--- id: model-01 title: "Lumped Spark Model Theory" section: "Advanced Modeling" difficulty: "advanced" estimated_time: 35 prerequisites: ["phys-09", "phys-10", "phys-11"] objectives: - 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 tags: ["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:** 1. **C_mut (Mutual Capacitance):** Capacitance between topload and spark channel - Typical range: 5-15 pF - Extracted from FEMM electrostatic analysis 2. **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 3. **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:** 1. **Short to Medium Sparks (<1-2 m)** - Uniform properties dominate - Single R approximation valid 2. **Impedance Matching Studies** - Quick evaluation of different topload sizes - Coil-level optimization - Matching network design 3. **First-Order Power Estimates** - Energy transfer calculations - Efficiency predictions - Quick design iterations 4. **Engineering Estimates** - Performance predictions - Component selection - Safety margins **Computational cost:** <1 second per simulation ### When Lumped Models Fail **Switch to distributed models when:** 1. **Long Sparks (>2-3 m)** - Base vs tip properties differ significantly - Leader/streamer transition critical - Current distribution non-uniform 2. **Current Distribution Matters** - Measuring actual current along spark - Validating against detailed measurements - Research applications 3. **Extreme Parameters** - Very low frequency (λ approaches L) - Very high voltage (breakdown physics critical) - Unusual geometries 4. **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:** ```spice * 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:** 1. **Loaded Q reduction:** ``` Q_loaded < Q_unloaded More resistive spark → lower Q → faster ringdown ``` 2. **Resonant frequency shift:** ``` f_loaded ≠ f₀ Spark adds capacitance → lowers frequency Magnitude: Δf ≈ 1-5 kHz typical ``` 3. **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** ```spice 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](02-femm-extraction-lumped.md)