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id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
model-01	Lumped Spark Model Theory	Advanced Modeling	advanced	35	[phys-09 phys-10 phys-11]	[Understand single-element lumped model structure and assumptions Learn when lumped models are appropriate vs distributed models Master the complete workflow for building lumped spark models Integrate lumped spark models with full Tesla coil circuit analysis]	[modeling lumped-model circuit-theory SPICE]

Lumped Spark Model Theory

The lumped spark model treats the entire spark as a single equivalent circuit element. This is the simplest and most computationally efficient approach for Tesla coil spark modeling, suitable for most practical engineering applications.

What is a Lumped Model?

Circuit Structure

The lumped spark model represents the spark channel as three components:

Topload (V_top)
    |
    +---[C_mut]---+---[R]---+---[C_sh]---+
                  |                       |
                Node              Node    GND

Components:

C_mut (Mutual Capacitance): Capacitance between topload and spark channel
- Typical range: 5-15 pF
- Extracted from FEMM electrostatic analysis
R (Plasma Resistance): Effective resistance of the entire spark
- Typical range: 10-500 kΩ at 200 kHz
- Optimized for maximum power transfer
- Variable, depends on plasma state
C_sh (Shunt Capacitance): Capacitance from spark to ground
- Typical rule: ~2 pF/foot of spark length
- Also extracted from FEMM
- Critical for capacitive divider effect

Physical Meaning

The lumped model assumes:

Uniform current distribution along spark
Single averaged resistance value
Quasi-static voltage distribution
Spark can be treated as electrically short at operating frequency

This works when:

λ >> L (wavelength much greater than spark length)
At 200 kHz: λ = 1500 m, sparks typically <3 m
Distributed effects are second-order corrections

When to Use Lumped Models

Appropriate Applications

Use lumped models for:

Short to Medium Sparks (<1-2 m)
- Uniform properties dominate
- Single R approximation valid
Impedance Matching Studies
- Quick evaluation of different topload sizes
- Coil-level optimization
- Matching network design
First-Order Power Estimates
- Energy transfer calculations
- Efficiency predictions
- Quick design iterations
Engineering Estimates
- Performance predictions
- Component selection
- Safety margins

Computational cost: <1 second per simulation

When Lumped Models Fail

Switch to distributed models when:

Long Sparks (>2-3 m)
- Base vs tip properties differ significantly
- Leader/streamer transition critical
- Current distribution non-uniform
Current Distribution Matters
- Measuring actual current along spark
- Validating against detailed measurements
- Research applications
Extreme Parameters
- Very low frequency (λ approaches L)
- Very high voltage (breakdown physics critical)
- Unusual geometries
Publication-Quality Results
- Peer review requires distributed model
- Detailed physics validation

Trade-off: Distributed models 1000-2000× slower

Complete Lumped Model Workflow

Step 1: FEMM Electrostatic Analysis

Setup requirements:

Geometry:
- Axisymmetric (r-z coordinates)
- Topload: toroid or sphere
- Spark: vertical cylinder
- Ground plane below

Problem type:
- Electrostatic (frequency = 0)
- Two conductors: topload (V=1V), spark (floating)
- Ground boundary condition

Solve:
- Extract 2×2 capacitance matrix [C]

Detailed FEMM procedure covered in next lesson.

Step 2: Extract Circuit Elements

From FEMM capacitance matrix:

       [Topload] [Spark]
[Top]  [  C₁₁     C₁₂  ]
[Spark][  C₂₁     C₂₂  ]

Where:
- C_ii > 0 (diagonal: self-capacitance)
- C_ij < 0 (off-diagonal: mutual capacitance, negative)
- C₁₂ = C₂₁ (symmetric)

Extraction formulas:

Mutual capacitance:

C_mut = |C₁₂| = |C₂₁|

Take absolute value of off-diagonal element.

Shunt capacitance:

C_sh = C₂₂ + C₂₁
     = C₂₂ - |C₁₂|    (since C₂₁ < 0)

This is spark-to-ground capacitance with topload present.

Step 3: Calculate Optimal Resistance

Power-optimal resistance formula:

R_opt_power = 1 / (ω × C_total)

Where:
  ω = 2πf (angular frequency)
  C_total = C_mut + C_sh

Physical basis: Hungry streamer theory

Plasma adjusts to maximize power extraction
R = 1/(ωC) gives optimal power transfer for capacitive load
Valid for streamer-dominated discharge

Apply physical bounds:

R_min = 5 kΩ    (hot leader, best case)
R_max = 500 kΩ  (cool streamer, worst case)

R_clipped = clip(R_opt_power, R_min, R_max)

Use R_clipped in final model.

Step 4: Build SPICE Netlist

Example SPICE implementation:

* Lumped spark model - Tesla coil discharge
.param freq=200k
.param omega={2*pi*freq}

* Operating frequency
* Angular frequency

* Test voltage source (or connect to coil model)
V_topload topload 0 AC 1V

* Spark circuit elements
C_mut topload spark_node {C_mut_value}
R_spark spark_node spark_r {R_value}
C_sh spark_r 0 {C_sh_value}

* AC analysis
.ac lin 1 {freq} {freq}

* Output admittance at topload
.print ac v(topload) i(V_topload) vp(topload) ip(V_topload)

.end

Step 5: Run AC Analysis and Extract Results

Calculate admittance:

Y = I / V    (complex admittance)

Re{Y} = real part (conductance)
Im{Y} = imaginary part (susceptance)

Convert to impedance if needed:

Z = 1/Y

|Z| = magnitude
φ_Z = phase angle

Calculate power (for actual operating voltage):

P_spark = 0.5 × |V_actual|² × Re{Y}

Example:
If V_actual = 320 kV, Re{Y} = 1.5 μS
P_spark = 0.5 × (320×10³)² × 1.5×10⁻⁶
        = 76.8 kW

Step 6: Validation Checks

1. Phase angle check:

Expected: φ_Z = -55° to -75°
(Capacitive-resistive, more capacitive than resistive)

If outside range:
- Check C values (FEMM errors?)
- Check R (unphysical value?)
- Review frequency

2. Resistance range check:

At 200 kHz:
- Short spark (0.5 m): R ≈ 50-150 kΩ
- Medium spark (1.5 m): R ≈ 100-300 kΩ
- Long spark (3 m): R ≈ 200-500 kΩ

If much higher: likely streamer-dominated (OK but low power)
If much lower: check calculations

3. Capacitance validation:

C_sh ≈ 2 pF/foot × L_spark

Within factor of 2 is acceptable:
- Higher: concentrated field near ground
- Lower: elevated geometry, less ground coupling

Exact match not expected (geometry dependent)

4. Compare to measurements:

If available:
- Ringdown frequency shift → Y_spark
- E-field probe + current probe → Z_spark

Adjust R within bounds to match measurements

Integration with Full Coil Model

Connection to Secondary Circuit

The lumped spark model appears as a load impedance at the topload terminal:

[Primary] → [Coupled Transformer] → [Secondary L_sec, R_sec] → [C_topload] → [Z_spark]
                                                                                  ↓
                                                                                 GND

Effects on coil performance:

Loaded Q reduction:

Q_loaded < Q_unloaded

More resistive spark → lower Q → faster ringdown

Resonant frequency shift:

f_loaded ≠ f₀

Spark adds capacitance → lowers frequency
Magnitude: Δf ≈ 1-5 kHz typical

Power extraction:

P_spark = fraction of total power

Well-matched: 50-70% to spark
Poorly matched: <30% to spark

Impedance Matching

For maximum power transfer:

Want: Z_spark ≈ Z_secondary*

Where Z_secondary* is complex conjugate of secondary impedance

Practical approach:
- Adjust C_topload to tune frequency
- Spark length determines Z_spark
- Iterate to find optimal balance

Trade-offs:

Larger topload: better coupling, heavier load
Smaller topload: higher voltage, weaker coupling
Spark impedance: fixed by physics (less control)

Worked Example: Complete Lumped Model

Given parameters:

Frequency: f = 190 kHz
FEMM results: C_mut = 9.5 pF, C_sh = 7.2 pF
Physical bounds: R_min = 5 kΩ, R_max = 500 kΩ

Step 1: Calculate R_opt_power

ω = 2π × 190×10³ = 1.194×10⁶ rad/s

C_total = C_mut + C_sh
        = 9.5 + 7.2
        = 16.7 pF

R_opt = 1/(ω × C_total)
      = 1/(1.194×10⁶ × 16.7×10⁻¹²)
      = 1/(1.994×10⁻⁵)
      = 50.2 kΩ

Step 2: Check bounds

R_min = 5 kΩ
R_opt = 50.2 kΩ    ✓ Within bounds
R_max = 500 kΩ

Use R = 50.2 kΩ

Step 3: Build SPICE model

V_test topload 0 AC 1V
C_mut topload n1 9.5p
R_spark n1 n2 50.2k
C_sh n2 0 7.2p

.ac lin 1 190k 190k
.end

Step 4: Simulate (example results)

Y = I/V = 5.23 μS ∠74.5°

Re{Y} = 5.23 × cos(74.5°) = 1.39 μS
Im{Y} = 5.23 × sin(74.5°) = 5.04 μS

Convert to Z:
|Z| = 1/5.23×10⁻⁶ = 191 kΩ
φ_Z = -74.5°

Step 5: Validate

✓ φ_Z = -74.5° in expected range (-55° to -75°)
✓ R_eq ≈ 51 kΩ close to R_opt = 50.2 kΩ
✓ Physical: Between 5-500 kΩ

C_sh check:
L ≈ 7.2 pF / (2 pF/ft) = 3.6 ft ≈ 1.1 m
✓ Reasonable for medium spark

Step 6: Power calculation (if V_topload = 320 kV actual)

P = 0.5 × |V|² × Re{Y}
  = 0.5 × (320×10³)² × 1.39×10⁻⁶
  = 71.2 kW

Model complete and ready for coil integration!

Key Takeaways

Lumped model treats spark as single R-C-C network: simple, fast, accurate for most cases
Use for: sparks <2 m, impedance matching, engineering estimates, quick iterations
FEMM extraction: C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂| from Maxwell matrix
Optimal resistance: R = 1/(ω × C_total) from hungry streamer theory, with physical bounds
Validation checks: phase angle, resistance range, C_sh ≈ 2 pF/ft, compare to measurements
Integration: appears as load impedance at topload, affects Q, frequency, power transfer
When to upgrade: long sparks (>2 m), current distribution needed, research applications

Practice

{exercise:model-ex-01}

Next Lesson: FEMM Extraction for Lumped Models

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