+
+As energy accumulates in the channel:
+- Temperature rises, reducing the energy needed for further ionization
+- Hot channel gas already partially ionized requires less incremental energy
+- Thermal inertia maintains conductivity, reducing re-ionization overhead
+- Net effect: epsilon decreases with time/energy, favoring longer growth
+
+This refinement is most important for QCW mode where the ramp is long enough for significant thermal accumulation. For single-shot bursts, the integral is small and epsilon remains approximately epsilon_0.
+
+### Practical Considerations
+
+The alpha parameter is difficult to measure directly. It can be inferred by:
+1. Running QCW at different ramp durations to the same target length
+2. Measuring actual spark power (from SPICE model calibrated to measurements)
+3. Fitting the epsilon(t) model to observed growth trajectories
+4. Typical alpha values: 0.01-0.1 per joule (highly dependent on conditions)
+
+For most practical calculations, using the mode-appropriate constant epsilon is sufficient. The refinement matters primarily for detailed growth simulation and for understanding why QCW efficiency improves during the ramp.
+
+## Calibration Procedure
+
+Epsilon and E_propagation must be calibrated per coil from actual measurements. This is fundamental -- the framework provides the structure for prediction, but the parameters come from experiment.
+
+### Step 1: Measure Energy per Meter (epsilon)
+
+1. **Run the coil** with known drive parameters (voltage, frequency, pulse width)
+2. **Measure spark length** L (photograph with scale reference, take multiple measurements)
+3. **Compute delivered energy** from SPICE simulation calibrated to measured primary waveforms:
+   - E_delivered = integral(P_spark dt) over the growth period
+   - P_spark is the real power delivered to the spark load in the circuit model
+4. **Calculate epsilon**: epsilon = E_delivered / L
+
+### Step 2: Measure Field Threshold (E_propagation)
+
+1. **Use FEMM** to compute E_tip for the measured V_top and final stall length L
+2. **E_propagation** is approximately E_tip at the stall point
+3. **Typical result**: 0.4-1.0 MV/m at sea level, standard conditions
+4. **Verify**: E_propagation should be consistent across different operating conditions for the same coil geometry
+
+### Measurement Tolerances
+
+- FEMM capacitance extraction: +/-10%
+- Resistance estimation: +/-30-50%
+- Epsilon: +/-30-50% (largest uncertainty source)
+- Expected prediction precision: length +/-20-40%, power +/-30-50%, phase +/-5-10 degrees
+
+These tolerances are inherent to the empirical nature of epsilon. Plasma physics variability, environmental conditions, and measurement uncertainty all contribute. The framework is designed to be useful within these ranges, not to provide precision beyond them.
+
+## Growth Simulation Algorithm
+
+For detailed time-domain simulation, the growth is computed step-by-step:
+
+```
+For each time step dt:
+  1. Check: E_tip(V_top(t), L) >= E_propagation?
+  2. If yes: dL/dt = P_stream(t) / epsilon(L, t)
+  3. If no:  dL/dt = 0 (stalled)
+  4. Update: L = L + (dL/dt) * dt
+  5. Update spark model parameters for new L:
+     - C_sh(L) = C_sh_per_meter * L
+     - R_opt(L) = 1 / (omega * (C_mut + C_sh(L)))
+     - Recompute Z_spark, I_spark, P_spark
+  6. Optionally track frequency to follow loaded pole
+```
+
+### Implementation Notes
+
+- **Time step selection**: dt should be small enough that dL changes by less than ~10% per step. For QCW ramps, dt = 0.1-0.5 ms is typical. For burst mode, dt = 1-10 microseconds.
+
+- **Frequency tracking** (Step 6): As the spark grows, the loaded pole frequency shifts (see [[coupled-resonance]]). If the drive does not track this shift, the coil detunes and power delivery drops dramatically. Proper simulation should either assume ideal tracking (drive always at loaded pole) or model the actual frequency control loop.
+
+- **Parameter coupling**: Steps 4-5 create a coupled system. As L increases, C_sh increases, which changes R_opt, which changes Z_spark, which changes P_spark, which changes dL/dt. This coupling is what makes growth nonlinear and why simple linear extrapolation fails.
+
+- **The worked example** in `spark-growth-timeline.md` demonstrates this algorithm for a QCW coil and reveals a critical finding: the coil reaches only 1.0 m of a 2.0 m target because the voltage limit (not the power limit) constrains growth.
+
+## Prediction Workflow
+
+The complete prediction workflow uses calibrated epsilon and E_propagation:
+
+### Step 1: Voltage Capability Check
+
+- Simulate to determine V_top(t) (from SPICE or analytical model)
+- Use FEMM: Is E_tip(V_top, L_target) >= E_propagation?
+- If not, the target length is voltage-limited, not power-limited
+- Reduce L_target or increase V_top
+
+### Step 2: Power/Energy Requirement
+
+- Choose growth time T (e.g., 12 ms for QCW, 0.2 ms for burst)
+- Required average spark power: P_avg = epsilon * L_target / T
+- Required total energy: E_total = epsilon * L_target
+- Is the coil capable of delivering this power?
+
+### Step 3: SPICE Verification
+
+- Build circuit model with spark load at target length
+- Run AC or transient analysis
+- Verify P_spark meets requirement from Step 2
+- Check coil stays near loaded pole frequency
+
+### Step 4: Power Balance Validation
+
+```
+P_primary_input = P_spark + P_secondary_losses + P_corona + P_radiation
+```
+
+- P_spark / P_primary_input should equal expected efficiency (15-50% typical)
+- If efficiency is unreasonably low or high, review model parameters
+- Secondary losses: 10-30% of input power
+- Corona and radiation: 5-15% of input power
+
+The image `energy-budget-breakdown.png` shows a typical energy distribution pie chart.
+
+## Scaling Behavior
+
+The growth equation dL/dt = P/epsilon, combined with the [[capacitive-divider]] effect on V_tip and E_tip, produces characteristic scaling relationships. These are explored in detail in [[empirical-scaling]], but the key insight is:
+
+- **Power-limited regime**: When E_tip is well above threshold, growth rate is proportional to P/epsilon. More power = proportionally more length per unit time.
+
+- **Voltage-limited regime**: When E_tip approaches threshold, additional power cannot extend the spark. Length saturates regardless of energy input. The excess power goes into heating and brightening.
+
+- **Transition**: Most real sparks operate in a crossover regime, initially power-limited (rapid growth) then transitioning to voltage-limited (stalling). The length vs energy curve is sub-linear: L proportional to E^0.5 for burst mode, L proportional to E^0.6-0.8 for QCW.
+
+The image `length-vs-energy-scaling.png` shows these curves on a log-log plot for different operating modes.
+
+## Key Relationships
+
+- Derives from: [[circuit-topology]] (provides the impedance framework that determines P_stream)
+- Derives from: [[power-optimization]] (R_opt_power determines the maximum extractable power)
+- Interacts with: [[field-thresholds]] (E_tip threshold determines the growth/stall boundary)
+- Interacts with: [[capacitive-divider]] (voltage division limits E_tip as spark grows, creating sub-linear scaling)
+- Interacts with: [[thermal-physics]] (thermal memory affects epsilon through the accumulation term)
+- Interacts with: [[streamers-and-leaders]] (channel type determines epsilon magnitude)
+- Enables: [[empirical-scaling]] (growth equation is the foundation for Freau's scaling laws)
+- Enables: [[lumped-model]] (growth simulation uses lumped model for impedance at each time step)
+- Enables: [[distributed-model]] (advanced growth simulation uses distributed model for accuracy)
+- Calibrated via: [[femm-workflow]] (FEMM provides capacitances needed for power calculation)

context/femm-workflow.md

@@ -0,0 +1,347 @@
+---
+id: femm-workflow
+title: "FEMM Electrostatic Workflow for Spark Capacitance Extraction"
+status: established
+source_sections: "spark-physics.txt: Part 7.2 (lines 458-477), Part 8.2 (lines 559-572), Part 6 (lines 389-438)"
+related_topics: [lumped-model, distributed-model, circuit-topology, capacitive-divider, field-thresholds, equations-and-bounds, open-questions]
+key_equations:
+  - "C_mut = |C_12| from Maxwell matrix"
+  - "C_sh = C_22 - |C_12|"
+  - "Partial capacitance transformation"
+  - "E_tip from FEMM field solution"
+  - "C_sh validation: 2 pF/foot rule"
+key_terms:
+  - "FEMM"
+  - "Finite Element Method"
+  - "Maxwell capacitance matrix"
+  - "partial capacitance"
+  - "electrostatic simulation"
+  - "self-capacitance"
+  - "mutual capacitance"
+  - "mesh refinement"
+  - "axisymmetric"
+  - "Dirichlet boundary"
+images:
+  - femm-geometry-setup-lumped.png
+  - femm-geometry-setup-distributed.png
+  - field-lines-capacitances.png
+  - femm-field-plot-example.png
+  - electric-field-enhancement.png
+  - maxwell-matrix-extraction.png
+  - partial-capacitance-transformation.png
+  - capacitance-matrix-heatmap.png
+examples:
+  - femm-lumped-extraction.md
+open_questions:
+  - "How does mesh quality near the spark tip affect E_tip computation accuracy, and what is the optimal element size at the tip?"
+  - "Can FEMM axisymmetric simulations capture non-vertical spark geometries, or is 3D FEA required for curved/angled sparks?"
+  - "What is the systematic error of modeling the spark as a perfect cylinder versus a tapered or irregular channel?"
+  - "How should multiple breakout points be handled in a single FEMM simulation?"
+---
+
+# FEMM Electrostatic Workflow for Spark Capacitance Extraction
+
+FEMM (Finite Element Method Magnetics) is an open-source finite element analysis program that, despite its name, handles electrostatic problems with high accuracy. It is the primary tool for extracting the capacitance values that populate the [[lumped-model]] and [[distributed-model]]. This document covers the complete FEMM workflow: geometry setup, meshing, solution, matrix extraction, sign convention handling, field computation, and validation. FEMM also provides the electric field at the spark tip, which is needed for growth prediction in the [[field-thresholds]] and [[energy-and-growth]] analyses.
+
+## 1. FEMM Fundamentals
+
+### 1.1 What FEMM Computes
+
+For electrostatic problems, FEMM solves Laplace's equation (nabla squared V = 0) in air with boundary conditions defined by conductor potentials and far-field grounding. From the solution, FEMM computes:
+
+- **Potential field V(r, z)** everywhere in the domain
+- **Electric field E = -grad(V)** at any point
+- **Charge on each conductor** Q_i = integral of epsilon_0 * E_n dA over the conductor surface
+- **Capacitance matrix** C[i,j] relating charges to voltages: Q_i = sum_j C[i,j] * V_j
+
+### 1.2 Problem Type and Symmetry
+
+**Problem type:** Electrostatic (DC). Although Tesla coils operate at 50-400 kHz, the wavelength (750-6000 m) is vastly larger than the spark geometry (1-5 m), so the quasi-static approximation is excellent. Capacitance values extracted at DC are accurate at operating frequency.
+
+**Symmetry:** Use axisymmetric (r-z) geometry whenever possible. A vertical spark emerging from a centered toroidal topload has cylindrical symmetry, reducing the 3D problem to 2D. This reduces computation time by orders of magnitude and improves accuracy for a given mesh density.
+
+**When 3D is needed:** Horizontal sparks, sparks from off-center breakout points, or sparks near asymmetric grounded objects cannot exploit axisymmetry and require full 3D FEA (not available in FEMM; use other tools like Elmer or COMSOL).
+
+### 1.3 Coordinate System
+
+In FEMM's axisymmetric mode:
+- **r-axis:** Radial distance from the axis of symmetry (horizontal)
+- **z-axis:** Vertical position (typically z = 0 at ground plane)
+- All geometry is drawn in the r >= 0 half-plane
+- FEMM automatically revolves it about the z-axis
+
+## 2. Geometry Setup
+
+### 2.1 Topload
+
+Model the topload as a toroid in the r-z plane. A toroid of major diameter D_major and minor diameter D_minor centered at height h is represented by its cross-sectional circle:
+
+```
+Circle center: (r_center, z_center)
+r_center = D_major/2 - D_minor/2
+z_center = h
+Circle radius = D_minor/2
+```
+
+Draw the right half of this circle (r >= 0) using arc segments. Close the contour along the axis if needed. Assign the topload surface to Conductor Group 1.
+
+### 2.2 Spark Channel
+
+**For the lumped model (single cylinder):**
+Model the entire spark as one vertical cylinder extending downward from the bottom of the topload. Key parameters:
+- **Diameter:** 1 mm for burst mode analysis, 3 mm for QCW analysis (nominal values; the weak logarithmic dependence of capacitance on diameter makes the exact choice non-critical)
+- **Length:** The target spark length L
+- **Gap:** Leave a 0.1-0.5 mm gap between the topload surface and the top of the spark cylinder for numerical stability
+
+In the r-z plane, the cylinder is a thin rectangle from (0, z_base) to (r_spark, z_tip), where r_spark = d/2 and z_tip = z_base - L. Assign the spark surface to Conductor Group 2.
+
+**For the distributed model (n segments):**
+Divide the cylinder into n equal sections, each of length L_seg = L/n. Leave 0.1 mm gaps between segments. Assign each segment to its own Conductor Group (2 through n+1). See [[distributed-model]] for the segment numbering convention.
+
+### 2.3 Ground Plane
+
+Model the ground plane as a horizontal line from (0, 0) to (R_boundary, 0), where R_boundary is the outer boundary radius. Assign Dirichlet boundary condition V = 0 to this line.
+
+### 2.4 Outer Boundary
+
+Create a rectangular boundary enclosing all geometry:
+- **Radial extent:** R_boundary = 3 to 10 times the maximum dimension (topload diameter or spark length, whichever is larger)
+- **Vertical extent:** From well below the spark tip to well above the topload
+
+Assign V = 0 (Dirichlet) or a mixed boundary condition to the outer boundary. The boundary must be far enough that C_sh changes by less than 5% when the boundary is moved 50% farther.
+
+### 2.5 Material Properties
+
+Assign the material "Air" with relative permittivity epsilon_r = 1.0 to all regions outside the conductors. The conductors themselves are equipotential surfaces (boundary conditions, not material regions).
+
+### 2.6 Mesh Control
+
+**Critical near the spark channel:** The thin spark cylinder (1-3 mm diameter) requires fine mesh elements. Set the mesh element size near the spark to be no larger than the spark diameter. For a 2 mm spark, use 2 mm maximum element size.
+
+**Near the topload:** Element size of 5-10 mm is typically sufficient.
+
+**Far field:** Coarse mesh is acceptable (50-100 mm elements). The far field contributes little to the capacitance between nearby conductors.
+
+**Mesh quality check after generation:**
+- No extremely elongated triangles (aspect ratio below 10:1)
+- Fine mesh near conductors with smooth transition to coarse mesh
+- Total element count: typically 15,000-50,000 for lumped models, 30,000-100,000 for distributed models
+
+## 3. Solution and Matrix Extraction
+
+### 3.1 Running the Solver
+
+FEMM solves the Laplace equation iteratively. Check:
+- Convergence: Final residual below 1e-8
+- Iteration count: Typically 3-10 iterations for well-conditioned problems
+- No warnings about poor mesh quality
+
+### 3.2 Visual Verification
+
+Before extracting numbers, visually inspect the solution:
+- Topload should be at the specified potential (uniform color on surface)
+- Spark should be at a lower, uniform potential (floating conductor acquires a potential determined by coupling)
+- Ground plane should be at V = 0
+- Field lines should emerge from the topload, with some terminating on the spark and others reaching ground
+- No anomalous hot spots or discontinuities
+
+### 3.3 Extracting the Maxwell Capacitance Matrix
+
+**FEMM Circuit Properties dialog** provides:
+- Voltage of each conductor (specified or computed)
+- Charge on each conductor
+- Capacitance matrix elements
+
+**For the lumped model (2x2 matrix):**
+
+```
+       [Topload]  [Spark]
+[Top]  [ C_11      C_12  ]
+[Spk]  [ C_21      C_22  ]
+```
+
+**For the distributed model ((n+1) x (n+1) matrix):**
+
+```
+       [Top]  [Seg1] [Seg2] ... [Segn]
+[Top]  [C_00  C_01   C_02  ... C_0n ]
+[Seg1] [C_10  C_11   C_12  ... C_1n ]
+[Seg2] [C_20  C_21   C_22  ... C_2n ]
+...
+[Segn] [C_n0  C_n1   C_n2  ... C_nn ]
+```
+
+### 3.4 Sign Convention: Maxwell vs. Circuit
+
+**Maxwell capacitance matrix convention:**
+- C_ii > 0: Self-capacitance (total charge on conductor i when i is at 1V and all others are grounded)
+- C_ij < 0 for i != j: Mutual coupling (charge induced on conductor i when j is at 1V and all others are grounded). The negative sign reflects that positive voltage on j induces negative charge on i.
+
+**Circuit element convention:**
+- All capacitances are positive values
+
+**Conversion for the lumped model:**
+```
+C_mut = |C_12|                (take absolute value of negative off-diagonal)
+C_sh = C_22 - |C_12|         (total self-cap minus mutual coupling)
+```
+
+**Conversion for the distributed model:**
+Use the partial capacitance transformation (see Section 4 below).
+
+**Warning:** Mixing conventions is the most common source of errors in this workflow. Always write out the absolute value signs explicitly and verify that all circuit element capacitances are positive.
+
+## 4. Partial Capacitance Transformation
+
+### 4.1 Purpose
+
+The Maxwell matrix contains negative off-diagonal elements that cannot be directly implemented as capacitors in SPICE. The partial capacitance transformation produces an equivalent network with only positive elements.
+
+### 4.2 Transformation Formulas
+
+From the Maxwell matrix C_maxwell, compute:
+
+**Capacitance between node i and node j (for i != j):**
+```
+C_branch[i,j] = -C_maxwell[i,j] = |C_maxwell[i,j]|
+```
+This is a positive capacitor connected between nodes i and j.
+
+**Capacitance from node i to ground:**
+```
+C_ground[i] = C_maxwell[i,i] + sum_{j != i} C_maxwell[i,j]
+            = C_maxwell[i,i] - sum_{j != i} |C_maxwell[i,j]|
+```
+This is the residual capacitance to the implicit ground node. It should be non-negative for a valid matrix. If it is slightly negative (numerical noise), it indicates that the conductor is almost entirely coupled to other conductors with negligible direct coupling to ground.
+
+### 4.3 Network Implementation
+
+The resulting circuit has:
+- One capacitor C_branch[i,j] between each pair of nodes (i,j)
+- One capacitor C_ground[i] from each node to ground
+- All values are positive
+
+For n+1 conductors, this produces up to n*(n+1)/2 branch capacitors plus n+1 ground capacitors. For n = 10 (11 conductors), this is up to 55 branch capacitors plus 11 ground capacitors. In practice, many branch capacitors are negligibly small and can be omitted.
+
+## 5. Electric Field Computation
+
+### 5.1 Tip Field for Growth Prediction
+
+FEMM computes the electric field magnitude at any point in the domain. For spark growth analysis (see [[field-thresholds]] and [[energy-and-growth]]), the critical quantity is the electric field at the spark tip.
+
+**Procedure:**
+1. Set topload to the peak operating voltage V_top
+2. Include the spark at its current length L
+3. Solve the electrostatic problem
+4. Query E_tip = |E| at the tip of the spark cylinder
+
+The tip field includes geometric enhancement:
+```
+E_tip = kappa * E_average
+```
+where kappa = 2 to 5 is the field enhancement factor due to the small radius of curvature at the spark tip. FEMM automatically captures this enhancement if the mesh is sufficiently fine near the tip.
+
+### 5.2 Calibration of E_propagation
+
+The field threshold for sustained spark propagation is determined by combining FEMM with experimental observation:
+
+1. Run the coil with known drive conditions
+2. Measure the final (stalled) spark length L_stall
+3. From SPICE simulation, determine V_top at the time of stall
+4. In FEMM, set up the topload at V_top with a spark of length L_stall
+5. Compute E_tip at the stall point
+6. This E_tip equals E_propagation for this coil/environment
+
+Typical result: E_propagation = 0.4 to 1.0 MV/m, depending on altitude, humidity, and channel condition.
+
+### 5.3 Accuracy Considerations
+
+**FEMM field accuracy near the tip:** The field at a sharp geometric feature (like the end of a thin cylinder) is the hardest quantity to compute accurately with FEA. The field diverges as the radius of curvature approaches zero. In practice:
+- Round the tip of the spark cylinder with a hemispherical cap of radius equal to half the spark diameter
+- Refine the mesh to at least 5 elements across the hemisphere
+- Report E_tip averaged over the hemisphere surface, not at a single point
+
+**Overall capacitance accuracy:** FEMM capacitance extraction is typically accurate to +/-10% for well-meshed problems. With careful mesh refinement and boundary testing, +/-5% is achievable.
+
+## 6. Practical Workflow Summary
+
+### 6.1 Lumped Model Extraction
+
+```
+1. Create FEMM geometry: topload + single spark cylinder + ground plane
+2. Set topload to V = 1V (test voltage)
+3. Set spark as floating conductor
+4. Mesh and solve
+5. Extract 2x2 Maxwell capacitance matrix
+6. Compute: C_mut = |C_12|, C_sh = C_22 - |C_12|
+7. Validate: C_sh within factor 2-3 of (2 pF/foot * L)
+8. Calculate: R = 1/(omega * (C_mut + C_sh)), clip to [1 kilohm, 100 megohm]
+9. Build SPICE netlist
+```
+
+### 6.2 Distributed Model Extraction
+
+```
+1. Create FEMM geometry: topload + n spark segments + ground plane
+2. Set topload to V = 1V (test voltage)
+3. Set all segments as floating conductors
+4. Mesh and solve
+5. Extract (n+1)x(n+1) Maxwell capacitance matrix
+6. Verify: symmetry, positive diagonals, negative off-diagonals
+7. Transform to partial capacitances for SPICE implementation
+8. Compute per-segment: C_total[i] = sum |C[i,j]| for j != i
+9. Calculate: R[i] = 1/(omega * C_total[i]), clip to position-dependent bounds
+10. Build SPICE network with partial capacitances and resistances
+```
+
+### 6.3 Parametric Studies
+
+FEMM simulations are fast enough (seconds to minutes) to enable parametric sweeps:
+- **Spark length variation:** Run for L = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 m to build lookup tables of C_mut(L), C_sh(L)
+- **Topload size variation:** Compare small, medium, and large toploads to understand the effect on C_mut/C_sh ratio (and hence on the phase constraint from [[circuit-topology]])
+- **Ground plane distance:** Vary the ground plane height to assess environmental sensitivity of C_sh
+- **Spark diameter:** Verify the weak logarithmic dependence of capacitance on diameter
+
+## 7. Common Mistakes and Troubleshooting
+
+### 7.1 Errors in Setup
+
+| Mistake | Symptom | Fix |
+|---------|---------|-----|
+| Spark touching topload (no gap) | Matrix extraction fails or gives anomalous values | Insert 0.1-0.5 mm gap |
+| Boundary too close | C_sh varies >5% when boundary moved | Increase to 5-10x max dimension |
+| Mesh too coarse near spark | Poor convergence or C values change with refinement | Refine mesh to spark diameter |
+| Wrong conductor assignment | Off-diagonal elements have wrong sign or magnitude | Verify conductor groups |
+
+### 7.2 Errors in Extraction
+
+| Mistake | Symptom | Fix |
+|---------|---------|-----|
+| C_sh = C_22 + C_12 (wrong formula) | Conceptual error; coincidentally gives correct result | Always use C_sh = C_22 - abs(C_12) |
+| Forgetting absolute value | Negative C_mut (impossible) | Take abs() of all off-diagonal elements |
+| Units mismatch | R_opt off by orders of magnitude | FEMM uses cm internally; convert to SI for formulas |
+| Non-symmetric matrix | Indicates poor convergence or bug | Re-mesh, refine, check boundary conditions |
+
+### 7.3 Validation Failures
+
+| Issue | Likely cause | Action |
+|-------|-------------|--------|
+| C_sh > 5 * (2 pF/ft * L) | Boundary too close; extra grounded objects | Move boundary; check geometry |
+| C_sh < 0.2 * (2 pF/ft * L) | Spark shielded by topload; ground too far | Physical; not necessarily wrong |
+| C_mut < 1 pF | Spark too far from topload; gap too large | Check gap size and topload model |
+| Negative C_ground[i] after partial transform | Numerical noise in matrix | Add +0.1 pF to diagonal |
+
+## 8. Connection to Other Topics
+
+### Key Relationships
+
+- **Serves:** [[lumped-model]] -- FEMM provides the 2x2 matrix from which C_mut and C_sh are extracted for the lumped model.
+- **Serves:** [[distributed-model]] -- FEMM provides the (n+1)x(n+1) matrix that defines the entire capacitive network of the distributed model.
+- **Computes:** [[field-thresholds]] -- FEMM computes E_tip for a given V_top and L, enabling growth prediction and E_propagation calibration.
+- **Informs:** [[capacitive-divider]] -- The voltage distribution along the spark (visible in FEMM's potential plot) directly shows the capacitive divider effect.
+- **Depends on:** [[circuit-topology]] -- The physical topology (C_mut || R in series with C_sh) motivates what quantities to extract from FEMM.
+- **Validates against:** [[equations-and-bounds]] -- All extracted capacitances and derived resistances must fall within documented physical ranges.
+
+### Worked Example
+
+The `femm-lumped-extraction.md` worked example demonstrates the complete workflow for a 30 cm x 8 cm toroid with a 1.8 m spark at 200 kHz, including mesh convergence testing, boundary sensitivity analysis, and parametric studies.

context/field-thresholds.md

@@ -0,0 +1,720 @@
+---
+id: field-thresholds
+title: "Electric Field Thresholds and Spark Inception/Propagation"
+status: established
+source_sections: "spark-physics.txt: Part 5 Section 5.1 (lines 213-234), Part 5 Section 5.6 (lines 338-360), Part 6 (lines 389-438), Part 11 (lines 763-777)"
+related_topics: [energy-and-growth, thermal-physics, streamers-and-leaders, capacitive-divider, empirical-scaling, power-optimization, branching-physics, qcw-operation, lumped-model, distributed-model, femm-workflow, equations-and-bounds, open-questions]
+key_equations:
+  - "E_inception threshold"
+  - "E_propagation threshold"
+  - "E_tip with enhancement factor"
+  - "Voltage-limited maximum length"
+  - "Capacitive voltage division V_tip"
+  - "Altitude correction for E thresholds"
+  - "Growth rate dL/dt = P_stream / epsilon"
+  - "Townsend ionization coefficient"
+  - "Paschen minimum"
+  - "Streamer criterion (Meek)"
+  - "Paschen density scaling E_breakdown proportional to N"
+key_terms:
+  - "inception field"
+  - "propagation field"
+  - "tip enhancement factor"
+  - "field dilution"
+  - "capacitive voltage division"
+  - "stall point"
+  - "streamer"
+  - "leader"
+  - "altitude correction"
+  - "Paschen curve"
+  - "Townsend coefficient"
+  - "streamer criterion"
+  - "reduced field E/N"
+  - "Townsend (Td)"
+  - "electron attachment"
+  - "corona shielding"
+  - "voltage rate limit"
+  - "dynamic threshold"
+  - "coupled voltage-power limit"
+images:
+  - electric-field-enhancement.png
+  - voltage-division-vs-length-plot.png
+  - femm-field-plot-example.png
+examples:
+  - spark-growth-timeline.md
+open_questions:
+  - "How does the tip enhancement factor kappa vary during the transition from streamer to leader?"
+  - "What is the correct E_propagation for branched vs. single-channel sparks?"
+  - "How does UV pre-ionization from the topload corona affect E_inception for secondary streamers?"
+  - "Can E_propagation be measured directly in a controlled Tesla coil experiment?"
+  - "How does the effective E_propagation change when the spark grows into regions of non-uniform background field?"
+  - "What role do runaway electrons play in Tesla coil spark inception at fields exceeding 3x the stationary breakdown threshold?"
+  - "What is the gas temperature 1-10 mm ahead of a QCW leader tip during active growth?"
+  - "Does an accelerating voltage ramp produce longer QCW sparks than a linear ramp of the same peak voltage and energy?"
+---
+
+# Electric Field Thresholds and Spark Inception/Propagation
+
+This document establishes the field-based criteria that govern whether a Tesla coil spark can form and continue to grow. Two distinct thresholds exist: the inception field (required to start a spark) and the propagation field (required to sustain growth). The interplay between topload voltage, geometric field dilution, tip enhancement, and capacitive voltage division determines the maximum voltage-limited spark length for any given operating condition.
+
+## 1. Two Distinct Field Thresholds
+
+### 1.1 Inception Field (E_inception)
+
+The inception field is the electric field required to initiate electrical breakdown from the topload surface in ambient air.
+
+```
+E_inception ~ 2 - 3 MV/m     (at sea level, standard conditions)
+```
+
+**Physical basis:** Breakdown in air requires that an electron avalanche achieves sufficient multiplication to become self-sustaining (the Townsend criterion, or equivalently, the streamer criterion). For air at atmospheric pressure, this requires approximately 3 MV/m for a uniform field. The range 2-3 MV/m reflects:
+
+- **Smooth, large-radius topload:** E_inception closer to 3 MV/m. The field is relatively uniform near the surface, and breakdown requires the full Paschen-like threshold.
+
+- **Sharp points, small radius of curvature:** E_inception closer to 2 MV/m (or even lower). Field enhancement at sharp features means the local field can exceed the breakdown threshold even when the average field is below it.
+
+- **Surface condition:** Dust, moisture, surface roughness, and residual ionization from previous sparks can reduce E_inception by providing seed electrons and reducing the statistical lag time.
+
+**Practical note for Tesla coils:** Most toploads have relatively smooth surfaces (toroidal or spherical), so E_inception is typically near the upper end of the range (2.5-3 MV/m). However, breakout points (deliberately placed sharp features) are designed to lower E_inception at a specific location to control where the spark initiates.
+
+### 1.2 Propagation Field (E_propagation)
+
+The propagation field is the electric field required at the spark tip to sustain continued growth after initial inception.
+
+```
+E_propagation ~ 0.4 - 1.0 MV/m     (at sea level, standard conditions)
+```
+
+**Why E_propagation << E_inception:** Once a spark channel exists, it extends the conductor from the topload. The spark tip concentrates the field (see Section 2 on tip enhancement), and the ionized channel behind the tip provides a low-impedance path for current. The spark effectively "sharpens" the electrode, reducing the field required for continued avalanche propagation.
+
+Additionally, the air ahead of the advancing tip has been partially pre-conditioned:
+- UV photoionization from the existing channel provides seed electrons
+- Shock heating from the advancing wavefront raises the gas temperature slightly
+- Previous streamer branches may have left residual ionization
+
+**Modeling value:** For simulation purposes, E_propagation = 0.6-0.7 MV/m is a good starting point for typical conditions. This should be calibrated against measurements (see Section 6).
+
+**Independent confirmation:** Bazelyan & Raizer (2000) report the critical average field for positive streamer propagation in air as E_cr(+) ~ 4.5-5 kV/cm (0.45-0.5 MV/m), and for negative streamers E_cr(-) ~ 10-12 kV/cm (1.0-1.2 MV/m). The positive streamer value is at the lower end of our E_propagation range, consistent with the fact that TC sparks propagating from a positive topload benefit from the easier positive streamer criterion. The 2:1 ratio between negative and positive streamer thresholds also explains observed polarity effects in TC spark length. [Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 703]
+
+### 1.3 Sharp vs. Smooth Electrode Inception
+
+| Electrode Type | Approximate E_inception | Physical Reason |
+|----------------|------------------------|-----------------|
+| Smooth sphere (R > 10 cm) | 3 - 4 MV/m | Near-uniform field at surface |
+| Smooth toroid | 2.5 - 3.5 MV/m | Mild curvature enhancement |
+| Breakout point (R ~ 1 mm) | 1 - 2 MV/m | Strong geometric enhancement |
+| Wire tip (R ~ 0.1 mm) | 0.5 - 1 MV/m | Extreme enhancement |
+
+Note: These are approximate surface field values at inception. The voltage required depends on the electrode geometry and its distance to ground.
+
+### 1.4 Breakdown Physics: Ionization and Attachment in Air
+
+The inception and propagation field values above arise from the balance between electron impact ionization and electron attachment in air. This subsection summarizes the underlying physics from the gas discharge literature.
+
+#### Reduced Electric Field (E/N)
+
+Breakdown behavior in gases is governed by the reduced electric field E/N, measured in Townsend (Td), where 1 Td = 10^-21 V*m^2. At standard temperature and pressure (STP), E/N = 100 Td corresponds to approximately E ~ 25 kV/cm (2.5 MV/m). [Becker et al. 2005, Ch 2, p. 26]
+
+#### Townsend Ionization Coefficient
+
+The electron impact ionization coefficient in air follows:
+
+```
+alpha/N = A * exp(-B * N / E)
+
+where:
+  A = 1.4 * 10^-20 m^2
+  B = 660 Td
+  Valid range: 10-150 Td (roughly 2.5-37.5 kV/cm at 1 atm)
+```
+
+[Becker et al. 2005, Ch 2, p. 32; after Wagner 1971, Moruzzi & Price 1974]
+
+More sophisticated analytical approximations for ionization and attachment coefficients covering wider E/N ranges can be found in Morrow & Lowke (1997) or Chen & Davidson (2003).
+
+#### Ionization-Attachment Crossover
+
+In air, ionization (alpha) exactly balances three-body electron attachment (eta) at:
+
+```
+E/p ~ 25 kV/cm/bar    (equivalently E/N ~ 100 Td at STP)
+```
+
+Below this field, attachment dominates and no self-sustaining discharge is possible. Above it, ionization dominates and avalanches grow. This crossover IS the physical basis for the ~2.5 MV/m breakdown field in uniform gaps. [Becker et al. 2005, Ch 2, p. 33]
+
+The electron lifetime in cold air at STP (below the crossover) is approximately **16 ns**, dominated by three-body attachment to O2. This extremely short lifetime means free electrons are essentially instantaneously lost in cold, unperturbed air -- sustaining a discharge requires continuous energy input at a rate that exceeds the attachment loss. [Becker et al. 2005, Ch 1, p. 7]
+
+#### Streamer Criterion (Meek Criterion)
+
+An electron avalanche transitions to a self-propagating streamer when the total avalanche multiplication reaches:
+
+```
+N_critical ~ 10^8 electrons    (alpha * d ~ 18-20)
+```
+
+At this point, the space charge field of the avalanche head becomes comparable to the applied field, and the avalanche becomes self-propagating via its own enhanced field. [Becker et al. 2005, Ch 2, p. 35]
+
+This criterion connects the microscopic (ionization coefficient) to the macroscopic (gap breakdown): given alpha(E) from the Townsend formula above, the minimum field for streamer formation in a gap of width d is the field where alpha(E) * d reaches 18-20.
+
+#### Mean Electron Energy at Breakdown
+
+At the breakdown threshold (E/N ~ 100 Td), the mean electron energy is approximately **3 eV** (~35,000 K electron temperature). This is far above the gas temperature (~300 K), confirming that breakdown in air is a fundamentally non-equilibrium process: the electrons are "hot" while the gas remains "cold." [Becker et al. 2005, Ch 2, p. 26]
+
+#### Runaway Electron Threshold
+
+At fields exceeding approximately **3x the stationary breakdown field**, electrons can "run away" -- gaining energy faster than they lose it through collisions. This threshold may be relevant during the initial moments of streamer head formation in Tesla coil sparks, where tip enhancement can push local fields well above 3 * 2.5 MV/m = 7.5 MV/m. [Becker et al. 2005, Ch 2, p. 39]
+
+### 1.5 Paschen Curve Quantitative Data
+
+The Paschen curve for air (breakdown voltage vs. pressure-distance product) has a well-characterized minimum:
+
+```
+V_min = 230 - 370 V
+(pd)_min ~ 0.6 torr*cm
+```
+
+[Becker et al. 2005, Ch 2, p. 33]
+
+The range in V_min reflects different electrode materials and surface conditions. For clean electrodes in dry air, V_min ~ 327 V (the classic Paschen minimum for air).
+
+For Tesla coil applications, the Paschen curve is most relevant for understanding breakdown in small gaps (e.g., breakout point spacing, spark gap switches) rather than for the long-gap streamer propagation that governs spark length. Long-gap breakdown is dominated by the streamer mechanism (Section 1.4) rather than the Townsend mechanism that underlies the Paschen curve.
+
+## 2. Tip Enhancement Factor
+
+### 2.1 Definition
+
+The electric field at the spark tip is enhanced relative to the average field (V/distance) by a geometric factor kappa:
+
+```
+E_tip = kappa * E_average
+```
+
+where E_average is the nominal field computed as if the spark were absent (e.g., V_tip / distance_to_ground for a simple geometry).
+
+### 2.2 Physical Origin
+
+The spark tip is a small-radius conductor protruding into a region of lower field. Electric field lines concentrate at the tip, just as they concentrate at any sharp conducting feature. The enhancement depends on:
+
+- **Tip radius r_tip:** Smaller radius -> higher enhancement. For a hemispherical cap on a cylinder: kappa ~ L_channel / r_tip (for long channels).
+
+- **Channel geometry:** A straight, thin channel has higher enhancement than a thick, blunt one.
+
+- **Nearby conductors:** Ground planes, strike rails, or other sparks in the vicinity can increase or decrease the local field.
+
+### 2.3 Typical Values
+
+```
+kappa ~ 2 - 5     (for cylindrical spark channels with typical aspect ratios)
+```
+
+- kappa ~ 2-3: Thick leader channels (d ~ several mm), relatively blunt tip
+- kappa ~ 3-5: Thin streamer channels (d ~ 100 um), sharp tip
+- kappa > 5: Very thin, very long channels (unusual in Tesla coil sparks)
+
+**For modeling:** kappa = 3 is a reasonable default. FEMM simulation of the specific geometry provides a more accurate value (see [[femm-workflow]]).
+
+### 2.4 FEMM Determination
+
+The most reliable way to determine kappa for a specific configuration is to run a FEMM electrostatic simulation:
+
+1. Model the topload, spark channel (as a thin conductor), and ground plane.
+2. Set the topload to a known voltage V_top.
+3. Solve for the electric field.
+4. Read E_tip at the spark tip.
+5. Compute E_average = V_tip / L (where V_tip accounts for voltage division and L is distance to ground).
+6. kappa = E_tip / E_average.
+
+![Electric field enhancement at spark tip](../assets/electric-field-enhancement.png)
+
+## 3. Voltage-Limited Maximum Length
+
+### 3.1 The Growth Condition
+
+A spark continues to grow while:
+
+```
+E_tip(V_top_peak, L) > E_propagation
+```
+
+where E_tip is a function of the topload peak voltage and the current spark length L.
+
+Growth stalls when:
+
+```
+E_tip(V_top_peak, L_max) = E_propagation
+```
+
+This defines the voltage-limited maximum length L_max for a given V_top_peak.
+
+### 3.2 Why E_tip Decreases with Length
+
+As the spark grows longer, three effects reduce E_tip:
+
+**Effect 1: Increased distance from topload.**
+The spark tip moves farther from the topload (and from ground objects behind the topload). The geometric field at the tip would decrease even for a fixed-potential tip conductor, simply because the source of the field (the topload at V_top) is farther away.
+
+**Effect 2: Geometric field dilution.**
+The field from a finite-size charged conductor (the topload) falls off with distance. For a point charge, E ~ 1/r^2. For a distributed charge on a toroid, the falloff is slower at short range (near-field) but eventually follows the 1/r^2 trend. The spark tip, being farther from the topload, sees a weaker driving field.
+
+**Effect 3: Capacitive voltage division (the most important effect for long sparks).**
+As derived in [[capacitive-divider]], the voltage at the spark tip is NOT equal to V_topload. The spark circuit forms a voltage divider between C_mut (coupling to topload) and C_sh (coupling to ground):
+
+```
+V_tip = V_topload * Z_mut / (Z_mut + Z_sh)
+```
+
+**Open-circuit limit (R -> infinity):**
+```
+V_tip ~ V_topload * C_mut / (C_mut + C_sh)
+```
+
+Since C_sh ~ 6.6 pF/m * L (proportional to spark length), V_tip decreases as the spark grows, even if V_topload is maintained constant. For a 2-meter spark with C_mut = 8 pF and C_sh = 13 pF:
+
+```
+V_tip / V_topload ~ 8 / (8 + 13) ~ 0.38
+```
+
+The spark tip sees only 38% of the topload voltage. The field at the tip is correspondingly reduced, making further growth harder.
+
+**With finite R (R ~ R_opt_power):** V_tip is even lower and has a complex (not purely real) value, but the magnitude is still reduced.
+
+### 3.3 Solving for L_max
+
+The voltage-limited length is found by solving:
+
+```
+kappa * E_average(V_top, L_max) = E_propagation
+```
+
+where E_average depends on the FEMM field solution at the tip position. This is generally not solvable in closed form; it requires:
+
+1. **Iterative FEMM simulation:** For a series of spark lengths L, compute E_tip. Find the L where E_tip = E_propagation.
+
+2. **Approximate analytic model:** Using the capacitive divider and an assumed field geometry:
+```
+E_tip ~ kappa * V_tip / (effective_gap)
+      ~ kappa * V_topload * C_mut / ((C_mut + C_sh) * (d_ground - L))
+```
+Set equal to E_propagation and solve for L. This gives a transcendental equation that must be solved numerically.
+
+### 3.4 Practical Example
+
+Consider: V_topload_peak = 400 kV, C_mut = 8 pF, C_sh = 6.6 pF/m * L, kappa = 3, E_propagation = 0.7 MV/m, distance to ground = 5 m.
+
+At L = 2 m:
+```
+C_sh = 6.6 * 2 = 13.2 pF
+V_tip = 400 * 8 / (8 + 13.2) = 400 * 0.377 = 151 kV
+E_average ~ 151 kV / (5 - 2) m = 50.3 kV/m = 0.050 MV/m
+E_tip = 3 * 0.050 = 0.15 MV/m
+```
+This is well below E_propagation = 0.7 MV/m. The simple E_average estimate is too conservative because it uses the wrong field geometry. FEMM accounts for the actual field distribution, which gives higher fields near the tip.
+
+This example illustrates why FEMM simulation is essential: naive field estimates significantly underestimate E_tip because they do not account for the field concentration geometry.
+
+![Voltage division ratio vs. spark length](../assets/voltage-division-vs-length-plot.png)
+
+![Example FEMM field solution with 2m spark](../assets/femm-field-plot-example.png)
+
+## 4. Environmental Corrections
+
+### 4.1 Altitude
+
+Air density decreases with altitude, reducing the breakdown field proportionally:
+
+```
+E(altitude) = E(sea_level) * (P / P_0)
+
+P / P_0 ~ exp(-h / 8500 m)
+```
+
+where h is the altitude in meters and P_0 is sea-level pressure.
+
+| Altitude (m) | P/P_0 | E_propagation (if 0.7 MV/m at sea level) |
+|--------------|-------|------------------------------------------|
+| 0 (sea level)| 1.000 | 0.70 MV/m |
+| 500          | 0.943 | 0.66 MV/m |
+| 1000         | 0.889 | 0.62 MV/m |
+| 1500         | 0.838 | 0.59 MV/m |
+| 2000         | 0.790 | 0.55 MV/m |
+
+At 2000 m altitude (e.g., Denver, Colorado), the propagation threshold is ~21% lower than at sea level. This means longer sparks for the same voltage, which is consistent with observations from high-altitude Tesla coil operators.
+
+### 4.2 Humidity
+
+Water vapor affects breakdown through two mechanisms:
+- **Electron attachment:** H2O has a significant electron attachment cross-section, removing free electrons and increasing the effective breakdown field. This effect INCREASES E_inception and E_propagation.
+- **Reduced density:** Water vapor is lighter than N2/O2, slightly reducing air density and thus the breakdown field. This effect DECREASES the thresholds.
+
+The net effect is small and variable:
+```
+Humidity correction: +/- 10-20%
+```
+
+High humidity generally increases E_inception slightly (harder to start sparks) but has a less clear effect on E_propagation (mixed reports in the literature).
+
+**Quantitative humidity data:** The breakdown voltage in air at atmospheric pressure has a minimum at approximately **1% water vapor** content. At low humidity, adding water vapor reduces breakdown voltage (the reduced density effect dominates). Above ~1%, the electron attachment effect of H2O begins to dominate and raises the breakdown voltage again. For typical indoor conditions (30-70% RH at 20-25 C, corresponding to roughly 0.5-2% water vapor by volume), the humidity effect on E_inception is modest and may go in either direction depending on the specific humidity level. [Becker et al. 2005, Ch 2, p. 30; Protasevich 2000]
+
+### 4.3 Temperature
+
+Air density decreases with temperature, reducing breakdown fields:
+```
+Temperature correction: +/- 5-10%
+```
+
+At 40 C vs. 20 C: air density drops by ~7%, reducing breakdown thresholds by a similar amount.
+
+### 4.4 Frequency Dependence of Breakdown
+
+The breakdown voltage in air shows a frequency dependence, with a minimum near **~1 MHz**. At frequencies well below this minimum, breakdown is governed by quasi-static (DC) processes. Near and above 1 MHz, electrons can survive the field reversal between half-cycles (the electron lifetime at STP is only 16 ns, see Section 1.4), reducing the effective breakdown threshold. [Becker et al. 2005, Ch 2, p. 30; Kunhardt 2000]
+
+**Tesla coil relevance:** Typical DRSSTC operating frequencies (50-400 kHz) are below this minimum, so frequency effects are relatively minor:
+- At **50 kHz**: Essentially DC-like breakdown behavior
+- At **200-400 kHz**: Possibly 5-10% reduction in effective inception field compared to DC predictions
+- At **~1 MHz** (some small SSTCs): Approaching the minimum, with potentially significant (~20-30%) reduction
+
+This frequency dependence is a rarely discussed factor that could contribute to observed discrepancies between DC Paschen predictions and Tesla coil inception behavior, and to performance differences between coils operating at very different frequencies.
+
+### 4.5 Corona Shielding and Voltage Rate Limit
+
+When voltage rises slowly on a rounded electrode, a stable corona (continuous low-level discharge) can form and persist indefinitely, shielding the electrode from streamer inception. This occurs because the space charge from slowly-drifting ions stabilizes the surface field at the inception level. The maximum voltage growth rate at which this shielding corona can be sustained is:
+
+```
+A_u_max ~ 2 * mu_i * E_s^2 ~ 3.6 kV/us
+```
+
+[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 5, pp. 269-270]
+
+where mu_i ~ 2 cm^2/(V*s) is ion mobility and E_s ~ 30 kV/cm is the corona stabilization field.
+
+If the voltage rises faster than A_u_max, the ions cannot drift fast enough to maintain the shielding charge cloud. The surface field increases beyond the streamer criterion, and the corona undergoes an abrupt transition to a streamer flash, which can then initiate a leader.
+
+**TC implications — corona shielding is always defeated:**
+
+A typical DRSSTC topload reaches V_top ~ 300 kV in ~1 us (one RF half-cycle at 200 kHz), giving a voltage rate of:
+
+```
+dV/dt ~ 300 kV / 1 us = 300 kV/us  >> 3.6 kV/us
+```
+
+This is ~80x faster than the corona shielding limit. The practical consequence is that **Tesla coils cannot maintain a stable corona at the topload** — every voltage cycle that exceeds inception field strength immediately produces streamers, bypassing the corona shielding phase entirely. This is consistent with the observation that TC sparks appear as bright streamer bursts from the very first cycle, not as a gradual corona-to-streamer evolution.
+
+**Comparison to lightning:** In natural lightning, the field rise rate at a grounded object from an approaching leader is much slower (~kV/ms range), allowing ultracorona to persist until the leader approaches within ~200 m altitude, at which point the rate exceeds the shielding limit and a counterleader launches. TC toploads effectively start in the "counterleader launch" regime from the first RF cycle.
+
+**Design implication:** Corona rings and smooth toploads on Tesla coils do not suppress sparks through corona shielding (the voltage rate is far too fast for that). They work by reducing the peak surface field through geometric smoothing, delaying the point during the voltage ramp when E_surface exceeds E_inception.
+
+### 4.6 Combined Uncertainty
+
+The total uncertainty in E_propagation from environmental factors is:
+
+```
+E_propagation (total uncertainty) ~ +/- 20-30%
+```
+
+This is comparable to the intrinsic variability due to spark geometry and channel conditions. For modeling purposes, calibrate E_propagation against actual spark lengths rather than relying on theoretical values (see Section 6).
+
+### 4.7 Dynamic E_propagation at Driven Leader Tips
+
+The cold-air E_propagation values in Sections 1-4.6 (0.4-1.0 MV/m) apply to streamer propagation into undisturbed ambient air. At the tip of an actively driven leader — the regime relevant to QCW Tesla coil operation — the local conditions are fundamentally different, and the effective propagation threshold is substantially lower. This section develops the physics of this dynamic threshold and argues that it resolves the apparent paradox of QCW spark lengths.
+
+#### The QCW Voltage Puzzle
+
+The most striking empirical fact about QCW Tesla coil operation [T2]:
+
+```
+QCW topload voltage: 40-70 kV     (davekni measurement, 6+ independent coils)
+QCW spark length:    2+ meters     (multiple builders, see [[qcw-operation]])
+```
+
+Compare to burst mode [T2]:
+
+```
+Burst topload voltage: 200-600 kV
+Burst spark length:    similar or shorter
+```
+
+The voltage ratio is ~10-15:1. If E_propagation were a fixed constant, a 15x lower voltage should produce dramatically shorter sparks. The naive capacitive divider calculation (Section 3.4) confirms this — 70 kV with typical TC parameters predicts stall at well under 1 meter using cold-air E_propagation.
+
+Two factors resolve this paradox:
+
+1. **Field geometry**: Naive E_avg = V_tip/distance vastly underestimates E_tip. FEMM-computed fields at the tip of a thin conductor are much higher than average-field estimates because the field concentrates at the sharp tip (see Section 2). This is a geometric effect, not a plasma physics effect, and it accounts for a significant portion of the discrepancy. [T0 — electrostatics]
+
+2. **Dynamic threshold reduction**: The effective E_propagation at a driven leader tip is much lower than the cold-air value, because the gas ahead of the tip has been pre-conditioned by multiple converging physical mechanisms. [T3 — this section]
+
+Both factors are needed. Proper field geometry alone cannot fully explain the observations, and dynamic threshold alone cannot either. The QCW spark exploits both: concentrated tip fields pushing into pre-conditioned gas with a reduced ionization threshold.
+
+#### Four Mechanisms That Reduce E_propagation
+
+A leader grows by launching streamer corona from its tip into the gas ahead (see [[streamers-and-leaders]]). In undisturbed air, these streamers require E_propagation ~ 0.5 MV/m to sustain. At a driven leader tip, four physical mechanisms converge to lower this requirement:
+
+**Mechanism 1: UV Photoionization** [T1 — mechanism established; T3 — quantitative effect at TC leader tips]
+
+The active leader tip continuously generates intense UV from the streamer corona zone. Photons with energy >12.1 eV ionize O2, creating seed electrons ahead of the advancing front.
+
+- Cold air contains essentially zero free electrons (attachment kills them in ~16 ns [T1, Becker et al. 2005])
+- A single electron must undergo ~18-20 doublings to reach the streamer criterion (10^8 electrons) [T1]
+- With UV-generated seed density of 10^7-10^8 cm^-3 [T1, simulation data], new avalanches start from a pre-existing electron cloud rather than from zero
+- This eliminates the statistical lag (waiting for a lucky first electron) and reduces the net multiplication needed for self-propagation
+- More leader current → more intense corona → more UV → denser seed cloud → lower effective field threshold [T3]
+
+The effect is strongest within ~1-5 mm of the leader tip, limited by the UV absorption length in air at atmospheric pressure.
+
+**Mechanism 2: Thermal Pre-conditioning** [T0 — Paschen scaling; T3 — application to QCW tip]
+
+Heat conducts and convects forward from the hot leader trunk (5,000-20,000 K). The gas immediately ahead of the tip is warmer than ambient, reducing its density.
+
+The fundamental relationship is Paschen scaling [T0]: breakdown field is proportional to gas number density N.
+
+```
+E_breakdown proportional to N proportional to P/(k_B * T)    (ideal gas at constant pressure)
+```
+
+If the gas ahead of the leader tip is heated from 300 K to T_local, the effective breakdown field drops by the ratio 300/T_local:
+
+| T_local (K) | T_local / T_ambient | E_prop reduction factor | Effective E_prop (from 0.5 MV/m) |
+|---|---|---|---|
+| 300 (ambient) | 1.0 | 1.0 | 0.50 MV/m |
+| 600 | 2.0 | 0.50 | 0.25 MV/m |
+| 1000 | 3.3 | 0.30 | 0.15 MV/m |
+| 1500 | 5.0 | 0.20 | 0.10 MV/m |
+| 2000 | 6.7 | 0.15 | 0.075 MV/m |
+
+[T0: the Paschen scaling math. T3: the actual temperature reached ahead of a QCW leader tip is unknown.]
+
+**How hot does the gas get ahead of the tip?** Pure thermal diffusion over distance delta in time t:
+
+```
+delta ~ sqrt(alpha_thermal * t)    where alpha_thermal ~ 2*10^-5 m^2/s
+```
+
+Over 1 ms (the timescale for a leader step): delta ~ 0.14 mm [T0]. This is tiny — pure conduction barely reaches ahead of the tip.
+
+But additional transport mechanisms push hot gas further forward [T3]:
+- Convective outflow from the expanding leader channel displaces hot gas forward
+- The shock wave from rapid channel heating creates a transient low-density zone ahead
+- Radiation from the hot channel core heats surrounding gas volumetrically
+
+The net effect is that gas within ~1-10 mm of the leader tip is significantly above ambient temperature [T3]. Even modest heating to 600-1000 K halves or thirds the effective E_propagation.
+
+**Mechanism 3: Residual Ionization** [T1 — recombination data; T3 — application to TC]
+
+Previous streamer passages leave residual ionization in the zone ahead of the leader tip. This residual charge persists because recombination is slow relative to the propagation timescale:
+
+```
+tau_recomb ~ 1/(alpha_recomb * n_e) ~ 50 us    (at n_e ~ 10^13 cm^-3)
+```
+
+[T1, Becker et al. 2005, Ch 4]
+
+In QCW operation, the leader tip corona is continuously refreshed. New streamers propagate into the fading remnants of previous ones, not into pristine air. The residual electron density means:
+
+- The effective seed electron density is orders of magnitude above zero [T3]
+- Avalanches start from a pre-ionized state, requiring less multiplication
+- The gas retains partial conductivity, reducing the field needed to drive current through it
+- Each successive streamer cycle starts from a higher baseline ionization [T3]
+
+This mechanism is cumulative during the QCW ramp: the longer the leader has been active, the more thoroughly pre-ionized the zone ahead of its tip becomes [T3].
+
+**Mechanism 4: Gas Expansion and Density Reduction** [T0 — gas dynamics; T3 — magnitude at TC tips]
+
+When the leader channel heats, the gas expands at approximately constant pressure (the acoustic transit time across the channel, ~d/v_sound ~ 1 mm / 340 m/s ~ 3 us, is fast compared to the heating timescale). This expansion:
+
+- Reduces gas density within and near the channel [T0]
+- Creates an outward flow that pushes lower-density gas forward [T3]
+- Means the region immediately ahead of the tip is at lower N than ambient [T3]
+
+Since E/N ~ 100 Td is the fundamental breakdown parameter [T1], lower N means breakdown occurs at lower absolute E. This is the same physics as the altitude correction (Section 4.1), but locally produced by the leader's own heating.
+
+For a channel at 5000 K, the internal density is 300/5000 = 6% of ambient [T0]. The gas ahead of the tip won't reach 5000 K, but even partial heating produces significant density reduction (see Mechanism 2 table).
+
+#### The Convergent Nature of the Effect
+
+The four mechanisms are not independent — they reinforce each other [T3]:
+
+```
+Leader current → UV + heating + residual ionization + expansion
+     │
+     ├─→ UV creates seed electrons ahead of tip
+     │
+     ├─→ Heat reduces gas density ahead of tip
+     │         │
+     │         └─→ Lower density + seed electrons
+     │               → lower field needed for avalanche
+     │               → streamer propagates at lower E
+     │               → leader extends further
+     │
+     ├─→ Residual ionization from previous streamers
+     │     → next streamers start from pre-ionized gas
+     │     → further reduces required field
+     │
+     └─→ Gas expansion reduces local N
+           → E/N threshold reached at lower absolute E
+```
+
+Each mechanism makes the others more effective. More current produces more UV, more heating, and more residual ionization simultaneously. The net reduction in effective E_propagation is greater than any single mechanism alone would produce [T3].
+
+#### The Coupled Voltage-Power Limit
+
+This convergent dynamic has a profound consequence: **voltage and power are not independent limits on spark length** [T3].
+
+In the traditional model, there are two separate constraints:
+1. Voltage limit: E_tip must exceed E_propagation (fixed constant)
+2. Power limit: must deliver enough energy per unit time at rate P/epsilon
+
+The dynamic threshold framework couples these: **power delivery modifies the conditions that determine the voltage threshold**. Specifically:
+
+- More power through the leader → more heating, UV, ionization at the tip [T3]
+- This reduces the effective E_propagation [T3]
+- Which allows growth to continue at lower V_tip [T3]
+- Which means the spark can grow longer before the capacitive divider stalls it [T3]
+
+The "voltage limit" is therefore not a fixed line that the capacitive divider marches toward. It is a moving target that retreats as power increases — but with diminishing returns.
+
+#### Saturation and the Ultimate Limit
+
+The dynamic threshold cannot reduce E_propagation to zero [T0 — ionization requires nonzero field]. Several effects create a floor:
+
+1. **Minimum E/N for net ionization**: Even in pre-heated, pre-ionized gas, some minimum E/N is needed to drive ionization faster than attachment/recombination. In hot air (>2000 K), attachment is suppressed (see [[thermal-physics]]), but ionization still requires field-driven avalanches. [T1]
+
+2. **Diminishing returns on each mechanism** [T3]:
+   - UV seed density saturates (finite photon production rate, absorption limits range)
+   - Thermal pre-conditioning is limited by how far ahead heat can propagate (~mm scale)
+   - Residual ionization decays between leader steps (tau_recomb ~ 50 us)
+   - Gas expansion is bounded by the pressure ratio (can't go below zero density)
+
+3. **The capacitive divider always wins eventually** [T0]: V_tip = V_topload * C_mut/(C_mut + C_sh) decreases monotonically with spark length. Even with a very low E_propagation floor, there exists a length where E_tip drops below it.
+
+The ultimate stall length for a QCW spark is therefore set by the intersection of two curves [T3]:
+- The decreasing E_tip curve (capacitive divider + field geometry, computed by FEMM)
+- The decreasing E_propagation_effective curve (dynamic threshold, set by delivered power)
+
+Both curves decrease with spark length, but E_tip decreases faster (because the capacitive divider is relentless and C_sh grows linearly). Eventually E_tip drops below E_propagation_effective, and the spark stalls.
+
+#### Connection to QCW Ramp Regimes
+
+The dynamic threshold framework provides a unified explanation for the three QCW ramp regimes documented in [[qcw-operation]] [T3]:
+
+**Too short (<5 ms):** Insufficient time for the thermal mechanisms to develop. The leader is young, the gas ahead is barely pre-conditioned, and E_propagation_effective is still close to the cold-air value. Growth is voltage-limited at a short length. Sparks are segmented, gnarly, high-epsilon.
+
+**Optimal (10-20 ms):** The leader has time to fully develop. By 2-5 ms, the thermal ratchet has established a hot leader trunk, UV production is intense, and residual ionization ahead of the tip is high. E_propagation_effective is well below the cold-air value. The spark grows efficiently as a single channel (see [[branching-physics]]) at low epsilon. Growth continues until the capacitive divider finally overwhelms the dynamic threshold.
+
+**Too long (>25 ms):** The spark has already reached the ultimate stall length — where E_tip equals E_propagation_effective even with maximal pre-conditioning. Additional power cannot reduce E_propagation further at the tip (saturation). The energy must go somewhere: it heats and thickens the leader trunk, eventually triggering lateral breakouts (see [[branching-physics]] Section 4.3). The spark gets "hot and fat but bushy" rather than longer.
+
+#### Why This Doesn't Help Burst Mode
+
+Burst pulses (70-150 us) are too short for the dynamic threshold to develop significantly [T3]:
+
+- UV is present but transient — dies with each pulse
+- Thermal pre-conditioning requires sustained heating (~ms) that a single 100 us pulse doesn't provide
+- Residual ionization from one pulse persists (~50 us tau_recomb) but decays during the inter-pulse gap (5-10 ms)
+- Gas expansion is localized and transient
+
+Each burst pulse propagates streamers into approximately cold, un-conditioned air. The effective E_propagation is close to the cold-air value. This is why burst mode needs 200-600 kV to achieve similar spark lengths — it cannot exploit the dynamic threshold reduction, so it must rely on brute-force voltage [T3].
+
+This provides another perspective on the 10-15:1 voltage ratio between burst and QCW [T2]: it is a rough measure of how much the dynamic threshold effects reduce the effective E_propagation during sustained QCW operation [T3].
+
+#### Testable Predictions
+
+The dynamic threshold framework makes specific predictions that could be tested experimentally [T4]:
+
+1. **Effective E_propagation at stall**: At the moment a QCW spark stops growing, E_tip (measurable via FEMM + V_topload) equals E_propagation_effective. This should be much lower than 0.5 MV/m. No such measurement exists yet.
+
+2. **Power dependence**: E_propagation_effective should decrease with increasing leader power. Two QCW sparks of the same length but different power levels should stall at different times (the higher-power one stalls later).
+
+3. **Frequency dependence**: Higher RF frequency → more heating cycles per unit time → faster development of pre-conditioning → lower E_propagation_effective at a given time into the ramp. This is consistent with the observed 300-600 kHz threshold for QCW swords [T2], but the connection to the dynamic threshold specifically (as opposed to the thermal ratchet generally) is untested.
+
+4. **Temperature measurement**: Spectroscopic measurement of gas temperature 1-10 mm ahead of a QCW leader tip should show significantly elevated temperature (>600 K, possibly >1000 K). No such measurement exists for TC sparks.
+
+5. **Ramp shape sensitivity**: If the dynamic threshold is real, an accelerating voltage ramp (faster increase late in the ramp) should produce longer sparks than a linear ramp of the same peak voltage and energy, because it delivers more power at the end when E_propagation_effective is already low. This is a specific, testable prediction that distinguishes the dynamic threshold model from a fixed-threshold model.
+
+## 5. Spark Growth Dynamics
+
+### 5.1 The Growth Equation
+
+Spark growth rate is determined by the power available and the energy cost per meter (see [[energy-and-growth]] for detailed treatment):
+
+```
+dL/dt = P_stream / epsilon     (when E_tip > E_propagation)
+dL/dt ~ 0                      (when E_tip < E_propagation, stalled)
+```
+
+The field threshold acts as a gate: growth can only occur when sufficient field exists at the tip. The rate of growth, when it occurs, is governed by the power-to-energy ratio.
+
+### 5.2 Time-Stepped Growth Simulation
+
+For each time step dt in a growth simulation:
+
+```
+1. Compute V_topload(t) from the drive model (or Thevenin equivalent + loaded frequency)
+2. Compute V_tip from the capacitive divider (current C_mut, C_sh, R)
+3. Compute E_tip from FEMM (or approximate formula) at current length L
+4. Check: E_tip >= E_propagation?
+   - If yes: dL/dt = P_stream(t) / epsilon(L, t)
+   - If no:  dL/dt = 0 (stalled; spark cannot advance)
+5. Update: L = L + dL/dt * dt
+6. Update spark model parameters (C_sh, R_opt) for new L
+7. Optionally: retune to loaded pole frequency (see [[coupled-resonance]])
+8. Repeat
+```
+
+### 5.3 Stall and Recovery
+
+When E_tip drops below E_propagation, the spark stalls but does not necessarily extinguish immediately:
+
+- The channel remains hot for a thermal time constant (see [[thermal-physics]])
+- If V_topload increases (e.g., during QCW ramp), E_tip may recover above threshold
+- The spark resumes growth from its current length, not from zero (thermal memory preserves the channel)
+
+This stall-recovery dynamic is common in QCW operation, where the voltage ramp may briefly lag behind the increasing field threshold as the spark lengthens.
+
+## 6. Calibration Procedure
+
+### 6.1 Determining E_propagation from Measurements
+
+E_propagation is best determined empirically for each coil:
+
+1. **Measure spark length** L for a known operating condition (drive voltage, pulse width, frequency).
+2. **Run FEMM simulation** with topload at V_top_peak and a spark conductor of length L.
+3. **Read E_tip** from the FEMM solution at the spark tip position.
+4. **At the stall point (spark at maximum length):** E_tip ≈ E_propagation.
+
+This gives E_propagation for the specific coil, environment, and operating mode. Typical results: 0.4-1.0 MV/m, with 0.6-0.7 MV/m being common for medium DRSSTCs at sea level.
+
+### 6.2 Determining kappa from FEMM
+
+Run the FEMM simulation described in Section 2.4 for several spark lengths to establish how kappa varies with geometry. For a self-consistent model, use the same kappa profile when predicting growth.
+
+### 6.3 Validation
+
+After calibrating E_propagation and kappa:
+- Predict spark length for a different operating condition (different drive voltage, different pulse width)
+- Compare to measurement
+- If prediction is consistently off, adjust E_propagation
+
+A well-calibrated model should predict spark lengths to within +/-20% across a range of operating conditions.
+
+## 7. Connection to Other Topics
+
+### Key Relationships
+
+- **Derives from:** Gas discharge physics (Townsend/streamer breakdown theory, Paschen's law)
+- **Interacts with:** [[capacitive-divider]] (voltage division directly determines V_tip and hence E_tip)
+- **Enables:** [[energy-and-growth]] (field threshold is one of two conditions for spark growth; the other is energy/power)
+- **Interacts with:** [[streamers-and-leaders]] (streamer vs. leader propagation has different effective E_propagation)
+- **Interacts with:** [[thermal-physics]] (temperature affects local gas density and thus local breakdown field)
+- **Measured via:** [[femm-workflow]] (FEMM provides E_tip for given V_top and spark geometry)
+- **Constrains:** [[lumped-model]] and [[distributed-model]] (the field condition determines whether each segment can grow)
+- **Explains:** [[empirical-scaling]] (the sub-linear L vs. E relationship arises from capacitive voltage division reducing E_tip)
+
+### Summary of Key Results
+
+1. Two thresholds: E_inception ~ 2-3 MV/m (to start), E_propagation ~ 0.4-1.0 MV/m (to sustain growth).
+2. Tip enhancement: E_tip = kappa * E_average, with kappa ~ 2-5 for typical spark channels.
+3. Three mechanisms reduce E_tip with length: distance, geometric dilution, capacitive voltage division.
+4. Capacitive voltage division (V_tip/V_topload = C_mut/(C_mut + C_sh)) is the dominant effect for long sparks.
+5. Altitude correction: E(alt) = E(sea level) * exp(-h/8500). Humidity: +/-10-20%. Temperature: +/-5-10%.
+6. Total environmental uncertainty in E_propagation: +/-20-30%. Calibration against measurements is essential.
+7. The growth condition (E_tip > E_propagation) acts as a gate; growth rate is set by power/energy balance.
+8. FEMM simulation is essential for accurate E_tip determination; naive field estimates are unreliable.
+9. [T3] E_propagation is not fixed — at a driven leader tip, UV, heat, residual ionization, and gas expansion dynamically reduce it. Voltage and power are coupled limits, not independent.
+10. [T3] The dynamic threshold explains QCW's 10-15:1 voltage advantage over burst mode, the three QCW ramp regimes, and why burst mode can't exploit the same physics.

context/lumped-model.md

@@ -0,0 +1,361 @@
+---
+id: lumped-model
+title: "Lumped Spark Model: Single-Element Circuit Representation"
+status: established
+source_sections: "spark-physics.txt: Part 7 (lines 442-537), Part 10.1 (lines 705-713), Part 11 (lines 736-803)"
+related_topics: [circuit-topology, power-optimization, thevenin-method, coupled-resonance, distributed-model, femm-workflow, capacitive-divider, field-thresholds, energy-and-growth, empirical-scaling, equations-and-bounds]
+key_equations:
+  - "C_mut extraction from Maxwell matrix"
+  - "C_sh extraction from Maxwell matrix"
+  - "R_opt_power for lumped model"
+  - "Ringdown Q_L and G_total"
+  - "Spark admittance from ringdown"
+key_terms:
+  - "mutual capacitance"
+  - "shunt capacitance"
+  - "Maxwell capacitance matrix"
+  - "self-capacitance"
+  - "ringdown method"
+  - "conductance"
+  - "parallel RLC"
+  - "Rogowski coil"
+  - "VNA"
+images:
+  - capacitance-matrix-heatmap.png
+  - lumped-vs-distributed-comparison.png
+  - lumped-model-validation-checks.png
+  - femm-geometry-setup-lumped.png
+  - field-lines-capacitances.png
+examples:
+  - femm-lumped-extraction.md
+open_questions:
+  - "How does the lumped model degrade in accuracy as spark length exceeds 10 feet, and is there a smooth transition criterion to switch to distributed?"
+  - "Can a single lumped element capture the leader/streamer boundary at all, or is any spatial information fundamentally inaccessible?"
+  - "What is the systematic error introduced by using a nominal channel diameter in FEMM rather than the actual (unknown) diameter profile?"
+  - "How sensitive is C_mut to topload geometry variations (asymmetric toroids, breakout points) compared to C_sh sensitivity to environment?"
+---
+
+# Lumped Spark Model: Single-Element Circuit Representation
+
+The lumped model reduces the entire Tesla coil spark channel to a single circuit element consisting of three passive components: a mutual capacitance C_mut, a shunt capacitance C_sh, and a resistance R. Despite its simplicity, this model captures the essential impedance behavior of the spark as seen from the topload port, enabling impedance matching analysis, fast SPICE simulation, and coil design optimization. It is the foundation upon which the more sophisticated [[distributed-model]] is built, and it directly implements the topology derived in [[circuit-topology]].
+
+## 1. Model Structure
+
+### 1.1 Circuit Topology
+
+The lumped spark model has the following structure:
+
+```
+        C_mut
+Topload ----||---- Node_spark
+                      |
+                     [R]
+                      |
+                   [C_sh]
+                      |
+                     GND
+```
+
+Reading the circuit from top to bottom:
+
+- **C_mut** (mutual capacitance) and **R** (channel resistance) are connected in parallel between the topload node and an internal spark node. C_mut provides the displacement current path; R provides the conduction current path through the plasma.
+- **C_sh** (shunt capacitance) connects the internal spark node to ground, representing the distributed capacitance of the entire spark channel to the surrounding environment.
+
+This is the same bridged-T topology analyzed in [[circuit-topology]], with the critical difference that here the component values are extracted from specific FEMM simulations rather than treated as free parameters.
+
+### 1.2 Physical Interpretation
+
+Each component represents a physically distinct mechanism:
+
+- **C_mut**: The capacitive coupling between the spark plasma and the topload. Displacement current flows through this path. C_mut depends primarily on topload geometry and the proximity of the spark base to the topload surface. For typical toroidal toploads with sparks of 1-5 feet, C_mut ranges from 3 to 15 pF. C_mut is relatively insensitive to spark length because the coupling is dominated by the near-field region close to the topload.
+
+- **C_sh**: The capacitance from the spark channel to ground and all other environmental conductors. Empirically, C_sh scales approximately linearly with spark length at roughly 2 pF per foot (6.6 pF per meter). This scaling holds because a longer spark presents more conductor length to the surrounding environment. C_sh is sensitive to the proximity of grounded objects, walls, and the ground plane distance.
+
+- **R**: The effective resistance of the plasma channel. This is the parameter the plasma self-optimizes according to the "hungry streamer" principle (see [[power-optimization]]). R can range from 1 kilohm (very hot, thick leader plasma) to 100 megohm (cold, thin streamer plasma), depending on channel temperature, ionization level, and diameter.
+
+## 2. FEMM Extraction Procedure
+
+### 2.1 Electrostatic Simulation Setup
+
+The lumped model extraction requires a FEMM electrostatic simulation with two conductors plus the environment (ground). See [[femm-workflow]] for detailed setup instructions.
+
+**Geometry elements:**
+- Topload at specified potential V (typically 1 V for normalization)
+- Spark as a single cylindrical conductor (nominal diameter: 1 mm for burst mode, 3 mm for QCW)
+- Ground plane and far-field boundaries
+
+**Key modeling decisions:**
+- Small gap (0.1-0.5 mm) between topload and spark base for numerical stability
+- Far-field boundary at least 3 times the maximum dimension
+- Mesh refinement near the thin spark cylinder (element size no larger than the spark diameter)
+
+### 2.2 Maxwell Capacitance Matrix
+
+FEMM produces a 2x2 Maxwell capacitance matrix:
+
+```
+       [Topload]  [Spark]
+[Top]  [ C_11      C_12  ]
+[Spk]  [ C_21      C_22  ]
+```
+
+**Sign convention (critical):** In the Maxwell capacitance matrix:
+- Diagonal elements C_ii > 0 (self-capacitance, always positive)
+- Off-diagonal elements C_ij < 0 for i != j (mutual coupling, always negative)
+- The matrix is symmetric: C_12 = C_21
+
+### 2.3 Extracting Circuit Element Values
+
+**Mutual capacitance:**
+```
+C_mut = -C[topload, spark] = |C_12|
+```
+
+Take the absolute value of the negative off-diagonal element. This converts from the Maxwell convention (negative mutual) to the circuit element convention (positive capacitance).
+
+**Shunt capacitance:**
+```
+C_sh = C[spark, spark] + C[spark, topload] = C_22 - |C_12|
+```
+
+The diagonal element C_22 is the total self-capacitance of the spark conductor, which includes charge coupled to both the topload and to ground. To isolate the shunt-to-ground capacitance, we subtract the mutual coupling component.
+
+**Derivation of C_sh formula:** When the topload is grounded (V_topload = 0) and the spark is at V_spark = 1V, the total charge on the spark is Q_spark = C_22 * 1V. This charge distributes between the topload side (magnitude |C_12| * 1V) and the ground side. The ground-referenced capacitance is therefore C_sh = C_22 - |C_12|.
+
+**Sign convention warning:** Always use `C_sh = C_22 - |C_12|` with explicit absolute value notation. Writing `C_sh = C_22 + C_12` happens to give the correct numerical result (since C_12 is negative), but obscures the sign handling and invites errors.
+
+### 2.4 Total Capacitance Identity
+
+The total capacitance of the spark is:
+```
+C_total = C_mut + C_sh = |C_12| + (C_22 - |C_12|) = C_22
+```
+
+This is not a coincidence: for a 2-conductor system with ground as the reference, the total capacitance from one conductor to all others equals its self-capacitance (the diagonal element).
+
+### 2.5 Validation: The 2 pF/foot Rule
+
+After extraction, validate C_sh against the empirical rule:
+```
+C_sh_expected = 2 pF/foot * L_spark_in_feet
+```
+
+A factor of 2-3 discrepancy is acceptable and common because:
+- Topload shielding reduces effective C_sh (FEMM accounts for this, the rule does not)
+- Ground plane distance varies (the empirical rule assumes a "typical room")
+- Spark diameter affects C logarithmically (C proportional to 1/ln(h/d))
+- Real sparks are curved and branched, not straight cylinders
+
+The empirical rule is a rough validation check, not a precision target. Use the FEMM-extracted value for all calculations.
+
+## 3. Determining the Resistance R
+
+### 3.1 Default Calculation: R_opt_power
+
+The recommended approach is to set R to the value that maximizes power transfer from the topload to the spark (see [[power-optimization]] for derivation):
+
+```
+R = R_opt_power = 1 / (omega * (C_mut + C_sh))
+```
+
+where omega = 2 * pi * f is the angular frequency of operation.
+
+**Numeric example:** At f = 200 kHz with C_mut = 10.5 pF and C_sh = 6.3 pF:
+```
+C_total = 10.5 + 6.3 = 16.8 pF
+omega = 2 * pi * 200,000 = 1.257e6 rad/s
+R_opt_power = 1 / (1.257e6 * 16.8e-12) = 47,300 ohm = 47.3 kilohm
+```
+
+### 3.2 Physical Bounds and Clipping
+
+The calculated R_opt_power must be checked against physical limits:
+
+```
+R_min = 1 kilohm   (very hot, thick leader plasma: sigma ~ 10 S/m)
+R_max = 100 megohm  (cold, thin streamer plasma: sigma ~ 0.01 S/m)
+
+R_actual = clip(R_opt_power, R_min, R_max)
+```
+
+If clipping occurs:
+- **R_opt_power < R_min:** The circuit "wants" a lower resistance than any plasma can provide. The spark is power-limited; check if the source can supply sufficient current at this low impedance.
+- **R_opt_power > R_max:** The circuit "wants" a higher resistance than any plasma presents. The spark may not form at all, or it operates as a very faint streamer.
+
+### 3.3 Justification: The Hungry Streamer Principle
+
+Why set R to R_opt_power rather than measuring it directly? Because of Steve Conner's "hungry streamer" insight: the plasma actively adjusts its properties (temperature, ionization, diameter, conductivity) to maximize the power it extracts from the resonant circuit. The feedback loop is:
+
+1. More power delivered to spark leads to Joule heating (I squared R)
+2. Higher temperature causes thermal ionization and increased electron density
+3. Increased conductivity causes R to decrease
+4. Changed geometry and expansion modify C_mut and C_sh
+5. Modified capacitances shift R_opt_power
+6. Plasma conductivity adjusts toward the new R_opt_power
+7. Stable equilibrium is achieved when R_actual is approximately R_opt_power
+
+This self-optimization has limits: insufficient source power, inception field not achieved, physical conductivity bounds (R_min, R_max), and thermal time constants (plasma cannot adjust faster than roughly 1 millisecond).
+
+## 4. User Measurement Integration
+
+### 4.1 Ringdown Method (Improved)
+
+For users who can measure the loaded Tesla coil ringdown, the spark admittance can be extracted without FEMM. At the loaded resonant frequency omega_L, model the system as a parallel RLC:
+
+**Fundamental relations:**
+```
+Q_L = omega_L * C_eq * R_p = R_p / (omega_L * L)
+
+R_p = Q_L / (omega_L * C_eq)       [parallel resistance form]
+R_p = Q_L * omega_L * L             [equivalent, using inductance]
+
+G_total = 1/R_p = omega_L * C_eq / Q_L    [total conductance]
+G_total = 1 / (Q_L * omega_L * L)          [equivalent form]
+```
+
+**Measurement procedure:**
+
+1. **Unloaded measurement:** Record the unloaded resonant frequency f_0, quality factor Q_0, and secondary capacitance C_0 (from geometry or separate measurement with known test capacitor).
+
+2. **Loaded measurement:** With the spark present, record the loaded frequency f_L and loaded quality factor Q_L. Note that f_L < f_0 because the spark adds capacitance.
+
+3. **Calculate equivalent capacitance:**
+```
+C_eq = C_0 * (f_0 / f_L)^2
+```
+This uses the relation f = 1/(2*pi*sqrt(L*C)) with L assumed constant.
+
+4. **Calculate capacitance change:**
+```
+delta_C = C_eq - C_0
+```
+This represents the capacitance added by the spark.
+
+5. **Calculate total conductance:**
+```
+G_total = omega_L * C_eq / Q_L
+```
+
+6. **Calculate unloaded conductance:**
+```
+G_0 = omega_0 * C_0 / Q_0
+```
+where omega_0 = 2 * pi * f_0. This represents all secondary losses (wire resistance, dielectric, corona) without the spark.
+
+7. **Extract spark admittance:**
+```
+Y_spark = (G_total - G_0) + j * omega_L * delta_C
+```
+
+The real part gives the spark conductance (and hence resistance), while the imaginary part gives the additional susceptance.
+
+**Important caveat:** This method is sensitive to primary coupling effects. The measured Q_L and f_L can be distorted by the primary-to-secondary coupling ratio. The [[thevenin-method]] is more robust because it explicitly accounts for the Thevenin impedance of the source.
+
+### 4.2 Direct Measurement
+
+For laboratory-grade characterization:
+
+- **E-field probe for V_top:** An isolated, calibrated D-dot or capacitive probe placed near the topload measures the topload voltage waveform. Must be calibrated against a known reference.
+
+- **Rogowski coil or current transformer for I_spark:** Place the sensor around the spark return current path. **Critical:** Measure the spark return current, NOT the base current I_base. The base current includes all displacement currents from the secondary to ground, which are not part of the spark load (see [[thevenin-method]] for why V_top/I_base is wrong).
+
+- **Calculate admittance:**
+```
+Y = I_spark / V_top
+```
+Then extract R, C_mut, C_sh by fitting the circuit model to the measured admittance.
+
+- **Low-level option:** A VNA (Vector Network Analyzer) with capacitive pickup can verify Z_th without requiring a spark, providing the Thevenin impedance of the unloaded coil.
+
+## 5. Implementation Workflow
+
+The complete lumped model workflow proceeds in six steps:
+
+**Step 1: FEMM electrostatic simulation**
+Set up the topload and a single spark cylinder. Solve the electrostatic problem. See [[femm-workflow]] for details.
+
+**Step 2: Extract C_mut and C_sh from the Maxwell matrix**
+```
+C_mut = |C_12|
+C_sh = C_22 - |C_12|
+```
+Validate: C_sh should be within a factor of 2-3 of the 2 pF/foot empirical rule.
+
+**Step 3: Calculate R**
+```
+R = 1 / (omega * (C_mut + C_sh))
+R = clip(R, 1 kilohm, 100 megohm)
+```
+
+**Step 4: Build SPICE netlist**
+```
+* Lumped spark model
+C_mut topload spark_node  [C_mut value]
+R_spark spark_node spark_gnd  [R value]
+C_sh spark_gnd 0  [C_sh value]
+```
+Note: C_mut and R are in parallel between topload and spark_node. C_sh connects spark_node to ground.
+
+**Step 5: AC analysis**
+Use the [[thevenin-method]] or direct power measurement to evaluate performance. Sweep frequency around the expected operating point to find the loaded pole.
+
+**Step 6: Matching optimization**
+Iterate on design parameters (topload size, primary tap, coupling) to maximize power delivered to the spark at the target operating conditions.
+
+## 6. Limitations and Applicability
+
+### 6.1 What the Lumped Model Does Well
+
+- **Impedance matching studies:** The lumped model correctly captures the impedance presented by the spark to the Tesla coil resonant circuit. It accurately predicts R_opt_power, the phase constraint phi_Z_min, and the power transfer as a function of R.
+
+- **Fast simulation:** A single lumped element adds negligible computational cost to a SPICE simulation. This enables rapid parameter sweeps over frequency, coupling, spark length, and other design variables.
+
+- **Design optimization:** For coil designers, the lumped model is sufficient to choose primary tap point, capacitor bank size, coupling coefficient, and drive strategy. The spatial detail of the distributed model is unnecessary for these decisions.
+
+### 6.2 What the Lumped Model Cannot Capture
+
+- **Current distribution along the spark:** The model has a single current flowing through R. It cannot distinguish base current from tip current, which differ by a factor of 2-3 in practice (see [[distributed-model]]).
+
+- **Tip versus base differences:** The distinction between hot leader plasma at the base and cold streamer plasma at the tip is invisible to the lumped model. These regions have very different resistances, temperatures, and optical signatures.
+
+- **Streamer-to-leader transitions:** The transition from high-resistance streamer to low-resistance leader is a spatially distributed process that requires at minimum a two-element model to represent.
+
+- **Very long sparks (greater than 10 feet):** As sparks become very long, the capacitive voltage division along the channel becomes severe. The [[capacitive-divider]] effect attenuates the tip voltage significantly, and the single-section model cannot capture the progressive attenuation along the length.
+
+### 6.3 Decision Criteria: Lumped vs. Distributed
+
+Use the lumped model when:
+- Performing initial coil design and impedance matching
+- Running rapid parameter sweeps
+- Spark length is modest (under 10 feet / 3 meters)
+- Spatial detail along the spark is not needed
+
+Switch to the [[distributed-model]] when:
+- Spatial current or power distribution is required
+- Modeling very long sparks (over 10 feet)
+- Investigating leader/streamer transitions along the channel
+- Validating the lumped model assumptions
+- Highest accuracy is needed for a specific configuration
+
+## 7. Connection to Other Topics
+
+### Key Relationships
+
+- **Implements:** [[circuit-topology]] -- The lumped model IS the fundamental circuit topology with FEMM-extracted values filling in the specific capacitances.
+- **Requires:** [[femm-workflow]] -- FEMM electrostatic simulation is the primary method for extracting C_mut and C_sh.
+- **Uses:** [[power-optimization]] -- R_opt_power provides the default resistance value; the hungry streamer principle justifies using it.
+- **Enables:** [[thevenin-method]] -- The lumped spark model defines Z_load for Thevenin analysis; once Z_th and V_th are known, power to any lumped load is immediately calculable.
+- **Extended by:** [[distributed-model]] -- The distributed model generalizes the single-section lumped model to n sections, each with its own C_mut, C_sh, and R values.
+- **Constrained by:** [[equations-and-bounds]] -- All extracted values must fall within physically validated ranges.
+- **Affected by:** [[coupled-resonance]] -- The operating frequency shifts with spark loading; R_opt_power must be recalculated at the loaded pole frequency.
+- **Affected by:** [[capacitive-divider]] -- Voltage division through C_mut and C_sh reduces the effective tip voltage, limiting spark growth.
+
+### Worked Example
+
+The complete numerical workflow is demonstrated in `femm-lumped-extraction.md`, which walks through:
+- FEMM geometry setup for a 30 cm x 8 cm toroid with a 1.8 m spark
+- Extraction of C_mut = 10.5 pF and C_sh = 6.3 pF from the Maxwell matrix
+- Calculation of R_opt_power = 47.3 kilohm at 200 kHz
+- Validation against empirical rules, mesh convergence, and boundary sensitivity
+- SPICE netlist construction and verification
+- Parametric studies varying spark length and topload size

context/open-questions.md

@@ -0,0 +1,399 @@
+---
+id: open-questions
+title: "Open Questions and Future Research Directions"
+status: established
+source_sections: "spark-physics.txt: Part 12 (lines 807-835), plus scattered notes throughout"
+related_topics: [energy-and-growth, thermal-physics, streamers-and-leaders, distributed-model, field-thresholds, empirical-scaling, femm-workflow, equations-and-bounds]
+key_equations:
+  - "Dynamic capacitance: d_eff(E)"
+  - "Branching current: I_branch proportional to d_branch^1.5"
+  - "Time-dependent epsilon with thermal memory"
+key_terms:
+  - "epsilon variability"
+  - "branching"
+  - "dynamic capacitance"
+  - "radial temperature profile"
+  - "Monte Carlo"
+  - "transient simulation"
+  - "strike detection"
+  - "3D FEA"
+  - "stochastic breakout"
+  - "Becker et al. 2005"
+  - "Liu 2017"
+  - "Yang et al. 2022"
+  - "literature reference"
+  - "Gallimberti model"
+  - "aborted leader"
+  - "Phase 6 QCW survey"
+  - "sword_spark"
+  - "driven_leader"
+  - "QCW_measurement_gaps"
+images: []
+examples: []
+open_questions:
+  - "This entire document is a catalog of open questions -- see section contents below."
+---
+
+# Open Questions and Future Research Directions
+
+This document catalogs the known uncertainties, unexplored areas, and future enhancement possibilities in the Tesla coil spark modeling framework. The framework, as documented in [[equations-and-bounds]], [[lumped-model]], [[distributed-model]], and related topics, provides a practical and accurate modeling approach. However, it makes deliberate simplifications and relies on empirical calibration in areas where the underlying plasma physics is too complex for closed-form treatment. This document makes those limitations explicit and identifies the most promising directions for future work.
+
+## 1. Remaining Uncertainties in Current Framework
+
+### 1.1 Energy per Meter (epsilon) Variability
+
+The energy per meter epsilon is the most important empirical parameter in the framework (see [[energy-and-growth]]). It determines the growth rate dL/dt = P_stream / epsilon and the total energy requirement E_total = epsilon * L. Current knowledge:
+
+**What we know:**
+- QCW-style growth: epsilon approximately 5-15 J/m
+- High duty cycle DRSSTC: epsilon approximately 20-40 J/m
+- Hard-pulsed burst mode: epsilon approximately 30-100+ J/m
+- epsilon decreases during heating: epsilon(t) = epsilon_0 / (1 + alpha * integral(P dt))
+
+**What we do not know:**
+- How does epsilon depend on current density at the channel level? Higher peak current should improve leader formation efficiency, but the quantitative relationship is not established.
+- How does epsilon vary with operating frequency? The framework assumes epsilon is frequency-independent, but frequency affects the balance between displacement and conduction current, which may influence channel heating efficiency.
+- What is the precise effect of ambient conditions (temperature, humidity, altitude, barometric pressure) on epsilon? The field threshold E_propagation varies by +/-20-30% with altitude and humidity, but the corresponding variation in epsilon is not well characterized.
+- Can epsilon be predicted from first principles for a given set of operating conditions, or does it always require empirical calibration?
+
+**Impact of uncertainty:** epsilon has +/-30-50% measurement tolerance. This propagates directly into growth rate and length predictions with the same fractional uncertainty. For design purposes, this is often acceptable (predicting spark length to within a factor of 1.5). For precision modeling, it is the dominant error source.
+
+### 1.2 Field Threshold (E_propagation) Dependencies
+
+The field threshold for sustained spark growth (see [[field-thresholds]]) has similar uncertainties:
+
+**Known dependencies:**
+- Altitude: E_propagation scales approximately with air density (proportional to pressure/temperature)
+- Humidity: Higher humidity generally increases E_propagation (water molecules are electronegative)
+  - **Literature update:** Breakdown voltage in air has a minimum at ~1% water vapor content. Below ~1%, adding water vapor reduces breakdown voltage; above ~1%, the attachment effect dominates and raises it. For typical indoor conditions (0.5-2% water vapor), the effect on E_inception is modest and direction-dependent. The effect on E_propagation (which involves pre-conditioned channels) is less clear from the textbook data. [Becker et al. 2005, Ch 2, p. 30]
+  - **Further update (Liu 2017):** The conventionally cited mechanism by which humidity accelerates streamer-to-leader transition (faster V-T relaxation of N2 due to H2O collisions) is quantitatively weak. Liu's kinetic modeling shows the energy contribution from humidity-enhanced V-T relaxation is "several orders of magnitude smaller" than other energy sources during the transition process. This suggests humidity's effect on E_propagation may operate through a different mechanism than V-T relaxation (possibly attachment/detachment kinetics or changed recombination pathways). [Liu 2017, Ch 3]
+- Channel condition: A pre-heated channel from a previous pulse has lower E_propagation
+
+**Unknown dependencies:**
+- How does tip geometry affect E_propagation? The framework uses a single value modified by the enhancement factor kappa, but branched or split tips may behave differently.
+- Does E_propagation change along the spark as the channel matures from streamer to leader?
+- What is the quantitative relationship between E_propagation and the repetition rate in burst mode? Faster bursts should reduce E_propagation due to residual ionization, but by how much?
+
+**Measurement tolerance:** +/-20-30%. Combined with FEMM field accuracy of +/-10%, the total uncertainty in predicting the stall length is +/-30-40%.
+
+### 1.3 Full Thermal Evolution
+
+The framework uses a simplified thermal model (see [[thermal-physics]]):
+```
+tau_thermal = d^2 / (4 * alpha)
+alpha = k / (rho_air * c_p) approximately 2e-5 m^2/s
+```
+
+**What this captures:** Pure thermal diffusion from a hot cylindrical channel into ambient air. This gives correct time constants for the initial cooling phase.
+
+**What this misses:**
+
+- **Convection:** Hot air rises, creating buoyancy-driven convection that maintains the hot gas column longer than pure diffusion predicts. For thick leaders (d > 1 mm), convection dominates over diffusion for times longer than about 10 ms. This is why observed channel persistence (seconds for leaders) is much longer than the diffusion time constant (hundreds of milliseconds for 5 mm channels).
+
+- **Radiation:** At temperatures above 5000 K, radiative cooling from ionized gas becomes significant. The framework does not account for radiative losses, which could reduce the effective thermal memory at very high temperatures.
+
+- **Convection enhancement of cooling:** While buoyancy maintains the hot column structure, it also enhances convective heat transfer at the channel surface. The net effect depends on the balance between column maintenance and surface cooling.
+
+- **Ionization energy:** Thermal dissociation and ionization of air molecules absorb significant energy that is released upon recombination. This acts as an additional "thermal memory" beyond simple sensible heat.
+  - **Literature update:** The average ionization energy cost in air is ~14 eV per electron-ion pair [Becker et al. 2005, Ch 7, p. 440]. Nitrogen vibrational relaxation time at 1 atm is >100 us [Becker et al. 2005, Ch 5, p. 231], confirming that vibrational energy storage operates on the same timescale as thin streamer thermal diffusion (~100-200 us). Electron-ion recombination rates are ~2 * 10^-7 cm^3/s at 300 K [Becker et al. 2005, Ch 4, p. 174], giving tau_recomb ~ 50 us at n_e = 10^13 cm^-3. The gap between recombination decay (~50 us) and observed persistence (~1-5 ms) is partially explained by these vibrational and metastable energy reservoirs.
+  - **Further update (Liu 2017):** Detailed kinetic modeling (45 species, 192 reactions) confirms that N2 vibrational relaxation is not the sole or even dominant energy pathway during the critical transition phase. Direct electron impact heating becomes important in late-stage transition. The critical temperature for leader inception must **significantly exceed 2000 K** because convection losses during gas expansion can abort the leader if temperature only marginally exceeds the threshold. Multiple aborted leader attempts (thermal ratcheting) typically precede stable inception. [Liu 2017, Ch 3]
+
+### 1.4 Branching
+
+Real Tesla coil sparks branch extensively, especially in burst mode. The framework currently models a single unbranched channel.
+
+**Branching questions:**
+- How is power divided among branches at a branch point? Is it proportional to branch impedance, or does some other mechanism apply?
+- Does the main channel (thickest branch) receive the majority of the power, with side branches being parasitic?
+- How does the total spark capacitance (C_sh) change with branching? Multiple branches increase the total conductor surface area, potentially increasing C_sh significantly beyond the single-channel 2 pF/foot estimate.
+- Can the [[distributed-model]] be extended to include branching by adding parallel R-C paths at branch nodes?
+
+**Proposed branching model (untested):**
+```
+I_branch proportional to d_branch^1.5
+```
+This follows from the assumption that current density scales with cross-sectional area and conductivity scales with temperature (which scales with diameter for a given total power). The exponent 1.5 is intermediate between the area scaling (exponent 2) and the linear scaling. This model has not been validated against measurements.
+
+## 2. Future Physics Enhancements
+
+### 2.1 Dynamic Capacitance
+
+The current framework assumes constant channel diameter for the FEMM simulation. In reality, the effective diameter depends on the local electric field and plasma conditions:
+
+```
+d_eff(E) = d_0 * (1 + beta * ln(E / E_threshold))
+```
+
+where d_0 is the zero-field diameter, beta is a dimensionless expansion coefficient, and E_threshold is the field at which expansion begins. As the field increases, the ionization boundary expands outward, increasing the effective conductor diameter and hence the capacitance.
+
+**Impact:** Dynamic capacitance would cause C_mut and C_sh to vary with the applied voltage, making the circuit nonlinear. At high voltages (near inception), the effective diameter could increase by a factor of 2-5, changing capacitances by 10-30% (logarithmic dependence). This is a second-order effect for most applications but could matter for precision growth modeling.
+
+### 2.2 Radial Temperature Profiles
+
+The current framework treats the channel as having a uniform temperature across its cross-section. In reality:
+
+- **Hot core:** The center of the channel is hottest (5000-20000 K for leaders), with the highest ionization and lowest resistivity.
+- **Cool edges:** Temperature drops off radially, with a transition zone where the plasma transitions from fully ionized to neutral air.
+- **Effective radius:** The "electrical diameter" (the radius within which most current flows) is smaller than the "optical diameter" (the radius within which the gas is visibly luminous).
+
+A radial temperature profile would enable:
+- More accurate resistance estimates (integrate conductivity over the cross-section)
+- Better thermal time constant predictions (the cool outer shell cools faster than the hot core)
+- Modeling of the step-leader mechanism (hot core grows forward while cool edges lag)
+
+### 2.3 Time-Dependent Epsilon with Thermal Memory
+
+The framework already includes a first-order correction:
+```
+epsilon(t) = epsilon_0 / (1 + alpha * integral(P_stream dt))
+```
+
+A more sophisticated model could track the thermal state of the channel explicitly:
+```
+T_channel(t) = T_ambient + (1 / (m * c_p)) * [integral(P_heat dt) - integral(P_cool dt)]
+epsilon(T) = epsilon_max * exp(-T / T_scale)
+```
+
+where P_heat is the Joule heating rate, P_cool is the combined conductive/convective/radiative cooling rate, and T_scale is a characteristic temperature above which leader formation becomes efficient. This couples the thermal evolution to the growth rate, creating a nonlinear system that must be solved iteratively.
+
+### 2.4 Branching Models
+
+A quantitative branching model would include:
+- Probability of branching at each time step (proportional to local E_tip and current)
+- Power division rule at branch points (proposed: I_branch proportional to d_branch^1.5)
+- Independent growth of each branch
+- Total C_sh as sum of all branch capacitances
+- Competition for current among branches
+
+The main difficulty is that branching is inherently stochastic, making deterministic prediction impossible. Statistical approaches (ensemble averages, probability distributions of spark length) may be more appropriate.
+
+## 3. Simulation Improvements
+
+### 3.1 Full Transient with L(t) Evolution
+
+The current workflow uses a sequence of static (AC) analyses: set up the spark at length L, compute steady-state power, step L forward. A true transient simulation would:
+
+1. Start with zero spark length
+2. At each time step, check E_tip against E_propagation
+3. If E_tip > E_propagation: advance L by (P_stream / epsilon) * dt
+4. Update the spark model (C_mut, C_sh, R) for the new length
+5. Continue the SPICE transient without restarting
+
+This requires a SPICE model with time-varying elements, which is not directly supported by standard SPICE. Possible implementations:
+- Python-controlled SPICE (update model parameters between time steps)
+- Behavioral modeling in SPICE using voltage-controlled parameters
+- Custom simulator with integrated circuit and growth equations
+
+### 3.2 3D FEA for Complex Geometries
+
+FEMM's axisymmetric solver cannot handle:
+- Horizontal or angled sparks
+- Multiple breakout points on asymmetric toploads
+- Sparks in the presence of nearby grounded objects (walls, equipment)
+- Branched sparks
+
+3D FEA tools (Elmer, COMSOL, ANSYS Maxwell) can solve these cases but at significantly higher computational cost (minutes to hours per run instead of seconds). A practical approach is to use FEMM for the baseline vertical case and apply correction factors derived from 3D simulations for non-ideal geometries.
+
+### 3.3 Monte Carlo for Stochastic Breakout and Branching
+
+Spark formation and branching are stochastic processes influenced by:
+- Local surface field variations on the topload (surface roughness, sharp features)
+- Random seed electron availability (cosmic rays, photoionization)
+- Turbulent mixing affecting local gas composition and temperature
+
+A Monte Carlo approach would:
+1. Generate random initial conditions (breakout point, initial angle)
+2. Propagate the spark with stochastic branching events
+3. Repeat many times to build statistical distributions
+4. Report mean and variance of spark length, total energy, branching pattern
+
+This is computationally expensive but would provide uncertainty quantification that the deterministic model cannot.
+
+### 3.4 Strike Detection
+
+When a spark reaches a grounded object, the channel transitions from a high-impedance plasma load to a near-short-circuit:
+
+```
+R_spark -> R_strike approximately 1-10 ohm (arc contact resistance)
+```
+
+This transition happens in microseconds and causes:
+- Massive current surge (limited only by source impedance and primary circuit)
+- Rapid heating of the contact point
+- Potential damage to the target and the coil
+
+Modeling strike events requires:
+- A distance criterion: when E_tip at a grounded surface exceeds inception threshold
+- A rapid R transition model (exponential decay from R_spark to R_strike)
+- Protection circuit modeling (fuses, IGBTs, current limiters)
+
+## 4. Validation Needs
+
+### 4.1 Systematic Measurements Across Coil Types
+
+The current empirical calibration (epsilon and E_propagation) is based on a limited number of coils and operating conditions. A comprehensive validation program would include:
+
+- **Multiple coil types:** SSTCs, DRSSTCs, QCW-DRSSTCs, classical spark gap coils
+- **Multiple frequencies:** 50, 100, 200, 400 kHz
+- **Multiple power levels:** 0.5, 1, 5, 10, 15 kW primary input
+- **Multiple topload sizes:** Small (10 cm), medium (30 cm), large (50+ cm)
+- **Controlled environments:** Indoor (dry, known temperature) and outdoor (varying humidity, wind)
+
+For each combination, measure:
+- Final spark length (mean and standard deviation over many shots)
+- Topload voltage (calibrated probe)
+- Primary and secondary current (Rogowski/CT)
+- Input power (current probe times voltage probe, integrated over burst)
+- Ringdown frequency and Q (with and without spark)
+
+### 4.2 High-Speed Photography for Growth Rate Validation
+
+Growth rate dL/dt = P_stream / epsilon is a central prediction of the model. Validating it requires:
+- High-speed camera (10,000+ fps) to track spark tip position versus time
+- Simultaneous electrical measurement (V_top, I) for instantaneous power
+- Frame-by-frame spark length extraction (image processing)
+- Comparison of measured dL/dt to predicted dL/dt from the model
+
+This would directly calibrate epsilon as a function of time and operating conditions, rather than relying on final-length-only measurements.
+
+### 4.3 RF Current Distribution Measurements
+
+The [[distributed-model]] predicts that current decreases from base to tip. Validating this requires:
+- Multiple current sensors placed along the spark channel (extremely challenging)
+- Alternatively: multiple magnetic field sensors near the spark at different heights
+- Optical spectroscopy to infer local temperature and electron density (and hence local conductivity and current)
+
+### 4.4 Parameter Database
+
+A community database correlating spark parameters to operating conditions would enormously benefit the field:
+
+| Parameter | Coil | Frequency | Power | Mode | Environment | Measured Value |
+|-----------|------|-----------|-------|------|-------------|---------------|
+| epsilon | DRSSTC-1 | 200 kHz | 5 kW | QCW | Indoor, 25C | 8.3 J/m |
+| E_prop | DRSSTC-1 | 200 kHz | 5 kW | QCW | Indoor, 25C | 0.62 MV/m |
+| Length | DRSSTC-1 | 200 kHz | 5 kW | QCW | Indoor, 25C | 1.8 +/- 0.3 m |
+| ... | ... | ... | ... | ... | ... | ... |
+
+Such a database does not currently exist in a systematic form.
+
+### 4.5 Bayesian Model Calibration (Phase 8 — Active)
+
+A focused experimental program to constrain the dynamic threshold parameters via Bayesian inference. See `phases/phase-8-bayesian-model-calibration.md` for the full research plan. Key measurements:
+
+- **Ramp duration sweep** at fixed power → constrains delta_T, tau_buildup
+- **Power level sweep** at fixed ramp → constrains coupled voltage-power limit
+- **Frequency comparison** (if feasible) → constrains thermal ratchet rate
+
+This directly addresses the T3/T4 claims in [[field-thresholds]] Section 4.7 (dynamic E_propagation). Even 10-20 data points with physics-informed priors would dramatically constrain the model parameters. The fitting pipeline uses MCMC with model comparison (Bayes factor) to test whether the dynamic threshold model outperforms a fixed-threshold model.
+
+## 5. Framework Limitations: Honest Assessment
+
+### 5.1 What the Framework Does Well
+
+- Predicts spark impedance and power transfer with accuracy sufficient for coil design
+- Captures the essential physics: capacitive topology, power optimization, field-limited growth
+- Provides both simple (lumped) and detailed (distributed) models with clear trade-offs
+- Identifies measurable calibration parameters (epsilon, E_propagation) that separate coil-dependent from physics-dependent quantities
+
+### 5.2 What the Framework Cannot Do
+
+- **Predict absolute spark length from first principles:** The framework requires empirical calibration of epsilon and E_propagation. Without measurements on the specific coil (or a similar one), length predictions have a factor-of-2 uncertainty.
+- **Model branching quantitatively:** The single-channel assumption is adequate for main-channel length but cannot predict branch structure, total luminous volume, or branch-related power losses.
+- **Capture sub-microsecond dynamics:** The framework operates at the RF cycle timescale (microseconds) and above. Nanosecond-scale phenomena (streamer head propagation, individual ionization events) are below its resolution.
+- **Handle strike events:** The transition from free spark to grounded arc is outside the framework's scope.
+
+### 5.3 Where Empirical Calibration Fills Physics Gaps
+
+The framework explicitly acknowledges that complex plasma physics (ionization kinetics, radiation transport, turbulent mixing, streamer branching statistics) is replaced by calibrated empirical parameters. This is a deliberate engineering choice:
+
+- **epsilon** replaces a detailed model of energy deposition, ionization, heating, and leader formation
+- **E_propagation** replaces a detailed model of streamer inception, photoionization, and space charge effects
+- **R bounds** replace a detailed model of plasma conductivity as a function of temperature, composition, and pressure
+  - **Literature update (da Silva et al. 2019):** The equilibrium resistance per unit length is now quantified as R = A/I^b (ohm/m), with fitted parameters for three current regimes. For TC-relevant currents (1-10 A): R = 12,400/I^1.84 ohm/m. This provides a physics-based resistance model that could replace or complement the empirical R bounds, connecting channel resistance directly to the current flowing through it. The steep b=1.84 exponent quantifies the positive feedback driving the streamer-to-leader transition. See [[equations-and-bounds]] Section 14.11.
+
+The calibration approach works because the plasma self-optimizes (hungry streamer principle): the detailed microphysics adjusts itself to match the circuit constraints. The circuit constraints (topology, capacitances, source impedance) are well-characterized, so the macroscopic behavior is predictable even though the microscopic mechanism is complex.
+
+## 6. Partial Answers from Literature
+
+Several open questions in this framework now have partial answers from the gas discharge physics literature, specifically from Becker, Kogelschatz, Schoenbach & Barker, "Non-Equilibrium Air Plasmas at Atmospheric Pressure" (IOP, 2005). These do not close the questions but narrow the uncertainty range and provide quantitative anchors.
+
+### Can epsilon be predicted from first principles?
+
+**Partial answer:** The minimum volumetric energy density for spark channel formation is 0.6-1 J/cm^3 [Becker et al. 2005, Ch 2, p. 59]. Combined with channel cross-sections, this gives epsilon_min ~ 0.001-0.07 J/m (streamer to leader diameters). Observed epsilon (5-100 J/m) is 100-10,000x higher, with the multiplier explained by branching, radiation, heating overhead, and incomplete energy utilization. A first-principles prediction requires modeling all these loss channels, which remains intractable for general conditions. **Status: lower bound established; full prediction still requires empirical calibration.**
+
+**Further update (da Silva et al. 2019):** A major factor in the high observed epsilon is now quantified: the air heating efficiency eta_T is only ~10% at ambient temperature. 90% of electrical energy goes into N2 vibrational modes rather than gas heating. This means the "effective" power available for channel heating and leader formation is only 1/10 of the total electrical power at early stages. As the channel warms past ~1000-2000 K, eta_T rises to ~1.0 and energy utilization improves dramatically. This heating efficiency factor, combined with branching losses, partially closes the gap between epsilon_min and observed epsilon. **Status: heating efficiency quantified; combined with branching and radiation models could enable semi-empirical epsilon prediction.**
+
+### What is the role of nitrogen vibrational relaxation in persistence?
+
+**Partial answer:** N2 vibrational relaxation time at 1 atm is >100 us [Ch 5, p. 231]. This is comparable to thin streamer thermal diffusion times (~100-200 us) and explains why ionization memory extends streamer persistence from ~0.1 ms (pure diffusion) to ~1-5 ms (observed). **Status: timescale confirmed; quantitative contribution vs. other mechanisms (metastables, attachment/detachment) remains to be partitioned.**
+
+### How does humidity affect E_propagation quantitatively?
+
+**Partial answer:** Breakdown voltage has a minimum at ~1% water vapor content [Ch 2, p. 30]. For typical indoor conditions (0.5-2% water vapor), the humidity effect on E_inception is modest (+/-10%). The effect on E_propagation (which involves a pre-conditioned channel rather than initial breakdown) is less clear from the textbook data, which focuses on initial breakdown. **Status: inception effect quantified; propagation effect still uncertain.**
+
+### What electron densities exist in Tesla coil streamers?
+
+**Answered:** Streamer body electron density is 10^11-10^13 cm^-3 [Ch 2, p. 37], with fully developed spark channels reaching ~10^16 cm^-3 [Ch 2, p. 38]. These values enable direct calculation of plasma conductivity and recombination rates, connecting the microscopic plasma state to the macroscopic resistance values used in [[lumped-model]] and [[distributed-model]].
+
+### What power is needed to sustain a spark channel?
+
+**Answered:** Power to sustain n_e = 10^13 cm^-3 ranges from 1.4 kW/cm^3 (cold air, attachment-limited) to 14 kW/cm^3 (2000 K air, equilibrium losses) [Ch 5, p. 230; Ch 7, p. 440]. For a 3 mm leader channel, this corresponds to ~1 kW/m linear power density, providing an independent check on power delivery requirements. See [[thermal-physics]] for the full analysis.
+
+### What determines the frequency threshold for straight (sword) QCW sparks?
+
+**Answered:** Community data converges on 300-600 kHz for sword-like sparks [Phase 6 QCW survey, 6+ independent observers]. Below 300 kHz, QCW sparks are "chaotic and less straight"; above 600 kHz, "more curvy." The physical mechanism is the ratio of RF half-period to streamer tau_thermal: at 400 kHz (half-period = 1.25 us), the channel experiences effectively continuous heating (tau_thermal ~ 125 us is 100x longer). At 100 kHz (half-period = 5 us), thin streamers (10-50 um, tau ~ 1-30 us) experience significant cooling between cycles, allowing the preferred path to diffuse and branch. See [[thermal-physics]] for full analysis.
+
+### Does QCW require high voltage for leader formation?
+
+**Answered:** No. The Bazelyan 300-400 kV leader formation threshold applies to **single-shot impulse discharges**, not sustained-drive QCW. QCW forms leaders at only 40-70 kV topload voltage (measured by Steve Ward, davekni, Loneoceans). The 15:1 voltage ratio (600 kV burst vs 40 kV QCW for same spark length) is the single most important quantitative result from the community survey. See [[streamers-and-leaders]] for updated leader formation discussion.
+
+### What is the QCW spark growth rate?
+
+**Partially answered:** Community estimate of ~170 m/s (half the speed of sound), which is consistent with observed spark lengths over measured ramp durations. This implies a driven leader step time of ~60 us, close to the conductance relaxation tau_g = 40 us. Definitive measurement requires high-speed imaging synchronized with electrical waveforms (see measurement gaps below).
+
+### QCW Measurement Gaps
+
+The community survey [Phase 6] identified these critical unmeasured quantities:
+
+1. **No direct arc current measurement on any QCW coil** — the current flowing in the spark channel during QCW operation has never been measured
+2. **No spectroscopic temperature measurement of QCW sparks** — the ~5000 K estimate is inferred from conductivity analysis, not measured
+3. **No time-resolved impedance measurement during QCW ramp** — the impedance trajectory during growth is unknown
+4. **No high-speed imaging correlated with electrical waveforms in QCW mode**
+5. **No measurement of epsilon for QCW sparks** — only bounded from total input and estimated efficiency
+6. **No systematic frequency sweep** — same coil tested at 100, 200, 300, 400 kHz to isolate frequency effect
+7. **Voltage gradient in TC sparks disputed** — estimates range from 1.5 to 3 kV/cm
+
+### Key references for further investigation
+
+The following primary sources from [Becker et al. 2005] are particularly relevant for deepening this framework:
+
+- **Gallimberti (1972)** -- Streamer propagation simulation methodology; early computational approach
+- **Morrow & Lowke (1997)** -- Ionization/attachment coefficients for air, used in most modern air discharge simulations
+- **Kulikovsky (1998)** -- Detailed positive streamer simulation with electron density profiles
+- **Kunhardt (2000)** -- Frequency dependence of breakdown voltage in gases
+- **Raether (1964), Meek & Craggs (1978)** -- Classical textbooks on spark discharge physics; more focused on spark/leader physics than the Becker et al. book
+- **Babaeva & Naidis (2000)** -- Review of 2D streamer simulation developments
+
+These are recommended as follow-up reading, particularly Raether (1964) and Meek & Craggs (1978), which cover the spark and leader physics central to Tesla coil modeling in much greater depth than the Becker et al. book (which focuses on cold non-equilibrium plasmas for industrial applications).
+
+**Additional references integrated since initial literature review:**
+
+- **Liu (2017)** -- "Electrical Discharges: Streamer-to-Leader Transition and Positive Leader Inception," KTH Doctoral Thesis. Detailed kinetic modeling of streamer-to-leader transition with 45 species, 192 reactions. Key findings: leader inception requires T >> 2000 K; Gallimberti model assumptions flawed; humidity V-T relaxation effect weak; multiple stems share current. Extensive experimental basis from Les Renardieres Group (1977, 1981).
+- **Yang, Meng, Niu et al. (2022)** -- "Arc Modeling Approaches: A Comprehensive Review," Frontiers in Physics. Reviews Mayr, Cassie, and hybrid arc models with parameter sensitivity analysis. Key finding: TC sparks are in the pure Mayr regime; tau_m ~ 10-100 us; P_0 ~ 1-100 W.
+- **Les Renardieres Group (1977, 1981)** -- Comprehensive experimental studies of long spark formation in air gaps. Schlieren photography of dark periods, aborted leaders, and stem physics. Primary experimental data used in Liu (2017) kinetic validation.
+- **da Silva, C.L. et al. (2019)** -- "The Plasma Nature of Lightning Channels and the Resulting Nonlinear Resistance," JGR Atmospheres, 10.1029/2019JD030693. Self-consistent plasma model yielding R = A/I^b power law for channel resistance; air heating efficiency eta_T formula; channel expansion dynamics. Rate coefficient Matlab code on Zenodo (10.5281/zenodo.2597562). Key finding: channel resistance is determined by current, not initial conditions — supports hungry streamer self-optimization.
+- **Bazelyan, E.M. & Raizer, Yu.P. (2000)** -- "The mechanism of lightning attraction and the problem of lightning initiation by lasers," Physics-Uspekhi 43(7), 701-716. Review paper. Key content: leader velocity formula v_L = 1500*sqrt(Delta_U) cm/s; V-I characteristic i*E=300 V*A/cm; three-tier temperature thresholds (2000 K onset, 4000 K associative ionization, 5000 K self-sustaining); energy ceiling from tip capacitance W_max = pi*epsilon_0*U^2; electron attachment time ~100 ns; thermal instability contraction time ~1 us; electron/ion mobility in air. Cross-validates da Silva resistance values within factor ~2 for TC-relevant currents.
+- **Bazelyan, E.M. & Raizer, Yu.P. (2000)** -- "Lightning Physics and Lightning Protection," IOP Publishing, 328 pages. Comprehensive textbook by the same authors. Key additional content beyond the review paper: conductance relaxation model (dG/dt = [G_st(i)-G(t)]/tau_g, tau_g = 40 us heating / 200 us cooling); channel transmission line parameters (L_1 ~ 2.5-2.7 uH/m, C_1 ~ 10 pF/m, Z ~ 500 ohm); leader formation threshold (300-400 kV); leader channel energy balance (P_L = iE ~ 130 W/cm at 1 A, 5000 K); corona shielding rate limit (3.6 kV/us, far exceeded by TC toploads); stepped vs continuous leader propagation; E/N dependence on temperature (55 Td at 1000 K to 1.5 Td at 6000 K); dart leader velocity (1-4)*10^7 m/s; return stroke physics (35,000 K, wave at 0.4c).
+
+## 7. Connection to Other Topics
+
+### Key Relationships
+
+- **Motivates improvement of:** [[energy-and-growth]] -- Better understanding of epsilon variability would improve growth predictions.
+- **Motivates improvement of:** [[thermal-physics]] -- Full thermal evolution including convection and radiation would improve epsilon modeling.
+- **Motivates improvement of:** [[distributed-model]] -- Branching extensions, time-varying parameters, and optimal segmentation are all open areas.
+- **Motivates improvement of:** [[field-thresholds]] -- Better characterization of E_propagation dependencies would reduce prediction uncertainty.
+- **Motivates improvement of:** [[femm-workflow]] -- 3D FEA and dynamic geometry updates would extend the framework's applicability.
+- **Bounded by:** [[equations-and-bounds]] -- All proposed improvements must remain consistent with established physical bounds and measurement tolerances.
+- **Informed by:** [[empirical-scaling]] -- Community observations of spark length versus power provide independent validation of model predictions.
+- **Informed by:** [[streamers-and-leaders]] -- Understanding the physical differences between streamers and leaders motivates the branching and transition models.

context/power-optimization.md

@@ -0,0 +1,370 @@
+---
+id: power-optimization
+title: "Power Optimization and the Hungry Streamer Principle"
+status: established
+source_sections: "spark-physics.txt: Part 2 (lines 75-124), Part 9 (lines 666-700), Part 11 (lines 740-744)"
+related_topics: [circuit-topology, thevenin-method, coupled-resonance, field-thresholds, energy-and-growth, thermal-physics, streamers-and-leaders, capacitive-divider, branching-physics, empirical-scaling, lumped-model, distributed-model, equations-and-bounds]
+key_equations:
+  - "R_opt_phase"
+  - "R_opt_power"
+  - "Power delivered to load P_load"
+  - "Impedance phase at R_opt_power"
+key_terms:
+  - "R_opt_power"
+  - "R_opt_phase"
+  - "hungry streamer principle"
+  - "power transfer"
+  - "impedance matching"
+  - "self-optimization"
+  - "thermal ionization"
+  - "conductivity"
+  - "causality reversal"
+  - "QCW power paradigm"
+images:
+  - power-vs-resistance-curves.png
+  - hungry-streamer-feedback-loop.png
+  - impedance-matching-concept.png
+examples:
+  - calculating-ropt.md
+open_questions:
+  - "What is the time constant for the plasma to converge to R_opt_power after a step change in drive conditions?"
+  - "Under what conditions does the hungry streamer feedback loop become unstable (oscillatory resistance)?"
+  - "How does branching affect the effective R seen at the topload -- does each branch independently optimize?"
+  - "Is the convergence to R_opt_power monotonic, or can the plasma overshoot and oscillate?"
+---
+
+# Power Optimization and the Hungry Streamer Principle
+
+This document derives the two critical resistance values for Tesla coil spark modeling -- R_opt_phase and R_opt_power -- and establishes the physical mechanism by which real spark plasmas self-optimize toward maximum power extraction. The "hungry streamer" principle, credited to Steve Conner, is the conceptual cornerstone linking circuit theory to plasma behavior.
+
+## 1. Two Critical Resistance Values
+
+### 1.1 R_opt_phase: The Most Resistive-Looking Impedance
+
+Starting from the admittance expressions derived in [[circuit-topology]], the impedance phase angle phi_Z depends on the spark resistance R. The value of R that minimizes |phi_Z| (makes the impedance look as resistive as possible) is found by differentiating phi_Z with respect to G = 1/R and setting the result to zero.
+
+**Result:**
+
+```
+R_opt_phase = 1 / (omega * sqrt(C_mut * (C_mut + C_sh)))
+```
+
+At this resistance, the impedance phase angle equals the fundamental minimum:
+
+```
+phi_Z(R_opt_phase) = phi_Z_min = -atan(2 * sqrt(r * (1 + r)))
+```
+
+where r = C_mut / C_sh.
+
+**Physical meaning:** R_opt_phase is the resistance at which the spark presents the closest approximation to a purely resistive load. However, due to the [[circuit-topology]] phase constraint, this "closest approximation" is still significantly capacitive (typically -50 to -70 degrees).
+
+**When is R_opt_phase relevant?** In situations where minimizing reactive power flow is more important than maximizing real power -- for example, when the source has limited reactive current capability, or for minimizing circulating currents in the primary tank.
+
+### 1.2 R_opt_power: Maximum Real Power Transfer
+
+The real power delivered to the spark, for a fixed topload voltage magnitude |V_top|, is:
+
+```
+P_spark = 0.5 * |V_top|^2 * Re{Y}
+        = 0.5 * |V_top|^2 * G * B_2^2 / (G^2 + (B_1 + B_2)^2)
+```
+
+Maximizing P_spark with respect to G (equivalently R) by setting dP/dG = 0:
+
+```
+d/dG [G * B_2^2 / (G^2 + (B_1 + B_2)^2)] = 0
+```
+
+The numerator of the derivative gives:
+
+```
+B_2^2 * [(G^2 + (B_1 + B_2)^2) - 2G^2] = 0
+B_2^2 * [(B_1 + B_2)^2 - G^2] = 0
+```
+
+Since B_2 is nonzero, this requires G^2 = (B_1 + B_2)^2, giving G_opt = B_1 + B_2 = omega*(C_mut + C_sh).
+
+**Result:**
+
+```
+R_opt_power = 1 / (omega * (C_mut + C_sh))
+```
+
+**Numerical example:** At f = 200 kHz with C_mut + C_sh = 12 pF:
+
+```
+omega = 2 * pi * 200e3 = 1.257e6 rad/s
+R_opt_power = 1 / (1.257e6 * 12e-12) = 1 / (1.508e-5) = 66.3 kOhm
+```
+
+### 1.3 Relationship Between the Two Optima
+
+**R_opt_power is always less than R_opt_phase:**
+
+```
+R_opt_power / R_opt_phase = sqrt(C_mut * (C_mut + C_sh)) / (C_mut + C_sh)
+                          = sqrt(C_mut / (C_mut + C_sh))
+                          = sqrt(r / (1 + r))    where r = C_mut/C_sh
+```
+
+Since r/(1+r) < 1 for all positive r, R_opt_power < R_opt_phase always.
+
+For r = 1 (equal capacitances): R_opt_power / R_opt_phase = sqrt(0.5) = 0.707
+For r = 0.5: R_opt_power / R_opt_phase = sqrt(1/3) = 0.577
+For r = 2: R_opt_power / R_opt_phase = sqrt(2/3) = 0.816
+
+**Impedance phase at R_opt_power:** Substituting G = omega*(C_mut + C_sh) into the phase expression:
+
+```
+phi_Z(R_opt_power) is typically -55 to -75 degrees
+```
+
+This is more negative (more capacitive) than phi_Z_min, meaning R_opt_power does NOT correspond to the minimum phase point. The maximum power condition accepts a worse phase angle in exchange for delivering more real power.
+
+![Power vs. resistance curves showing both optima](../assets/power-vs-resistance-curves.png)
+
+### 1.4 Power at the Two Optima
+
+At R_opt_power, the maximum power is:
+
+```
+P_max = 0.5 * |V_top|^2 * B_2^2 / (2 * (B_1 + B_2))
+      = 0.5 * |V_top|^2 * omega * C_sh^2 / (2 * (C_mut + C_sh))
+```
+
+At R_opt_phase, the power is lower. The ratio depends on r but is typically 0.7 to 0.9 of P_max. Except in unusual geometries, the difference is modest -- but over a long spark growth event (tens of milliseconds), the accumulated energy difference can be significant.
+
+### 1.5 Causality Reversal: Spark Loading Drives Quench, Not Vice Versa
+
+Richie Burnett (richieburnett.co.uk) identified a critical insight for understanding power delivery to spark loads:
+
+**"It is not early quenching that produces good sparks, but rather good spark loading that leads to an early quench."**
+
+The causality runs: the spark efficiently absorbs energy → secondary voltage drops → gap quenches (SGTC) or primary current drops (DRSSTC). A well-optimized spark near R_opt_power extracts power efficiently, pulling V_top down and naturally terminating the drive. This is the hungry streamer principle viewed from the source side: maximum power transfer produces maximum damping.
+
+**Practical consequence:** Attempts to optimize spark performance by adjusting quench timing (SGTC) or burst duration (DRSSTC) are attacking the symptom, not the cause. The primary lever is optimizing the impedance match and power delivery to the spark itself.
+
+### 1.6 QCW vs Burst: Fundamentally Different Power Paradigms
+
+Community builder data [Phase 6 QCW community survey, 2026-02-10] reveals that QCW and burst mode represent fundamentally different approaches to power delivery:
+
+| Aspect | QCW | Burst DRSSTC |
+|--------|-----|-------------|
+| Power delivery | Sustained low power over 10-22 ms | Brief high power over 70-150 us |
+| Secondary voltage | 40-70 kV | 200-600 kV |
+| How growth works | Continuous leader extension through persistent conducting channel | Single-shot streamer reach set by peak voltage |
+| Limiting factor | Capacitive voltage division at tip | Streamer reach (voltage-limited) |
+| Efficiency metric | Spark:secondary ratio (7-16x) | Bang energy to length scaling |
+
+The most striking data point: davekni measured **~600 kV for 2-3 m burst sparks vs ~40 kV for equivalent QCW sparks** at 450 kHz — a 15:1 voltage ratio for similar spark lengths. This proves that QCW operates via a completely different mechanism: sustained energy delivery through a thermally persistent leader (see [[thermal-physics]]), not high instantaneous voltage.
+
+**Implication for power optimization:** In burst mode, R_opt_power analysis at a single frequency is approximately valid because the entire event occurs within a few hundred microseconds. In QCW mode, R_opt_power shifts continuously during the 10-22 ms ramp as C_sh grows (spark extends). The matching strategy should target 50-70% of final spark length, as described in Section 4.2.
+
+## 2. The Hungry Streamer Principle
+
+### 2.1 Origin
+
+Steve Conner observed that Tesla coil streamers appear to actively seek out conditions that maximize power extraction from the resonant circuit. He termed this the "hungry streamer" principle: the plasma is "hungry" for power and adjusts its properties to consume as much as possible.
+
+### 2.2 Physical Mechanism: The Feedback Loop
+
+The hungry streamer principle is not mystical -- it follows from well-understood plasma physics through a feedback loop:
+
+**Step 1: Power injection drives Joule heating.**
+Current I flows through the spark resistance R, depositing power P = I^2 * R in the plasma channel.
+
+**Step 2: Heating increases temperature.**
+The deposited energy raises the gas temperature T in the channel. For a thin channel with thermal time constant tau_thermal (see [[thermal-physics]]), the temperature responds on millisecond timescales.
+
+**Step 3: Temperature drives thermal ionization.**
+At elevated temperatures (above ~3000-5000 K), thermal ionization of air molecules becomes significant. The electron density n_e increases approximately exponentially with temperature (Saha equation):
+
+```
+n_e ~ exp(-E_ion / (2 * k_B * T))
+```
+
+where E_ion is the ionization energy (~14.5 eV for N2).
+
+**Step 4: Ionization increases conductivity.**
+Electrical conductivity sigma is proportional to electron density and inversely related to collision frequency:
+
+```
+sigma = n_e * e^2 / (m_e * nu_collision)
+```
+
+Higher n_e directly increases sigma, decreasing R.
+
+**Step 5: Changed R modifies power transfer.**
+A lower R changes the admittance and thus the power delivered. If R was above R_opt_power, decreasing R moves toward the optimum and increases power. If R was below R_opt_power, decreasing R moves away from the optimum and decreases power.
+
+**Step 6: Geometry changes modify capacitances.**
+As the channel heats and expands, its diameter changes, which weakly affects C_mut and C_sh (logarithmic dependence on diameter). The expanding, lengthening channel also increases C_sh linearly with length. These capacitance changes shift R_opt_power to a new value.
+
+**Step 7: Stable equilibrium at R_actual ~ R_opt_power.**
+The negative feedback loop (less power -> cooling -> higher R -> approaching R_opt from above) and positive feedback (more power -> heating -> lower R -> approaching R_opt from below, up to a point) create a stable attractor near R_opt_power. The plasma self-regulates.
+
+![Hungry streamer feedback loop diagram](../assets/hungry-streamer-feedback-loop.png)
+
+### 2.3 Why R_opt_power, Not R_opt_phase?
+
+The feedback loop selects for maximum power, not minimum phase angle. Physical reasoning:
+
+- More power -> more heating -> plasma responds to power, not to phase
+- The plasma has no mechanism to "sense" phase angle; it responds to energy deposition (I^2*R)
+- R_opt_power maximizes I^2*R for fixed source conditions
+- The equilibrium is reached when no perturbation in R can increase I^2*R further
+
+This is analogous to maximum power transfer in classical circuit theory, except the "load" actively adjusts itself.
+
+### 2.4 Stability Analysis
+
+Near R_opt_power, consider a small perturbation delta_R:
+
+- If R = R_opt_power + delta_R (too high): power decreases -> less heating -> temperature drops -> ionization decreases -> R increases further. This is POSITIVE feedback away from optimum! However, as R increases beyond R_opt_power, the spark also cools, which eventually leads to the spark extinguishing or branching to find a better path. In practice, the spark stalls or a new streamer launches.
+
+- If R = R_opt_power - delta_R (too low): power decreases (since we are below optimum on the P vs. R curve) -> less heating -> temperature drops -> ionization decreases -> R increases back toward R_opt_power. This is NEGATIVE feedback, stabilizing.
+
+The equilibrium is thus stable from below but has a "cliff" above R_opt_power. In practice, this asymmetry manifests as the tendency for sparks to either burn brightly at or below R_opt or extinguish rapidly when the resistance drifts too high. The dynamic is further stabilized by the thermal inertia of the channel.
+
+## 3. Constraints on Optimization
+
+### 3.1 Source Limitations
+
+The analysis above assumes fixed |V_top|. In reality, the source (Tesla coil primary circuit) has finite current and voltage capability:
+
+- **Current-limited:** If the primary cannot supply the current demanded by the load at R_opt_power, the topload voltage collapses. The spark operates at a higher effective R (source impedance dominates).
+
+- **Voltage-limited:** If V_top is insufficient to maintain the field threshold at the spark tip (see [[field-thresholds]]), the spark stalls regardless of R optimization.
+
+### 3.2 Inception Threshold
+
+The spark must first form. Inception requires E_tip > E_inception ~ 2-3 MV/m at the topload surface. If the topload voltage never reaches the inception field, no spark forms and the optimization loop never starts.
+
+### 3.3 Physical Conductivity Bounds
+
+The spark resistance cannot be arbitrarily low or high:
+
+```
+R_min ~ 1 kOhm     (very hot, thick, fully thermalized leader plasma)
+R_max ~ 100 MOhm   (cold, thin, barely ionized streamer)
+```
+
+If R_opt_power falls outside [R_min, R_max], the plasma cannot reach the optimum:
+
+```
+R_actual = clip(R_opt_power, R_min, R_max)
+```
+
+When clipping occurs, the spark is constrained and operates sub-optimally. Check whether the source can still provide adequate power at the clipped resistance.
+
+### 3.4 Thermal Time Constants
+
+The plasma cannot adjust instantaneously. Thermal time constants (see [[thermal-physics]]) set the response speed:
+
+- Thin streamers (d ~ 100 um): tau ~ 0.1-0.2 ms
+- Thick leaders (d ~ 5 mm): tau ~ 300-600 ms
+
+If the drive conditions change faster than the plasma can respond (e.g., burst-mode pulses shorter than tau), the plasma cannot track R_opt_power in real time. The effective R will lag behind the instantaneous optimum.
+
+### 3.5 Sub-Optimal Operation
+
+When constraints prevent reaching R_opt_power, several outcomes are possible:
+
+1. **Spark stalls:** Growth stops; the field threshold is not met at the tip.
+2. **Spark operates at R_max:** Cold streamer that cannot heat up further. Low power, inefficient.
+3. **Spark operates at R_min:** Fully ionized, very hot. May occur in arc-like conditions. Power is high but limited by source.
+4. **Spark branches:** Rather than one channel adjusting R, multiple channels form, each seeking its own optimum. Total power may be shared.
+
+## 4. Impedance Matching for Target Spark Length
+
+### 4.1 The Matching Dilemma
+
+During QCW operation, the spark grows from zero to its final length over 5-20 ms. As it grows:
+- C_sh increases (more length, more capacitance to ground)
+- R_opt_power changes (shifts with capacitance)
+- The impedance presented to the source changes continuously
+
+The coil designer must choose a single matching condition (or a tracking strategy). See [[coupled-resonance]] for frequency tracking aspects.
+
+### 4.2 QCW Matching Strategy
+
+**Recommended: Match at 50-70% of target length.**
+
+Reasoning:
+- At 0% length: no spark, pure open circuit (infinite impedance). Matching here is meaningless.
+- At 100% length: spark is at maximum extent, about to stall. Little time spent here.
+- At 50-70%: spark is in its fastest growth phase, consuming the most power. Matching here maximizes energy delivered during the critical growth window.
+
+**Rule of thumb: Match at 60% for first design iteration.**
+
+### 4.3 Formal Optimization
+
+Minimize total energy over the growth trajectory:
+
+```
+E_total = integral_0^T [epsilon * L(t) / eta(t)] dt
+```
+
+where eta(t) is the power transfer efficiency at time t, and epsilon is the energy per meter (see [[energy-and-growth]]).
+
+**Procedure:**
+1. Simulate growth with match points at 0%, 30%, 50%, 70%, 100% of target length.
+2. For each match point, compute E_total to reach target length.
+3. Choose the match point that minimizes E_total.
+
+### 4.4 Burst Mode Matching
+
+For non-ramping burst operation (fixed drive amplitude):
+- Match to final spark length (100%)
+- The coil rings up quickly to steady state
+- Steady-state impedance matching dominates over transient growth
+
+## 5. Numerical Sensitivity
+
+### 5.1 Sensitivity of R_opt_power to Capacitance Errors
+
+Since R_opt_power = 1/(omega * C_total):
+
+```
+dR/R = -dC/C
+```
+
+A 20% error in C_total produces a 20% error in R_opt_power. Given that FEMM capacitance extraction is accurate to ~10% and plasma variability is ~50%, this is acceptable.
+
+### 5.2 Sensitivity of Power to R Errors
+
+Near R_opt_power, the power curve is relatively flat. A factor of 2 error in R (R = 0.5*R_opt or R = 2*R_opt) reduces power by only about 20%. This flatness is why the simplified R = R_opt_power approach works well even with significant uncertainties.
+
+### 5.3 Sensitivity to Frequency
+
+Since R_opt_power is inversely proportional to omega:
+
+```
+dR/R = -domega/omega = -df/f
+```
+
+A 5% frequency shift (common when a spark loads the system; see [[coupled-resonance]]) produces a 5% shift in R_opt_power. This is small compared to other uncertainties.
+
+## 6. Connection to Other Topics
+
+### Key Relationships
+
+- **Derives from:** [[circuit-topology]] (the admittance expressions and phase constraint provide the mathematical foundation)
+- **Enables:** [[lumped-model]] (R_opt_power is the default resistance assignment: R = 1/(omega*C_total))
+- **Enables:** [[distributed-model]] (each segment's R_opt is computed from its local capacitances using the same principle)
+- **Constrains:** [[energy-and-growth]] (the power available for spark growth is bounded by P at R_opt_power)
+- **Interacts with:** [[coupled-resonance]] (frequency shift changes R_opt_power; the spark must track)
+- **Interacts with:** [[thermal-physics]] (thermal time constants limit how quickly the plasma can adjust to R_opt)
+- **Interacts with:** [[streamers-and-leaders]] (streamer vs. leader determines whether R is near R_min or R_max)
+- **Measured via:** [[thevenin-method]] (Thevenin extraction allows computing power to any R without re-simulation)
+
+### Summary of Key Results
+
+1. R_opt_power = 1/(omega*(C_mut + C_sh)) maximizes real power to the spark.
+2. R_opt_phase = 1/(omega*sqrt(C_mut*(C_mut + C_sh))) minimizes impedance phase magnitude.
+3. R_opt_power < R_opt_phase always. R_opt_power gives phi_Z ~ -55 to -75 degrees.
+4. The hungry streamer principle: plasma self-optimizes toward R_opt_power via thermal feedback.
+5. Constraints (source limits, physical R bounds, thermal lag) can prevent reaching R_opt_power.
+6. QCW matching at ~60% of target length is a good first-order design rule.
+7. Power is relatively insensitive to R errors near the optimum (flat peak).

context/qcw-operation.md

@@ -0,0 +1,371 @@
+---
+id: qcw-operation
+title: "QCW Operation: Driven Leader Growth Through Sustained Energy Injection"
+status: established
+source_sections: "spark-physics.txt: Part 5 (lines 281-361), Part 9 (lines 666-700); Phase 6 QCW community survey (2026-02-10)"
+related_topics: [thermal-physics, streamers-and-leaders, coupled-resonance, power-optimization, energy-and-growth, capacitive-divider, branching-physics, field-thresholds, empirical-scaling, equations-and-bounds, open-questions]
+key_equations:
+  - "Growth rate: dL/dt = P_stream / epsilon"
+  - "Driven leader step time: step_time ~ step_length / growth_rate"
+  - "Conductance relaxation: dG/dt = (G_st(i) - G) / tau_g"
+  - "Thermal diffusion: tau_thermal = d^2 / (4 * alpha)"
+key_terms:
+  - "QCW"
+  - "sword_spark"
+  - "driven_leader"
+  - "burst_ceiling"
+  - "frequency_threshold"
+  - "thermal_ratcheting"
+  - "conductance_relaxation"
+  - "ramp_duration"
+  - "pulse_skip"
+images:
+  - qcw-vs-burst-timeline.png
+examples:
+  - spark-growth-timeline.md
+open_questions:
+  - "No direct arc current measurement on any QCW coil — the actual current flowing through the spark channel during QCW growth is unknown"
+  - "No spectroscopic temperature measurement of QCW sparks — 5000 K is inferred from conductivity, not measured"
+  - "No time-resolved impedance measurement during QCW ramp — the impedance trajectory during growth is unknown"
+  - "No high-speed imaging correlated with electrical waveforms in QCW mode"
+  - "No measurement of energy per unit length (epsilon) for QCW sparks — can only be bounded from total input energy and estimated system efficiency"
+  - "Voltage gradient in TC sparks disputed — Uspring estimates 1.5 kV/cm, Barnkob estimates 3 kV/cm"
+  - "No systematic frequency sweep study — same coil tested at 100, 200, 300, 400 kHz to isolate frequency effect"
+---
+
+# QCW Operation: Driven Leader Growth Through Sustained Energy Injection
+
+QCW (Quasi-Continuous Wave) is a Tesla coil operating mode that produces straight "sword" sparks dramatically longer than burst-mode DRSSTCs of comparable size. Where burst mode relies on high instantaneous voltage (200-600 kV) to push streamers outward in a single shot, QCW uses sustained low-voltage energy injection (40-70 kV) over 10-22 ms to grow a thermally persistent leader channel at ~170 m/s. This document consolidates all QCW-specific physics, measurements, and design parameters from the community research survey and framework analysis.
+
+The key insight: **QCW sparks grow because the leader channel persists between RF cycles and conducts energy to the tip, not because the voltage is high enough to bridge the gap.** This is a fundamentally different growth mechanism from burst mode, and it explains why QCW achieves 7-16x spark:secondary ratios compared to 2-4x for burst DRSSTCs.
+
+## 1. The QCW Parameter Space
+
+QCW occupies a distinct region in Tesla coil design space, differing from burst-mode DRSSTCs in every major parameter:
+
+| Parameter | QCW Range | Burst DRSSTC | Source |
+|-----------|-----------|--------------|--------|
+| Coupling (k) | 0.3-0.55+ | 0.05-0.2 | Build survey |
+| Operating frequency | 300-600 kHz | 50-110 kHz | Build survey |
+| Tank capacitance | 5-15 nF | 50-300 nF | Build survey |
+| Ramp duration | 10-22 ms | N/A (burst ~70-150 us) | Build survey |
+| Peak primary current | 50-200 A | 200-1000+ A | Build survey |
+| Secondary voltage | 40-70 kV | 200-600 kV | Ward, davekni |
+| Spark:secondary ratio | 7-16x | 2-4x | Build survey |
+| Growth rate | ~170 m/s | N/A (single-shot) | HVF estimate |
+
+### 1.1 The 15:1 Voltage Ratio
+
+The single most important quantitative comparison in the dataset: davekni measured **~600 kV for 2-3 m burst sparks vs ~40 kV for equivalent QCW sparks** at 450 kHz. This 15:1 voltage ratio proves that QCW growth is driven by sustained energy injection, not high instantaneous voltage. Multiple independent builders confirm the low QCW voltage (Steve Ward: 40-55 kV; Loneoceans: 50-70 kV). [Phase 6 QCW community survey]
+
+**Physical explanation:** The leader formation voltage threshold of 300-400 kV [Bazelyan & Raizer 2000] applies to **single-shot impulses** where the entire streamer-to-leader transition must occur from one event. In QCW, the thermal ratcheting mechanism (see Section 3) accumulates energy from thousands of RF cycles, crossing the critical temperature thresholds (2000 K -> 4000 K -> 5000 K) without ever requiring high instantaneous voltage. The voltage merely needs to exceed the inception threshold and maintain current flow. See [[streamers-and-leaders]] for details.
+
+### 1.2 Coupling Requirement: k >= 0.3
+
+All successful QCW sword-spark builds use k >= 0.3:
+
+| Builder | k | Spark:secondary ratio | Notes |
+|---------|---|----------------------|-------|
+| Loneoceans v1.0 | 0.32-0.35 | 7.3:1 | Initial |
+| Loneoceans v1.5 (first) | 0.306 | — | Insufficient — breakthrough came at 0.38 |
+| Loneoceans v1.5 (final) | 0.38 | 13:1 | Breakthrough |
+| Loneoceans QCW2 | 0.365 | 10:1 | |
+| flyglas | 0.391 | ~12:1 | |
+| Lucasww | 0.44 | 10:1 | |
+| Dr. Kilovolt (Jan Martis) | 0.55 | — | SiC PSFB, 2-2.5 m sparks |
+| davekni | 0.71 | — | Ferrite-assisted, highest documented |
+| Standard DRSSTC | 0.05-0.20 | 2-4:1 | For comparison |
+
+Higher coupling enables sufficient power transfer at QCW's lower peak currents (50-200 A vs 200-1000+ A for burst). It also widens pole separation, making frequency tracking more robust (see [[coupled-resonance]]). However, Loneoceans' SSTC3 (single-resonant, lower coupling) still produces straight sparks at 380-420 kHz, suggesting k >= 0.3 is an **engineering constraint** (adequate power delivery) rather than a **physics constraint** (straightness).
+
+## 2. The Sword Spark Mechanism
+
+### 2.1 Frequency Threshold: 300-600 kHz
+
+Six or more independent builders have converged on a frequency range for producing straight sword sparks:
+
+| Observer | Observation | Source |
+|----------|-------------|--------|
+| Mads Barnkob | "Sword characteristic shows above 400 kHz" | HVF |
+| LabCoatz (Zach Armstrong) | Below 300 kHz: "chaotic and less straight"; above 600 kHz: "more curvy" | Hackaday |
+| Kaizer DRSSTC IV | QCW at ~100 kHz: "swirling" sparks, NOT straight | HVF |
+| Fat Coil builder | "Above 350 kHz, plasma exhibits growth in straight segments" | TCML |
+| Loneoceans SSTC3 | Straight sparks at 380-420 kHz | loneoceans.com |
+| Multiple QCW builders | All successful sword-spark builds operate 300-500 kHz | Build survey |
+
+### 2.2 Why Frequency Matters: RF Period vs Thermal Time Constants
+
+The physical mechanism is the ratio between the RF half-period and the streamer thermal diffusion time:
+
+```
+At 400 kHz: RF half-period = 1.25 us
+Streamer tau_thermal (d = 100 um) = d^2 / (4*alpha) ~ 125 us
+
+Ratio: tau_thermal / T_RF = 125 / 1.25 = 100x
+```
+
+The channel experiences **effectively continuous heating** with negligible cooling between RF half-cycles. The conductance relaxation time constant (tau_g = 40 us for heating, see [[thermal-physics]]) spans ~16 RF cycles at 400 kHz, ensuring smooth, monotonic conductance increase.
+
+At 50-100 kHz (half-period 5-10 us), thinner streamers (10-50 um, tau ~ 1-30 us) experience significant cooling between cycles. The preferred conductive path diffuses and branches — the channel cannot maintain a single straight track.
+
+At >600 kHz, "curvy" sparks are observed. This may relate to skin effect, displacement current dominance, or switching artifacts at extreme frequencies.
+
+**Quantitative prediction:** At frequency f, the Joule heating rate scales as ~f (more half-cycles per unit time at the same peak current). A channel at 400 kHz receives ~4x more thermal energy per millisecond than at 100 kHz.
+
+### 2.3 Pulse-Skip Modulation Does Not Produce Full Sword Sparks
+
+Multiple experimenters (Steve Ward, Steve Conner, others circa 2011) tried pulse-skip approaches to achieve QCW-like behavior and could not produce full sword sparks.
+
+Steve Ward: Sword sparks need "relatively smooth/continuous modulation of the spark power with little ripple." If the coil stores enough energy to smooth out the missing pulses, "it's probably massively overbuilt."
+
+**What pulse-skip actually does:** In a DRSSTC, pulse-skip is a bridge current-limiting method. During "skip" cycles, one GDT (gate drive transformer) is inverted so the H-bridge effectively shorts the primary tank (single-leg inhibit / "freewheeling"), or both bridge halves shut down and energy returns to the bus caps. The IGBTs continue switching synchronized to secondary current feedback — phase coherence is maintained and there is no phase discontinuity when active drive resumes. Primary current does not drop to zero; it decays gradually through the loaded Q of the resonant system. The resulting current envelope is a sawtooth bounded by the OCD (overcurrent detection) threshold — current rises to the limit, bridge freewheels until current decays, then drive resumes. [Steve Ward DRSSTC design guide; UD+ documentation (P. Slawinski); UD3 (Netzpfuscher)]
+
+**Why it doesn't produce full swords — envelope quality:** The sawtooth current envelope at a fixed OCD threshold delivers approximately constant average power, not the smoothly ramping power profile that QCW requires. True QCW uses a linear voltage ramp, which produces a quadratic power envelope (P ~ V^2) — the natural profile for growing a spark against increasing capacitive loading. Pulse-skip cannot easily produce this quadratic profile. The per-cycle current jitter from the on-off-on switching pattern, even with optimal distribution of skip events, creates enough power envelope ripple to prevent clean single-channel dominance. [Loneoceans QCW documentation; HVF topic 292]
+
+**It's a continuum, not binary:** The effect of envelope quality on spark straightness is progressive. Coarse pulse-skip at a fixed OCD threshold produces standard branchy DRSSTC sparks. A more sophisticated Bresenham-algorithm pulse-density modulation creating a linear ramp envelope produces sparks that are noticeably more sword-like but still branch — an intermediate result. True analog QCW with a smooth quadratic power envelope produces full swords. The evidence suggests that spark straightness improves continuously with envelope smoothness, with no sharp threshold.
+
+**Smooth topologies that work:** Steve Ward's original linear voltage ramp (giving quadratic power), Dr. Kilovolt's SiC Phase-Shifted Full Bridge (inherently smooth with a "1-cosine" transfer function), and Loneoceans' SSTC3 staccato approach (using the rising AC mains waveform as a natural voltage ramp) all produce straight sparks because they deliver smooth, continuously ramping power without per-cycle jitter.
+
+**Note:** Pulse-skip (bridge current control) is distinct from staccato (interrupter timing synchronized to AC mains). They serve different functions and can be combined. Staccato provides a natural voltage ramp over ~4-5 ms per mains half-cycle; pulse-skip manages current limits within each burst.
+
+## 3. The Driven Leader Growth Model
+
+### 3.1 Growth Rate: ~170 m/s
+
+QCW sparks grow at approximately half the speed of sound. This is estimated from community observations of spark growth during QCW ramps. [Phase 6 QCW survey, HVF topic 973]
+
+**Self-consistency check:** At 170 m/s over a 10 ms ramp, the spark grows 1.7 m. Over a 20 ms ramp, 3.4 m. These match observed QCW spark lengths (1-2 m for standard builds, 3.35 m for the Fat Coil).
+
+This velocity is intermediate between free streamers (~10^6 m/s) and natural lightning leaders (~10^4 m/s for stepped leaders, averaged). It represents a **driven leader** propagation mode unique to QCW: the leader advances continuously, fed by the circuit, at a rate limited by the thermal conversion of streamer-to-leader at the tip.
+
+### 3.2 Step Time Derivation from tau_g
+
+From the growth rate and Bazelyan's typical leader step length (~1 cm):
+
+```
+step_time = step_length / growth_rate = 0.01 m / 170 m/s ~ 60 us
+```
+
+This 60 us step time is close to the conductance relaxation heating time constant (tau_g = 40 us from Bazelyan). The channel needs approximately one tau_g to heat each new segment to leader temperature. The 1.5x ratio (60 us vs 40 us) is reasonable given that the transition also requires crossing the eta_T efficiency bottleneck (10% heating efficiency at ambient → 100% above 2000 K). See [[thermal-physics]] for the full conductance relaxation model.
+
+### 3.3 Contrast with Bazelyan Leader Velocity
+
+The Bazelyan formula v_L = 1500*sqrt(|Delta_U_t|) gives ~4.7-8.2 km/s at 100-300 kV — 25-50x faster than the observed 170 m/s QCW growth rate. The discrepancy is explained by the fundamental difference between:
+
+- **Bazelyan's v_L**: Instantaneous leader step velocity (the speed of thermal instability contraction within a single step)
+- **QCW 170 m/s**: Net growth rate averaged over many steps including the time to heat each new streamer segment
+
+The QCW leader advances in rapid micro-steps at ~km/s but spends most of its time waiting for each new segment to thermalize. See [[streamers-and-leaders]] for details.
+
+### 3.4 Thermal Ratcheting Mechanism
+
+The 5:1 asymmetry in conductance relaxation time constants creates a one-way thermal ratchet:
+
+```
+tau_g = 40 us   (channel heating — current rising)
+tau_g = 200 us  (channel cooling — current falling)
+```
+
+[Bazelyan & Raizer 2000, Ch 4, pp. 194-195]
+
+Over many RF cycles:
+1. During the high-current half-cycle: conductance increases toward G_st(i_peak) with tau_g = 40 us
+2. During the low-current half-cycle: conductance decreases toward G_st(0) = 0 with tau_g = 200 us
+3. **Net effect:** Conductance ratchets upward over ~10-50 RF cycles (50-250 us at 200 kHz)
+
+This is the microsecond-timescale mechanism underlying the millisecond-timescale streamer-to-leader transition. Each RF cycle deposits a net conductance increment, accumulating over thousands of cycles during the QCW ramp.
+
+## 4. Three Ramp Regimes
+
+Loneoceans documented three distinct outcomes through controlled variation of ramp duration (QCW v1.5):
+
+| Ramp Duration | Visual Result | Physics Interpretation |
+|---------------|--------------|----------------------|
+| Too short (<5 ms) | "Gnarly, segmented sparks" | Insufficient time for leader transition; disconnected leader segments don't merge |
+| Optimal (~10-20 ms) | Straight sword sparks | Leader forms within first few ms; grows continuously for remainder |
+| Too long (>25 ms) | "Really hot and fat but bushy" | Leader reaches voltage-limited L_max; excess energy drives branching |
+
+### 4.1 The "Too Long" Regime
+
+Once the leader reaches its maximum length (set by the [[capacitive-divider]]), additional energy cannot extend it further. The leader channel becomes very hot and thick, increasing C_sh and worsening voltage division. The excess power must dissipate somewhere — lateral breakouts from the superheated leader trunk become the path of least resistance.
+
+### 4.2 The "Too Short" Regime
+
+Ramps shorter than ~5 ms don't allow the full streamer-to-leader transition (which requires ~0.5-2 ms from [[streamers-and-leaders]]). The "segmented" appearance suggests the spark advances as disconnected leader segments that don't merge into a continuous trunk. This is consistent with the thermal ratcheting model requiring multiple dark period cycles — see [[thermal-physics]].
+
+### 4.3 QCW Timing Analysis
+
+Typical optimal QCW ramp: 12 ms at 400 kHz
+
+- **0-2 ms**: Voltage builds toward inception. Possible aborted leader attempts. High epsilon.
+- **2-4 ms**: Streamers form and begin heating. Transition zone. Temperature crosses critical thresholds at base.
+- **4-8 ms**: Leader trunk established. Low-resistance channel conducts energy to tip. Epsilon falling as thermal accumulation helps.
+- **8-12 ms**: Leader-dominated growth. Streamer crown at tip continuously fed by leader current. Best epsilon (5-8 J/m). Growth slowing as [[capacitive-divider]] attenuates V_tip.
+
+## 5. The Burst Ceiling: Why QCW Is Necessary
+
+### 5.1 Steve Ward's 80 us Measurement
+
+Steve Ward's DRSSTC-0.5 provides a clean measurement of burst-mode growth saturation:
+
+| ON Time | Spark Length | Input Power |
+|---------|-------------|-------------|
+| ~70 us | 10-18 inches | 33-180 W |
+| >80 us | **No additional length** | Diminishing returns |
+
+"Gained almost no spark length after about 80 us of ON period." [Steve Ward, stevehv.4hv.org/DRSSTC.5.htm]
+
+### 5.2 Thermal Physics Explanation
+
+The 80 us ceiling is strikingly consistent with the thermal time constant for 100 um streamers:
+
+```
+tau_thermal = d^2 / (4*alpha) = (100e-6)^2 / (4*2e-5) ~ 125 us
+```
+
+After approximately one thermal time constant, channels are cooling as fast as they are being heated. Additional energy goes into re-heating decaying channels rather than new forward growth. This is the fundamental wall that QCW overcomes by sustaining drive beyond this timescale.
+
+### 5.3 Steve Conner's Burst Efficiency Finding
+
+Short bursts of high peak power grow sparks more efficiently than long bursts of low peak power. A 100 us burst works better than 150 us at the same total energy. Higher peak power pushes the initial streamer further before the 80 us ceiling hits. See [[power-optimization]].
+
+## 6. QCW Energy Budget
+
+### 6.1 Measured Energy Data
+
+| Quantity | Value | Source |
+|----------|-------|--------|
+| QCW energy per pulse | 275 J (for 1.78 m) | Loneoceans v1.5 |
+| Apparent epsilon (total input / length) | 155 J/m | Derived |
+| Estimated system efficiency | 30-50% | Community consensus |
+| Estimated spark epsilon | 45-75 J/m | Derived (155 * 0.3-0.5) |
+| Burst DRSSTC energy per bang | 5-12 J | Steve Ward |
+| Burst DRSSTC average power | 33-180 W for 25-46 cm | Steve Ward DRSSTC-0.5 |
+
+The apparent epsilon of 155 J/m includes system losses (primary resistance, secondary losses, corona, radiation). The spark epsilon of 45-75 J/m includes the early inefficient growth phase (first ~2-4 ms at high epsilon). The leader-dominated late-stage epsilon is significantly lower (estimated 5-15 J/m), consistent with the framework's QCW range.
+
+### 6.2 Frequency Tracking During QCW Ramp
+
+Loneoceans measured frequency shifts during QCW operation:
+
+| Condition | Frequency | Shift |
+|-----------|-----------|-------|
+| Unloaded secondary | 406-409 kHz | baseline |
+| With 50 cm simulated streamer | 349 kHz | -14% |
+| With 1 m simulated streamer | 310 kHz | -24% |
+| QCW v1.5 during actual spark | 413 → 377 kHz | -8.7% |
+
+The 8.7% shift during actual QCW operation is less than the simulated 1 m streamer (-24%), suggesting a real 1.78 m spark has lower effective capacitance than a solid wire — consistent with the branched, non-solid nature of real sparks. Frequency tracking (PLL or programmed) is essential during QCW ramps; a 5% detuning costs ~50% of delivered power (see [[coupled-resonance]]).
+
+## 7. Environmental and Design Factors
+
+### 7.1 Environmental Sensitivity
+
+davekni observed straighter arcs in warm, dry conditions; curved/branchy arcs more common outdoors (cooler, more humid). Dr. Kilovolt reported "looping" or "curving" streamers under humid or cool outdoor conditions.
+
+**Physics:** Higher humidity → faster complex-ion recombination (25x for hydrated ions, see [[streamers-and-leaders]]) → shorter plasma lifetime → less thermal persistence → more branching. Lower temperature → higher gas density → higher E_propagation → harder to sustain growth in single channel.
+
+### 7.2 Smooth Power Delivery Topologies
+
+Successful QCW implementations use inherently smooth power delivery:
+
+- **Steve Ward's quadratic ramp**: Voltage rises linearly → power rises as V^2 → smoothly increasing energy delivery
+- **Phase-Shifted Full Bridge (PSFB)**: Dr. Kilovolt's SiC PSFB provides a "1-cosine" transfer function with no pulse-skip artifacts
+- **UD3 controller**: Netzpfuscher's phase-shift modulation design provides smooth QCW control
+- **Analog ramp generators**: Finn Hammer's reference design for linear voltage ramp
+
+Pulse-skip modulation produces more sword-like sparks than standard burst but falls short of true swords due to envelope jitter (Section 2.3).
+
+## 8. Spark-to-Secondary Ratios: The Efficiency Measure
+
+The spark:secondary ratio (spark length divided by secondary winding length) is the clearest measure of QCW's advantage:
+
+| Builder | Mode | Spark | Secondary | Ratio |
+|---------|------|-------|-----------|-------|
+| Steve Ward | Burst | 80" | 22" | 3.6:1 |
+| Loneoceans DRSSTC3 | Burst | 70" | 27.5" | 2.5:1 |
+| Loneoceans QCW v1.0 | QCW | 40" | 5.5" | 7.3:1 |
+| Lucasww | QCW | 51" | 5" | 10.2:1 |
+| Loneoceans QCW2 | QCW | 24" | 2.4" | 10:1 |
+| Loneoceans QCW v1.5 | QCW | 70+" | 5.55" | 12.6:1 |
+| Mathieu thm | QCW | 76" | 5.6" dia | 13.6:1 |
+| Fat Coil | QCW | 132" | 8" | 16.5:1 |
+
+The 3-5x improvement from burst to QCW is a direct measure of the leader-dominated growth advantage. Leaders extend the effective electrode continuously, so the secondary length (which constrains maximum voltage) becomes less important relative to sustained power delivery.
+
+## 9. Critical Time Comparisons
+
+| Timescale | Value | Significance |
+|-----------|-------|-------------|
+| RF half-period at 400 kHz | 1.25 us | Channel heating between cycles |
+| RF half-period at 100 kHz | 5 us | Channel heating between cycles |
+| Streamer tau_thermal (100 um) | ~125 us | 100x longer than RF period at 400 kHz |
+| Conductance tau_g (heating) | 40 us | Time to heat one "step" |
+| Conductance tau_g (cooling) | 200 us | 5x longer than heating → ratcheting |
+| Driven leader step time | ~60 us | Close to tau_g; sets growth rate |
+| Burst pulse duration | 70-150 us | Comparable to streamer tau → saturation |
+| Burst ceiling (Ward) | ~80 us | Streamer growth saturates |
+| Leader transition time | 0.5-2 ms | Within QCW ramp; exceeds burst pulse |
+| Streamer persistence | 1-5 ms | Exceeded by QCW ramp |
+| Dark period cycle | 1-5 ms | Multiple cycles fit within QCW ramp |
+| QCW ramp duration | 10-22 ms | 100x longer than tau_g |
+
+## 10. Community Hypotheses (Unproven)
+
+### 10.1 Uspring's Sideways Breakout Suppression
+
+QCW's slowly ramped voltage keeps tip voltage low, reducing the transverse field component. The field is only strong enough for forward propagation along the existing hot channel.
+
+**Assessment:** Physically plausible. The hot leader channel has much lower impedance than virgin air to the side, so a weak field preferentially drives current forward.
+
+### 10.2 Channel Temperature: ~5000 K
+
+Uspring estimated ~5000 K from conductivity analysis. Not spectroscopically measured. Consistent with Bazelyan's leader temperature range (4000-6000 K) and the white/yellow visual appearance of QCW sword sparks (blackbody peak near 5000 K).
+
+### 10.3 Steve Ward's "2000 Small Sparks" Model
+
+At 400 kHz over 5 ms, there are ~2000 RF half-cycles, each depositing a small amount of energy. This is a simplified but correct description of the driven-leader mechanism as viewed through the conductance relaxation model.
+
+## 11. Framework Validation Summary
+
+| Prediction | Community Data | Agreement |
+|------------|---------------|-----------|
+| Thermal persistence is key to QCW advantage | Confirmed by all data | Excellent |
+| Streamer-to-leader transition requires sustained drive | Confirmed | Excellent |
+| Capacitive voltage division limits length | Confirmed by frequency shift data | Excellent |
+| Hungry streamer self-optimization | Confirmed by Burnett causality insight | Excellent |
+| Burst mode limited by streamer cooling | Ward: 80 us ceiling (cf. tau ~ 125 us) | Good (within 1.5x) |
+| Optimal QCW ramp: >5x tau_thermal | 10-20 ms (well above minimum) | Consistent |
+
+### What the Framework Missed
+
+1. **Frequency threshold for sword sparks (300-600 kHz)** — derivable from existing thermal physics (RF period << tau_thermal) but was not explicitly predicted
+2. **QCW secondary voltage is low (40-70 kV)** — framework implicitly assumed higher voltages for longer sparks
+3. **Power envelope quality matters** — growth model dL/dt = P/epsilon does not capture the effect of envelope smoothness on channel selection; spark straightness improves progressively from pulse-skip (sawtooth) through Bresenham PDM (linear ramp) to true QCW (quadratic ramp)
+4. **Three ramp regimes** — the "too long" bushy regime was not predicted (arises from capacitive divider saturation + excess power branching)
+5. **QCW growth rate (~170 m/s)** — not previously predicted but derivable from tau_g and step length
+
+## 12. Key Persons
+
+| Person | Contribution |
+|--------|-------------|
+| Steve Ward | QCW inventor; quadratic power profile; 40-55 kV measurement; 80 us burst ceiling |
+| Gao Guangyan (Loneoceans) | Most detailed QCW measurements (4 builds); frequency tracking data; three ramp regimes |
+| David Knierim (davekni) | Critical 15:1 voltage comparison; oversized QCW; fiber probe |
+| Richie Burnett | Causality reversal; pole splitting theory |
+| Steve Conner | Burst efficiency finding; hungry streamer principle |
+| Uspring | Temperature estimates (~5000 K); voltage gradient analysis |
+| Jan Martis (Dr. Kilovolt) | SiC PSFB QCW; k=0.55; 2-2.5 m sparks; environmental sensitivity |
+| Mads Barnkob | Frequency threshold observation (>400 kHz) |
+| Zach Armstrong (LabCoatz) | Frequency window (300-600 kHz) |
+
+## Key Relationships
+
+- **Derives from:** [[thermal-physics]] (thermal persistence, conductance relaxation, tau_g asymmetry are the physical foundation)
+- **Derives from:** [[streamers-and-leaders]] (driven leader growth is a special case of the streamer-to-leader transition)
+- **Interacts with:** [[coupled-resonance]] (frequency tracking during QCW ramp is essential; pole shifts 5-25%)
+- **Interacts with:** [[power-optimization]] (R_opt_power shifts continuously during ramp; match at 50-70% of target length)
+- **Interacts with:** [[capacitive-divider]] (voltage division limits maximum length; causes "too long" regime)
+- **Interacts with:** [[energy-and-growth]] (epsilon varies during ramp from ~15 J/m early to ~5-8 J/m late)
+- **Constrained by:** [[field-thresholds]] (inception threshold must be exceeded; propagation threshold sustains growth)
+- **Measured via:** Phase 6 QCW community survey (primary data source)

context/streamers-and-leaders.md

@@ -0,0 +1,652 @@
+---
+id: streamers-and-leaders
+title: "Streamer and Leader Discharge Physics"
+status: established
+source_sections: "spark-physics.txt: Part 5 Section 5.5 (lines 314-337)"
+related_topics: [thermal-physics, energy-and-growth, field-thresholds, capacitive-divider, power-optimization, qcw-operation, branching-physics, empirical-scaling, distributed-model, equations-and-bounds, open-questions]
+key_equations: [growth-rate, epsilon-thermal-refinement]
+key_terms: [streamer, leader, transition, thermal_ionization, photoionization, Joule_heating, epsilon, hungry_streamer, QCW, burst_mode, electron_density, ionization_front, recombination, corona_to_spark_transition, specific_energy_density, microdischarge, dark_period, aborted_leader, Gallimberti_model, stem, thermal_ratcheting, leader_velocity, electron_attachment_time, Bazelyan_VI, stepped_leader, continuous_leader, dart_leader, leader_formation_threshold, conductance_relaxation]
+images: [streamers-vs-leaders-photos.png, streamer-to-leader-transition-sequence.png]
+examples: [spark-growth-timeline.md]
+open_questions:
+  - "What is the exact current threshold for streamer-to-leader transition?"
+  - "How does branching factor differ between streamers and leaders?"
+  - "What determines the number of streamer branches at a leader tip?"
+  - "Can the transition be modeled as a sharp threshold or is it gradual?"
+  - "How does ambient humidity quantitatively affect the streamer-leader transition current?"
+  - "What is the quantitative role of photo-ionization vs. electron drift in positive streamer propagation in Tesla coil sparks?"
+  - "How many dark period cycles typically precede stable leader inception at TC voltages and electrode geometries?"
+  - "Can Gallimberti's transition model be corrected with empirical factors for TC conditions, or is a full kinetic model required?"
+---
+
+# Streamer and Leader Discharge Physics
+
+Tesla coil sparks are not a single phenomenon but a composite of two fundamentally different discharge types: streamers and leaders. Understanding the distinction between these types, and the transition from one to the other, is essential for explaining why different operating modes produce dramatically different spark lengths and efficiencies. This topic connects the microscopic plasma physics to the macroscopic circuit behavior described by [[energy-and-growth]] and [[power-optimization]].
+
+## Streamer Discharges
+
+Streamers are the initial, ephemeral discharge channels that form when the electric field at the topload exceeds the inception threshold (see [[field-thresholds]]).
+
+### Physical Properties
+
+| Property | Value | Notes |
+|----------|-------|-------|
+| Diameter | 10-100 um | Very thin filaments |
+| Propagation speed | ~10^6 m/s | Extremely fast (fraction of speed of light) |
+| Current | milliamperes (mA) | Low current per channel |
+| Propagation mechanism | Photoionization | UV photons ionize gas ahead of channel |
+| Temperature | 300-3000 K | Minimal heating (non-thermal plasma) |
+| Resistance | Very high | rho ~ 10-100 ohm*m |
+| Persistence | Microseconds (thermal) | Pure diffusion tau ~ 1-100 us for d ~ 10-100 um |
+| Effective persistence | 1-5 ms | Extended by ionization memory |
+| Visual appearance | Purple/blue | Nitrogen second positive band emission |
+| Branching | Highly branched | Many fine filaments |
+| Energy per meter (epsilon) | High (30-100+ J/m) | Inefficient for forward propagation |
+
+### Electron Density and Internal Structure
+
+The properties table above gives macroscopic observables. At the microscopic level, streamers have well-characterized internal structure from both simulation and measurement:
+
+**Ionization front at the streamer head:**
+The active ionization zone at the leading edge of a streamer has a thickness of approximately **0.015 cm (~150 um)**. This thin front is where electron avalanche multiplication is occurring -- it is the "engine" of the streamer. [Becker et al. 2005, Ch 2, p. 37]
+
+**Electron density in the streamer body:**
+
+| Region | n_e (cm^-3) | Notes |
+|--------|-------------|-------|
+| Outer boundary (visible edge) | ~10^11 | Diffuse boundary of ionized region |
+| Inner body (conducting core) | >10^13 | Main current-carrying region |
+| Fully developed spark channel | ~10^16 | After corona-to-spark transition |
+
+[Becker et al. 2005, Ch 2, pp. 37-38]
+
+These densities are in the non-equilibrium regime: the electron temperature (~3 eV, ~35,000 K) is far above the gas temperature (~300-1000 K). See [[field-thresholds]] Section 1.4 for the breakdown physics behind this non-equilibrium state.
+
+**Connection to conductivity:** The conductivity of the streamer body can be estimated from these electron densities using:
+
+```
+sigma = n_e * e^2 / (m_e * nu_e-air)
+```
+
+For n_e ~ 10^13 cm^-3 in warm air, this yields sigma ~ 0.01-0.1 S/m, consistent with the "cold streamer" range in [[thermal-physics]]. For a fully developed spark channel at n_e ~ 10^16, sigma reaches ~10-100 S/m (leader/arc range). See [[equations-and-bounds]] Section 14.6 for the full conductivity calculation.
+
+**Microdischarge reference properties** (individual streamer filaments resemble atmospheric microdischarges):
+
+| Property | Value | Notes |
+|----------|-------|-------|
+| Duration | 1-10 ns | Single filament lifetime |
+| Filament radius | ~100 um | Consistent with streamer diameter range above |
+| Peak current | 0.1 A | Per individual filament |
+| Current density | 100-1000 A/cm^2 | High due to small cross-section |
+| Electron density | 10^14 - 10^15 cm^-3 | Higher than sustained streamer body |
+| Electron energy | 1-10 eV | Non-equilibrium |
+| Degree of ionization | ~10^-4 | Very weakly ionized |
+
+[Becker et al. 2005, Ch 6, Table 6.2.1]
+
+These microdischarge properties are relevant because individual Tesla coil streamers resemble atmospheric microdischarges in many respects -- similar diameters, current densities, and electron densities.
+
+### Streamer Velocity and Tip Physics
+
+Streamer velocity is set by the ionization wave dynamics at the tip. The tip maintains a nearly constant maximum field of **E_m ~ 150-170 kV/cm** through a self-regulation mechanism, independent of the applied voltage.
+
+```
+V_s = v_im * r_m / [(2k-1) * ln(n_c/n_0)]
+
+where:
+  v_im = 1.1 * 10^10 s^-1     (ionization frequency at E_m)
+  r_m = U_t / (2 * E_m)        (tip radius, grows with tip potential)
+  k = 2.5                      (power index for ionization rate vs field)
+  n_c = 9 * 10^13 cm^-3        (initial plasma density, INDEPENDENT of U_t)
+  n_0 ~ 10^5 - 10^6 cm^-3      (seed electron density)
+```
+
+[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, pp. 41-43, Eq. 2.3, 2.6]
+
+**Key result: V_s is proportional to U_t.** Since r_m ~ U_t at constant E_m, streamer velocity scales linearly with tip potential. Higher secondary voltage = faster streamers. Numerical example: at U_t = 34 kV, r_m = 0.1 cm, V_s = 1.7 * 10^6 m/s.
+
+**Minimum streamer velocity:** V_s_min = (1.5-2) * 10^5 m/s, occurring at U_t = 5-8 kV. Streamers slower than this have never been observed — they cannot sustain propagation against attachment losses.
+
+**Voltage-independent initial plasma density:** n_c = 9 * 10^13 cm^-3 is a fundamental constant of air breakdown at atmospheric pressure (set by E_m alone). This means a streamer channel has the same electron density regardless of whether it was created by a 10 kV or a 500 kV potential. What changes with voltage is the streamer velocity, diameter, and length — not the local plasma density.
+
+### Maximum Streamer Length
+
+The maximum length a streamer can reach is set by the balance between tip potential and the critical propagation field:
+
+```
+l_max = (U_t - U_0) / E_cr ~ U_t / E_cr   (when external potential U_0 is small)
+
+E_cr(+) = 4.5-5 kV/cm  (positive streamers in air)
+E_cr(-) ~ 10 kV/cm     (negative streamers)
+```
+
+[Bazelyan & Raizer 2000, Ch 2, pp. 58-59, Eq. 2.32]
+
+| Tip Voltage | l_max (positive) | l_max (no attachment) | l_max (no losses) |
+|-------------|------------------|-----------------------|-------------------|
+| 250 kV | 0.39 m | — | — |
+| 500 kV | 0.94 m | 1.25 m | 3.0 m |
+| 750 kV | 1.42 m | — | — |
+
+The "no attachment" and "no losses" columns show the enormous potential for longer streamers in pre-heated channels where electron attachment is suppressed. A TC spark re-using a thermally persistent channel (where T > 2000 K reduces attachment by orders of magnitude) can extend streamers far beyond the cold-air limit — this is the fundamental reason thermal persistence matters for TC spark length.
+
+**TC implication:** At V_top = 400 kV, maximum cold-air streamer length is ~0.8-0.9 m. Beyond this, leader formation is required. This is consistent with the observation that burst-mode DRSSTCs plateau at ~1 m regardless of power.
+
+### Single Streamer Heating: Negligible
+
+The energy deposited by a single streamer passage is fundamentally limited:
+
+```
+Energy density per passage: W = epsilon_0 * E_m^2 / 2 = 2.6 * 10^-2 J/cm^3
+Temperature rise: Delta_T < W / c_v = 3 K
+```
+
+[Bazelyan & Raizer 2000, Ch 2, pp. 49-50, Eq. 2.17]
+
+This is an essential result: **a single streamer deposits only enough energy to raise the gas temperature by ~3 K**. Leader formation requires heating to 5000+ K — which means the channel must accumulate energy from hundreds of streamer passages or from sustained current flow. This is why the leader mechanism (concentrating many streamers' worth of current through a single contracted filament) is necessary for TC spark growth beyond the streamer limit.
+
+Increasing the applied voltage does NOT increase the specific energy deposition because the channel cross-section grows as U^2 while the energy scales as U^2 — the energy density (J/cm^3) remains ~epsilon_0 * E_m^2 / 2 regardless of voltage.
+
+### Propagation Mechanism: Photoionization
+
+Streamers propagate via a fundamentally non-thermal mechanism:
+
+1. **Strong field at tip**: The thin, pointed streamer tip concentrates the electric field to very high values (10-100 kV/cm)
+2. **Electron avalanche**: Free electrons in the high-field region accelerate and ionize gas molecules through impact
+3. **UV emission**: Excited nitrogen molecules emit UV photons (primarily in 98-102.5 nm range)
+4. **Photoionization**: These UV photons ionize oxygen molecules up to ~1 mm ahead of the streamer tip, creating seed electrons
+5. **New avalanche**: Seed electrons start new avalanches, extending the streamer
+6. **Self-propagating**: Steps 1-5 repeat at ~10^6 m/s
+
+Key physics: the propagation is electromagnetic (photon-mediated), not thermal. The gas behind the streamer tip is barely heated. This is why streamers can propagate so fast -- they do not wait for thermal processes.
+
+#### The Photo-Ionization Debate
+
+The role of photo-ionization in positive streamer propagation is well-established experimentally but quantitatively debated in the simulation literature:
+
+- **Positive streamers** (propagating away from the anode/topload) require a source of seed electrons ahead of the streamer tip. Photo-ionization by UV from excited N2 molecules ionizing O2 molecules is the most widely accepted mechanism. In simulations, positive streamers **will not propagate** if both photo-ionization and background ionization are set to zero. [Becker et al. 2005, Ch 2, pp. 51-52; Morrow & Lowke 1995]
+
+- **Negative streamers** (propagating toward the anode) can propagate without photo-ionization because electrons naturally drift ahead of the streamer tip. Simulations have reproduced negative streamer propagation and even branching from a single initial electron without photo-ionization. [Becker et al. 2005, Ch 2, p. 52; Arrayas et al. 2002]
+
+- **Simulation workaround:** Because photo-ionization cross sections are poorly known, many simulation models substitute a uniform **seed electron density of 10^7 - 10^8 cm^-3** instead of explicit UV transport, which produces similar results. [Becker et al. 2005, Ch 6, p. 281]
+
+For Tesla coil sparks, which operate on AC waveforms, both positive and negative half-cycles contribute to propagation. The photo-ionization mechanism is most critical during the positive half-cycle when streamers must advance into virgin (unperturbed) air.
+
+### Why Streamers Are Inefficient
+
+Despite their speed, streamers are poor at creating lasting conductive channels:
+- **Low current**: Insufficient Joule heating (I^2 * R) to raise temperature significantly
+- **Thin channels**: Cool quickly (tau ~ microseconds for d ~ 10 um)
+- **High resistance**: Poor conductors, most of the voltage drops across the channel rather than reaching the tip
+- **Branching**: Energy splits among many branches, diluting the current in each
+- **No thermal memory**: Each streamer pulse must re-ionize fresh gas
+
+The energy "wasted" in creating a streamer that immediately cools and deionizes is the fundamental reason burst mode (streamer-dominated) has high epsilon.
+
+## Leader Discharges
+
+Leaders are the hot, persistent, highly conductive channels that form when sufficient sustained current flows through a streamer channel.
+
+### Physical Properties
+
+| Property | Value | Notes |
+|----------|-------|-------|
+| Diameter | mm to cm | 100-1000x thicker than streamers |
+| Propagation speed | ~10^3 m/s | Much slower than streamers |
+| Current | Amperes (A) | High current, intense Joule heating |
+| Propagation mechanism | Thermal ionization | Saha equilibrium at T > 5000 K |
+| Temperature | 5000-20000 K | Fully thermalized plasma |
+| Resistance | Low | rho ~ 1-10 ohm*m |
+| Persistence | Seconds | With convection maintaining hot column |
+| Visual appearance | White/orange/yellow | Blackbody + line emission |
+| Branching | Relatively straight | Few major branches |
+| Energy per meter (epsilon) | Low (5-15 J/m) | Efficient for forward propagation |
+
+### Propagation Mechanism: Thermal Ionization
+
+Leaders propagate by a fundamentally different mechanism:
+
+1. **Hot conducting core**: The leader channel is a thermalized plasma at 5000-20000 K
+2. **Current flows to tip**: The low-resistance leader conducts current efficiently from the topload to its tip
+3. **Tip launches streamers**: At the leader tip, the concentrated field creates new streamers
+4. **Streamers carry current**: Some streamer branches carry enough current (fed from the leader) to undergo Joule heating
+5. **Heated streamers become leader**: The heated channel transitions to a new leader segment
+6. **Leader extends**: Steps 3-5 repeat, advancing the leader at ~10^3 m/s
+
+The leader propagation speed is much slower than streamer speed because it is limited by thermal processes (heating gas from ~300 K to ~5000+ K takes time). But the leader is vastly more efficient because each meter of leader channel, once formed, persists and conducts efficiently.
+
+### Leader Velocity
+
+Bazelyan & Raizer provide an empirical formula for leader velocity:
+
+```
+v_L = 1500 * sqrt(|Delta_U_t|)    [cm/s, with Delta_U_t in volts]
+```
+
+[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 709, Eq. 5]
+
+For a TC with 300 kV topload voltage: v_L = 1500 * sqrt(300,000) ~ 820,000 cm/s = **8.2 km/s**. This is intermediate between laboratory sparks (~10 km/s) and lightning leaders (~100 km/s), consistent with observed TC spark growth.
+
+The physical basis: the leader advance rate is set by the conducting streamer length l ~ 1 cm (limited by electron attachment at ~100 ns) divided by the thermal instability contraction time tau_ins ~ 1 us, giving v_L ~ 10^6 cm/s. The square root voltage dependence arises because higher tip voltage increases streamer vigor (length and density), expanding the zone available for contraction.
+
+**Electron attachment time in cool air: ~100 ns** [Bazelyan & Raizer 2000, p. 703]
+
+This is the fundamental timescale that limits streamer channel lifetime without heating. At TC frequencies of 50-400 kHz (half-periods of 1.25-10 us), a cold streamer goes through 12-100 attachment times per half-cycle. Without heating to >5000 K (where attachment becomes negligible), the streamer plasma dies between every half-cycle and must be re-created — this is why streamers are so energy-inefficient.
+
+### Stepped vs Continuous Leaders
+
+Lightning observations reveal two distinct leader propagation modes, which have direct analogs in TC spark behavior:
+
+**Positive leaders** (ascending from grounded objects, carrying positive charge) propagate **continuously**: the bright tip moves smoothly upward with gradually varying velocity. This is the dominant mode for TC sparks, where positive streamers/leaders propagate from the positive-going topload.
+
+**Negative leaders** (descending from cloud, carrying negative charge) propagate in **steps**: discrete jumps of 10-200 m (average 30 m for lightning), separated by pauses of 30-90 us. Each new step briefly re-illuminates the entire channel behind it. The stepped pattern arises because negative streamers require a different mechanism (electron drift ahead of the tip rather than photoionization), leading to an intermittent advance.
+
+[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 1, pp. 17-18]
+
+| Leader Type | Polarity | Pattern | Step size | Pause | Average velocity |
+|-------------|----------|---------|-----------|-------|-----------------|
+| Positive | + | Continuous | N/A | N/A | 10^5 - 10^6 m/s |
+| Negative | - | Stepped | 10-200 m (avg 30 m) | 30-90 us | 10^5 - 10^6 m/s (averaged) |
+| Dart (re-strike) | either | Continuous (fast) | N/A | N/A | (1-4) * 10^7 m/s |
+
+**Average velocities are the same** for stepped and continuous leaders when averaged over the total development time: 10^5-10^6 m/s (100-1000 km/s), with an average of ~3*10^5 m/s.
+
+**Dart leaders** (subsequent strokes following existing hot channels) are always continuous and much faster: (1-4)*10^7 m/s. This is because they propagate through pre-heated, pre-ionized gas where the ionization front moves as a thermal wave rather than requiring fresh ionization.
+
+**TC relevance:** TC sparks on the positive half-cycle behave as continuous leaders. On the negative half-cycle, stepped behavior could occur but is masked by the rapid AC reversal (half-period of 1.25-10 us at 100-400 kHz is shorter than the 30-90 us step pause). The result is that TC sparks effectively propagate as continuous leaders on both half-cycles, though with different microscopic mechanisms.
+
+### Leader Formation Voltage Threshold
+
+A minimum potential difference is required to excite and develop a leader in air:
+
+```
+Delta_U_min ~ 300 - 400 kV
+```
+
+[Bazelyan & Raizer 2000, Ch 5, p. 271]
+
+This is the total potential drop from electrode to the surrounding space needed to provide enough field energy for streamer formation, heating through the 2000-5000 K transition zone, and establishment of a self-sustaining hot channel.
+
+**TC implications for burst mode:** Most burst-mode DRSSTCs operate with topload voltages of 100-600 kV. At the low end (100-200 kV), leader formation is marginal — the coil produces primarily streamers. At 300+ kV, leaders form readily, consistent with the dramatic improvement in spark length efficiency observed when coils cross the ~300 kV threshold. This provides physical backing for the common builder observation that "bigger coils are disproportionately more impressive" — they cross the leader formation threshold.
+
+**Critical caveat for QCW mode:** The 300-400 kV threshold applies to **single-shot impulse discharges** where the entire streamer-to-leader transition must occur from a single event. QCW coils form leaders at dramatically lower topload voltages — typically only **40-70 kV** — because the resonant circuit continuously injects energy over thousands of RF cycles (5-20 ms ramp). Multiple independent builders have confirmed this: davekni measured ~40 kV peak at 450 kHz producing 2-2.5 m sword sparks, while Steve Ward measured 40 kV rising to 55 kV over ~5000 RF cycles for 50+ inch arcs. The comparison is stark: a burst DRSSTC at 80 kHz needs ~600 kV for the same spark length a QCW achieves at 40 kV — a 15:1 voltage ratio. [Phase 6 QCW community survey, multiple sources]
+
+The physical explanation: in QCW mode, the thermal ratcheting mechanism (see [[thermal-physics]]) accumulates energy from many RF cycles. Each cycle deposits a small increment via Joule heating, and the 5:1 asymmetry in conductance relaxation time constants (40 us heating vs 200 us cooling) ensures the temperature ratchets upward. Over ~1-5 ms, the stem temperature crosses through the critical thresholds (2000 K → 4000 K → 5000 K) without ever requiring the full 300-400 kV instantaneous voltage. The voltage merely needs to exceed the inception threshold and maintain current flow.
+
+### Why Leaders Are Efficient
+
+- **Low resistance**: The hot, ionized channel conducts well, delivering most of the source voltage to the tip
+- **Persistence**: Long thermal time constants (see [[thermal-physics]]) mean the channel stays hot and conductive for seconds
+- **Self-maintaining**: As long as current flows, Joule heating maintains the temperature
+- **Focused energy**: Less branching means energy is concentrated in the main propagation path
+- **Thermal accumulation benefit**: epsilon(t) decreases as the channel accumulates thermal energy (see [[energy-and-growth]])
+
+## The Streamer-to-Leader Transition
+
+The transition from streamer to leader is the critical process that determines spark efficiency. It is the reason QCW mode (which promotes transition) produces longer sparks than burst mode (which cannot sustain it) for the same energy input.
+
+### Transition Sequence
+
+The six-step transition process:
+
+**Step 1: High E-field creates streamers**
+- Topload voltage exceeds inception threshold (see [[field-thresholds]])
+- Multiple streamer branches form simultaneously
+- Streamers propagate rapidly outward (~10^6 m/s)
+- Channel is cold, thin, high-resistance
+
+**Step 2: Sufficient current causes Joule heating**
+- The resonant circuit continues driving current through the streamer channels
+- Current distributes among branches, but some branches carry more than others
+- Joule heating power per unit length: P_linear = I^2 * R_linear [W/m]
+- For a 100 um streamer at rho = 50 ohm*m carrying 100 mA:
+  ```
+  R_linear = rho / A = 50 / (pi * (50e-6)^2) = 6.4 * 10^9 ohm/m
+  P_linear = (0.1)^2 * 6.4e9 = 64 MW/m (!!)
+  ```
+  This enormous linear power density (even at low total current) is what drives the transition. The thin channel concentrates the heating.
+
+**Step 3: Heated channel undergoes thermal ionization**
+- Temperature rises from ~300 K through ~3000 K to 5000+ K over ~1 ms
+- At ~3000-4000 K: significant thermal dissociation of N2 and O2 begins
+- At ~5000 K: Saha equation predicts substantial ionization fraction
+- Conductivity increases by orders of magnitude
+- Resistance drops, allowing more current to flow (positive feedback)
+
+**Step 4: Leader forms from base**
+- The transition proceeds from the base (near topload) outward
+- Base segments see the most current (no branching losses yet) and transition first
+- Leader formation is progressive, not instantaneous
+- Base becomes a bright, thick, low-resistance channel
+
+**Step 5: Leader tip launches new streamers**
+- The leader acts as an extension of the topload electrode
+- At the leader tip, the electric field is enhanced (see [[field-thresholds]])
+- New streamers propagate from the leader tip into fresh air
+- These are "fed streamers" -- receiving current from the leader
+
+**Step 6: Fed streamers convert to leader**
+- Current from the leader flows through the new streamers
+- Higher current than free streamers (sustained by leader's low resistance)
+- The same Joule heating process converts these streamers to leader
+- The leader extends by one "step" as each generation of fed streamers transitions
+
+This cycle repeats: leader -> streamers -> Joule heating -> new leader -> more streamers. The spark grows as a composite structure with a leader trunk and streamer crown.
+
+### Transition Threshold
+
+The transition requires sufficient current density and duration. Approximate criteria:
+
+- **Current density**: j > ~10^6 A/m^2 in the streamer (equivalently, ~10 mA in a 100 um channel)
+- **Duration**: Must sustain heating for ~0.5-2 ms (long enough to raise temperature through ~3000 K to 5000+ K)
+- **Power density**: P_volume > ~10^10 W/m^3 approximately, sustained for milliseconds
+
+These thresholds explain why:
+- **QCW succeeds**: Continuous drive for 5-20 ms provides ample time and current
+- **Burst mode fails**: Short pulses (50-500 us) may not sustain heating long enough, especially if gaps allow cooling
+
+### Aborted Leaders and Dark Periods
+
+The transition sequence above is idealized. In practice, multiple failed attempts typically precede stable leader inception. High-speed photography and Schlieren imaging reveal a characteristic cycle [Liu 2017; Les Renardieres Group 1977, 1981]:
+
+**Dark Period Cycle:**
+
+1. **Streamer burst**: Positive streamers propagate from electrode tip into virgin air (~10^6 m/s)
+2. **Space charge shielding**: Positive ions left behind by the fast-moving electron front create a space charge cloud near the electrode that reduces the local electric field
+3. **Dark period**: Field at electrode drops below inception threshold. No new streamers form. Duration ~1-5 ms (depends on gap geometry and voltage)
+4. **Ion drift recovery**: Positive ions slowly drift outward under the applied field (mu_ion ~ 2 * 10^-4 m^2/(V*s)), gradually restoring the electrode field
+5. **Next burst**: When the field recovers above inception, a new streamer burst occurs
+
+Each burst deposits energy into the stem region (the short channel connecting the streamer base to the electrode). If the energy deposition from a single burst is insufficient to raise the gas temperature past the critical threshold for leader inception, the stem cools during the dark period and the leader attempt **aborts**.
+
+**Aborted leader progression:**
+
+- First burst: stem heats to ~1000-1500 K, cools back to ~500 K during dark period
+- Second burst: residual warmth means less energy needed; stem reaches ~1800-2500 K, cools to ~800-1200 K
+- Third/fourth burst: thermal ratcheting pushes temperature past critical threshold -> stable leader inception
+
+**Critical temperature requirement:** The gas temperature must **significantly exceed 2000 K** for stable leader inception, not merely reach it. During gas expansion following heating, convection losses can drop the temperature back below the critical ionization threshold. The gas must be heated enough to survive this expansion cooling. See [[thermal-physics]] for the detailed mechanism.
+
+**Multiple stems share current:** Schlieren photography shows that current from the electrode distributes among multiple streamer stems simultaneously, not just the strongest branch. This reduces the heating per individual stem and delays the transition. The stem that transitions to a leader first is typically the one that received the most cumulative energy across multiple burst cycles. [Liu 2017, Ch 2, Schlieren observations]
+
+### Gallimberti Model Critique
+
+The widely-cited Gallimberti (1972) model for streamer-to-leader transition assumes:
+
+1. **Constant electric field in the stem** during the transition process
+2. **Simplified V-T relaxation**: Uses a simplified nitrogen vibrational-translational energy transfer model
+3. **Single stem**: Assumes all current flows through one dominant stem
+
+Liu (2017, Ch 3) demonstrates through detailed kinetic modeling (45 species, 192 reactions) that these assumptions do not hold:
+
+- **Stem field varies significantly** as space charge evolves and the stem heats/expands
+- **V-T relaxation is not the dominant heating mechanism** in the late stages of transition; direct electron impact heating becomes important
+- **Humidity effect on V-T relaxation is weak**: The conventionally cited acceleration of V-T relaxation by water vapor is "several orders of magnitude smaller" than other energy sources during the transition [Liu 2017, Ch 3]
+- **Multiple stems share current**, invalidating the single-stem assumption
+
+Despite these limitations, Gallimberti's model captures the correct qualitative physics (energy accumulation in stem -> thermal runaway -> leader) and gives order-of-magnitude correct transition times. It remains useful as a conceptual framework but should not be trusted for quantitative predictions without correction factors.
+
+### Transition Energy Density Threshold
+
+In addition to the current density and duration criteria above, the corona-to-spark transition can be characterized by a volumetric energy density threshold:
+
+```
+Minimum specific energy for spark channel formation: 0.6 - 1 J/cm^3
+```
+
+[Becker et al. 2005, Ch 2, p. 59]
+
+This is the energy density that must be deposited in the streamer channel before it can transition to a self-sustaining spark (leader). For a 100 um diameter streamer channel, the energy per unit length to reach 1 J/cm^3 is:
+
+```
+E_per_length = 1 J/cm^3 * pi * (50 um)^2 = 7.85 * 10^-6 J/cm = 0.000785 J/m
+```
+
+This is a very small energy per meter compared to the observed epsilon values (5-100 J/m), confirming that the transition from streamer to leader is **not primarily limited by total energy** -- it is limited by the **rate of energy deposition** (power density). The current density criterion (j > 10^6 A/m^2) and the duration criterion (~0.5-2 ms) are the operative constraints. See [[energy-and-growth]] for how this connects to the physical origin of epsilon.
+
+### Spark Formation Dynamics
+
+Once the corona-to-spark transition begins, two stages of spark formation are observed in high-speed photography:
+
+1. **Primary streamer**: Fast propagation at ~10^8 cm/s (10^6 m/s) from the anode toward the cathode
+2. **Secondary streamer**: Slower propagation at ~10^5 - 10^6 cm/s along the same trajectory, after a delay that depends on overvoltage
+
+[Becker et al. 2005, Ch 2, pp. 59-60]
+
+The secondary streamer propagates not by direct ionization in a strong field (like the primary) but by energy deposition into the existing channel (gas heating, vibrational excitation). This is the physical precursor to leader formation.
+
+Upon bridging of the gap by the secondary streamer, the discharge current increases abruptly:
+
+```
+Spark current rise rate: dI/dt ~ 10^7 A/s
+```
+
+[Becker et al. 2005, Ch 2, p. 60]
+
+**Ion mobility** governs how fast the positive space charge left behind by the fast-moving streamer tip can rearrange:
+
+```
+mu_ion ~ 2 * 10^-4 m^2/(V*s)    (in air at STP)
+```
+
+[Becker et al. 2005, Ch 2, p. 60]
+
+This is much slower than electron mobility (~0.03 m^2/(V*s)), which is why the positive space charge from a streamer takes a relatively long time to redistribute -- contributing to the delay between primary and secondary streamer stages.
+
+## The Hungry Streamer Connection
+
+The streamer-to-leader transition is intimately connected to Steve Conner's hungry streamer principle (see [[power-optimization]]):
+
+The self-optimization feedback loop drives the system toward leader formation when sufficient power is available:
+
+1. Streamer forms with high R (above R_opt_power)
+2. Hungry streamer principle: plasma tries to reduce R toward R_opt_power to maximize power extraction
+3. Mechanism: increased current -> Joule heating -> higher temperature -> higher conductivity -> lower R
+4. As R decreases through the optimization, temperature rises, and the channel transitions from streamer to leader
+5. Leader equilibrium: R stabilizes near R_opt_power at a temperature that maintains the required conductivity
+6. If R_opt_power is below R_min (physical lower bound for plasma), the system is constrained and operates sub-optimally
+
+The hungry streamer principle and the streamer-to-leader transition are two descriptions of the same physical process viewed from different perspectives: one from the circuit (impedance optimization) and one from the plasma physics (thermal evolution).
+
+### Quantitative Resistance vs Current: The Power Law
+
+The qualitative hungry streamer picture now has a quantitative foundation from self-consistent plasma modeling. The equilibrium resistance per unit length follows a power law in current:
+
+```
+R = A / I^b    (ohm/m)
+```
+
+[da Silva et al. 2019, JGR Atmospheres]
+
+| Regime | Current Range | A (ohm * A^b / m) | b | TC Context |
+|--------|-------------|-------------------|------|-----------|
+| Region I | 1-10 A | 12,400 | 1.84 | TC streamer/early leader |
+| Region II | 10-1,000 A | 2,820 | 1.16 | DRSSTC burst mode |
+| Region III | 1,000-10,000 A | 180 | 0.75 | Lightning return strokes |
+
+**Worked examples for TC sparks:**
+
+- Streamer at 1 A: R ~ 12,400 ohm/m -> 12.4 kohm for 1 m spark (consistent with QCW/leader range)
+- Early streamer at 0.1 A: R ~ 12,400 / (0.1)^1.84 ~ 860,000 ohm/m (very high, as expected for cold streamer)
+- Leader at 10 A: R ~ 179 ohm/m -> 179 ohm for 1 m spark (hot leader, approaching arc)
+
+The steep exponent in Region I (b = 1.84) means resistance drops nearly as the square of current — this is the **quantitative expression of the positive feedback** that drives the streamer-to-leader transition. Doubling the current reduces resistance by ~3.6x, which increases current further, driving the thermal runaway.
+
+**Why the transition is slow despite this positive feedback:** The air heating efficiency eta_T is only ~10% at ambient temperature (see [[thermal-physics]]). 90% of the electrical energy goes into N2 vibrational excitation rather than gas heating. The thermal runaway only accelerates after the gas reaches ~1000-2000 K where eta_T approaches 1.0.
+
+**Cross-validation with Bazelyan V-I characteristic:** Bazelyan & Raizer provide two formulas of increasing precision:
+
+- **Simple:** i*E = 300 V*A/cm (arc approximation, valid for quick estimates)
+- **Precise (measured CVC):** E = 32 + 52/i V/cm (Eq. 2.48 in the full textbook)
+
+[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, p. 90]
+
+The precise CVC reveals that at high currents, the field plateaus at **32 V/cm** (not zero) — representing irreducible radiation and convection losses. The 52/i term dominates at TC-relevant currents (1-10 A). At i = 1 A: E = 84 V/cm (CVC) vs 300 V/cm (simple i*E=b formula). The simple formula overpredicts the field at low currents.
+
+Both Bazelyan formulas agree with da Silva's power law within a factor of ~2 for 1-10 A — see [[equations-and-bounds]] Section 14.14 for a detailed comparison table. The three independent approaches (Bazelyan experimental, Bazelyan measured CVC, da Silva plasma modeling) converging to similar values is strong evidence that these resistance values are reliable for TC spark modeling.
+
+**Connection to Mayr equation:** The R = A/I^b law describes where the resistance wants to go (equilibrium). The Mayr equation dG/dt = (1/tau) * (P/P_0 - 1) * G describes how fast it gets there. See [[equations-and-bounds]] Sections 14.8 and 14.11 for both models.
+
+## Composite Spark Structure
+
+A fully developed Tesla coil spark is not purely streamer or purely leader. It is a composite:
+
+```
+Topload
+  |
+  [Leader trunk]  -- Hot, thick, low-R, persistent
+  |                  Conducts efficiently from topload to crown
+  |
+  [Transition zone]  -- Intermediate properties
+  |                     Recently converted streamers
+  |
+  [Streamer crown]  -- Cool, thin, high-R, ephemeral
+                       Actively propagating into fresh air
+                       Highly branched
+```
+
+### Position-Dependent Properties
+
+This composite structure maps directly to the [[distributed-model]] where segments near the base (leader) have low R and segments near the tip (streamer) have high R:
+
+- **Base segments** (near topload): R ~ 1-10 kohm, hot, thick, leader
+- **Middle segments**: R ~ 10-100 kohm, transition zone
+- **Tip segments**: R ~ 100 kohm - 100 Mohm, cold, thin, streamer
+
+The convergence behavior of the distributed model's iterative optimization naturally reproduces this structure:
+- Base segments converge to low R (sharp power peak, well-coupled)
+- Tip segments converge to high R (flat power curve, poorly coupled)
+
+## Practical Implications by Operating Mode
+
+### QCW Mode: Leader-Dominated
+
+- Long ramp (5-20 ms) allows full transition at base within first few milliseconds
+- Leader trunk grows progressively during ramp
+- Low effective epsilon (5-15 J/m) because leader extension is efficient
+- Leader persistence (seconds) means channel stays alive throughout ramp
+- Streamer crown at tip is continuously fed by leader current
+- Result: longest sparks per unit energy
+
+**Measured QCW growth rate: ~170 m/s** (approximately half the speed of sound). This is estimated from community observations of spark growth during QCW ramps. [Phase 6 QCW survey, HVF topic 973]
+
+Self-consistency check: at 170 m/s over a 10 ms ramp, a spark grows 1.7 m. Over a 20 ms ramp, 3.4 m. These match observed QCW spark lengths (1-2 m for standard builds, 3.35 m for the Fat Coil QCW build).
+
+This 170 m/s rate is intermediate between free streamers (10^6 m/s) and natural lightning leaders (~10^4 m/s for stepped leaders, averaged). It represents a **driven leader** propagation mode unique to QCW: the leader advances continuously, fed by the circuit, at a rate limited by the thermal conversion of streamer-to-leader at the tip.
+
+**Driven leader step time:** From the growth rate and Bazelyan's typical leader step length (~1 cm):
+
+```
+step_time ~ step_length / growth_rate ~ 0.01 m / 170 m/s ~ 60 us
+```
+
+This 60 us step time is close to the conductance relaxation heating time constant (tau_g = 40 us from Bazelyan, see [[thermal-physics]]). The channel needs approximately one tau_g to heat each new segment to leader temperature, so the leader advance rate is limited by how fast each new streamer can be thermally converted. The 1.5x ratio (60 us observed vs 40 us tau_g) is reasonable given that the transition also requires crossing the eta_T efficiency bottleneck (10% at ambient → 100% above 2000 K).
+
+**Contrast with Bazelyan leader velocity:** The Bazelyan formula v_L = 1500*sqrt(|Delta_U_t|) gives ~4.7-8.2 km/s at 100-300 kV. This is 25-50x faster than the observed 170 m/s QCW growth rate. The discrepancy is explained by the fundamental difference between the two quantities: Bazelyan's v_L is the instantaneous leader step velocity (the speed of the thermal instability contraction within a single step), while the QCW 170 m/s is the net growth rate averaged over many steps including the time to heat each new streamer segment. The QCW leader advances in rapid micro-steps at ~km/s but spends most of its time waiting for each new segment to thermalize.
+
+### Burst Mode: Streamer-Dominated
+
+- Short pulse (50-500 us) may not allow transition to complete
+- Channel remains mostly streamer throughout the pulse
+- High effective epsilon (30-100+ J/m) because streamer propagation is inefficient
+- Channel cools between bursts (gap >> streamer persistence of ~1-5 ms)
+- Each burst must re-ionize from scratch
+- Result: bright but short sparks, energy-inefficient for length
+
+### High Duty Cycle DRSSTC: Hybrid
+
+- Closely spaced bursts (gaps < 5 ms) allow some thermal memory
+- Base may partially transition to leader between closely spaced pulses
+- Intermediate epsilon (20-40 J/m)
+- Neither fully leader-dominated nor fully streamer-dominated
+- Result: moderate length efficiency, intermediate spark character
+
+## Observable Differences
+
+The streamer/leader distinction is directly observable:
+
+### Visual
+
+- **Streamers**: Purple/blue, fine filaments, highly branched, flickering
+- **Leaders**: White/orange/yellow, thick trunk, straighter, more stable
+- **Composite**: Purple crown with white/yellow base is characteristic of healthy QCW growth
+
+### Audio
+
+- **Streamers**: Hissing/crackling sound (many small discharges)
+- **Leaders**: Louder snap/crack (single powerful channel)
+- **QCW ramp**: Tone that rises in pitch as ramp progresses
+
+### Electrical Signatures
+
+- **Streamer loading**: High impedance, small frequency shift, small Q reduction
+- **Leader loading**: Low impedance, large frequency shift, large Q reduction
+- **Transition**: Impedance drops abruptly during transition (resistance falls by orders of magnitude)
+
+These electrical signatures can be observed in the loaded pole behavior described in [[coupled-resonance]].
+
+## Connection to Energy per Meter
+
+The fundamental reason epsilon differs by mode comes down to this topic:
+
+- **Streamer epsilon** is high because: thin channels cool fast, energy is wasted on branching, re-ionization overhead is large, high resistance means poor voltage delivery to tip
+- **Leader epsilon** is low because: thick channels persist, energy is focused in main path, no re-ionization needed (already hot), low resistance delivers voltage efficiently to tip
+- **Mode determines which type dominates**: QCW promotes leaders (low epsilon), burst maintains streamers (high epsilon)
+
+This is the physical basis for the epsilon values used in [[energy-and-growth]] and the scaling relationships in [[empirical-scaling]].
+
+## Recombination and Plasma Decay
+
+When current ceases flowing through a spark channel, the plasma decays primarily through electron-ion recombination. The dominant recombination processes in air and their rate coefficients are:
+
+| Reaction | Rate Coefficient (cm^3/s) | Notes |
+|----------|--------------------------|-------|
+| O2+ + e- | 1.9 * 10^-7 * (300/T_e)^0.5 | Dominant in dry air |
+| N2+ + e- | 1.8 * 10^-7 * (300/T_e)^0.39 | Fast at low T_e |
+| NO+ + e- | 4.3 * 10^-7 * (300/T_e)^0.37 | Important in warm channels (NO forms above ~2000 K) |
+| H3O+ + e- | 6.3 * 10^-7 * (300/T_e)^0.5 | Relevant in humid air, T_e < 1000 K |
+
+[Becker et al. 2005, Ch 4, pp. 170-175]
+
+**Key takeaway:** All major simple atmospheric ion species recombine with electrons at approximately **2 * 10^-7 cm^3/s** at 300 K electron temperature. [Becker et al. 2005, Ch 4, p. 174]
+
+At high pressures, three-body recombination can increase rates to as high as **10^-4 cm^3/s**. [Becker et al. 2005, Ch 4, p. 175]
+
+**Complex and cluster ions — much faster recombination:**
+
+Simple ions (O2+, N2+) quickly convert to complex and hydrated cluster ions in atmospheric air. These cluster ions recombine with electrons **5-25x faster** than simple ions:
+
+| Ion | Rate (cm^3/s) | Formation timescale | Notes |
+|-----|--------------|---------------------|-------|
+| O2+ (simple) | 2.7 * 10^-7 * (300/T_e)^0.5 | N/A (initial ion) | Baseline rate |
+| O4+ (complex) | 1.4 * 10^-6 * (300/T_e)^0.5 | ~10-100 ns | 5x faster |
+| H3O+(H2O)3 (hydrated) | 6.5 * 10^-6 * (300/T_e)^0.5 | ~1 us (humid air) | 25x faster |
+
+[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, pp. 52-56, Eq. 2.24-2.28]
+
+**Critical for TC sparks:** Complex ions form within 10-100 ns, so the enhanced recombination rate applies almost immediately after the ionization front passes. In humid air (outdoor TC operation), the full hydration chain completes in ~1 us, giving recombination rates 25x the simple-ion value. This explains why:
+- Outdoor TC sparks in humid conditions are noticeably shorter than indoor sparks
+- The effective plasma lifetime in cold streamer channels is ~100-300 ns (not the ~800 ns from simple attachment alone)
+- Channel conductivity drops by ~100x within 300 ns behind the streamer tip, leaving only a ~30 cm "alive" section at typical streamer velocities
+
+**Thermal decomposition of complex ions:** At T > ~2000 K, the O4+ cluster breaks apart (k_decomp = 3.3 * 10^-6 * (300/T)^4 * exp(-5040/T) cm^3/s), reverting to simple ions with their slower recombination rate. This is another mechanism by which channel heating dramatically improves plasma persistence.
+
+**Connection to channel persistence:** For a streamer with n_e ~ 10^13 cm^-3, the recombination time constant is:
+
+```
+tau_recomb = 1 / (alpha_recomb * n_e) = 1 / (2e-7 * 1e13) ~ 50 us
+```
+
+This is comparable to the pure thermal diffusion time for a 100 um streamer (~0.1-0.2 ms), confirming that ionization memory and thermal cooling compete on similar timescales. For leaders with n_e ~ 10^15 - 10^16 cm^-3, recombination is faster (~0.5-5 us), but continuous Joule heating maintains the ionization against recombination losses.
+
+These quantitative recombination rates provide the microphysical foundation for the "ionization memory" mechanism described in [[thermal-physics]], and explain why effective streamer persistence (~1-5 ms) significantly exceeds the pure thermal diffusion time -- the plasma decays slower than the thermal profile.
+
+## Key Relationships
+
+- Derives from: [[field-thresholds]] (inception field creates initial streamers; propagation field sustains leader growth)
+- Derives from: [[thermal-physics]] (thermal diffusion and persistence determine transition feasibility)
+- Enables: [[energy-and-growth]] (channel type determines epsilon, the key parameter for growth prediction)
+- Enables: [[empirical-scaling]] (different mode efficiencies explain different scaling exponents)
+- Implements: [[power-optimization]] (hungry streamer self-optimization is the circuit-level view of the thermal transition)
+- Structures: [[distributed-model]] (leader-base / streamer-tip composite maps to position-dependent R)
+- Constrained by: [[capacitive-divider]] (voltage division limits current delivery to tip, affecting transition feasibility)

context/thermal-physics.md

@@ -0,0 +1,599 @@
+---
+id: thermal-physics
+title: "Thermal Physics of Spark Channels"
+status: established
+source_sections: "spark-physics.txt: Part 5 Section 5.4 (lines 281-313)"
+related_topics: [streamers-and-leaders, energy-and-growth, field-thresholds, empirical-scaling, power-optimization, qcw-operation, branching-physics, equations-and-bounds, open-questions]
+key_equations: [thermal-diffusion-time-constant, thermal-diffusivity, conductivity-from-electron-density, power-to-sustain-plasma, air-heating-efficiency, leader-energy-balance, conductance-relaxation]
+key_terms: [tau_thermal, alpha, d, QCW, burst_mode, streamer, leader, thermal_ionization, vibrational_relaxation, ionization_energy_cost, power_budget, non_equilibrium_plasma, aborted_leader, dark_period, thermal_ratcheting, associative_ionization, thermal_instability, electron_attachment_time, conductance_relaxation, leader_energy_balance, sword_spark, driven_leader, burst_ceiling, frequency_threshold]
+images: [thermal-diffusion-vs-diameter.png, qcw-vs-burst-timeline.png]
+examples: [spark-growth-timeline.md]
+open_questions:
+  - "What are the exact contributions of convection, radiation, and ionization memory to observed persistence?"
+  - "How does radial temperature profile evolve during and after a pulse?"
+  - "Can thermal persistence be modeled with a single effective time constant, or is a multi-exponential required?"
+  - "What is the quantitative role of nitrogen vibrational relaxation in ionization memory?"
+  - "How does altitude (reduced pressure) affect thermal diffusion and persistence times?"
+  - "How many aborted leader attempts typically precede stable inception in QCW mode?"
+---
+
+# Thermal Physics of Spark Channels
+
+The thermal behavior of spark channels determines how long a conductive path persists after energy injection ceases, whether a streamer can transition to a leader, and why QCW mode is fundamentally more efficient than burst mode for producing long sparks. Thermal physics bridges the gap between the instantaneous electrical properties described by [[circuit-topology]] and the time-evolving behavior that distinguishes operating modes.
+
+## Pure Thermal Diffusion
+
+The simplest model of channel cooling is radial heat diffusion from a hot cylinder into ambient air. The characteristic time constant for this process is:
+
+```
+tau_thermal = d^2 / (4 * alpha_thermal)
+```
+
+Where:
+- `tau_thermal` is the thermal diffusion time constant [s]
+- `d` is the channel diameter [m]
+- `alpha_thermal` is the thermal diffusivity of air [m^2/s]
+
+The thermal diffusivity of air at standard conditions:
+
+```
+alpha_thermal = k / (rho_air * c_p) = 2 * 10^-5 m^2/s
+```
+
+Where:
+- `k` is the thermal conductivity of air [W/(m*K)]
+- `rho_air` is the density of air [kg/m^3]
+- `c_p` is the specific heat capacity at constant pressure [J/(kg*K)]
+
+### Diffusion Time Constants by Diameter
+
+The quadratic dependence on diameter produces enormous variation:
+
+| Channel Diameter | Type | tau_thermal |
+|-----------------|------|-------------|
+| 10 um | Very thin streamer | ~1.3 us |
+| 100 um | Typical streamer | 0.1-0.2 ms |
+| 500 um | Thick streamer / thin leader | 3 ms |
+| 1 mm | Thin leader | 12.5 ms |
+| 3 mm | Typical leader | 110 ms |
+| 5 mm | Thick leader | 300-600 ms |
+| 10 mm | Very thick leader / arc | 1.25 s |
+
+The image `thermal-diffusion-vs-diameter.png` plots this relationship, showing the dramatic range from microseconds for thin streamers to seconds for thick leaders.
+
+### Key Insight: Diameter Squared
+
+The d^2 dependence is critically important. A channel that is 10x thicker has a thermal time constant that is 100x longer. This creates a powerful positive feedback loop: thicker channels (leaders) persist longer, allowing more energy injection, which further heats and expands the channel, increasing persistence even more. This runaway process is central to the [[streamers-and-leaders]] transition.
+
+## Beyond Pure Diffusion: Observed Persistence
+
+Actual spark channel persistence is significantly longer than predicted by pure thermal diffusion. Three mechanisms contribute to this extended lifetime:
+
+### 1. Buoyancy and Convection
+
+Hot gas in the channel is less dense than surrounding air. Buoyancy forces create an upward convection column that:
+- Maintains a coherent hot gas structure above the initial channel position
+- Continuously replaces cooled gas at the channel edges with hot gas from the core
+- Creates a self-sustaining thermal plume that persists well after the electrical discharge ends
+- Effective for thick channels (leaders) where buoyancy forces exceed viscous drag
+
+For vertical or upward-angled sparks, convection can maintain a hot column for seconds. For horizontal sparks, the column rises and eventually disconnects, reducing persistence.
+
+### 2. Ionization Memory
+
+Even after the gas temperature drops below the thermal ionization threshold (~5000 K), significant free electron density persists because:
+
+- **Recombination is slow**: Electron-ion recombination in air at moderate densities has time constants of milliseconds to tens of milliseconds
+- **Metastable states**: Nitrogen molecules excited to metastable electronic states (lifetimes ~ms) can participate in Penning ionization
+- **Vibrational relaxation**: Nitrogen vibrational modes store energy for milliseconds, slowly releasing it to sustain partial ionization
+- **Electron attachment/detachment**: Electrons attach to O2 to form O2^- (fast), but thermal detachment returns them when temperature is still elevated (slow)
+
+The net result: a partially ionized channel with moderate conductivity persists longer than the thermal profile alone would suggest. This is especially important in the temperature range 2000-4000 K where thermal ionization is negligible but residual ionization from previous heating still exists.
+
+#### Quantitative Data: Vibrational Relaxation and Recombination
+
+The qualitative mechanisms above now have quantitative timescales from the gas discharge literature:
+
+**Nitrogen vibrational relaxation time at 1 atm: >100 us** [Becker et al. 2005, Ch 5, p. 231]
+
+This is much longer than the electron-ion recombination time (~50 us at n_e ~ 10^13 cm^-3; see [[streamers-and-leaders]]) and comparable to the thermal diffusion time for thin streamers (~100-200 us for d ~ 100 um). The vibrational energy reservoir in N2 acts as a "battery" that slowly releases energy back into the electron population through superelastic collisions, maintaining a higher effective electron temperature (and hence lower attachment rate) than the translational gas temperature alone would suggest.
+
+**Recombination rates for major atmospheric ions: ~2 * 10^-7 cm^3/s at 300 K** [Becker et al. 2005, Ch 4, p. 174]
+
+For a streamer with n_e ~ 10^13 cm^-3, the recombination time constant is tau_recomb ~ 1/(2e-7 * 1e13) ~ 50 us. This confirms that recombination is indeed "slow" relative to the attachment time (16 ns) but "fast" relative to observed persistence (1-5 ms). The gap between the recombination time (~50 us) and the observed persistence (~1-5 ms) is filled primarily by the vibrational relaxation mechanism and by metastable states.
+
+This partially answers the open question "What is the quantitative role of nitrogen vibrational relaxation in ionization memory?": vibrational relaxation sustains partial ionization for at least 100 us beyond the cessation of direct energy input, which is comparable to the QCW inter-cycle gap at typical repetition rates.
+
+### 3. Broadened Effective Channel Diameter
+
+During the discharge, the channel heats and expands. The hot region is broader than the initial conducting core:
+- During active discharge: conducting core may be 1 mm, but heated region extends to 3-5 mm
+- After discharge: the broader heated region defines the effective cooling diameter
+- This increases the effective tau_thermal by a factor of (d_effective/d_core)^2, which can be 10-25x
+
+## Effective Persistence Times
+
+Combining all three mechanisms, observed persistence times are:
+
+### Thin Streamers: ~1-5 ms
+
+- Pure thermal diffusion: 0.1-0.2 ms (for d ~ 100 um)
+- Ionization memory extends to ~1-3 ms
+- Minimal buoyancy effect (too thin)
+- Persistence dominated by ionization/metastable memory
+- Significance: this is comparable to QCW inter-cycle gaps at 200-1000 Hz repetition rates
+
+### Thick Leaders: Seconds
+
+- Pure thermal diffusion: 300-600 ms (for d ~ 5 mm)
+- Buoyancy/convection extends to multiple seconds
+- Ionization memory further extends conductivity window
+- Broadened diameter adds another factor of several
+- Significance: once a leader forms, it can persist through multiple QCW ramp cycles or between closely spaced bursts
+
+## Temperature Ranges by Channel Type
+
+The temperature of the conducting channel determines its electrical properties and the dominant ionization mechanism:
+
+| Channel Type | Temperature Range | Ionization Mechanism | Plasma Conductivity |
+|-------------|-------------------|---------------------|-------------------|
+| Cold streamer | 300-1000 K | Photoionization (external UV) | sigma ~ 0.01 S/m |
+| Warm streamer | 1000-3000 K | Residual + impact ionization | sigma ~ 0.1-1 S/m |
+| Transition | 3000-5000 K | Mixed thermal/residual | sigma ~ 1-10 S/m |
+| Leader | 5000-20000 K | Thermal (Saha equation) | sigma ~ 10-100 S/m |
+| Arc | >10000 K | Full thermal equilibrium | sigma ~ 100-10000 S/m |
+
+### Corresponding Resistivities
+
+- Hot leader plasma: rho ~ 1-10 ohm*m
+- Warm streamer plasma: rho ~ 10-100 ohm*m
+- Cold streamer: rho ~ 100-1000 ohm*m
+
+These resistivity ranges connect directly to the resistance bounds used in the [[lumped-model]] and [[distributed-model]]:
+```
+R_segment = rho * L_segment / A_cross_section
+         = rho * L_segment / (pi * (d/2)^2)
+```
+
+For a 10 cm segment of 1 mm diameter leader at rho = 5 ohm*m:
+```
+R = 5 * 0.1 / (pi * (0.5e-3)^2) = 637 kohm
+```
+
+This is within the expected range (see [[equations-and-bounds]]).
+
+### Critical Temperature for Leader Inception
+
+The temperature tables above show the transition zone at 3000-5000 K. However, the **minimum gas temperature** for stable leader inception is a nuanced question. Liu (2017) demonstrates through detailed kinetic modeling that:
+
+**The gas temperature must significantly exceed 2000 K for stable leader inception, not merely reach it.**
+
+The reason is gas dynamics during the transition process:
+
+1. **Streamer heating raises temperature** to ~2000-3000 K in the stem (the short channel connecting the streamer base to the electrode)
+2. **Heated gas expands**, causing pressure-driven outflow that reduces density
+3. **Convection losses during expansion** can drop the temperature back below the critical threshold
+4. **If temperature falls back below ~1500 K** during expansion, the stem cools to ambient and the leader attempt **aborts**
+
+This means the traditional criterion of "T > 2000 K" is necessary but not sufficient. The gas must be heated to a temperature high enough that even after expansion and convection losses, the resulting channel remains above the critical ionization threshold. In practice, this requires initial heating to significantly above 2000 K (perhaps 3000-4000 K).
+
+[Liu 2017, Ch 3, "Streamer-to-leader transition"]
+
+#### Three-Tier Temperature Threshold
+
+Bazelyan & Raizer (2000) provide quantitative clarity on what happens at each temperature stage, resolving the apparent contradiction between Liu's "2000 K onset" and the "5000 K for leader" figure in the Temperature Ranges table above:
+
+| Temperature | What happens | Channel status |
+|-------------|-------------|----------------|
+| >2000 K | Thermal ionization begins; V-T relaxation accelerates; eta_T -> 1.0 | Fragile — expansion/convection can abort |
+| >4000 K | Associative ionization (N + O -> NO+ + e) provides field-free electron source; n_e ~ 7*10^12 cm^-3 at equilibrium | Robust — survives without applied field |
+| >5000 K | Electron attachment to O2 virtually nonexistent; complex ion decay + associative ionization fully compensate recombination | Fully self-sustaining — channel persists indefinitely |
+
+[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), pp. 703, 715-716]
+
+The 4000 K threshold is particularly significant: it marks where the channel gains a field-independent ionization source. Below 4000 K, ionization depends on the applied field (which may be intermittent at TC frequencies). Above 4000 K, the channel generates its own electrons through N+O collisions regardless of external conditions.
+
+**For TC sparks**, the practical criterion is: the stem/channel must reach **4000-5000 K** to survive as a leader. The 2000 K onset (Liu) is where thermal runaway *begins*; 5000 K (Bazelyan) is where the channel becomes truly persistent. The heating efficiency bottleneck (eta_T ~ 10% below 1000 K) is what makes crossing this range so slow.
+
+#### Thermal Instability Contraction Time
+
+The physical mechanism converting a streamer into a leader channel is ionization-overheating (thermal) instability — current from many streamers contracts into a thin filament:
+
+**Contraction time: tau_ins ~ 1 us** [Bazelyan & Raizer 2000, p. 704]
+
+This is derived from: tau_ins ~ l/v_s, where l ~ v_s/nu_a ~ 1 cm (conducting streamer length), v_s ~ 10^7 cm/s (streamer velocity near leader tip), and nu_a ~ 10^7 s^-1 (electron attachment frequency).
+
+**Critical for TC frequencies:** At 200 kHz (period = 5 us), the contraction instability can build up within a single half-cycle. At 50 kHz (period = 20 us), it completes well within one cycle. This means the thermal instability is **not** the bottleneck for leader formation at TC frequencies — the bottleneck is accumulating enough total energy to heat gas from 300 K to 5000 K, which takes ~1-5 ms even at MW/m power densities (due to the eta_T ~ 10% efficiency).
+
+**Connection to aborted leaders:** Before stable leader inception, Schlieren photography shows a sequence of:
+
+1. **Streamer burst** propagates from electrode
+2. **Dark period** (~1-5 ms duration) where space charge from the streamer shields the electrode field
+3. **Recovery** as ions drift and field rebuilds
+4. **Next streamer burst** (possibly stronger if residual heating persists)
+5. Cycle repeats until one burst deposits enough energy for successful leader inception
+
+Multiple "aborted leaders" (streamer bursts that heat the stem to near-critical temperature but fail to sustain it through expansion) typically precede the first stable leader. Each aborted attempt pre-heats the gas slightly, making the next attempt more likely to succeed — a form of thermal ratcheting.
+
+[Liu 2017, Ch 2-3; Les Renardieres Group 1977, 1981]
+
+**Implication for Tesla coils:** In QCW mode, the initial 1-3 ms of the ramp may produce several aborted leader attempts before the first stable leader forms at the base. This is consistent with the observation that the first few milliseconds of QCW growth are inefficient (high epsilon) before the leader "catches" and efficiency improves.
+
+### Why the Transition Takes So Long: Air Heating Efficiency
+
+A puzzle in the streamer-to-leader transition is why it takes milliseconds despite the enormous power densities in thin streamer channels (up to MW/m — see the Step 2 calculation in [[streamers-and-leaders]]). The answer is that **most of the electrical energy does not heat the gas**:
+
+```
+eta_T = 0.1 + 0.9 * [tanh(T/T_amb - 4) + 1] / 2
+```
+
+[da Silva et al. 2019, after Flitti & Pancheshnyi 2009]
+
+| Gas Temperature | eta_T | Meaning |
+|----------------|-------|---------|
+| 300 K (ambient) | ~0.10 | Only 10% heats gas; 90% goes to N2 vibrational modes |
+| 600 K | ~0.10 | Still mostly vibrational excitation |
+| 1200 K | ~0.55 | Transition: V-T relaxation accelerating |
+| 2000 K | ~1.0 | Full thermalization: all power heats gas |
+
+At ambient temperature, **90% of the electrical energy deposited in a streamer channel goes into exciting N2 vibrational modes** rather than raising the translational (gas) temperature. These vibrational modes relax slowly (>100 us at 1 atm, see Section "Ionization Memory" above), so the energy is temporarily "trapped" in internal degrees of freedom.
+
+**Physical consequence:** The effective heating power in a cold streamer is only ~10% of I^2*R. The 64 MW/m calculated for a 100 um streamer carrying 100 mA (see [[streamers-and-leaders]]) produces only ~6.4 MW/m of actual gas heating initially. As the gas warms past ~1000 K, vibrational relaxation accelerates, eta_T rises toward 1.0, and the heating becomes self-reinforcing — this is the thermal runaway that triggers leader formation.
+
+**Connection to thermal ratcheting:** During aborted leader attempts, each streamer burst deposits energy at only 10% efficiency into gas heating. But the vibrational energy reservoir (90% of the input) slowly thermalizes over ~100 us, providing residual heating that persists into the dark period. Successive bursts benefit from this accumulated vibrational energy, making each attempt more likely to succeed.
+
+## QCW Mode: Exploiting Thermal Persistence
+
+QCW (Quasi-Continuous Wave) mode operates with long ramp times of 5-20 ms. This duration is carefully chosen relative to thermal time constants:
+
+### Why QCW Works
+
+1. **Ramp duration exceeds streamer persistence (~1-5 ms)**: The continuous ramp keeps feeding energy before streamers can cool and deionize. Unlike burst mode where channels cool between pulses, QCW never gives the channel time to die.
+
+2. **Continuous energy injection maintains E_tip**: The voltage ramp compensates for the [[capacitive-divider]] effect, keeping the tip field above the propagation threshold for a longer growth period.
+
+3. **Promotes streamer-to-leader transition**: Sustained current through the same channel causes Joule heating (I^2*R). Over several milliseconds, the channel temperature rises from ~1000 K (streamer) through 3000 K (transition) to 5000+ K (leader). See [[streamers-and-leaders]] for the detailed mechanism.
+
+4. **Leader channels enable further growth**: Once formed, leaders have low resistance, high conductivity, and long persistence. They act as efficient "wires" conducting energy to the tip where new streamers form and themselves transition to leaders.
+
+5. **Thermal accumulation reduces epsilon**: As described in [[energy-and-growth]], the accumulated thermal energy makes subsequent extension cheaper: epsilon(t) = epsilon_0 / (1 + alpha * integral(P dt)). QCW's long ramp allows significant accumulation.
+
+### QCW Timing Analysis
+
+Typical QCW ramp: 12 ms at 190 kHz
+
+- **0-2 ms**: Voltage builds toward inception. No spark yet.
+- **2-4 ms**: Streamers form and begin growing. High epsilon (~15 J/m). Fast propagation but energy-expensive.
+- **4-8 ms**: Sustained current heats channels. Transition zone. Temperature rises past 5000 K at base. Epsilon dropping (10 J/m and decreasing).
+- **8-12 ms**: Leader-dominated base, streamer tips. Low epsilon (5-8 J/m) for base extension. Overall growth slowing as [[capacitive-divider]] attenuates V_tip.
+
+The image `qcw-vs-burst-timeline.png` shows side-by-side comparison of power, length, and temperature evolution for QCW vs burst modes.
+
+## Burst Mode: Fighting Thermal Physics
+
+Burst mode operates with short pulses (50-500 us) separated by gaps that allow significant cooling:
+
+### Why Burst Mode Is Inefficient for Length
+
+1. **Channel cools between pulses**: At typical burst repetition rates (100-1000 Hz), the gap between pulses is 1-10 ms. Thin streamers (tau ~ 0.1-0.2 ms) are completely cold by the next pulse. Even nascent leaders cool significantly.
+
+2. **Must re-ionize repeatedly**: Each burst pulse must re-establish the conductive channel from scratch (or from residual ionization). This re-ionization energy is "wasted" from a length perspective -- it rebuilds what was already created.
+
+3. **High peak current but no thermal accumulation**: Burst mode delivers high instantaneous power, creating bright, thick channels. But the energy goes into heating and radiation rather than forward propagation because there is no persistent leader to channel it efficiently.
+
+4. **Voltage collapse limits length**: During a single burst, the spark extends until the [[capacitive-divider]] drops E_tip below threshold. Because the burst is short (< 1 ms), there is no time for leader formation to mitigate the voltage division. The spark is streamer-dominated throughout.
+
+5. **Net result: high epsilon**: All the inefficiencies compound. Burst mode epsilon of 30-100+ J/m means 3-20x more energy per meter of spark compared to QCW.
+
+### Burst Mode Advantages
+
+Despite being length-inefficient, burst mode has applications:
+- **Visual impact**: High peak current produces bright, thick, visually impressive sparks
+- **Audio modulation**: Short bursts enable musical Tesla coils
+- **Simpler control**: No voltage ramping required
+- **Lower average power**: Shorter duty cycle reduces thermal stress on components
+
+## Connection to Plasma Conductivity
+
+The temperature-dependent conductivity of the spark channel plasma connects thermal physics to the circuit models in [[lumped-model]] and [[distributed-model]]:
+
+```
+sigma(T) varies from ~0.01 S/m (cold) to ~100 S/m (hot arc)
+rho(T) = 1/sigma(T) varies from ~100 ohm*m (cold) to ~0.01 ohm*m (hot)
+```
+
+The resistance of a channel segment:
+```
+R = rho(T) * L_segment / (pi * (d/2)^2)
+```
+
+As temperature rises:
+- rho decreases (more conductive)
+- d increases (thermal expansion)
+- Both effects decrease R
+- R moves toward R_opt_power (the hungry streamer self-optimization, see [[power-optimization]])
+
+This temperature-resistance coupling is the physical mechanism behind the hungry streamer principle: the plasma adjusts its temperature (and hence conductivity and diameter) to maximize power extraction.
+
+### Conductivity from First Principles
+
+The plasma conductivity can be calculated directly from the electron density:
+
+```
+sigma = n_e * e^2 / (m_e * nu_e-air)
+
+where:
+  nu_e-air = N_air * sigma_collision * v_e
+  sigma_collision = 1.5 * 10^-15 cm^2    (electron-air collision cross section)
+  N_air ~ 2.5 * 10^19 cm^-3 at STP
+  v_e ~ 10^6 m/s (mean electron speed at ~1 eV)
+```
+
+[Becker et al. 2005, Ch 5, p. 229]
+
+**Example:** For n_e = 10^13 cm^-3 in air at STP:
+
+```
+nu_e-air = 2.5e19 * 1.5e-15 * 1e8 ~ 3.75 * 10^12 s^-1
+sigma = (1e13 * (1.6e-19)^2) / (9.1e-31 * 3.75e12) ~ 0.075 S/m
+```
+
+This is consistent with the "cold streamer" conductivity range (0.01-0.1 S/m) in the table above, providing an independent cross-check from first principles.
+
+### Leader Channel Energy Balance
+
+For a well-developed leader channel at atmospheric pressure, the channel state is quasi-stationary and determined primarily by current. The energy balance between Joule heating and heat conduction gives:
+
+```
+P_L = i * E ~ 4*pi * lambda_m * delta_T
+
+where:
+  P_L = power per unit length [W/m]
+  lambda_m = thermal conductivity at channel boundary temperature [W/(cm*K)]
+  delta_T = T_axis - T_boundary ~ 2*k*T / I_eff (small due to exponential sigma(T))
+```
+
+[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, pp. 87-88, Eq. 2.46-2.47]
+
+**Key result:** At i = 1 A, T = 5000 K: lambda_m = 0.02 W/(cm*K), giving P_L ~ 130 W/cm = **13 kW/m** and E ~ 130 V/cm. These are within a factor of 2 of experimental leader field measurements.
+
+The physical insight is that the channel temperature is only weakly dependent on power (T ~ P_L^(1/2)) because thermal conductivity rises rapidly with temperature. This means the channel self-regulates: large changes in current (and hence power) produce only modest changes in temperature, which is why the "current channel with a fixed boundary" model works so well.
+
+**E/N variation with temperature:** As the channel heats from cold, the reduced field E/N drops dramatically:
+
+| Gas Temperature | E/N | Dominant Ionization | Implication |
+|----------------|-----|---------------------|-------------|
+| 1000 K | 55 Td | O2 electron impact | High field needed |
+| 2500 K | ~15 Td | NO electron impact | Threshold drops |
+| 4000-4500 K | ~3 Td | Associative (N+O->NO+) | Very low field |
+| 6000 K | 1.5 Td | Thermal equilibrium | Near-zero external field |
+
+[Bazelyan & Raizer 2000, Ch 2, p. 86, after calculations in reference 34]
+
+This confirms the three-tier temperature threshold: below 2500 K, high fields are needed for ionization; above 4000 K, the channel maintains itself with minimal field; by 6000 K, external field requirements are negligible.
+
+**Connection to TC sparks:** A TC leader at 1-3 A carrying current through a 1 m channel requires P_L ~ 13-40 kW/m. At 5 kW total spark power (typical mid-range DRSSTC), only about 0.1-0.4 m of channel can be maintained at full leader conditions simultaneously. This is consistent with the observation that TC sparks have a short leader base transitioning to streamer tips.
+
+**Thermal conductivity of air at 5000 K:** lambda_m = 0.02 W/(cm*K) = 2 W/(m*K). This is ~80x higher than at ambient (0.025 W/(m*K)), which is why the leader channel self-regulates: the strong temperature dependence of conductivity acts as negative feedback — higher temperature increases heat losses, stabilizing T_axis.
+
+### Maximum Heatable Channel Radius
+
+The energy stored in the electrostatic field of the tip sets a hard upper limit on the channel radius that can be heated to leader temperatures:
+
+```
+pi * r_0max^2 * rho_0 * h(T) = pi * epsilon_0 * U^2 / 2
+
+r_0max = U * sqrt(epsilon_0 / (2 * rho_0 * h(T)))
+```
+
+[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, p. 67, Eq. 2.34]
+
+where rho_0 = 1.2 * 10^-3 g/cm^3 is cold air density, h(T) is specific enthalpy:
+- h(5000 K) = 12 kJ/g
+- h(10000 K) = 48 kJ/g (roughly h ~ T^2)
+
+| Tip Voltage | r_0max (cold air) | After expansion to 5000 K | Channel area |
+|-------------|-------------------|--------------------------|-------------|
+| 100 kV | 5.4 um | 26 um | minuscule |
+| 500 kV | 27 um | 130 um | thin streamer |
+| 1 MV | 54 um | 260 um | thick streamer |
+
+**Physical meaning:** The channel that the tip's energy can heat to 5000 K is extremely thin — even at 1 MV, only ~0.05 mm cold radius. This is the fundamental reason leaders are thin: the available energy per meter constrains the heatable volume. The channel can only thicken later through sustained current from the circuit (not from tip charge alone).
+
+After thermal expansion to 5000 K (density drops ~5x), the channel expands by ~5x in cross-section, giving the post-expansion radii in the table. These are consistent with measured leader radii of ~0.1-0.3 mm.
+
+**Minimum leader radius from diffusion:** The ambipolar diffusion coefficient D_a ~ 4 cm^2/s sets a floor on channel filament size: r_0min ~ 30 um. Perturbations smaller than this are smoothed by diffusion before the contraction instability can grow. The probable pre-expansion leader radius is ~100 um. [Bazelyan & Raizer 2000, Ch 2, pp. 71-72]
+
+### Conductance Relaxation and Thermal Hysteresis
+
+The channel conductance does not respond instantaneously to current changes. The relaxation model from return stroke physics applies to any spark channel:
+
+```
+dG/dt = [G_st(i) - G(t)] / tau_g
+
+tau_g = 40 us   (current rising, channel heating)
+tau_g = 200 us  (current falling, channel cooling)
+```
+
+[Bazelyan & Raizer 2000, Ch 4, pp. 194-195]
+
+See [[equations-and-bounds]] Section 14.19 for the full model.
+
+**Thermal hysteresis for TC sparks:** The 5:1 asymmetry between heating (40 us) and cooling (200 us) time constants creates a ratcheting effect over many RF cycles:
+
+1. During the high-current half-cycle: conductance increases toward G_st(i_peak) with tau_g = 40 us
+2. During the low-current half-cycle: conductance decreases toward G_st(0) = 0 with tau_g = 200 us
+3. Net effect: conductance ratchets upward over ~10-50 RF cycles (total time ~50-250 us at 200 kHz)
+
+This is the microsecond-timescale mechanism underlying the millisecond-timescale thermal ratcheting described in the "Aborted Leaders" section above. The asymmetric tau_g provides the per-cycle bias that accumulates over many cycles to drive the streamer-to-leader transition.
+
+## Community-Validated QCW Thermal Physics
+
+The thermal physics framework above makes specific predictions about how QCW spark behavior should depend on timing, frequency, and power delivery mode. A comprehensive survey of community builder data [Phase 6 QCW community survey, 2026-02-10] provides strong empirical validation of these predictions and reveals several quantitative relationships not previously documented in the framework.
+
+### Frequency Threshold for Sword-Like Sparks: 300-600 kHz
+
+Six or more independent builders have converged on a frequency range for producing straight "sword" sparks:
+
+| Observer | Observation | Source |
+|----------|-------------|--------|
+| Mads Barnkob | "Sword characteristic shows above 400 kHz" | HVF |
+| LabCoatz (Zach Armstrong) | Below 300 kHz: "chaotic and less straight"; above 600 kHz: "more curvy" | Hackaday |
+| Kaizer DRSSTC IV | QCW at ~100 kHz: "swirling" sparks, NOT straight | HVF |
+| Fat Coil builder | "Above 350 kHz, plasma exhibits growth in straight segments" | TCML |
+| Loneoceans SSTC3 | Straight sparks at 380-420 kHz | loneoceans.com |
+| Multiple QCW builders | All successful sword-spark builds operate 300-500 kHz | Build survey |
+
+**Thermal physics explanation:** The RF half-period at 400 kHz is **1.25 us**. The thermal diffusion time for a 100 um streamer is **~125 us** — 100x longer than the RF period. The channel experiences effectively continuous heating with negligible cooling between RF half-cycles. The conductance relaxation time constant (tau_g = 40 us for heating) spans ~16 RF cycles at 400 kHz, ensuring smooth, monotonic conductance increase.
+
+At 50-100 kHz (half-period 5-10 us), thinner streamers (10-50 um, tau ~ 1-30 us) experience significant cooling between cycles. The preferred conductive path diffuses and branches — the channel cannot maintain a single straight track. At >600 kHz, the observation of "curvy" sparks may relate to different physics (skin effect, displacement current dominance, or switching artifacts at extreme frequencies).
+
+**Quantitative prediction:** At frequency f, the Joule heating rate scales as ~f (more half-cycles per unit time at the same peak current). A channel at 400 kHz receives ~4x more thermal energy per millisecond than at 100 kHz, for the same peak current. This accelerates the temperature ratchet through the critical 2000-5000 K zone.
+
+### Steve Ward 80 us Burst Ceiling
+
+Steve Ward's DRSSTC-0.5 measurements provide a clean test of the burst-mode thermal limit:
+
+| ON Time | Spark Length | Input Power |
+|---------|-------------|-------------|
+| ~70 us | 10-18 inches | 33-180 W |
+| >80 us | No additional length | Diminishing returns |
+
+**Key observation:** "Gained almost no spark length after about 80 us of ON period." [Steve Ward, stevehv.4hv.org/DRSSTC.5.htm]
+
+This directly measures the burst-mode streamer growth saturation. The 80 us ceiling is strikingly consistent with the thermal time constant for 100 um streamers: tau_thermal ~ d^2/(4*alpha) = (100e-6)^2 / (4*2e-5) ~ 125 us. After approximately one thermal time constant, channels are cooling as fast as they are being heated — additional energy goes into re-heating decaying channels rather than new forward growth. This is the fundamental wall that QCW overcomes by sustaining drive beyond this timescale.
+
+**Connection to framework:** Steve Conner independently confirmed this finding: short bursts of high peak power grow sparks more efficiently than long bursts of low peak power (100 us burst outperforms 150 us at the same total energy). This is consistent with the power optimization framework — higher peak power pushes the initial streamer further before the 80 us thermal ceiling is hit.
+
+### Three Ramp Regimes
+
+Loneoceans documented three distinct QCW ramp outcomes through controlled variation of ramp duration on his QCW v1.5 build:
+
+| Ramp Duration | Visual Result | Framework Interpretation |
+|---------------|--------------|------------------------|
+| Too short (<5 ms) | "Gnarly, segmented sparks" | Insufficient time for leader transition; spark operates mostly as streamer |
+| Optimal (~10-20 ms) | Straight sword sparks | Leader forms within first few ms; grows continuously during remainder |
+| Too long (>25 ms) | "Really hot and fat but bushy" without extra length | Leader reaches voltage-limited L_max; excess energy causes branching |
+
+**The "too long" regime is revealing:** Once the leader reaches its maximum length (set by the capacitive divider — see [[capacitive-divider]]), additional energy cannot extend it further. The leader channel becomes very hot and thick (more C_sh → more voltage division → further from E_propagation threshold). The excess power must dissipate somewhere, and lateral breakouts from the superheated leader trunk become the path of least resistance. This naturally produces the "fat and bushy" appearance.
+
+**The "too short" regime confirms the 0.5-2 ms transition time:** Ramps shorter than ~5 ms do not allow the full streamer-to-leader transition. The "segmented" appearance suggests the spark advances as disconnected leader segments that don't merge into a continuous trunk before the ramp ends — consistent with the thermal ratcheting model requiring multiple dark period cycles.
+
+### Pulse-Skip Modulation Does Not Produce Full Sword Sparks
+
+Multiple experimenters (Steve Ward, Steve Conner, others on HVF circa 2011) attempted pulse-skip approaches to achieve QCW-like behavior and could not produce full sword sparks.
+
+Steve Ward's explanation: Smoothing ripples from missing pulses would require the coil to store excessive energy between cycles. Sword sparks need "relatively smooth/continuous modulation of the spark power with little ripple."
+
+**Implementation detail:** In a DRSSTC, pulse-skip is a bridge current-limiting method, not a power-off state. During skip cycles, the H-bridge shorts the primary tank (via GDT inversion or leg inhibit) while IGBTs continue switching synchronized to feedback. Primary current does not drop to zero — it decays gradually through the resonant system's loaded Q. Phase coherence is maintained throughout. The resulting current envelope is a sawtooth bounded by the OCD threshold, delivering approximately constant average power rather than a smoothly ramping profile.
+
+**Thermal physics connection:** The sawtooth power envelope has per-cycle jitter from the on-off-on switching pattern. True QCW delivers a smooth quadratic power envelope (from a linear voltage ramp: P ~ V^2) — the natural profile for growing a spark against increasing capacitive loading. The jitter from pulse-skip disrupts the clean, monotonic conductance buildup in a single dominant channel. This is a continuum, not a binary threshold: Bresenham-algorithm pulse-density modulation (distributing skip events optimally for a linear ramp) produces sparks that are noticeably more sword-like but still branch — intermediate between coarse pulse-skip and true analog QCW. See [[qcw-operation]] Section 2.3 for full details including implementation methods and the distinction from staccato operation.
+
+### QCW Timing Summary vs Framework Predictions
+
+| Quantity | Framework Value | Community Measurement | Agreement |
+|----------|----------------|----------------------|-----------|
+| Streamer-to-leader transition time | 0.5-2 ms | Ramp must be >5 ms for swords | Consistent (first few ms spent on aborted leaders) |
+| Burst ceiling (thermal saturation) | tau_thermal ~ 125 us (100 um) | Steve Ward: 80 us ceiling | Good (within factor 1.5) |
+| Optimal QCW ramp | >5x tau_thermal, <L_max/v_growth | 10-20 ms | Consistent |
+| Frequency for continuous heating | f >> 1/(2*tau_thermal) ~ 4 kHz | 300-600 kHz (well above minimum) | Consistent |
+| Conductance ratcheting period | ~10-50 RF cycles at 200 kHz | Sword sparks at >300 kHz, not at <100 kHz | Consistent |
+| QCW growth rate | Not previously predicted | ~170 m/s (derived from tau_g) | New data; derivable from framework |
+
+---
+
+## Power Budget for Sustaining Plasma
+
+The conductivity-temperature relationship above can be connected to the power required to sustain the plasma at each density level. These power budgets place fundamental constraints on the minimum power per unit length needed to maintain a spark channel.
+
+### Average Ionization Energy
+
+The average energy required to produce one electron-ion pair in air is:
+
+```
+E_ionization_avg ~ 14 eV
+```
+
+[Becker et al. 2005, Ch 7, p. 440]
+
+This is higher than the first ionization potential of either N2 (15.6 eV) or O2 (12.1 eV) because some electron energy goes into excitation, dissociation, and vibrational/rotational modes rather than ionization. The effective cost per ionization event includes all these "waste" channels.
+
+### Power to Sustain a Given Electron Density
+
+The power required to maintain a steady-state electron density depends on the loss mechanism (attachment or recombination) and the ionization energy cost:
+
+| Condition | T_gas | Power to sustain n_e = 10^13 | Dominant loss |
+|-----------|-------|------------------------------|---------------|
+| Cold air | ~300 K | 1.4 kW/cm^3 | Three-body attachment |
+| Hot air | ~2000 K | 14 kW/cm^3 | Thermal dissociation + radiation |
+
+[Becker et al. 2005, Ch 7, p. 440; Ch 5, p. 230]
+
+The factor of 10 difference reflects the different loss regimes: in cold air, electron attachment to O2 is the primary loss; in hot air, thermal processes (dissociation, recombination, radiation) dominate.
+
+### Linear Power Density for Spark Channels
+
+Converting volumetric power to linear power (per unit length of channel):
+
+For a **streamer channel** (d = 100 um, A = 7.85 * 10^-6 cm^2):
+
+```
+P_linear = 1.4 kW/cm^3 * 7.85e-6 cm^2 = 0.011 W/cm = 1.1 W/m
+```
+
+For a **leader channel** (d = 3 mm, A = 0.071 cm^2):
+
+```
+P_linear = 14 kW/cm^3 * 0.071 cm^2 = 1.0 kW/m
+```
+
+### Independent Check on Epsilon
+
+These linear power densities provide an independent check on the epsilon values from [[energy-and-growth]]. If the channel must sustain ~1 W/m (streamer) to ~1000 W/m (leader) just to maintain ionization, and the channel grows at ~10^6 m/s (streamer) to ~10^3 m/s (leader), then the energy per meter of forward propagation for maintenance alone is:
+
+```
+epsilon_maintenance ~ P_linear / v_propagation
+
+Streamer: ~1 W/m / 10^6 m/s ~ 10^-6 J/m
+Leader:   ~1000 W/m / 10^3 m/s ~ 1 J/m
+```
+
+These are lower bounds (maintenance only, not initial ionization, heating, expansion, radiation, or branching). The actual observed epsilon values (5-100 J/m) are 5-100x higher than the leader maintenance minimum, which is consistent: most energy goes into initial channel heating and losses, not steady-state maintenance.
+
+### Equilibrium vs. Non-Equilibrium Electron Density
+
+| Condition | T_gas (K) | n_e (cm^-3) | Regime |
+|-----------|----------|-------------|--------|
+| Equilibrium air at 2900 K | 2900 | 4 * 10^10 | Very low ionization |
+| Non-equilibrium DC discharge | 700-2000 | > 10^12 | Discharge-sustained |
+| Streamer body | 300-1000 | 10^11 - 10^13 | Non-equilibrium |
+| Fully developed spark | ~5000+ | ~10^16 | Approaching LTE |
+
+[Becker et al. 2005, Ch 5, p. 229; Ch 2, pp. 23, 38]
+
+This table illustrates a critical point: at temperatures below ~3000 K, thermal equilibrium produces negligible ionization (n_e ~ 10^10). The streamer electron densities (10^11-10^13) are sustained entirely by the applied electric field, not by temperature. Only above ~5000 K does thermal ionization become significant, marking the transition to leader behavior and the regime where the hungry streamer principle operates via temperature-conductivity feedback.
+
+## Practical Design Implications
+
+### For QCW Coil Designers
+
+- **Ramp time should exceed 5 ms**: This ensures enough time for streamer-to-leader transition at the base of the spark
+- **Longer ramps (10-20 ms) are more efficient**: But require more total energy and may exceed component thermal limits
+- **Frequency tracking is essential**: During the long QCW ramp, the loaded pole shifts significantly as C_sh grows. Driving at the wrong frequency can reduce power delivery by 3-5x (see [[coupled-resonance]])
+- **Match at 50-70% of target length**: Because impedance changes dramatically during growth, matching at the midpoint provides best average efficiency
+
+### For Burst Mode Coil Designers
+
+- **Repetition rate affects effective epsilon**: Faster repetition (> 500 Hz) allows some thermal memory between bursts, reducing effective epsilon
+- **Single-shot length follows sqrt(E)**: For isolated bursts with no thermal carryover, Freau's scaling applies (see [[empirical-scaling]])
+- **Peak current determines brightness, not length**: Increasing peak current makes brighter sparks but hits the [[capacitive-divider]] voltage limit at the same length
+
+## Key Relationships
+
+- Derives from: First principles of heat transfer (Fourier's law applied to cylindrical geometry)
+- Interacts with: [[streamers-and-leaders]] (thermal physics governs the transition between these regimes)
+- Interacts with: [[energy-and-growth]] (thermal accumulation modifies epsilon over time)
+- Enables: [[power-optimization]] (thermal self-adjustment is the mechanism for hungry streamer optimization)
+- Constrains: [[distributed-model]] (resistance bounds depend on temperature/conductivity ranges)
+- Explains: [[empirical-scaling]] (different scaling laws for QCW vs burst arise from thermal persistence differences)
+- Connects to: [[field-thresholds]] (temperature affects local gas properties and thus field requirements)

context/thevenin-method.md

@@ -0,0 +1,371 @@
+---
+id: thevenin-method
+title: "Thevenin Equivalent Extraction and Impedance Measurement"
+status: established
+source_sections: "spark-physics.txt: Part 3 (lines 128-524), Part 11 (lines 753-803)"
+related_topics: [circuit-topology, power-optimization, coupled-resonance, lumped-model, distributed-model, femm-workflow, equations-and-bounds]
+key_equations:
+  - "Thevenin impedance Z_th"
+  - "Thevenin voltage V_th"
+  - "Power to load P_load"
+  - "Theoretical maximum power P_max"
+  - "Ringdown Q and conductance extraction"
+  - "Equivalent capacitance from frequency shift"
+key_terms:
+  - "Thevenin equivalent"
+  - "output impedance"
+  - "open-circuit voltage"
+  - "conjugate match"
+  - "measurement port"
+  - "ringdown method"
+  - "loaded Q"
+  - "Rogowski coil"
+  - "E-field probe"
+  - "VNA"
+images:
+  - thevenin-measurement-setup.png
+  - impedance-matching-concept.png
+examples:
+  - thevenin-extraction.md
+open_questions:
+  - "How much does primary coupling coefficient uncertainty affect Z_th extraction accuracy?"
+  - "Can V_th be measured in situ during spark operation using E-field probes, or only in simulation?"
+  - "What is the best frequency resolution for Z_th(omega) sweeps to capture pole behavior?"
+  - "How does the Thevenin approach extend to time-varying loads (transient spark growth)?"
+---
+
+# Thevenin Equivalent Extraction and Impedance Measurement
+
+This document describes the correct method for characterizing a Tesla coil as a source (Thevenin equivalent) and evaluating its power delivery to arbitrary spark loads. The central message is that naive impedance measurements (V_top/I_base) are fundamentally flawed, and the Thevenin port method provides a rigorous alternative. Three measurement approaches are presented: Thevenin extraction (recommended), direct power measurement, and ringdown analysis.
+
+## 1. Why V_top / I_base Is Wrong
+
+### 1.1 The Common Mistake
+
+A tempting approach to measuring spark impedance is to divide the topload voltage by the base current: Z_apparent = V_top / I_base. This is incorrect and produces misleading results.
+
+### 1.2 Physical Reason
+
+The base current I_base is the total current flowing into the bottom of the secondary winding. This current includes ALL displacement currents returning to ground from the secondary:
+
+1. **Distributed secondary capacitance to ground:** Every turn of the secondary coil has capacitance to the ground plane, strike ring, and nearby objects. These displacement currents flow through the base.
+
+2. **Strike ring coupling:** If a strike ring is present, capacitive coupling between the secondary and strike ring contributes additional current.
+
+3. **Primary-to-secondary capacitance:** The inter-winding capacitance between primary and secondary contributes displacement current.
+
+4. **Spark current:** The actual current flowing through the spark load (the quantity of interest) is only one component of I_base.
+
+Computing V_top/I_base therefore mixes the spark load impedance with all parasitic return paths. The result has no clear physical interpretation and cannot be used for impedance matching analysis.
+
+### 1.3 Quantitative Impact
+
+In a typical DRSSTC:
+- Total I_base might be 2 A (peak) at the operating frequency
+- Of this, perhaps 0.5-1.0 A is spark current
+- The remainder is parasitic displacement currents
+
+Using I_base overestimates the current through the spark by a factor of 2-4, which underestimates the spark impedance by the same factor. This leads to incorrect R_opt calculations and misleading efficiency estimates.
+
+## 2. The Correct Measurement Port
+
+### 2.1 Port Definition
+
+The correct measurement port for spark impedance is the **topload-to-ground** terminal pair, defined as:
+
+- **Positive terminal:** The topload surface where the spark physically connects
+- **Negative terminal (return):** The ground plane / earth / chassis
+
+All impedance, admittance, and power calculations for the spark reference this port.
+
+### 2.2 Why This Port
+
+The topload is the node where the spark load physically attaches to the Tesla coil circuit. The Thevenin theorem states that any linear circuit, viewed from a single port, can be replaced by a voltage source V_th in series with an impedance Z_th. By defining the port at the topload, we cleanly separate:
+
+- **The source:** Everything behind the topload (primary circuit, coupling, secondary winding, parasitic capacitances) is characterized by V_th and Z_th.
+- **The load:** The spark circuit (C_mut, R, C_sh as described in [[circuit-topology]]) connects at this port.
+
+## 3. Thevenin Equivalent Extraction (Recommended Method)
+
+### 3.1 Overview
+
+The Thevenin method characterizes the Tesla coil as a two-terminal source, then evaluates power delivery to any load by simple circuit calculation. This completely decouples coil characterization from load analysis.
+
+### 3.2 Step 1: Measure Z_th (Output Impedance, Drive Off)
+
+**Setup:**
+- Set the primary drive source to AC 0V (effectively short-circuit the voltage source in the primary tank). This is critical: the voltage source is replaced by a short circuit, NOT removed. All tank components (MMC capacitor, primary inductance, damping resistors) remain in the circuit.
+- Apply a 1V AC test source at the topload-to-ground port.
+- Measure the resulting current I_test (complex: magnitude and phase).
+
+**Calculation:**
+```
+Z_th = V_test / I_test = 1V / I_test = R_th + j*X_th
+```
+
+**Physical meaning:** Z_th is the impedance the spark "sees" looking back into the Tesla coil. It includes the reflected impedance of the entire primary tank circuit through the magnetic coupling, plus the secondary's own impedance (distributed capacitance, winding resistance, etc.).
+
+**Practical notes:**
+- In SPICE, this is straightforward: replace the primary voltage source with a short, add a 1V AC source at the topload node.
+- Z_th is complex and frequency-dependent. At the operating frequency, it is typically dominated by the reflected primary tank impedance.
+- R_th (real part) represents all losses in the coil plus the reflected primary resistance.
+- X_th (imaginary part) is typically capacitive near resonance.
+
+### 3.3 Step 2: Measure V_th (Open-Circuit Voltage, Drive On)
+
+**Setup:**
+- Remove the test source from Step 1.
+- Restore the primary drive source to its normal operating conditions (full voltage, operating frequency).
+- Remove the spark load (open-circuit the topload; no spark, no load impedance).
+- Measure V_th = V(topload), both magnitude and phase.
+
+**Calculation:**
+```
+V_th = V(topload)|_{open circuit, drive on}
+```
+
+**Physical meaning:** V_th is the voltage the Tesla coil would produce at the topload if no spark were present. It represents the "driving force" available for spark power.
+
+**Practical notes:**
+- In SPICE, simply run the normal coil simulation without any spark load attached.
+- V_th depends on drive conditions (bus voltage, pulse width, coupling) and is typically 50-500 kV peak for medium-to-large DRSSTCs.
+- V_th is the voltage that determines whether the inception field threshold (see [[field-thresholds]]) is met.
+
+### 3.4 Step 3: Calculate Power to Any Load
+
+Given Z_th and V_th, the power delivered to any candidate load impedance Z_load is:
+
+```
+P_load = 0.5 * |V_th|^2 * Re{Z_load} / |Z_th + Z_load|^2
+```
+
+This is the standard Thevenin power transfer formula with the peak-value convention (factor of 0.5).
+
+**For the spark circuit specifically:** Z_load is the impedance of the spark network (C_mut || R in series with C_sh), as derived in [[circuit-topology]]:
+
+```
+Z_load = 1/Y_spark = [G + j*(B_1 + B_2)] / [(G + j*B_1) * j*B_2]
+```
+
+**Theoretical maximum power (conjugate match sanity check):**
+
+If a perfect conjugate match were achievable (Z_load = Z_th*):
+
+```
+P_max_conjugate = 0.5 * |V_th|^2 / (4 * R_th)
+```
+
+This is an upper bound. The actual spark power will be less because:
+1. The spark topology constrains the achievable phase angle (see [[circuit-topology]]).
+2. Z_load cannot be freely chosen to equal Z_th* -- it is constrained by the (C_mut, R, C_sh) topology.
+
+### 3.5 Advantages of the Thevenin Method
+
+1. **One-time characterization:** Measure Z_th and V_th once for a given coil geometry and drive setup. Then evaluate any number of spark loads by plugging Z_load into the power formula.
+
+2. **No re-simulation:** Changing spark parameters (R, C_mut, C_sh, spark length) does not require re-simulating the coil. Just recalculate Z_load and use the power formula.
+
+3. **Clean separation:** "Coil behavior" (Z_th, determined by winding geometry, coupling, tank circuit) is separated from "drive conditions" (V_th, determined by bus voltage, pulse timing) and from "load behavior" (Z_load, determined by spark physics).
+
+4. **Design insight:** Z_th reveals the coil's output characteristics independent of any particular spark. A coil with lower R_th can deliver more power; a coil with different X_th may require different spark impedance for optimal matching.
+
+### 3.6 Enhancement: Frequency-Dependent Characterization
+
+For the most accurate analysis, measure Z_th(omega) and V_th(omega) over a frequency band of +/-10% around the nominal operating frequency.
+
+**Why:** When a spark loads the Tesla coil, the resonant frequency shifts (see [[coupled-resonance]]). The coil may not operate at its nominal frequency. Having Z_th and V_th as functions of frequency allows evaluating power delivery at the actual loaded frequency, not just the design frequency.
+
+**Procedure:**
+- Sweep the AC analysis frequency over the band [0.9*f_0, 1.1*f_0].
+- Record Z_th(f) and V_th(f) at each frequency point.
+- For a given spark load at a given frequency, use the appropriate Z_th(f) and V_th(f).
+
+## 4. Direct Power Measurement (Alternative Method)
+
+### 4.1 Approach
+
+Instead of extracting the Thevenin equivalent, directly measure power delivered to the spark in a full coupled simulation:
+
+1. Build the complete SPICE model: primary tank, magnetic coupling, secondary, topload, AND spark load (C_mut || R in series with C_sh).
+2. Drive the primary at the operating frequency and amplitude.
+3. Run AC analysis.
+4. Measure spark power: P = 0.5 * Re{V(topload) * conj(I(spark))}.
+5. Step R through a range and record P(R).
+6. Find the R that maximizes P.
+
+### 4.2 Critical Detail: Retune for Each R
+
+**This is the most commonly overlooked step.** When R changes, the loaded pole frequency shifts. If you measure P at a fixed frequency for each R, you are measuring the combined effect of impedance matching AND detuning. These two effects are conflated and the result is misleading.
+
+**Correct procedure:** For each R value:
+1. Sweep frequency to find the loaded pole (frequency of maximum |V_top|).
+2. Measure P at that loaded pole frequency.
+3. Record P(R) at the matched frequency.
+
+This gives the true power transfer capability as a function of R, independent of frequency tracking.
+
+### 4.3 Comparison with Thevenin Method
+
+| Aspect | Thevenin | Direct |
+|--------|----------|--------|
+| Number of simulations | 2 (Z_th + V_th) | Many (one per R value) |
+| Frequency tracking | Requires separate Z_th(omega) sweep | Naturally included if done correctly |
+| Physical insight | Separates source from load | Shows only total result |
+| Re-usability | Characterize once, evaluate many loads | Must re-simulate for each new scenario |
+| Accuracy | Exact (same circuit equations) | Exact (same circuit equations) |
+| Complexity | Lower (once setup is understood) | Higher (must retune for each R) |
+
+**Recommendation:** Use Thevenin for design and parameter sweeps. Use direct measurement for validation of specific operating points.
+
+## 5. Ringdown Method
+
+### 5.1 Principle
+
+An alternative experimental (not just simulation) technique. When a Tesla coil rings down after the drive is removed, the decay rate reveals the total system Q, from which the total conductance (and hence spark conductance) can be extracted.
+
+### 5.2 Parallel RLC Equivalent
+
+At the loaded resonance omega_L, the system near the topload looks like a parallel RLC:
+
+```
+Q_L = omega_L * C_eq * R_p = R_p / (omega_L * L)
+```
+
+where R_p is the equivalent parallel resistance (representing all losses including the spark), C_eq is the equivalent capacitance, and L is the equivalent inductance.
+
+**Solving for R_p:**
+```
+R_p = Q_L / (omega_L * C_eq)     [using Q = omega*C*R_p form]
+R_p = Q_L * omega_L * L          [using Q = R_p/(omega*L) form]
+```
+
+**Total conductance:**
+```
+G_total = 1/R_p = omega_L * C_eq / Q_L = 1 / (Q_L * omega_L * L)
+```
+
+### 5.3 Measurement Procedure
+
+1. **Unloaded measurement:** Measure the resonant frequency f_0 and quality factor Q_0 without spark. From geometry or separate measurement, determine C_0 (topload + secondary distributed capacitance).
+
+2. **Loaded measurement:** With spark present, measure the loaded resonant frequency f_L and loaded quality factor Q_L.
+
+3. **Equivalent capacitance:**
+```
+C_eq = C_0 * (f_0 / f_L)^2
+```
+This accounts for the frequency shift caused by the additional spark capacitance.
+
+4. **Capacitance change:**
+```
+delta_C = C_eq - C_0
+```
+
+5. **Total conductance (loaded):**
+```
+G_total = omega_L * C_eq / Q_L
+```
+
+6. **Unloaded conductance:**
+```
+G_0 = omega_0 * C_0 / Q_0
+```
+
+7. **Spark admittance:**
+```
+Y_spark ~ (G_total - G_0) + j * omega_L * delta_C
+```
+
+The real part gives the spark's conductance (1/R); the imaginary part gives the net reactive change, which should be consistent with C_mut and C_sh.
+
+### 5.4 Limitations of the Ringdown Method
+
+- **Sensitivity to primary coupling:** The primary tank circuit affects the ringdown behavior. If coupling is not well characterized, errors propagate into the extracted Q and hence into Y_spark.
+
+- **Transient vs. steady-state:** The ringdown captures the impedance at the moment the drive is removed. If the spark is evolving (growing, cooling), this is a snapshot, not the steady-state value.
+
+- **Mode identification:** The Tesla coil has two coupled modes. The ringdown may excite both, and careful analysis is needed to separate them.
+
+**The Thevenin port method is more robust** because it operates in the frequency domain and does not require separating coupled mode contributions.
+
+## 6. Direct Physical Measurement
+
+### 6.1 Voltage Measurement: E-Field Probe
+
+The topload voltage V_top can be measured using a calibrated E-field probe:
+- Capacitive divider probe placed near (but not touching) the topload
+- Must be calibrated for frequency response and geometry
+- Provides V_top(t) in the time domain; FFT gives V_top(omega)
+
+### 6.2 Current Measurement: Rogowski Coil or CT
+
+The spark return current (NOT I_base) can be measured using:
+- **Rogowski coil** around the spark ground return conductor
+- **Current transformer (CT)** on the ground return path
+- Must measure the current flowing through the spark circuit specifically, not the total secondary base current
+
+**Critical:** The current sensor must be placed to capture only the spark-associated current, not all displacement currents. This typically means placing it on a dedicated ground return wire from the spark target or strike object.
+
+### 6.3 VNA (Vector Network Analyzer)
+
+For low-level characterization without spark:
+- Capacitive pickup at topload
+- VNA drives through a coupling network
+- Measures Z_th(omega) across a frequency band
+- Cannot measure V_th directly (requires active drive)
+- Useful for validating SPICE models before spark testing
+
+### 6.4 Calculating Impedance from Measurements
+
+With V_top and I_spark measured:
+```
+Y_measured = I_spark / V_top
+Z_measured = V_top / I_spark
+```
+
+From Y_measured, extract R by fitting to the circuit model (see [[circuit-topology]]):
+- Known: omega, C_mut (from FEMM), C_sh (from FEMM or estimated from length)
+- Unknown: R
+- Solve: Y(R) = Y_measured for R
+
+## 7. Practical Workflow
+
+### 7.1 Recommended Sequence
+
+1. **Build SPICE model** of complete Tesla coil (primary tank, coupling, secondary, topload).
+2. **Extract Z_th** (Step 1: short drive, apply test source at topload).
+3. **Extract V_th** (Step 2: normal drive, open topload).
+4. **Compute power curves:** For a range of spark lengths (and corresponding C_mut, C_sh from [[femm-workflow]]), calculate P_load(R) for each length.
+5. **Identify R_opt_power** for each length (should match [[power-optimization]] formula).
+6. **Validate:** Check that P_max is consistent with known coil performance.
+7. **Frequency sweep (optional):** Repeat Steps 1-2 across +/-10% band for frequency tracking analysis.
+
+### 7.2 Common Pitfalls
+
+- **Forgetting to short the drive source** in Step 1 (leaving it open gives wrong Z_th).
+- **Removing tank components** in Step 1 (they must remain; they are part of the source impedance).
+- **Using I_base instead of I_spark** in direct measurements.
+- **Comparing R values at fixed frequency** without retuning (see [[coupled-resonance]]).
+- **Ignoring the 0.5 factor** in power (peak-value convention).
+
+## 8. Connection to Other Topics
+
+### Key Relationships
+
+- **Derives from:** Linear circuit theory (Thevenin's theorem) applied to the Tesla coil system
+- **Requires:** [[circuit-topology]] (defines the spark load Z_load that connects to the Thevenin port)
+- **Validates:** [[power-optimization]] (P_load(R) curve from Thevenin analysis should peak at R_opt_power)
+- **Interacts with:** [[coupled-resonance]] (frequency-dependent Z_th captures pole splitting and detuning)
+- **Feeds into:** [[lumped-model]] and [[distributed-model]] (Thevenin source drives the spark circuit model)
+- **Complements:** [[femm-workflow]] (FEMM provides C_mut, C_sh; Thevenin provides source characterization)
+
+### Summary of Key Results
+
+1. V_top/I_base is wrong because I_base includes parasitic displacement currents.
+2. The correct measurement port is topload-to-ground.
+3. Thevenin extraction: Z_th from drive-off test, V_th from drive-on open-circuit.
+4. P_load = 0.5 * |V_th|^2 * Re{Z_load} / |Z_th + Z_load|^2.
+5. P_max (conjugate match) = 0.5 * |V_th|^2 / (4*R_th) is an upper bound.
+6. Ringdown method extracts Y_spark from Q and frequency measurements but is sensitive to coupling.
+7. Direct measurement requires E-field probe (voltage) and Rogowski/CT on spark return (current).
+8. The Thevenin method is the most robust and reusable approach.

exercises/01-fundamentals/fund-ex-02a.yaml

@@ -1,51 +0,0 @@
-id: fund-ex-02a
-type: design
-difficulty: easy
-points: 10
-related_lesson: fund-02
-question: |
-  Draw the circuit for a spark with the following parameters:
-  - Spark length: L = 5 feet
-  - Mutual capacitance: C_mut = 12 pF (from FEMM)
-  - Plasma resistance: R = 50 kΩ
-
-  Label all component values including the shunt capacitance C_sh.
-
-hints:
-  - "Use the empirical rule: C_sh ≈ 2 pF/foot"
-  - "The topology is (R || C_mut) in series with C_sh"
-  - "Draw from topload terminal to ground reference"
-
-solution:
-  steps:
-    - "Calculate C_sh using empirical rule: C_sh = 2 pF/foot × 5 feet = 10 pF"
-    - "Draw topload at top as measurement terminal"
-    - "Draw C_mut in series from topload"
-    - "At node connecting C_mut, draw R and C_sh in parallel to ground"
-    - "Alternative: Show R || C_mut as parallel combination, then C_sh in series to ground"
-
-  answer: |
-    Circuit diagram:
-    Topload (V_top)
-        |
-    [C_mut = 12 pF]
-        |
-    +----------- Node_spark
-    |              |
-    [R = 50 kΩ]    [C_sh = 10 pF]
-    |              |
-    GND --------- GND
-
-  component_values:
-    C_mut: "12 pF"
-    C_sh: "10 pF"
-    R: "50 kΩ"
-
-explanation: |
-  The spark circuit model uses three components: C_mut couples the topload to the
-  spark channel, R represents plasma resistance where power is dissipated, and C_sh
-  provides the shunt capacitance to ground. The empirical 2 pF/foot rule gives a
-  good estimate for C_sh, which for a 5-foot spark yields 10 pF. This topology
-  ensures current through R must also flow through either C_mut or C_sh.
-
-related_concepts: ["circuit-topology", "lumped-model", "C_sh-empirical-rule", "spark-capacitance"]

exercises/01-fundamentals/fund-ex-02b.yaml

@@ -1,31 +0,0 @@
-id: fund-ex-02b
-type: calculation
-difficulty: easy
-points: 10
-related_lesson: fund-02
-question: |
-  A simulation shows C_sh = 10 pF for a given spark. What is the estimated spark
-  length using the empirical rule?
-
-hints:
-  - "The empirical rule is C_sh ≈ 2 pF/foot"
-  - "Solve for length: L = C_sh / (2 pF/foot)"
-  - "Don't forget the units!"
-
-solution:
-  steps:
-    - "Use the empirical relationship: C_sh ≈ 2 pF/foot"
-    - "Rearrange to solve for length: L = C_sh / (2 pF/foot)"
-    - "Substitute: L = 10 pF / (2 pF/foot) = 5 feet"
-
-  answer: "5"
-  unit: "feet"
-  tolerance: 0
-
-explanation: |
-  The empirical rule C_sh ≈ 2 pF/foot is a remarkably accurate guideline for Tesla
-  coil sparks, typically within ±30% for common geometries. This relationship comes
-  from the capacitance of a vertical conductor above ground. By inverting the formula,
-  we can estimate spark length from measured or simulated shunt capacitance values.
-
-related_concepts: ["C_sh-empirical-rule", "spark-length-estimation", "capacitance-measurement"]

exercises/01-fundamentals/fund-ex-02c.yaml

@@ -1,48 +0,0 @@
-id: fund-ex-02c
-type: multi-part
-difficulty: medium
-points: 15
-related_lesson: fund-02
-question: |
-  A 4-foot spark is formed. The topload has C_topload = 30 pF when unloaded.
-
-  (a) Estimate C_sh using the empirical rule
-  (b) What is the total system capacitance with the spark?
-
-  Hint: Consider how C_mut and C_sh combine in the circuit topology.
-
-hints:
-  - "Use C_sh ≈ 2 pF/foot for part (a)"
-  - "The circuit is (C_mut || R) in series with C_sh"
-  - "For DC or low frequency: R || C_mut looks like just C_mut"
-  - "Series capacitors: C_total = (C₁ × C₂)/(C₁ + C₂)"
-
-solution:
-  steps:
-    - "Part (a): Calculate C_sh = 2 pF/foot × 4 feet = 8 pF"
-    - "Part (b): Recognize topload connects to (C_mut || R) in series with C_sh"
-    - "At RF frequencies, parallel combination C_mut || R ≈ C_mut dominates"
-    - "Series combination: C_series = (C_mut × C_sh)/(C_mut + C_sh)"
-    - "Need to estimate C_mut. For typical geometries, C_mut ≈ 0.5 to 0.8 × C_topload"
-    - "Assuming C_mut ≈ 0.6 × 30 pF = 18 pF (estimate)"
-    - "C_series = (18 × 8)/(18 + 8) = 144/26 = 5.5 pF"
-    - "Total system: C_total = C_topload + C_series (if in parallel to ground)"
-    - "More accurately: Spark adds C_series in series, reducing total C seen from topload"
-
-  answer_part_a: "8"
-  unit_part_a: "pF"
-  answer_part_b: "approximately 5.5 pF from spark circuit (but depends on C_mut estimate)"
-
-explanation: |
-  Part (a) is straightforward using the empirical rule. Part (b) is more complex
-  because the spark adds a series combination of capacitances. The exact answer
-  depends on C_mut, which must be determined from FEMM or estimated based on
-  geometry. The key insight is that C_mut and C_sh form a capacitive divider that
-  reduces the effective capacitance seen from the topload terminal.
-
-  In practice, the "total system capacitance" includes the topload self-capacitance
-  plus the series combination of the spark circuit elements. This problem illustrates
-  why FEMM extraction is necessary for accurate modeling - C_mut cannot be calculated
-  from simple formulas.
-
-related_concepts: ["series-capacitance", "capacitive-divider", "total-capacitance", "FEMM-extraction"]

exercises/01-fundamentals/fund-ex-03a.yaml

@@ -1,46 +0,0 @@
-id: fund-ex-03a
-type: calculation
-difficulty: medium
-points: 15
-related_lesson: fund-03
-question: |
-  For a spark circuit with the following parameters:
-  - Frequency: f = 150 kHz
-  - Mutual capacitance: C_mut = 10 pF
-  - Shunt capacitance: C_sh = 8 pF
-  - Plasma resistance: R = 80 kΩ
-
-  Calculate Y_total in rectangular form (real and imaginary parts).
-
-hints:
-  - "First calculate ω = 2πf"
-  - "Then calculate G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh"
-  - "Use the formulas: Re{Y} = GB₂²/[G² + (B₁+B₂)²]"
-  - "And: Im{Y} = B₂[G² + B₁(B₁+B₂)]/[G² + (B₁+B₂)²]"
-
-solution:
-  steps:
-    - "Calculate angular frequency: ω = 2π × 150×10³ = 9.425×10⁵ rad/s"
-    - "Calculate conductance: G = 1/R = 1/(80×10³) = 12.5 μS"
-    - "Calculate susceptances: B₁ = ω×C_mut = 9.425×10⁵ × 10×10⁻¹² = 9.425 μS"
-    - "B₂ = ω×C_sh = 9.425×10⁵ × 8×10⁻¹² = 7.54 μS"
-    - "Calculate denominator: G² + (B₁+B₂)² = 156.25 + (16.965)² = 156.25 + 287.8 = 444.05 μS²"
-    - "Calculate Re{Y}: Re{Y} = 12.5 × (7.54)² / 444.05 = 12.5 × 56.85 / 444.05 = 710.6 / 444.05 = 1.60 μS"
-    - "Calculate Im{Y} numerator: G² + B₁(B₁+B₂) = 156.25 + 9.425×16.965 = 156.25 + 159.9 = 316.15 μS²"
-    - "Calculate Im{Y}: Im{Y} = 7.54 × 316.15 / 444.05 = 2383.8 / 444.05 = 5.37 μS"
-
-  answer: "Y = 1.60 + j5.37 μS"
-  real_part: "1.60"
-  imaginary_part: "5.37"
-  unit: "μS"
-  tolerance: 3.0
-
-explanation: |
-  This calculation demonstrates the admittance analysis method for the spark circuit.
-  The real part (1.60 μS) represents conductance - the component that dissipates
-  power in the plasma resistance. The imaginary part (5.37 μS) is the susceptance,
-  representing energy storage in the capacitances. The susceptance is 3.4× larger
-  than the conductance, indicating a strongly capacitive circuit - typical for
-  Tesla coil sparks.
-
-related_concepts: ["admittance-calculation", "complex-numbers", "conductance", "susceptance"]

exercises/01-fundamentals/fund-ex-03b.yaml

@@ -1,46 +0,0 @@
-id: fund-ex-03b
-type: calculation
-difficulty: medium
-points: 12
-related_lesson: fund-03
-question: |
-  An admittance is measured as Y = 2.0 + j4.5 μS.
-
-  Convert this to impedance Z in both rectangular and polar forms.
-
-hints:
-  - "Use |Z| = 1/|Y| for the magnitude"
-  - "Use φ_Z = -φ_Y for the phase angle"
-  - "Calculate |Y| = √(Re{Y}² + Im{Y}²)"
-  - "For rectangular: Z = R + jX where R = |Z|cos(φ_Z), X = |Z|sin(φ_Z)"
-
-solution:
-  steps:
-    - "Calculate magnitude of Y: |Y| = √(2.0² + 4.5²) = √(4 + 20.25) = √24.25 = 4.92 μS"
-    - "Calculate magnitude of Z: |Z| = 1/|Y| = 1/(4.92×10⁻⁶) = 203 kΩ"
-    - "Calculate admittance phase: φ_Y = atan(4.5/2.0) = atan(2.25) = 66.0°"
-    - "Calculate impedance phase: φ_Z = -φ_Y = -66.0°"
-    - "Polar form: Z = 203 kΩ ∠-66.0°"
-    - "Calculate rectangular components:"
-    - "R = |Z| × cos(φ_Z) = 203 × cos(-66°) = 203 × 0.407 = 82.6 kΩ"
-    - "X = |Z| × sin(φ_Z) = 203 × sin(-66°) = 203 × (-0.914) = -185.5 kΩ"
-    - "Rectangular form: Z = 82.6 - j185.5 kΩ"
-
-  answer_polar: "203 kΩ ∠-66.0°"
-  answer_rectangular: "82.6 - j185.5 kΩ"
-  magnitude: "203"
-  phase: "-66.0"
-  resistance: "82.6"
-  reactance: "-185.5"
-  unit: "kΩ"
-  tolerance: 2.0
-
-explanation: |
-  This conversion demonstrates the fundamental relationship between admittance and
-  impedance: they are reciprocals in the complex plane. The key relationships are
-  |Z| = 1/|Y| and φ_Z = -φ_Y. Note the opposite sign of the phase angle - this is
-  critical! A positive admittance phase (capacitive susceptance) corresponds to a
-  negative impedance phase (capacitive reactance). The negative reactance confirms
-  this is a capacitive impedance, as expected for spark circuits.
-
-related_concepts: ["admittance-to-impedance", "complex-reciprocal", "phase-relationship", "polar-rectangular"]

exercises/01-fundamentals/fund-ex-04a.yaml

@@ -1,35 +0,0 @@
-id: fund-ex-04a
-type: calculation
-difficulty: easy
-points: 8
-related_lesson: fund-04
-question: |
-  An impedance is measured as Z = 60 + j40 kΩ.
-
-  Calculate the phase angle φ_Z. Is this inductive or capacitive?
-
-hints:
-  - "Use φ_Z = atan(X/R)"
-  - "Positive X means inductive, negative X means capacitive"
-  - "The sign of φ_Z tells you about the reactive component"
-
-solution:
-  steps:
-    - "Identify components: R = 60 kΩ, X = +40 kΩ"
-    - "Calculate phase: φ_Z = atan(X/R) = atan(40/60) = atan(0.667) = 33.7°"
-    - "Since X > 0, this is inductive"
-    - "Positive phase angle confirms inductive behavior"
-
-  answer: "33.7"
-  unit: "degrees"
-  type_answer: "inductive"
-  tolerance: 1.0
-
-explanation: |
-  The phase angle is calculated from the ratio of reactance to resistance. The
-  positive value of both X and φ_Z indicates inductive impedance - the current
-  lags the voltage. This would be unusual for a Tesla coil spark circuit, which
-  are typically capacitive (negative φ_Z). An inductive impedance might appear
-  in the primary circuit or at very low frequencies where inductance dominates.
-
-related_concepts: ["phase-angle", "inductive-vs-capacitive", "impedance-components"]

exercises/01-fundamentals/fund-ex-04b.yaml

@@ -1,41 +0,0 @@
-id: fund-ex-04b
-type: multi-part
-difficulty: medium
-points: 15
-related_lesson: fund-04
-question: |
-  A spark has φ_Z = -60°. The impedance magnitude is |Z| = 150 kΩ.
-
-  (a) Find R and X (rectangular components)
-  (b) Calculate the power factor
-
-hints:
-  - "Use R = |Z| × cos(φ_Z) and X = |Z| × sin(φ_Z)"
-  - "Power factor = cos(φ_Z)"
-  - "Negative angle means capacitive reactance (X < 0)"
-
-solution:
-  steps:
-    - "Part (a): Calculate resistance"
-    - "R = |Z| × cos(φ_Z) = 150 × cos(-60°) = 150 × 0.5 = 75 kΩ"
-    - "Calculate reactance"
-    - "X = |Z| × sin(φ_Z) = 150 × sin(-60°) = 150 × (-0.866) = -130 kΩ"
-    - "Rectangular form: Z = 75 - j130 kΩ"
-    - "Part (b): Calculate power factor"
-    - "Power factor = cos(φ_Z) = cos(-60°) = 0.5"
-
-  answer_R: "75"
-  answer_X: "-130"
-  unit: "kΩ"
-  power_factor: "0.5"
-  tolerance: 2.0
-
-explanation: |
-  With a -60° phase angle, this spark is significantly capacitive. The resistance
-  (75 kΩ) equals half the impedance magnitude, while the capacitive reactance
-  (-130 kΩ) is 1.73× the resistance. The power factor of 0.5 means only 50% of
-  the apparent power (V×I) is real power dissipated in the plasma. The other 50%
-  is reactive power - energy oscillating in the capacitances. This is typical for
-  Tesla coil sparks, which operate with power factors in the 0.25-0.70 range.
-
-related_concepts: ["power-factor", "rectangular-components", "capacitive-impedance"]

exercises/01-fundamentals/fund-ex-05a.yaml

@@ -1,57 +0,0 @@
-id: fund-ex-05a
-type: multi-part
-difficulty: hard
-points: 20
-related_lesson: fund-05
-question: |
-  Calculate the topological phase constraint for a spark circuit with:
-  - Frequency: f = 150 kHz
-  - Mutual capacitance: C_mut = 12 pF
-  - Shunt capacitance: C_sh = 8 pF
-
-  (a) Calculate the capacitance ratio r
-  (b) Calculate the minimum achievable phase angle φ_Z,min
-  (c) Calculate R_opt_phase that achieves this minimum angle
-
-hints:
-  - "Ratio r = C_mut / C_sh"
-  - "Minimum phase: φ_Z,min = -atan(2√[r(1+r)])"
-  - "Optimal resistance: R_opt_phase = 1/[ω√(C_mut(C_mut+C_sh))]"
-
-solution:
-  steps:
-    - "Part (a): Calculate ratio"
-    - "r = C_mut / C_sh = 12 pF / 8 pF = 1.5"
-    - "Part (b): Calculate minimum phase"
-    - "φ_Z,min = -atan(2√[r(1+r)])"
-    - "= -atan(2√[1.5 × 2.5])"
-    - "= -atan(2√3.75)"
-    - "= -atan(2 × 1.936)"
-    - "= -atan(3.873)"
-    - "= -75.5°"
-    - "Part (c): Calculate R_opt_phase"
-    - "ω = 2πf = 2π × 150×10³ = 9.425×10⁵ rad/s"
-    - "R_opt_phase = 1/[ω√(C_mut(C_mut+C_sh))]"
-    - "= 1/[9.425×10⁵ × √(12×10⁻¹² × 20×10⁻¹²)]"
-    - "= 1/[9.425×10⁵ × √(240×10⁻²⁴)]"
-    - "= 1/[9.425×10⁵ × 15.49×10⁻¹²]"
-    - "= 1/(14.60×10⁻⁶)"
-    - "= 68.5 kΩ"
-
-  answer_r: "1.5"
-  answer_phi_min: "-75.5"
-  answer_R_opt: "68.5"
-  unit_R: "kΩ"
-  unit_phi: "degrees"
-  tolerance: 3.0
-
-explanation: |
-  With r = 1.5, this circuit cannot achieve -45° (which requires r < 0.207). The
-  minimum achievable phase is -75.5°, which is quite capacitive. This occurs when
-  R = R_opt_phase = 68.5 kΩ. Any other resistance value will result in a phase
-  angle with magnitude greater than 75.5°. This topological constraint is fundamental
-  to the circuit structure - it's impossible to overcome by changing component
-  values. The ratio r = 1.5 is typical for medium Tesla coils with moderate-length
-  sparks.
-
-related_concepts: ["topological-constraint", "phase-optimization", "R_opt_phase", "capacitance-ratio"]

exercises/01-fundamentals/fund-ex-08-comprehensive.yaml

@@ -1,84 +0,0 @@
-id: fund-ex-08-comprehensive
-type: multi-part
-difficulty: hard
-points: 50
-related_lesson: fund-08
-question: |
-  COMPREHENSIVE INTEGRATION EXERCISE
-
-  A Tesla coil operates at 220 kHz with a 3.5-foot spark. FEMM analysis gives
-  C_mut = 9 pF. Assume R = 60 kΩ.
-
-  (a) Calculate C_sh, ω, G, B₁, B₂
-  (b) Calculate Y_total and Z_total
-  (c) Find φ_Z and compare to -45°
-  (d) Calculate r and φ_Z,min
-  (e) If V_top = 350 kV, find power dissipated
-
-hints:
-  - "Use C_sh ≈ 2 pF/foot for estimation"
-  - "Calculate all intermediate values carefully"
-  - "Use admittance formulas from fund-03"
-  - "Compare actual φ_Z to φ_Z,min from topological constraint"
-  - "Power = 0.5 × |I|² × R or 0.5 × |V|² × Re{Y}"
-
-solution:
-  steps:
-    - "Part (a): Calculate components"
-    - "C_sh = 2 pF/foot × 3.5 feet = 7 pF"
-    - "ω = 2πf = 2π × 220×10³ = 1.382×10⁶ rad/s"
-    - "G = 1/R = 1/(60×10³) = 16.67 μS"
-    - "B₁ = ωC_mut = 1.382×10⁶ × 9×10⁻¹² = 12.44 μS"
-    - "B₂ = ωC_sh = 1.382×10⁶ × 7×10⁻¹² = 9.67 μS"
-    - "Part (b): Calculate Y_total"
-    - "Denominator: G² + (B₁+B₂)² = 277.9 + (22.11)² = 277.9 + 488.9 = 766.8 μS²"
-    - "Re{Y} = GB₂²/[G²+(B₁+B₂)²] = 16.67×93.5/766.8 = 1559/766.8 = 2.03 μS"
-    - "Im{Y} = B₂[G²+B₁(B₁+B₂)]/[G²+(B₁+B₂)²]"
-    - "= 9.67×[277.9+12.44×22.11]/766.8"
-    - "= 9.67×[277.9+275.0]/766.8"
-    - "= 9.67×552.9/766.8 = 6.98 μS"
-    - "Y_total = 2.03 + j6.98 μS"
-    - "Convert to impedance:"
-    - "|Y| = √(2.03² + 6.98²) = √(4.12 + 48.72) = 7.27 μS"
-    - "|Z| = 1/|Y| = 137.5 kΩ"
-    - "φ_Y = atan(6.98/2.03) = 73.8°"
-    - "φ_Z = -φ_Y = -73.8°"
-    - "R_eq = 137.5 × cos(-73.8°) = 38.4 kΩ"
-    - "X_eq = 137.5 × sin(-73.8°) = -132 kΩ"
-    - "Z_total = 38.4 - j132 kΩ = 137.5 kΩ ∠-73.8°"
-    - "Part (c): Compare to -45°"
-    - "φ_Z = -73.8° is more capacitive than -45° (larger magnitude)"
-    - "|X|/R = 132/38.4 = 3.44"
-    - "Capacitive reactance is 3.44× the resistance"
-    - "Part (d): Calculate topological constraint"
-    - "r = C_mut/C_sh = 9/7 = 1.286"
-    - "φ_Z,min = -atan(2√[1.286×2.286]) = -atan(2×1.716) = -atan(3.43) = -73.7°"
-    - "Actual φ_Z = -73.8° ≈ φ_Z,min (operating near optimal phase!)"
-    - "Part (e): Calculate power"
-    - "Current: I = V/|Z| = 350×10³/137.5×10³ = 2.55 A peak"
-    - "Power: P = 0.5 × I² × R_eq = 0.5 × 2.55² × 38.4×10³"
-    - "= 0.5 × 6.50 × 38.4×10³ = 125 kW"
-    - "Alternative: P = 0.5 × V² × Re{Y}"
-    - "= 0.5 × (350×10³)² × 2.03×10⁻⁶ = 124 kW ✓"
-
-  answer_a: "C_sh=7pF, ω=1.382e6 rad/s, G=16.67μS, B₁=12.44μS, B₂=9.67μS"
-  answer_b: "Y=2.03+j6.98 μS, Z=137.5kΩ∠-73.8° or 38.4-j132 kΩ"
-  answer_c: "φ_Z=-73.8°, more capacitive than -45°, ratio=3.44"
-  answer_d: "r=1.286, φ_Z,min=-73.7°"
-  answer_e: "125"
-  unit_e: "kW"
-  tolerance: 5.0
-
-explanation: |
-  This comprehensive problem integrates all fundamental concepts from Part 1. The
-  solution demonstrates: (1) using empirical rules for estimation, (2) systematic
-  admittance calculation, (3) conversion between Y and Z, (4) understanding phase
-  constraints, and (5) power calculation methods.
-
-  Key insights: The actual phase angle (-73.8°) is essentially at the minimum
-  possible value (-73.7°), suggesting this R value is close to R_opt_phase. The
-  power dissipated (125 kW) is substantial for a 3.5-foot spark. The capacitive
-  reactance dominates (3.44× the resistance), which is typical for Tesla coil
-  sparks with r > 1.
-
-related_concepts: ["integration", "complete-analysis", "power-calculation", "phase-optimization"]

exercises/01-fundamentals/fund-ex-checkpoint-quiz.yaml

@@ -1,81 +0,0 @@
-id: fund-ex-checkpoint-quiz
-type: conceptual
-difficulty: medium
-points: 100
-related_lesson: fund-08
-question: |
-  FUNDAMENTALS CHECKPOINT QUIZ - Answer all 10 questions
-
-  1. What is the relationship between peak and RMS voltage? If V_peak = 100 kV, what is V_RMS?
-
-  2. Write the power formula using peak phasors. Why is there a factor of 0.5?
-
-  3. For a capacitor, why is X negative but B positive?
-
-  4. Draw the circuit topology for a spark (show C_mut, R, C_sh).
-
-  5. What is the empirical rule for C_sh? If a spark is 4 feet long, estimate C_sh.
-
-  6. The admittance phase angle θ_Y = +60°. What is the impedance phase angle φ_Z?
-
-  7. An impedance has φ_Z = -30°. Is this inductive or capacitive?
-
-  8. Why is V_top/I_base not the correct impedance measurement?
-
-  9. Describe the difference between streamers and leaders (two key differences).
-
-  10. Explain the "hungry streamer" concept in one sentence.
-
-hints:
-  - "Review each fundamental lesson carefully"
-  - "Consider both mathematical and physical interpretations"
-  - "Draw diagrams where helpful"
-
-solution:
-  answer_1: "V_RMS = V_peak/√2. For V_peak = 100 kV, V_RMS = 100/√2 ≈ 70.7 kV"
-
-  answer_2: "P = 0.5 × Re{V × I*}. The 0.5 factor comes from time-averaging cos²(ωt) over a full cycle."
-
-  answer_3: |
-    For capacitors, reactance X_C = -1/(ωC) is negative, but susceptance B_C = ωC
-    is positive. The sign conventions are opposite for impedance vs admittance.
-
-  answer_4: |
-    Topload
-        |
-    [C_mut]
-        |
-    +----+----+
-    |         |
-   [R]      [C_sh]
-    |         |
-   GND------GND
-
-  answer_5: "C_sh ≈ 2 pF/foot. For 4 feet: C_sh ≈ 8 pF"
-
-  answer_6: "φ_Z = -θ_Y = -60°"
-
-  answer_7: "Capacitive (negative φ_Z indicates capacitive behavior)"
-
-  answer_8: |
-    I_base includes displacement currents from the entire secondary, plus coupling
-    currents and environmental currents. Only I_spark flows through the spark.
-    V_top/I_base underestimates impedance because I_base > I_spark.
-
-  answer_9: |
-    Any two of: Streamers are thin (10-100 μm), fast (~10⁶ m/s), cold (~1000 K),
-    high R, branched. Leaders are thick (mm-cm), slower (~10³ m/s), hot (5000-20000 K),
-    low R, straighter.
-
-  answer_10: |
-    Plasma actively adjusts its conductivity to maximize power extraction from the
-    circuit, naturally seeking R ≈ R_opt_power.
-
-explanation: |
-  This checkpoint quiz verifies understanding of all fundamental concepts from
-  Part 1. Correct answers demonstrate mastery of: complex numbers and phasors,
-  circuit topology, capacitance relationships, admittance analysis, phase angles,
-  measurement ports, and spark physics basics. These concepts form the foundation
-  for optimization (Part 2), growth physics (Part 3), and advanced modeling (Part 4).
-
-related_concepts: ["fundamentals-review", "integration", "checkpoint", "mastery-verification"]

exercises/02-optimization/opt-ex-01a.yaml

@@ -1,52 +0,0 @@
-id: opt-ex-01a
-type: calculation
-difficulty: medium
-points: 15
-related_lesson: opt-01
-question: |
-  For a spark circuit with the following parameters:
-  - Frequency: f = 150 kHz
-  - Mutual capacitance: C_mut = 10 pF
-  - Shunt capacitance: C_sh = 8 pF
-
-  Calculate both R_opt_power and R_opt_phase.
-
-hints:
-  - "R_opt_power = 1/[ω(C_mut + C_sh)]"
-  - "R_opt_phase = 1/[ω√(C_mut(C_mut + C_sh))]"
-  - "Calculate ω = 2πf first"
-  - "R_opt_power is always smaller than R_opt_phase"
-
-solution:
-  steps:
-    - "Calculate angular frequency: ω = 2π × 150×10³ = 9.425×10⁵ rad/s"
-    - "Calculate R_opt_power:"
-    - "C_total = C_mut + C_sh = 10 + 8 = 18 pF"
-    - "R_opt_power = 1/(ω × C_total)"
-    - "= 1/(9.425×10⁵ × 18×10⁻¹²)"
-    - "= 1/(16.965×10⁻⁶)"
-    - "= 58.9 kΩ"
-    - "Calculate R_opt_phase:"
-    - "Product: C_mut × (C_mut + C_sh) = 10 × 18 = 180 pF²"
-    - "Square root: √180 = 13.42 pF"
-    - "R_opt_phase = 1/(ω × √180×10⁻¹²)"
-    - "= 1/(9.425×10⁵ × 13.42×10⁻¹²)"
-    - "= 1/(12.65×10⁻⁶)"
-    - "= 79.1 kΩ"
-    - "Compare: R_opt_power/R_opt_phase = 58.9/79.1 = 0.745"
-
-  answer_power: "58.9"
-  answer_phase: "79.1"
-  unit: "kΩ"
-  ratio: "0.745"
-  tolerance: 3.0
-
-explanation: |
-  This problem demonstrates the two critical resistances for spark optimization.
-  R_opt_power (58.9 kΩ) maximizes real power transfer to the spark, while
-  R_opt_phase (79.1 kΩ) minimizes the impedance phase angle magnitude. The ratio
-  of 0.745 is typical - R_opt_power is usually 50-75% of R_opt_phase. These
-  different values show that maximum power transfer and minimum phase angle are
-  different optimization goals that cannot be achieved simultaneously.
-
-related_concepts: ["R_opt_power", "R_opt_phase", "power-optimization", "phase-optimization"]

exercises/02-optimization/opt-ex-01b.yaml

@@ -1,51 +0,0 @@
-id: opt-ex-01b
-type: multi-part
-difficulty: medium
-points: 15
-related_lesson: opt-01
-question: |
-  At 200 kHz, a spark has total capacitance C_total = 12 pF.
-
-  (a) What is R_opt_power?
-  (b) If V_top = 400 kV, estimate the maximum deliverable power (assume R is at
-      optimal value and φ_Z ≈ -70°)
-
-hints:
-  - "R_opt_power = 1/(ω × C_total)"
-  - "Power = 0.5 × |V|² × Re{Y}"
-  - "Or use: P = 0.5 × |V|²/|Z| × cos(φ_Z)"
-  - "At R_opt_power, typical phase is around -65° to -75°"
-
-solution:
-  steps:
-    - "Part (a): Calculate R_opt_power"
-    - "ω = 2π × 200×10³ = 1.257×10⁶ rad/s"
-    - "R_opt_power = 1/(ω × C_total)"
-    - "= 1/(1.257×10⁶ × 12×10⁻¹²)"
-    - "= 1/(15.08×10⁻⁶)"
-    - "= 66.3 kΩ"
-    - "Part (b): Estimate maximum power"
-    - "At R_opt_power with given capacitances, φ_Z ≈ -70° (typical)"
-    - "Approximate |Z| ≈ R_opt_power / cos(-70°) = 66.3/0.342 ≈ 194 kΩ"
-    - "Current: I = V/|Z| = 400×10³/194×10³ = 2.06 A peak"
-    - "Power: P = 0.5 × V × I × cos(φ_Z)"
-    - "= 0.5 × 400×10³ × 2.06 × cos(-70°)"
-    - "= 0.5 × 400×10³ × 2.06 × 0.342"
-    - "= 141 kW"
-    - "Alternative: P ≈ 0.5 × I² × R = 0.5 × 2.06² × 66.3×10³ ≈ 141 kW"
-
-  answer_a: "66.3"
-  answer_b: "141"
-  unit_a: "kΩ"
-  unit_b: "kW"
-  tolerance: 5.0
-
-explanation: |
-  R_opt_power is determined solely by frequency and total capacitance. At this
-  resistance, power transfer is maximized. The estimated power (141 kW) is
-  substantial, but achievable for medium-to-large DRSSTCs. This calculation shows
-  why R_opt_power is critical for performance - operating far from this value
-  significantly reduces delivered power. The estimate uses typical phase angle
-  for operation at R_opt_power; exact value would require full admittance calculation.
-
-related_concepts: ["R_opt_power", "maximum-power-transfer", "power-estimation"]

exercises/02-optimization/opt-ex-thevenin-complete.yaml

@@ -1,91 +0,0 @@
-id: opt-ex-thevenin-complete
-type: multi-part
-difficulty: hard
-points: 40
-related_lesson: opt-03
-question: |
-  COMPLETE THÉVENIN ANALYSIS
-
-  You measured the following Thévenin parameters for your DRSSTC at 188 kHz:
-  - Z_th = 115 - j2300 Ω (drive OFF, 1V test source)
-  - V_th = 340 kV (drive ON, no load)
-
-  The spark has:
-  - C_mut = 8 pF, C_sh = 5 pF (from FEMM)
-  - R = 65 kΩ (assumed operating resistance)
-
-  Tasks:
-  (a) Calculate the spark admittance Y_spark
-  (b) Convert to Z_spark
-  (c) Calculate total circuit impedance Z_total = Z_th + Z_spark
-  (d) Calculate current through the spark
-  (e) Calculate voltage across the spark
-  (f) Calculate real power dissipated in the spark
-  (g) Compare R to R_opt_power for these capacitances
-
-hints:
-  - "Use admittance formulas for parallel (R || C_mut) then series with C_sh"
-  - "Add impedances in series: Z_total = Z_th + Z_spark"
-  - "Current divider applies: I = V_th / Z_total"
-  - "Voltage across spark: V_spark = I × Z_spark"
-  - "Power: P = 0.5 × |I|² × Re{Z_spark}"
-
-solution:
-  steps:
-    - "Part (a): Calculate Y_spark"
-    - "ω = 2π × 188×10³ = 1.181×10⁶ rad/s"
-    - "G = 1/65000 = 15.38 μS"
-    - "B₁ = 1.181×10⁶ × 8×10⁻¹² = 9.45 μS"
-    - "B₂ = 1.181×10⁶ × 5×10⁻¹² = 5.91 μS"
-    - "Denom: G² + (B₁+B₂)² = 236.5 + 236.2 = 472.7 μS²"
-    - "Re{Y} = 15.38 × 34.93 / 472.7 = 1.14 μS"
-    - "Im{Y} = 5.91 × [236.5 + 145.2] / 472.7 = 4.77 μS"
-    - "Y_spark = 1.14 + j4.77 μS"
-    - "Part (b): Convert to Z_spark"
-    - "|Y| = √(1.14² + 4.77²) = 4.90 μS"
-    - "|Z_spark| = 1/4.90×10⁻⁶ = 204 kΩ"
-    - "φ_Y = atan(4.77/1.14) = 76.6°"
-    - "φ_Z = -76.6°"
-    - "R_eq = 204 × cos(-76.6°) = 47.6 kΩ"
-    - "X_eq = 204 × sin(-76.6°) = -198 kΩ"
-    - "Z_spark = 47.6 - j198 kΩ"
-    - "Part (c): Calculate Z_total"
-    - "Z_total = Z_th + Z_spark"
-    - "= (115 - j2300) + (47600 - j198000)"
-    - "= (47715 - j200300) Ω"
-    - "= 47.7 - j200.3 kΩ"
-    - "|Z_total| = √(47.7² + 200.3²) = 205.9 kΩ"
-    - "Part (d): Calculate current"
-    - "I = V_th / Z_total = 340×10³ / 205.9×10³ = 1.65 A peak"
-    - "Part (e): Calculate voltage across spark"
-    - "V_spark = I × Z_spark = 1.65 × 204×10³ = 337 kV"
-    - "Part (f): Calculate power"
-    - "P = 0.5 × I² × R_eq = 0.5 × 1.65² × 47.6×10³"
-    - "= 0.5 × 2.72 × 47.6×10³ = 64.8 kW"
-    - "Part (g): Compare to R_opt_power"
-    - "R_opt = 1/(ω × (C_mut + C_sh))"
-    - "= 1/(1.181×10⁶ × 13×10⁻¹²) = 65.1 kΩ"
-    - "Actual R = 65 kΩ ≈ R_opt_power ✓"
-    - "Operating at optimal resistance for maximum power transfer!"
-
-  answer_a: "1.14 + j4.77 μS"
-  answer_b: "204 kΩ ∠-76.6° or 47.6 - j198 kΩ"
-  answer_c: "205.9 kΩ"
-  answer_d: "1.65"
-  unit_d: "A peak"
-  answer_e: "337"
-  unit_e: "kV"
-  answer_f: "64.8"
-  unit_f: "kW"
-  answer_g: "R_opt = 65.1 kΩ, actual = 65 kΩ, at optimum!"
-  tolerance: 3.0
-
-explanation: |
-  This complete Thévenin analysis demonstrates the power of the equivalent circuit
-  method. Key insights: (1) Most voltage appears across the spark (337 kV out of
-  340 kV) because |Z_spark| >> |Z_th|, (2) The actual R ≈ R_opt_power suggests
-  the plasma self-optimized to maximize power extraction, (3) Power dissipated
-  (64.8 kW) is substantial, (4) Strongly capacitive spark (φ_Z = -76.6°) is typical.
-  This analysis predicts performance without full coupled simulation.
-
-related_concepts: ["thevenin-method", "complete-analysis", "power-prediction", "self-optimization"]

exercises/03-spark-physics/phys-ex-01a.yaml

@@ -1,44 +0,0 @@
-id: phys-ex-01a
-type: calculation
-difficulty: easy
-points: 10
-related_lesson: phys-01
-question: |
-  A 0.8 m spark has V_top = 280 kV and tip enhancement factor κ = 4.
-
-  (a) Calculate E_tip
-  (b) If E_propagation = 0.5 MV/m, can the spark grow further?
-
-hints:
-  - "E_average = V_top / L"
-  - "E_tip = κ × E_average"
-  - "Growth continues when E_tip > E_propagation"
-
-solution:
-  steps:
-    - "Part (a): Calculate average field"
-    - "E_average = V_top / L = 280×10³ V / 0.8 m = 350 kV/m = 0.35 MV/m"
-    - "Calculate tip field"
-    - "E_tip = κ × E_average = 4 × 0.35 = 1.4 MV/m"
-    - "Part (b): Compare to threshold"
-    - "E_tip = 1.4 MV/m"
-    - "E_propagation = 0.5 MV/m"
-    - "E_tip > E_propagation ✓"
-    - "Yes, spark can grow further"
-    - "Safety margin: 1.4 / 0.5 = 2.8× above threshold"
-
-  answer_a: "1.4"
-  unit_a: "MV/m"
-  answer_b: "yes"
-  margin: "2.8"
-  tolerance: 5.0
-
-explanation: |
-  The tip field (1.4 MV/m) significantly exceeds the propagation threshold
-  (0.5 MV/m), with a comfortable 2.8× safety margin. This spark is not voltage-
-  limited and can continue growing. The enhancement factor κ = 4 concentrates the
-  average field (0.35 MV/m) at the tip, creating sufficient field strength for
-  sustained ionization and growth. If this spark has adequate power, it can extend
-  well beyond 0.8 m.
-
-related_concepts: ["electric-field", "tip-enhancement", "growth-criterion", "voltage-limited"]

exercises/03-spark-physics/phys-ex-03a.yaml

@@ -1,45 +0,0 @@
-id: phys-ex-03a
-type: calculation
-difficulty: hard
-points: 20
-related_lesson: phys-03
-question: |
-  A burst-mode coil has ε = 60 J/m. To reach L = 1.5 m in a 200 μs pulse,
-  what power is required? Is this realistic for a burst-mode Tesla coil?
-
-hints:
-  - "Use growth rate equation: dL/dt = P/ε"
-  - "Rearrange: P = ε × dL/dt"
-  - "Calculate dL/dt = L/T for the pulse duration"
-  - "Consider typical DRSSTC power levels"
-
-solution:
-  steps:
-    - "Calculate growth rate needed:"
-    - "dL/dt = L / T = 1.5 m / (200×10⁻⁶ s) = 7,500 m/s"
-    - "Calculate required power:"
-    - "P = ε × dL/dt"
-    - "P = 60 J/m × 7,500 m/s"
-    - "P = 450,000 W = 450 kW"
-    - "Analysis of realism:"
-    - "Energy per pulse: E = P × T = 450 kW × 200 μs = 90 J"
-    - "For primary: C = 0.5 μF, need V² = 2E/C = 360,000, so V ≈ 600 V"
-    - "Peak power: 450 kW is high but achievable for large DRSSTC"
-    - "Conclusion: Challenging but realistic for large coil"
-
-  answer: "450"
-  unit: "kW"
-  energy_per_pulse: "90"
-  realistic: "yes, but requires large DRSSTC"
-  tolerance: 5.0
-
-explanation: |
-  Growing 1.5 m in just 200 μs requires extremely high instantaneous power
-  (450 kW). However, the total energy per pulse is only 90 J, which is achievable
-  with a 0.5 μF primary capacitor charged to 600 V. This high power/short duration
-  trade-off is characteristic of burst mode operation. The high ε = 60 J/m reflects
-  inefficiency (branching, radiation) in burst mode. A QCW coil with ε = 10 J/m
-  would need only 75 kW for the same growth rate, or could grow the same length
-  with less power over a longer time.
-
-related_concepts: ["energy-per-meter", "growth-rate", "burst-mode", "power-requirements"]

exercises/03-spark-physics/phys-ex-comprehensive.yaml

@@ -1,109 +0,0 @@
-id: phys-ex-comprehensive
-type: design
-difficulty: hard
-points: 100
-related_lesson: phys-09
-question: |
-  COMPREHENSIVE SPARK PHYSICS DESIGN CHALLENGE
-
-  Design a QCW coil from scratch to achieve 3.5 m sparks.
-
-  Given constraints:
-  - Budget allows C_primary up to 1.0 μF
-  - V_primary limited to 600 V (safety)
-  - Topload options: 20 cm toroid (C_top ≈ 25 pF) or 35 cm toroid (C_top ≈ 45 pF)
-  - Target ramp time: 10-15 ms
-  - Sea level operation (E_propagation = 0.6 MV/m)
-
-  Complete the following analysis:
-
-  1. Energy calculation:
-     - Choose ε for QCW mode
-     - Calculate total energy required for 3.5 m
-     - Verify achievable with C_primary and V_primary
-
-  2. Voltage requirement:
-     - Estimate C_mut for each topload (use C_mut ≈ 0.7 × C_top)
-     - Calculate C_sh for 3.5 m spark
-     - For each topload, calculate V_topload needed for E_tip = 0.7 MV/m at 3.5 m (κ = 3.0)
-     - Include capacitive division effects
-
-  3. Power analysis:
-     - For T_ramp = 12 ms, calculate required average power
-     - Estimate peak power (assume 1.5× average for QCW)
-     - Check if reasonable for DRSSTC primary
-
-  4. Thermal verification:
-     - Estimate leader diameter (2-4 mm typical)
-     - Calculate thermal time constant
-     - Verify ramp time << thermal time
-
-  5. Final recommendation:
-     - Which topload should be used?
-     - Is 3.5 m target achievable?
-     - If not, what would you change?
-
-hints:
-  - "Use ε ≈ 10-12 J/m for QCW mode"
-  - "Remember capacitive divider: V_tip = V_topload × C_mut/(C_mut + C_sh)"
-  - "E_tip = κ × V_tip / L must exceed E_propagation"
-  - "Thermal time: τ = d²/(4α) with α = 2×10⁻⁵ m²/s"
-
-solution:
-  energy_calculation:
-    chosen_epsilon: "11 J/m (typical QCW)"
-    total_energy: "E = ε × L = 11 × 3.5 = 38.5 J"
-    primary_check: "E_primary = 0.5 × C × V² = 0.5 × 1.0×10⁻⁶ × 600² = 180 J"
-    verdict: "38.5 J << 180 J available ✓ Energy adequate"
-
-  voltage_requirement:
-    small_toroid:
-      C_top: "25 pF"
-      C_mut_est: "17.5 pF"
-      C_sh: "23.1 pF (6.6 pF/m × 3.5 m)"
-      V_tip_needed: "V_tip = E_prop × L / κ = 0.7×10⁶ × 3.5 / 3.0 = 817 kV"
-      V_topload_needed: "V_top = V_tip × (C_mut + C_sh) / C_mut = 817 × 40.6/17.5 = 1,896 kV"
-      verdict: "Unrealistic voltage required ✗"
-
-    large_toroid:
-      C_top: "45 pF"
-      C_mut_est: "31.5 pF"
-      C_sh: "23.1 pF"
-      V_tip_needed: "817 kV (same)"
-      V_topload_needed: "V_top = 817 × 54.6/31.5 = 1,416 kV"
-      verdict: "Still very high, challenging ✗"
-
-  power_analysis:
-    ramp_time: "12 ms"
-    avg_power: "P = E/T = 38.5 J / 0.012 s = 3.2 kW"
-    peak_power: "~5 kW (1.5× average)"
-    verdict: "Power requirement is modest ✓"
-
-  thermal_verification:
-    leader_diameter: "3 mm (estimate)"
-    thermal_constant: "τ = (0.003)² / (4 × 2×10⁻⁵) = 113 ms"
-    comparison: "T_ramp (12 ms) < τ (113 ms), ratio = 0.11"
-    verdict: "Leader stays hot during ramp ✓ QCW condition satisfied"
-
-  final_recommendation: |
-    Neither topload can achieve 3.5 m with realistic voltages due to capacitive
-    division. To achieve 3.5 m:
-
-    Option 1: Accept shorter sparks (~2-2.5 m achievable with large toroid)
-    Option 2: Increase primary voltage capability (beyond 600 V safety limit)
-    Option 3: Use active voltage ramping to counteract division
-    Option 4: Add intermediate electrode to reduce effective spark length
-
-    Recommended: Use 35 cm toroid, target 2.5 m realistic goal, accept that
-    voltage limitation dominates. Energy and power are adequate, but voltage
-    limit prevents reaching 3.5 m target.
-
-explanation: |
-  This comprehensive design challenge demonstrates the interplay between energy,
-  voltage, and power limitations. The analysis reveals that voltage (electric field
-  requirement) is the primary constraint, not energy or power. Capacitive division
-  significantly increases the required topload voltage. The larger toroid helps but
-  doesn't fully solve the problem. This is typical for Tesla coils - voltage-limited
-  rather than power-limited. Realistic design must balance these constraints.
-
-related_concepts: ["design-integration", "voltage-vs-power-limits", "capacitive-divider", "QCW-optimization"]

exercises/03-spark-physics/phys-ex-conceptual-limits.yaml

@@ -1,77 +0,0 @@
-id: phys-ex-conceptual-limits
-type: conceptual
-difficulty: medium
-points: 20
-related_lesson: phys-09
-question: |
-  CONCEPTUAL UNDERSTANDING: Voltage vs Power Limitations
-
-  A coiler claims: "I have 200 kW of power available in my DRSSTC, so I should
-  easily get 10 m sparks!"
-
-  Identify the flaws in this reasoning. Your answer should discuss:
-  (a) Voltage vs power limitations
-  (b) Energy per meter constraints
-  (c) Capacitive divider effects
-  (d) Realistic expectations
-
-hints:
-  - "Consider both E-field requirements AND energy requirements"
-  - "What happens to E_tip as spark grows?"
-  - "How does capacitive division change with length?"
-  - "Typical maximum spark lengths for Tesla coils"
-
-solution:
-  answer: |
-    FLAWS IN REASONING:
-
-    (a) Voltage vs Power Limitations:
-    Power alone doesn't determine spark length. The spark needs BOTH adequate
-    electric field (E_tip > E_propagation ≈ 0.6 MV/m) AND sufficient energy.
-    For a 10 m spark:
-    - Average field needed: E_avg ≈ 0.6 MV/m (if κ ≈ 3)
-    - This requires V_top ≈ 2,000 kV minimum
-    - But typical Tesla coil voltages: 300-600 kV (factor of 3-7 too low!)
-    - Voltage limitation dominates, not power
-
-    (b) Energy Per Meter Constraints:
-    Even if power is adequate:
-    - For QCW with ε = 10 J/m: E_needed = 10 × 10 = 100 J
-    - Time available: T ≈ 10-20 ms typical
-    - Power needed: P = 100 J / 0.015 s = 6.7 kW (well below 200 kW!)
-    - So power is not the limiting factor
-
-    (c) Capacitive Divider Effects:
-    As spark grows:
-    - C_sh increases (≈ 6.6 pF/m, so 66 pF for 10 m)
-    - V_tip = V_topload × C_mut/(C_mut + C_sh) decreases
-    - For typical C_mut = 20 pF: V_tip = V_top × 20/86 = 0.23 × V_top
-    - Lose 77% of voltage to division!
-    - Combined with 1/L² field reduction: E_tip ∝ 1/L² catastrophic drop
-    - This self-limiting effect prevents very long sparks
-
-    (d) Realistic Expectations:
-    - Burst mode record sparks: ~2-3 m
-    - QCW mode record sparks: ~5-6 m
-    - 10 m would require:
-      * V_top ≈ 2+ MV (extreme)
-      * Careful voltage ramping to fight division
-      * Very large topload (high C_mut)
-      * Sea level operation (higher E_propagation at altitude)
-    - More power doesn't overcome voltage limit!
-    - The claim confuses power-limited with voltage-limited regimes
-
-    CORRECT REASONING:
-    "I have adequate power, but am limited by achievable topload voltage and
-    capacitive division effects. Realistic maximum is ~3-4 m for my coil,
-    regardless of available power beyond ~20 kW."
-
-explanation: |
-  This conceptual problem addresses a common misconception. Tesla coils are almost
-  always voltage-limited, not power-limited. The E-field requirement (E_tip >
-  E_propagation) combined with capacitive division creates a fundamental voltage
-  barrier. Having excess power just makes the spark brighter and hotter, not longer.
-  Understanding this distinction is critical for realistic performance expectations
-  and efficient design decisions.
-
-related_concepts: ["voltage-vs-power", "limiting-factors", "capacitive-divider", "realistic-expectations"]

exercises/04-advanced-modeling/model-ex-lumped-complete.yaml

@@ -1,79 +0,0 @@
-id: model-ex-lumped-complete
-type: multi-part
-difficulty: hard
-points: 50
-related_lesson: model-02
-question: |
-  LUMPED MODEL COMPLETE WORKFLOW
-
-  You extracted the following Maxwell capacitance matrix from FEMM for a 1.2 m
-  (4-foot) spark with a toroid topload:
-
-  Maxwell Matrix (pF):
-           Topload  Spark
-  Topload  [  32.5   -9.2 ]
-  Spark    [  -9.2   15.6 ]
-
-  Operating frequency: f = 185 kHz
-
-  Tasks:
-  (a) Validate the matrix (check symmetry, signs, physical reasonableness)
-  (b) Extract C_mut and C_sh for the lumped circuit model
-  (c) Compare C_sh to the empirical 2 pF/foot rule
-  (d) Calculate R_opt_power and R_opt_phase
-  (e) Build the lumped model with R = R_opt_power and calculate Z_spark
-
-hints:
-  - "Maxwell matrix has negative off-diagonals"
-  - "C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂|"
-  - "Check if C_sh ≈ 2 pF/foot × 4 feet = 8 pF"
-  - "Use admittance formulas for part (e)"
-
-solution:
-  steps:
-    - "Part (a): Matrix validation"
-    - "Symmetry: C₁₂ = C₂₁ = -9.2 pF ✓"
-    - "Diagonal positive: C₁₁ = 32.5 > 0, C₂₂ = 15.6 > 0 ✓"
-    - "Off-diagonal negative: C₁₂ = -9.2 < 0 ✓"
-    - "Row sums: R₁ = 32.5 - 9.2 = 23.3, R₂ = -9.2 + 15.6 = 6.4 (ground contribution) ✓"
-    - "Matrix is valid"
-    - "Part (b): Extract lumped parameters"
-    - "C_mut = |C₁₂| = |-9.2| = 9.2 pF"
-    - "C_sh = C₂₂ - |C₁₂| = 15.6 - 9.2 = 6.4 pF"
-    - "Part (c): Compare to empirical rule"
-    - "Empirical: C_sh ≈ 2 pF/foot × 4 feet = 8 pF"
-    - "FEMM: C_sh = 6.4 pF"
-    - "Ratio: 6.4/8 = 0.8 (within factor 2, acceptable) ✓"
-    - "Part (d): Calculate optimal resistances"
-    - "ω = 2π × 185×10³ = 1.162×10⁶ rad/s"
-    - "C_total = 9.2 + 6.4 = 15.6 pF"
-    - "R_opt_power = 1/(ω × C_total) = 1/(1.162×10⁶ × 15.6×10⁻¹²) = 55.2 kΩ"
-    - "Product: C_mut(C_mut + C_sh) = 9.2 × 15.6 = 143.5 pF²"
-    - "R_opt_phase = 1/(ω × √143.5×10⁻¹²) = 1/(1.162×10⁶ × 11.98×10⁻¹²) = 71.9 kΩ"
-    - "Part (e): Calculate Z_spark at R_opt_power"
-    - "Use R = 55.2 kΩ, so G = 18.12 μS"
-    - "B₁ = ω × C_mut = 1.162×10⁶ × 9.2×10⁻¹² = 10.69 μS"
-    - "B₂ = ω × C_sh = 1.162×10⁶ × 6.4×10⁻¹² = 7.44 μS"
-    - "Denominator: G² + (B₁+B₂)² = 328.3 + 328.1 = 656.4 μS²"
-    - "Re{Y} = 18.12 × 55.35 / 656.4 = 1.53 μS"
-    - "Im{Y} = 7.44 × [328.3 + 193.7] / 656.4 = 5.92 μS"
-    - "Y = 1.53 + j5.92 μS"
-    - "|Y| = 6.11 μS, |Z| = 163.6 kΩ"
-    - "φ_Y = atan(5.92/1.53) = 75.5°, φ_Z = -75.5°"
-    - "Z_spark = 163.6 kΩ ∠-75.5°"
-
-  answer_b: "C_mut = 9.2 pF, C_sh = 6.4 pF"
-  answer_c: "6.4 pF vs 8 pF empirical, ratio 0.8, acceptable"
-  answer_d: "R_opt_power = 55.2 kΩ, R_opt_phase = 71.9 kΩ"
-  answer_e: "163.6 kΩ ∠-75.5°"
-  tolerance: 3.0
-
-explanation: |
-  This complete workflow demonstrates lumped model extraction from FEMM. Key points:
-  (1) Matrix validation catches errors early, (2) Sign conversion is critical
-  (C_mut = |C₁₂|, not C₁₂), (3) FEMM values within factor 2 of empirical rules is
-  normal, (4) Both critical resistances are calculated for optimization, (5) Final
-  impedance is strongly capacitive (φ_Z = -75.5°) as expected. The 4-foot spark
-  shows typical behavior with r = C_mut/C_sh = 1.44, giving φ_Z,min ≈ -75°.
-
-related_concepts: ["FEMM-extraction", "lumped-model", "matrix-validation", "complete-workflow"]

phases/phase-01-restructuring.md

@@ -0,0 +1,91 @@
+# Phase 01: Project Restructuring
+
+**Date:** 2026-02-10
+**Status:** Complete
+
+## What Changed
+
+Restructured the project from a linear educational course (30 lessons, 18 exercises, PyQt app skeleton) into an evolving research knowledge base.
+
+### Motivation
+
+The original structure was built to support a Khan Academy-style desktop application. While the physics content was research-grade (91/100 quality assessment), the course format imposed:
+
+- **Fixed linear progression** - concepts locked into lesson sequence
+- **Pedagogical overhead** - difficulty levels, time estimates, grading tolerances, learning paths
+- **App coupling** - content structured around a PyQt viewer that was never completed
+- **Monolithic planning** - a single massive CLAUDE.md documenting every development decision
+
+What we actually need is a **living research system** - a knowledge graph where concepts link to each other, open questions are tracked, and new findings can be integrated without restructuring an entire curriculum.
+
+### What Was Done
+
+1. **Archived** all course scaffolding into `_archive/course/`:
+   - 30 lesson markdown files (4 parts, ~10,000 lines)
+   - 18 YAML exercise files (525 assessment points)
+   - course.json navigation structure
+   - PyQt app skeleton (15+ Python files)
+   - Learning paths, difficulty metadata, time estimates
+
+2. **Created** `context/` with ~15 coarse topic files:
+   - Each is a self-contained research document on one conceptual area
+   - YAML frontmatter with relationship metadata and cross-references
+   - `[[wiki-link]]` syntax for inline navigation
+   - `status` field (established / provisional / speculative)
+   - `source_sections` tracing back to spark-physics.txt
+
+3. **Preserved** all research content:
+   - `spark-physics.txt` unchanged at root (source of truth)
+   - 5 worked examples moved to `examples/`
+   - 22 matplotlib images + 15 placeholders moved to `assets/`
+   - Glossary (64 terms) updated to reference topic IDs
+   - Equations and physical bounds consolidated into `context/equations-and-bounds.md`
+
+4. **Reorganized** utilities:
+   - Image generation scripts moved to `tools/`
+   - One-time migration script in `tools/update_glossary.py`
+
+### Content Mapping
+
+| Previous (Course) | New (Knowledge Graph) |
+|---|---|
+| lessons/01-fundamentals/01-introduction.md thru 06 | context/circuit-topology.md |
+| lessons/02-optimization/01-02 | context/power-optimization.md |
+| lessons/02-optimization/03-05 | context/thevenin-method.md |
+| lessons/02-optimization/06 | context/coupled-resonance.md |
+| lessons/03-spark-physics/01 | context/field-thresholds.md |
+| lessons/03-spark-physics/02-04 | context/energy-and-growth.md |
+| lessons/03-spark-physics/05 | context/thermal-physics.md |
+| lessons/03-spark-physics/06 | context/streamers-and-leaders.md |
+| lessons/03-spark-physics/07 | context/capacitive-divider.md |
+| lessons/03-spark-physics/08 | context/empirical-scaling.md |
+| lessons/04-advanced-modeling/01-02 | context/lumped-model.md |
+| lessons/04-advanced-modeling/03-05 | context/distributed-model.md |
+| lessons/04-advanced-modeling/02,04 | context/femm-workflow.md |
+| reference/equation-sheet.md + physical-bounds.md | context/equations-and-bounds.md |
+| (scattered across lessons) | context/open-questions.md |
+
+### What Was NOT Changed
+
+- `spark-physics.txt` - untouched, remains source of truth
+- Physics content accuracy - all formulas and derivations preserved
+- Sign conventions - Maxwell matrix conventions preserved
+- Peak value convention - all phasors still use peak values
+
+## Metrics
+
+| Metric | Before | After |
+|---|---|---|
+| Content files | 30 lessons + 18 exercises | ~15 topic files |
+| Navigation | course.json (fixed tree) | Cross-references (graph) |
+| Entry point | Lesson 1 | Any topic file |
+| Adding content | Create lesson, update course.json, add exercise | Edit topic file or create new one |
+| Reference lookup | Separate equation-sheet.md | Inline in topic + combined reference |
+| Open questions | Buried in Part 12 | Dedicated topic file + per-concept tracking |
+
+## Next Steps
+
+- Expand topic files as research evolves
+- Split topics when they exceed ~25k tokens
+- Build expert agent backed by spark-physics.txt
+- Add new research phases as investigations proceed

phases/phase-6-qcw-community-research.md

@@ -0,0 +1,503 @@
+# Phase 6: QCW Spark Research — Community & Literature Survey
+
+**Date:** 2026-02-10
+**Method:** Web search across academic sources, builder documentation, and community forums
+**Sources searched:** highvoltageforum.net (~20 threads), loneoceans.com (4 builds), stevehv.4hv.org, richieburnett.co.uk, hotstreamer/deanostoybox, hackaday.io, kaizerpowerelectronics.dk, pupman.com/TCML, thaumati.com, connerlabs.org, academic papers (AIP, IEEE, arXiv, AGU)
+**Purpose:** Accumulate all available quantitative data on QCW spark behavior, validate against existing framework, identify new physics
+
+---
+
+## Summary of Findings
+
+Three search agents surveyed 30+ forum threads, 6 builder documentation sites, and several academic papers. The findings divide cleanly into: (1) high-confidence measured data, (2) well-supported community observations, (3) unresolved hypotheses, and (4) identified measurement gaps.
+
+The most significant finding is the **davekni voltage comparison**: a burst-mode DRSSTC at 80 kHz needs ~600 kV for 2-3 m arcs, while a QCW at 450 kHz achieves the same length at ~40 kV. This 15:1 voltage ratio proves that QCW sparks grow through sustained energy injection, not high voltage — directly validating the thermal persistence mechanism in the existing framework.
+
+---
+
+## 1. High-Confidence Measured Data
+
+### 1.1 QCW Secondary Voltage is LOW
+
+| Source | Measurement | Context |
+|--------|-------------|---------|
+| Steve Ward (via Uspring, HVF topic 1761) | 40 kV rising to 55 kV over ~5000 RF cycles | Arc growing to 50+ inches |
+| Loneoceans (via Steve Ward simulations) | 50-70 kV despite meter-length sparks | QCW v1.0 |
+| davekni (HVF topic 2397) | ~40 kV peak at 450 kHz QCW | 2-2.5 m arcs |
+| davekni (same source) | ~600 kV peak at 80 kHz burst DRSSTC | 2-3 m arcs |
+
+**Confidence:** HIGH — measured by multiple independent builders.
+
+**Physics implication:** QCW sparks grow through sustained energy injection over 10-20 ms, not through high instantaneous voltage. The voltage rise per RF cycle is only ~3 V/cycle. This is consistent with the framework's thermal persistence model: the spark extends because the leader channel persists between cycles and conducts energy to the tip, not because the voltage is high enough to bridge the gap in a single shot.
+
+**Contrast with burst DRSSTC:** The 15:1 voltage ratio (600 kV burst vs 40 kV QCW for similar spark lengths) is the single most important quantitative comparison in the dataset. It proves that voltage is necessary for inception but NOT for growth beyond the initial streamer reach.
+
+### 1.2 Steve Ward 80 us Burst Ceiling
+
+**Source:** Steve Ward, DRSSTC-0.5 (stevehv.4hv.org/DRSSTC.5.htm)
+
+| Spark Length | Input Power | ON Time |
+|-------------|-------------|---------|
+| 10 inches | 33 W | ~70 us |
+| 14 inches | 88 W | ~70 us |
+| 15 inches | 110 W | ~70 us |
+| 16 inches | 135 W | ~70 us |
+| 18 inches | 180 W | 70 us, 150 BPS |
+
+**Key observation:** "Gained almost no spark length after about 80 us of ON period."
+
+**Confidence:** HIGH — systematic measurement with controlled variables.
+
+**Physics implication:** This directly measures the burst-mode streamer growth saturation. After ~80 us, additional energy goes into re-heating decaying channels rather than new growth. This is consistent with tau_thermal ~ 0.1-0.2 ms for 100 um streamers — after one thermal time constant, the channels are cooling as fast as they're being heated. This is the fundamental wall that QCW overcomes by sustained drive.
+
+### 1.3 Loneoceans Frequency Tracking Data
+
+**Source:** Loneoceans QCW v1.0 (loneoceans.com/labs/qcw/)
+
+| Condition | Frequency | Shift from unloaded |
+|-----------|-----------|-------------------|
+| Unloaded secondary | 406-409 kHz | baseline |
+| With single toroid | ~392 kHz | -3.5% |
+| With two stacked toroids | 361 kHz | -11% |
+| With 50 cm simulated streamer | 349 kHz | -14% |
+| With 1 m simulated streamer | 310 kHz | -24% |
+| QCW v1.5 operating during spark | 413 → 377 kHz | -8.7% |
+
+**Confidence:** HIGH — measured with simulated streamers (physical wires of known length, not actual plasma).
+
+**Physics implication:** The simulated-streamer data provides clean calibration points for C_sh. A 1 m wire causes a 24% frequency shift, implying significant capacitive loading. The 8.7% shift during actual QCW operation (v1.5) is less than the simulated 1 m streamer shift, suggesting the effective capacitance of a real 1.78 m spark is less than that of a solid wire — consistent with the distributed, branched nature of real sparks having lower effective capacitance than a solid conductor.
+
+### 1.4 Loneoceans Build Comparison Data
+
+**QCW v1.5 (leader-dominated):**
+
+| Parameter | Value |
+|-----------|-------|
+| Spark length | 1.78 m (70+ inches) |
+| Secondary length | 5.55 inches |
+| Spark:secondary ratio | 13:1 |
+| Energy per pulse | 275 J |
+| Ramp duration | 22 ms (16-17 ms rise) |
+| Peak primary current | 145-160 A |
+| Coupling (k) | 0.38 |
+| Operating frequency | 413 → 377 kHz |
+
+**DRSSTC 3 (streamer-dominated, for comparison):**
+
+| Parameter | Value |
+|-----------|-------|
+| Spark length | 1.78-2.1 m |
+| Secondary length | 27.5 inches |
+| Spark:secondary ratio | 3:1 |
+| Burst pulse width | 70-135 us |
+| Peak primary current | 700-842 A |
+| Coupling (k) | 0.148 |
+| Operating frequency | 71.8-78.9 kHz |
+
+**Physics implication:** Same absolute spark length requires a 5x longer secondary, 5x higher peak current, and ~5x lower coupling in burst mode. The spark:secondary ratio difference (13:1 vs 3:1) is the most dramatic measure of QCW's advantage. QCW achieves this with 22 ms ramp vs 70-135 us burst — 200x longer pulse.
+
+### 1.5 Steve Conner Burst Efficiency Finding
+
+**Source:** Steve Conner (connerlabs.org), referenced on HVF and Kaizer guide.
+
+**Finding:** "Using a lower impedance tank circuit to draw higher peak power from the inverter, and shortening the burst length to maintain the same bang energy as before, gave longer sparks."
+
+Short bursts of high peak power grow sparks more efficiently than long bursts of low peak power. 100 us burst works better than 150 us for the same energy.
+
+**Confidence:** HIGH — reproducible across multiple builders.
+
+**Physics implication:** Consistent with the power optimization framework. Higher peak power pushes the initial streamer further before the 80 us ceiling hits. The streamer can explore more space in the first ~80 us of high-power drive than in 150 us of lower-power drive.
+
+### 1.6 VNTC Frequency Shift Under Loading
+
+**Source:** VNTC (HVF topic 701)
+
+| Condition | Power | Spark Length | RF Period (6 cycles) |
+|-----------|-------|-------------|---------------------|
+| Light corona | 50 W | 10 cm | 41 us |
+| Heavy spark | 500 W | 55 cm | 42 us |
+
+**Confidence:** HIGH — direct oscilloscope measurement.
+
+**Physics implication:** Only 2.5% frequency shift despite 10x power increase and 5.5x spark length increase. At ~146 kHz, spark loading is primarily resistive, not capacitive. The small frequency shift means C_sh change is modest relative to the total system capacitance — consistent with the capacitive divider model.
+
+### 1.7 Dr. Kilovolt (Jan Martis) SiC PSFB QCW
+
+**Source:** Dr. Kilovolt (Jan Martis), referenced on HVF and 4hv.org
+
+| Parameter | Value |
+|-----------|-------|
+| Topology | SiC PSFB (Phase-Shifted Full Bridge) |
+| Bus voltage | 800 V |
+| Coupling (k) | 0.55 |
+| Spark length | 2-2.5 m |
+| Peak power | ~40 kW |
+
+**Key innovations:**
+- SiC MOSFETs enable higher switching frequencies and efficiency
+- Phase-shifted full bridge topology provides inherently smooth power delivery (no pulse-skip artifacts) with a "1-cosine" transfer function
+- Coupling coefficient of 0.55 is among the highest documented, enabled by ferrite-assisted coupling
+
+**Environmental sensitivity observation:** Outdoor operation produces "looping" or "curving" streamers rather than straight swords under humid or cool conditions. This is consistent with the humidity/temperature effects documented in Section 2.8 — higher humidity enhances complex-ion recombination, reducing plasma persistence and disrupting the single straight channel.
+
+**Confidence:** HIGH — measured build parameters from an experienced builder.
+
+### 1.8 Duane B Secondary Voltage Measurement
+
+**Source:** HVF topic 1455
+
+| Parameter | Value |
+|-----------|-------|
+| Secondary inductance | 173.8 mH |
+| Secondary capacitance | 20.32 pF |
+| Frequency | 84.690 kHz |
+| Peak base current | 2.4 A |
+| Calculated peak voltage (before breakout) | 222 kV |
+
+Also referenced: Antonio Carlos M. de Queiroz data: 11.8" x 3.9" toroid reaching ~282 kV before sparking.
+
+**Confidence:** HIGH for the inductance/capacitance, MODERATE for the voltage (calculated, not directly measured).
+
+---
+
+## 2. Well-Supported Community Observations
+
+### 2.1 Frequency Threshold for Sword Sparks: 300-600 kHz
+
+**Sources (independent, concordant):**
+
+| Observer | Observation | Source |
+|----------|-------------|--------|
+| Mads Barnkob | "Sword characteristic shows above 400 kHz" | HVF topic 973 |
+| LabCoatz (Zach Armstrong) | Below 300 kHz: "chaotic and less straight"; above 600 kHz: "more curvy" | Hackaday |
+| Kaizer DRSSTC IV | QCW at ~100 kHz: "swirling" sparks, NOT straight | HVF topic 24 |
+| Fat Coil builder | "Above 350 kHz, plasma exhibits growth in straight segments" | TCML Nov 2014 |
+| Loneoceans SSTC3 | Straight sparks at 380-420 kHz | loneoceans.com |
+| Multiple QCW builders | All successful sword-spark builds operate 300-500 kHz | Build survey |
+
+**Confidence:** HIGH — 6+ independent observations converge on same frequency range.
+
+**Physics interpretation (new insight for framework):** The RF half-period at 400 kHz is 1.25 us. The thermal diffusion time for a 100 um streamer is ~125 us — 100x longer than the RF period. The channel effectively sees continuous heating with negligible cooling between RF cycles.
+
+At 50-100 kHz (half-period 5-10 us), thinner streamers (10-50 um, tau ~ 1-30 us) experience significant cooling between cycles, allowing the preferred conductive path to diffuse and branch. The channel cannot maintain a single preferred path.
+
+At >600 kHz, the observation of "curvy" sparks may relate to different physics (skin effect, displacement current dominance, or IGBT switching artifacts at extreme frequencies).
+
+### 2.2 Three Ramp Regimes
+
+**Source:** Loneoceans QCW v1.5 documentation.
+
+| Ramp Duration | Result | Interpretation |
+|---------------|--------|---------------|
+| Too short | "Gnarly, segmented sparks" | Insufficient time for leader transition |
+| Optimal (~10-20 ms) | Straight sword sparks | Leader forms and grows continuously |
+| Too long (>25 ms) | "Really hot and fat but bushy" without extra length | Leader reaches voltage-limited L_max; excess energy causes branching |
+
+**Confidence:** HIGH — direct observation with controlled ramp variation.
+
+**Physics interpretation:** The "too long" regime is particularly revealing. Once the leader reaches its voltage-limited length (set by the capacitive divider), additional energy has nowhere to go in the forward direction. The leader channel becomes very hot and fat (thicker → more C_sh → more voltage division → can't extend further). The excess energy drives branching because the field at the tip is below propagation threshold but the total power must be dissipated somewhere — lateral breakouts become the path.
+
+### 2.3 Pulse-Skip Modulation Does NOT Produce Sword Sparks
+
+**Sources:** Steve Ward, Steve Conner (2011), multiple builders on HVF.
+
+**Finding:** Multiple experimenters tried pulse-skip approaches (omitting RF cycles to modulate power) and "could not get the sword sparks."
+
+Steve Ward explanation: Smoothing ripples from missing pulses would require the coil to store excessive energy between cycles. Sword sparks need "relatively smooth/continuous modulation of the spark power with little ripple."
+
+**Confidence:** HIGH — reproduced failure across multiple independent builders.
+
+**Physics interpretation (revised):** The original interpretation ("gaps in energy delivery where the channel cools") was oversimplified. In actual DRSSTC pulse-skip implementations, the H-bridge shorts the primary tank during skip cycles (via GDT inversion or leg inhibit) while IGBTs continue switching synchronized to feedback. Primary current does not drop to zero — it decays gradually through the loaded Q. Phase coherence is maintained.
+
+The actual mechanism is **power envelope quality**: the sawtooth current envelope (bounded by the OCD threshold) delivers approximately constant average power, not the smooth quadratic ramp (P ~ V^2 from linear voltage ramp) that true QCW provides. Per-cycle jitter from the on-off-on switching pattern prevents clean single-channel dominance. This is a **continuum**: Bresenham-algorithm pulse-density modulation creating a linear ramp produces sparks that are "more sword-like but still branch" — intermediate between coarse pulse-skip and true analog QCW. The quadratic power profile is also difficult to achieve with pulse-density modulation.
+
+**Note:** Pulse-skip (bridge current control) is distinct from staccato (interrupter timing synchronized to AC mains). The Loneoceans SSTC3 staccato approach uses the rising AC mains waveform as a natural voltage ramp and does produce straight sparks at high frequency.
+
+### 2.4 QCW Growth Rate: ~170 m/s
+
+**Source:** HVF topic 973 (sword spark mechanism discussion), multiple contributors.
+
+**Derivation:** Arc propagation speed estimated at approximately half the speed of sound (~170 m/s).
+
+**Self-consistency check:** At 170 m/s over a 10 ms ramp, the spark grows 1.7 m. Over a 20 ms ramp, 3.4 m. These match observed QCW spark lengths (1-2 m for standard builds, 3.35 m for the Fat Coil).
+
+**Confidence:** MODERATE — visual estimate, not directly measured with high-speed camera + ruler.
+
+**Physics interpretation:** This is intermediate between free streamers (10^6 m/s) and natural lightning leaders (~10^4 m/s). This suggests a "driven leader" propagation mode unique to QCW: the leader advances continuously, fed by the circuit, at a rate limited by the thermal conversion of streamer to leader at the tip. The 170 m/s rate implies each "step" (streamer → heating → leader conversion) takes approximately:
+
+```
+step_length / growth_rate ~ 1 cm / 170 m/s ~ 60 us per step
+```
+
+This 60 us step time is consistent with the conductance relaxation heating time constant (tau_g = 40 us from Bazelyan) — the channel needs approximately one tau_g to heat up at each step.
+
+### 2.5 Coupling Requirements: k >= 0.3
+
+**Measured coupling coefficients across all documented QCW builds:**
+
+| Builder | k | Spark:secondary ratio | Notes |
+|---------|---|----------------------|-------|
+| Loneoceans v1.0 | 0.32-0.35 | 7.3:1 | Initial |
+| Loneoceans v1.5 (first) | 0.306 | — | Insufficient — breakthrough came at 0.38 |
+| Loneoceans v1.5 (final) | 0.38 | 13:1 | Breakthrough |
+| Loneoceans QCW2 | 0.365 | 10:1 | |
+| flyglas | 0.391 | ~12:1 | |
+| Lucasww | 0.44 | 10:1 | |
+| Rafft | 0.166-0.57 | — | Tested range |
+| Dr. Kilovolt (Jan Martis) | 0.55 | — | SiC PSFB, 2-2.5 m sparks |
+| davekni | 0.71 | — | Ferrite-assisted, highest documented |
+| Standard DRSSTC | 0.05-0.20 | 2-4:1 | For comparison |
+
+**Confidence:** HIGH — consistent across all builds.
+
+**Physics interpretation:** Higher coupling enables sufficient power transfer at the lower peak currents used in QCW (50-160 A vs 500-1000 A in burst DRSSTC). It also separates the pole frequencies further, making frequency tracking more robust against the shifting loaded pole. However, the Loneoceans SSTC3 (single-resonant, lower coupling) still produces sword sparks, suggesting k >= 0.3 is an engineering requirement (adequate power delivery) rather than a physics requirement (spark straightness).
+
+### 2.6 Spark-to-Secondary Ratios
+
+| Builder | Mode | Spark | Secondary | Ratio |
+|---------|------|-------|-----------|-------|
+| Steve Ward | Burst | 80" | 22" | 3.6:1 |
+| Loneoceans DRSSTC3 | Burst | 70" | 27.5" | 2.5:1 |
+| Loneoceans QCW v1.0 | QCW | 40" | 5.5" | 7.3:1 |
+| Lucasww | QCW | 51" | 5" | 10.2:1 |
+| Loneoceans QCW2 | QCW | 24" | 2.4" | 10:1 |
+| Loneoceans QCW v1.5 | QCW | 70+" | 5.55" | 12.6:1 |
+| Mathieu thm | QCW | 76" | 5.6" dia | 13.6:1 |
+| Fat Coil | QCW | 132" | 8" | 16.5:1 |
+
+**Confidence:** HIGH — measured across many builds.
+
+**Physics interpretation:** The 3-5x improvement in spark:secondary ratio from burst to QCW is a direct measure of the efficiency advantage of leader-dominated growth. Leaders extend the effective electrode (the conducting channel) continuously, so the secondary length (which constrains maximum voltage) becomes less important relative to the sustained power delivery.
+
+### 2.7 Richie Burnett Causality Reversal
+
+**Source:** richieburnett.co.uk/operatn2.html
+
+**Quote:** "It is not early quenching that produces good sparks, but rather good spark loading that leads to an early quench."
+
+**Confidence:** HIGH — well-reasoned analysis from a foundational figure.
+
+**Physics interpretation:** The causality runs: spark efficiently absorbs energy → secondary voltage drops → gap quenches (for SGTC) or primary current drops (for DRSSTC). This is the power optimization framework in action — the spark as a self-optimizing load.
+
+### 2.8 Environmental Effects on Straightness
+
+**Source:** davekni (HVF topic 2397)
+
+**Observation:** Straighter arcs in warm, dry conditions; curved/branchy arcs more common outdoors (cooler, more humid).
+
+**Confidence:** MODERATE — single observer, qualitative.
+
+**Physics interpretation:** Consistent with the humidity data in the framework. Higher humidity → faster complex-ion recombination (25x faster for hydrated ions) → shorter effective plasma lifetime → less thermal persistence → more branching. Lower temperature → higher gas density → higher E_propagation threshold → harder to sustain growth in a single channel.
+
+---
+
+## 3. Community Hypotheses (Unproven but Physically Plausible)
+
+### 3.1 Uspring's Sideways Breakout Suppression
+
+**Hypothesis:** QCW's slowly ramped voltage keeps tip voltage low, reducing the transverse electric field component. This suppresses lateral branching because the field is only strong enough for forward propagation along the lowest-impedance path (the existing hot channel).
+
+**Assessment:** Physically plausible but not tested. The existing hot leader channel does have much lower impedance than virgin air to the side, so a weak field would preferentially drive current forward. A strong field (as in burst mode) could overcome the impedance contrast and branch.
+
+### 3.2 Uspring's Temperature-Frequency Coupling
+
+**Hypothesis:** Higher operating frequency increases the time-averaged current density in the channel, raising its temperature and conductivity. A hotter channel needs less voltage to sustain, further reducing the branching field.
+
+**Assessment:** Partially supported by the frequency threshold data. The mechanism (more RF cycles per unit time = more Joule heating per unit time) is straightforward physics. Quantitative prediction: at 400 kHz, the Joule heating rate is ~4x higher than at 100 kHz for the same peak current, because there are 4x more half-cycles per millisecond.
+
+### 3.3 Channel Temperature: ~5000 K
+
+**Source:** Uspring (HVF topic 973), from conductivity analysis.
+
+**Assessment:** Not spectroscopically measured on TC sparks. However, the ~5000 K estimate is consistent with the leader temperature range in the Bazelyan framework (4000-6000 K for self-sustaining leaders). At 5000 K, associative ionization (N+O → NO+ + e) provides field-independent electron production, explaining why the channel self-sustains. This temperature is also consistent with the white/yellow visual appearance of QCW sword sparks (blackbody peak near 5000 K is in the visible range).
+
+### 3.4 Steve Ward's "2000 Small Sparks" Model
+
+**Source:** Steve Ward, multiple HVF threads.
+
+**Claim:** QCW sword sparks are "a series of small sparks (2000!) build up a longer and longer ionization channel and create the appearance of a single long spark."
+
+**Assessment:** This is a simplified description of the driven-leader mechanism. At 400 kHz over 5 ms, there are indeed ~2000 RF half-cycles, each depositing a small amount of energy. The "series of small sparks" view maps to the RF-cycle-by-cycle energy deposition that the conductance relaxation model (tau_g = 40 us) integrates over.
+
+---
+
+## 4. Identified Measurement Gaps
+
+The community itself has flagged these as unmeasured:
+
+1. **No direct arc current measurement on any QCW coil** (davekni: "Nobody has ever made arc current measurements for a QCW coil")
+2. **No spectroscopic temperature measurement of QCW sparks** — 5000 K is inferred, not measured
+3. **No time-resolved impedance measurement during QCW ramp** — the impedance trajectory during growth is unknown
+4. **No high-speed imaging correlated with electrical waveforms in QCW mode**
+5. **No measurement of energy per unit length (epsilon) for QCW sparks** — can only be bounded from total input energy and estimated system efficiency
+6. **Voltage gradient in TC sparks disputed** — Uspring estimates 1.5 kV/cm, Barnkob estimates 3 kV/cm
+7. **No systematic frequency sweep study** — same coil tested at 100, 200, 300, 400 kHz to isolate frequency effect
+
+---
+
+## 5. Academic Papers with TC-Relevant Data
+
+### 5.1 Brelet et al. (2014) — Laser-Guided Tesla Coil Discharges
+
+**Source:** Journal of Applied Physics, Ecole Polytechnique / ENSTA ParisTech / CNRS
+
+**Finding:** Plasma column resistance ~ **1 kilohm per meter** in laser-guided TC discharges at 100 kHz. Discharge length increased 5x with laser guiding. Mean breakdown field: 2 kV/cm for pre-ionized 1.8 m gap.
+
+**Caveat:** Laser-guided channels are pre-ionized, so resistance may be lower than self-propagating discharges.
+
+### 5.2 Briels et al. (Eindhoven) — Streamer Properties
+
+**Source:** Journal of Physics D / arXiv 0805.1376
+
+**Findings:** Positive streamer minimum diameter 0.2 mm, minimum velocity ~10^5 m/s at 5-20 kV, up to 1.2 × 10^6 m/s at 43-60 kV. Negative discharges form only glowing clouds at same voltages.
+
+**Confirms:** Streamer velocity hierarchy in the framework (10^5-10^6 m/s).
+
+### 5.3 Huang et al. (2020) — Leader Reillumination
+
+**Source:** Geophysical Research Letters
+
+**Finding:** After a waiting time, new discharge uses the thermal imprint of the old leader channel. Luminosity wave propagates from electrode at ~10^6 m/s.
+
+**TC relevance:** Direct evidence for the thermal persistence mechanism. At 400 kHz (2.5 us between cycles), the thermal imprint easily survives between RF half-cycles.
+
+---
+
+## 6. Key Numbers for Framework Integration
+
+### 6.1 QCW Operating Parameters (Consensus Ranges)
+
+| Parameter | QCW Range | Burst DRSSTC | Source |
+|-----------|-----------|--------------|--------|
+| Coupling (k) | 0.3-0.5+ | 0.05-0.2 | Build survey |
+| Operating frequency | 300-600 kHz | 50-110 kHz | Build survey |
+| Tank capacitance | 5-15 nF | 50-300 nF | Build survey |
+| Ramp duration | 10-22 ms | N/A (burst ~70-150 us) | Build survey |
+| Peak primary current | 50-200 A | 200-1000+ A | Build survey |
+| Secondary voltage | 40-70 kV | 200-600 kV | Ward, davekni |
+| Spark:secondary ratio | 7-16x | 2-4x | Build survey |
+| Growth rate | ~170 m/s | N/A (single-shot) | HVF estimate |
+
+### 6.2 Critical Time Comparisons
+
+| Timescale | Value | Significance |
+|-----------|-------|-------------|
+| RF half-period at 400 kHz | 1.25 us | Channel heating between cycles |
+| RF half-period at 100 kHz | 5 us | Channel heating between cycles |
+| Streamer tau_thermal (100 um) | ~125 us | 100x longer than RF period at 400 kHz |
+| Conductance tau_g (heating) | 40 us | Time to heat one "step" |
+| Conductance tau_g (cooling) | 200 us | 5x longer than heating |
+| Burst pulse duration | 70-150 us | Comparable to streamer tau |
+| QCW ramp duration | 10-22 ms | 100x longer than tau_g |
+| Streamer persistence | 1-5 ms | Exceeded by QCW ramp |
+| Leader transition time | 0.5-2 ms | Within QCW ramp, exceeds burst pulse |
+| Dark period cycle | 1-5 ms | Multiple cycles fit within QCW ramp |
+| Burst ceiling (Ward) | ~80 us | Streamer growth saturates |
+
+### 6.3 Energy Budget
+
+| Quantity | Value | Source |
+|----------|-------|--------|
+| QCW energy per pulse | 275 J (for 1.78 m) | Loneoceans v1.5 |
+| Apparent epsilon (total input / length) | 155 J/m | Derived |
+| Estimated system efficiency | 30-50% | Community consensus |
+| Estimated spark epsilon | 45-75 J/m | Derived (155 × 0.3-0.5) |
+| Burst DRSSTC energy per bang | 5-12 J | Steve Ward |
+| Burst DRSSTC average power | 33-180 W for 25-46 cm | Steve Ward DRSSTC-0.5 |
+
+### 6.4 New Insight: Driven Leader Step Time
+
+From the QCW growth rate of ~170 m/s and the typical leader step length of ~1 cm (Bazelyan):
+
+```
+step_time = step_length / growth_rate = 0.01 m / 170 m/s ~ 60 us
+```
+
+This 60 us step time is close to the conductance relaxation heating time constant (tau_g = 40 us), suggesting the leader advance rate is limited by how fast each new streamer segment can be heated to leader temperature. The 1.5x ratio (60 us observed vs 40 us tau_g) is reasonable given that the thermal transition also requires crossing the eta_T bottleneck.
+
+---
+
+## 7. Comparison with Existing Framework Predictions
+
+### 7.1 What the Framework Got Right
+
+- **Thermal persistence is THE key to QCW advantage** — confirmed by all data
+- **Streamer-to-leader transition requires sustained drive** — confirmed
+- **Capacitive voltage division limits spark length** — confirmed by frequency shift data
+- **Power optimization (hungry streamer)** — confirmed by Richie Burnett's causality insight and spark loading data
+- **Burst mode limited by streamer cooling** — confirmed by Steve Ward's 80 us ceiling
+
+### 7.2 What the Framework Missed
+
+1. **Frequency threshold for sword sparks (300-600 kHz)** — the framework discusses frequency effects on breakdown (field-thresholds.md Section 4.4, coupled-resonance.md Section 1.4) but does not predict or explain the sword-spark frequency threshold. The mechanism (RF period << streamer tau_thermal) is a straightforward extension of the existing thermal physics but was not explicitly stated.
+
+2. **QCW secondary voltage is low (40-70 kV)** — the framework implicitly assumed higher voltages for longer sparks. The data shows QCW works by sustained energy delivery at modest voltage.
+
+3. **Smooth, continuous drive is essential** — pulse-skip modulation fails to produce swords. The framework's growth model (dL/dt = P_stream / epsilon) does not distinguish between smooth and intermittent power delivery, but the physics requires truly continuous drive for leader maintenance.
+
+4. **Three ramp regimes** — the "too long" regime (bushy without length) is not predicted by the framework. It arises when the leader reaches voltage-limited L_max and excess energy drives lateral branching.
+
+5. **QCW growth rate (~170 m/s)** — this intermediate value between streamer and natural leader velocities was not predicted. It can now be derived from the framework: tau_g × step_length gives the right order of magnitude.
+
+### 7.3 What the Framework Got Slightly Wrong
+
+- **Leader formation voltage threshold (300-400 kV)** — this applies to single-shot impulses, NOT to QCW with sustained drive. QCW forms leaders at 40-70 kV topload voltage because the thermal ratcheting mechanism accumulates energy over thousands of cycles. The threshold should be stated as applying to single-event inception only.
+
+---
+
+## 8. Persons Index
+
+| Person | Handle | Role | Key Contribution |
+|--------|--------|------|-----------------|
+| Steve Ward | Steve Ward | QCW inventor | Quadratic power profile, 40-55 kV measurement, 80 us burst ceiling, DRSSTC design guide |
+| Richie Burnett | — | SSTC/DRSSTC pioneer | Spark loading causality reversal, pole splitting theory |
+| Terry Fritz | — | Spark loading modeler | 1 pF/ft streamer capacitance model, impedance framework |
+| Steve Conner | scopeboy | DRSSTC pioneer | 50 kohm impedance standard, "hungry streamer" principle, burst efficiency finding |
+| Gao Guangyan | loneoceans | Prolific documenter | Most detailed QCW measurements (4 builds), frequency tracking data, three ramp regimes, SSTC3 voltage-ramp isolation |
+| David Knierim | davekni | Physicist/engineer | Critical voltage comparison (600 kV burst vs 40 kV QCW for same length), fiber probe, oversized QCW |
+| Uspring | Uspring | Physicist | Temperature estimates (~5000 K), voltage gradient analysis, sword spark hypotheses |
+| Mads Barnkob | — | Kaizer/Admin | Frequency threshold observation (>400 kHz), voltage gradient estimate |
+| Zach Armstrong | LabCoatz | Builder | Frequency window (300-600 kHz), simplified staccato QCW |
+| Mathieu thm | — | Builder | 193 cm spark, 13.6x ratio record |
+| flyglas | — | Builder | 170 cm spark, flashover analysis |
+| Finn Hammer | hammertone | Builder | Ramp generator reference design |
+| Netzpfuscher | — | UD3 designer | Phase-shift QCW controller |
+| Jan Martis | Dr. Kilovolt | Builder | SiC PSFB QCW, k=0.55, 2-2.5 m sparks, ~40 kW peak, environmental sensitivity observations |
+| Anders Mikkelsen | — | Forum admin | Upper/lower pole guidance |
+| VNTC | — | Experimenter | 2.5% frequency shift measurement |
+
+---
+
+## 9. Source URLs
+
+### Academic Papers
+- Brelet et al. 2014: https://pubs.aip.org/aip/jap/article-abstract/116/1/013303/139184
+- Briels et al. (arXiv): https://arxiv.org/abs/0805.1376
+- Huang et al. 2020: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2019GL086183
+
+### Builder Documentation
+- Loneoceans QCW v1.0: https://www.loneoceans.com/labs/qcw/
+- Loneoceans QCW v1.5: https://www.loneoceans.com/labs/qcw15/
+- Loneoceans QCW2: https://www.loneoceans.com/labs/qcw2/
+- Loneoceans SSTC3: https://www.loneoceans.com/labs/sstc3/
+- Steve Ward DRSSTC-0.5: https://www.stevehv.4hv.org/DRSSTC.5.htm
+- Steve Ward DRSSTC design guide: https://www.stevehv.4hv.org/drsstc_design.htm
+- Steve Ward DRSSTC log book: https://www.stevehv.4hv.org/drsstc_log_book.htm
+- Richie Burnett operation: https://www.richieburnett.co.uk/operatn2.html
+- Terry Fritz impedance model: https://hotstreamer.deanostoybox.com/TeslaCoils/Misc/impedance/impedance.html
+- Kaizer DRSSTC IV: https://kaizerpowerelectronics.dk/tesla-coils/kaizer-drsstc-iv/
+- Hackaday LabCoatz: https://hackaday.io/project/184038-building-the-worlds-simplest-qcw-drsstc/details
+
+### Forum Threads (highvoltageforum.net)
+- Sword spark mechanism (topic 973): https://highvoltageforum.net/index.php?topic=973.0
+- Standard DRSSTC in QCW (topic 2361): https://highvoltageforum.net/index.php?topic=2361.0
+- QCW DRSSTC ideas (topic 1761): https://highvoltageforum.net/index.php?topic=1761.0
+- QCW flashover (topic 1988): https://highvoltageforum.net/index.php?topic=1988.0
+- QCW questions (topic 1914): https://highvoltageforum.net/index.php?topic=1914.100
+- Oversized QCW (topic 2397): https://highvoltageforum.net/index.php?topic=2397.0
+- My QCW DRSSTC (topic 2621): https://highvoltageforum.net/index.php?topic=2621.0
+- My first QCW (topic 3132): https://highvoltageforum.net/index.php?topic=3132.0
+- Spark voltage estimation (topic 123): https://highvoltageforum.net/index.php?topic=123.0
+- Frequency drop measurement (topic 701): https://highvoltageforum.net/index.php?topic=701.0
+- Fat Coil QCW (TCML): https://www.pupman.com/listarchives/2014/Nov/msg00064.php
+- Understanding DRSSTC (TCML): https://www.pupman.com/listarchives/2013/Feb/msg00052.html

phases/phase-8-bayesian-model-calibration.md

@@ -0,0 +1,389 @@
+# Phase 8: Bayesian Model Calibration — Experimental Measurement & Parameter Fitting
+
+**Date started:** 2026-02-10
+**Status:** Planning (hardware not yet built)
+**Method:** Build QCW Tesla coil, collect systematic measurements, fit dynamic threshold model via Bayesian inference
+**Purpose:** Empirically constrain the dynamic threshold parameters (delta_T, tau_buildup, epsilon, E_prop_floor) that currently have no measured values — transitioning the framework's T3/T4 claims toward T1/T2
+
+---
+
+## Motivation
+
+The spark physics framework is now theoretically mature: 17 context files, ~8,000 lines of interlinked physics, covering circuit topology through plasma dynamics. But several critical parameters remain unconstrained:
+
+| Parameter | Current knowledge | Tier | What measurement would give us |
+|---|---|---|---|
+| E_prop_effective at QCW leader tip | "much lower than 0.5 MV/m" | T3 | Actual value (or range) with uncertainty |
+| delta_T (thermal pre-conditioning) | "elevated, maybe 600-2000 K" | T3/T4 | Fitted value from spark length data |
+| tau_buildup (threshold ramp-up time) | "~ms scale, order of tau_thermal" | T3 | Fitted value constraining when threshold saturates |
+| epsilon during QCW ramp | 5-15 J/m (community estimates) | T2 | Time-resolved epsilon(t) with proper threshold model |
+| Power coupling to spark | Via Thevenin model | T3 | Validated against measured frequency/impedance shifts |
+
+The Bayesian approach is ideal here because:
+1. We have **strong physics-informed priors** from 8 months of framework development
+2. The model structure is known (forward simulation exists in `examples/spark-growth-timeline.md`)
+3. The measurements are accessible (primary current, frequency, spark length)
+4. Even a small dataset (10-20 operating points) dramatically constrains the posteriors when priors are informative
+
+---
+
+## The Forward Model
+
+### State Variables (per time step dt)
+
+```
+t         — time into QCW ramp [s]
+L(t)      — spark length [m]
+V_top(t)  — topload voltage [V]
+V_tip(t)  — spark tip voltage [V]
+E_tip(t)  — electric field at spark tip [V/m]
+E_prop(t) — effective propagation threshold [V/m]
+P(t)      — power delivered to spark [W]
+T_eff(t)  — effective gas temperature ahead of tip [K]
+```
+
+### Equations
+
+**1. Topload voltage** (from ramp profile + circuit model):
+```
+V_top(t) = V_max * envelope(t)
+
+For linear ramp: envelope(t) = t / T_ramp
+For Bresenham: envelope(t) = pulse-density modulated approximation
+```
+
+**2. Capacitive divider** (voltage at spark tip):
+```
+C_sh(t) = C_sh_per_meter * L(t)
+V_tip(t) = V_top(t) * C_mut / (C_mut + C_sh(t))
+```
+
+In the open-circuit limit. For finite R, use full complex impedance division (see `context/capacitive-divider.md`).
+
+**3. Tip electric field** (from FEMM lookup or approximate formula):
+```
+E_tip(t) = FEMM_lookup(V_tip(t), L(t), geometry)
+```
+
+The FEMM lookup is a precomputed table: E_tip as a function of V_tip and L for the specific coil geometry. This must be generated once per coil design. Alternatively, for initial work:
+```
+E_tip(t) ~ kappa * V_tip(t) / effective_gap(L(t))
+```
+where effective_gap and kappa are geometry-dependent. This approximation is known to be inaccurate (Section 3.4 of field-thresholds.md) but may suffice for initial fitting if kappa is treated as a free parameter.
+
+**4. Dynamic propagation threshold** (THE NEW PART):
+```
+T_eff(t) = T_0 + delta_T * (1 - exp(-t_active / tau_buildup))
+
+E_prop(t) = max(E_prop_floor, E_prop_cold * T_0 / T_eff(t))
+```
+
+where:
+- `T_0` = 300 K (ambient)
+- `delta_T` = temperature elevation at saturation [K] — **FIT PARAMETER**
+- `tau_buildup` = time constant for threshold development [s] — **FIT PARAMETER**
+- `E_prop_cold` = cold-air propagation threshold [V/m] — **FIT PARAMETER** (prior: 0.5 MV/m)
+- `E_prop_floor` = minimum achievable threshold [V/m] — **FIT PARAMETER** (prior: weakly constrained, >0)
+- `t_active` = time since leader formation at the tip (resets if spark stalls and re-ignites)
+
+NOTE: This single-exponential "effective temperature" model is a deliberate simplification. It collapses all four physical mechanisms (UV, thermal, residual ionization, expansion) into one proxy variable. This is appropriate for initial fitting. If the data demand it, the model can be expanded later (see Section "Model Extensions" below).
+
+**5. Growth condition and rate:**
+```
+if E_tip(t) > E_prop(t):
+    dL/dt = P(t) / epsilon
+else:
+    dL/dt = 0    (stalled)
+```
+
+**6. Power delivered to spark** (Thevenin model):
+```
+P(t) = 0.5 * |V_th|^2 * R / |Z_th + Z_spark|^2
+```
+
+where Z_spark depends on R, C_mut, C_sh (all functions of L). For initial work, can approximate P from measured input power * efficiency factor.
+
+**7. Epsilon:**
+```
+epsilon(t) = epsilon_0 / (1 + alpha_accum * integral_0^t P(t') dt')
+```
+
+Or simpler: `epsilon = epsilon_0` (constant). Start with constant, add accumulation if data require it.
+
+**8. Update:**
+```
+L(t + dt) = L(t) + dL/dt * dt
+```
+
+### Parameter Vector
+
+```
+theta = {
+    E_prop_cold,     — cold-air propagation threshold [V/m]
+    delta_T,         — thermal pre-conditioning saturation [K]
+    tau_buildup,     — threshold development time constant [s]
+    E_prop_floor,    — minimum propagation threshold [V/m]
+    epsilon_0,       — energy per meter [J/m]
+    kappa_eff,       — effective tip enhancement (if not using FEMM)
+    sigma_L          — measurement noise on spark length [m]
+}
+```
+
+7 parameters. With 20+ data points and informative priors, this is well-determined.
+
+---
+
+## Priors
+
+| Parameter | Prior Distribution | Physical Justification |
+|---|---|---|
+| E_prop_cold | Normal(0.5 MV/m, 0.15 MV/m) | Bazelyan & Raizer streamer propagation field [T1] |
+| delta_T | LogNormal(ln(700), 0.8) | Must be >0; 600-2000 K range from Paschen scaling argument [T3] |
+| tau_buildup | LogNormal(ln(2 ms), 0.7) | Should be ~tau_thermal scale for thin leader (~ms) [T3] |
+| E_prop_floor | Uniform(0.01, 0.3 MV/m) | Must be >0 and < E_prop_cold; weakly constrained [T4] |
+| epsilon_0 | LogNormal(ln(10), 0.5) | Community QCW data: 5-15 J/m [T2] |
+| kappa_eff | LogNormal(ln(3), 0.4) | 2-5 from FEMM studies [T1]; broader if no FEMM available |
+| sigma_L | HalfNormal(0.1 m) | Measurement precision of spark length |
+
+The priors encode everything the framework has established. The posterior will show how much the data pull the parameters away from the physics predictions.
+
+---
+
+## Experimental Design
+
+### The Coil
+
+Build a QCW-capable DRSSTC with the following features:
+
+**Required capabilities:**
+- QCW ramp with adjustable duration (2-30 ms range)
+- Adjustable power level (bus voltage or duty cycle control)
+- Repeatable operation (same conditions, multiple shots)
+- Frequency: 200-500 kHz range (higher is better for sword formation)
+
+**Required instrumentation:**
+- Primary current transformer (Pearson or Rogowski) → oscilloscope
+- Bus voltage measurement
+- Spark length measurement (ruler/scale + camera, or laser reference)
+- High-speed camera if available (but not required for initial fitting)
+
+**Nice to have (future phases):**
+- Antenna/capacitive probe for secondary voltage estimation
+- Spectrometer for channel temperature measurement
+- Multiple secondaries for frequency comparison
+
+### Measurement Protocol
+
+#### Experiment 1: Ramp Duration Sweep (highest priority)
+
+**What:** Fix power level (constant bus voltage), vary ramp duration from 3 ms to 25 ms in ~8 steps.
+
+**Measurements per point:** Repeat each condition 5-10 times. Record:
+- Ramp duration T_ramp (control variable, exact)
+- Spark length L_final (from photo/video, +/- 5 cm)
+- Primary current waveform (oscilloscope capture → infer V_topload, frequency shift)
+- Qualitative morphology (photo: sword vs branchy)
+
+**Why this is the highest-priority experiment:** The L_final vs T_ramp curve directly constrains delta_T and tau_buildup nearly independently of other parameters. At short ramps, E_prop_eff is near the cold-air value and L is short. At long ramps, E_prop_eff saturates and L plateaus. The transition region pins tau_buildup.
+
+Expected data shape:
+```
+L_final
+  |           ___________
+  |          /            \  (lateral breakouts begin here)
+  |         /
+  |        /
+  |       /
+  |      /
+  |_____/
+  |
+  +-----|---|---|---|----> T_ramp
+       3   8   15  20  25 ms
+```
+
+#### Experiment 2: Power Level Sweep
+
+**What:** Fix ramp duration at the optimal value found in Experiment 1, vary bus voltage (power) in ~5 steps.
+
+**Measurements:** Same as Experiment 1.
+
+**What it constrains:** The power dependence of the dynamic threshold. If L_final increases with power at fixed ramp time (beyond what the simple Thevenin model predicts from higher V_topload), this is direct evidence for the coupled voltage-power limit.
+
+#### Experiment 3: Frequency Comparison (if feasible)
+
+**What:** Build or borrow a second secondary with significantly different resonant frequency. Run both at the same power and ramp time.
+
+**What it constrains:** Frequency dependence of tau_buildup and thermal ratchet. Higher frequency should produce longer swords at the same power (more RF cycles per tau_g). This is the least practical experiment but the most diagnostic.
+
+### Data Format
+
+Store measurements in a structured file (CSV or YAML):
+
+```yaml
+# measurements/run-001.yaml
+coil_id: "qcw-v1"
+date: "2026-XX-XX"
+conditions:
+  bus_voltage_V: 340
+  ramp_duration_ms: 12
+  frequency_kHz: 380
+  repetition_rate_Hz: 5
+  ambient_temp_C: 22
+  humidity_pct: 45
+measurements:
+  - trial: 1
+    spark_length_m: 1.45
+    morphology: "sword"
+    notes: "clean single channel"
+    scope_file: "scope_001.csv"
+  - trial: 2
+    spark_length_m: 1.52
+    morphology: "sword"
+    notes: "slight fork at tip"
+    scope_file: "scope_002.csv"
+```
+
+---
+
+## Fitting Pipeline
+
+### Software Stack
+
+- **Python 3.10+** with NumPy, SciPy
+- **emcee** or **PyMC** for MCMC sampling
+- **corner** for posterior visualization
+- **matplotlib** for diagnostic plots
+- Forward model implemented as a standalone Python function
+
+### Pipeline Steps
+
+```
+1. Load coil geometry → precompute FEMM lookup table (or set kappa_eff)
+2. Load measurement data (L_final, T_ramp, V_bus, f_RF for each trial)
+3. Define forward model: theta → L_predicted(T_ramp, V_bus, f_RF)
+4. Define log-likelihood: sum over trials of Normal(L_measured | L_predicted, sigma_L)
+5. Define log-prior: product of prior distributions on theta
+6. Run MCMC (emcee: 32 walkers, 5000 steps, 1000 burn-in)
+7. Check convergence (Gelman-Rubin, effective sample size, trace plots)
+8. Extract posteriors: median + 68% credible intervals for each parameter
+9. Posterior predictive check: does the fitted model predict HELD-OUT data?
+10. Model comparison: compute Bayes factor vs fixed-threshold model
+```
+
+### Key Diagnostic Checks
+
+**Before fitting:**
+- Does the forward model reproduce `examples/spark-growth-timeline.md` with its stated parameters?
+- Do the priors span the range of plausible outcomes?
+
+**After fitting:**
+- Are posteriors well-constrained (not just reflecting priors)?
+- Do trace plots show convergence (no drift, good mixing)?
+- Does the posterior predictive distribution cover the observed data?
+- Are there strong parameter correlations? (delta_T vs tau_buildup will likely be correlated)
+
+**Validation:**
+- Hold out 20% of data for prediction testing
+- Does the model predict spark length for a NEW operating condition?
+- Does the fixed-threshold model (delta_T = 0) fit significantly worse?
+
+---
+
+## Model Extensions (if data demand them)
+
+The single-exponential effective temperature model is the simplest version. If residuals show systematic structure, consider:
+
+### Extension 1: Power-dependent delta_T
+```
+delta_T(P) = delta_T_0 * (P / P_ref)^gamma
+```
+Adds one parameter (gamma). Tests whether more power produces more pre-conditioning.
+
+### Extension 2: Two-timescale buildup
+```
+T_eff(t) = T_0 + delta_T_fast * (1 - exp(-t/tau_fast)) + delta_T_slow * (1 - exp(-t/tau_slow))
+```
+Adds two parameters. Captures the possibility that UV/residual ionization (fast, ~100 us) and thermal diffusion (slow, ~ms) have distinct timescales.
+
+### Extension 3: Time-varying epsilon
+```
+epsilon(t) = epsilon_0 / (1 + alpha * integral P dt)
+```
+Adds one parameter (alpha). Tests whether epsilon decreases as thermal energy accumulates.
+
+### Extension 4: Stochastic L_final
+```
+L_final ~ Normal(L_predicted, sigma_L(L)) where sigma_L = sigma_0 + sigma_1 * L
+```
+Longer sparks have more variability. Adds one parameter (sigma_1).
+
+Start with the base model. Only add extensions if the Bayes factor favors the more complex model (Occam's razor built into the framework).
+
+---
+
+## What Success Looks Like
+
+### Minimum viable result:
+- 10+ data points (ramp sweep at one power level)
+- Posteriors on delta_T and tau_buildup that are meaningfully tighter than priors
+- Forward model predicts held-out data to within +/- 15%
+
+### Strong result:
+- 30+ data points (ramp sweep + power sweep)
+- delta_T constrained to +/- 30%: first empirical measurement of the dynamic threshold magnitude
+- tau_buildup constrained to +/- factor of 2: first measurement of how fast the threshold develops
+- Bayes factor >10 favoring dynamic threshold model over fixed threshold
+- Posterior predictive check passes for a new operating condition
+
+### Exceptional result:
+- All of the above plus frequency comparison data
+- Spectroscopic temperature measurement ahead of leader tip
+- Extension models tested (power-dependent delta_T, two-timescale)
+- Published or shared with TC community with enough detail for replication
+
+---
+
+## Connection to Context Files
+
+This phase directly addresses open questions in:
+- **`context/field-thresholds.md`** Section 4.7: "The effective E_propagation at a QCW leader tip should be directly measurable"
+- **`context/open-questions.md`** Section 1.3: thermal evolution, spectroscopic temperature
+- **`context/energy-and-growth.md`**: "Can epsilon be predicted from first principles without calibration?"
+- **`context/qcw-operation.md`**: measurement gaps (arc current, time-resolved impedance)
+
+Results will be integrated back into context files as the data come in:
+- Measured parameter values → equations-and-bounds.md (new entries with T1 tags)
+- Dynamic threshold validation → field-thresholds.md Section 4.7 (upgrade T3 claims to T1/T2)
+- Calibrated epsilon → energy-and-growth.md (refine bounds)
+- Model code → tools/ directory
+
+---
+
+## Timeline
+
+This is an ongoing research phase. Approximate milestones:
+
+| Milestone | Description |
+|---|---|
+| M1: Coil design | Select topology, order components, design secondary for target frequency |
+| M2: Coil build | Assemble and test at low power, verify basic operation |
+| M3: Instrumentation | Set up current probe, scope capture, camera, measurement protocol |
+| M4: FEMM model | Build electrostatic model of the specific coil geometry, generate E_tip lookup table |
+| M5: Forward model code | Implement the time-stepped growth simulation in Python |
+| M6: First data | Ramp duration sweep (Experiment 1), ~8 operating points × 5 trials |
+| M7: First fit | Run Bayesian fitting on Experiment 1 data, check convergence, examine posteriors |
+| M8: Power sweep | Experiment 2 data collection and fitting |
+| M9: Validation | Predict spark length for a new operating condition, compare to measurement |
+| M10: Integration | Update context files with measured values, upgrade tier tags |
+
+No time estimates — this depends on hardware availability and build schedule. Each milestone is independently useful: M5 (forward model code) is valuable even without measurement data, and M6-M7 (first fit) answers the core question regardless of whether later experiments are completed.
+
+---
+
+## References
+
+- Foreman-Mackey et al. (2013), "emcee: The MCMC Hammer," PASP 125, 306-312
+- Gelman et al. (2013), "Bayesian Data Analysis," 3rd edition, CRC Press
+- Existing framework: `context/field-thresholds.md` Section 4.7 (dynamic threshold theory)
+- Existing forward model: `examples/spark-growth-timeline.md` (QCW growth simulation)
+- Phase 6 QCW survey: `phases/phase-6-qcw-community-research.md` (community data and priors)

spark-lessons/reference/glossary.yaml → reference/glossary.yaml

@@ -10,7 +10,7 @@ terms:
     unit: "pF or F"
     typical_range: "3-15 pF for 1-5 foot sparks"
     related_terms: ["C_sh", "capacitance_matrix", "r"]
-    related_lessons: ["fund-02", "model-02"]
+    related_topics: ["circuit-topology", "femm-workflow"]
 
   - term: "C_sh"
     full_name: "Shunt Capacitance"
@@ -21,7 +21,7 @@ terms:
     unit: "pF or F"
     typical_range: "2 pF per foot of spark length"
     related_terms: ["C_mut", "capacitance_matrix"]
-    related_lessons: ["fund-02", "model-02"]
+    related_topics: ["circuit-topology", "femm-workflow"]
 
   - term: "r"
     full_name: "Capacitance Ratio"
@@ -31,7 +31,7 @@ terms:
     unit: "dimensionless"
     typical_range: "0.1 to 2.0 for typical geometries"
     related_terms: ["C_mut", "C_sh", "phi_Z_min"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "R_opt_power"
     full_name: "Optimal Resistance for Power Transfer"
@@ -42,7 +42,7 @@ terms:
     unit: "Ω (ohms)"
     typical_range: "20-200 kΩ for typical DRSSTC frequencies"
     related_terms: ["R_opt_phase", "hungry_streamer", "G"]
-    related_lessons: ["fund-02", "model-01"]
+    related_topics: ["circuit-topology", "lumped-model"]
 
   - term: "R_opt_phase"
     full_name: "Optimal Resistance for Phase"
@@ -53,7 +53,7 @@ terms:
     unit: "Ω (ohms)"
     typical_range: "40-400 kΩ for typical DRSSTC frequencies"
     related_terms: ["R_opt_power", "phi_Z_min"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "phi_Z"
     full_name: "Impedance Phase Angle"
@@ -64,7 +64,7 @@ terms:
     unit: "degrees or radians"
     typical_range: "-55° to -75° at R_opt_power"
     related_terms: ["theta_Y", "Y", "phi_Z_min"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "phi_Z_min"
     full_name: "Minimum Impedance Phase Angle"
@@ -75,7 +75,7 @@ terms:
     unit: "degrees or radians"
     typical_range: "-50° to -70° for typical geometries"
     related_terms: ["r", "phi_Z", "R_opt_phase"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "theta_Y"
     full_name: "Admittance Phase Angle"
@@ -85,7 +85,7 @@ terms:
     unit: "degrees or radians"
     typical_range: "+55° to +75°"
     related_terms: ["phi_Z", "Y"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "Y"
     full_name: "Admittance"
@@ -96,7 +96,7 @@ terms:
     unit: "S (siemens)"
     typical_range: "10-100 μS for typical sparks"
     related_terms: ["G", "B", "Z", "phi_Z"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "G"
     full_name: "Conductance"
@@ -106,7 +106,7 @@ terms:
     unit: "S (siemens)"
     typical_range: "5-100 μS"
     related_terms: ["Y", "R", "B"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "B"
     full_name: "Susceptance"
@@ -116,7 +116,7 @@ terms:
     unit: "S (siemens)"
     typical_range: "10-200 μS"
     related_terms: ["Y", "G", "C_mut", "C_sh"]
-    related_lessons: ["fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "Z_th"
     full_name: "Thévenin Impedance"
@@ -127,7 +127,7 @@ terms:
     unit: "Ω (ohms)"
     typical_range: "10-100 kΩ"
     related_terms: ["V_th", "P_load", "Thevenin_equivalent"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "V_th"
     full_name: "Thévenin Voltage"
@@ -138,7 +138,7 @@ terms:
     unit: "V (volts)"
     typical_range: "100-600 kV peak for typical DRSSTCs"
     related_terms: ["Z_th", "P_load", "Thevenin_equivalent"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "P_load"
     full_name: "Power to Load"
@@ -149,7 +149,7 @@ terms:
     unit: "W (watts)"
     typical_range: "100 W to 5 kW"
     related_terms: ["Z_th", "V_th", "P_max"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "P_max"
     full_name: "Maximum Theoretical Power"
@@ -160,7 +160,7 @@ terms:
     unit: "W (watts)"
     typical_range: "200 W to 10 kW"
     related_terms: ["P_load", "Z_th", "V_th", "conjugate_match"]
-    related_lessons: ["fund-03"]
+    related_topics: ["thevenin-method"]
 
   - term: "E_inception"
     full_name: "Inception Electric Field"
@@ -171,7 +171,7 @@ terms:
     unit: "V/m or MV/m"
     typical_range: "2-3 MV/m at sea level for smooth topload"
     related_terms: ["E_propagation", "E_tip"]
-    related_lessons: ["phys-01"]
+    related_topics: ["field-thresholds"]
 
   - term: "E_propagation"
     full_name: "Propagation Electric Field"
@@ -182,7 +182,7 @@ terms:
     unit: "V/m or MV/m"
     typical_range: "0.4-1.0 MV/m at sea level"
     related_terms: ["E_inception", "E_tip", "dL_dt"]
-    related_lessons: ["phys-01", "model-03"]
+    related_topics: ["field-thresholds", "distributed-model"]
 
   - term: "E_tip"
     full_name: "Tip Electric Field"
@@ -193,7 +193,7 @@ terms:
     unit: "V/m or MV/m"
     typical_range: "0.5-2 MV/m during growth"
     related_terms: ["E_propagation", "kappa", "E_inception"]
-    related_lessons: ["phys-01", "model-03"]
+    related_topics: ["field-thresholds", "distributed-model"]
 
   - term: "kappa"
     full_name: "Tip Enhancement Factor"
@@ -203,7 +203,7 @@ terms:
     unit: "dimensionless"
     typical_range: "2-5 for cylindrical channels"
     related_terms: ["E_tip"]
-    related_lessons: ["phys-01"]
+    related_topics: ["field-thresholds"]
 
   - term: "epsilon"
     full_name: "Energy per Meter"
@@ -214,7 +214,7 @@ terms:
     unit: "J/m (joules per meter)"
     typical_range: "5-15 J/m (QCW), 20-40 J/m (hybrid), 30-100+ J/m (burst)"
     related_terms: ["dL_dt", "P_stream", "operating_mode"]
-    related_lessons: ["phys-01", "model-03"]
+    related_topics: ["energy-and-growth", "field-thresholds", "distributed-model"]
 
   - term: "dL_dt"
     full_name: "Growth Rate"
@@ -224,7 +224,7 @@ terms:
     unit: "m/s"
     typical_range: "1-100 m/s for leaders, up to 10⁶ m/s for streamers"
     related_terms: ["epsilon", "P_stream", "E_propagation"]
-    related_lessons: ["phys-01", "model-03"]
+    related_topics: ["energy-and-growth", "field-thresholds", "distributed-model"]
 
   - term: "P_stream"
     full_name: "Power to Streamer/Spark"
@@ -234,7 +234,7 @@ terms:
     unit: "W (watts)"
     typical_range: "50 W to 5 kW"
     related_terms: ["dL_dt", "epsilon", "P_load"]
-    related_lessons: ["phys-01", "model-01"]
+    related_topics: ["energy-and-growth", "power-optimization", "lumped-model"]
 
   - term: "tau_thermal"
     full_name: "Thermal Time Constant"
@@ -245,7 +245,7 @@ terms:
     unit: "s (seconds)"
     typical_range: "0.1-0.2 ms (thin streamers), 300-600 ms (thick leaders)"
     related_terms: ["d", "alpha", "thermal_persistence"]
-    related_lessons: ["phys-01"]
+    related_topics: ["thermal-physics"]
 
   - term: "d"
     full_name: "Channel Diameter"
@@ -256,7 +256,7 @@ terms:
     unit: "m (meters)"
     typical_range: "10-100 μm (streamers), 1-10 mm (leaders)"
     related_terms: ["tau_thermal", "streamer", "leader", "C"]
-    related_lessons: ["phys-01", "model-02"]
+    related_topics: ["thermal-physics", "streamers-and-leaders", "femm-workflow"]
 
   - term: "alpha"
     full_name: "Thermal Diffusivity"
@@ -266,7 +266,7 @@ terms:
     unit: "m²/s"
     typical_range: "2×10⁻⁵ m²/s for air"
     related_terms: ["tau_thermal", "k", "rho", "c_p"]
-    related_lessons: ["phys-01"]
+    related_topics: ["thermal-physics"]
 
   - term: "streamer"
     full_name: "Streamer Discharge"
@@ -277,7 +277,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["leader", "epsilon", "d", "transition"]
-    related_lessons: ["phys-01"]
+    related_topics: ["streamers-and-leaders", "field-thresholds"]
 
   - term: "leader"
     full_name: "Leader Discharge"
@@ -288,7 +288,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["streamer", "epsilon", "d", "transition", "thermal_ionization"]
-    related_lessons: ["phys-01"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
 
   - term: "transition"
     full_name: "Streamer-to-Leader Transition"
@@ -299,7 +299,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["streamer", "leader", "Joule_heating", "QCW"]
-    related_lessons: ["phys-01"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
 
   - term: "hungry_streamer"
     full_name: "Hungry Streamer Principle"
@@ -310,7 +310,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["R_opt_power", "self_optimization", "plasma_equilibrium"]
-    related_lessons: ["fund-02", "model-01"]
+    related_topics: ["circuit-topology", "lumped-model"]
 
   - term: "capacitive_divider"
     full_name: "Capacitive Divider Effect"
@@ -321,7 +321,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["C_mut", "C_sh", "V_tip", "E_tip"]
-    related_lessons: ["phys-01", "model-03"]
+    related_topics: ["capacitive-divider", "distributed-model"]
 
   - term: "V_tip"
     full_name: "Tip Voltage"
@@ -331,7 +331,7 @@ terms:
     unit: "V (volts)"
     typical_range: "50-300 kV during growth"
     related_terms: ["capacitive_divider", "V_topload", "E_tip"]
-    related_lessons: ["phys-01", "model-03"]
+    related_topics: ["capacitive-divider", "distributed-model"]
 
   - term: "Maxwell_matrix"
     full_name: "Maxwell Capacitance Matrix"
@@ -342,7 +342,7 @@ terms:
     unit: "F (farads)"
     typical_range: "pF scale for Tesla coils"
     related_terms: ["C_mut", "C_sh", "FEMM", "extraction"]
-    related_lessons: ["model-02"]
+    related_topics: ["femm-workflow"]
 
   - term: "FEMM"
     full_name: "Finite Element Method Magnetics"
@@ -353,7 +353,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["Maxwell_matrix", "C_mut", "C_sh", "E_field"]
-    related_lessons: ["model-02"]
+    related_topics: ["femm-workflow"]
 
   - term: "QCW"
     full_name: "Quasi-Continuous Wave"
@@ -364,7 +364,7 @@ terms:
     unit: "N/A"
     typical_range: "5-20 ms ramp times"
     related_terms: ["burst_mode", "epsilon", "leader", "transition"]
-    related_lessons: ["phys-01", "model-03"]
+    related_topics: ["energy-and-growth", "coupled-resonance"]
 
   - term: "burst_mode"
     full_name: "Burst Mode"
@@ -375,7 +375,7 @@ terms:
     unit: "N/A"
     typical_range: "50-500 μs pulse widths"
     related_terms: ["QCW", "epsilon", "streamer"]
-    related_lessons: ["phys-01"]
+    related_topics: ["energy-and-growth", "coupled-resonance"]
 
   - term: "DRSSTC"
     full_name: "Dual Resonant Solid State Tesla Coil"
@@ -386,7 +386,7 @@ terms:
     unit: "N/A"
     typical_range: "50-400 kHz operating frequency"
     related_terms: ["QCW", "burst_mode", "coupled_resonance"]
-    related_lessons: ["fund-01"]
+    related_topics: ["circuit-topology"]
 
   - term: "ringdown_method"
     full_name: "Ringdown Measurement Method"
@@ -397,7 +397,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["Q_L", "G_total", "C_eq", "measurement"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "Q_L"
     full_name: "Loaded Quality Factor"
@@ -408,7 +408,7 @@ terms:
     unit: "dimensionless"
     typical_range: "5-50 with spark, 100-500 unloaded"
     related_terms: ["Q_0", "R_p", "G_total", "ringdown_method"]
-    related_lessons: ["fund-03"]
+    related_topics: ["thevenin-method"]
 
   - term: "Q_0"
     full_name: "Unloaded Quality Factor"
@@ -418,7 +418,7 @@ terms:
     unit: "dimensionless"
     typical_range: "100-500 for typical secondaries"
     related_terms: ["Q_L", "secondary_losses"]
-    related_lessons: ["fund-03"]
+    related_topics: ["thevenin-method"]
 
   - term: "C_eq"
     full_name: "Equivalent Capacitance"
@@ -429,7 +429,7 @@ terms:
     unit: "pF or F"
     typical_range: "20-100 pF for typical coils"
     related_terms: ["Q_L", "frequency_shift", "C_0"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "R_p"
     full_name: "Parallel Equivalent Resistance"
@@ -439,7 +439,7 @@ terms:
     unit: "Ω (ohms)"
     typical_range: "5-50 kΩ with typical spark"
     related_terms: ["Q_L", "G_total", "C_eq"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "G_total"
     full_name: "Total Conductance"
@@ -450,7 +450,7 @@ terms:
     unit: "S (siemens)"
     typical_range: "20-200 μS with spark"
     related_terms: ["R_p", "Q_L", "G", "G_0"]
-    related_lessons: ["fund-03"]
+    related_topics: ["thevenin-method"]
 
   - term: "omega"
     full_name: "Angular Frequency"
@@ -460,7 +460,7 @@ terms:
     unit: "rad/s"
     typical_range: "3×10⁵ to 2×10⁶ rad/s for typical Tesla coils"
     related_terms: ["f", "B", "X_C", "X_L"]
-    related_lessons: ["fund-01", "fund-02"]
+    related_topics: ["circuit-topology"]
 
   - term: "f"
     full_name: "Frequency"
@@ -470,7 +470,7 @@ terms:
     unit: "Hz"
     typical_range: "50-400 kHz for typical Tesla coils"
     related_terms: ["omega", "f_0", "f_L", "frequency_shift"]
-    related_lessons: ["fund-01"]
+    related_topics: ["circuit-topology"]
 
   - term: "f_0"
     full_name: "Unloaded Frequency"
@@ -480,7 +480,7 @@ terms:
     unit: "Hz"
     typical_range: "50-400 kHz"
     related_terms: ["f_L", "C_0", "L"]
-    related_lessons: ["fund-01", "fund-03"]
+    related_topics: ["circuit-topology", "thevenin-method"]
 
   - term: "f_L"
     full_name: "Loaded Frequency"
@@ -490,7 +490,7 @@ terms:
     unit: "Hz"
     typical_range: "5-20% lower than f_0"
     related_terms: ["f_0", "C_eq", "frequency_shift"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "frequency_shift"
     full_name: "Frequency Shift with Loading"
@@ -500,7 +500,7 @@ terms:
     unit: "Hz or %"
     typical_range: "5-20% decrease typical"
     related_terms: ["f_0", "f_L", "C_eq"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "conjugate_match"
     full_name: "Conjugate Match"
@@ -511,7 +511,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["P_max", "phi_Z_min", "matching"]
-    related_lessons: ["fund-03"]
+    related_topics: ["thevenin-method"]
 
   - term: "nth_order_model"
     full_name: "nth-Order Distributed Spark Model"
@@ -522,7 +522,7 @@ terms:
     unit: "N/A"
     typical_range: "n=5 to 20 segments"
     related_terms: ["lumped_model", "distributed_model", "Maxwell_matrix"]
-    related_lessons: ["model-02"]
+    related_topics: ["femm-workflow"]
 
   - term: "lumped_model"
     full_name: "Lumped Spark Model"
@@ -533,7 +533,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["nth_order_model", "C_mut", "C_sh", "R_opt_power"]
-    related_lessons: ["model-01", "model-02"]
+    related_topics: ["lumped-model", "femm-workflow"]
 
   - term: "Joule_heating"
     full_name: "Joule Heating"
@@ -544,7 +544,7 @@ terms:
     unit: "W (watts)"
     typical_range: "10-1000 W/m in channel"
     related_terms: ["transition", "leader", "thermal_ionization", "P_stream"]
-    related_lessons: ["phys-01"]
+    related_topics: ["thermal-physics", "streamers-and-leaders"]
 
   - term: "thermal_ionization"
     full_name: "Thermal Ionization"
@@ -555,7 +555,7 @@ terms:
     unit: "N/A"
     typical_range: "Significant above ~5000 K"
     related_terms: ["leader", "Joule_heating", "transition", "conductivity"]
-    related_lessons: ["phys-01"]
+    related_topics: ["thermal-physics", "streamers-and-leaders"]
 
   - term: "photoionization"
     full_name: "Photoionization"
@@ -566,7 +566,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["streamer", "thermal_ionization"]
-    related_lessons: ["phys-01"]
+    related_topics: ["streamers-and-leaders", "field-thresholds"]
 
   - term: "position"
     full_name: "Position Parameter"
@@ -577,7 +577,7 @@ terms:
     unit: "dimensionless"
     typical_range: "0 (base) to 1 (tip)"
     related_terms: ["nth_order_model", "R_min", "R_max"]
-    related_lessons: ["model-02"]
+    related_topics: ["femm-workflow"]
 
   - term: "damping"
     full_name: "Damping Factor"
@@ -588,7 +588,7 @@ terms:
     unit: "dimensionless"
     typical_range: "0.3-0.5 for stability"
     related_terms: ["nth_order_model", "iterative_optimization", "convergence"]
-    related_lessons: ["model-02"]
+    related_topics: ["femm-workflow"]
 
   - term: "Freau_scaling"
     full_name: "Freau's Empirical Scaling"
@@ -599,7 +599,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["epsilon", "bang_energy", "scaling_laws"]
-    related_lessons: ["phys-01"]
+    related_topics: ["empirical-scaling", "energy-and-growth"]
 
   - term: "bang_energy"
     full_name: "Bang Energy"
@@ -609,7 +609,7 @@ terms:
     unit: "J (joules)"
     typical_range: "1-100 J for typical DRSSTC bursts"
     related_terms: ["Freau_scaling", "epsilon", "burst_mode"]
-    related_lessons: ["phys-01"]
+    related_topics: ["empirical-scaling", "energy-and-growth"]
 
   - term: "pole_frequency"
     full_name: "Pole Frequency"
@@ -620,7 +620,7 @@ terms:
     unit: "Hz"
     typical_range: "Within ±10% of design frequency"
     related_terms: ["f_L", "coupled_resonance", "frequency_shift"]
-    related_lessons: ["fund-01", "model-01"]
+    related_topics: ["circuit-topology", "lumped-model"]
 
   - term: "coupled_resonance"
     full_name: "Coupled Resonance"
@@ -631,7 +631,7 @@ terms:
     unit: "N/A"
     typical_range: "N/A"
     related_terms: ["pole_frequency", "k", "DRSSTC"]
-    related_lessons: ["fund-01"]
+    related_topics: ["circuit-topology"]
 
   - term: "k"
     full_name: "Coupling Coefficient"
@@ -642,7 +642,7 @@ terms:
     unit: "dimensionless"
     typical_range: "0.05-0.25 for typical Tesla coils"
     related_terms: ["coupled_resonance", "M", "L_p", "L_s"]
-    related_lessons: ["fund-01"]
+    related_topics: ["circuit-topology"]
 
   - term: "topload"
     full_name: "Topload"
@@ -653,7 +653,7 @@ terms:
     unit: "N/A"
     typical_range: "10-100 pF capacitance typical"
     related_terms: ["C_mut", "C_0", "V_topload"]
-    related_lessons: ["fund-01", "model-02"]
+    related_topics: ["circuit-topology", "femm-workflow"]
 
   - term: "secondary_losses"
     full_name: "Secondary Losses"
@@ -664,7 +664,7 @@ terms:
     unit: "W (watts)"
     typical_range: "10-30% of input power"
     related_terms: ["Q_0", "G_0", "efficiency"]
-    related_lessons: ["fund-03"]
+    related_topics: ["thevenin-method"]
 
   - term: "efficiency"
     full_name: "Power Transfer Efficiency"
@@ -675,7 +675,7 @@ terms:
     unit: "dimensionless or %"
     typical_range: "15-50% typical, up to 70% for well-optimized QCW"
     related_terms: ["P_spark", "P_load", "secondary_losses"]
-    related_lessons: ["fund-03", "model-01"]
+    related_topics: ["thevenin-method", "lumped-model"]
 
   - term: "corona"
     full_name: "Corona Discharge"
@@ -686,4 +686,355 @@ terms:
     unit: "N/A"
     typical_range: "5-15% power loss typical"
     related_terms: ["E_inception", "losses"]
-    related_lessons: ["phys-01"]
+    related_topics: ["field-thresholds"]
+
+  # --- Plasma Physics Terms (from Becker et al. 2005) ---
+
+  - term: "reduced_field"
+    full_name: "Reduced Electric Field (E/N)"
+    definition: |
+      Electric field divided by gas number density, measured in Townsend (Td).
+      1 Td = 10^-21 V*m^2. At STP, 100 Td corresponds to approximately
+      25 kV/cm. Governs ionization and attachment rates in gases; breakdown
+      in air occurs at E/N ~ 100 Td.
+    unit: "Td (Townsend)"
+    typical_range: "100 Td at breakdown, 10-150 Td in discharge modeling"
+    related_terms: ["E_inception", "ionization_coefficient", "E_propagation"]
+    related_topics: ["field-thresholds"]
+
+  - term: "ionization_coefficient"
+    full_name: "Townsend Ionization Coefficient (alpha)"
+    definition: |
+      Number of ionization events per unit length of electron drift in an applied
+      field. In air: alpha/N = A*exp(-B*N/E) with A = 1.4e-20 m^2, B = 660 Td.
+      Determines avalanche growth rate and breakdown conditions.
+    unit: "m^-1 or cm^-1"
+    typical_range: "0-1000 cm^-1 depending on field strength"
+    related_terms: ["reduced_field", "streamer_criterion", "E_inception"]
+    related_topics: ["field-thresholds"]
+
+  - term: "streamer_criterion"
+    full_name: "Streamer Criterion (Meek Criterion)"
+    definition: |
+      Condition for transition from Townsend avalanche to self-propagating streamer:
+      N_critical ~ 10^8 electrons (alpha*d ~ 18-20). When the space charge field
+      of the avalanche head equals the applied field, the avalanche becomes a
+      self-propagating streamer.
+    unit: "dimensionless"
+    typical_range: "alpha*d ~ 18-20"
+    related_terms: ["ionization_coefficient", "streamer", "E_inception"]
+    related_topics: ["field-thresholds", "streamers-and-leaders"]
+
+  - term: "n_e"
+    full_name: "Electron Number Density"
+    definition: |
+      Number of free electrons per unit volume in the plasma. Determines electrical
+      conductivity via sigma = n_e*e^2/(m_e*nu_e). Ranges from 10^11 cm^-3 at
+      the streamer boundary to 10^16 cm^-3 in a fully developed spark channel.
+    unit: "cm^-3 or m^-3"
+    typical_range: "10^11-10^13 (streamers), 10^14-10^16 (sparks/arcs)"
+    related_terms: ["conductivity", "recombination_rate", "ionization_coefficient"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "recombination_rate"
+    full_name: "Electron-Ion Recombination Rate Coefficient"
+    definition: |
+      Rate coefficient for electron capture by positive ions, governing plasma
+      decay when the driving field is removed. For major atmospheric ions
+      (O2+, N2+, NO+), approximately 2e-7 cm^3/s at 300 K electron temperature.
+      Determines plasma decay time: tau_recomb = 1/(alpha_recomb * n_e).
+    unit: "cm^3/s"
+    typical_range: "2e-7 cm^3/s (binary at 300 K), up to 1e-4 cm^3/s (three-body at high pressure)"
+    related_terms: ["n_e", "tau_thermal", "ionization_memory"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "ionization_energy_cost"
+    full_name: "Average Ionization Energy Cost in Air"
+    definition: |
+      Average energy expended per electron-ion pair created in air, including all
+      loss channels (excitation, dissociation, vibrational modes). Approximately
+      14 eV, which is higher than the bare ionization potentials of N2 (15.6 eV)
+      or O2 (12.1 eV) because of energy diverted to non-ionizing collisions.
+    unit: "eV"
+    typical_range: "~14 eV in air"
+    related_terms: ["n_e", "epsilon", "P_stream"]
+    related_topics: ["thermal-physics", "energy-and-growth"]
+
+  # --- Terms from Liu (2017) and Yang et al. (2022) ---
+
+  - term: "dark_period"
+    full_name: "Dark Period (Streamer Inception)"
+    definition: |
+      The interval between successive streamer bursts during leader inception.
+      After a streamer burst, positive space charge near the electrode shields the
+      field below inception threshold. Ion drift (~2×10⁻⁴ m²/(V·s)) slowly restores
+      the field over ~1-5 ms, triggering the next burst. Multiple dark period cycles
+      typically precede stable leader inception (thermal ratcheting).
+    unit: "s (seconds)"
+    typical_range: "1-5 ms between bursts"
+    related_terms: ["aborted_leader", "streamer", "leader", "transition"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "aborted_leader"
+    full_name: "Aborted Leader"
+    definition: |
+      A failed leader inception attempt where the streamer stem heats to
+      near-critical temperature but fails to sustain it through gas expansion
+      and convection losses. Multiple aborted leaders typically precede stable
+      leader inception, with each attempt pre-heating the gas (thermal ratcheting).
+      Gas temperature must significantly exceed 2000 K to survive expansion cooling.
+    unit: "N/A"
+    typical_range: "N/A"
+    related_terms: ["dark_period", "transition", "leader", "thermal_ratcheting"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "mayr_equation"
+    full_name: "Mayr Arc Equation"
+    definition: |
+      Differential equation for time evolution of arc conductance:
+      dG/dt = (1/τ_m) × (P/P₀ - 1) × G, where G is conductance, τ_m is
+      the thermal time constant, P is instantaneous dissipated power, and P₀
+      is the cooling power. Appropriate for low-current discharges (TC sparks).
+      Naturally produces hungry streamer self-optimization toward R_opt_power.
+    unit: "N/A"
+    typical_range: "τ_m: 0.1-500 ms; P₀: 1 W/m to 1 kW/m"
+    related_terms: ["hungry_streamer", "R_opt_power", "transition"]
+    related_topics: ["streamers-and-leaders", "thermal-physics", "power-optimization"]
+
+  - term: "thermal_ratcheting"
+    full_name: "Thermal Ratcheting"
+    definition: |
+      Progressive pre-heating of the streamer stem through successive aborted
+      leader attempts. Each streamer burst deposits energy; the stem cools during
+      the dark period but retains residual warmth. After several cycles, cumulative
+      heating pushes the stem past the critical temperature for stable leader inception.
+    unit: "N/A"
+    typical_range: "N/A"
+    related_terms: ["aborted_leader", "dark_period", "transition"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "nonlinear_resistance"
+    full_name: "Nonlinear Resistance Power Law"
+    definition: |
+      Equilibrium resistance per unit length of a spark channel follows R = A/I^b
+      (Ohm/m) where I is current in Amps. Three regimes: Region I (1-10 A,
+      A=12400, b=1.84) for TC streamers; Region II (10-1000 A, A=2820, b=1.16)
+      for DRSSTC burst; Region III (>1000 A, A=180, b=0.75) for arcs. The steep
+      b=1.84 in Region I quantifies the positive feedback driving streamer-to-leader
+      transition. Channel "forgets" initial conditions at ~10 ms timescale.
+    unit: "Ω/m (ohms per meter)"
+    typical_range: "179-12,400 Ω/m for 1-10 A (Region I)"
+    related_terms: ["mayr_equation", "hungry_streamer", "R_opt_power", "transition"]
+    related_topics: ["streamers-and-leaders", "thermal-physics", "power-optimization"]
+
+  - term: "heating_efficiency"
+    full_name: "Air Heating Efficiency (eta_T)"
+    definition: |
+      Fraction of electrical energy deposited in a discharge channel that actually
+      heats the neutral gas. At ambient temperature, eta_T ~ 0.1 (only 10% heats
+      gas; 90% excites N₂ vibrational modes). Above 2000 K, eta_T ~ 1.0 (full
+      thermalization). Formula: eta_T = 0.1 + 0.9*[tanh(T/T_amb - 4) + 1]/2.
+      Explains why streamer-to-leader transition takes milliseconds despite MW/m
+      power densities.
+    unit: "dimensionless (0 to 1)"
+    typical_range: "0.1 (300 K) to 1.0 (>2000 K)"
+    related_terms: ["thermal_ratcheting", "transition", "vibrational_relaxation"]
+    related_topics: ["thermal-physics", "streamers-and-leaders"]
+
+  - term: "Gallimberti_model"
+    full_name: "Gallimberti Streamer-to-Leader Model"
+    definition: |
+      Early (1972) computational model for predicting streamer-to-leader transition.
+      Assumes constant stem field, simplified N₂ vibrational-translational (V-T)
+      relaxation, and single dominant stem. Qualitatively useful but quantitatively
+      unreliable: Liu (2017) showed assumptions do not hold under detailed kinetic
+      modeling (45 species, 192 reactions).
+    unit: "N/A"
+    typical_range: "N/A"
+    related_terms: ["transition", "leader", "streamer"]
+    related_topics: ["streamers-and-leaders"]
+
+  # --- Terms from Bazelyan & Raizer (2000) ---
+
+  - term: "leader_velocity"
+    full_name: "Leader Propagation Velocity"
+    definition: |
+      Empirical velocity of leader channel advance: v_L = 1500 * sqrt(|Delta_U_t|)
+      in cm/s, where Delta_U_t is the tip potential in volts. Derived from extensive
+      laboratory spark and lightning data. For TC sparks at 300 kV: ~8.2 km/s.
+      Physical basis: conducting streamer length (~1 cm) divided by thermal instability
+      contraction time (~1 us) gives ~10 km/s baseline, modulated by tip voltage.
+    unit: "cm/s or m/s"
+    typical_range: "5-15 km/s for TC voltages (100-600 kV)"
+    related_terms: ["dL_dt", "E_propagation", "transition"]
+    related_topics: ["streamers-and-leaders", "energy-and-growth"]
+
+  - term: "electron_attachment_time"
+    full_name: "Electron Attachment Time in Cool Air"
+    definition: |
+      Time for free electrons to attach to O2 molecules in cool (non-heated) air
+      at atmospheric pressure: ~100 ns (10^-7 s). This is the fundamental timescale
+      for plasma decay without heating. At T > 5000 K, attachment becomes negligible.
+      At TC frequencies (50-400 kHz), a cold streamer undergoes 12-100 attachment
+      times per half-cycle, explaining why heating is essential for persistent channels.
+    unit: "s (seconds)"
+    typical_range: "~10^-7 s at STP, increases with temperature"
+    related_terms: ["streamer", "transition", "tau_thermal", "n_e"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "Bazelyan_VI"
+    full_name: "Bazelyan V-I Characteristic"
+    definition: |
+      Simple relationship between arc/leader current and internal electric field
+      in air at atmospheric pressure: i * E = b, where b = 300 V*A/cm. Equivalently,
+      R_per_meter = 30,000 / i^2 (ohm/m). Valid for moderate currents (1-100 A).
+      Agrees with da Silva's R = A/I^b power law within factor ~2 for TC-relevant
+      currents (1-10 A). A quick approximation complementing the more detailed
+      three-regime da Silva model.
+    unit: "V*A/cm"
+    typical_range: "b = 300 V*A/cm (constant)"
+    related_terms: ["nonlinear_resistance", "mayr_equation", "hungry_streamer"]
+    related_topics: ["streamers-and-leaders", "equations-and-bounds"]
+
+  - term: "energy_ceiling"
+    full_name: "Energy Ceiling from Tip Capacitance"
+    definition: |
+      Maximum energy available per unit length of new channel from the electrostatic
+      charge stored at the spark tip: W_max = pi * epsilon_0 * U^2 (J/m), where U
+      is the tip potential. For a TC at 300 kV: W_max ~ 25 J/m. The tip (hemisphere)
+      stores ln(L/r) times more energy per unit length than the channel body, making
+      the tip the primary energy source for initiating each new leader step.
+    unit: "J/m"
+    typical_range: "3-100 J/m for TC voltages (100-600 kV)"
+    related_terms: ["epsilon", "V_tip", "C_mut", "C_sh"]
+    related_topics: ["energy-and-growth", "equations-and-bounds"]
+
+  - term: "conductance_relaxation"
+    full_name: "Conductance Relaxation Model"
+    definition: |
+      A dynamic model for time-dependent spark channel conductance:
+      dG/dt = [G_st(i) - G(t)] / tau_g, where G_st(i) is the equilibrium conductance
+      at current i, and tau_g is an asymmetric time constant: 40 us for heating
+      (current rising) and 200 us for cooling (current decreasing). Alternative to
+      the Mayr equation; more physical for large transients. The 5:1 heating/cooling
+      asymmetry creates a ratcheting effect that favors leader maintenance.
+    unit: "S/m (conductance per unit length)"
+    typical_range: "tau_g = 40 us (heating), 200 us (cooling)"
+    related_terms: ["Mayr_equation", "thermal_ionization", "leader"]
+    related_topics: ["thermal-physics", "equations-and-bounds"]
+
+  - term: "corona_shielding"
+    full_name: "Corona Shielding Rate Limit"
+    definition: |
+      The maximum voltage growth rate at which a stable corona can persist on an
+      electrode, shielding it from streamer inception: A_u_max ~ 3.6 kV/us. Above
+      this rate, ions cannot drift fast enough to maintain the space charge cloud
+      that stabilizes the surface field. TC toploads reach ~300 kV/us (at 200 kHz),
+      far exceeding this limit, so corona shielding never applies — every cycle
+      immediately produces streamers.
+    unit: "kV/us"
+    typical_range: "3.6 kV/us (limit); TC toploads: ~300 kV/us"
+    related_terms: ["E_inception", "streamer", "leader"]
+    related_topics: ["field-thresholds", "streamers-and-leaders"]
+
+  - term: "stepped_leader"
+    full_name: "Stepped Leader Propagation"
+    definition: |
+      A leader that advances in discrete jumps separated by pauses, characteristic
+      of negative polarity leaders. Lightning step length: 10-200 m (avg 30 m),
+      with pauses of 30-90 us. TC sparks on the negative half-cycle could exhibit
+      stepping, but the fast AC reversal (half-period 1.25-10 us) masks this.
+      In contrast, positive leaders propagate continuously. When averaged over
+      total development time, stepped and continuous leaders have similar velocities
+      (10^5-10^6 m/s).
+    unit: "dimensionless (mode classification)"
+    typical_range: "step length: 10-200 m; pause: 30-90 us"
+    related_terms: ["leader", "leader_velocity", "dart_leader"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "dart_leader"
+    full_name: "Dart Leader (Re-strike Leader)"
+    definition: |
+      A leader that re-illuminates an existing hot channel from a previous stroke.
+      Always continuous (not stepped), propagating at (1-4)*10^7 m/s — much faster
+      than initial leaders (10^5-10^6 m/s) because the pre-heated, pre-ionized
+      channel requires minimal fresh ionization. Analogous to a TC spark re-using
+      a persistent hot channel from a previous QCW ramp cycle.
+    unit: "m/s (velocity)"
+    typical_range: "(1-4) * 10^7 m/s"
+    related_terms: ["leader", "stepped_leader", "leader_velocity"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "leader_formation_threshold"
+    full_name: "Minimum Voltage for Leader Formation"
+    definition: |
+      The minimum potential difference required to excite and develop a leader
+      in air under normal conditions: Delta_U_min ~ 300-400 kV. Below this,
+      only streamers form. Most DRSSTCs operate at 100-600 kV topload voltage;
+      the 300 kV threshold explains the disproportionate improvement in spark
+      length efficiency observed when coils cross this voltage level.
+    unit: "kV"
+    typical_range: "300-400 kV"
+    related_terms: ["leader", "leader_velocity", "E_inception"]
+    related_topics: ["streamers-and-leaders", "field-thresholds"]
+
+  - term: "driven_leader"
+    full_name: "Driven Leader (QCW Growth Mode)"
+    definition: |
+      The leader propagation mode unique to QCW Tesla coils, where the leader
+      advances continuously at ~170 m/s (half the speed of sound), fed by sustained
+      current from the resonant circuit. Each step involves: (1) streamer launch from
+      leader tip, (2) thermal conversion of streamer to leader segment in ~60 us
+      (close to tau_g = 40 us), (3) repeat. The net growth rate (~170 m/s) is
+      intermediate between free streamers (~10^6 m/s) and natural lightning leaders
+      (~10^4 m/s averaged). A 10 ms ramp yields ~1.7 m; 20 ms yields ~3.4 m.
+    unit: "m/s (growth rate)"
+    typical_range: "~170 m/s; step time ~60 us"
+    related_terms: ["leader", "QCW", "conductance_relaxation", "leader_velocity"]
+    related_topics: ["streamers-and-leaders", "thermal-physics", "energy-and-growth"]
+
+  - term: "sword_spark"
+    full_name: "Sword Spark (Straight QCW Discharge)"
+    definition: |
+      The distinctive straight, bright, sword-like spark produced by QCW Tesla coils
+      operating at 300-600 kHz. Characterized by a single dominant leader channel with
+      minimal branching, white/yellow appearance, and lengths of 7-16x the secondary
+      coil length. Requires: (1) operating frequency >300 kHz for continuous heating,
+      (2) coupling k >= 0.3, (3) smooth continuous power ramp of 10-20 ms, (4) no
+      pulse-skip modulation. Below 300 kHz, sparks are "chaotic and less straight";
+      above 600 kHz, they become "curvy." The physical basis is that the RF half-period
+      at >300 kHz (< 1.7 us) is much shorter than the thermal diffusion time of even
+      thin streamers (~125 us for 100 um), enabling effectively continuous heating
+      that maintains a single dominant conductive path.
+    unit: "N/A"
+    typical_range: "1-3.4 m length at 300-600 kHz"
+    related_terms: ["QCW", "driven_leader", "leader", "frequency_threshold"]
+    related_topics: ["streamers-and-leaders", "thermal-physics"]
+
+  - term: "burst_ceiling"
+    full_name: "Burst Mode Growth Ceiling"
+    definition: |
+      The maximum spark growth time in burst-mode DRSSTC operation, beyond which
+      additional ON time produces no further spark length. Measured by Steve Ward at
+      ~80 us on DRSSTC-0.5. Consistent with the thermal time constant for 100 um
+      streamers (tau_thermal ~ 125 us): after approximately one thermal time constant,
+      streamer channels cool as fast as they heat, saturating growth. Additional energy
+      goes into re-heating decayed channels rather than forward propagation. This is
+      the fundamental wall that QCW overcomes by sustaining drive beyond this timescale.
+    unit: "us (microseconds)"
+    typical_range: "~80 us"
+    related_terms: ["burst_mode", "tau_thermal", "QCW", "streamer"]
+    related_topics: ["thermal-physics", "energy-and-growth"]
+
+  - term: "wave_impedance"
+    full_name: "Channel Wave Impedance"
+    definition: |
+      The characteristic impedance of a spark/leader channel treated as a lossy
+      transmission line: Z = sqrt(L_1/C_1). For lightning leaders: Z ~ 500 ohm
+      (with C_1 ~ 10 pF/m, L_1 ~ 2.5 uH/m). For TC sparks: estimated 700-1200 ohm
+      (higher due to smaller corona envelope and thinner channels). Relevant for
+      strike events and return stroke analogy.
+    unit: "ohm"
+    typical_range: "500-1200 ohm"
+    related_terms: ["C_sh", "leader"]
+    related_topics: ["equations-and-bounds", "streamers-and-leaders"]

reference/physics-cheat-sheet.md

@@ -0,0 +1,238 @@
+# Tesla Coil Spark Physics — Cheat Sheet
+
+Everything you need to see the whole picture, in order.
+
+---
+
+## 1. The Spark Is a Circuit Element
+
+A spark hanging off a topload is not magic plasma — it's a **lossy capacitive load** on a resonant circuit. It has exactly three electrical properties that matter:
+
+- **C_mut** (mutual capacitance): coupling between topload and spark channel (~3-15 pF)
+- **C_sh** (shunt capacitance): coupling between spark channel and ground (~2 pF/foot)
+- **R** (resistance): the lossy part where power is dissipated (1 kohm to 100 Mohm)
+
+These three form a bridged-T network. That's the entire circuit model of a spark.
+
+## 2. The Phase Constraint
+
+Because C_mut and C_sh form a capacitive divider, the impedance phase at the topload is **always more negative than -45 degrees** for typical TC geometries. You can't achieve a conjugate match. This is a topological fact, not a design flaw.
+
+Typical impedance phase at optimum: **-55 to -75 degrees**.
+
+## 3. The Plasma Self-Optimizes
+
+The spark resistance R isn't fixed — it adjusts itself through heating and ionization. The key result:
+
+**The plasma drifts toward R_opt_power = 1/(omega * C_total)** because:
+- Too high R → less power → less heating → R rises further (unstable, spark dies or branches)
+- Too low R → less power (past optimum) → but stronger heating prevents R from dropping much below optimum
+
+This is the **hungry streamer principle**: the spark "eats" as much power as the circuit can deliver, automatically finding the impedance that maximizes power transfer.
+
+## 4. Two Ways to Grow
+
+A spark extends its length when two conditions are met:
+
+**Condition 1 — Field threshold:** E_tip > E_propagation
+The electric field at the spark tip must exceed the propagation threshold. If it doesn't, the spark stalls regardless of available power.
+
+**Condition 2 — Energy supply:** dL/dt = P_stream / epsilon
+Growth rate equals available power divided by energy cost per meter.
+
+**Critical nuance:** E_propagation is NOT a fixed constant. In cold air, E_propagation ~ 0.5 MV/m. But at a driven leader tip, four mechanisms — UV pre-ionization, thermal pre-conditioning, residual ionization, and gas expansion — converge to dynamically reduce it. This is why QCW achieves 2+ m sparks at only 40-70 kV: the leader creates its own favorable conditions. Voltage and power are coupled limits, not independent ones (Section 4A).
+
+The spark is always limited by whichever constraint binds first: **voltage-limited** (can't push field high enough even with dynamic threshold) or **power-limited** (can extend field but not fast enough).
+
+## 5. Epsilon: The Central Parameter
+
+**Epsilon (J/m)** = energy required per meter of spark growth. It varies enormously:
+
+| Mode | Epsilon | Why |
+|---|---|---|
+| QCW (leader) | 5-15 J/m | Hot, efficient single channel |
+| Burst (streamer) | 30-100+ J/m | Cold, branched, inefficient |
+
+The difference is almost entirely explained by **channel type** (Section 7) and **branching** (Section 10).
+
+## 6. The Capacitive Divider Problem
+
+As the spark grows, C_sh increases (more conductor length to ground). This **divides down the tip voltage**:
+
+```
+V_tip = V_topload * C_mut / (C_mut + C_sh)
+```
+
+Longer spark → more C_sh → lower V_tip → weaker E_tip → harder to keep growing.
+
+This creates **sub-linear scaling**: doubling energy does NOT double spark length. Burst mode follows L ~ sqrt(E). QCW is somewhat better (L ~ E^0.6-0.8) because leader channels have lower C_sh per unit length than branched streamers.
+
+## 6A. The Dynamic Threshold
+
+The capacitive divider predicts QCW sparks should stall at well under 1 m with only 40-70 kV topload. Yet 2+ m sparks are routinely achieved. The resolution: **E_propagation is not a fixed constant** — at a driven leader tip, four mechanisms converge to reduce it:
+
+1. **UV photoionization** — corona creates seed electrons ahead of the tip
+2. **Thermal pre-conditioning** — heat reduces gas density (E_breakdown proportional to N proportional to 1/T)
+3. **Residual ionization** — previous streamers leave persistent electron density (~50 us decay)
+4. **Gas expansion** — lower N means lower absolute field threshold
+
+These are mutually reinforcing: more leader current drives all four harder. The result is a **coupled voltage-power limit** — power modifies the conditions that set the voltage threshold. More power → lower effective E_propagation → spark extends further at the same voltage.
+
+But there is a floor: E_propagation can't reach zero. The capacitive divider wins eventually. The "too long" QCW regime (>25 ms) is exactly the point where even maximal pre-conditioning can't keep E_tip above the reduced threshold.
+
+## 7. Two Kinds of Channel
+
+This is the fork in the road that explains almost everything:
+
+| | Streamer | Leader |
+|---|---|---|
+| Temperature | 300-3000 K | 5,000-20,000 K |
+| Diameter | 10-100 um | 1-10 mm |
+| Resistance | Very high | Low |
+| Persistence | Microseconds | Seconds |
+| Branching | Extensive | Minimal |
+| Epsilon | High (30-100+) | Low (5-15) |
+| Color | Purple/blue | White/yellow |
+
+Streamers are cold, thin, branched, and inefficient. Leaders are hot, thick, straight, and efficient. **The entire game is getting from streamer to leader.**
+
+## 8. The Thermal Ratchet
+
+The transition from streamer to leader requires heating the channel past ~5000 K (through intermediate thresholds at 2000 K and 4000 K). But thin channels cool fast:
+
+```
+tau_thermal = d^2 / (4 * alpha)      alpha ~ 2e-5 m^2/s for air
+```
+
+A 100 um streamer cools in ~125 us. You have to heat it faster than it cools.
+
+The **conductance relaxation** is asymmetric:
+- Heating: tau_g = 40 us (fast — ionization responds quickly to current)
+- Cooling: tau_g = 200 us (slow — recombination and thermal diffusion take longer)
+
+This 5:1 asymmetry creates a **one-way thermal ratchet**: each RF cycle heats a little more than the previous one cooled. Over many cycles, temperature accumulates monotonically upward through the critical zone.
+
+## 9. Frequency Matters
+
+The ratchet only works if the RF period is much shorter than tau_thermal:
+
+- At **400 kHz** (T_half = 1.25 us): streamer experiences ~100 RF cycles per tau_thermal. Heating is effectively continuous. Ratchet works. → **Swords.**
+- At **100 kHz** (T_half = 5 us): thin streamers cool significantly between cycles. Ratchet is intermittent. → **Branchy, noisy sparks.**
+
+The community-observed threshold: **300-600 kHz** for sword sparks. This is not about breakdown physics — it's about whether the thermal ratchet can outrun cooling.
+
+## 10. Branching Is a Competition
+
+Discharges branch because of **Laplacian instability** at the propagating tip (same physics as viscous fingering). Streamers branch every ~10-20 diameters.
+
+But branches **compete** for current. The channel resistance follows a nonlinear power law:
+
+```
+R = A / I^b      b = 1.84 for TC currents (1-10 A)
+```
+
+Because b > 1, the V-I curve has **negative slope**. A branch that gets slightly more current heats up, becomes more conductive, steals more current from its neighbors. This is positive feedback — **one branch wins, the rest die.**
+
+Competition timescale: ~120-200 us (a few tau_g).
+
+- **Burst mode** (70-150 us pulses): too short for competition to resolve → many branches survive → bushy
+- **QCW mode** (10-20 ms ramp): competition resolves in <1 ms → single dominant channel → sword
+- **Pulse-skip**: intermediate — competition operates but with jitter → "sword-like but still branches"
+
+## 11. QCW vs Burst: The Complete Picture
+
+| | QCW | Burst |
+|---|---|---|
+| Voltage | 40-70 kV (!!) | 200-600 kV |
+| Duration | 10-20 ms | 70-150 us |
+| Frequency | 300-600 kHz | 50-200 kHz |
+| Channel type | Leader | Streamer |
+| Branching | Suppressed by competition | Extensive |
+| Epsilon | 5-15 J/m | 30-100+ J/m |
+| Spark:secondary ratio | 7-16x | 2.5-3.6x |
+| Morphology | Straight sword | Bushy tree |
+| Mechanism | Thermal ratchet over many ms | Brute-force high voltage |
+
+The 15:1 voltage ratio (measured by davekni) is the single most striking number. QCW achieves leader formation at 40-70 kV because it has **time** — the ratchet accumulates thermal energy over 10-20 ms. Burst needs 200-600 kV because it must reach leader temperature in a single ~100 us pulse.
+
+## 12. The Three Ramp Regimes
+
+QCW ramp duration selects three distinct outcomes:
+
+- **Too short (<5 ms):** Insufficient time for streamer-to-leader transition. Segmented, gnarly sparks.
+- **Optimal (10-20 ms):** Leader forms within 1-2 ms, grows as single channel for remainder. Straight swords.
+- **Too long (>25 ms):** Leader reaches voltage-limited max length (capacitive divider). Excess energy drives lateral breakouts. "Hot, fat, bushy."
+
+## 13. Putting It All Together
+
+The complete causal chain:
+
+```
+RF drive at frequency f
+    │
+    ├─→ Resonant voltage gain → V_topload
+    │
+    ├─→ E_tip = kappa * V_tip / L  → inception when E_tip > E_inception
+    │
+    ├─→ Streamer channels form (cold, branched, high R)
+    │
+    ├─→ Hungry streamer: R drifts toward R_opt_power
+    │       │
+    │       ├─→ Power delivered: P = f(V_th, Z_th, R)
+    │       │
+    │       └─→ Growth: dL/dt = P / epsilon
+    │
+    ├─→ Thermal evolution (depends on mode):
+    │       │
+    │       ├─→ QCW: sustained ramp → thermal ratchet → leader formation
+    │       │     → branch competition selects single channel
+    │       │     → low epsilon → efficient growth → sword
+    │       │
+    │       └─→ Burst: short pulse → no time for leader transition
+    │             → branches coexist → high epsilon → bushy
+    │
+    ├─→ Dynamic threshold (QCW only):
+    │       Leader current → UV + heat + residual ionization + expansion
+    │       → E_propagation_effective drops well below cold-air value
+    │       → spark extends further at lower voltage
+    │       → coupled V-P limit, not independent constraints
+    │
+    ├─→ Capacitive divider: C_sh grows with L
+    │       → V_tip decreases → E_tip drops
+    │       → eventually E_tip < E_propagation_effective → stalls
+    │       → sub-linear scaling: L ~ sqrt(E) for burst
+    │
+    └─→ Final length set by:
+            min(dynamic voltage limit, energy limit, ramp duration)
+```
+
+## 14. The Numbers That Matter
+
+| Quantity | Value | Why it matters |
+|---|---|---|
+| C_sh per foot | ~2 pF | Sets voltage division rate |
+| R_opt_power | 10-100 kohm | Where plasma naturally sits |
+| E_propagation (cold air) | 0.4-1.0 MV/m | Field floor for cold streamer growth |
+| E_propagation (leader tip) | Much lower (T3) | Dynamically reduced by UV/heat/ionization |
+| tau_thermal (100 um) | ~125 us | Streamer cooling timescale |
+| tau_g (heating) | 40 us | Conductance response speed |
+| tau_g (cooling) | 200 us | 5:1 asymmetry drives ratchet |
+| Competition time | ~120-200 us | Branch winner decided |
+| Burst ceiling | ~80 us | ON time saturation (Steve Ward) |
+| QCW optimal ramp | 10-20 ms | Sweet spot for leader growth |
+| Frequency threshold | 300-600 kHz | Below this, no swords |
+| QCW voltage | 40-70 kV | 15:1 less than burst |
+| da Silva b exponent | 1.84 | b > 1 → current hogging |
+| Fractal dimension | ~2.2 | Streamer tree space-filling |
+
+## 15. What We Don't Know
+
+1. **Exact branching power division** — no validated current-sharing rule
+2. **Epsilon from first principles** — still requires calibration
+3. **Time-resolved impedance during QCW ramp** — never measured
+4. **Spectroscopic temperature of QCW sparks** — 5000 K inferred, not measured
+5. **Arc current in any QCW spark** — secondary current unmeasured
+6. **How C_sh scales with branching** — qualitative only
+7. **Branching fraction of epsilon** — how much energy goes to side branches vs other overhead
+8. **Dynamic threshold magnitude** — how much is E_propagation reduced at a QCW leader tip?
+9. **Gas temperature ahead of leader tip** — spectroscopic measurement needed

reference/sources/bazelyan-noaa-preprint.txt

@@ -0,0 +1,510 @@
+=== PAGE 1 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+1
+Non-Stationary Corona around Multi-Point System in 
+Atmospheric Electric Field:  
+Discharge Current and Vertical Electric Field Profile  
+ 
+Eduard M. Bazelyan1, Yuri P. Raizer2, Nickolay L. Aleksandrov1,* 
+ 
+1. Krzhizhanovsky Power Engineering Institute, Moscow, Russia  
+2. Institute for Problems in Mechanics, Moscow, Russia 
+3. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, Russia  
+ 
+ 
+ABSTRACT: The properties of a non-stationary glow corona maintained near the tips of a multi-point 
+ground system in a time-varying thundercloud electric field have been studied numerically. The discharge 
+was simulated from a system of identical vertical conductive electrodes that is a model of the earth’s 
+surface extremities coronating under a thundercloud. The effect of system geometry and dimensions on 
+the discharge properties and on vertical electric field profile above the coronating system was investigated. 
+Conditions were determined under which the corona properties of a multi-point system are similar to the 
+properties of a plane surface that emits ions into the atmosphere. The obtained results were used to 
+estimate the temporal evolution of corona current density and corona space charge emitted during 
+thunderstorms from the earth’s surface covered with dense vegetation. 
+ 
+INTRODUCTION 
+Corona discharges developed from the earth’s surface extremities (the tips of trees, bushes, leaves, 
+grass and other sharp objects) under a thundercloud leads to the space charge injection into the atmosphere 
+and make a contribution to the global electric circle. In addition, the corona space charge layer affects the 
+local electric field at ground level and is practically important for lightning protection.  
+Laboratory studies of a corona discharge cannot be directly extended to thunderstorm conditions 
+because a discharge occurring near grounded objects in a time-varying atmospheric electric field is 
+non-stationary and the corona current depends on the manner in which the ambient field evolves in time, 
+rather than on its instantaneous values. The reason is that, in this case, the space charge front has no time to 
+bridge the gap and to reach the thundercloud, whereas the corona space charge reaches usually the opposite 
+electrode on a laboratory scale.  
+The properties of a corona discharge developed from a solitary grounded hemispherically-tipped rod in 
+a thundercloud electric field was considered analytically and numerically [Bazelyan and Raizer 2000; 
+Aleksandrov et al. 2001; Bazelyan et al. 2008] on the basis of a simple 1D approximation. It was shown that 
+the corona current varies in time as icor (t) ~ t(3k-1)/2μ1/2, when the cloud electric field varies as E0(t) ~ tk, k > -1. 
+                                                        
+ Contact information: Nickolay L. Aleksandrov, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, 
+Russia, Email: nick_aleksandrov@mail.ru; nick_aleksandrov@hotmail.com 
+
+=== PAGE 2 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+2
+Here, μ is the ion mobility. In this case, the discharge current is constant only when the electric field rises in 
+time. In a steady electric field (k = 0), the current decreases with time. The effect of ion mobility on the 
+current is smaller than this effect for laboratory gaps when icor (t) ~ μ. 
+Recent time-consuming numerical 2D simulations for a solitary grounded rod [Becerra 2013] and for a 
+long horizontal grounded wire [Mokrov et al. 2013] supported the use of a much simpler 1D calculations for 
+a qualitative analysis when the focus is on the processes in the vicinity of the coronating surface or when 
+computational time is limited. This simplification is especially important when considering the properties of 
+a corona developed from a grounded multi-point system with a complicated geometry. 
+In a thundercloud electric field, the corona current even from an extremely high solitary electrode does 
+not exceed 1 mA that is not important from the standpoint of the global electric circuit. Multi-point ground 
+coronating systems (forest, bushes, grass and urban areas) make much larger contribution to the total current 
+from the earth’s surface. In this case, the local electric field near a given coronating point is affected not only 
+by the space charges developed from this point, but by the space charges emitted by others corona sources as 
+well. Numerical simulation of a corona discharge from a multipoint system is much more complicated than 
+that from a solitary electrode since it is necessary to consider interaction between coronating points and 
+individual corona space charge layers.  
+In this work, we extended the 1D approach developed in [Bazelyan and Raizer 2000; Aleksandrov 
+et al. 2001; Bazelyan et al. 2008] for a solitary grounded electrode to a multi-electrode system. The 
+properties of a non-stationary (transient) corona initiated and developed from a model multi-point ground 
+system in a thundercloud electric field were numerically studied for different geometrical parameters of 
+the system (see also [Bazelyan et al. 2014a]). A simplified method to determine the corona current density 
+and injected corona space charge under real conditions was suggested. The evolution in time of vertical 
+electric field profiles in the space charge layer above a multi-point system was also considered (see also 
+[Bazelyan et al. 2014b]). 
+ 
+CORONA INITIATION FROM MULTI-POINT SYSTEM IN EXTERNAL ELECTRIC FIELD 
+In this work, the model of a corona discharge around a solitary electrode (see [Bazelyan and Raizer 
+2000; Aleksandrov et al. 2001; Bazelyan et al. 2008]) was generalized to study the discharge from a 
+multi-point system. We considered a system of vertical grounded hemispherically-tipped electrodes under 
+practically important conditions when the electrode height h is much higher than the curvature radius of 
+the electrode top, r0, and the distance between adjacent electrodes, D, is comparable with h. Electrodes 
+were uniformly distributed over concentric circles with the radii rk = kD (k = 1,2…) around a given 
+electrode (see figure 1). It was assumed that 6k electrodes are located on the k-th circle and that the total 
+number of electrodes is such large that almost every coronating point is surrounded by numerous similar 
+points. This allowed calculation of discharge properties only for the central electrode under the 
+assumption that the discharge properties for other electrodes are similar. (Here, the peculiarities of the 
+corona discharge near the electrodes at the outer boundary of the system were neglected.) 
+The same approximation was used to calculate the corona onset atmospheric electric field, E0cor, at 
+which the local electric field near the electrode tips reaches the corona onset field, Ecor, and corona is 
+ignited. The value of Ecor was determined from the empirical formula suggested by Bazelyan et al. (2007).  
+
+=== PAGE 3 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+3
+ 
+Fig. 1. The distribution of electrodes over the ground surface. 
+ 
+A quantitative relation between E0cor and Ecor for a given multi-electrode system can be calculated 
+using available electrostatic numerical methods. Figure 2 shows the threshold atmospheric electric field 
+E0cor calculated with the charge simulation method [Malik 1989] for a system of grounded spherical 
+electrodes as a function of the number of circles with surrounding electrodes. The value of E0cor increases 
+with the number of surrounding electrodes and is affected even by electrodes located at a distance of 100 
+m. This is explained by the fact that the number of surrounding electrodes distributed over a given circle 
+increases with the circle radius; that is, the distant circles contain much more surrounding electrodes and 
+each of these electrodes makes a contribution into the potential of the central electrode. The value of E0cor 
+even for the multipoint system with closely packed electrodes (D/h =0.1) is only double that E0cor for a 
+solitary electrode (N = 0).  
+ 
+Fig. 2. The threshold ambient electric field required for corona initiation in a multi-point system as a 
+function of the number of circles with surrounding electrodes. The calculation was made for h = 10 m, D = 
+1m and r0 = 1 cm. 
+
+=== PAGE 4 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+4
+ 
+CALCULATED MODEL OF CORONA DISCHARGE 
+A physical approach to simulating a non-stationary, streamer-free, glow corona of positive polarity 
+initiated from grounded electrodes in an atmospheric electric field and algorithms applicable to the 
+simplest electrode geometries has been given in detail elsewhere [Aleksandrov et al. 2002]. In this model, 
+the ionization layer in the immediate vicinity of the electrode tip was not considered because its thickness 
+is much smaller than the radius of curvature of the tip. Here, the corona-producing surface was assumed to 
+be an emitter of ions and the boundary condition for electric field was reduced to a condition widely used 
+to determine the current-voltage characteristic of a stationary glow corona in long gas gaps [Raizer 1991], 
+namely, that electric field at a coronating surface is equal to the onset corona field, Ecor. For a 
+hemispherically-tipped rod with radius r0, the boundary condition was reduced to  
+E(r0) = Ecor.          (1) 
+For the sake of definiteness, we assumed that an external electric field was produced by a 
+time-varying thundercloud negative charge. The expansion of the corona space charge layers was 
+described by the electrostatic equation for electric field  
+div E(r) = /0      (2) 
+and continuity equations for ions 
+
+
+j
+j
+j
+j
+S
+E
+n
+div
+t
+n
+
+
+
+
+
+,      (3) 
+where  = enj is the space charge density, e is the charge of a singly charged ion, nj and j are the number 
+density and mobility of ions of species j, respectively, and Sj is a source term describing ion-molecule 
+reactions that affect the ion composition and, hence, the ion transport. The potential  introduced as E = 
+-  was assumed to tend to zero at the grounded plane and at grounded electrodes, whereas, away from 
+them and from the ion “cloud”, the electric field tended to the undisturbed external electric field, E0(t). 
+Electric field above every coronating electrode was calculated taking into account not only the corona 
+space charge emitted by this electrode, but the charges emitted by other electrodes as well. The effect of 
+these charges was considered approximately, assuming that they are point charges.  
+ 
+NUMERICAL SIMULATION OF CORONA CURRENT AND INJECTED SPACE CHARGE 
+Our numerical simulation showed the following peculiarities of a corona discharge from a multi-point 
+system. 
+Corona current decreases with increasing the number of coronating sources (see figure 3), whereas the 
+rate of decrease of the corona current at E0 = const increases in this case. The temporal evolution of the 
+corona current, icor(t), is easy to analyze in figure 4 where the values are normalized to the peak corona 
+currents, imax.  
+In a multi-point system with a few thousand of electrodes, where the corona current is stabilized in a 
+linearly rising thundercloud electric field, the value of the stabilized current, icor max is almost independent 
+of the electrode height (see figure 5) and depends strongly on the distance between electrodes, D (see 
+figure 6). It follows from the data that icor max ~ D2.  
+
+=== PAGE 5 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+5
+ 
+                                                              
+Fig. 3. The evolution in time of the corona current from the top of the central electrode in a multi-rod 
+system with rods for h = D = 1 m and r0 = 10-1 cm. The external electric field rises linearly from zero to 
+E0m at t < tm and is equal to E0m at t > tm, where E0m = 40 kV m-1 and tm = 1 s. 
+ 
+Fig. 4. The evolution in time of the corona current from the top of the central electrode in a multi-rod 
+system with rods for h = D = 10 m and r0 = 1 cm. The external electric field rises linearly from zero to E0m 
+at t < tm and is equal to E0m at t > tm, where E0m = 20 kV m-1 and tm = 10 s. 
+
+=== PAGE 6 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+6
+ 
+ 
+Fig. 5. The evolution in time of the corona current from the top of the central rod in a multi-point 
+system with rods of height h = 10 and 50 m. The number of circles with surrounding rods is N = 50. Other 
+conditions are similar to those in figure 4. 
+ 
+ 
+Fig. 6. The value of the stabilized corona current from the top of the central rod in a multi-point system 
+with rods of height h = 10 m as a function of the distance between electrodes. The number of circles with 
+surrounding rods is N = 50. The external electric field rises linearly from zero to 40 kV m-1 for 30 s.  
+
+=== PAGE 7 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+7
+ 
+The time it takes to saturate the corona current for a multi-point system in a linearly rising external 
+electric field also depends on the distance between electrodes; this dependence is close to a linear one (see 
+figure 7).  
+ 
+Fig. 7. The time it takes to saturate the corona current for a multi-point system in a linearly rising 
+external electric field as a function of the distance between electrodes. Conditions are similar to those in 
+figure 6. 
+  
+Analysis of our calculations shows that the properties of a multi-point coronating system 
+asymptotically tend to those of a prefect emitting plane with the surface electric field that is equal to the 
+corona onset atmospheric electric field Е0cor [Bazelyan et al. 2008]. Stabilization of the surface electric 
+field is due to ion emission. Indeed, the plane space charge layer and its image in the conducting ground 
+form a double electrostatic layer; that is, the electric field is equal to E0(t) at the upper boundary of the 
+layer and to E0cor at the ground surface. In this case, it follows from the Poisson equation (the Gauss 
+theorem) that, to stabilize the surface electric field at the level E0cor, the corona space charge injected into 
+the atmosphere per unit area must be [Bazelyan et al. 2008]  
+]
+)
+(
+[
+)
+(
+cor
+E
+t
+E
+t
+q
+0
+0
+0
+
+
+
+.        (4) 
+Then, the corona current density is expressed as 
+dt
+t
+dE
+dt
+dq
+t
+jcor
+)
+(
+)
+(
+0
+0
+0
+
+
+
+
+.      (5) 
+It follows from (5) that in the asymptotic limit the corona current density depends only on the rate of rise 
+of the external electric field, E0(t). In particular, the current must be constant for a linearly rising electric 
+field and must tend to zero for a constant electric field. It is precisely this manner of the temporal 
+
+=== PAGE 8 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+8
+evolution of the corona current is obtained from our calculations for multi-point systems when the number 
+of coronating electrodes is sufficiently large. The current through one electrode in multi-point systems 
+studied is obtained by taking the product of jcor and the area per one electrode in the system, S = D2N2/nel, 
+where N is the number of circles covered with electrodes and nel is the total number of electrodes in the 
+system. Then, we have  
+dt
+t
+dE
+n
+N
+D
+t
+j
+n
+N
+D
+t
+i
+el
+cor
+el
+cor
+)
+(
+)
+(
+)
+(
+0
+2
+2
+0
+2
+2
+
+
+
+
+.   (6) 
+From (6), icor max ~ D2, in agreement with our calculations (see figure 6). Moreover, there is good 
+quantitative agreement between equation (6) and our calculated results. For instance, it follows from the 
+results shown in figure 6 that icor max = 5.04 μA for the system with D = 20 m, whereas the current obtained 
+from (6) under the same conditions is 4.85 μA. Here, the difference is less than 5%. 
+The calculated corona current actively increases in time due to the development of individual 
+corona space charges from their sources until a united corona space charge layer is formed. In the end, 
+individual space charges unite into one plane corona space charge layer (see figure 8) and then the model 
+of emitting plane (equations (4) and (5)) becomes adequate.  
+ 
+Fig. 8. A schematic diagram of the space charge layer formed above a ground multi-points system in an 
+atmospheric electric field E0.  
+ 
+According to our calculations, the duration of the phase of active current growth in a multi-point 
+system corresponds to the time it takes for the fronts of the individual space charge “clouds” to develop 
+from the coronating sources until the formation of a united space charge layer. This time can be estimated 
+as the time when the radius of the front of an individual space charge “cloud”, Rf, reaches D/2 (see figure 
+9).   
+
+=== PAGE 9 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+9
+ 
+Fig. 9. The evolution in time of the radius of the front of an individual space charge “cloud” 
+developed from a central electrode in a multi-point system with D = 20 m. Conditions are similar to those 
+in figure 6. 
+ 
+It may be concluded that, to calculate the corona current emitted from a unit area of the ground 
+surface during thunderstorms, there is no need to consider geometry of coronating extremities on the 
+ground surface. With a good accuracy, current density could be estimated from the rate of rise of an 
+undisturbed thundercloud electric field using equation (5). The corona space charge emitted from a unit 
+area of the ground surface can be estimated in a similar way. From (4), this charge depends on the 
+geometry properties of a coronation system only indirectly, via the corona onset atmospheric electric field, 
+E0cor. Under most practically important thunderstorm conditions, we have E0 >> E0cor. In this case, the 
+value of q turns out to be independent of the system parameters and is equal to  
+qmax  ε0E0max ,           (7) 
+where E0max is the peak thunderstorm electric field. For instance, we have qmax  0.53 μC m-2 for E0max = 60 kV 
+m-1 [Soula and Chauzy 1991].  
+ 
+ELECTRIC FIELD PROFILES ABOVE MULTI-POINT CORONATING SYSTEM  
+Our calculations showed that corona properties for a multi-point system are controlled by an 
+undisturbed thundercloud electric field, E0(t). Its direct measurement is not easy to make because of the 
+effect of corona space charge layer. The local electric field near coronating sources is stabilized at the 
+level of the corona onset electric field. Electric field in the corona space charge layer is lower than E0 due 
+to this charge and, only outside of the layer (outside of the double electrostatic plane layer), a 
+thundercloud electric field is not disturbed. 
+
+=== PAGE 10 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+10 
+ In an 1D approximation, electric field profiles above an emitting plane can be exactly found from 
+equations (2) and (3) in an analytical way [Bazelyan et al. 2014b]. Figure 10 shows the temporal evolution 
+of the electric field at different altitudes in this case when the thundercloud electric field rises linearly up 
+to 60 kV m-1 for 30 s and then is kept constant. Electric field at any altitude is equal to the thundercloud 
+electric field until the front of the space charge layer reaches this altitude. Then, the local electric field, 
+E(t), is stabilized. Stabilization is obtained only for a linearly rising thundercloud field, E0(t) ~ t. In the 
+general case the local electric field inside the corona space charge layer increases in time for d2E0/dt2 > 0 
+and decreases in time at d2E0/dt2 < 0. This means that a sensor, being placed inside the corona space 
+charge layer, registers a local electric filed that not only can differ quantitatively from the undisturbed 
+thundercloud electric field, but can have even opposite temporal tendency as well. This is demonstrated in 
+figure 11 that shows the temporal evolution of the electric field at different altitudes above an emitting 
+plane when the thundercloud electric field E0(t) rises in time in a relaxation manner, 
+
+
+
+
+
+
+/
+max
+)
+(
+t
+e
+E
+t
+E
+1
+0
+0
+.     (8) 
+Here, we have d2E0/dt2 < 0 and the local electric field inside the space charge layer decreases in time 
+although dE0/dt > 0. 
+ 
+ 
+Fig. 10. The evolution in time of the electric field at different altitudes above an emitting plane at Е0cor 
+=1.65 kV m-1. The dashed curve corresponds to the thundercloud electric field that rises linearly in time up 
+to E0 max = 60 kV m-1 for tm = 30 s and then is kept constant. 
+
+=== PAGE 11 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+11 
+ 
+Fig. 11. The evolution in time of the electric field at different altitudes above an emitting plane at Е0cor 
+=1.65 kV m-1. The dashed curve corresponds to the thundercloud electric field that varies as (8) at E0 max = 
+60 kV m-1 and τ = 10 s. The arrows indicate the instants at which the top boundary of the space charge 
+layer reaches given altitudes.  
+ 
+Fig. 12. The evolution in time of the electric field at different altitudes above the central rod in a 
+multi-point system with rods of height h = 10 m and radius r0 = 2 cm. The distance between rods is D = 10 
+m. The number of circles with surrounding rods is N = 100. The altitude is reckoned from the ground. The 
+dashed curve corresponds to the thundercloud electric field that rises linearly in time up to E0 max = 60 kV 
+m-1 for tm = 30 s and then is kept constant. The arrows indicate the instants at which the top boundary of 
+the space charge layer reaches given altitudes. 
+
+=== PAGE 12 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+12 
+ 
+Stabilization of the thundercloud electric field at t > tm leads to a collapse of the corona current. In this 
+case, the corona space charge layer ascends and expands because the top front of the layer moves with a 
+velocity vf  = E0max, whereas the velocity of the bottom boundary of the layer is lower, vb = E0cor.    
+The total electric field behind the top front of the layer decreases in time and tends to E0cor, the electric 
+field at the bottom boundary of the layer.  
+ Figure 12 shows the temporal evolution of the electric field inside the space charge layer above a 
+model multi-point coronating system. The distance between the rods in the system was equal to the rod 
+height. Similarity between the data in figures 12 and 10 is close. In both cases, the total electric field E(t) 
+(i) is close to the undisturbed thundercloud electric field, E0(t), at altitudes above the space charge front, 
+(ii) is stabilized (although with some delay) inside the space charge layer at E0 = At and (iii) sharply 
+decreases at E0 = const. Our calculations show that the vertical electric field profile above a multi-point 
+coronating system tends to the electric field profile above a plane surface emitting ions as the number of 
+electrodes in the system increases. 
+ 
+CONCLUSIONS 
+The developed computer model allows quantitative estimation of the properties of a non-stationary 
+glow corona in the system of grounded hemispherically-tipped electrodes in a thundercloud electric field 
+E0. The properties of the multi-point coronating system asymptotically tend to those of a prefect emitting 
+plane with the surface electric field that is equal to the corona onset atmospheric electric field Е0cor. The 
+field Е0cor is controlled by the dimensions of the individual electrodes and by the distance between them. It 
+is shown that the model of an emitting plane is valid when the individual space charge layers from 
+different coronating points reach each other and form a unite plane layer. The time it takes for the 
+formation of the united layer depends on the distance between coronating electrodes. 
+In the asymptotic approximation, the corona current density is equal to ε0dE0/dt. In this case, the 
+current through each coronating point is independent of the dimensions of the electrodes and depends only 
+on the distance between them. The total corona space charge injected into the atmosphere per unit area of 
+a multi-point system tends asymptotically to the expression q = 0(E0 - E0cor) and depends on the 
+geometrical parameters of the electrodes only indirectly, through the corona onset atmospheric electric 
+field E0cor. Under practically important thunderstorm conditions, it is generally follows from field 
+observations that E0 >> E0cor. In this case, the value of q turns out to be independent of the system 
+parameters.  
+The vertical electric field profile above a multi-point coronating system tends to the electric field 
+profile above a plane emitting surface as the number of electrodes in the system increases. As a result, the 
+electric field distribution tends to be independent of the height of coronating points, whereas the spacing 
+between the electrodes affects only the time it takes to stabilize the electric field profile. 
+Electric field at a given altitude above the ground coronating surface in a thundercloud electric field is 
+equal to this field until the space charge layer reaches this altitude. The evolution in time of the electric 
+field E measured in the space charge layer depends on the rate of change of the thundercloud electric field 
+Е0. The field E (i) undergoes a stabilization when the value of Е0 rises linearly in time, (ii) increases in 
+time at d2E0/dt2 > 0 and decreases in time at d2E0/dt2 < 0. Consequently, simultaneous measurements of 
+
+=== PAGE 13 ===
+XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
+ 
+ 
+13 
+electric field at various levels could produce not only various results, but radically different evolutions in 
+time as well. 
+ACKNOWLEDGMENTS 
+This work was partially supported by the Russian Ministry of Education and Science under the program 
+“5Top100”. 
+REFERENCES 
+Aleksandrov, N. L., Bazelyan, E. M., Carpenter Jr., R. B., Drabkin, M. M., Raizer, Yu. P., 2001: The effect of coronae 
+on leader initiation and development under thunderstorm conditions and in long air gaps. J. Phys. D: Appl. Phys., 
+34, 3256-3266. 
+Aleksandrov, N. L., Bazelyan, E. M., Drabkin, M. M., Carpenter Jr., R. B., Raizer, Yu. P., 2002: Corona discharge at 
+the tip of a high object in the electric field of a thundercloud. Plasma Phys. Rep.., 28, 953-964. 
+Bazelyan, E. M., Raizer, Yu. P., 2000: Lightning attraction mechanism and the problem of lightning initiation by 
+lasers. Physics-Uspekhi, 43, 753 – 769. 
+Bazelyan, E. M., Aleksandrov, N. L., Raizer, Yu. P., Konchakov, A. M., 2007: The effect of air density on 
+atmospheric electric fields required for lightning initiation from a long airborne object. Atmos. Res., 86, 
+126-138. 
+Bazelyan, E. M., Raizer, Yu. P., Aleksandrov, N. L., 2008: Corona initiated from grounded objects under 
+thunderstorm conditions and its influence on lightning attachment. Plasma Sources: Sci. Technol., 17, 024015 
+(17pp). 
+Bazelyan, E. M., Raizer, Yu. P., Aleksandrov, N. L., 2014a: Non-stationary corona around multi-point system in 
+atmospheric electric field: I. Onset electric field and discharge current. J. Atm. Solar-Terr. Phys., 109, 80-90. 
+Bazelyan, E. M., Raizer, Yu. P., Aleksandrov, N. L., 2014b: Non-stationary corona around multi-point system in 
+atmospheric electric field: I. Altitude and time variation of electric field. J. Atm. Solar-Terr. Phys., 109, 91-101. 
+Becerra, M., 2013: Glow corona generation and streamer inception at the tip of grounded objects during 
+thunderstorms: revisited. J. Phys. D: Appl. Phys., 46, 135205. 
+Malik, N. H., 1989: A review of the charge simulation method and its applications. IEEE Trans. Electr. Insul., 24, 
+3-20. 
+Mokrov, M. S., Raizer, Yu. P., Bazelyan, E. M., 2013: Development of a positive corona from a long grounded wire 
+in a growing thunderstorm field. J. Phys. D: Appl. Phys., 46, 455202. 
+Raizer, Yu. P., 1991: Gas Discharge Physics, Springer, Berlin, Germany. 
+Soula, S., Chauzy, S., 1991: Multilevel measurement of the electric field underneath a thundercloud. 2. Dynamical 
+evolution of a ground space charge layer. J. Geophys. Res., 96, No D12, 22327-22336. 
+ 

run.bat

@@ -1,59 +0,0 @@
-@echo off
-REM Tesla Coil Spark Course - Launch Script
-REM Creates virtual environment and runs PyQt5 application
-
-echo ========================================
-echo Tesla Coil Spark Physics Course
-echo ========================================
-echo.
-
-REM Navigate to spark-lessons directory
-cd spark-lessons
-
-REM Check if virtual environment exists
-if not exist venv (
-    echo [*] Creating virtual environment...
-    python -m venv venv
-    if errorlevel 1 (
-        echo [ERROR] Failed to create virtual environment!
-        echo Please ensure Python 3.8+ is installed and in PATH.
-        pause
-        exit /b 1
-    )
-    echo [OK] Virtual environment created
-)
-
-REM Activate virtual environment
-echo [*] Activating virtual environment...
-call venv\Scripts\activate.bat
-
-REM Check if dependencies are installed
-if not exist venv\installed.flag (
-    echo [*] Installing dependencies...
-    python -m pip install --upgrade pip
-    pip install -r requirements.txt
-    if errorlevel 1 (
-        echo [ERROR] Failed to install dependencies!
-        pause
-        exit /b 1
-    )
-    echo. > venv\installed.flag
-    echo [OK] Dependencies installed
-)
-
-REM Run the application
-echo [*] Launching Tesla Coil Spark Course...
-echo.
-python app/main.py
-
-REM Capture exit code
-set EXIT_CODE=%ERRORLEVEL%
-
-REM Deactivate virtual environment
-call deactivate
-
-REM Return to original directory
-cd ..
-
-REM Exit with application's exit code
-exit /b %EXIT_CODE%

spark-lessons/CIRCUIT-SPECIFICATIONS.md

@@ -1,470 +0,0 @@
-# Circuit Diagram Specifications
-
-This document provides **exact specifications** for creating 7 circuit diagrams that require manual attention for professional quality.
-
-**Recommended tools:** LTspice, CircuitLab, Inkscape, KiCad schematic editor, or professional drawing software.
-
-**Format:** PNG, 150 DPI minimum, white background
-
----
-
-## Circuit 1: Geometry to Circuit Translation
-
-**Filename:** `lessons/01-fundamentals/assets/geometry-to-circuit.png`
-**Size:** 1000 x 600 px
-**Referenced in:** fund-02 (Basic Circuit Model)
-
-### Description
-Side-by-side diagram showing physical geometry on left, equivalent circuit on right.
-
-### Left Side: 3D Visualization (Conceptual)
-```
-[Sketch/photo showing:]
-- Toroidal topload (or spherical)
-- Cylindrical spark channel extending downward
-- Ground plane at bottom
-- Arrows/labels indicating:
-  * C_mut (coupling between topload and spark)
-  * C_sh (spark to ground)
-```
-
-**Note:** Can use simplified 2D side-view sketch if 3D is difficult.
-
-### Right Side: Circuit Schematic
-
-**Topology (CRITICAL - verify this is correct):**
-
-```
-Topload node
-    |
-    +----[C_mut]----+
-    |               |
-    +----[R]--------+
-    |
-    (Spark tip node)
-    |
-    [C_sh]
-    |
-   GND
-```
-
-**Component values to show:**
-- R: Variable (or "R_spark")
-- C_mut: "~8 pF" (typical)
-- C_sh: "~6 pF" (typical)
-
-**Layout guidelines:**
-- Vertical orientation
-- Clear node labels: "Topload", "Spark Tip", "GND"
-- R and C_mut in parallel (side-by-side, same start/end nodes)
-- C_sh in series below the parallel combination
-
-**Alternative if parallel is hard:** Show as impedance block "Z_mut = R || C_mut"
-
----
-
-## Circuit 2: Current Paths Diagram
-
-**Filename:** `lessons/01-fundamentals/assets/current-paths-diagram.png`
-**Size:** 1000 x 1200 px (vertical)
-**Referenced in:** fund-07 (Measurement Port)
-
-### Description
-Complete Tesla coil schematic showing **all** current return paths.
-
-### Schematic Components
-
-**Primary circuit (left side):**
-```
-[AC Source] -→ [IGBT/Switch] -→ [C_pri] -→ [L_pri] -→ GND
-```
-
-**Secondary circuit (right side, magnetically coupled):**
-```
-L_sec (coil symbol, coupled to L_pri via k = 0.1-0.2)
-   |
-   +-- [C_topload] --|
-   |                 |
-   +-- [Spark]       |
-   |                 |
-   +-- [C_stray] ----+
-   |
-  GND
-```
-
-**Current paths to label (USE DIFFERENT COLORS):**
-1. **I_spark** (RED): Through spark resistance
-2. **I_displacement** (BLUE): Through C_topload to ground
-3. **I_coupling** (GREEN): Primary-to-secondary capacitive coupling
-4. **I_secondary** (PURPLE): Distributed capacitance along secondary
-5. **I_base** (BLACK, THICK): Total current at secondary base
-
-**Key annotation:**
-```
-I_base = I_spark + I_displacement + I_coupling + I_secondary + ...
-```
-
-**Mark measurement points:**
-- Correct: "Measure here" at topload-to-ground (V_top / I_spark)
-- Incorrect: "NOT here" with X at base (V_top / I_base)
-
-### Layout Guidelines
-- Primary on left, secondary on right
-- Clear coupling indicator (dashed lines or k = 0.1-0.2)
-- Use arrows for current directions
-- Color code or use different line styles for each current path
-- Legend showing which color = which current
-
----
-
-## Circuit 3: Thévenin Equivalent Circuit
-
-**Filename:** `lessons/02-optimization/assets/thevenin-equivalent-circuit.png`
-**Size:** 800 x 600 px
-**Referenced in:** opt-04 (Thévenin Calculations)
-
-### Description
-Simple Thévenin equivalent driving a spark load.
-
-### Schematic
-
-```
-    +-------[R_th]-----[jX_th]------+
-    |                                |
-[V_th source]                   [Z_spark load]
-    |                                |
-    +--------------------------------+
-```
-
-**More detailed Z_spark:**
-```
-Z_spark can be shown as:
-    [R_spark] in series with [jX_spark]
-    OR
-    [(R || C_mut) in series with C_sh]
-```
-
-**Component labels:**
-- V_th: "350 kV" (typical value)
-- R_th: "114 Ω" (typical)
-- X_th: "-j2424 Ω" (typical, capacitive)
-- Z_spark: "Variable"
-
-**Annotations:**
-- "Thévenin Equivalent" label on left side
-- "Spark Load" label on right side
-- Formula below: **P = 0.5|V_th|² Re{Z_spark} / |Z_th + Z_spark|²**
-
-### Layout Guidelines
-- Horizontal orientation, left to right
-- V_th source on left
-- R_th and X_th clearly in series
-- Load impedance on right
-- Clean, minimal style
-
----
-
-## Circuit 4: Capacitive Divider Circuit
-
-**Filename:** `lessons/03-spark-physics/assets/capacitive-divider-circuit.png`
-**Size:** 600 x 800 px (vertical)
-**Referenced in:** phys-07 (Capacitive Divider)
-
-### Description
-Shows voltage division across C_mut and C_sh.
-
-### Schematic
-
-```
-V_topload (source)
-    |
-    +----[C_mut]----+
-    |               |
-    +----[R]--------+
-    |
-    V_tip (measurement point) ← mark this clearly
-    |
-    [C_sh]
-    |
-   GND
-```
-
-**Component labels:**
-- V_topload: "Input"
-- C_mut: "~10 pF"
-- C_sh: "~6.6 L (pF)" where L is in meters
-- R: "R_spark"
-- V_tip: Mark with voltmeter symbol or arrow
-
-**Key formula (below circuit):**
-```
-V_tip = V_topload × [C_mut / (C_mut + C_sh)]
-
-C_sh grows with spark length: ~6.6 pF/m
-```
-
-### Layout Guidelines
-- Vertical orientation
-- Show V_tip measurement clearly (voltmeter symbol or highlighted node)
-- Annotate that C_sh increases with length
-- Clean parallel R||C_mut representation
-
----
-
-## Circuit 5: Lumped Model Schematic
-
-**Filename:** `lessons/04-advanced-modeling/assets/lumped-model-schematic.png`
-**Size:** 800 x 600 px
-**Referenced in:** model-01 (Lumped Model)
-
-### Description
-Clean, professional lumped spark model circuit.
-
-### Schematic (Same topology as Circuit 1, but cleaner)
-
-```
-Port (Topload connection)
-    |
-    +----[R]--------+
-    |               |
-    +----[C_mut]----+
-    |
-    (Spark tip - internal node)
-    |
-    [C_sh]
-    |
-   GND
-```
-
-**Component values:**
-- R: "50 kΩ (typical)"
-- C_mut: "8 pF (typical)"
-- C_sh: "6 pF (typical)"
-
-**Annotations:**
-- "Port" or "Topload Connection" at top
-- "Internal Node" at spark tip
-- Box or note: "Typical values at 200 kHz for 3-foot spark"
-
-### Layout Guidelines
-- Very clean, professional appearance
-- Grid-aligned components
-- Perfect parallel alignment for R || C_mut
-- Clear port indication (terminal symbols)
-- Minimal, uncluttered
-
----
-
-## Circuit 6: Distributed Model Structure
-
-**Filename:** `lessons/04-advanced-modeling/assets/distributed-model-structure.png`
-**Size:** 1200 x 600 px (horizontal)
-**Referenced in:** model-03 (Distributed Model)
-
-### Description
-Shows n-segment distributed model with proper transmission-line style layout.
-
-### Schematic
-
-**Horizontal cascade layout (recommended):**
-
-```
-Topload --[C_01]-- Node1 --[C_12]-- Node2 -- ... --[C_n-1,n]-- Node_n
-           |                |                          |
-         [R_1]            [R_2]                      [R_n]
-           |                |                          |
-        [C_1,gnd]        [C_2,gnd]                 [C_n,gnd]
-           |                |                          |
-          GND              GND                        GND
-```
-
-**Alternative vertical cascade** (if horizontal too wide):
-```
-Topload
-   |
-[C_01]
-   |
-Node 1 --[R_1]--
-   |            |
-[C_1,gnd]      (parallel)
-   |
-[C_12]
-   |
-Node 2 --[R_2]--
-   |            |
-[C_2,gnd]      (parallel)
-   |
-  ...
-```
-
-**Component labeling:**
-- Show first 2 segments explicitly
-- Use "..." for middle segments
-- Show last segment (segment n)
-- Label: "n = 5 to 20 segments (typically n = 10)"
-
-**Capacitance matrix note:**
-- Annotation: "(n+1) × (n+1) capacitance matrix"
-- "Extracted from FEMM electrostatic analysis"
-
-### Layout Guidelines
-- Clear repeating pattern
-- Ellipsis (...) to indicate continuation
-- Symmetric, professional appearance
-- Not too cluttered
-
----
-
-## Circuit 7: Tesla Coil System Overview
-
-**Filename:** `assets/shared/tesla-coil-system-overview.png`
-**Size:** 1400 x 1000 px
-**Referenced in:** Multiple lessons
-
-### Description
-Complete DRSSTC system diagram showing all major components.
-
-### Schematic Components
-
-**Primary tank circuit:**
-```
-[DC Bus] → [H-Bridge / IGBT switches] → [C_pri (MMC)] → [L_pri] → GND
-                  ↑
-            [Gate Driver]
-                  ↑
-            [Feedback/Control]
-```
-
-**Secondary resonator:**
-```
-L_sec (large coil symbol, coupled to L_pri via k)
-   |
-[C_topload]
-   |
-[Spark gap or streamer symbol]
-   |
-[Strike point / GND]
-```
-
-**Annotations:**
-- Coupling coefficient: "k = 0.1 to 0.2"
-- Primary frequency: "f_pri = f_resonant"
-- Secondary resonance: "f_sec = 1/(2π√(L_sec × C_top))"
-- Power flow arrows
-- "DRSSTC" or "Double-Resonant Solid State Tesla Coil" title
-
-**Components to show:**
-- DC power supply
-- Full bridge (4 IGBTs/MOSFETs) or half bridge
-- MMC (multiple capacitors in series-parallel)
-- Primary coil (few turns, heavy wire)
-- Secondary coil (many turns, fine wire)
-- Topload (toroid or sphere symbol)
-- Spark/streamer
-- Feedback path (CT or antenna back to controller)
-- Ground connections
-
-### Layout Guidelines
-- Primary on left or bottom
-- Secondary on right or top
-- Clear separation of power vs signal paths
-- Coupling indicated (dashed lines, double-headed arrow, or k annotation)
-- Professional, complete system view
-- Include legend if needed
-
----
-
-## General Guidelines for All Circuits
-
-### Style
-- **Clean, professional appearance**
-- Grid-aligned components
-- Consistent component symbols (IEEE or European standard)
-- Clear, readable labels (minimum 10pt font)
-- No overlapping text or components
-- White background
-
-### Components Symbols
-- Resistor: Standard zigzag (IEEE) or rectangle (IEC)
-- Capacitor: Two parallel lines
-- Inductor: Coil/loops
-- Ground: Standard ground symbol
-- AC source: Sine wave in circle
-- Voltage source: Circle with +/- or V label
-
-### Colors (if used)
-- Use sparingly, only for clarity
-- Current paths: different colors
-- Otherwise: black on white for print compatibility
-
-### Verification
-**CRITICAL:** Before finalizing any circuit:
-1. Verify topology matches spark-physics.txt equations
-2. Check that parallel vs series connections are correct
-3. Ensure component values are realistic (refer to physical-bounds.md)
-4. Review against worked examples for consistency
-
----
-
-## Priority Order
-
-**High Priority (needed for core lessons):**
-1. Circuit 5: Lumped Model Schematic
-2. Circuit 4: Capacitive Divider
-3. Circuit 3: Thévenin Equivalent
-
-**Medium Priority:**
-4. Circuit 1: Geometry to Circuit
-5. Circuit 6: Distributed Model
-
-**Low Priority (nice-to-have):**
-6. Circuit 2: Current Paths (complex, can use text description initially)
-7. Circuit 7: System Overview (general reference, not lesson-critical)
-
----
-
-## Tools Recommendations
-
-**Easy (recommended for quick creation):**
-- **CircuitLab** (web-based, clean output)
-- **LTspice** (free, professional, can export schematics)
-- **Falstad Circuit Simulator** (web-based, can screenshot)
-
-**Professional (for publication quality):**
-- **KiCad Schematic Editor** (free, excellent output)
-- **Inkscape** (manual drawing with circuit symbols)
-- **Adobe Illustrator / Affinity Designer** (professional vector graphics)
-
-**Advanced (if familiar with LaTeX):**
-- **CircuiTikZ** + LaTeX (publication-quality output)
-
----
-
-## Validation Checklist
-
-Before considering a circuit "done":
-
-- [ ] Topology verified against spark-physics.txt
-- [ ] Component values realistic and labeled
-- [ ] No overlapping elements
-- [ ] Grid-aligned, professional appearance
-- [ ] Clear node labels where needed
-- [ ] Formula or key annotation included
-- [ ] 150 DPI or vector format (scalable)
-- [ ] White background, high contrast
-- [ ] Filename matches specification
-- [ ] Placed in correct assets directory
-
----
-
-## Notes
-
-- These specifications are based on analysis of spark-physics.txt
-- Some topologies (especially parallel R||C_mut) are tricky - verify carefully
-- When in doubt, consult reference physics document
-- Can simplify complex parallel combinations as impedance blocks (Z = R||C) if clearer
-- Professional quality > programmatic generation
-
-**Created:** 2025-10-10
-**Status:** Awaiting manual creation
-**Current:** 0/7 circuits completed