plasma-expert/context/lumped-model.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
lumped-model	Lumped Spark Model: Single-Element Circuit Representation	established	spark-physics.txt: Part 7 (lines 442-537), Part 10.1 (lines 705-713), Part 11 (lines 736-803)	[circuit-topology power-optimization thevenin-method coupled-resonance distributed-model femm-workflow capacitive-divider field-thresholds energy-and-growth empirical-scaling equations-and-bounds]	[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]	[mutual capacitance shunt capacitance Maxwell capacitance matrix self-capacitance ringdown method conductance parallel RLC Rogowski coil VNA]	[capacitance-matrix-heatmap.png lumped-vs-distributed-comparison.png lumped-model-validation-checks.png femm-geometry-setup-lumped.png field-lines-capacitances.png]	[femm-lumped-extraction.md]	[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/(2pisqrt(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