plasma-expert/context/capacitive-divider.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
capacitive-divider	The Capacitive Divider Problem	established	spark-physics.txt: Part 5 Section 5.6 (lines 338-361)	[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]	[voltage-division-complex voltage-division-open-circuit capacitive-divider-impedances]	[capacitive_divider V_tip C_mut C_sh E_tip E_propagation R_opt_power]	[voltage-division-vs-length-plot.png]	[spark-growth-timeline.md]	[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/(jomegaC_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/(jomegaC_mut)) has lower magnitude than 1/(jomegaC_mut) alone
4. V_tip magnitude decreases and acquires a phase shift

The quantitative effect depends on the ratio omegaC_mutR:

• When omegaC_mutR >> 1 (R large, streamer-like): approaches open-circuit limit
• When omegaC_mutR ~ 1 (R near R_opt_power): V_tip reduced by ~20-30%
• When omegaC_mutR << 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)