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7327 lines
183 KiB
7327 lines
183 KiB
# Tesla Coil Spark Modeling - Complete Lesson Plan Index
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## Overview
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This lesson plan is designed to take someone from basic circuit concepts through advanced Tesla coil spark modeling. Each part builds progressively, with worked examples, visual aids descriptions, and practice problems.
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---
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## **Part 1: Foundation - Circuits, Impedance, and Basic Spark Behavior**
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*Target: 2-3 hours of study*
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### Module 1.1: AC Circuit Fundamentals Review
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- Peak vs RMS values (why we use peak)
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- Complex numbers and phasor notation (j, magnitude, phase)
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- Resistance (R), Reactance (X), Impedance (Z)
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- Conductance (G), Susceptance (B), Admittance (Y)
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- Power in AC circuits: P = 0.5 × Re{V × I*}
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- **Worked Example 1.1:** Calculate power with peak phasors
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- **Practice Problems:** 3 problems on complex impedance calculations
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### Module 1.2: Capacitance in Tesla Coils
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- What is capacitance physically?
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- Self-capacitance vs mutual capacitance
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- Capacitance to ground (shunt capacitance)
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- The 2 pF/foot empirical rule
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- **Worked Example 1.2:** Estimate C_sh for a 2-meter spark
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- **Visual Aid:** Diagram showing field lines for C_mut and C_sh
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- **Practice Problems:** 2 problems on capacitance estimation
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### Module 1.3: The Basic Spark Circuit Topology
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- Why spark has TWO capacitances (C_mut and C_sh)
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- Drawing the circuit: parallel R||C_mut in series with C_sh
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- Where is "ground" in a Tesla coil?
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- The topload port (measurement reference)
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- **Worked Example 1.3:** Draw circuit for given geometry
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- **Visual Aid:** 3D geometry → circuit schematic translation
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- **Practice Problems:** 2 problems on circuit topology
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### Module 1.4: Admittance Analysis of the Spark Circuit
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- Why use admittance (Y) instead of impedance (Z)?
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- Parallel combinations are easy in Y
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- Deriving Y_total = ((G+jB₁)·jB₂)/(G+j(B₁+B₂))
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- Real and imaginary parts
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- Converting back to impedance
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- **Worked Example 1.4:** Calculate Y and Z for specific values
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- **Visual Aid:** Complex plane plots showing Y and Z
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- **Practice Problems:** 3 problems on admittance calculations
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### Module 1.5: Phase Angles and What They Mean
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- Impedance phase φ_Z vs admittance phase θ_Y
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- Why φ_Z = -θ_Y
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- The "famous -45°" myth
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- Physical meaning: how much does load look resistive?
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- **Worked Example 1.5:** Calculate φ_Z from given R, C_mut, C_sh
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- **Visual Aid:** Phase angle on complex plane
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- **Practice Problems:** 2 problems on phase angle interpretation
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### Module 1.6: Introduction to Spark Physics
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- What is a spark? (brief non-mathematical overview)
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- Streamers vs leaders (qualitative)
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- Why sparks need voltage AND power
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- The "hungry streamer" principle (conceptual introduction)
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- **Visual Aid:** Photos/diagrams of streamers vs leaders
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- **Discussion Questions:** 3 conceptual questions
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### Part 1 Summary & Integration
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- Checkpoint quiz (10 questions, multiple choice + short answer)
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- Concept map connecting all Module 1 topics
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- Preview of Part 2
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**Estimated Token Count: ~15,000-18,000**
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---
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## **Part 2: Optimization and Power Transfer - Making Sparks Efficient**
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*Target: 2-3 hours of study*
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### Module 2.1: The Topological Phase Constraint
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- What is a topological constraint?
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- Deriving φ_Z,min = -atan(2√(r(1+r)))
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- Why r = C_mut/C_sh matters
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- The critical value r = 0.207
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- When is -45° impossible?
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- **Worked Example 2.1:** Calculate φ_Z,min for typical geometries
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- **Visual Aid:** Graph of φ_Z,min vs r
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- **Practice Problems:** 3 problems on phase constraints
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### Module 2.2: The Two Critical Resistances
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- R_opt_power: maximum power transfer
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- R_opt_phase: closest to resistive
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- Why R_opt_power < R_opt_phase always
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- Deriving R_opt_power = 1/(ω(C_mut + C_sh))
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- Deriving R_opt_phase = 1/(ω√(C_mut(C_mut + C_sh)))
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- **Worked Example 2.2:** Calculate both for f=200 kHz, various capacitances
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- **Visual Aid:** Power vs R curves showing optima
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- **Practice Problems:** 4 problems on optimal resistances
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### Module 2.3: The "Hungry Streamer" - Self-Optimization
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- How plasma conductivity changes with power
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- Temperature → ionization → conductivity loop
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- Why sparks naturally seek R_opt_power
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- Constraints that prevent optimization
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- Physical limits: R_min and R_max
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- **Worked Example 2.3:** Trace through optimization process
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- **Visual Aid:** Flowchart of self-optimization mechanism
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- **Discussion Questions:** 3 questions on optimization limits
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### Module 2.4: Power Calculations
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- Power to a load: P = 0.5|V|²Re{Z_load}/|Z_th+Z_load|²
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- Why V_top/I_base is wrong
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- Displacement current problem
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- Correct measurement at topload port
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- **Worked Example 2.4:** Calculate power with correct vs incorrect method
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- **Visual Aid:** Current flow diagram showing displacement currents
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- **Practice Problems:** 3 problems on power calculations
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### Module 2.5: Thévenin Equivalent Method (Part A)
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- What is a Thévenin equivalent?
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- Why it separates coil from load
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- Measuring Z_th (output impedance)
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- Step-by-step procedure
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- **Worked Example 2.5A:** Extract Z_th from simulation
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- **Visual Aid:** Circuit diagrams for measurement setup
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- **Practice Problems:** 2 problems on Z_th measurement
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### Module 2.6: Thévenin Equivalent Method (Part B)
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- Measuring V_th (open-circuit voltage)
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- Using Z_th and V_th to predict any load
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- Theoretical maximum power (conjugate match)
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- Why actual spark power is less
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- **Worked Example 2.6:** Complete Thévenin analysis
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- **Visual Aid:** Load line analysis
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- **Practice Problems:** 3 problems on load power prediction
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### Module 2.7: Quality Factor and Ringdown Measurements
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- What is Q? (energy storage vs loss)
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- Q₀ (unloaded) vs Q_L (loaded)
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- Measuring Q from ringdown waveform
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- Extracting spark admittance from Q_L, f_L measurements
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- **Worked Example 2.7:** Q measurement from oscilloscope capture
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- **Visual Aid:** Annotated ringdown waveform
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- **Practice Problems:** 3 problems on Q measurements
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### Part 2 Summary & Integration
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- Checkpoint quiz (12 questions)
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- Worked example combining all of Part 2
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- Design exercise: optimize R for a given coil
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- Preview of Part 3
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**Estimated Token Count: ~18,000-20,000**
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---
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## **Part 3: Growth Physics and FEMM Modeling - Where Sparks Come From**
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*Target: 3-4 hours of study*
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### Module 3.1: Electric Fields and Breakdown
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- Electric field basics (V/m)
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- Field concentration at sharp points
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- E_inception: initial breakdown (~2-3 MV/m)
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- E_propagation: sustained growth (~0.4-1.0 MV/m)
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- Tip enhancement factor κ
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- **Worked Example 3.1:** Calculate E_tip for given voltage and geometry
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- **Visual Aid:** Field line diagram with enhancement
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- **Practice Problems:** 3 problems on field calculations
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### Module 3.2: Energy Requirements for Growth
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- Energy per meter (ε) concept
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- Why different operating modes have different ε
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- QCW: 5-15 J/m (efficient)
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- Burst: 30-100 J/m (inefficient)
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- Physical mechanisms behind ε
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- **Worked Example 3.2:** Calculate energy needed for target length
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- **Visual Aid:** Energy budget breakdown
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- **Practice Problems:** 2 problems on energy requirements
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### Module 3.3: Growth Rate Equation
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- dL/dt = P_stream/ε (when E_tip > E_propagation)
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- Voltage limit vs power limit
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- When does growth stall?
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- Time to reach target length
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- **Worked Example 3.3:** Predict growth time for QCW ramp
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- **Visual Aid:** Length vs time curves for different modes
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- **Practice Problems:** 3 problems on growth dynamics
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### Module 3.4: Thermal Physics of Plasma Channels
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- Temperature in streamers vs leaders
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- Thermal diffusion time constant τ_thermal = d²/(4α)
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- Why observed persistence is longer
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- Convection and ionization memory
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- QCW advantage: maintaining hot channels
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- **Worked Example 3.4:** Calculate thermal time constants
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- **Visual Aid:** Temperature profile cross-section
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- **Practice Problems:** 2 problems on thermal dynamics
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### Module 3.5: The Capacitive Divider Problem
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- How V_tip < V_topload due to C_sh
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- V_tip = V_topload × C_mut/(C_mut+C_sh) (open circuit)
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- Effect of finite R
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- As spark grows, C_sh grows, V_tip drops
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- Why length scales sub-linearly with energy
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- **Worked Example 3.5:** Calculate V_tip for growing spark
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- **Visual Aid:** Equivalent circuit with divider highlighted
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- **Practice Problems:** 3 problems on voltage division
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### Module 3.6: Introduction to FEMM
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- What is FEMM? (Finite Element Method Magnetics)
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- Electrostatic analysis for capacitances
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- Setting up a problem: geometry, boundaries, materials
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- Meshing and solving
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- Extracting results
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- **Worked Example 3.6:** Step-by-step FEMM tutorial (simple geometry)
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- **Visual Aid:** Screenshots of FEMM interface
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- **Practice Problems:** 1 guided FEMM exercise
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### Module 3.7: Extracting Capacitances from FEMM
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- The Maxwell capacitance matrix [C]
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- Diagonal elements: self-capacitances (positive)
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- Off-diagonal: mutual capacitances (negative)
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- For 2-body problem: C_mut = |C₁₂|, C_sh = C₂₂ - |C₁₂|
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- Validation: C_sh ≈ 2 pF/foot check
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- **Worked Example 3.7:** Extract values from FEMM output
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- **Visual Aid:** Annotated capacitance matrix
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- **Practice Problems:** 2 problems on matrix interpretation
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### Module 3.8: Building the Lumped Spark Model
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- Using FEMM capacitances in circuit
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- Choosing R = R_opt_power
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- Clipping to physical bounds (R_min, R_max)
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- Implementing in SPICE
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- Running AC analysis
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- **Worked Example 3.8:** Complete lumped model simulation
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- **Visual Aid:** Flowchart from FEMM to SPICE
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- **Practice Problems:** 1 complete modeling exercise
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### Part 3 Summary & Integration
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- Checkpoint quiz (15 questions)
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- Complete design project: predict spark length for given coil
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- Comparison exercise: simulation vs empirical rules
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- Preview of Part 4
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**Estimated Token Count: ~20,000-22,000**
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---
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## **Part 4: Advanced Topics - Distributed Models and Real-World Application**
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*Target: 3-4 hours of study*
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### Module 4.1: Why Distributed Models?
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- Limitations of lumped model
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- Current distribution along spark
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- Tip vs base differences
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- When is distributed model necessary?
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- **Visual Aid:** Comparison showing where lumped fails
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- **Discussion Questions:** 3 questions on model selection
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### Module 4.2: nth-Order Model Structure
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- Dividing spark into n segments (typically n=10)
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- Circuit topology with multiple segments
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- Capacitance matrix grows to (n+1)×(n+1)
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- Including all segment-to-segment couplings
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- Optional: inductance matrix
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- **Worked Example 4.2:** Draw 3-segment distributed model
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- **Visual Aid:** Progressive complexity (n=1, 3, 5, 10)
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- **Practice Problems:** 2 problems on model structure
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### Module 4.3: FEMM for Distributed Models
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- Multi-body electrostatic analysis
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- Defining n cylindrical segments
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- Extracting large capacitance matrix
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- Matrix properties: symmetric, semi-definite
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- Numerical stability and passivity
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- **Worked Example 4.3:** FEMM setup for n=5 model
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- **Visual Aid:** FEMM geometry with labeled segments
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- **Practice Problems:** 1 FEMM exercise with multiple bodies
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### Module 4.4: Implementing Capacitance Matrices in SPICE
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- Challenge: negative off-diagonal elements
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- Solution 1: Partial capacitance transformation
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- Solution 2: Controlled sources (MNA approach)
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- Solution 3: Nearest-neighbor approximation
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- Validation and stability
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- **Worked Example 4.4:** Convert 3×3 Maxwell to SPICE
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- **Visual Aid:** Circuit comparison of methods
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- **Practice Problems:** 2 problems on matrix implementation
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### Module 4.5: Resistance Optimization - Iterative Method
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- Initialization: tapered R profile
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- Iterative power maximization algorithm
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- Damping for stability (α_damp ≈ 0.3-0.5)
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- Position-dependent bounds: R_min[i], R_max[i]
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- Convergence criteria
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- **Worked Example 4.5:** Hand-trace 3 iterations for small model
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- **Visual Aid:** Flowchart of optimization algorithm
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- **Pseudo-code:** Python-style implementation
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- **Practice Problems:** 2 problems on optimization
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### Module 4.6: Resistance Optimization - Simplified Method
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- Circuit-determined resistance: R[i] = 1/(ω×C_total[i])
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- Weak diameter dependence (logarithmic)
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- When is this good enough?
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- Comparison with iterative method
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- **Worked Example 4.6:** Calculate R distribution for n=10 model
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- **Visual Aid:** Comparison plot: iterative vs simplified
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- **Practice Problems:** 2 problems on simplified method
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### Module 4.7: Diameter and Self-Consistency
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- Nominal diameter choice (1 mm burst, 3 mm QCW)
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- Back-calculating implied diameter from R
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- Self-consistency iteration (usually 1-2 steps)
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- Why it matters (and when it doesn't)
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- **Worked Example 4.7:** Self-consistency check
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- **Visual Aid:** Iteration convergence diagram
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- **Practice Problems:** 1 problem on diameter calculation
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### Module 4.8: Complete Simulation Workflow
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- Step 1: FEMM electrostatic analysis
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- Step 2: Extract capacitance matrix
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- Step 3: Choose/optimize resistances
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- Step 4: Build SPICE model
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- Step 5: Run analysis (AC or transient)
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- Step 6: Validate results
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- **Worked Example 4.8:** End-to-end simulation project
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- **Visual Aid:** Comprehensive workflow diagram
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- **Practice Problems:** 1 complete simulation exercise
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### Module 4.9: Validation and Physical Checks
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- Power balance: P_in = P_spark + P_losses
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- Total R in expected range (5-300 kΩ at 200 kHz)
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- R distribution: base < tip
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- C_sh validation: 2 pF/foot rule
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- Convergence tests: n=5 vs n=10 vs n=20
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- **Worked Example 4.9:** Validate a questionable simulation
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- **Visual Aid:** Checklist with pass/fail criteria
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- **Practice Problems:** 2 validation exercises
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### Module 4.10: Calibration from Real Measurements
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- Measuring ε: known drive, measure final length
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- Measuring E_propagation: V_top and L at stall
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- Using ringdown for Y_spark
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- Iterative refinement of model parameters
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- Building a calibration database
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- **Worked Example 4.10:** Calibrate ε from test data
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- **Visual Aid:** Calibration workflow
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- **Practice Problems:** 2 calibration problems
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### Module 4.11: Advanced Topics Preview
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- Frequency tracking during growth
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- Branching models (power division)
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- Strike event simulation (R collapse)
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- 3D FEA for complex geometries
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- Monte Carlo for stochastic effects
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- **Visual Aid:** Gallery of advanced scenarios
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- **Further Reading:** Resources for each topic
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### Module 4.12: Complete Design Case Study
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- Given: Coil specifications (f, L_secondary, C_topload, etc.)
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- Goal: Predict spark length for QCW operation
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- Work through entire process step-by-step
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- Compare prediction to empirical rules
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- Discuss uncertainties and limitations
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- **Comprehensive Example:** Full documentation
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- **Visual Aid:** Annotated results presentation
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### Part 4 Summary & Final Integration
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- Comprehensive final quiz (20 questions)
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- Capstone project: Design and simulate your own coil
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- Troubleshooting guide: Common errors and fixes
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- Resources for continued learning
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- Community and collaboration suggestions
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**Estimated Token Count: ~22,000-25,000**
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---
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## Appendices (Reference Material - Brief)
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*Can be included at end of Part 4 or as separate quick-reference*
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### Appendix A: Complete Variable Reference Table
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- All variables with units and definitions (condensed)
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### Appendix B: Formula Quick Reference
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- All key equations organized by topic
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### Appendix C: Physical Constants
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- Standard values for air properties, field thresholds, etc.
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### Appendix D: SPICE Component Reference
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- How to implement various elements
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### Appendix E: FEMM Quick Start Guide
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- Installation, basic navigation, common tasks
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### Appendix F: Troubleshooting Guide
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- Common problems and solutions organized by symptom
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**Estimated Token Count: ~5,000-6,000**
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---
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## Teaching Philosophy Embedded in This Plan
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1. **Spiral learning:** Concepts introduced simply, then revisited with more depth
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2. **Worked examples:** Every mathematical concept has at least one complete example
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3. **Visual aids:** Descriptions provided so you can create diagrams/graphs
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4. **Practice problems:** Incremental difficulty, answers can be provided separately
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5. **Checkpoints:** Regular assessment to ensure understanding before proceeding
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6. **Real-world connection:** Every module ties back to actual Tesla coil behavior
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---
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# Tesla Coil Spark Modeling - Complete Lesson Plan
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## Part 1: Foundation - Circuits, Impedance, and Basic Spark Behavior
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---
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## Module 1.1: AC Circuit Fundamentals Review
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### Peak vs RMS Values
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In AC circuits, voltage and current vary sinusoidally with time. We can express them in two ways:
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**Time domain:**
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```
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v(t) = V_peak × cos(ωt + φ)
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```
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**Two amplitude conventions:**
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- **Peak value:** The maximum value reached (V_peak)
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- **RMS value:** Root-Mean-Square, V_RMS = V_peak/√2 ≈ 0.707 × V_peak
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**For this entire framework, we use PEAK VALUES exclusively.**
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**Why peak values?**
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1. Tesla coils are concerned with maximum voltage (breakdown, field stress)
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2. Consistent with phasor notation in engineering
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3. Power formula becomes: P = 0.5 × V_peak × I_peak × cos(θ)
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**Example:** If your oscilloscope shows a 100 kV peak-to-peak waveform:
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- V_peak-to-peak = 100 kV
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- V_peak = 50 kV (one-sided amplitude)
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- V_RMS = 50 kV / √2 ≈ 35.4 kV
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### Complex Numbers and Phasors
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AC circuit analysis uses complex numbers to represent magnitude and phase simultaneously.
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**Rectangular form:**
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```
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Z = R + jX
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where j = √(-1) (imaginary unit, engineers use 'j' instead of 'i')
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R = real part (resistance)
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X = imaginary part (reactance)
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```
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**Polar form:**
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```
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Z = |Z| ∠φ = |Z| × e^(jφ)
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where |Z| = √(R² + X²) (magnitude)
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φ = atan(X/R) (phase angle)
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```
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**Conversion:**
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```
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R = |Z| × cos(φ)
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X = |Z| × sin(φ)
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```
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**Phasor notation:** A complex number representing sinusoidal amplitude and phase:
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```
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V = V_peak ∠φ_v
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I = I_peak ∠φ_i
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```
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**Complex conjugate:** Used in power calculations
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```
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If I = a + jb, then I* = a - jb (flip sign of imaginary part)
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```
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### Resistance, Reactance, Impedance
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|
|
**Resistance (R):** Opposition to current that dissipates energy as heat
|
|
- Units: Ω (ohms)
|
|
- Always real and positive
|
|
- V = I × R (Ohm's law)
|
|
|
|
**Reactance (X):** Opposition to current that stores energy (no dissipation)
|
|
- Units: Ω (ohms)
|
|
- Can be positive (inductive) or negative (capacitive)
|
|
- **Capacitive reactance:** X_C = -1/(ωC) where ω = 2πf
|
|
- **Inductive reactance:** X_L = ωL
|
|
|
|
**Impedance (Z):** Total opposition to AC current
|
|
```
|
|
Z = R + jX (complex)
|
|
|Z| = √(R² + X²)
|
|
φ_Z = atan(X/R)
|
|
```
|
|
|
|
**Sign conventions:**
|
|
- X > 0: inductive (current lags voltage)
|
|
- X < 0: capacitive (current leads voltage)
|
|
- φ_Z > 0: inductive
|
|
- φ_Z < 0: capacitive
|
|
|
|
### Conductance, Susceptance, Admittance
|
|
|
|
For parallel circuits, **admittance (Y)** is more convenient than impedance.
|
|
|
|
**Conductance (G):** Inverse of resistance
|
|
```
|
|
G = 1/R
|
|
Units: S (siemens)
|
|
```
|
|
|
|
**Susceptance (B):** Inverse of reactance (BUT with opposite sign convention!)
|
|
```
|
|
For capacitor: B_C = ωC (positive!)
|
|
For inductor: B_L = -1/(ωL) (negative)
|
|
```
|
|
|
|
**Important:** Susceptance sign convention is OPPOSITE of reactance:
|
|
- Capacitor: X_C < 0, but B_C > 0
|
|
- Inductor: X_L > 0, but B_L < 0
|
|
|
|
**Admittance (Y):** Inverse of impedance
|
|
```
|
|
Y = G + jB = 1/Z
|
|
|Y| = 1/|Z|
|
|
φ_Y = -φ_Z (opposite sign!)
|
|
```
|
|
|
|
**Conversion between Z and Y:**
|
|
```
|
|
Y = 1/Z = 1/(R + jX) = R/(R² + X²) - jX/(R² + X²)
|
|
|
|
Therefore:
|
|
G = R/(R² + X²)
|
|
B = -X/(R² + X²)
|
|
```
|
|
|
|
### Power in AC Circuits
|
|
|
|
**Using peak phasors:**
|
|
```
|
|
P = 0.5 × Re{V × I*}
|
|
|
|
where V and I are complex peak phasors
|
|
I* is the complex conjugate of I
|
|
Re{·} means "real part of"
|
|
```
|
|
|
|
**Why the 0.5 factor?**
|
|
- Average power over a full AC cycle
|
|
- Comes from time-averaging cos²(ωt), which equals 0.5
|
|
- If you used RMS values, formula would be P = V_RMS × I_RMS × cos(θ), NO 0.5
|
|
|
|
**Expanded form:**
|
|
```
|
|
If V = V_peak ∠φ_v and I = I_peak ∠φ_i, then:
|
|
P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
|
|
```
|
|
|
|
The angle difference (φ_v - φ_i) is the power factor angle.
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 1.1: Power Calculation with Peak Phasors
|
|
|
|
**Given:**
|
|
- Voltage: V = 50 kV ∠0° (peak, using 0° as reference)
|
|
- Impedance: Z = 100 kΩ ∠-60° (capacitive load)
|
|
|
|
**Find:** Real power dissipated
|
|
|
|
**Solution:**
|
|
|
|
Step 1: Calculate current using Ohm's law
|
|
```
|
|
I = V/Z = (50 kV ∠0°)/(100 kΩ ∠-60°)
|
|
I = 0.5 A ∠(0° - (-60°)) = 0.5 A ∠60°
|
|
```
|
|
|
|
Step 2: Calculate power
|
|
```
|
|
P = 0.5 × Re{V × I*}
|
|
P = 0.5 × Re{(50 kV ∠0°) × (0.5 A ∠-60°)}
|
|
P = 0.5 × Re{25 kW ∠-60°}
|
|
```
|
|
|
|
Step 3: Convert to rectangular to get real part
|
|
```
|
|
25 kW ∠-60° = 25 kW × (cos(-60°) + j×sin(-60°))
|
|
= 25 kW × (0.5 - j×0.866)
|
|
= 12.5 kW - j×21.65 kW
|
|
```
|
|
|
|
Step 4: Extract real part and apply 0.5 factor
|
|
```
|
|
P = 0.5 × 12.5 kW = 6.25 kW
|
|
```
|
|
|
|
**Alternative method:** Using power factor angle
|
|
```
|
|
P = 0.5 × V_peak × I_peak × cos(φ_v - φ_i)
|
|
P = 0.5 × 50 kV × 0.5 A × cos(0° - 60°)
|
|
P = 0.5 × 25 kW × cos(-60°)
|
|
P = 0.5 × 25 kW × 0.5
|
|
P = 6.25 kW
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 1.1
|
|
|
|
**Problem 1:** A capacitor has reactance X_C = -80 kΩ at 200 kHz. What is its capacitance? What is its susceptance?
|
|
|
|
**Problem 2:** An impedance Z = 50 kΩ - j75 kΩ has current I = 0.2 A ∠30° (peak). Calculate: (a) Voltage magnitude and phase, (b) Real power
|
|
|
|
**Problem 3:** An admittance Y = 0.00001 + j0.00002 S. Convert to impedance Z = R + jX.
|
|
|
|
---
|
|
|
|
## Module 1.2: Capacitance in Tesla Coils
|
|
|
|
### What is Capacitance Physically?
|
|
|
|
**Definition:** Capacitance (C) is the ability to store electric charge for a given voltage:
|
|
```
|
|
Q = C × V
|
|
Units: Farads (F), typically pF (10⁻¹² F) for Tesla coils
|
|
```
|
|
|
|
**Physical picture:**
|
|
- Electric field between two conductors stores energy
|
|
- Higher field → more stored energy → more capacitance
|
|
- Capacitance depends on geometry, NOT on voltage
|
|
|
|
**For parallel plates:**
|
|
```
|
|
C = ε₀ × A / d
|
|
|
|
where ε₀ = 8.854×10⁻¹² F/m (permittivity of free space)
|
|
A = plate area (m²)
|
|
d = separation distance (m)
|
|
```
|
|
|
|
**Key insight:** Capacitance increases with:
|
|
- Larger conductor area (more field lines)
|
|
- Smaller separation (stronger field concentration)
|
|
|
|
### Self-Capacitance vs Mutual Capacitance
|
|
|
|
**Self-capacitance:** Capacitance of a single conductor to infinity (or ground)
|
|
- Topload has self-capacitance to ground
|
|
- Depends on size and shape
|
|
- Toroid: C ≈ 4πε₀√(D×d) where D = major diameter, d = minor diameter
|
|
|
|
**Mutual capacitance:** Capacitance between two conductors
|
|
- Energy stored in field between them
|
|
- Both conductors at different potentials
|
|
- Can be positive or negative in matrix formulation
|
|
|
|
**For Tesla coils with sparks:**
|
|
- **C_mut:** mutual capacitance between topload and spark channel
|
|
- **C_sh:** capacitance from spark to ground (shunt capacitance)
|
|
|
|
### Capacitance to Ground (Shunt Capacitance)
|
|
|
|
Any conductor elevated above ground has capacitance to ground.
|
|
|
|
**For vertical wire above ground plane:**
|
|
```
|
|
C ≈ 2πε₀L / ln(2h/d)
|
|
|
|
where L = wire length
|
|
h = height above ground
|
|
d = wire diameter
|
|
```
|
|
|
|
**For Tesla coil sparks:** Empirical rule based on community measurements:
|
|
```
|
|
C_sh ≈ 2 pF per foot of spark length
|
|
|
|
Examples:
|
|
1 foot (0.3 m) spark: C_sh ≈ 2 pF
|
|
3 feet (0.9 m) spark: C_sh ≈ 6 pF
|
|
6 feet (1.8 m) spark: C_sh ≈ 12 pF
|
|
```
|
|
|
|
This rule is surprisingly accurate (±30%) for typical Tesla coil geometries.
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 1.2: Estimating C_sh for a Spark
|
|
|
|
**Given:** A 2-meter (6.6 foot) spark
|
|
|
|
**Find:** Estimated shunt capacitance
|
|
|
|
**Solution:**
|
|
```
|
|
C_sh ≈ 2 pF/foot × 6.6 feet
|
|
C_sh ≈ 13.2 pF
|
|
```
|
|
|
|
**Refined estimate using cylinder formula:**
|
|
|
|
Assume spark is vertical cylinder:
|
|
- Length L = 2 m
|
|
- Diameter d = 2 mm (typical for bright spark)
|
|
- Height above ground h = L/2 = 1 m (average height)
|
|
|
|
```
|
|
C ≈ 2πε₀L / ln(2h/d)
|
|
C ≈ 2π × 8.854×10⁻¹² × 2 / ln(2×1/0.002)
|
|
C ≈ 1.112×10⁻¹⁰ / ln(1000)
|
|
C ≈ 1.112×10⁻¹⁰ / 6.91
|
|
C ≈ 16 pF
|
|
```
|
|
|
|
The empirical rule (13 pF) and formula (16 pF) agree reasonably well.
|
|
|
|
---
|
|
|
|
### VISUAL AID 1.2: Field Lines for C_mut and C_sh
|
|
|
|
```
|
|
[Describe for drawing:]
|
|
|
|
Side view of Tesla coil with spark:
|
|
|
|
Spark tip (pointed)
|
|
|
|
|
| C_sh field lines radiate from
|
|
| spark to ground plane horizontally
|
|
Spark | (curved lines going left/right to ground)
|
|
body |
|
|
|
|
|
|
|
|
Topload (toroid)
|
|
|
|
|
Secondary
|
|
|
|
C_mut field lines: Connect topload surface to spark channel
|
|
- Start on topload outer surface
|
|
- End on spark channel surface
|
|
- Concentrated near base of spark
|
|
- These store mutual electric field energy
|
|
|
|
C_sh field lines: Connect spark to remote ground
|
|
- Start on spark surface
|
|
- Radiate outward to walls, floor, ceiling
|
|
- Distributed along entire spark length
|
|
- These store shunt field energy
|
|
|
|
Key observation: Same spark channel participates in BOTH capacitances!
|
|
This is why we need parallel C_mut || R, then series C_sh
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 1.2
|
|
|
|
**Problem 1:** A 4-foot spark is formed. Estimate C_sh using the empirical rule. If the topload has C_topload = 30 pF unloaded, what is the total system capacitance with the spark?
|
|
|
|
**Problem 2:** Using the cylinder formula, calculate C_sh for a spark with: L = 1.5 m, d = 3 mm, average height h = 0.75 m. Compare to the empirical rule.
|
|
|
|
---
|
|
|
|
## Module 1.3: The Basic Spark Circuit Topology
|
|
|
|
### Why Sparks Have TWO Capacitances
|
|
|
|
A spark channel is a conductor in space with:
|
|
1. **Proximity to the topload** → mutual capacitance C_mut
|
|
2. **Proximity to ground/environment** → shunt capacitance C_sh
|
|
|
|
**Both exist simultaneously** because the spark interacts with multiple conductors.
|
|
|
|
**Analogy:** A wire near two metal plates
|
|
- Capacitance to plate 1: C₁
|
|
- Capacitance to plate 2: C₂
|
|
- Both must be included in the circuit model
|
|
|
|
### The Correct Circuit Topology
|
|
|
|
```
|
|
Topload (measurement reference)
|
|
|
|
|
[C_mut] ← Mutual capacitance between topload and spark
|
|
|
|
|
+---------+--------- Node_spark
|
|
| |
|
|
[R] [C_sh] ← Shunt capacitance spark-to-ground
|
|
| |
|
|
GND ------------ GND
|
|
```
|
|
|
|
**Equivalent description:**
|
|
- C_mut and R in parallel
|
|
- That parallel combination in series with C_sh
|
|
- All connected between topload and ground
|
|
|
|
**Why this topology?**
|
|
1. C_mut couples topload voltage to spark
|
|
2. R represents plasma resistance (where power is dissipated)
|
|
3. C_sh provides current return path to ground
|
|
4. Current through R must also flow through either C_mut or C_sh (series connection)
|
|
|
|
### Where is "Ground" in a Tesla Coil?
|
|
|
|
**Earth ground:** Actual connection to soil/building ground
|
|
**Circuit ground (reference):** Arbitrary 0V reference point
|
|
|
|
**For Tesla coils:**
|
|
- Primary circuit: Chassis/mains ground is reference
|
|
- Secondary base: Usually connected to primary ground via RF ground
|
|
- **Practical ground:** Floor, walls, nearby objects, you standing nearby
|
|
- **Measurement ground:** Choose ONE point as 0V reference (usually secondary base)
|
|
|
|
**Important:** "Ground" in spark model means "remote return path" - could be walls, floor, strike ring, or actual earth.
|
|
|
|
### The Topload Port
|
|
|
|
**Definition:** The two-terminal measurement point between topload and ground where we characterize impedance and power.
|
|
|
|
```
|
|
Port definition:
|
|
Terminal 1: Topload terminal (high voltage)
|
|
Terminal 2: Ground reference (0V)
|
|
```
|
|
|
|
**All impedance measurements reference this port:**
|
|
- Z_spark: impedance looking into spark from topload
|
|
- Z_th: Thévenin impedance of coil at this port
|
|
- V_th: Open-circuit voltage at this port
|
|
|
|
**Not the same as:**
|
|
- V_top / I_base (includes displacement currents from entire secondary)
|
|
- Any two-point measurement along the secondary winding
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 1.3: Drawing the Circuit
|
|
|
|
**Given:**
|
|
- Spark is 3 feet long
|
|
- FEMM analysis gives C_mut = 8 pF (between topload and spark)
|
|
- Estimate C_sh using empirical rule
|
|
- Assume R = 100 kΩ
|
|
|
|
**Task:** Draw complete circuit diagram
|
|
|
|
**Solution:**
|
|
|
|
Step 1: Calculate C_sh
|
|
```
|
|
C_sh ≈ 2 pF/foot × 3 feet = 6 pF
|
|
```
|
|
|
|
Step 2: Draw topology
|
|
```
|
|
Topload (V_top)
|
|
|
|
|
[C_mut = 8 pF]
|
|
|
|
|
+-------- Node_spark
|
|
| |
|
|
[R = 100 kΩ] [C_sh = 6 pF]
|
|
| |
|
|
GND -------- GND
|
|
```
|
|
|
|
Step 3: Simplify to show parallel/series structure
|
|
```
|
|
Topload
|
|
|
|
|
+---- [C_mut = 8 pF] ----+
|
|
| |
|
|
+---- [R = 100 kΩ] ------+ Node_spark
|
|
|
|
|
[C_sh = 6 pF]
|
|
|
|
|
GND
|
|
```
|
|
|
|
This is the basic lumped model for a Tesla coil spark.
|
|
|
|
---
|
|
|
|
### VISUAL AID 1.3: 3D Geometry → Circuit Schematic
|
|
|
|
```
|
|
[Describe for drawing:]
|
|
|
|
Panel 1: Physical 3D view
|
|
- Toroidal topload at top (labeled "Topload")
|
|
- Vertical spark channel extending downward (labeled "Spark, length L")
|
|
- Ground plane at bottom (labeled "Ground")
|
|
- Dashed lines showing C_mut field (topload to spark)
|
|
- Dotted lines showing C_sh field (spark to ground)
|
|
|
|
Panel 2: Conceptual extraction
|
|
- Topload → single node
|
|
- Spark → two elements: resistance R and capacitances
|
|
- Ground → common reference
|
|
- Arrows showing "Extract C_mut from field between topload and spark"
|
|
- Arrows showing "Extract C_sh from field between spark and ground"
|
|
|
|
Panel 3: Circuit schematic (as drawn above)
|
|
- Proper circuit symbols
|
|
- Component values labeled
|
|
- Ground symbol at bottom
|
|
- Clear port definition marked
|
|
|
|
Annotation: "Same physics, different representations"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 1.3
|
|
|
|
**Problem 1:** Draw the circuit for a spark with: L = 5 feet, C_mut = 12 pF (from FEMM), R = 50 kΩ. Label all component values.
|
|
|
|
**Problem 2:** A simulation shows C_sh = 10 pF for a given spark. What is the estimated spark length using the empirical rule?
|
|
|
|
---
|
|
|
|
## Module 1.4: Admittance Analysis of the Spark Circuit
|
|
|
|
### Why Use Admittance?
|
|
|
|
For the spark circuit topology (parallel R||C_mut, in series with C_sh), admittance simplifies calculations.
|
|
|
|
**Parallel elements:** Add admittances directly
|
|
```
|
|
Y_total = Y₁ + Y₂ + Y₃ + ...
|
|
vs impedances: 1/Z_total = 1/Z₁ + 1/Z₂ + ... (messy!)
|
|
```
|
|
|
|
**Our circuit:**
|
|
```
|
|
Y_mut_R = Y_Cmut + Y_R (parallel: C_mut || R)
|
|
Then series with C_sh requires impedance: Z = Z_mut_R + Z_Csh
|
|
Then convert back: Y_total = 1/Z_total
|
|
```
|
|
|
|
### Deriving the Total Admittance Formula
|
|
|
|
**Step 1:** Admittance of R and C_mut in parallel
|
|
|
|
```
|
|
Y_R = G = 1/R
|
|
Y_Cmut = jωC_mut = jB₁ (where B₁ = ωC_mut)
|
|
|
|
Y_mut_R = G + jB₁
|
|
```
|
|
|
|
**Step 2:** Convert to impedance for series combination
|
|
```
|
|
Z_mut_R = 1/(G + jB₁)
|
|
```
|
|
|
|
**Step 3:** Add impedance of C_sh in series
|
|
```
|
|
Z_Csh = 1/(jωC_sh) = -j/(ωC_sh) = 1/(jB₂) (where B₂ = ωC_sh)
|
|
|
|
Z_total = Z_mut_R + Z_Csh
|
|
Z_total = 1/(G + jB₁) + 1/(jB₂)
|
|
```
|
|
|
|
**Step 4:** Find common denominator
|
|
```
|
|
Z_total = [jB₂ + (G + jB₁)] / [(G + jB₁) × jB₂]
|
|
Z_total = [G + j(B₁ + B₂)] / [jB₂(G + jB₁)]
|
|
```
|
|
|
|
**Step 5:** Invert to get admittance
|
|
```
|
|
Y_total = 1/Z_total = [jB₂(G + jB₁)] / [G + j(B₁ + B₂)]
|
|
|
|
Y_total = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
|
|
```
|
|
|
|
This is the **fundamental admittance equation** for the spark circuit.
|
|
|
|
### Extracting Real and Imaginary Parts
|
|
|
|
Multiply numerator:
|
|
```
|
|
(G + jB₁) × jB₂ = jGB₂ + j²B₁B₂ = jGB₂ - B₁B₂
|
|
= -B₁B₂ + jGB₂
|
|
```
|
|
|
|
So:
|
|
```
|
|
Y = [-B₁B₂ + jGB₂] / [G + j(B₁ + B₂)]
|
|
```
|
|
|
|
To separate real and imaginary parts, multiply numerator and denominator by complex conjugate of denominator:
|
|
|
|
```
|
|
Denominator conjugate: G - j(B₁ + B₂)
|
|
Denominator magnitude squared: G² + (B₁ + B₂)²
|
|
```
|
|
|
|
After algebra (multiply out and simplify):
|
|
|
|
```
|
|
Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
|
|
|
|
Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
|
|
```
|
|
|
|
These are the **working formulas** for calculating admittance from R, C_mut, C_sh.
|
|
|
|
### Converting to Impedance
|
|
|
|
From Y = G_total + jB_total:
|
|
|
|
```
|
|
Z = 1/Y = 1/(G_total + jB_total)
|
|
|
|
Multiply by conjugate:
|
|
Z = (G_total - jB_total) / (G_total² + B_total²)
|
|
|
|
R_total = G_total / (G_total² + B_total²)
|
|
X_total = -B_total / (G_total² + B_total²)
|
|
|
|
Or directly:
|
|
|Z| = 1/|Y|
|
|
φ_Z = -φ_Y (opposite sign!)
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 1.4: Complete Y and Z Calculation
|
|
|
|
**Given:**
|
|
- Frequency: f = 200 kHz → ω = 2π × 200×10³ = 1.257×10⁶ rad/s
|
|
- C_mut = 8 pF = 8×10⁻¹² F
|
|
- C_sh = 6 pF = 6×10⁻¹² F
|
|
- R = 100 kΩ = 10⁵ Ω
|
|
|
|
**Find:** Y_total (rectangular), Z_total (rectangular and polar)
|
|
|
|
**Solution:**
|
|
|
|
Step 1: Calculate component values
|
|
```
|
|
G = 1/R = 1/(10⁵) = 10⁻⁵ S = 10 μS
|
|
B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06×10⁻⁶ S = 10.06 μS
|
|
B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54×10⁻⁶ S = 7.54 μS
|
|
```
|
|
|
|
Step 2: Calculate Re{Y}
|
|
```
|
|
Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
|
|
|
|
Numerator: 10 × (7.54)² = 10 × 56.85 = 568.5 μS²
|
|
Denominator: (10)² + (10.06 + 7.54)² = 100 + (17.6)² = 100 + 309.8 = 409.8 μS²
|
|
|
|
Re{Y} = 568.5 / 409.8 = 1.387 μS
|
|
```
|
|
|
|
Step 3: Calculate Im{Y}
|
|
```
|
|
Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
|
|
|
|
Numerator inner: G² + B₁(B₁ + B₂) = 100 + 10.06×17.6 = 100 + 177.1 = 277.1 μS²
|
|
Numerator: 7.54 × 277.1 = 2089.3 μS³
|
|
Denominator: 409.8 μS² (same as before)
|
|
|
|
Im{Y} = 2089.3 / 409.8 = 5.10 μS
|
|
```
|
|
|
|
Step 4: Admittance result
|
|
```
|
|
Y_total = 1.387 + j5.10 μS
|
|
|Y| = √(1.387² + 5.10²) = √(1.92 + 26.01) = √27.93 = 5.28 μS
|
|
φ_Y = atan(5.10/1.387) = atan(3.68) = 74.8°
|
|
```
|
|
|
|
Step 5: Convert to impedance
|
|
```
|
|
|Z| = 1/|Y| = 1/(5.28×10⁻⁶) = 189 kΩ
|
|
φ_Z = -φ_Y = -74.8°
|
|
|
|
In rectangular:
|
|
R_total = |Z| × cos(φ_Z) = 189 × cos(-74.8°) = 189 × 0.263 = 49.7 kΩ
|
|
X_total = |Z| × sin(φ_Z) = 189 × sin(-74.8°) = 189 × (-0.965) = -182 kΩ
|
|
|
|
Z_total = 49.7 - j182 kΩ = 189 kΩ ∠-74.8°
|
|
```
|
|
|
|
**Interpretation:**
|
|
- Impedance is strongly capacitive (φ_Z = -74.8°)
|
|
- Equivalent resistance ≈ 50 kΩ (half of actual R due to capacitive divider)
|
|
- Large capacitive reactance dominates
|
|
|
|
---
|
|
|
|
### VISUAL AID 1.4: Complex Plane Plots
|
|
|
|
```
|
|
[Describe for drawing:]
|
|
|
|
Two plots side-by-side:
|
|
|
|
LEFT: Admittance plane (Y = G + jB)
|
|
- Horizontal axis: G (conductance, μS), 0 to 2
|
|
- Vertical axis: B (susceptance, μS), 0 to 6
|
|
- Plot point at (1.387, 5.10) labeled "Y_total"
|
|
- Vector from origin to point
|
|
- Angle φ_Y = 74.8° marked from horizontal
|
|
- Length |Y| = 5.28 μS labeled
|
|
- Note: "Positive B means capacitive in admittance"
|
|
|
|
RIGHT: Impedance plane (Z = R + jX)
|
|
- Horizontal axis: R (kΩ), 0 to 60
|
|
- Vertical axis: X (kΩ), -200 to 0
|
|
- Plot point at (49.7, -182) labeled "Z_total"
|
|
- Vector from origin to point
|
|
- Angle φ_Z = -74.8° marked from horizontal (below axis)
|
|
- Length |Z| = 189 kΩ labeled
|
|
- Note: "Negative X means capacitive in impedance"
|
|
|
|
Connection between plots:
|
|
- Arrow showing "Invert Y → Z"
|
|
- Note: "Angles are opposite: φ_Z = -φ_Y"
|
|
- Note: "Magnitude inverts: |Z| = 1/|Y|"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 1.4
|
|
|
|
**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF, R = 80 kΩ, calculate Y_total (real and imaginary parts).
|
|
|
|
**Problem 2:** An admittance Y = 2.0 + j4.5 μS. Convert to impedance Z in both rectangular and polar forms.
|
|
|
|
**Problem 3:** Show algebraically that if R → ∞ (open circuit), the formula reduces to Y = jωC_mut × C_sh/(C_mut + C_sh), which is two capacitors in series.
|
|
|
|
---
|
|
|
|
## Module 1.5: Phase Angles and What They Mean
|
|
|
|
### Impedance Phase vs Admittance Phase
|
|
|
|
**Impedance phase angle φ_Z:**
|
|
```
|
|
φ_Z = atan(X/R) = atan(Im{Z}/Re{Z})
|
|
|
|
Interpretation:
|
|
φ_Z > 0: inductive (current lags voltage)
|
|
φ_Z = 0: purely resistive (in phase)
|
|
φ_Z < 0: capacitive (current leads voltage)
|
|
```
|
|
|
|
**Admittance phase angle θ_Y:**
|
|
```
|
|
θ_Y = atan(B/G) = atan(Im{Y}/Re{Y})
|
|
|
|
Relationship: θ_Y = -φ_Z (OPPOSITE SIGNS!)
|
|
```
|
|
|
|
**Why opposite?** Because Y = 1/Z, so angles subtract:
|
|
```
|
|
If Z = |Z|∠φ_Z, then Y = (1/|Z|)∠(-φ_Z)
|
|
```
|
|
|
|
**Convention in this framework:** We primarily discuss **impedance phase φ_Z** because that's what measurements typically report.
|
|
|
|
### The "Famous -45°" and Why It's Special (Sort Of)
|
|
|
|
In power electronics, a load with φ_Z = -45° is sometimes called "well-matched" because:
|
|
- Equal resistive and capacitive components: |R| = |X_C|
|
|
- Power factor = cos(-45°) = 0.707 (reasonable power transfer)
|
|
- Not maximum power transfer, but balanced
|
|
|
|
**Formula:** For φ_Z = -45°:
|
|
```
|
|
tan(-45°) = -1 = X/R
|
|
Therefore: R = |X| = 1/(ωC) for capacitive load
|
|
Or: R ≈ |X_c| = 1/(ωC_total) approximately
|
|
```
|
|
|
|
This is why you'll see "spark resistance should equal capacitive reactance" in old Tesla coil literature.
|
|
|
|
**BUT:** As we'll see in Part 2, achieving exactly -45° is **impossible** for many Tesla coil geometries due to topological constraints!
|
|
|
|
### Physical Meaning of Phase Angle
|
|
|
|
**φ_Z = 0° (purely resistive):**
|
|
- All power dissipated
|
|
- No energy storage/return
|
|
- Voltage and current in phase
|
|
|
|
**φ_Z = -90° (purely capacitive):**
|
|
- No power dissipated
|
|
- All energy stored and returned each cycle
|
|
- Current leads voltage by 90°
|
|
|
|
**φ_Z = -45° (mixed):**
|
|
- Some power dissipated (cos(-45°) ≈ 71% of |V||I|)
|
|
- Some energy stored
|
|
- Current leads voltage by 45°
|
|
|
|
**For Tesla coil sparks:** Typical φ_Z = -55° to -75°
|
|
- Significant capacitive component (energy storage in C_mut, C_sh)
|
|
- Moderate power dissipation (plasma heating)
|
|
- More capacitive than the "ideal" -45°
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 1.5: Calculating Phase Angle
|
|
|
|
**Given:** (from Example 1.4)
|
|
- Z_total = 49.7 - j182 kΩ
|
|
|
|
**Find:** φ_Z and interpret
|
|
|
|
**Solution:**
|
|
|
|
Step 1: Calculate phase angle
|
|
```
|
|
φ_Z = atan(X/R) = atan(-182/49.7)
|
|
φ_Z = atan(-3.66) = -74.8°
|
|
```
|
|
|
|
Step 2: Verify with magnitude and components
|
|
```
|
|
|Z| = √(49.7² + 182²) = √(2470 + 33124) = √35594 = 189 kΩ ✓
|
|
|
|
cos(φ_Z) = R/|Z| = 49.7/189 = 0.263
|
|
φ_Z = arccos(0.263) = 74.8°, but X is negative, so φ_Z = -74.8° ✓
|
|
```
|
|
|
|
Step 3: Interpret
|
|
- **Strongly capacitive:** |φ_Z| = 74.8° is much larger than 45°
|
|
- **Comparison:** |R| = 49.7 kΩ, but |X| = 182 kΩ
|
|
- Capacitive reactance is 3.66× larger than resistance
|
|
- Far from "balanced" -45° condition
|
|
- **Power factor:** cos(-74.8°) = 0.263
|
|
- Only 26.3% of |V||I| is real power
|
|
- Most current is reactive (charging/discharging capacitances)
|
|
|
|
This is typical for Tesla coil sparks: strongly capacitive impedance.
|
|
|
|
---
|
|
|
|
### VISUAL AID 1.5: Phase Angle on Complex Plane
|
|
|
|
```
|
|
[Describe for drawing:]
|
|
|
|
Impedance plane (Z = R + jX):
|
|
- Horizontal axis: R (resistance, kΩ), 0 to 100
|
|
- Vertical axis: X (reactance, kΩ), -200 to +200
|
|
|
|
Three vectors from origin:
|
|
|
|
1. Resistive (φ_Z = 0°):
|
|
- Point at (50, 0)
|
|
- Horizontal vector, angle = 0°
|
|
- Label: "Pure resistance, φ_Z = 0°"
|
|
|
|
2. Balanced (φ_Z = -45°):
|
|
- Point at (50, -50)
|
|
- Vector at -45° angle
|
|
- Dashed line showing equal R and |X|
|
|
- Label: "Balanced, φ_Z = -45°, R = |X|"
|
|
|
|
3. Typical spark (φ_Z = -75°):
|
|
- Point at (50, -186)
|
|
- Vector at -75° angle
|
|
- Label: "Typical spark, φ_Z = -75°"
|
|
- Annotation: "Strongly capacitive, |X| >> R"
|
|
|
|
Additional marks:
|
|
- φ_Z = -90° line (vertical downward): "Pure capacitor"
|
|
- Shaded region between -45° and -90°: "Typical Tesla coil spark range"
|
|
- Note: "More negative φ_Z = more capacitive"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 1.5
|
|
|
|
**Problem 1:** An impedance Z = 60 + j40 kΩ. Calculate φ_Z. Is this inductive or capacitive?
|
|
|
|
**Problem 2:** A spark has φ_Z = -60°. If |Z| = 150 kΩ, find R and X. Calculate the power factor.
|
|
|
|
---
|
|
|
|
## Module 1.6: Introduction to Spark Physics
|
|
|
|
### What is a Spark? (Qualitative)
|
|
|
|
**Definition:** A spark is a transient electrical breakdown of air, creating a conducting plasma channel between two electrodes.
|
|
|
|
**Basic process:**
|
|
1. High electric field ionizes air molecules (electrons stripped from atoms)
|
|
2. Free electrons accelerate, collide with more atoms → avalanche
|
|
3. Plasma forms: mixture of electrons, ions, neutral atoms
|
|
4. Plasma conducts electricity (lower resistance than air)
|
|
5. Current heats plasma → thermal ionization → sustained conduction
|
|
6. When voltage removed, plasma cools and recombines
|
|
|
|
**Key point:** Plasma is not a simple resistor! Its properties change dynamically:
|
|
- Temperature: 1000 K (cool streamers) to 20,000 K (hot leaders)
|
|
- Conductivity: varies with temperature and ionization
|
|
- Geometry: diameter, length change during growth
|
|
|
|
### Streamers vs Leaders (Qualitative)
|
|
|
|
**Streamers:**
|
|
- **Thin:** 10-100 μm diameter (thinner than human hair)
|
|
- **Fast:** Propagate at ~10⁶ m/s (1% speed of light!)
|
|
- **Cold:** Low temperature, weakly ionized
|
|
- **Mechanism:** Photoionization (UV from excited atoms ionizes ahead)
|
|
- **Appearance:** Purple/blue, highly branched, brief flashes
|
|
- **Resistance:** High (MΩ range)
|
|
- **Energy inefficient:** Much energy → light/heat, little → length
|
|
|
|
**Leaders:**
|
|
- **Thick:** mm to cm diameter (visible as bright core)
|
|
- **Slower:** Propagate at ~10³ m/s (walking speed to car speed)
|
|
- **Hot:** 5,000-20,000 K, fully ionized plasma
|
|
- **Mechanism:** Thermal ionization (Joule heating)
|
|
- **Appearance:** White/orange, straighter, persistent glow
|
|
- **Resistance:** Low (kΩ range)
|
|
- **Energy efficient:** More energy → length extension
|
|
|
|
**Transition:** Streamers can become leaders if sufficient current flows → heating → thermal ionization. This requires power and time.
|
|
|
|
### Why Sparks Need Voltage AND Power
|
|
|
|
**Voltage requirement (field threshold):**
|
|
```
|
|
E_tip > E_propagation ≈ 0.4-1.0 MV/m
|
|
|
|
For spark to grow, tip field must exceed threshold
|
|
If E_tip drops below threshold, growth stalls
|
|
```
|
|
|
|
**Power requirement (energy per meter):**
|
|
```
|
|
To extend spark by ΔL, need energy: ΔE ≈ ε × ΔL
|
|
where ε ≈ 5-100 J/m depending on mode
|
|
|
|
Power determines growth rate: dL/dt ≈ P/ε
|
|
```
|
|
|
|
**Analogy:** Starting a fire
|
|
- Voltage = temperature of match (need minimum to ignite)
|
|
- Power = fuel supply rate (determines how fast fire spreads)
|
|
- Both are necessary: hot match but no fuel → small flame dies
|
|
- Lots of fuel but no ignition heat → no fire
|
|
|
|
**For Tesla coils:**
|
|
- Insufficient voltage → spark won't start or grows slowly
|
|
- Insufficient power → spark stalls before reaching potential length
|
|
- **Both must be adequate** for target spark length
|
|
|
|
### The "Hungry Streamer" Principle (Conceptual)
|
|
|
|
**Key insight:** Plasma is not passive! It actively adjusts its properties to maximize power extraction from the circuit.
|
|
|
|
**Mechanism (simplified):**
|
|
1. More power → more Joule heating (I²R)
|
|
2. Higher temperature → more ionization
|
|
3. More ionization → higher conductivity → lower R
|
|
4. Changed geometry → modified capacitances
|
|
5. Circuit has new optimal R for max power transfer
|
|
6. Plasma conductivity adjusts toward this new optimal R
|
|
7. Equilibrium when R_actual ≈ R_optimal_for_max_power
|
|
|
|
**Physical limits:**
|
|
- R cannot be infinite (some conductivity always present)
|
|
- R cannot be zero (finite electron mobility)
|
|
- Source has limited voltage/current
|
|
- Takes time to adjust (thermal time constants)
|
|
|
|
**Result:** In steady state, plasma R tends toward the value that maximizes power transfer, within physical constraints.
|
|
|
|
**Why this matters:** We can model spark as "choosing" R = R_opt_power without detailed plasma chemistry! The physics self-optimizes.
|
|
|
|
---
|
|
|
|
### VISUAL AID 1.6: Streamers vs Leaders
|
|
|
|
```
|
|
[Describe for photo/diagram annotations:]
|
|
|
|
Two-panel comparison:
|
|
|
|
LEFT PANEL: Streamer
|
|
- Photo/drawing of thin, branched, purple discharge
|
|
- Annotations:
|
|
* Diameter: 10-100 μm (draw scale bar)
|
|
* Temperature: ~1000 K
|
|
* Speed: ~1,000,000 m/s
|
|
* Color: Purple/blue (label spectrum)
|
|
* Structure: Highly branched (mark branching points)
|
|
* Duration: <1 μs per event
|
|
* Resistance: High (MΩ)
|
|
|
|
RIGHT PANEL: Leader
|
|
- Photo/drawing of thick, straight, white discharge
|
|
- Annotations:
|
|
* Diameter: 1-10 mm (draw scale bar)
|
|
* Temperature: 5,000-20,000 K
|
|
* Speed: ~1,000 m/s
|
|
* Color: White/orange (label spectrum)
|
|
* Structure: Straighter channel (mark path)
|
|
* Duration: Seconds with sustained power
|
|
* Resistance: Low (kΩ)
|
|
|
|
BOTTOM: Transition diagram
|
|
- Timeline showing streamer → leader conversion
|
|
- Labels: "Initial: streamers form at tip"
|
|
"Current flows → Joule heating"
|
|
"Channel heats → thermal ionization"
|
|
"Leader forms from base, grows toward tip"
|
|
"Leader tip launches new streamers"
|
|
"Cycle repeats for continued growth"
|
|
```
|
|
|
|
---
|
|
|
|
### DISCUSSION QUESTIONS 1.6
|
|
|
|
**Question 1:** If a Tesla coil produces high voltage but very low current, would you expect long streamers or short leaders? Why?
|
|
|
|
**Question 2:** A coil generates 500 kV but only 100 mA. Another generates 200 kV but 1 A. Which is more likely to produce longer sparks? (Consider both voltage and power requirements.)
|
|
|
|
**Question 3:** Explain in your own words why the spark plasma can be modeled as a resistance that "optimizes itself" rather than as a fixed resistance value.
|
|
|
|
---
|
|
|
|
## Part 1 Summary: Concepts Checklist
|
|
|
|
Before proceeding to Part 2, ensure you understand:
|
|
|
|
### Circuit Fundamentals
|
|
- [ ] Difference between peak and RMS values
|
|
- [ ] Complex number representation: rectangular (R+jX) and polar (|Z|∠φ)
|
|
- [ ] Power calculation: P = 0.5 × Re{V × I*} with peak phasors
|
|
- [ ] Impedance Z = R + jX and admittance Y = G + jB
|
|
- [ ] Relationship: Y = 1/Z, and φ_Y = -φ_Z
|
|
|
|
### Capacitances
|
|
- [ ] Physical meaning of capacitance (charge storage)
|
|
- [ ] Self-capacitance vs mutual capacitance
|
|
- [ ] Shunt capacitance C_sh ≈ 2 pF/foot for sparks
|
|
- [ ] Both C_mut and C_sh exist simultaneously
|
|
|
|
### Circuit Topology
|
|
- [ ] Spark circuit: (R || C_mut) in series with C_sh
|
|
- [ ] Topload port as measurement reference (topload-to-ground)
|
|
- [ ] Why V_top/I_base is incorrect
|
|
|
|
### Admittance Analysis
|
|
- [ ] Advantages of Y for parallel circuits
|
|
- [ ] Formula: Y = [(G+jB₁)×jB₂]/[G+j(B₁+B₂)]
|
|
- [ ] Extracting Re{Y} and Im{Y}
|
|
- [ ] Converting Y ↔ Z
|
|
|
|
### Phase Angles
|
|
- [ ] φ_Z = atan(X/R) for impedance
|
|
- [ ] Negative φ_Z means capacitive
|
|
- [ ] The -45° "balanced" condition: R = |X|
|
|
- [ ] Typical sparks: φ_Z ≈ -55° to -75° (more capacitive than -45°)
|
|
|
|
### Spark Physics (Qualitative)
|
|
- [ ] Streamers: thin, fast, cold, high R, branched
|
|
- [ ] Leaders: thick, slower, hot, low R, straighter
|
|
- [ ] Need both voltage (E-field) and power (energy/time)
|
|
- [ ] "Hungry streamer": plasma self-optimizes R
|
|
|
|
---
|
|
|
|
## Integration Exercise: Putting It All Together
|
|
|
|
**Scenario:** You have a Tesla coil operating at 180 kHz with a 2-foot spark.
|
|
|
|
**Given data:**
|
|
- C_mut = 7 pF (from FEMM)
|
|
- Assume R = 75 kΩ (plasma resistance)
|
|
- Estimate C_sh using empirical rule
|
|
|
|
**Tasks:**
|
|
1. Calculate ω, B₁, B₂, G
|
|
2. Calculate Y_total (real and imaginary parts)
|
|
3. Convert to Z_total (magnitude and phase)
|
|
4. Calculate φ_Z and interpret (is it more or less capacitive than -45°?)
|
|
5. If V_top = 300 kV peak, calculate power dissipated
|
|
|
|
**Work through this problem completely before checking the solution below.**
|
|
|
|
---
|
|
|
|
### Integration Exercise Solution
|
|
|
|
**Step 1:** Calculate C_sh
|
|
```
|
|
C_sh ≈ 2 pF/foot × 2 feet = 4 pF
|
|
```
|
|
|
|
**Step 2:** Calculate ω and component values
|
|
```
|
|
ω = 2πf = 2π × 180×10³ = 1.131×10⁶ rad/s
|
|
|
|
G = 1/R = 1/(75×10³) = 13.33 μS
|
|
B₁ = ωC_mut = 1.131×10⁶ × 7×10⁻¹² = 7.92 μS
|
|
B₂ = ωC_sh = 1.131×10⁶ × 4×10⁻¹² = 4.52 μS
|
|
```
|
|
|
|
**Step 3:** Calculate Y_total
|
|
```
|
|
Re{Y} = GB₂²/[G² + (B₁+B₂)²]
|
|
= 13.33 × (4.52)² / [13.33² + (7.92+4.52)²]
|
|
= 13.33 × 20.43 / [177.7 + 154.4]
|
|
= 272.3 / 332.1
|
|
= 0.82 μS
|
|
|
|
Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
|
|
= 4.52 × [177.7 + 7.92×12.44] / 332.1
|
|
= 4.52 × [177.7 + 98.5] / 332.1
|
|
= 4.52 × 276.2 / 332.1
|
|
= 3.76 μS
|
|
|
|
Y_total = 0.82 + j3.76 μS
|
|
```
|
|
|
|
**Step 4:** Convert to impedance
|
|
```
|
|
|Y| = √(0.82² + 3.76²) = √(0.67 + 14.14) = √14.81 = 3.85 μS
|
|
|
|
|Z| = 1/|Y| = 1/(3.85×10⁻⁶) = 260 kΩ
|
|
|
|
φ_Y = atan(3.76/0.82) = atan(4.59) = 77.7°
|
|
φ_Z = -φ_Y = -77.7°
|
|
|
|
Z_total = 260 kΩ ∠-77.7°
|
|
|
|
In rectangular:
|
|
R_eq = 260 × cos(-77.7°) = 260 × 0.213 = 55.4 kΩ
|
|
X_eq = 260 × sin(-77.7°) = 260 × (-0.977) = -254 kΩ
|
|
|
|
Z_total = 55.4 - j254 kΩ
|
|
```
|
|
|
|
**Step 5:** Interpret phase
|
|
```
|
|
φ_Z = -77.7° is more capacitive than -45° (larger magnitude)
|
|
Ratio: |X|/R = 254/55.4 = 4.6
|
|
Capacitive reactance is 4.6× the resistance
|
|
Very capacitive load!
|
|
```
|
|
|
|
**Step 6:** Calculate power
|
|
```
|
|
Current: I = V/Z = (300 kV)/(260 kΩ) = 1.15 A peak
|
|
|
|
Power: P = 0.5 × V × I × cos(φ_Z)
|
|
= 0.5 × 300×10³ × 1.15 × cos(-77.7°)
|
|
= 0.5 × 345×10³ × 0.213
|
|
= 36.7 kW
|
|
|
|
Alternative: P = 0.5 × I² × R_eq
|
|
= 0.5 × 1.15² × 55.4×10³
|
|
= 0.5 × 1.32 × 55.4×10³
|
|
= 36.6 kW ✓ (checks!)
|
|
```
|
|
|
|
**Result:** 36.7 kW dissipated in the spark plasma.
|
|
|
|
---
|
|
|
|
## Preview of Part 2
|
|
|
|
In Part 2, we'll discover:
|
|
|
|
- **Why -45° is often impossible:** The topological phase constraint
|
|
- **Two critical resistances:** R_opt_power and R_opt_phase
|
|
- **Thévenin method:** Properly characterizing the Tesla coil
|
|
- **Power optimization:** How the "hungry streamer" finds R_opt_power
|
|
- **Measurements:** Extracting spark parameters from real coils
|
|
|
|
These concepts build directly on the circuit analysis and phase relationships you've learned in Part 1.
|
|
|
|
---
|
|
|
|
## CHECKPOINT QUIZ - Part 1
|
|
|
|
Answer these questions to verify your understanding:
|
|
|
|
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.
|
|
|
|
---
|
|
|
|
**END OF PART 1**
|
|
|
|
---
|
|
|
|
# Tesla Coil Spark Modeling - Complete Lesson Plan
|
|
## Part 2: Optimization and Power Transfer - Making Sparks Efficient
|
|
|
|
---
|
|
|
|
## Module 2.1: The Topological Phase Constraint
|
|
|
|
### What is a Topological Constraint?
|
|
|
|
**Definition:** A limitation imposed by the **structure** of the circuit itself, independent of component values.
|
|
|
|
**Example:** Series RLC circuit
|
|
- Can only have impedance phase between -90° (pure C) and +90° (pure L)
|
|
- Cannot have φ_Z = +120° no matter what component values you choose
|
|
- This is a topological constraint
|
|
|
|
**For spark circuits:** The specific arrangement (R||C_mut) in series with C_sh creates a fundamental limit on how resistive the impedance can appear.
|
|
|
|
### Deriving the Minimum Phase Angle
|
|
|
|
From Part 1, we have:
|
|
```
|
|
Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
|
|
|
|
where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh
|
|
```
|
|
|
|
The impedance phase is:
|
|
```
|
|
φ_Z = atan(-Im{Y}/Re{Y})
|
|
```
|
|
|
|
**Question:** For fixed C_mut and C_sh, which R value minimizes |φ_Z| (makes most resistive)?
|
|
|
|
**Mathematical result:** Taking derivative ∂φ_Z/∂G = 0 and solving:
|
|
```
|
|
G_opt = ω√[C_mut(C_mut + C_sh)]
|
|
|
|
Therefore:
|
|
R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
|
|
```
|
|
|
|
At this resistance, the phase angle magnitude is minimized to:
|
|
```
|
|
φ_Z,min = -atan(2√[r(1 + r)])
|
|
|
|
where r = C_mut/C_sh (capacitance ratio)
|
|
```
|
|
|
|
### The Critical Ratio r = 0.207
|
|
|
|
Let's find when φ_Z,min = -45° is achievable:
|
|
```
|
|
-45° = -atan(2√[r(1 + r)])
|
|
tan(45°) = 1 = 2√[r(1 + r)]
|
|
0.5 = √[r(1 + r)]
|
|
0.25 = r(1 + r) = r + r²
|
|
r² + r - 0.25 = 0
|
|
|
|
Using quadratic formula:
|
|
r = [-1 ± √(1 + 1)] / 2 = [-1 ± √2] / 2
|
|
|
|
Taking positive root:
|
|
r = (√2 - 1) / 2 ≈ 0.207
|
|
```
|
|
|
|
**Critical insight:**
|
|
- If r < 0.207: Can achieve φ_Z = -45° (with appropriate R)
|
|
- If r > 0.207: **Cannot achieve φ_Z = -45° no matter what R you choose!**
|
|
- If r ≥ 0.207: φ_Z,min is more negative than -45°
|
|
|
|
### Typical Tesla Coil Values
|
|
|
|
**Large topload, short spark:**
|
|
```
|
|
C_mut = 10 pF, C_sh = 4 pF (2 feet)
|
|
r = 10/4 = 2.5
|
|
|
|
φ_Z,min = -atan(2√[2.5 × 3.5]) = -atan(2 × 2.96) = -atan(5.92) = -80.4°
|
|
```
|
|
|
|
**Small topload, long spark:**
|
|
```
|
|
C_mut = 6 pF, C_sh = 12 pF (6 feet)
|
|
r = 6/12 = 0.5
|
|
|
|
φ_Z,min = -atan(2√[0.5 × 1.5]) = -atan(2 × 0.866) = -atan(1.732) = -60.0°
|
|
```
|
|
|
|
**Common range:** r = 0.5 to 2.0, giving φ_Z,min ≈ -60° to -80°
|
|
|
|
**Conclusion:** For most Tesla coil geometries, -45° is **mathematically impossible**!
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 2.1: Calculate Minimum Phase Angle
|
|
|
|
**Given:**
|
|
- Frequency: f = 200 kHz
|
|
- C_mut = 8 pF
|
|
- C_sh = 6 pF
|
|
|
|
**Find:**
|
|
(a) Capacitance ratio r
|
|
(b) Minimum achievable phase angle φ_Z,min
|
|
(c) R_opt_phase that achieves this angle
|
|
|
|
**Solution:**
|
|
|
|
**Part (a):** Capacitance ratio
|
|
```
|
|
r = C_mut / C_sh = 8 / 6 = 1.333
|
|
```
|
|
|
|
**Part (b):** Minimum phase angle
|
|
```
|
|
φ_Z,min = -atan(2√[r(1 + r)])
|
|
= -atan(2√[1.333 × 2.333])
|
|
= -atan(2√3.11)
|
|
= -atan(2 × 1.764)
|
|
= -atan(3.528)
|
|
= -74.2°
|
|
```
|
|
|
|
**Part (c):** Resistance for minimum phase
|
|
```
|
|
ω = 2πf = 2π × 200×10³ = 1.257×10⁶ rad/s
|
|
|
|
R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
|
|
= 1 / [1.257×10⁶ × √(8×10⁻¹² × 14×10⁻¹²)]
|
|
= 1 / [1.257×10⁶ × √(112×10⁻²⁴)]
|
|
= 1 / [1.257×10⁶ × 10.58×10⁻¹²]
|
|
= 1 / (13.30×10⁻⁶)
|
|
= 75.2 kΩ
|
|
```
|
|
|
|
**Interpretation:**
|
|
- With r = 1.333, cannot achieve -45°
|
|
- Best possible is -74.2° (much more capacitive)
|
|
- This requires R = 75.2 kΩ
|
|
- Any other R value gives |φ_Z| > 74.2°
|
|
|
|
---
|
|
|
|
### VISUAL AID 2.1: Graph of φ_Z,min vs r
|
|
|
|
```
|
|
[Describe for plotting:]
|
|
|
|
Graph with:
|
|
- X-axis: r = C_mut/C_sh (log scale), range 0.1 to 10
|
|
- Y-axis: φ_Z,min (degrees), range -90° to -40°
|
|
|
|
Plot curve: φ_Z,min = -atan(2√[r(1+r)])
|
|
|
|
Key points marked:
|
|
- r = 0.207, φ_Z,min = -45° (mark with horizontal dashed line)
|
|
- Shaded region r < 0.207: "Can achieve -45°"
|
|
- Shaded region r > 0.207: "Cannot achieve -45°"
|
|
- Typical Tesla coil range r = 0.5 to 2.0 highlighted
|
|
- Example points:
|
|
* r = 0.5, φ_Z = -60°
|
|
* r = 1.0, φ_Z = -70.5°
|
|
* r = 2.0, φ_Z = -79.7°
|
|
|
|
Annotations:
|
|
- "Larger r → more capacitive minimum"
|
|
- "Large topload + short spark → high r"
|
|
- "Small topload + long spark → low r"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 2.1
|
|
|
|
**Problem 1:** For C_mut = 12 pF, C_sh = 8 pF at f = 180 kHz:
|
|
(a) Calculate r
|
|
(b) Find φ_Z,min
|
|
(c) Can this circuit achieve -45°?
|
|
|
|
**Problem 2:** A designer wants φ_Z,min = -50°. What maximum value of r is allowed? If C_sh = 10 pF, what is the maximum C_mut?
|
|
|
|
**Problem 3:** Explain physically why larger r (more C_mut relative to C_sh) makes the impedance more capacitive.
|
|
|
|
---
|
|
|
|
## Module 2.2: The Two Critical Resistances
|
|
|
|
### R_opt_phase: Closest to Resistive (Revisited)
|
|
|
|
From Module 2.1:
|
|
```
|
|
R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
|
|
```
|
|
|
|
**Purpose:** Minimizes |φ_Z| to achieve φ_Z,min
|
|
|
|
**Use case:** If you want the "most resistive-looking" impedance possible
|
|
|
|
### R_opt_power: Maximum Power Transfer
|
|
|
|
**Different question:** Which R maximizes real power delivered to the spark for a given topload voltage?
|
|
|
|
**Setup:** Fixed voltage source V_top, variable load resistance R
|
|
|
|
**Power to load:**
|
|
```
|
|
P = 0.5 × |V_top|² × Re{Y(R)}
|
|
```
|
|
|
|
where Y(R) depends on R through G = 1/R.
|
|
|
|
**Mathematical derivation:** Take ∂P/∂G = 0, solve for G:
|
|
|
|
After calculus (see framework document for full derivation):
|
|
```
|
|
R_opt_power = 1 / [ω(C_mut + C_sh)]
|
|
```
|
|
|
|
**Simpler formula!** Just total capacitance, not geometric mean.
|
|
|
|
### Comparing the Two
|
|
|
|
**Relationship:**
|
|
```
|
|
R_opt_power = 1 / [ω(C_mut + C_sh)]
|
|
R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
|
|
|
|
Since √(C_mut(C_mut + C_sh)) < (C_mut + C_sh):
|
|
|
|
R_opt_power < R_opt_phase ALWAYS
|
|
```
|
|
|
|
**Numerical relationship:** For typical r = 0.5 to 2:
|
|
```
|
|
R_opt_power ≈ (0.5 to 0.7) × R_opt_phase
|
|
```
|
|
|
|
**Phase angle at R_opt_power:**
|
|
- Always more negative than φ_Z,min
|
|
- Typically φ_Z ≈ -55° to -75° at R_opt_power
|
|
- More capacitive than R_opt_phase, but delivers more power
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 2.2: Calculating Both Critical Resistances
|
|
|
|
**Given:**
|
|
- Frequency: f = 200 kHz → ω = 1.257×10⁶ rad/s
|
|
- C_mut = 8 pF = 8×10⁻¹² F
|
|
- C_sh = 6 pF = 6×10⁻¹² F
|
|
|
|
**Find:** R_opt_phase, R_opt_power, and compare
|
|
|
|
**Solution:**
|
|
|
|
**Part 1:** R_opt_phase (from Example 2.1)
|
|
```
|
|
R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
|
|
= 75.2 kΩ
|
|
```
|
|
|
|
**Part 2:** R_opt_power
|
|
```
|
|
C_total = C_mut + C_sh = 8 + 6 = 14 pF = 14×10⁻¹² F
|
|
|
|
R_opt_power = 1 / (ωC_total)
|
|
= 1 / (1.257×10⁶ × 14×10⁻¹²)
|
|
= 1 / (17.60×10⁻⁶)
|
|
= 56.8 kΩ
|
|
```
|
|
|
|
**Part 3:** Comparison
|
|
```
|
|
Ratio: R_opt_power / R_opt_phase = 56.8 / 75.2 = 0.755
|
|
|
|
R_opt_power is 75.5% of R_opt_phase
|
|
```
|
|
|
|
**Part 4:** Phase angle at R_opt_power
|
|
|
|
Calculate admittance with R = 56.8 kΩ:
|
|
```
|
|
G = 1/56800 = 17.61 μS
|
|
B₁ = ωC_mut = 1.257×10⁶ × 8×10⁻¹² = 10.06 μS
|
|
B₂ = ωC_sh = 1.257×10⁶ × 6×10⁻¹² = 7.54 μS
|
|
|
|
Re{Y} = GB₂²/[G² + (B₁+B₂)²]
|
|
= 17.61 × 56.85 / [310 + 309.8]
|
|
= 1001.2 / 619.8
|
|
= 1.615 μS
|
|
|
|
Im{Y} = 7.54[310 + 176.9] / 619.8
|
|
= 7.54 × 486.9 / 619.8
|
|
= 5.928 μS
|
|
|
|
φ_Y = atan(5.928/1.615) = atan(3.67) = 74.7°
|
|
φ_Z = -74.7°
|
|
```
|
|
|
|
**Summary:**
|
|
- R_opt_phase = 75.2 kΩ gives φ_Z = -74.2° (minimum)
|
|
- R_opt_power = 56.8 kΩ gives φ_Z = -74.7° (slightly more capacitive)
|
|
- Power is maximized at R_opt_power despite not having minimum phase
|
|
- Difference is small: both are strongly capacitive
|
|
|
|
---
|
|
|
|
### VISUAL AID 2.2: Power vs Resistance Curves
|
|
|
|
```
|
|
[Describe for plotting:]
|
|
|
|
Two overlaid plots sharing X-axis:
|
|
|
|
X-axis: R (kΩ), range 20 to 150, log scale
|
|
|
|
TOP PLOT - Power:
|
|
Y-axis: P (kW), normalized to max = 1.0
|
|
Curve: Bell-shaped, peaks at R_opt_power
|
|
- Peak marked at 56.8 kΩ, height = 1.0
|
|
- Label: "R_opt_power = 56.8 kΩ"
|
|
- Width shows power drops to 0.5 at ±50% R
|
|
- Annotation: "Maximum power transfer"
|
|
|
|
BOTTOM PLOT - Phase angle:
|
|
Y-axis: φ_Z (degrees), range -90° to -40°
|
|
Curve: Rises from -90° (R→0), peaks at R_opt_phase, falls back
|
|
- Peak (least negative) marked at 75.2 kΩ, φ_Z = -74.2°
|
|
- Label: "R_opt_phase = 75.2 kΩ, φ_Z,min = -74.2°"
|
|
- -45° reference line (dashed)
|
|
- Annotation: "Most resistive phase"
|
|
|
|
Vertical lines:
|
|
- At R_opt_power (56.8 kΩ): shows φ_Z = -74.7° on bottom plot
|
|
- At R_opt_phase (75.2 kΩ): shows lower power on top plot
|
|
|
|
Key insight box: "R_opt_power ≠ R_opt_phase"
|
|
"R_opt_power delivers more power but is more capacitive"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 2.2
|
|
|
|
**Problem 1:** For f = 150 kHz, C_mut = 10 pF, C_sh = 8 pF:
|
|
Calculate R_opt_power and R_opt_phase.
|
|
|
|
**Problem 2:** At 200 kHz, a spark has C_total = 12 pF. What is R_opt_power? If V_top = 400 kV, estimate the maximum deliverable power.
|
|
|
|
**Problem 3:** Prove algebraically that R_opt_power < R_opt_phase always (hint: compare 1/(C_mut+C_sh) with 1/√(C_mut(C_mut+C_sh))).
|
|
|
|
**Problem 4:** A measurement shows φ_Z = -68° at the operating point. Is R likely above or below R_opt_phase? Above or below R_opt_power?
|
|
|
|
---
|
|
|
|
## Module 2.3: The "Hungry Streamer" - Self-Optimization
|
|
|
|
### The Feedback Loop
|
|
|
|
Plasma conductivity changes dynamically with power:
|
|
|
|
**1. More power → Joule heating**
|
|
```
|
|
Heating rate: dT/dt ∝ I²R
|
|
Higher current → faster heating
|
|
```
|
|
|
|
**2. Higher temperature → ionization**
|
|
```
|
|
Thermal ionization: fraction ∝ exp(-E_ionization / kT)
|
|
Hotter plasma → more free electrons
|
|
```
|
|
|
|
**3. More electrons → higher conductivity**
|
|
```
|
|
σ = n_e × e × μ_e
|
|
where n_e = electron density, μ_e = electron mobility
|
|
σ ∝ n_e ∝ exp(-E_ionization / kT)
|
|
```
|
|
|
|
**4. Higher conductivity → lower R**
|
|
```
|
|
R = ρL/A = L/(σA)
|
|
σ increases → R decreases
|
|
```
|
|
|
|
**5. Changed R → new circuit behavior**
|
|
```
|
|
New R changes Y_spark, power transfer changes
|
|
If R < R_opt_power: reducing R further decreases power
|
|
If R > R_opt_power: reducing R increases power
|
|
```
|
|
|
|
**6. Stable equilibrium at R ≈ R_opt_power**
|
|
```
|
|
When R approaches R_opt_power:
|
|
- Small decrease → power decreases → cooling → R rises
|
|
- Small increase → power increases → heating → R falls
|
|
- Negative feedback stabilizes at R_opt_power
|
|
```
|
|
|
|
### Time Scales
|
|
|
|
**Thermal response:** ~0.1-1 ms for thin channels
|
|
- Heat diffusion time: τ = d²/(4α) ≈ 0.1 ms for d = 100 μm
|
|
- Fast enough to track AC envelope (kHz modulation)
|
|
- Too slow to track RF oscillation (hundreds of kHz)
|
|
|
|
**Ionization response:** ~μs to ms
|
|
- Recombination time varies with density and temperature
|
|
- Can follow slower modulation
|
|
|
|
**Result:** Plasma adjusts R on timescales of 0.1-10 ms, tracking power delivery changes.
|
|
|
|
### Physical Constraints
|
|
|
|
**Lower bound R_min:**
|
|
- Maximum conductivity limited by electron-ion collision frequency
|
|
- Typical: R_min ≈ 1-10 kΩ for hot, dense leaders
|
|
- If R_opt_power < R_min: plasma stuck at R_min (can't optimize)
|
|
|
|
**Upper bound R_max:**
|
|
- Minimum conductivity of partially ionized gas
|
|
- Typical: R_max ≈ 100 kΩ to 100 MΩ for cool streamers
|
|
- If R_opt_power > R_max: plasma stuck at R_max
|
|
|
|
**Source limitations:**
|
|
- Insufficient voltage: spark won't form at all
|
|
- Insufficient current: can't heat enough to reach R_opt_power
|
|
- Power supply impedance: limits available power
|
|
|
|
**When optimization fails:**
|
|
- Source too weak: spark operates at whatever R it can sustain
|
|
- Thermal time too long: can't adjust fast enough (burst mode)
|
|
- Branching: power divides, none optimizes well
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 2.3: Tracing Optimization Process
|
|
|
|
**Scenario:** Spark initially forms with R = 200 kΩ (cold streamer). Circuit has R_opt_power = 60 kΩ.
|
|
|
|
**Trace the evolution:**
|
|
|
|
**Initial state (t = 0):**
|
|
```
|
|
R = 200 kΩ >> R_opt_power
|
|
Power delivered: P_initial (suboptimal, low)
|
|
Temperature: T_initial (cool)
|
|
```
|
|
|
|
**Early phase (0 < t < 1 ms):**
|
|
```
|
|
Current flows → Joule heating: dT/dt = I²R/c_p
|
|
R is high → voltage division favorable → some heating occurs
|
|
Temperature rises → ionization begins → n_e increases
|
|
Conductivity σ ∝ n_e increases → R decreases
|
|
R drops toward 150 kΩ
|
|
```
|
|
|
|
**Middle phase (1 ms < t < 5 ms):**
|
|
```
|
|
R approaches 100 kΩ range
|
|
Now closer to R_opt_power → power transfer improves
|
|
More power → faster heating → faster ionization
|
|
Positive feedback: lower R → more power → lower R
|
|
R drops rapidly: 100 kΩ → 80 kΩ → 70 kΩ → 65 kΩ
|
|
```
|
|
|
|
**Approach to equilibrium (5 ms < t < 10 ms):**
|
|
```
|
|
R approaches R_opt_power = 60 kΩ
|
|
Power maximized at this R
|
|
If R < 60 kΩ: power would decrease → cooling → R rises
|
|
If R > 60 kΩ: power would increase → heating → R falls
|
|
Negative feedback stabilizes around R ≈ 60 kΩ
|
|
```
|
|
|
|
**Steady state (t > 10 ms):**
|
|
```
|
|
R oscillates around 60 kΩ ± 10%
|
|
Temperature stable at equilibrium
|
|
Power maximized and stable
|
|
Spark is "optimized"
|
|
```
|
|
|
|
**If constraints active:**
|
|
```
|
|
If R_opt_power = 30 kΩ but R_min = 50 kΩ:
|
|
Plasma can only reach R = 50 kΩ (not optimal)
|
|
Power is less than theoretical maximum
|
|
Spark is "starved" - wants more current than physics allows
|
|
```
|
|
|
|
---
|
|
|
|
### DISCUSSION QUESTIONS 2.3
|
|
|
|
**Question 1:** Why does the optimization work? Why doesn't the plasma just pick a random R value?
|
|
|
|
**Question 2:** In burst mode (short pulses, <100 μs), thermal time constants are longer than pulse duration. Would you expect the plasma to reach R_opt_power? Why or why not?
|
|
|
|
**Question 3:** A coil produces sparks with measured R ≈ 20 kΩ, but calculations show R_opt_power = 80 kΩ. What might explain this discrepancy?
|
|
|
|
---
|
|
|
|
## Module 2.4: Power Calculations and Common Errors
|
|
|
|
### Correct Power Formula
|
|
|
|
For AC circuit with peak phasors:
|
|
```
|
|
P = 0.5 × Re{V × I*}
|
|
|
|
Expanded:
|
|
P = 0.5 × |V| × |I| × cos(φ_v - φ_i)
|
|
|
|
For impedance Z:
|
|
I = V/Z
|
|
P = 0.5 × |V|² × Re{1/Z} = 0.5 × |V|² × Re{Y}
|
|
```
|
|
|
|
Or using impedance directly:
|
|
```
|
|
P = 0.5 × |I|² × Re{Z} = 0.5 × I² × R
|
|
```
|
|
|
|
### Why V_top/I_base is Wrong
|
|
|
|
**The problem:** Current at secondary base (I_base) includes ALL return currents:
|
|
|
|
1. **Capacitance to ground** along entire secondary
|
|
- Each turn has C to ground
|
|
- AC current: I_C = jωC × V
|
|
- Sum of all displacement currents
|
|
|
|
2. **Primary-to-secondary coupling**
|
|
- Displacement current through C_ps
|
|
- Part of transformer action
|
|
|
|
3. **Strike ring/environment coupling**
|
|
- Any nearby grounded object
|
|
|
|
4. **The spark current** (what we actually want)
|
|
|
|
**Result:**
|
|
```
|
|
I_base = I_spark + I_displacement_secondary + I_primary_coupling + I_environment
|
|
|
|
V_top/I_base = wrong because denominator includes parasitic currents!
|
|
```
|
|
|
|
**Measured impedance is too low** (I_base too high).
|
|
|
|
### Correct Measurement Port
|
|
|
|
**Definition:** Topload-to-ground is the correct measurement port.
|
|
|
|
**Current measurement:** Only the current **through the spark path** from topload.
|
|
|
|
**Methods:**
|
|
1. Measure I_spark return current separately (Rogowski/CT on spark ground return)
|
|
2. Use circuit analysis (know V_top, calculate I_spark from model)
|
|
3. Thévenin extraction (next modules)
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 2.4: Correct vs Incorrect Power Calculation
|
|
|
|
**Given:**
|
|
- V_top = 300 kV peak
|
|
- I_base (measured at secondary base) = 5 A peak
|
|
- I_spark (actual spark current) = 1.5 A peak
|
|
- Spark impedance phase: φ_Z = -70°
|
|
|
|
**Find:** Power using incorrect method, power using correct method
|
|
|
|
**Solution:**
|
|
|
|
**Incorrect method:** Using V_top/I_base
|
|
```
|
|
Z_apparent = V_top / I_base = 300 kV / 5 A = 60 kΩ
|
|
|
|
This is NOT the spark impedance!
|
|
|
|
If we naively calculated power:
|
|
P_wrong = 0.5 × 300 kV × 5 A × cos(-70°)
|
|
= 0.5 × 1500 kW × 0.342
|
|
= 257 kW
|
|
|
|
This is way too high!
|
|
```
|
|
|
|
**Correct method:** Using actual spark current
|
|
```
|
|
I_spark = 1.5 A peak
|
|
|
|
Real spark impedance:
|
|
Z_spark = V_top / I_spark = 300 kV / 1.5 A = 200 kΩ
|
|
|
|
Power:
|
|
P_correct = 0.5 × V_top × I_spark × cos(φ_Z)
|
|
= 0.5 × 300 kV × 1.5 A × cos(-70°)
|
|
= 0.5 × 450 kW × 0.342
|
|
= 77 kW
|
|
|
|
Or using resistance directly:
|
|
R = |Z| × cos(φ_Z) = 200 kΩ × 0.342 = 68.4 kΩ
|
|
P = 0.5 × I² × R = 0.5 × 1.5² × 68.4 kΩ = 77 kW ✓
|
|
```
|
|
|
|
**Error analysis:**
|
|
```
|
|
P_wrong / P_correct = 257 / 77 = 3.3×
|
|
|
|
The incorrect method overestimates power by 330%!
|
|
```
|
|
|
|
---
|
|
|
|
### VISUAL AID 2.4: Current Flow Diagram
|
|
|
|
```
|
|
[Describe for drawing:]
|
|
|
|
Side view of Tesla coil showing current paths:
|
|
|
|
PRIMARY:
|
|
- Primary coil at bottom (multi-turn)
|
|
- Current I_primary flowing
|
|
- Capacitor C_primary
|
|
- Ground connection
|
|
|
|
SECONDARY:
|
|
- Tall helical coil
|
|
- Multiple current paths illustrated with arrows:
|
|
|
|
Path 1 (RED): Spark current
|
|
- Flows from topload through spark to remote ground
|
|
- Returns through earth/floor to secondary base
|
|
- Labeled: "I_spark" (what we want to measure)
|
|
|
|
Path 2 (BLUE): Displacement currents along secondary
|
|
- From each turn to ground
|
|
- Many small arrows radiating outward
|
|
- Labeled: "I_displacement = Σ(jωC_turn × V_turn)"
|
|
|
|
Path 3 (GREEN): Primary-secondary coupling
|
|
- From primary through C_ps to secondary
|
|
- Labeled: "I_coupling"
|
|
|
|
Path 4 (YELLOW): Environmental coupling
|
|
- To nearby objects, walls, strike ring
|
|
- Labeled: "I_environment"
|
|
|
|
AT SECONDARY BASE:
|
|
- Large arrow labeled "I_base = I_spark + I_displacement + I_coupling + I_environment"
|
|
- RED path continues to ground separately
|
|
|
|
Key insight box: "I_base ≠ I_spark! Cannot use V_top/I_base for spark impedance!"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 2.4
|
|
|
|
**Problem 1:** A simulation shows V_top = 250 kV, I_base = 3.5 A, but the spark circuit model predicts Z_spark = 180 kΩ. Calculate the actual spark current and power.
|
|
|
|
**Problem 2:** Explain why displacement current is proportional to frequency (ω). If frequency doubles, what happens to I_displacement?
|
|
|
|
**Problem 3:** An experimenter measures I_base = 4 A and calculates Z = V_top/I_base = 75 kΩ. Another measurement with a Rogowski coil on the spark return path shows I_spark = 1.2 A. What is the true spark impedance?
|
|
|
|
---
|
|
|
|
## Module 2.5: Thévenin Equivalent Method - Part A (Measuring Z_th)
|
|
|
|
### What is a Thévenin Equivalent?
|
|
|
|
**Thévenin's Theorem:** Any linear two-terminal network can be replaced by:
|
|
- A voltage source V_th (open-circuit voltage)
|
|
- In series with an impedance Z_th (output impedance)
|
|
|
|
```
|
|
[Complex network] ≡ [V_th]---[Z_th]---o Output
|
|
|
|
|
GND
|
|
```
|
|
|
|
**Advantage:** Characterize the coil **once**, then predict behavior with **any load** instantly.
|
|
|
|
### Measuring Z_th: Output Impedance
|
|
|
|
**Procedure:**
|
|
|
|
**Step 1:** Turn OFF primary drive
|
|
- Set drive voltage to 0V (AC short circuit)
|
|
- Keep all tank components in place (MMC, L_primary, damping resistors)
|
|
- Tank circuit still present, just not driven
|
|
|
|
**Step 2:** Apply test source
|
|
- Apply 1V AC at operating frequency to topload-to-ground port
|
|
- Use small-signal AC source (simulation or actual)
|
|
|
|
**Step 3:** Measure current
|
|
```
|
|
I_test = current into topload port with 1V applied
|
|
```
|
|
|
|
**Step 4:** Calculate Z_th
|
|
```
|
|
Z_th = V_test / I_test = 1V / I_test
|
|
|
|
Z_th = R_th + jX_th (complex impedance)
|
|
```
|
|
|
|
**Physical meaning:**
|
|
- R_th: resistive losses (secondary winding, topload, damping)
|
|
- X_th: reactive component (usually capacitive from topload)
|
|
|
|
**Typical values at 200 kHz:**
|
|
- R_th: 10-100 Ω (depends on Q and coil size)
|
|
- X_th: -500 to -3000 Ω (capacitive)
|
|
- |Z_th|: 500-3000 Ω
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 2.5A: Extracting Z_th from Simulation
|
|
|
|
**Simulation setup:**
|
|
- DRSSTC at f = 185 kHz
|
|
- Primary drive set to 0V
|
|
- All components remain (L_primary, C_MMC, secondary, topload)
|
|
- AC test source: 1V ∠0° at topload-to-ground
|
|
|
|
**Simulation results:**
|
|
- I_test = 0.000412 ∠87.3° A = 0.412 mA ∠87.3°
|
|
|
|
**Calculate Z_th:**
|
|
|
|
**Step 1:** Impedance magnitude
|
|
```
|
|
|Z_th| = |V| / |I| = 1 V / 0.000412 A = 2427 Ω
|
|
```
|
|
|
|
**Step 2:** Impedance phase
|
|
```
|
|
φ_Z_th = φ_V - φ_I = 0° - 87.3° = -87.3°
|
|
```
|
|
|
|
**Step 3:** Polar form
|
|
```
|
|
Z_th = 2427 Ω ∠-87.3°
|
|
```
|
|
|
|
**Step 4:** Convert to rectangular
|
|
```
|
|
R_th = |Z_th| × cos(φ_Z_th) = 2427 × cos(-87.3°) = 2427 × 0.0471 = 114 Ω
|
|
X_th = |Z_th| × sin(φ_Z_th) = 2427 × sin(-87.3°) = 2427 × (-0.9989) = -2424 Ω
|
|
|
|
Z_th = 114 - j2424 Ω
|
|
```
|
|
|
|
**Interpretation:**
|
|
- **R_th = 114 Ω:** Secondary losses (winding resistance, dielectric losses)
|
|
- **X_th = -2424 Ω:** Strongly capacitive (topload dominates)
|
|
- **Phase ≈ -87°:** Nearly pure capacitor with small series resistance
|
|
- **Quality factor estimate:** Q ≈ |X_th|/R_th = 2424/114 ≈ 21
|
|
|
|
---
|
|
|
|
### VISUAL AID 2.5A: Thévenin Measurement Setup
|
|
|
|
```
|
|
[Describe for drawing:]
|
|
|
|
Two circuit diagrams side-by-side:
|
|
|
|
LEFT: Full Tesla coil circuit (complex)
|
|
- Primary side: Driver → L_primary → C_MMC → Ground
|
|
- Magnetic coupling to secondary
|
|
- Secondary: Base grounded, many turns, topload at top
|
|
- All parasitics shown (C to ground, etc.)
|
|
- Output port marked at topload
|
|
- Label: "Complex original circuit"
|
|
|
|
RIGHT: Thévenin equivalent (simple)
|
|
- Just two components:
|
|
* Voltage source V_th
|
|
* Series impedance Z_th = 114 - j2424 Ω
|
|
- Output port (same as left)
|
|
- Label: "Thévenin equivalent"
|
|
|
|
Arrow between them: "Extraction process"
|
|
|
|
BOTTOM: Measurement configuration
|
|
- Primary drive: OFF (0V symbol)
|
|
- Test source: 1V AC at topload
|
|
- Ammeter measuring I_test
|
|
- Calculation: Z_th = 1V / I_test
|
|
- Note: "All tank components remain in circuit"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 2.5A
|
|
|
|
**Problem 1:** A test measurement gives I_test = 0.00035 ∠82° A for V_test = 1 ∠0° V at f = 200 kHz. Calculate Z_th in rectangular form.
|
|
|
|
**Problem 2:** If Z_th = 85 - j1800 Ω, what is the unloaded Q of the secondary circuit?
|
|
|
|
---
|
|
|
|
## Module 2.6: Thévenin Equivalent Method - Part B (Using V_th and Z_th)
|
|
|
|
### Measuring V_th: Open-Circuit Voltage
|
|
|
|
**Procedure:**
|
|
|
|
**Step 1:** Remove load
|
|
- Disconnect spark (or set spark to not break out)
|
|
- Topload is open-circuit
|
|
|
|
**Step 2:** Turn ON primary drive
|
|
- Normal operating frequency and amplitude
|
|
- Drive as you would for spark operation
|
|
|
|
**Step 3:** Measure topload voltage
|
|
```
|
|
V_th = V(topload) with no load (complex magnitude and phase)
|
|
```
|
|
|
|
**Typical:** V_th = 200-500 kV peak for medium coils
|
|
|
|
### Predicting Power to Any Load
|
|
|
|
With Z_th and V_th known, calculate power to any load impedance Z_load:
|
|
|
|
**Circuit with load:**
|
|
```
|
|
[V_th] --- [Z_th] --- [Z_load] --- GND
|
|
|
|
Total impedance: Z_total = Z_th + Z_load
|
|
Current: I = V_th / (Z_th + Z_load)
|
|
Voltage across load: V_load = I × Z_load
|
|
Power in load: P_load = 0.5 × |I|² × Re{Z_load}
|
|
```
|
|
|
|
**Direct formula:**
|
|
```
|
|
P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
|
|
```
|
|
|
|
**No re-simulation needed!** Just plug in different Z_load values.
|
|
|
|
### Theoretical Maximum Power
|
|
|
|
**Conjugate match condition:** Maximum power transfer occurs when:
|
|
```
|
|
Z_load = Z_th* (complex conjugate)
|
|
|
|
If Z_th = R_th + jX_th, then Z_load = R_th - jX_th
|
|
```
|
|
|
|
**Maximum power:**
|
|
```
|
|
P_max = |V_th|² / (8 × R_th)
|
|
```
|
|
|
|
**BUT:** For spark loads, conjugate match is usually not achievable due to topological constraints (Module 2.1).
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 2.6: Complete Thévenin Analysis
|
|
|
|
**Given:**
|
|
- Z_th = 114 - j2424 Ω (from Example 2.5A)
|
|
- V_th = 350 kV ∠0° (measured with drive on, no load)
|
|
- Candidate spark load: Z_spark = 60 kΩ - j160 kΩ (from lumped model)
|
|
|
|
**Find:**
|
|
(a) Current through spark
|
|
(b) Voltage across spark
|
|
(c) Power dissipated in spark
|
|
(d) Theoretical maximum power (conjugate match)
|
|
|
|
**Solution:**
|
|
|
|
**Part (a):** Current
|
|
```
|
|
Z_total = Z_th + Z_spark
|
|
= (114 - j2424) + (60000 - j160000)
|
|
= (60114 - j162424) Ω
|
|
|
|
|Z_total| = √(60114² + 162424²) = √(3.614×10⁹ + 2.638×10¹⁰) = √3.00×10¹⁰ = 173 kΩ
|
|
|
|
I = V_th / Z_total = (350 kV) / (173 kΩ) = 2.02 A peak
|
|
```
|
|
|
|
**Part (b):** Voltage across spark
|
|
```
|
|
Voltage divider:
|
|
V_spark = V_th × [Z_spark / (Z_th + Z_spark)]
|
|
|
|
|V_spark| = 350 kV × (170 kΩ / 173 kΩ) = 350 kV × 0.983 = 344 kV
|
|
|
|
Most voltage appears across spark (Z_spark >> Z_th)
|
|
```
|
|
|
|
**Part (c):** Power in spark
|
|
```
|
|
P_spark = 0.5 × I² × Re{Z_spark}
|
|
= 0.5 × (2.02)² × 60000
|
|
= 0.5 × 4.08 × 60000
|
|
= 122 kW
|
|
```
|
|
|
|
**Part (d):** Theoretical maximum
|
|
```
|
|
Conjugate match: Z_load = Z_th* = 114 + j2424 Ω
|
|
|
|
P_max = |V_th|² / (8 × R_th)
|
|
= (350×10³)² / (8 × 114)
|
|
= 1.225×10¹¹ / 912
|
|
= 134 MW
|
|
|
|
Wait, this seems way too high! Let me recalculate...
|
|
|
|
P_max = 0.5 × |V_th|² / (4 × R_th) [Correct formula]
|
|
= 0.5 × (350×10³)² / (4 × 114)
|
|
= 0.5 × 1.225×10¹¹ / 456
|
|
= 134 MW
|
|
|
|
This is still huge because R_th is so small (114 Ω).
|
|
```
|
|
|
|
**Reality check:**
|
|
- Actual spark power: 122 kW
|
|
- Theoretical maximum: 134 MW
|
|
- Spark extracts: 122/134000 = 0.09% of theoretical maximum
|
|
|
|
**Why such a huge difference?**
|
|
- Conjugate match would require Z_load = 114 + j2424 Ω (very low resistance!)
|
|
- Actual spark: Z_spark = 60000 - j160000 Ω (much higher resistance, wrong phase)
|
|
- Topological constraints prevent achieving conjugate match
|
|
- This is normal for Tesla coils!
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 2.6
|
|
|
|
**Problem 1:** Given Z_th = 95 - j1850 Ω, V_th = 280 kV, and a spark model with Z_spark = 50 kΩ - j140 kΩ:
|
|
(a) Calculate power delivered to spark
|
|
(b) What percentage of theoretical maximum is this?
|
|
|
|
**Problem 2:** A load Z_load = 200 + j200 Ω is connected. If Z_th = 100 - j2000 Ω and V_th = 300 kV, calculate the power. Is this inductive or capacitive load?
|
|
|
|
---
|
|
|
|
## Module 2.7: Quality Factor and Ringdown Measurements
|
|
|
|
### What is Quality Factor (Q)?
|
|
|
|
**Definition:** Ratio of energy stored to energy dissipated per cycle:
|
|
```
|
|
Q = 2π × (Energy stored) / (Energy dissipated per cycle)
|
|
|
|
For series RLC: Q = ωL/R = 1/(ωRC)
|
|
For parallel RLC at resonance: Q = R/(ωL) = ωRC
|
|
```
|
|
|
|
**Physical meaning:**
|
|
- High Q: oscillation persists many cycles (low damping)
|
|
- Low Q: oscillation decays quickly (high damping)
|
|
|
|
### Measuring Q from Ringdown
|
|
|
|
**Procedure:**
|
|
1. Excite coil (burst of AC at resonance)
|
|
2. Turn off drive
|
|
3. Measure voltage decay
|
|
|
|
**Exponential envelope:**
|
|
```
|
|
V(t) = V₀ × exp(-t/τ) × cos(ωt)
|
|
|
|
where τ = 2Q/ω = decay time constant
|
|
```
|
|
|
|
**From consecutive peaks:**
|
|
```
|
|
Ratio of amplitudes n cycles apart:
|
|
A(t + nT) / A(t) = exp(-nT/τ) = exp(-nπ/Q)
|
|
|
|
Solving for Q:
|
|
Q = nπ / ln[A(t) / A(t + nT)]
|
|
```
|
|
|
|
**Practical:** Measure peak-to-peak over several cycles:
|
|
```
|
|
Q ≈ πf × Δt / ln(A₁/A₂)
|
|
|
|
where Δt = time between measured peaks
|
|
```
|
|
|
|
### Extracting Spark Parameters from Q Measurements
|
|
|
|
**Unloaded (no spark):**
|
|
- Measure f₀, Q₀
|
|
- Represents coil losses only
|
|
|
|
**Loaded (with spark):**
|
|
- Measure f_L, Q_L
|
|
- Spark adds resistance and capacitance
|
|
|
|
**At resonance:**
|
|
```
|
|
Q_L = ω_L × C_eq × R_p
|
|
|
|
where R_p = equivalent parallel resistance at resonance
|
|
C_eq = total capacitance = C₀ + ΔC
|
|
```
|
|
|
|
**Solving for conductance:**
|
|
```
|
|
G_total = 1/R_p = ω_L × C_eq / Q_L
|
|
|
|
Spark contribution:
|
|
G_spark ≈ G_total - G_0 = ω_L C_eq / Q_L - ω₀ C₀ / Q₀
|
|
```
|
|
|
|
**Capacitance from frequency shift:**
|
|
```
|
|
Frequency ratio: f₀/f_L = √(C_eq/C₀)
|
|
|
|
Therefore: C_eq = C₀ × (f₀/f_L)²
|
|
|
|
Spark capacitance: ΔC = C_eq - C₀
|
|
```
|
|
|
|
**Spark admittance:**
|
|
```
|
|
Y_spark ≈ G_spark + jω_L ΔC
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 2.7: Q Measurement and Spark Extraction
|
|
|
|
**Given measurements:**
|
|
|
|
**Unloaded:**
|
|
- f₀ = 200 kHz
|
|
- Q₀ = 80 (from ringdown)
|
|
- C₀ = 28 pF (calculated from geometry)
|
|
|
|
**With spark:**
|
|
- f_L = 185 kHz (frequency dropped)
|
|
- Q_L = 25 (from ringdown with spark)
|
|
|
|
**Find:** Spark admittance Y_spark
|
|
|
|
**Solution:**
|
|
|
|
**Step 1:** Calculate loaded capacitance
|
|
```
|
|
C_eq = C₀ × (f₀/f_L)²
|
|
= 28 pF × (200/185)²
|
|
= 28 pF × (1.081)²
|
|
= 28 pF × 1.169
|
|
= 32.7 pF
|
|
|
|
ΔC = C_eq - C₀ = 32.7 - 28 = 4.7 pF
|
|
```
|
|
|
|
**Step 2:** Calculate conductances
|
|
```
|
|
ω₀ = 2π × 200×10³ = 1.257×10⁶ rad/s
|
|
ω_L = 2π × 185×10³ = 1.162×10⁶ rad/s
|
|
|
|
G₀ = ω₀ C₀ / Q₀
|
|
= 1.257×10⁶ × 28×10⁻¹² / 80
|
|
= 35.2×10⁻⁶ / 80
|
|
= 0.44 μS
|
|
|
|
G_total = ω_L C_eq / Q_L
|
|
= 1.162×10⁶ × 32.7×10⁻¹² / 25
|
|
= 38.0×10⁻⁶ / 25
|
|
= 1.52 μS
|
|
|
|
G_spark = G_total - G₀ = 1.52 - 0.44 = 1.08 μS
|
|
```
|
|
|
|
**Step 3:** Construct spark admittance
|
|
```
|
|
B_spark = ω_L ΔC = 1.162×10⁶ × 4.7×10⁻¹² = 5.46 μS
|
|
|
|
Y_spark = G_spark + jB_spark
|
|
= 1.08 + j5.46 μS
|
|
```
|
|
|
|
**Step 4:** Convert to impedance
|
|
```
|
|
|Y_spark| = √(1.08² + 5.46²) = √(1.17 + 29.8) = 5.56 μS
|
|
|
|
Z_spark = 1/Y_spark
|
|
|Z_spark| = 1/(5.56×10⁻⁶) = 180 kΩ
|
|
|
|
φ_Y = atan(5.46/1.08) = atan(5.06) = 78.8°
|
|
φ_Z = -78.8°
|
|
|
|
Z_spark = 180 kΩ ∠-78.8°
|
|
|
|
In rectangular:
|
|
R = 180 × cos(-78.8°) = 180 × 0.194 = 35 kΩ
|
|
X = 180 × sin(-78.8°) = 180 × (-0.981) = -177 kΩ
|
|
|
|
Z_spark = 35 - j177 kΩ
|
|
```
|
|
|
|
**Interpretation:**
|
|
- Spark added 4.7 pF capacitance (consistent with ~2.4 foot spark)
|
|
- R ≈ 35 kΩ at 185 kHz
|
|
- Strongly capacitive: φ_Z = -78.8°
|
|
- Q dropped from 80 to 25 (spark loading dominates)
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 2.7
|
|
|
|
**Problem 1:** A ringdown shows voltage dropping from 100 kV to 50 kV in 8 cycles at f = 195 kHz. Calculate Q.
|
|
|
|
**Problem 2:** Measurements show: f₀ = 210 kHz, Q₀ = 65, f_L = 198 kHz (with spark), Q_L = 30. If C₀ = 25 pF, calculate the spark's added capacitance and equivalent resistance.
|
|
|
|
**Problem 3:** Why does frequency decrease when a spark forms? Explain in terms of capacitance.
|
|
|
|
---
|
|
|
|
## Part 2 Summary & Integration
|
|
|
|
### Key Concepts Checklist
|
|
|
|
- [ ] **Topological phase constraint:** φ_Z,min = -atan(2√[r(1+r)])
|
|
- [ ] **Critical ratio:** r ≥ 0.207 makes φ_Z = -45° impossible
|
|
- [ ] **R_opt_phase:** Minimizes |φ_Z|, gives φ_Z,min
|
|
- [ ] **R_opt_power:** Maximizes power transfer to load
|
|
- [ ] **Relationship:** R_opt_power < R_opt_phase always
|
|
- [ ] **Hungry streamer:** Plasma self-adjusts toward R_opt_power
|
|
- [ ] **Physical limits:** R_min (hot plasma) to R_max (cold plasma)
|
|
- [ ] **Why V_top/I_base fails:** Includes displacement currents
|
|
- [ ] **Correct port:** Topload-to-ground
|
|
- [ ] **Thévenin Z_th:** Output impedance (drive off, test on)
|
|
- [ ] **Thévenin V_th:** Open-circuit voltage (drive on, no load)
|
|
- [ ] **Power formula:** P = 0.5|V_th|²Re{Z_load}/|Z_th+Z_load|²
|
|
- [ ] **Conjugate match:** Usually unachievable due to constraints
|
|
- [ ] **Q from ringdown:** Q = πfΔt/ln(A₁/A₂)
|
|
- [ ] **Extract Y_spark:** From frequency shift and Q change
|
|
|
|
---
|
|
|
|
## Comprehensive Design Exercise
|
|
|
|
**Scenario:** Design matching for a DRSSTC
|
|
|
|
**Given:**
|
|
- Operating frequency: f = 190 kHz
|
|
- Topload: C_topload = 30 pF
|
|
- Target spark: 3 feet (estimate C_sh)
|
|
- FEMM analysis: C_mut = 9 pF for 3-foot spark
|
|
- Thévenin equivalent (measured): Z_th = 105 - j2100 Ω, V_th = 320 kV
|
|
|
|
**Tasks:**
|
|
|
|
1. **Calculate capacitance ratio and phase constraint:**
|
|
- Find r = C_mut/C_sh
|
|
- Calculate φ_Z,min
|
|
- Can this achieve -45°?
|
|
|
|
2. **Determine optimal resistances:**
|
|
- Calculate R_opt_power
|
|
- Calculate R_opt_phase
|
|
- What is typical φ_Z at R_opt_power?
|
|
|
|
3. **Build lumped spark model:**
|
|
- Draw circuit with C_mut, R, C_sh
|
|
- Use R = R_opt_power
|
|
- Calculate Y_spark
|
|
|
|
4. **Predict performance with Thévenin:**
|
|
- Calculate Z_spark from Y_spark
|
|
- Find total impedance Z_th + Z_spark
|
|
- Calculate spark current
|
|
- Calculate power delivered to spark
|
|
|
|
5. **Compare to theoretical maximum:**
|
|
- Calculate P_max (conjugate match)
|
|
- What percentage is actually delivered?
|
|
- Explain the difference
|
|
|
|
**Work through this completely, then check solutions in appendix.**
|
|
|
|
---
|
|
|
|
**END OF PART 2**
|
|
|
|
---
|
|
|
|
# Tesla Coil Spark Modeling - Complete Lesson Plan
|
|
## Part 3: Growth Physics and FEMM Modeling - Where Sparks Come From
|
|
|
|
---
|
|
|
|
## Module 3.1: Electric Fields and Breakdown
|
|
|
|
### Electric Field Basics
|
|
|
|
**Definition:** Electric field E is force per unit charge:
|
|
```
|
|
E = F/q [units: N/C or V/m]
|
|
|
|
Related to voltage:
|
|
E = -dV/dx (field is voltage gradient)
|
|
|
|
For uniform field:
|
|
E ≈ V/d (voltage divided by distance)
|
|
```
|
|
|
|
**Field at spark tip is NOT uniform** - concentrated by geometry.
|
|
|
|
### Breakdown Field Thresholds
|
|
|
|
**E_inception:** Field required to initiate breakdown from smooth electrode
|
|
```
|
|
E_inception ≈ 2-3 MV/m (at sea level, dry air)
|
|
|
|
Physical process:
|
|
- Natural cosmic rays create seed electrons
|
|
- Strong field accelerates electrons
|
|
- Collisions ionize more atoms
|
|
- Avalanche breakdown begins
|
|
```
|
|
|
|
**E_propagation:** Field required to sustain spark growth
|
|
```
|
|
E_propagation ≈ 0.4-1.0 MV/m (for leader propagation)
|
|
|
|
Lower than inception because:
|
|
- Channel already partially ionized
|
|
- Hot gas easier to ionize
|
|
- Photoionization helps (UV from plasma)
|
|
```
|
|
|
|
**Altitude/humidity effects:**
|
|
- Lower air density (altitude) → lower E_threshold (±20-30%)
|
|
- Humidity adds water vapor → changes breakdown (~10%)
|
|
- Temperature affects density → small effect
|
|
|
|
### Tip Enhancement Factor κ
|
|
|
|
Sharp tips concentrate field:
|
|
|
|
```
|
|
E_tip = κ × E_average
|
|
|
|
where E_average = V/L (voltage divided by length)
|
|
κ = enhancement factor ≈ 2-5 typical
|
|
```
|
|
|
|
**Physical origin:**
|
|
- Charge accumulates at sharp points
|
|
- Field lines concentrate at high curvature
|
|
- Smaller radius → higher κ
|
|
|
|
**FEMM calculates E_tip directly** from geometry and voltage.
|
|
|
|
### Growth Criterion
|
|
|
|
Spark continues growing when:
|
|
```
|
|
E_tip > E_propagation
|
|
|
|
If E_tip drops below E_propagation:
|
|
- Growth stalls
|
|
- Spark cannot extend further
|
|
- "Voltage-limited"
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.1: Field Calculation
|
|
|
|
**Given:**
|
|
- Spark length: L = 1.5 m
|
|
- Topload voltage: V_top = 400 kV
|
|
- Tip enhancement: κ = 3.5 (from FEMM or estimate)
|
|
|
|
**Find:**
|
|
(a) Average field
|
|
(b) Tip field
|
|
(c) Can spark grow if E_propagation = 0.6 MV/m?
|
|
|
|
**Solution:**
|
|
|
|
**Part (a):** Average field
|
|
```
|
|
E_average = V_top / L
|
|
= 400×10³ V / 1.5 m
|
|
= 267 kV/m
|
|
= 0.267 MV/m
|
|
```
|
|
|
|
**Part (b):** Tip field
|
|
```
|
|
E_tip = κ × E_average
|
|
= 3.5 × 0.267 MV/m
|
|
= 0.93 MV/m
|
|
```
|
|
|
|
**Part (c):** Compare to threshold
|
|
```
|
|
E_tip = 0.93 MV/m
|
|
E_propagation = 0.6 MV/m
|
|
|
|
E_tip > E_propagation ✓
|
|
|
|
Yes, spark can continue growing.
|
|
Margin: 0.93/0.6 = 1.55× above threshold
|
|
```
|
|
|
|
**If voltage drops to 300 kV:**
|
|
```
|
|
E_average = 300 kV / 1.5 m = 0.2 MV/m
|
|
E_tip = 3.5 × 0.2 = 0.7 MV/m
|
|
|
|
Still above 0.6 MV/m, but margin reduced to 1.17×
|
|
```
|
|
|
|
**If voltage drops to 250 kV:**
|
|
```
|
|
E_average = 250 kV / 1.5 m = 0.167 MV/m
|
|
E_tip = 3.5 × 0.167 = 0.58 MV/m
|
|
|
|
Below 0.6 MV/m - growth stalls!
|
|
```
|
|
|
|
---
|
|
|
|
### VISUAL AID 3.1: Field Enhancement
|
|
|
|
```
|
|
[Describe for drawing:]
|
|
|
|
Two panels side-by-side:
|
|
|
|
LEFT: Uniform field (parallel plates)
|
|
- Two flat plates, voltage V between them
|
|
- Evenly spaced field lines (vertical)
|
|
- Formula: E = V/d (constant everywhere)
|
|
- Label: "No enhancement, κ = 1"
|
|
|
|
RIGHT: Point-to-plane (spark geometry)
|
|
- Spherical topload at top (voltage V)
|
|
- Sharp spark tip pointing down
|
|
- Ground plane at bottom
|
|
- Field lines:
|
|
* Sparse near topload (low density)
|
|
* Dense at tip (concentrated)
|
|
* Spread out below tip
|
|
- Color gradient showing field strength:
|
|
* Blue (low) far from tip
|
|
* Red (high) at tip
|
|
- Annotations:
|
|
* E_average = V/L marked along spark
|
|
* E_tip at very tip (red zone)
|
|
* "Enhancement: E_tip = κ × E_average, κ = 2-5"
|
|
|
|
Inset graph: E vs distance from tip
|
|
- Sharp peak at tip (E_tip)
|
|
- Drops rapidly with distance
|
|
- Approaches E_average far from tip
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.1
|
|
|
|
**Problem 1:** A 0.8 m spark has V_top = 280 kV, κ = 4. Calculate E_tip. If E_propagation = 0.5 MV/m, can it grow?
|
|
|
|
**Problem 2:** A spark stalls at 2.0 m length with V_top = 500 kV and κ = 3. Estimate E_propagation for these conditions.
|
|
|
|
**Problem 3:** Why is E_inception > E_propagation? Explain the physical difference.
|
|
|
|
---
|
|
|
|
## Module 3.2: Energy Requirements for Growth
|
|
|
|
### Energy Per Meter (ε)
|
|
|
|
**Concept:** Extending spark by 1 meter requires approximately constant energy:
|
|
|
|
```
|
|
Energy to grow from L₁ to L₂:
|
|
ΔE ≈ ε × (L₂ - L₁)
|
|
|
|
where ε [J/m] depends on operating mode
|
|
```
|
|
|
|
**Not just ionization energy** - includes:
|
|
1. Initial ionization (breaking molecular bonds)
|
|
2. Heating to operating temperature
|
|
3. Work against pressure (channel expansion)
|
|
4. Radiation losses (light, UV, RF)
|
|
5. Branching (wasted energy in short branches)
|
|
6. Inefficiency (non-productive heating)
|
|
|
|
### Typical ε Values by Operating Mode
|
|
|
|
**QCW (Quasi-Continuous Wave):**
|
|
```
|
|
ε ≈ 5-15 J/m
|
|
|
|
Characteristics:
|
|
- Long ramp times (5-20 ms)
|
|
- Channel stays hot throughout growth
|
|
- Efficient leader formation
|
|
- Minimal re-ionization
|
|
```
|
|
|
|
**Hybrid DRSSTC (moderate duty cycle):**
|
|
```
|
|
ε ≈ 20-40 J/m
|
|
|
|
Characteristics:
|
|
- Medium pulses (1-5 ms)
|
|
- Mix of streamers and leaders
|
|
- Some thermal accumulation
|
|
- Moderate efficiency
|
|
```
|
|
|
|
**Burst mode (hard-pulsed):**
|
|
```
|
|
ε ≈ 30-100+ J/m
|
|
|
|
Characteristics:
|
|
- Short pulses (<500 μs)
|
|
- Channel cools between pulses
|
|
- Mostly streamers, bright but short
|
|
- Must re-ionize repeatedly
|
|
- Poor length efficiency
|
|
```
|
|
|
|
### Why Different Modes Have Different ε
|
|
|
|
**QCW efficiency (low ε):**
|
|
- Continuous power → channel stays ionized
|
|
- Thermal ionization maintained
|
|
- Leaders form efficiently
|
|
- Each Joule goes into extension
|
|
|
|
**Burst inefficiency (high ε):**
|
|
- Peak power → brightening, branching
|
|
- Channel cools between bursts
|
|
- Energy into light, heat, not length
|
|
- Must restart from cold each time
|
|
|
|
**Analogy:** Boiling water
|
|
- Low ε: Keep burner on, maintain simmer (efficient)
|
|
- High ε: Pulse burner on/off, water cools (inefficient)
|
|
|
|
### Theoretical Minimum Energy
|
|
|
|
**Just ionization:**
|
|
```
|
|
Ionization energy per molecule ≈ 15 eV
|
|
Air density ≈ 2.5×10²⁵ molecules/m³
|
|
Channel volume ≈ π(d/2)² × L
|
|
|
|
For d = 1 mm, L = 1 m:
|
|
E_ionize = 15 eV × 2.5×10²⁵ × π×(0.5×10⁻³)² × 1
|
|
≈ 0.3 J/m (theoretical minimum)
|
|
```
|
|
|
|
**Why ε >> 0.3 J/m?**
|
|
- Heating to 5000-20000 K (thermal energy)
|
|
- Radiation (visible light, UV, IR)
|
|
- Expansion work (push air aside)
|
|
- Branching losses (many failed attempts)
|
|
- Inefficiencies (not all current goes to useful ionization)
|
|
|
|
**Result:** Real ε is 20-300× theoretical minimum.
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.2: Energy Budget
|
|
|
|
**Given:**
|
|
- Target spark: L = 2 m
|
|
- Operating mode: QCW with ε = 10 J/m
|
|
- Growth time: T = 12 ms
|
|
|
|
**Find:**
|
|
(a) Total energy required
|
|
(b) Average power required
|
|
(c) If only 80 kW available, what happens?
|
|
|
|
**Solution:**
|
|
|
|
**Part (a):** Total energy
|
|
```
|
|
E_total = ε × L
|
|
= 10 J/m × 2 m
|
|
= 20 J
|
|
```
|
|
|
|
**Part (b):** Average power
|
|
```
|
|
P_avg = E_total / T
|
|
= 20 J / 0.012 s
|
|
= 1667 W
|
|
≈ 1.7 kW
|
|
```
|
|
|
|
**Part (c):** With limited power
|
|
```
|
|
Available: P = 80 kW (much more than needed!)
|
|
|
|
This is 80/1.7 = 47× the required power.
|
|
|
|
Options:
|
|
1. Grow much faster: T = 20 J / 80 kW = 0.25 ms (burst-like)
|
|
2. Grow to longer length: L = P × T / ε
|
|
For same 12 ms: L = 80 kW × 0.012 s / 10 J/m = 96 m (unrealistic!)
|
|
|
|
Reality: Voltage limit kicks in first
|
|
- Cannot maintain E_tip > E_propagation for 96 m
|
|
- Spark stalls at voltage-limited length
|
|
```
|
|
|
|
**Key insight:** Need BOTH adequate power AND adequate voltage!
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.2
|
|
|
|
**Problem 1:** A burst-mode coil has ε = 60 J/m. To reach 1.5 m in a 200 μs pulse, what power is required?
|
|
|
|
**Problem 2:** Two coils both deliver 50 kW. Coil A (QCW, ε = 8 J/m) vs Coil B (burst, ε = 50 J/m). For 10 ms operation, which produces longer sparks?
|
|
|
|
---
|
|
|
|
## Module 3.3: Growth Rate and Stalling
|
|
|
|
### The Growth Rate Equation
|
|
|
|
When field threshold is met:
|
|
```
|
|
dL/dt = P_stream / ε [units: m/s]
|
|
|
|
where P_stream = power delivered to spark [W]
|
|
ε = energy per meter [J/m]
|
|
```
|
|
|
|
**Physical meaning:**
|
|
- More power → faster growth
|
|
- Higher ε (inefficiency) → slower growth
|
|
|
|
**When growth stops:**
|
|
```
|
|
If E_tip < E_propagation:
|
|
dL/dt = 0 (stalled)
|
|
|
|
Cannot grow regardless of available power
|
|
```
|
|
|
|
### Voltage-Limited vs Power-Limited
|
|
|
|
**Voltage-limited:**
|
|
```
|
|
E_tip < E_propagation
|
|
- Field too weak at tip
|
|
- Spark cannot extend
|
|
- More power doesn't help (without more voltage)
|
|
- Common for small topload, long target
|
|
```
|
|
|
|
**Power-limited:**
|
|
```
|
|
E_tip > E_propagation, but P_stream < ε × (dL/dt)_desired
|
|
- Field adequate, but not enough energy
|
|
- Spark grows slowly or stalls before reaching potential
|
|
- More voltage doesn't help (without more power)
|
|
- Common for high-Q coils, weak drive
|
|
```
|
|
|
|
### Predicting Growth Time
|
|
|
|
For constant power during ramp:
|
|
```
|
|
L(t) = (P_stream / ε) × t
|
|
|
|
Time to reach L_target:
|
|
T = ε × L_target / P_stream
|
|
```
|
|
|
|
**More realistic:** Power changes as spark grows (loading changes)
|
|
```
|
|
T = ∫₀^L_target (ε / P_stream(L)) dL
|
|
|
|
Requires simulation or numerical integration
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.3: Growth Prediction
|
|
|
|
**Given:**
|
|
- QCW coil, ε = 12 J/m
|
|
- Target: L = 1.8 m
|
|
- Power profile: P_stream = 100 kW (constant during ramp)
|
|
- κ = 3.2, E_propagation = 0.7 MV/m
|
|
- V_top ramps linearly: V(t) = 50 kV/ms × t
|
|
|
|
**Find:**
|
|
(a) Growth time if power-limited
|
|
(b) Growth time if voltage-limited
|
|
(c) Actual growth (considering both limits)
|
|
|
|
**Solution:**
|
|
|
|
**Part (a):** Power-limited case (assume infinite voltage)
|
|
```
|
|
T_power = ε × L / P_stream
|
|
= 12 J/m × 1.8 m / 100000 W
|
|
= 21.6 J / 100000 W
|
|
= 0.000216 s
|
|
= 0.216 ms
|
|
```
|
|
|
|
**Part (b):** Voltage-limited case
|
|
|
|
At length L, need E_tip > E_propagation:
|
|
```
|
|
E_tip = κ × V(t) / L > E_propagation
|
|
V(t) > E_propagation × L / κ
|
|
|
|
For L = 1.8 m:
|
|
V_required > 0.7×10⁶ × 1.8 / 3.2
|
|
V_required > 0.394 MV = 394 kV
|
|
|
|
With ramp V(t) = 50 kV/ms × t:
|
|
T_voltage = 394 kV / (50 kV/ms) = 7.88 ms
|
|
```
|
|
|
|
**Part (c):** Actual growth (limited by slowest)
|
|
```
|
|
T_power = 0.216 ms (very fast if voltage available)
|
|
T_voltage = 7.88 ms (slower, limited by ramp rate)
|
|
|
|
Actual: T ≈ 7.88 ms (voltage-limited)
|
|
|
|
The spark grows as fast as voltage ramps allow.
|
|
Power is MORE than sufficient (100 kW available, only need ~2.7 kW)
|
|
```
|
|
|
|
**Verification of power requirement:**
|
|
```
|
|
P_needed = ε × L / T_actual
|
|
= 12 × 1.8 / 0.00788
|
|
= 2.74 kW
|
|
|
|
100 kW available >> 2.74 kW needed ✓
|
|
Confirms voltage-limited, not power-limited
|
|
```
|
|
|
|
---
|
|
|
|
### VISUAL AID 3.3: Growth Curves
|
|
|
|
```
|
|
[Describe for plotting:]
|
|
|
|
Graph: Spark length L vs time t
|
|
|
|
Three curves:
|
|
|
|
CURVE 1 (Blue): Power-limited
|
|
- Linear growth: L(t) = (P/ε) × t
|
|
- Steep slope (fast growth)
|
|
- Reaches target quickly (0.2 ms)
|
|
- Label: "Power-limited: unlimited voltage"
|
|
|
|
CURVE 2 (Red): Voltage-limited
|
|
- Curved growth: L(t) must satisfy E_tip(V(t),L) > E_prop
|
|
- Slower, follows voltage ramp capability
|
|
- Reaches target at 7.88 ms
|
|
- Label: "Voltage-limited: slow ramp"
|
|
|
|
CURVE 3 (Green): Actual (realistic)
|
|
- Follows faster curve initially
|
|
- Transitions to limiting constraint
|
|
- Usually voltage-limited for Tesla coils
|
|
- Label: "Actual: limited by slowest constraint"
|
|
|
|
Shaded regions:
|
|
- Below curves: "Achieved length"
|
|
- Above: "Not yet reached"
|
|
|
|
Annotations:
|
|
- "QCW: usually voltage-limited"
|
|
- "Burst: can be power-limited"
|
|
- "Need both P and V adequate"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.3
|
|
|
|
**Problem 1:** A spark grows at 2 m/s when P = 40 kW and ε = 20 J/m. Verify this is consistent with dL/dt = P/ε.
|
|
|
|
**Problem 2:** If E_propagation = 0.5 MV/m, κ = 3, and voltage is fixed at V = 300 kV, what is the maximum length the spark can reach (voltage-limited)?
|
|
|
|
**Problem 3:** A coil delivers 30 kW to a spark with ε = 15 J/m. How long to reach 2.5 m? If this time is longer than the voltage ramp allows, which limit dominates?
|
|
|
|
---
|
|
|
|
## Module 3.4: Thermal Physics of Plasma Channels
|
|
|
|
### Temperature Regimes
|
|
|
|
**Streamers (cold):**
|
|
```
|
|
T ≈ 1000-3000 K
|
|
- Weakly ionized
|
|
- Mostly neutral gas with some ions/electrons
|
|
- Purple/blue color (N₂ emission)
|
|
```
|
|
|
|
**Leaders (hot):**
|
|
```
|
|
T ≈ 5000-20000 K
|
|
- Fully ionized plasma
|
|
- White/orange color (blackbody + line emission)
|
|
- Approaching temperatures of stellar photospheres!
|
|
```
|
|
|
|
### Thermal Diffusion Time
|
|
|
|
Heat diffuses radially from hot channel core:
|
|
```
|
|
τ_thermal = d² / (4α_thermal)
|
|
|
|
where d = channel diameter
|
|
α_thermal ≈ 2×10⁻⁵ m²/s for air
|
|
```
|
|
|
|
**Examples:**
|
|
```
|
|
Thin streamer (d = 100 μm):
|
|
τ = (100×10⁻⁶)² / (4 × 2×10⁻⁵)
|
|
= 10⁻⁸ / (8×10⁻⁵)
|
|
= 0.125 ms
|
|
|
|
Thick leader (d = 5 mm):
|
|
τ = (5×10⁻³)² / (4 × 2×10⁻⁵)
|
|
= 25×10⁻⁶ / (8×10⁻⁵)
|
|
= 312 ms
|
|
```
|
|
|
|
### Why Observed Persistence is Longer
|
|
|
|
**Pure thermal diffusion** predicts cooling in 0.1-300 ms, but channels persist longer due to:
|
|
|
|
**1. Convection (buoyancy):**
|
|
```
|
|
Hot gas rises: v ≈ √(g × d × ΔT/T_amb)
|
|
|
|
For d = 2 mm, ΔT = 10000 K:
|
|
v ≈ √(9.8 × 0.002 × 10000/300)
|
|
≈ √(0.65) ≈ 0.8 m/s
|
|
|
|
Rising column remains hot longer than conduction alone
|
|
```
|
|
|
|
**2. Ionization memory:**
|
|
```
|
|
Recombination time: τ_recomb = 1/(α_recomb × n_e)
|
|
Can be 10 μs to 10 ms depending on density
|
|
Ions/electrons persist after thermal cooling begins
|
|
```
|
|
|
|
**Effective persistence:**
|
|
```
|
|
Streamers: ~1-5 ms (convection + ionization)
|
|
Leaders: seconds (buoyant column maintained)
|
|
```
|
|
|
|
### QCW Advantage
|
|
|
|
**QCW ramp times (5-20 ms) exploit channel persistence:**
|
|
```
|
|
1. Initial streamers form (t = 0)
|
|
2. Power heats channel → leader begins (t = 1 ms)
|
|
3. Leader maintained by continuous power (t = 1-20 ms)
|
|
4. Channel stays hot entire time
|
|
5. New growth builds on existing ionization
|
|
6. Efficient energy use
|
|
```
|
|
|
|
**Burst mode problem:**
|
|
```
|
|
1. Pulse creates bright streamer (t = 0-0.1 ms)
|
|
2. Pulse ends, channel cools (t = 0.1-1 ms)
|
|
3. Next pulse must re-ionize cold gas (t = 1 ms)
|
|
4. Energy wasted re-heating
|
|
5. Inefficient (high ε)
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.4: Thermal Time Constants
|
|
|
|
**Given:**
|
|
- Channel diameter: d = 2 mm (typical leader)
|
|
- Air thermal diffusivity: α = 2×10⁻⁵ m²/s
|
|
|
|
**Find:**
|
|
(a) Pure thermal diffusion time
|
|
(b) Estimate convection velocity if ΔT = 8000 K
|
|
(c) QCW ramp time recommendation
|
|
|
|
**Solution:**
|
|
|
|
**Part (a):** Thermal diffusion
|
|
```
|
|
τ_thermal = d² / (4α)
|
|
= (2×10⁻³)² / (4 × 2×10⁻⁵)
|
|
= 4×10⁻⁶ / (8×10⁻⁵)
|
|
= 0.05 s
|
|
= 50 ms
|
|
```
|
|
|
|
**Part (b):** Convection velocity
|
|
```
|
|
v ≈ √(g × d × ΔT/T_amb)
|
|
≈ √(9.8 × 0.002 × 8000/300)
|
|
≈ √(0.523)
|
|
≈ 0.72 m/s
|
|
|
|
Upward velocity helps maintain hot column
|
|
```
|
|
|
|
**Part (c):** QCW ramp recommendation
|
|
```
|
|
τ_thermal = 50 ms
|
|
|
|
Good QCW ramp: T_ramp << τ_thermal (finish before significant cooling)
|
|
Reasonable: T_ramp = 5-20 ms (10-40% of τ)
|
|
|
|
If T_ramp >> τ_thermal:
|
|
- Channel cools during ramp
|
|
- Must reheat repeatedly
|
|
- Loses QCW efficiency advantage
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.4
|
|
|
|
**Problem 1:** A streamer has d = 150 μm. Calculate τ_thermal. If burst pulse is 500 μs, does channel cool significantly during pulse?
|
|
|
|
**Problem 2:** Why do thick leaders persist longer than thin streamers? Give two physical reasons.
|
|
|
|
---
|
|
|
|
## Module 3.5: The Capacitive Divider Problem
|
|
|
|
### Voltage Division Along Spark
|
|
|
|
From Part 1, spark circuit:
|
|
```
|
|
[C_mut]
|
|
Topload ----||---- Spark
|
|
|
|
|
[R]
|
|
|
|
|
[C_sh]
|
|
|
|
|
GND
|
|
```
|
|
|
|
**Voltage divider:** V_tip depends on impedance ratio:
|
|
```
|
|
V_tip = V_topload × Z_mut / (Z_mut + Z_sh)
|
|
|
|
where Z_mut = (1/jωC_mut) || R (parallel combination)
|
|
Z_sh = 1/(jωC_sh)
|
|
```
|
|
|
|
### Open-Circuit Limit (No Current)
|
|
|
|
When R → ∞ (no conduction), only capacitances matter:
|
|
```
|
|
V_tip = V_topload × C_mut / (C_mut + C_sh)
|
|
```
|
|
|
|
**Problem:** As spark grows, C_sh increases (∝ length):
|
|
```
|
|
C_sh ≈ 2 pF/foot × L
|
|
|
|
As L increases → C_sh increases → V_tip decreases!
|
|
```
|
|
|
|
**Example:**
|
|
```
|
|
V_topload = 400 kV (constant)
|
|
C_mut = 8 pF (approximately constant)
|
|
|
|
Short spark (1 ft): C_sh = 2 pF
|
|
V_tip = 400 × 8/(8+2) = 320 kV (80%)
|
|
|
|
Medium spark (3 ft): C_sh = 6 pF
|
|
V_tip = 400 × 8/(8+6) = 229 kV (57%)
|
|
|
|
Long spark (6 ft): C_sh = 12 pF
|
|
V_tip = 400 × 8/(8+12) = 160 kV (40%)
|
|
```
|
|
|
|
**Tip voltage drops to 40% even with constant topload voltage!**
|
|
|
|
### With Finite Resistance
|
|
|
|
Real case with R = R_opt_power ≈ 1/(ω(C_mut+C_sh)):
|
|
|
|
```
|
|
Z_mut = R || (1/jωC_mut) ≈ complex value
|
|
V_tip is lower and phase-shifted
|
|
|
|
Effect is similar but worse:
|
|
- Magnitude division (as above)
|
|
- Plus current-dependent voltage drop across R
|
|
- V_tip drops faster than capacitive case alone
|
|
```
|
|
|
|
### Impact on Growth
|
|
|
|
```
|
|
E_tip = κ × V_tip / L
|
|
|
|
As L increases:
|
|
- Numerator (V_tip) decreases (capacitive division)
|
|
- Denominator (L) increases (geometry)
|
|
- E_tip decreases as L²
|
|
|
|
Growth becomes progressively harder!
|
|
```
|
|
|
|
**Why sub-linear scaling:**
|
|
```
|
|
If energy scales as E ∝ L², but division effect makes
|
|
V_tip ∝ 1/L, then achievable length L ∝ √E
|
|
|
|
This explains Freau's empirical observation: L ∝ √E for burst mode
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.5: Voltage Division
|
|
|
|
**Given:**
|
|
- V_topload = 350 kV (maintained constant)
|
|
- C_mut = 10 pF
|
|
- Spark grows from 0 to 4 feet
|
|
|
|
**Find:** V_tip at L = 1, 2, 3, 4 feet (open-circuit approximation)
|
|
|
|
**Solution:**
|
|
|
|
**At L = 1 ft:**
|
|
```
|
|
C_sh = 2 pF/ft × 1 ft = 2 pF
|
|
|
|
V_tip = 350 kV × 10/(10+2)
|
|
= 350 × 10/12
|
|
= 292 kV (83% of V_topload)
|
|
```
|
|
|
|
**At L = 2 ft:**
|
|
```
|
|
C_sh = 4 pF
|
|
|
|
V_tip = 350 × 10/14
|
|
= 250 kV (71%)
|
|
```
|
|
|
|
**At L = 3 ft:**
|
|
```
|
|
C_sh = 6 pF
|
|
|
|
V_tip = 350 × 10/16
|
|
= 219 kV (63%)
|
|
```
|
|
|
|
**At L = 4 ft:**
|
|
```
|
|
C_sh = 8 pF
|
|
|
|
V_tip = 350 × 10/18
|
|
= 194 kV (55%)
|
|
```
|
|
|
|
**Summary table:**
|
|
|
|
| Length | C_sh | V_tip | % of V_top |
|
|
|--------|------|-------|------------|
|
|
| 1 ft | 2 pF | 292 kV| 83% |
|
|
| 2 ft | 4 pF | 250 kV| 71% |
|
|
| 3 ft | 6 pF | 219 kV| 63% |
|
|
| 4 ft | 8 pF | 194 kV| 55% |
|
|
|
|
**Voltage drops almost linearly with length, making further extension difficult.**
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.5
|
|
|
|
**Problem 1:** V_top = 300 kV, C_mut = 12 pF. Calculate V_tip for L = 2 ft and L = 5 ft. What percentage is lost?
|
|
|
|
**Problem 2:** If E_propagation = 0.6 MV/m and κ = 3, what V_tip is needed for 2 m spark? Using C_mut = 8 pF, what V_topload is required?
|
|
|
|
---
|
|
|
|
## Module 3.6: Introduction to FEMM
|
|
|
|
### What is FEMM?
|
|
|
|
**FEMM = Finite Element Method Magnetics**
|
|
- Free, open-source electromagnetic FEA software
|
|
- 2D planar and axisymmetric problems
|
|
- Electrostatic, magnetostatic, AC magnetic, thermal analysis
|
|
|
|
**For Tesla coils:** Use electrostatic solver to extract capacitances
|
|
|
|
**Download:** www.femm.info
|
|
|
|
### Basic Workflow
|
|
|
|
**1. Define geometry:**
|
|
- Draw conductors (spark, topload, ground)
|
|
- Define materials (air, metal)
|
|
- Set boundaries (Dirichlet, Neumann)
|
|
|
|
**2. Assign properties:**
|
|
- Conductor potentials (voltages)
|
|
- Material properties (permittivity)
|
|
- Boundary conditions
|
|
|
|
**3. Mesh:**
|
|
- Automatic triangulation
|
|
- Refinement near conductors
|
|
|
|
**4. Solve:**
|
|
- Numerical solution of Laplace's equation
|
|
- ∇²V = 0 in free space
|
|
|
|
**5. Post-process:**
|
|
- Extract capacitance matrix
|
|
- Calculate electric fields
|
|
- Visualize field lines
|
|
|
|
### Problem Setup for Spark
|
|
|
|
**Geometry:**
|
|
```
|
|
- Toroidal topload (axisymmetric)
|
|
- Cylindrical spark channel (vertical)
|
|
- Ground plane (large boundary)
|
|
- Air region (surrounds everything)
|
|
```
|
|
|
|
**Materials:**
|
|
```
|
|
- Air: ε_r = 1.0
|
|
- Conductors: Set potentials, not material
|
|
```
|
|
|
|
**Boundaries:**
|
|
```
|
|
- Outer boundary: V = 0 (grounded, far from coil)
|
|
- Axisymmetric boundary: special condition (mirror)
|
|
```
|
|
|
|
**Potentials:**
|
|
```
|
|
- Topload: 1 V (arbitrary, will scale)
|
|
- Spark: floating (capacitance extraction)
|
|
- Ground: 0 V
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.6: FEMM Tutorial (Conceptual)
|
|
|
|
**Task:** Extract C_mut and C_sh for 1 m spark from 30 cm toroid
|
|
|
|
**Step 1: Geometry (axisymmetric)**
|
|
```
|
|
r-z coordinates (cylindrical)
|
|
- Toroid: major radius 15 cm, minor radius 5 cm, center at z = 0
|
|
- Spark: cylinder radius 1 mm, extends from z = -5 cm to z = -105 cm
|
|
- Ground plane: z = -120 cm (large disk)
|
|
- Outer boundary: r = 200 cm, z = ±150 cm (large region)
|
|
```
|
|
|
|
**Step 2: Materials**
|
|
```
|
|
- Everything is "Air" (ε_r = 1)
|
|
- Will assign potentials, not conductivities
|
|
```
|
|
|
|
**Step 3: Boundaries**
|
|
```
|
|
- r = 0: Axisymmetric boundary (axis of symmetry)
|
|
- Outer box: V = 0 (Dirichlet)
|
|
```
|
|
|
|
**Step 4: Conductors**
|
|
```
|
|
Create 3 conductor groups:
|
|
- Conductor 1: Topload surface, V = 1V
|
|
- Conductor 2: Spark surface, floating (no fixed potential)
|
|
- Conductor 3: Ground plane, V = 0V
|
|
```
|
|
|
|
**Step 5: Mesh and solve**
|
|
```
|
|
- Auto mesh: ~5000 elements typical
|
|
- Solve electrostatic problem
|
|
- Convergence <0.001%
|
|
```
|
|
|
|
**Step 6: Extract capacitance matrix**
|
|
```
|
|
FEMM outputs 3×3 Maxwell capacitance matrix [C]:
|
|
|
|
Top Spark Ground
|
|
Top [ 30 -8 -22 ] pF
|
|
Spark [ -8 14 -6 ] pF
|
|
Ground[ -22 -6 28 ] pF
|
|
|
|
(Values are example)
|
|
```
|
|
|
|
**Step 7: Calculate C_mut and C_sh**
|
|
```
|
|
C_mut = |C[Top, Spark]| = |-8| = 8 pF
|
|
|
|
C_sh = C[Spark, Spark] + C[Spark, Top]
|
|
= 14 + (-8)
|
|
= 6 pF
|
|
|
|
Validation: 6 pF ≈ 2 pF/ft × 3.3 ft ✓
|
|
```
|
|
|
|
---
|
|
|
|
### VISUAL AID 3.6: FEMM Interface
|
|
|
|
```
|
|
[Describe for screenshot annotation:]
|
|
|
|
FEMM main window with four panels:
|
|
|
|
UPPER LEFT: Geometry editor
|
|
- Drawing tools (point, line, arc, circle)
|
|
- Coordinate display (r, z in cm)
|
|
- Toroid drawn as rotated circle
|
|
- Spark as vertical line segment
|
|
- Ground as horizontal line
|
|
- All in r-z plane (axisymmetric)
|
|
|
|
UPPER RIGHT: Problem definition
|
|
- Properties: Frequency = 0 (electrostatic)
|
|
- Length units: centimeters
|
|
- Problem type: Axisymmetric
|
|
- Precision: 1e-8
|
|
|
|
LOWER LEFT: Mesh view
|
|
- Triangle mesh covering domain
|
|
- Refined near conductors (smaller triangles)
|
|
- Coarse far away (larger triangles)
|
|
- Color = element size
|
|
|
|
LOWER RIGHT: Solution view
|
|
- Filled contours: equipotential lines (V)
|
|
- Field vectors: E field (arrows)
|
|
- Concentrated at topload and spark tip
|
|
- Circuit property window showing capacitances
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.6
|
|
|
|
**Problem 1:** Why do we use V = 1 V instead of actual voltage (400 kV)? (Hint: electrostatics is linear)
|
|
|
|
**Problem 2:** A FEMM simulation with 2 m spark gives C_sh = 14 pF. Does this match the empirical 2 pF/ft rule? (Show calculation)
|
|
|
|
---
|
|
|
|
## Module 3.7: Extracting Capacitances from FEMM
|
|
|
|
### The Maxwell Capacitance Matrix
|
|
|
|
FEMM outputs matrix [C] where:
|
|
```
|
|
[Q] = [C] × [V]
|
|
|
|
Q_i = charge on conductor i
|
|
V_i = potential of conductor i
|
|
|
|
Matrix properties:
|
|
- Symmetric: C_ij = C_ji
|
|
- Diagonal positive: C_ii > 0
|
|
- Off-diagonal negative: C_ij < 0 for i≠j
|
|
- Row sums to zero: Σ_j C_ij = 0
|
|
```
|
|
|
|
**Physical meaning:**
|
|
- C_ii: self-capacitance (conductor i to infinity)
|
|
- C_ij (i≠j): mutual capacitance (coupling between i and j, negative)
|
|
|
|
### Two-Body System (Topload + Spark)
|
|
|
|
Matrix for topload (1), spark (2), ground (implicit):
|
|
```
|
|
[1] [2]
|
|
[1] [ C₁₁ C₁₂ ]
|
|
[2] [ C₂₁ C₂₂ ]
|
|
|
|
Example values:
|
|
[Top] [Spark]
|
|
[Top] [ 30 -8 ] pF
|
|
[Spark][ -8 14 ] pF
|
|
```
|
|
|
|
### Extraction Formulas
|
|
|
|
**C_mut (mutual capacitance):**
|
|
```
|
|
C_mut = |C₁₂| = |C₂₁|
|
|
|
|
Take absolute value of off-diagonal element
|
|
```
|
|
|
|
**C_sh (spark to ground):**
|
|
|
|
Method 1 - From row sum:
|
|
```
|
|
Ground capacitance = -(C₂₁ + C₂₂)
|
|
But we want spark-to-ground only: C_sh
|
|
|
|
C_sh = C₂₂ + C₂₁
|
|
= C₂₂ - |C₁₂| (since C₂₁ = C₁₂ < 0)
|
|
```
|
|
|
|
Method 2 - Direct measurement:
|
|
```
|
|
Run second simulation with topload grounded
|
|
Measure spark capacitance to ground directly
|
|
```
|
|
|
|
**Validation check:**
|
|
```
|
|
C_sh ≈ 2 pF/foot × L_spark
|
|
|
|
If ratio is 1.5-2.5 pF/foot: good
|
|
If significantly different: check geometry/mesh
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.7: Matrix Interpretation
|
|
|
|
**Given FEMM output:**
|
|
```
|
|
Conductor properties:
|
|
Conductor 1 (Topload): 35.2 pF to ground
|
|
Conductor 2 (Spark): 16.8 pF to ground
|
|
|
|
Circuit properties:
|
|
C[1,1] = 35.2 pF
|
|
C[1,2] = -10.5 pF
|
|
C[2,1] = -10.5 pF (symmetry)
|
|
C[2,2] = 16.8 pF
|
|
|
|
Spark length: 1.8 m = 5.9 ft
|
|
```
|
|
|
|
**Extract:**
|
|
(a) C_mut
|
|
(b) C_sh
|
|
(c) Validate against empirical rule
|
|
|
|
**Solution:**
|
|
|
|
**Part (a):** Mutual capacitance
|
|
```
|
|
C_mut = |C[1,2]| = |-10.5| = 10.5 pF
|
|
```
|
|
|
|
**Part (b):** Shunt capacitance
|
|
```
|
|
C_sh = C[2,2] + C[2,1]
|
|
= 16.8 + (-10.5)
|
|
= 6.3 pF
|
|
```
|
|
|
|
**Part (c):** Validation
|
|
```
|
|
Empirical prediction:
|
|
C_sh_predicted = 2 pF/ft × 5.9 ft = 11.8 pF
|
|
|
|
FEMM result:
|
|
C_sh_FEMM = 6.3 pF
|
|
|
|
Ratio: 6.3 / 11.8 = 0.53
|
|
|
|
This is LOWER than expected (by factor ~2)
|
|
```
|
|
|
|
**Possible explanations:**
|
|
```
|
|
1. Empirical rule assumes straight vertical spark
|
|
- If spark is angled or curved, less capacitance
|
|
|
|
2. Empirical rule from community measurements
|
|
- May include some C_mut in "measured" value
|
|
- Pure C_sh might be lower
|
|
|
|
3. Ground plane distance matters
|
|
- FEMM has specific ground geometry
|
|
- Empirical rule assumes "typical" room
|
|
|
|
4. Diameter assumption
|
|
- Thinner diameter → lower C_sh (logarithmic)
|
|
|
|
For modeling: Use FEMM value (more accurate for specific geometry)
|
|
```
|
|
|
|
---
|
|
|
|
### VISUAL AID 3.7: Capacitance Matrix Interpretation
|
|
|
|
```
|
|
[Describe for diagram:]
|
|
|
|
Left: Physical picture
|
|
- Topload (labeled "1")
|
|
- Spark channel (labeled "2")
|
|
- Ground plane (labeled "0" or implicit)
|
|
- Field lines showing:
|
|
* C₁₁: Topload to infinity (self)
|
|
* C₂₂: Spark to infinity (self)
|
|
* C₁₂: Topload to spark (mutual, shown in green)
|
|
|
|
Center: Matrix representation
|
|
```
|
|
[C] = [ 35.2 -10.5 ]
|
|
[-10.5 16.8 ]
|
|
```
|
|
- Diagonal highlighted (positive)
|
|
- Off-diagonal highlighted (negative)
|
|
- Symmetry shown with arrows
|
|
|
|
Right: Circuit extraction
|
|
- C_mut = |C₁₂| = 10.5 pF (between topload and spark)
|
|
- C_sh = C₂₂ - |C₁₂| = 6.3 pF (spark to ground)
|
|
- Circuit diagram showing extracted values
|
|
|
|
Bottom: Key points
|
|
- "Off-diagonal → mutual capacitance"
|
|
- "Diagonal - mutual → shunt capacitance"
|
|
- "Always check symmetry: C₁₂ = C₂₁"
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.7
|
|
|
|
**Problem 1:** FEMM gives C[1,1]=40 pF, C[1,2]=-12 pF, C[2,2]=20 pF for a 2 m spark. Extract C_mut and C_sh. Does C_sh match the empirical rule?
|
|
|
|
**Problem 2:** Why are off-diagonal elements negative in the Maxwell matrix? What would happen if they were positive?
|
|
|
|
---
|
|
|
|
## Module 3.8: Building the Lumped Spark Model
|
|
|
|
### Complete Workflow
|
|
|
|
**Step 1: FEMM electrostatic analysis**
|
|
```
|
|
- Geometry: topload + spark + ground
|
|
- Axisymmetric 2D
|
|
- Solve at frequency = 0 (electrostatic)
|
|
- Extract [C] matrix
|
|
```
|
|
|
|
**Step 2: Calculate circuit elements**
|
|
```
|
|
C_mut = |C₁₂| from matrix
|
|
C_sh = C₂₂ - |C₁₂| from matrix
|
|
R = R_opt_power = 1/(ω(C_mut + C_sh))
|
|
Clip to physical bounds: R = clip(R, R_min, R_max)
|
|
```
|
|
|
|
**Step 3: Build SPICE netlist**
|
|
```
|
|
* Lumped spark model
|
|
.param freq=200k
|
|
.param omega={2*pi*freq}
|
|
|
|
V_topload topload 0 AC 1 ; 1V test source
|
|
|
|
C_mut topload spark_node {C_mut}
|
|
R_spark spark_node spark_r {R}
|
|
C_sh spark_r 0 {C_sh}
|
|
|
|
.ac lin 1 {freq} {freq}
|
|
.print ac v(topload) i(V_topload)
|
|
.end
|
|
```
|
|
|
|
**Step 4: Run AC analysis**
|
|
```
|
|
- Calculate Y = I/V at topload port
|
|
- Extract Re{Y}, Im{Y}
|
|
- Convert to Z if needed
|
|
- Calculate power: P = 0.5 × |V|² × Re{Y}
|
|
```
|
|
|
|
**Step 5: Validate**
|
|
```
|
|
- Check φ_Z in expected range (-55° to -75°)
|
|
- Check R in physical range (kΩ to hundreds of kΩ)
|
|
- Check C_sh ≈ 2 pF/ft ± factor of 2
|
|
- Compare to measurements if available
|
|
```
|
|
|
|
### Integration with Full Coil Model
|
|
|
|
```
|
|
[Primary circuit] → [Coupled transformer] → [Secondary] → [Topload] → [Spark model]
|
|
|
|
Spark model appears as:
|
|
- Load impedance at topload port
|
|
- Affects loaded Q, resonant frequency
|
|
- Extracts power from secondary
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 3.8: Complete Lumped Model
|
|
|
|
**Given:**
|
|
- 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
|
|
|
|
**Build and analyze model:**
|
|
|
|
**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_power = 1/(ω × C_total)
|
|
= 1/(1.194×10⁶ × 16.7×10⁻¹²)
|
|
= 1/(19.94×10⁻⁶)
|
|
= 50.2 kΩ
|
|
```
|
|
|
|
**Step 2:** Check bounds
|
|
```
|
|
R_min = 5 kΩ
|
|
R_opt = 50.2 kΩ
|
|
R_max = 500 kΩ
|
|
|
|
5 < 50.2 < 500 ✓
|
|
|
|
Use R = 50.2 kΩ
|
|
```
|
|
|
|
**Step 3:** Build SPICE model
|
|
```
|
|
* Spark lumped model - 190 kHz
|
|
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
|
|
.print ac v(topload) i(V_test) vp(topload) ip(V_test)
|
|
.end
|
|
```
|
|
|
|
**Step 4:** Simulate and extract (example results)
|
|
```
|
|
Simulation output:
|
|
V(topload) = 1.000 V ∠0°
|
|
I(V_test) = 5.23×10⁻⁶ A ∠74.5°
|
|
|
|
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°
|
|
|
|
R_eq = 191 × cos(-74.5°) = 51 kΩ
|
|
X_eq = 191 × sin(-74.5°) = -184 kΩ
|
|
```
|
|
|
|
**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 validation:
|
|
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
|
|
= 0.5 × 1.024×10¹¹ × 1.39×10⁻⁶
|
|
= 71.2 kW
|
|
```
|
|
|
|
Model is complete and ready for coil integration!
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 3.8
|
|
|
|
**Problem 1:** Build lumped model for: f=200 kHz, C_mut=11 pF, C_sh=9 pF. Calculate all component values and expected φ_Z.
|
|
|
|
**Problem 2:** If SPICE simulation gives φ_Z=-85° (more capacitive than expected), what might be wrong with the model?
|
|
|
|
---
|
|
|
|
## Part 3 Summary & Integration
|
|
|
|
### Key Concepts Checklist
|
|
|
|
- [ ] **E_inception:** ~2-3 MV/m to start breakdown
|
|
- [ ] **E_propagation:** ~0.4-1.0 MV/m to sustain growth
|
|
- [ ] **Tip enhancement:** E_tip = κ × E_avg, κ ≈ 2-5
|
|
- [ ] **Growth criterion:** E_tip > E_propagation required
|
|
- [ ] **Energy per meter ε:** 5-15 (QCW), 30-100 (burst) J/m
|
|
- [ ] **Growth rate:** dL/dt = P/ε when field adequate
|
|
- [ ] **Voltage vs power limited:** Both constraints exist
|
|
- [ ] **Thermal time:** τ = d²/(4α), but persistence longer
|
|
- [ ] **QCW advantage:** Maintains hot channel (low ε)
|
|
- [ ] **Capacitive divider:** V_tip drops as C_sh grows
|
|
- [ ] **Sub-linear scaling:** L ∝ √E for voltage-limited
|
|
- [ ] **FEMM workflow:** Geometry → solve → extract [C]
|
|
- [ ] **Maxwell matrix:** Diagonal positive, off-diagonal negative
|
|
- [ ] **C_mut extraction:** |C₁₂| from off-diagonal
|
|
- [ ] **C_sh extraction:** C₂₂ - |C₁₂|
|
|
- [ ] **Validation:** C_sh ≈ 2 pF/ft ± factor 2
|
|
- [ ] **Lumped model:** (R||C_mut) + C_sh
|
|
- [ ] **R = R_opt_power:** For hungry streamer assumption
|
|
|
|
---
|
|
|
|
## Final Integration Exercise
|
|
|
|
**Complete design challenge:**
|
|
|
|
**Given:**
|
|
- DRSSTC at 185 kHz
|
|
- Toroid: 40 cm major diameter, 10 cm minor
|
|
- Target: 2 m spark
|
|
- Thévenin: Z_th = 120 - j2200 Ω, V_th = 380 kV
|
|
|
|
**Tasks:**
|
|
|
|
1. **FEMM analysis (describe setup):**
|
|
- Draw geometry for 2 m spark
|
|
- What boundaries to use?
|
|
- Expected C_sh range?
|
|
|
|
2. **Assume FEMM gives:** C_mut = 11 pF, C_sh = 13 pF
|
|
- Validate C_sh (empirical rule)
|
|
- Calculate R_opt_power at 185 kHz
|
|
- Is R within 5-500 kΩ bounds?
|
|
|
|
3. **Build lumped model:**
|
|
- Calculate Y_spark
|
|
- Convert to Z_spark
|
|
- What is φ_Z?
|
|
|
|
4. **Predict performance:**
|
|
- Calculate Z_total = Z_th + Z_spark
|
|
- Find current I
|
|
- Calculate power to spark
|
|
- Compare to theoretical max (conjugate match)
|
|
|
|
5. **Growth analysis:**
|
|
- Assume QCW, ε = 10 J/m
|
|
- How long to reach 2 m?
|
|
- Check voltage requirement: E_prop = 0.6 MV/m, κ = 3.5
|
|
- Is growth voltage-limited or power-limited?
|
|
|
|
**This exercise integrates all of Part 3!**
|
|
|
|
---
|
|
|
|
**END OF PART 3**
|
|
|
|
---
|
|
|
|
# Tesla Coil Spark Modeling - Complete Lesson Plan
|
|
## Part 4: Advanced Topics - Distributed Models and Real-World Application
|
|
|
|
---
|
|
|
|
## Module 4.1: Why Distributed Models?
|
|
|
|
### Limitations of Lumped Models
|
|
|
|
**Lumped model treats entire spark as single R, C_mut, C_sh:**
|
|
|
|
**Works well for:**
|
|
- Short sparks (<1 m)
|
|
- Impedance matching studies
|
|
- Quick optimization
|
|
- First-order power estimates
|
|
|
|
**Fails to capture:**
|
|
```
|
|
1. Current distribution along spark
|
|
- Base carries full current
|
|
- Tip may have much less (capacitive shunting)
|
|
|
|
2. Voltage distribution
|
|
- Not linear drop from top to tip
|
|
- Capacitive divider effects at each point
|
|
|
|
3. Tip vs base differences
|
|
- Base: hot, well-coupled, low R
|
|
- Tip: cool, weakly-coupled, high R
|
|
|
|
4. Streamer/leader transitions
|
|
- Base forms leader (low R)
|
|
- Tip remains streamer (high R)
|
|
- Lumped model averages this out
|
|
|
|
5. Very long sparks (>3 m)
|
|
- Distributed effects dominate
|
|
- Single lumped R is poor approximation
|
|
```
|
|
|
|
### When to Use Distributed Model
|
|
|
|
**Use distributed when:**
|
|
- Spark length > 1-2 meters
|
|
- Need current distribution (for measurements)
|
|
- Studying leader/streamer physics
|
|
- Validating against detailed measurements
|
|
- Research/publication quality results
|
|
|
|
**Stick with lumped when:**
|
|
- Quick design iterations
|
|
- Coil-level optimization (matching)
|
|
- Spark length < 1 meter
|
|
- Engineering estimates sufficient
|
|
|
|
**Computational cost:**
|
|
- Lumped: <1 second
|
|
- Distributed (n=10): ~10-30 seconds
|
|
- Distributed (n=20): ~1-5 minutes
|
|
|
|
---
|
|
|
|
### VISUAL AID 4.1: Lumped vs Distributed Comparison
|
|
|
|
```
|
|
[Describe for diagram:]
|
|
|
|
Two-panel comparison:
|
|
|
|
LEFT: Lumped model
|
|
- Single box representing entire spark
|
|
- Three components: C_mut, R, C_sh
|
|
- Simple circuit
|
|
- One current value
|
|
- One voltage drop
|
|
- Label: "Good for <1m, fast computation"
|
|
|
|
RIGHT: Distributed model (n=5 shown)
|
|
- Spark divided into 5 segments
|
|
- Each segment has: C_mutual[i], R[i], C_shunt[i]
|
|
- Coupling between segments shown
|
|
- Current arrows varying in size (large at base, small at tip)
|
|
- Voltage nodes at each junction
|
|
- Gradient showing R: low (blue) at base, high (red) at tip
|
|
- Label: "Captures physics, slower computation"
|
|
|
|
BOTTOM: Feature comparison table
|
|
| Feature | Lumped | Distributed |
|
|
|----------------------|--------|-------------|
|
|
| Setup time | Fast | Slow |
|
|
| Computation | <1s | 10s-min |
|
|
| Current distribution| No | Yes |
|
|
| Tip/base difference | No | Yes |
|
|
| Accuracy <1m | Good | Excellent |
|
|
| Accuracy >3m | Poor | Good |
|
|
```
|
|
|
|
---
|
|
|
|
### DISCUSSION QUESTIONS 4.1
|
|
|
|
**Question 1:** A 0.5 m spark shows good agreement between lumped model and measurements. A 3 m spark shows poor agreement. Why?
|
|
|
|
**Question 2:** If you only care about total power delivered to spark (not distribution), when would distributed model still be necessary?
|
|
|
|
**Question 3:** In what situation might even a distributed model fail? (Hint: think about branching)
|
|
|
|
---
|
|
|
|
## Module 4.2: nth-Order Model Structure
|
|
|
|
### Segmentation Strategy
|
|
|
|
**Divide spark into n equal-length segments:**
|
|
```
|
|
n = number of segments (typically 5-20)
|
|
L_segment = L_total / n
|
|
|
|
Segment numbering:
|
|
i = 1: Base (connected to topload)
|
|
i = 2, 3, ..., n-1: Middle sections
|
|
i = n: Tip (furthest from topload)
|
|
```
|
|
|
|
**Why equal lengths?**
|
|
- Simplifies FEMM geometry
|
|
- Uniform discretization
|
|
- Easy to implement
|
|
- Non-uniform possible but more complex
|
|
|
|
### Circuit Topology
|
|
|
|
**Each segment i has:**
|
|
```
|
|
1. Resistance R[i]
|
|
- Plasma resistance of that segment
|
|
- Variable, to be optimized
|
|
|
|
2. Mutual capacitances C[i,j]
|
|
- Coupling to all other segments j≠i
|
|
- And to topload (j=0)
|
|
- Extracted from FEMM
|
|
|
|
3. Shunt capacitance to ground
|
|
- Included in capacitance matrix
|
|
- Not a separate component
|
|
```
|
|
|
|
**Full network:**
|
|
```
|
|
Topload (node 0)
|
|
|
|
|
+-- C[0,1] -- Node 1 (base segment)
|
|
| |
|
|
| R[1]
|
|
| |
|
|
+-- C[0,2] ----+-- Node 2
|
|
| |
|
|
| R[2]
|
|
| |
|
|
...
|
|
|
|
|
+-- C[0,n] ----+-- Node n (tip segment)
|
|
|
|
|
R[n]
|
|
|
|
|
|
|
Plus C[i,j] between all segment pairs
|
|
Plus C[i,ground] for each segment to ground
|
|
```
|
|
|
|
**Complexity:** For n segments + topload:
|
|
- (n+1)×(n+1) capacitance matrix
|
|
- n resistance values
|
|
- Total unknowns: n (resistances)
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 4.2: Draw 3-Segment Model
|
|
|
|
**Given:**
|
|
- Total spark: 1.5 m
|
|
- Divide into n = 3 equal segments
|
|
- Each segment: 0.5 m
|
|
|
|
**Task:** Draw circuit topology (conceptual)
|
|
|
|
**Solution:**
|
|
|
|
```
|
|
Topload (V_top, node 0)
|
|
|
|
|
+---[C[0,1]]---+---[C[0,2]]---+---[C[0,3]]---+
|
|
| | | |
|
|
| | | |
|
|
Node 1 -------[R[1]]-------------|--------------|
|
|
(base) | | |
|
|
[C[1,2]] [C[1,3]] |
|
|
| | |
|
|
Node 2 -----------[R[2]]--------[C[2,3]]
|
|
(middle) | |
|
|
[C_sh,2] |
|
|
| |
|
|
Node 3 --------[R[3]]
|
|
(tip) |
|
|
[C_sh,3]
|
|
|
|
|
GND
|
|
|
|
Where:
|
|
- C[i,j] = mutual capacitance between segments
|
|
- C_sh[i] = shunt capacitance segment i to ground
|
|
- R[i] = resistance of segment i
|
|
```
|
|
|
|
**Note:** This is conceptual. Actual implementation uses full (n+1)×(n+1) matrix.
|
|
|
|
**Typical values (estimated):**
|
|
```
|
|
Segment 1 (base): R[1] = 10 kΩ (hot, well-coupled)
|
|
Segment 2 (mid): R[2] = 30 kΩ (moderate)
|
|
Segment 3 (tip): R[3] = 100 kΩ (cool, weak coupling)
|
|
|
|
C[0,1] > C[0,2] > C[0,3] (coupling decreases with distance)
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 4.2
|
|
|
|
**Problem 1:** A 2.4 m spark is divided into n=6 segments. What is the length of each segment? Number them from base to tip.
|
|
|
|
**Problem 2:** For n=10 segments, how many capacitance matrix elements are there? (Count all C[i,j] including diagonal)
|
|
|
|
**Problem 3:** Why might R[1] (base) be much smaller than R[10] (tip)? Give two physical reasons.
|
|
|
|
---
|
|
|
|
## Module 4.3: FEMM for Distributed Models
|
|
|
|
### Multi-Body Electrostatic Setup
|
|
|
|
**Geometry definition:**
|
|
```
|
|
For n segments + topload → (n+1) conductors
|
|
|
|
Example n=5:
|
|
- Body 0: Toroid topload
|
|
- Body 1: Cylinder, length L/5, base at topload
|
|
- Body 2: Cylinder, length L/5, above body 1
|
|
- Body 3: Cylinder, length L/5, above body 2
|
|
- Body 4: Cylinder, length L/5, above body 3
|
|
- Body 5: Cylinder, length L/5, top segment (tip)
|
|
- Ground plane at bottom
|
|
```
|
|
|
|
**Axisymmetric setup:**
|
|
```
|
|
r-z coordinates
|
|
All bodies as cylindrical sections
|
|
Diameter: 1-3 mm typical (uniform for simplicity)
|
|
Spacing: slight gap (~0.1 mm) between segments for FEMM
|
|
```
|
|
|
|
**Conductor properties:**
|
|
```
|
|
Group each body as separate conductor:
|
|
- Conductor 0: Topload, V = 1V
|
|
- Conductors 1-n: Spark segments, floating potential
|
|
- Ground: V = 0V (boundary condition)
|
|
```
|
|
|
|
### Solving and Extraction
|
|
|
|
**Mesh requirements:**
|
|
```
|
|
- Finer mesh near conductors
|
|
- Refinement at segment junctions
|
|
- Typical: 10,000-50,000 elements for n=10
|
|
- Convergence: <0.01% error
|
|
```
|
|
|
|
**Capacitance matrix output:**
|
|
```
|
|
FEMM circuit properties → Capacitance matrix
|
|
|
|
(n+1)×(n+1) symmetric matrix [C]:
|
|
|
|
[0] [1] [2] ... [n]
|
|
[0] [ C₀₀ C₀₁ C₀₂ ... C₀ₙ ]
|
|
[1] [ C₁₀ C₁₁ C₁₂ ... C₁ₙ ]
|
|
[2] [ C₂₀ C₂₁ C₂₂ ... C₂ₙ ]
|
|
...
|
|
[n] [ Cₙ₀ Cₙ₁ Cₙ₂ ... Cₙₙ ]
|
|
|
|
Properties:
|
|
- Symmetric: Cᵢⱼ = Cⱼᵢ
|
|
- Diagonal positive: Cᵢᵢ > 0
|
|
- Off-diagonal negative: Cᵢⱼ < 0 for i≠j
|
|
- Row sum = 0: Σⱼ Cᵢⱼ = 0
|
|
```
|
|
|
|
### Matrix Validation
|
|
|
|
**Check 1: Symmetry**
|
|
```
|
|
|C[i,j] - C[j,i]| / |C[i,j]| < 0.01
|
|
If not symmetric: numerical error, refine mesh
|
|
```
|
|
|
|
**Check 2: Positive definite**
|
|
```
|
|
All eigenvalues should be ≥ 0
|
|
One eigenvalue = 0 (ground reference freedom)
|
|
Rest positive
|
|
```
|
|
|
|
**Check 3: Physical values**
|
|
```
|
|
Nearby segments: larger |C[i,j]|
|
|
Distant segments: smaller |C[i,j]|
|
|
Base segments: larger C[i,0] (topload coupling)
|
|
Tip segments: smaller C[n,0]
|
|
```
|
|
|
|
**Check 4: Total shunt capacitance**
|
|
```
|
|
C_sh_total = Σᵢ (Cᵢᵢ - |Cᵢ₀|) for all spark segments
|
|
|
|
Should be approximately:
|
|
C_sh_total ≈ 2 pF/foot × L_total
|
|
|
|
Within factor of 2 is reasonable
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 4.3: FEMM Setup for n=5
|
|
|
|
**Given:**
|
|
- Spark length: 2.0 m = 6.56 feet
|
|
- Diameter: 2 mm
|
|
- n = 5 segments → each 0.4 m long
|
|
- Topload: 30 cm toroid
|
|
|
|
**FEMM procedure:**
|
|
|
|
**Step 1: Geometry (r-z coordinates)**
|
|
```
|
|
Topload:
|
|
- Major radius: 15 cm, minor radius: 5 cm
|
|
- Center at z = 0
|
|
- Lowest point: z = -5 cm
|
|
|
|
Segment 1 (base):
|
|
- r = 1 mm (0.1 cm)
|
|
- z from -5 cm to -45 cm
|
|
- Length: 40 cm
|
|
|
|
Segment 2:
|
|
- z from -45 cm to -85 cm
|
|
|
|
Segment 3:
|
|
- z from -85 cm to -125 cm
|
|
|
|
Segment 4:
|
|
- z from -125 cm to -165 cm
|
|
|
|
Segment 5 (tip):
|
|
- z from -165 cm to -205 cm
|
|
|
|
Ground plane:
|
|
- z = -220 cm (15 cm below tip)
|
|
- r = 0 to 300 cm (large)
|
|
|
|
Outer boundary:
|
|
- r = 300 cm, z = ±250 cm
|
|
```
|
|
|
|
**Step 2: Materials and conductors**
|
|
```
|
|
All regions: Air (ε_r = 1)
|
|
|
|
Define 6 conductor groups:
|
|
Group 0: Topload surface, V = 1V
|
|
Groups 1-5: Segment surfaces, floating
|
|
Ground: Boundary at z = -220 cm, V = 0V
|
|
```
|
|
|
|
**Step 3: Meshing**
|
|
```
|
|
Auto mesh with refinement:
|
|
- Triangle size near conductors: 0.5 mm
|
|
- Triangle size at boundaries: 50 mm
|
|
- ~25,000 elements total
|
|
```
|
|
|
|
**Step 4: Solve**
|
|
```
|
|
Problem type: Electrostatic, axisymmetric
|
|
Frequency: 0 Hz
|
|
Precision: 1e-8
|
|
```
|
|
|
|
**Step 5: Extract matrix (example results)**
|
|
```
|
|
Matrix [C] in pF:
|
|
|
|
[0] [1] [2] [3] [4] [5]
|
|
[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
|
|
[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
|
|
[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
|
|
[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
|
|
[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
|
|
[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ]
|
|
|
|
(Values are illustrative)
|
|
```
|
|
|
|
**Step 6: Validate**
|
|
```
|
|
Symmetry check: C[1,2] = C[2,1] = -2.8 ✓
|
|
|
|
Total shunt capacitance (approximate):
|
|
C_sh ≈ Σᵢ₌₁⁵ (Cᵢᵢ - |Cᵢ₀|)
|
|
= (14.8-9.2) + (10.4-3.1) + (8.6-1.2) + (7.4-0.6) + (5.8-0.3)
|
|
= 5.6 + 7.3 + 7.4 + 6.8 + 5.5
|
|
= 32.6 pF
|
|
|
|
Expected: 2 pF/ft × 6.56 ft = 13.1 pF
|
|
|
|
Ratio: 32.6/13.1 = 2.5
|
|
|
|
Higher than expected, but within factor of 2-3 (acceptable)
|
|
Difference due to matrix interpretation method
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 4.3
|
|
|
|
**Problem 1:** For n=10 segments, 3 m total, what is each segment length? What is the z-coordinate range for segment 5 if topload bottom is at z=0?
|
|
|
|
**Problem 2:** A capacitance matrix shows C[3,7] = -0.4 pF and C[3,4] = -2.1 pF. Which segments are closer to segment 3? Does this make physical sense?
|
|
|
|
---
|
|
|
|
## Module 4.4: Implementing Capacitance Matrices in SPICE
|
|
|
|
### The Challenge
|
|
|
|
**Maxwell matrix has negative off-diagonals:**
|
|
```
|
|
Literal SPICE capacitor implementation:
|
|
C_12 node1 node2 10p ← OK, positive value
|
|
C_12 node1 node2 -10p ← ERROR! Negative capacitance unphysical
|
|
```
|
|
|
|
**Problem:** Cannot directly use C[i,j] < 0 as SPICE capacitors
|
|
|
|
### Solution 1: Partial Capacitance Transformation
|
|
|
|
**Convert Maxwell → Partial (all-positive):**
|
|
|
|
**Partial capacitance:** Capacitance with all other nodes grounded
|
|
|
|
```
|
|
For node i:
|
|
C_partial[i,j] = -C_Maxwell[i,j] for i≠j (flip sign!)
|
|
C_partial[i,i] = Σⱼ |C_Maxwell[i,j]| (sum of magnitudes)
|
|
|
|
All C_partial > 0 → can implement as SPICE capacitors
|
|
```
|
|
|
|
**SPICE implementation:**
|
|
```
|
|
* Partial capacitance method
|
|
* Between every node pair i,j (i<j):
|
|
C_i_j nodei nodej {-C_Maxwell[i,j]}
|
|
|
|
* Each node to ground:
|
|
C_i_gnd nodei 0 {Σⱼ |C_Maxwell[i,j]| - (sum of partial caps)}
|
|
```
|
|
|
|
**Validation:** Check total capacitance matrix matches original
|
|
|
|
### Solution 2: Controlled Sources (MNA)
|
|
|
|
**Use voltage-controlled current sources:**
|
|
|
|
```
|
|
For each node i:
|
|
I[i] = Σⱼ C[i,j] × d(V[j])/dt
|
|
|
|
In Laplace (AC): I[i] = jω × Σⱼ C[i,j] × V[j]
|
|
```
|
|
|
|
**SPICE implementation:**
|
|
```
|
|
* For node i, contribution from node j:
|
|
G_i_j nodei 0 nodej 0 {j*omega*C[i,j]}
|
|
(frequency-dependent controlled source)
|
|
|
|
Or use behavioral source:
|
|
B_i nodei 0 V = {j*omega*sum(C[i,j]*V(nodej))}
|
|
```
|
|
|
|
**Advantage:** Directly implements matrix, handles negatives
|
|
|
|
**Disadvantage:** More complex, some SPICE versions limited
|
|
|
|
### Solution 3: Nearest-Neighbor Approximation
|
|
|
|
**Simplification:** Ignore weak couplings
|
|
|
|
```
|
|
Keep only:
|
|
1. Topload coupling: C[0,i] for all i
|
|
2. Adjacent segments: C[i,i+1]
|
|
3. Self terms: diagonal
|
|
|
|
Discard: C[i,j] for |i-j| > 1 (distant segments)
|
|
```
|
|
|
|
**When acceptable:**
|
|
- Large n (>10): distant couplings small
|
|
- Quick estimates
|
|
- Weak segment-to-segment coupling
|
|
|
|
**Validation:** Compare full vs approximate impedance
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 4.4: Partial Capacitance Conversion (3×3)
|
|
|
|
**Given Maxwell matrix (topload + 2 segments):**
|
|
```
|
|
[0] [1] [2]
|
|
[0] [ 30.0 -8.0 -2.0 ] pF
|
|
[1] [ -8.0 14.0 -3.0 ] pF
|
|
[2] [ -2.0 -3.0 9.0 ] pF
|
|
```
|
|
|
|
**Convert to partial (all-positive) for SPICE:**
|
|
|
|
**Step 1:** Between-node capacitances (flip signs)
|
|
```
|
|
C_partial[0,1] = -C_Maxwell[0,1] = -(-8.0) = 8.0 pF
|
|
C_partial[0,2] = -C_Maxwell[0,2] = -(-2.0) = 2.0 pF
|
|
C_partial[1,2] = -C_Maxwell[1,2] = -(-3.0) = 3.0 pF
|
|
```
|
|
|
|
**Step 2:** Ground capacitances
|
|
|
|
For each node, start with diagonal, subtract partial caps:
|
|
|
|
**Node 0:**
|
|
```
|
|
C[0,0] = 30.0 pF
|
|
Sum of partials leaving node 0: 8.0 + 2.0 = 10.0 pF
|
|
C_partial[0,gnd] = 30.0 - 10.0 = 20.0 pF
|
|
```
|
|
|
|
**Node 1:**
|
|
```
|
|
C[1,1] = 14.0 pF
|
|
Partials: 8.0 (to 0) + 3.0 (to 2) = 11.0 pF
|
|
C_partial[1,gnd] = 14.0 - 11.0 = 3.0 pF
|
|
```
|
|
|
|
**Node 2:**
|
|
```
|
|
C[2,2] = 9.0 pF
|
|
Partials: 2.0 (to 0) + 3.0 (to 1) = 5.0 pF
|
|
C_partial[2,gnd] = 9.0 - 5.0 = 4.0 pF
|
|
```
|
|
|
|
**Step 3:** SPICE netlist
|
|
```
|
|
* Partial capacitance implementation
|
|
* Between nodes
|
|
C_0_1 node0 node1 8.0p
|
|
C_0_2 node0 node2 2.0p
|
|
C_1_2 node1 node2 3.0p
|
|
|
|
* To ground
|
|
C_0_gnd node0 0 20.0p
|
|
C_1_gnd node1 0 3.0p
|
|
C_2_gnd node2 0 4.0p
|
|
|
|
* Resistances (to be determined)
|
|
R1 node1 node1_r {R1_value}
|
|
R2 node2 node2_r {R2_value}
|
|
```
|
|
|
|
**Validation:** Verify total capacitance node0→gnd matches:
|
|
```
|
|
With node1, node2 grounded:
|
|
C_total = C_0_gnd + C_0_1 || C_1_gnd + C_0_2 || C_2_gnd
|
|
|
|
Should equal approximately 30 pF (check numerically)
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 4.4
|
|
|
|
**Problem 1:** Given C_Maxwell = [25, -6; -6, 10] pF (2×2), convert to partial capacitances. Draw the SPICE circuit.
|
|
|
|
**Problem 2:** Why can't we just use "negative capacitors" in SPICE? What would it physically mean?
|
|
|
|
**Problem 3:** In nearest-neighbor approximation for n=10, how many capacitances are kept vs full matrix? Calculate percentage reduction.
|
|
|
|
---
|
|
|
|
## Module 4.5: Resistance Optimization - Iterative Method
|
|
|
|
### Algorithm Overview
|
|
|
|
**Goal:** Find R[i] for each segment that maximizes total power
|
|
|
|
**Challenge:** R[i] values are coupled (changing one affects power in others)
|
|
|
|
**Solution:** Iterative optimization with damping
|
|
|
|
### Initialization: Tapered Profile
|
|
|
|
**Physical expectation:**
|
|
- Base: hot, well-coupled → low R
|
|
- Tip: cool, weakly-coupled → high R
|
|
|
|
**Initialize with gradient:**
|
|
```
|
|
For i = 1 to n:
|
|
position = (i-1)/(n-1) # 0 at base, 1 at tip
|
|
R[i] = R_base + (R_tip - R_base) × position^2
|
|
|
|
Typical starting values:
|
|
R_base = 10 kΩ
|
|
R_tip = 1 MΩ
|
|
|
|
Quadratic taper gives smooth transition
|
|
```
|
|
|
|
### Iterative Optimization Loop
|
|
|
|
```
|
|
iteration = 0
|
|
converged = False
|
|
|
|
While not converged and iteration < max_iterations:
|
|
|
|
For i = 1 to n:
|
|
# Sweep R[i] while keeping other R[j] fixed
|
|
R_test = logspace(R_min[i], R_max[i], 20 points)
|
|
|
|
For each R_test_value:
|
|
Set R[i] = R_test_value
|
|
Run AC analysis
|
|
Calculate P[i] = power in segment i
|
|
|
|
Find R_optimal[i] = R_test that maximizes P[i]
|
|
|
|
# Apply damping for stability
|
|
R_new[i] = α * R_optimal[i] + (1-α) * R_old[i]
|
|
|
|
# Clip to physical bounds
|
|
R[i] = clip(R_new[i], R_min[i], R_max[i])
|
|
|
|
# Check convergence
|
|
max_change = max(|R_new[i] - R_old[i]| / R_old[i])
|
|
If max_change < 0.01: # 1% threshold
|
|
converged = True
|
|
|
|
iteration = iteration + 1
|
|
```
|
|
|
|
**Damping factor α:**
|
|
```
|
|
α = 0.3 to 0.5 typical
|
|
- Lower α: more stable, slower convergence
|
|
- Higher α: faster, may oscillate
|
|
- Start with α=0.3 for safety
|
|
```
|
|
|
|
### Position-Dependent Bounds
|
|
|
|
**Physical limits vary with position:**
|
|
```
|
|
position = (i-1)/(n-1)
|
|
|
|
R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
|
|
= 1 kΩ at base → 10 kΩ at tip
|
|
|
|
R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position^2
|
|
= 100 kΩ at base → 100 MΩ at tip
|
|
```
|
|
|
|
**Rationale:**
|
|
- Base can achieve very low R (hot leader)
|
|
- Tip unlikely to reach low R (cool, weak coupling)
|
|
- Prevents unphysical solutions
|
|
|
|
### Convergence Behavior
|
|
|
|
**Well-coupled base segments:**
|
|
- Sharp power peak at optimal R
|
|
- Fast convergence (2-3 iterations)
|
|
- Stable solution
|
|
|
|
**Weakly-coupled tip segments:**
|
|
- Flat power curve (many R values similar power)
|
|
- Slow/no convergence to unique value
|
|
- May stay at high R (physical - streamer regime)
|
|
|
|
**Expected result:**
|
|
```
|
|
R[1] ≈ 5-20 kΩ (base leader)
|
|
R[2] ≈ 10-40 kΩ
|
|
...
|
|
R[n-1] ≈ 50-200 kΩ
|
|
R[n] ≈ 100 kΩ - 10 MΩ (tip streamer)
|
|
|
|
Total: Σ R[i] should be in expected range (5-300 kΩ at 200 kHz)
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 4.5: Iterative Optimization (n=3, simplified)
|
|
|
|
**Given:**
|
|
- 3 segments, f = 200 kHz
|
|
- Capacitance matrix (from FEMM, simplified)
|
|
- Initial: R[1]=50k, R[2]=100k, R[3]=500k
|
|
|
|
**Iteration 1:**
|
|
|
|
**Optimize R[1] (keeping R[2], R[3] fixed):**
|
|
```
|
|
Sweep R[1] = [10k, 20k, 30k, 40k, 50k, 60k, 80k, 100k]
|
|
|
|
Results (example):
|
|
R[1]=10k → P[1]=5.2 kW
|
|
R[1]=20k → P[1]=8.1 kW
|
|
R[1]=30k → P[1]=9.4 kW ← maximum
|
|
R[1]=40k → P[1]=8.9 kW
|
|
R[1]=50k → P[1]=7.8 kW (current value)
|
|
...
|
|
|
|
R_optimal[1] = 30 kΩ
|
|
```
|
|
|
|
**Apply damping (α=0.4):**
|
|
```
|
|
R_new[1] = 0.4 × 30k + 0.6 × 50k
|
|
= 12k + 30k
|
|
= 42 kΩ
|
|
```
|
|
|
|
**Optimize R[2]:**
|
|
```
|
|
With R[1]=42k (updated), R[3]=500k (fixed)
|
|
|
|
Sweep R[2], find R_optimal[2] = 60 kΩ
|
|
Current: R[2] = 100 kΩ
|
|
|
|
R_new[2] = 0.4 × 60k + 0.6 × 100k
|
|
= 24k + 60k
|
|
= 84 kΩ
|
|
```
|
|
|
|
**Optimize R[3]:**
|
|
```
|
|
With R[1]=42k, R[2]=84k
|
|
|
|
Sweep R[3], power curve is FLAT:
|
|
R[3]=200k → P[3]=0.8 kW
|
|
R[3]=500k → P[3]=0.85 kW
|
|
R[3]=1M → P[3]=0.83 kW
|
|
|
|
Weakly coupled! Peak not well-defined.
|
|
Keep at R[3] = 500 kΩ (within bounds, acceptable)
|
|
```
|
|
|
|
**After iteration 1:**
|
|
```
|
|
R[1]: 50k → 42k (change = -16%)
|
|
R[2]: 100k → 84k (change = -16%)
|
|
R[3]: 500k → 500k (change = 0%)
|
|
|
|
Max change = 16% > 1% → not converged, continue
|
|
```
|
|
|
|
**Iteration 2:**
|
|
|
|
Repeat process with new R values...
|
|
(typically 3-5 iterations to converge for base/middle segments)
|
|
|
|
**Final converged result (example):**
|
|
```
|
|
R[1] = 35 kΩ (leader, base)
|
|
R[2] = 75 kΩ (transition)
|
|
R[3] = 500 kΩ (streamer, tip - weakly determined)
|
|
|
|
Total: 610 kΩ at 200 kHz
|
|
Check: Within expected range ✓
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 4.5
|
|
|
|
**Problem 1:** Initial R=[100k, 200k], optimal found R=[60k, 150k]. With α=0.3, what are the damped updates?
|
|
|
|
**Problem 2:** Why use damping factor α<1 instead of just setting R=R_optimal directly? What could go wrong?
|
|
|
|
**Problem 3:** After 10 iterations, base segment converged (0.5% change) but tip segment still changing 5% per iteration. What should you do?
|
|
|
|
---
|
|
|
|
## Module 4.6: Resistance Optimization - Simplified Method
|
|
|
|
### Circuit-Determined Resistance
|
|
|
|
**Key insight:** If plasma always seeks R_opt_power, and C depends weakly on diameter:
|
|
|
|
```
|
|
For each segment i:
|
|
C_total[i] = sum of all capacitances involving segment i
|
|
R[i] = 1 / (ω × C_total[i])
|
|
R[i] = clip(R[i], R_min[i], R_max[i])
|
|
```
|
|
|
|
**Extracting C_total from matrix:**
|
|
```
|
|
C_total[i] = |C[i,0]| + Σⱼ₌₁ⁿ |C[i,j]| (sum of absolute values)
|
|
|
|
This is total capacitance "seen" by segment i
|
|
```
|
|
|
|
### Why This Works
|
|
|
|
**Physical argument:**
|
|
|
|
1. Hungry streamer seeks R = 1/(ωC_total) for max power
|
|
2. C depends on diameter: C ∝ 1/ln(h/d)
|
|
3. Logarithmic dependence: 2× diameter → ~10% capacitance change
|
|
4. R_opt also changes ~10% for diameter change
|
|
5. Diameter adjusts to match R_opt (self-consistent)
|
|
6. Error from fixed C is comparable to other uncertainties
|
|
|
|
**Typical uncertainties:**
|
|
```
|
|
FEMM extraction: ±5-10%
|
|
Plasma physics (ε, E_prop): ±30-50%
|
|
Empirical calibration: ±20-30%
|
|
|
|
Diameter approximation: ±10-15%
|
|
|
|
Diameter error is SMALL compared to physics uncertainties!
|
|
```
|
|
|
|
### When to Use
|
|
|
|
**Good for:**
|
|
- Standard cases (typical geometries, frequencies)
|
|
- First-pass analysis
|
|
- Quick evaluation of many designs
|
|
- Educational purposes
|
|
|
|
**Use iterative when:**
|
|
- Research/validation
|
|
- Extreme parameters (very long, very short, very low frequency)
|
|
- Measurement comparison requires highest accuracy
|
|
- Publishing results
|
|
|
|
**Computational savings:**
|
|
```
|
|
Iterative: 5-10 iterations × 20 R-sweep points × n segments = 1000-2000 AC analyses
|
|
Simplified: 1 AC analysis
|
|
|
|
Speedup: 1000-2000× faster!
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 4.6: Simplified R Calculation (n=5)
|
|
|
|
**Given:**
|
|
- f = 190 kHz, ω = 1.194×10⁶ rad/s
|
|
- Capacitance matrix from Example 4.3 (repeated):
|
|
|
|
```
|
|
[0] [1] [2] [3] [4] [5]
|
|
[0] [ 32.5 -9.2 -3.1 -1.2 -0.6 -0.3 ]
|
|
[1] [ -9.2 14.8 -2.8 -0.9 -0.4 -0.2 ]
|
|
[2] [ -3.1 -2.8 10.4 -2.1 -0.7 -0.3 ]
|
|
[3] [ -1.2 -0.9 -2.1 8.6 -1.8 -0.5 ]
|
|
[4] [ -0.6 -0.4 -0.7 -1.8 7.4 -1.4 ]
|
|
[5] [ -0.3 -0.2 -0.3 -0.5 -1.4 5.8 ]
|
|
```
|
|
|
|
**Calculate R[i] for each segment:**
|
|
|
|
**Segment 1 (base):**
|
|
```
|
|
C_total[1] = |C[1,0]| + |C[1,2]| + |C[1,3]| + |C[1,4]| + |C[1,5]|
|
|
= 9.2 + 2.8 + 0.9 + 0.4 + 0.2
|
|
= 13.5 pF
|
|
|
|
R[1] = 1 / (ω × C_total[1])
|
|
= 1 / (1.194×10⁶ × 13.5×10⁻¹²)
|
|
= 1 / (16.12×10⁻⁶)
|
|
= 62.0 kΩ
|
|
```
|
|
|
|
**Segment 2:**
|
|
```
|
|
C_total[2] = |C[2,0]| + |C[2,1]| + |C[2,3]| + |C[2,4]| + |C[2,5]|
|
|
= 3.1 + 2.8 + 2.1 + 0.7 + 0.3
|
|
= 9.0 pF
|
|
|
|
R[2] = 1 / (1.194×10⁶ × 9.0×10⁻¹²)
|
|
= 93.0 kΩ
|
|
```
|
|
|
|
**Segment 3:**
|
|
```
|
|
C_total[3] = 1.2 + 0.9 + 2.1 + 1.8 + 0.5
|
|
= 6.5 pF
|
|
|
|
R[3] = 1 / (1.194×10⁶ × 6.5×10⁻¹²)
|
|
= 129 kΩ
|
|
```
|
|
|
|
**Segment 4:**
|
|
```
|
|
C_total[4] = 0.6 + 0.4 + 0.7 + 1.8 + 1.4
|
|
= 4.9 pF
|
|
|
|
R[4] = 1 / (1.194×10⁶ × 4.9×10⁻¹²)
|
|
= 171 kΩ
|
|
```
|
|
|
|
**Segment 5 (tip):**
|
|
```
|
|
C_total[5] = 0.3 + 0.2 + 0.3 + 0.5 + 1.4
|
|
= 2.7 pF
|
|
|
|
R[5] = 1 / (1.194×10⁶ × 2.7×10⁻¹²)
|
|
= 310 kΩ
|
|
```
|
|
|
|
**Summary:**
|
|
```
|
|
R[1] = 62 kΩ (base - lowest)
|
|
R[2] = 93 kΩ
|
|
R[3] = 129 kΩ
|
|
R[4] = 171 kΩ
|
|
R[5] = 310 kΩ (tip - highest)
|
|
|
|
Total: R_total = 765 kΩ
|
|
```
|
|
|
|
**Validation:**
|
|
```
|
|
At 190 kHz for 2 m spark:
|
|
Expected total: 50-300 kΩ (from Part 2 guidelines)
|
|
|
|
765 kΩ is higher than typical.
|
|
|
|
Possible reasons:
|
|
- Long spark (2 m), distributed effects significant
|
|
- Tip resistance (310k) is high (streamer-dominated)
|
|
- If measured, could be lower (iterative optimization might find lower R)
|
|
|
|
Within factor of 2-3 of expectations - acceptable for first pass
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 4.6
|
|
|
|
**Problem 1:** Given C_total[i] = [15, 10, 8, 6, 4] pF for n=5 at f=200 kHz, calculate R[i] for all segments.
|
|
|
|
**Problem 2:** Compare simplified method: one calculation (1 second) vs iterative: 10 iterations × 20 points × 5 segments = 1000 AC analyses (~100 seconds). For engineering design, which is more appropriate?
|
|
|
|
---
|
|
|
|
## Module 4.7: Quick Validation Checks
|
|
|
|
### Power Balance
|
|
|
|
**Energy conservation:**
|
|
```
|
|
P_input = P_spark + P_secondary_losses + P_corona + P_radiation + P_other
|
|
|
|
Check: P_spark should be 30-70% of P_input for typical coil
|
|
```
|
|
|
|
**If P_spark > 90% of P_input:**
|
|
- Secondary losses too low (unrealistic Q)
|
|
- Check winding resistance, dielectric losses
|
|
|
|
**If P_spark < 20% of P_input:**
|
|
- Excessive secondary losses
|
|
- Or spark model R too high (not optimized)
|
|
|
|
### Total Resistance Range Check
|
|
|
|
**Expected at 200 kHz for 1-3 m sparks:**
|
|
```
|
|
Burst/streamer-dominated: 50-300 kΩ
|
|
QCW/leader-dominated: 5-50 kΩ
|
|
Very low frequency (<100 kHz) or very long: 1-10 kΩ
|
|
|
|
R_total = Σ R[i] should fall in expected range
|
|
```
|
|
|
|
**If outside range:**
|
|
- Check frequency (R ∝ 1/f)
|
|
- Check optimization convergence
|
|
- Verify capacitance matrix extraction
|
|
- Consider if mode is truly different (all-leader vs all-streamer)
|
|
|
|
### Resistance Distribution Check
|
|
|
|
**Physical expectation:**
|
|
```
|
|
R[1] < R[2] < R[3] < ... < R[n]
|
|
|
|
Base should be lowest (hot, coupled)
|
|
Tip should be highest (cool, weakly coupled)
|
|
|
|
Monotonic increase expected
|
|
```
|
|
|
|
**If non-monotonic:**
|
|
- Check capacitance matrix (may have errors)
|
|
- Verify optimization didn't get stuck
|
|
- Physical interpretation: local heating/cooling variation
|
|
|
|
### Phase Angle Check
|
|
|
|
**Total impedance phase:**
|
|
```
|
|
Calculate Z_total at topload port
|
|
φ_Z should be -55° to -75° typical
|
|
|
|
If φ_Z > -45°: Too resistive (check if topological constraint violated)
|
|
If φ_Z < -85°: Too capacitive (R values too high, not optimized)
|
|
```
|
|
|
|
### Convergence Check
|
|
|
|
**For distributed models with n=5, 10, 20:**
|
|
```
|
|
Run same problem with different n:
|
|
- n=5 → Z_total, P_spark
|
|
- n=10 → Z_total, P_spark
|
|
- n=20 → Z_total, P_spark
|
|
|
|
Should converge: changes <10% from n=10 to n=20
|
|
|
|
If still changing >20%: need finer discretization
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 4.7: Validation Exercise
|
|
|
|
**Given simulation results:**
|
|
```
|
|
Coil: DRSSTC at 185 kHz
|
|
P_primary_input = 150 kW
|
|
P_spark = 105 kW (from distributed model n=10)
|
|
Spark: 2.5 m
|
|
|
|
Distributed R values [kΩ]:
|
|
[18, 25, 35, 48, 65, 88, 120, 165, 230, 320]
|
|
|
|
Z_total = 185 kΩ ∠-68°
|
|
```
|
|
|
|
**Validate:**
|
|
|
|
**Check 1: Power balance**
|
|
```
|
|
P_spark / P_input = 105 / 150 = 0.70 = 70%
|
|
|
|
Expected: 30-70% typical ✓
|
|
Reasonable - some secondary losses, but spark dominates
|
|
```
|
|
|
|
**Check 2: Total resistance**
|
|
```
|
|
R_total = Σ R[i] = 18+25+35+48+65+88+120+165+230+320
|
|
= 1114 kΩ
|
|
|
|
At 185 kHz, expected: 50-300 kΩ for typical
|
|
Actual: 1114 kΩ
|
|
|
|
High, but this is 2.5 m spark (long)
|
|
Factor of 3-4× over typical
|
|
Could indicate:
|
|
- Very streamer-dominated (burst mode?)
|
|
- Or optimization not fully converged
|
|
- Or long spark genuinely has higher R
|
|
|
|
Flag for investigation, but not necessarily wrong ✓?
|
|
```
|
|
|
|
**Check 3: Resistance distribution**
|
|
```
|
|
R[1]=18 < R[2]=25 < R[3]=35 < ... < R[10]=320
|
|
|
|
Monotonic increasing ✓
|
|
Expected pattern (base lower, tip higher) ✓
|
|
```
|
|
|
|
**Check 4: Phase angle**
|
|
```
|
|
φ_Z = -68°
|
|
|
|
Expected range: -55° to -75°
|
|
Actual: -68°
|
|
|
|
Right in the middle ✓
|
|
Indicates reasonable capacitive loading
|
|
```
|
|
|
|
**Check 5: Compare to lumped model**
|
|
```
|
|
Lumped model (from earlier): R ≈ 600 kΩ at similar conditions
|
|
|
|
Distributed: R_total = 1114 kΩ
|
|
|
|
Distributed is higher (factor ~2)
|
|
This can happen:
|
|
- Distributed captures tip streamer high-R better
|
|
- Lumped averages to middle value
|
|
- For long sparks, distributed more accurate
|
|
|
|
Consistent with expectations ✓
|
|
```
|
|
|
|
**Overall assessment:**
|
|
- Most checks pass
|
|
- Total R is high but potentially physical for long streamer spark
|
|
- Recommend: compare to measurement if available
|
|
- Model is usable for predictions
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 4.7
|
|
|
|
**Problem 1:** Simulation shows P_spark = 180 kW but P_input = 150 kW. What's wrong?
|
|
|
|
**Problem 2:** Distributed model gives R = [50, 45, 40, 35, 30] kΩ (decreasing from base to tip). Is this physical? What might be wrong?
|
|
|
|
**Problem 3:** At 150 kHz, 1.8 m spark, you get R_total = 2 kΩ. Check against expected range. Is this reasonable?
|
|
|
|
---
|
|
|
|
## Module 4.8: Complete Simulation Summary
|
|
|
|
### Workflow Checklist
|
|
|
|
**Phase 1: Geometry and FEMM**
|
|
- [ ] Define spark length L_total
|
|
- [ ] Choose n segments (typically 10)
|
|
- [ ] Create FEMM geometry (axisymmetric)
|
|
- [ ] Set up conductors (topload + n segments)
|
|
- [ ] Mesh and solve electrostatic
|
|
- [ ] Extract (n+1)×(n+1) capacitance matrix [C]
|
|
- [ ] Validate: symmetry, positive definite, C_sh ≈ 2 pF/ft
|
|
|
|
**Phase 2: Resistance Determination**
|
|
- [ ] Choose method: iterative or simplified
|
|
- [ ] If simplified: R[i] = 1/(ω × C_total[i])
|
|
- [ ] If iterative: initialize R[i], run optimization loop
|
|
- [ ] Apply position-dependent bounds R_min[i], R_max[i]
|
|
- [ ] Check convergence (<1% change)
|
|
- [ ] Validate: R distribution monotonic, total in expected range
|
|
|
|
**Phase 3: SPICE Implementation**
|
|
- [ ] Convert [C] matrix to SPICE-compatible form (partial or controlled sources)
|
|
- [ ] Add resistance elements R[i]
|
|
- [ ] Define topload voltage source (or integrate with full coil model)
|
|
- [ ] Set up AC analysis at operating frequency
|
|
|
|
**Phase 4: Analysis**
|
|
- [ ] Run AC simulation
|
|
- [ ] Extract V, I at each node
|
|
- [ ] Calculate P[i] in each segment: P[i] = 0.5 × I[i]² × R[i]
|
|
- [ ] Calculate total P_spark = Σ P[i]
|
|
- [ ] Calculate Y_spark or Z_spark at topload port
|
|
|
|
**Phase 5: Validation**
|
|
- [ ] Power balance: P_spark reasonable fraction of P_input
|
|
- [ ] Total R in expected range for frequency and length
|
|
- [ ] Phase angle φ_Z in typical range
|
|
- [ ] Resistance distribution physical (increasing base→tip)
|
|
- [ ] Compare to lumped model (should be similar order of magnitude)
|
|
- [ ] Compare to measurements if available
|
|
|
|
**Phase 6: Iteration (if needed)**
|
|
- [ ] If validation fails, identify issue
|
|
- [ ] Adjust and re-run
|
|
- [ ] Document assumptions and uncertainties
|
|
|
|
---
|
|
|
|
## Module 4.9: Calibration and Measurement Integration
|
|
|
|
### Calibrating ε (Energy Per Meter)
|
|
|
|
**Procedure:**
|
|
|
|
**Step 1: Controlled test**
|
|
```
|
|
Run coil with known drive conditions
|
|
Measure final spark length L_measured
|
|
```
|
|
|
|
**Step 2: Simulation**
|
|
```
|
|
Simulate same conditions
|
|
Calculate E_delivered = ∫ P_spark dt over growth time
|
|
```
|
|
|
|
**Step 3: Extract ε**
|
|
```
|
|
ε_calibrated = E_delivered / L_measured
|
|
|
|
Example:
|
|
E_delivered = 18 J (from simulation)
|
|
L_measured = 1.5 m (from photograph/measurement)
|
|
|
|
ε = 18 J / 1.5 m = 12 J/m
|
|
```
|
|
|
|
**Step 4: Build database**
|
|
```
|
|
Repeat for different operating modes:
|
|
- QCW long ramp: ε_QCW
|
|
- Burst mode: ε_burst
|
|
- Intermediate: ε_hybrid
|
|
|
|
Use appropriate ε for future predictions
|
|
```
|
|
|
|
### Calibrating E_propagation
|
|
|
|
**Procedure:**
|
|
|
|
**Step 1: Measure stall condition**
|
|
```
|
|
Ramp voltage slowly
|
|
Observe maximum length L_max when growth stops
|
|
Measure V_topload at stall
|
|
```
|
|
|
|
**Step 2: FEMM field analysis**
|
|
```
|
|
Set up geometry with spark length = L_max
|
|
Apply V = V_topload
|
|
Calculate E_tip at tip using FEMM
|
|
```
|
|
|
|
**Step 3: Extract threshold**
|
|
```
|
|
E_propagation ≈ E_tip at stall
|
|
|
|
Typical: 0.4-1.0 MV/m
|
|
Calibrate for your specific conditions (altitude, humidity, geometry)
|
|
```
|
|
|
|
### Using Measurements to Refine Model
|
|
|
|
**Ringdown method (from Part 2):**
|
|
```
|
|
1. Measure f₀, Q₀ (unloaded)
|
|
2. Measure f_L, Q_L (with spark)
|
|
3. Extract Y_spark from frequency shift and Q change
|
|
4. Compare to model prediction
|
|
5. Adjust R values if significant discrepancy (>factor of 2)
|
|
```
|
|
|
|
**Direct impedance measurement:**
|
|
```
|
|
If you have:
|
|
- Calibrated E-field probe (V_topload)
|
|
- Calibrated current probe on spark return path (I_spark, not I_base!)
|
|
|
|
Then:
|
|
Z_measured = V_topload / I_spark
|
|
|
|
Compare to model Z_spark
|
|
Adjust R values to match
|
|
```
|
|
|
|
**Iterative refinement:**
|
|
```
|
|
1. Initial model from FEMM + simplified R
|
|
2. Simulate → predict Z_spark, power
|
|
3. Measure actual Z_spark, power
|
|
4. Adjust R distribution (proportionally) to match measured total R
|
|
5. Validate that distribution shape is still physical
|
|
6. Use refined model for future predictions
|
|
```
|
|
|
|
---
|
|
|
|
### WORKED EXAMPLE 4.9: Calibrating ε
|
|
|
|
**Measurement:**
|
|
```
|
|
QCW coil, 12 ms ramp
|
|
Final spark length: L = 2.2 m
|
|
```
|
|
|
|
**Simulation:**
|
|
```
|
|
Full model with distributed spark
|
|
Calculate power to spark over time:
|
|
P_spark(t) varies from 20 kW to 80 kW during ramp
|
|
|
|
Total energy:
|
|
E_delivered = ∫₀^0.012 P_spark(t) dt
|
|
= 26 J (numerical integration)
|
|
```
|
|
|
|
**Calibration:**
|
|
```
|
|
ε = E_delivered / L_measured
|
|
= 26 J / 2.2 m
|
|
= 11.8 J/m
|
|
```
|
|
|
|
**Interpretation:**
|
|
```
|
|
This is at low end of QCW range (5-15 J/m)
|
|
Indicates efficient leader formation
|
|
Consistent with long ramp time (12 ms)
|
|
|
|
Use ε = 12 J/m for future predictions with this coil in QCW mode
|
|
```
|
|
|
|
**Validation:**
|
|
```
|
|
Predict different condition:
|
|
New ramp: 8 ms, available energy: E = 30 J
|
|
|
|
Expected length: L = E/ε = 30/12 = 2.5 m
|
|
|
|
Run test, measure actual length, compare
|
|
If within ±20%: calibration good
|
|
If >30% error: investigate (different mode? voltage limited?)
|
|
```
|
|
|
|
---
|
|
|
|
### PRACTICE PROBLEMS 4.9
|
|
|
|
**Problem 1:** Simulation shows E = 40 J delivered, measurement shows L = 2.8 m. Calculate ε. Is this more consistent with QCW or burst mode?
|
|
|
|
**Problem 2:** A calibration at sea level gives E_propagation = 0.5 MV/m. At 2000 m altitude (air density ~80% of sea level), estimate new E_propagation.
|
|
|
|
---
|
|
|
|
## Part 4 Conclusion: Practical Guidelines
|
|
|
|
### Decision Tree: Which Model to Use?
|
|
|
|
```
|
|
START
|
|
|
|
|
└─ Spark length < 1 m?
|
|
├─ YES → Use LUMPED model
|
|
| * Fast, accurate enough
|
|
| * R = R_opt_power
|
|
|
|
|
└─ NO → Spark length < 3 m?
|
|
├─ YES → Choice:
|
|
| * Quick answer: LUMPED
|
|
| * Best accuracy: DISTRIBUTED (n=10)
|
|
|
|
|
└─ NO (>3 m) → Use DISTRIBUTED (n=15-20)
|
|
* Essential for accuracy
|
|
* Captures tip/base differences
|
|
|
|
Research/validation? → Always use DISTRIBUTED
|
|
```
|
|
|
|
### Typical Simulation Times
|
|
|
|
```
|
|
Lumped model:
|
|
- FEMM: 2 min (single geometry)
|
|
- SPICE: <1 sec
|
|
- Total: ~3 minutes
|
|
|
|
Distributed (n=10), simplified R:
|
|
- FEMM: 5 min (multi-body)
|
|
- SPICE: 1 sec (one analysis)
|
|
- Total: ~6 minutes
|
|
|
|
Distributed (n=10), iterative R:
|
|
- FEMM: 5 min
|
|
- SPICE: 100 sec (100 iterations × 1 sec)
|
|
- Total: ~7 minutes
|
|
|
|
Distributed (n=20), iterative R:
|
|
- FEMM: 10 min (larger matrix)
|
|
- SPICE: 300 sec (more elements)
|
|
- Total: ~15 minutes
|
|
```
|
|
|
|
### Accuracy Expectations
|
|
|
|
```
|
|
Lumped model:
|
|
- Impedance: ±20%
|
|
- Power: ±30%
|
|
- Good enough for: matching studies, coil optimization
|
|
|
|
Distributed (simplified R):
|
|
- Impedance: ±15%
|
|
- Power: ±25%
|
|
- Current distribution: ±30%
|
|
|
|
Distributed (iterative R):
|
|
- Impedance: ±10%
|
|
- Power: ±20%
|
|
- Current distribution: ±20%
|
|
- Best available without plasma modeling
|
|
|
|
Measurement comparison:
|
|
- ±20-50% agreement is GOOD (plasma variability)
|
|
- ±factor of 2: acceptable (many unknowns)
|
|
- Better than factor of 2: excellent!
|
|
```
|
|
|
|
### Final Recommendations
|
|
|
|
**For hobbyist design:**
|
|
- Use lumped model
|
|
- Calibrate ε from one measurement
|
|
- Predict new conditions
|
|
|
|
**For research:**
|
|
- Use distributed model (n=10-15)
|
|
- Iterative optimization
|
|
- Document all assumptions
|
|
- Compare to measurements
|
|
- Report uncertainties
|
|
|
|
**For publications:**
|
|
- Distributed model required
|
|
- Validation against measurements
|
|
- Sensitivity analysis
|
|
- Clear methodology section
|
|
|
|
---
|
|
|
|
## Final Comprehensive Problem
|
|
|
|
**Design Challenge: Predict Performance of New Coil**
|
|
|
|
**Given:**
|
|
- DRSSTC, f = 195 kHz
|
|
- Topload: 35 cm toroid (major diameter)
|
|
- Target: 2 m spark, QCW mode (10 ms ramp)
|
|
- Primary input: P_input = 120 kW
|
|
- Thévenin: Z_th = 110 - j2300 Ω, V_th = 340 kV
|
|
|
|
**Required:**
|
|
|
|
**Part 1: Distributed Model Setup**
|
|
- Choose n (justify)
|
|
- Describe FEMM geometry
|
|
- What validation checks after extracting [C]?
|
|
|
|
**Part 2: Resistance Calculation**
|
|
- Choose method (iterative or simplified, justify)
|
|
- Estimate expected R_total range
|
|
- What bounds for R[i]?
|
|
|
|
**Part 3: Performance Prediction**
|
|
- Calculate Z_spark
|
|
- Find current and power
|
|
- What % of theoretical max?
|
|
|
|
**Part 4: Growth Analysis**
|
|
- Assume ε = 12 J/m (from calibration)
|
|
- Can 2 m be reached in 10 ms with available power?
|
|
- Check voltage: κ = 3.2, E_prop = 0.7 MV/m
|
|
- Is growth voltage-limited or power-limited?
|
|
|
|
**Part 5: Validation Plan**
|
|
- What measurements would you take?
|
|
- How would you refine the model?
|
|
- What accuracy do you expect?
|
|
|
|
**This problem integrates all four parts of the course!**
|
|
|
|
---
|
|
|
|
## Course Summary: Master Checklist
|
|
|
|
### Part 1 Concepts
|
|
- [ ] Peak vs RMS phasor convention
|
|
- [ ] Complex impedance and admittance
|
|
- [ ] Power formula: P = 0.5 × Re{V × I*}
|
|
- [ ] C_mut and C_sh in spark circuit
|
|
- [ ] Circuit topology: (R||C_mut) + C_sh
|
|
- [ ] Phase angles and capacitive loading
|
|
|
|
### Part 2 Concepts
|
|
- [ ] Topological phase constraint φ_Z,min
|
|
- [ ] R_opt_power maximizes power transfer
|
|
- [ ] Hungry streamer self-optimization
|
|
- [ ] Why V_top/I_base is wrong
|
|
- [ ] Thévenin equivalent extraction and use
|
|
- [ ] Q measurement and ringdown analysis
|
|
|
|
### Part 3 Concepts
|
|
- [ ] E_inception and E_propagation thresholds
|
|
- [ ] Energy per meter ε by mode
|
|
- [ ] Growth rate dL/dt = P/ε
|
|
- [ ] Thermal time constants and persistence
|
|
- [ ] Capacitive divider problem
|
|
- [ ] FEMM electrostatic analysis
|
|
- [ ] Maxwell capacitance matrix extraction
|
|
- [ ] Lumped model construction
|
|
|
|
### Part 4 Concepts
|
|
- [ ] When distributed models needed
|
|
- [ ] nth-order segmentation
|
|
- [ ] Multi-body FEMM analysis
|
|
- [ ] Capacitance matrix in SPICE (partial capacitance)
|
|
- [ ] Iterative R optimization with damping
|
|
- [ ] Simplified R = 1/(ωC_total) method
|
|
- [ ] Validation checks (power balance, R range, distribution)
|
|
- [ ] Calibration from measurements (ε, E_prop)
|
|
|
|
---
|
|
|
|
## Resources for Continued Learning
|
|
|
|
**Software:**
|
|
- FEMM: www.femm.info (free)
|
|
- LTSpice: www.analog.com/ltspice (free)
|
|
- Python + NumPy/SciPy for automation
|
|
|
|
**Tesla Coil Communities:**
|
|
- 4hv.org forums (active community)
|
|
- highvoltageforum.net
|
|
- teslamap.com (coil database)
|
|
|
|
**Further Reading:**
|
|
- "The Spark Gap" magazine (archived)
|
|
- Lightning physics textbooks (Uman, Rakov)
|
|
- Plasma physics introductions (Chen)
|
|
- High voltage engineering (Kuffel)
|
|
|
|
**This framework:**
|
|
- Original document for full mathematical details
|
|
- Implement in stages (lumped → distributed)
|
|
- Calibrate to YOUR coil
|
|
- Share results with community!
|
|
|
|
---
|
|
|
|
**END OF PART 4**
|
|
|
|
**END OF COMPLETE LESSON PLAN**
|
|
|
|
---
|
|
|
|
**Congratulations!** You now have a complete framework to:
|
|
1. Understand Tesla coil spark physics
|
|
2. Extract parameters from FEMM
|
|
3. Build circuit models (lumped and distributed)
|
|
4. Predict performance
|
|
5. Validate against measurements
|
|
6. Iterate and improve
|
|
|
|
**Next steps:**
|
|
- Work through practice problems
|
|
- Build your first model
|
|
- Compare to real coil
|
|
- Refine and calibrate
|
|
|
|
# Tesla Coil Spark Modeling - Complete Lesson Plan
|
|
## Appendices: Quick Reference Materials
|
|
|
|
---
|
|
|
|
## Appendix A: Complete Variable Reference Table
|
|
|
|
### Circuit Variables
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **C_mut** | F (pF) | Mutual capacitance between topload and spark | 5-15 pF |
|
|
| **C_sh** | F (pF) | Shunt capacitance spark-to-ground | 2 pF/foot × length |
|
|
| **C_total** | F (pF) | Total capacitance: C_mut + C_sh | 10-30 pF |
|
|
| **C_eq** | F (pF) | Equivalent loaded capacitance | Calculated from f shift |
|
|
| **R** | Ω (kΩ) | Spark plasma resistance | 5-500 kΩ @ 200 kHz |
|
|
| **R_opt_power** | Ω | Resistance for maximum power transfer | 1/(ω(C_mut+C_sh)) |
|
|
| **R_opt_phase** | Ω | Resistance for minimum phase angle | 1/(ω√(C_mut(C_mut+C_sh))) |
|
|
| **R_min** | Ω | Minimum physical resistance (hot leader) | 1-10 kΩ |
|
|
| **R_max** | Ω | Maximum physical resistance (cold streamer) | 100 kΩ - 100 MΩ |
|
|
| **G** | S (μS) | Conductance: 1/R | 1-100 μS typical |
|
|
| **B₁** | S (μS) | Susceptance of C_mut: ωC_mut | Positive (capacitive) |
|
|
| **B₂** | S (μS) | Susceptance of C_sh: ωC_sh | Positive (capacitive) |
|
|
| **Y** | S (μS) | Complex admittance: G + jB | - |
|
|
| **Z** | Ω (kΩ) | Complex impedance: R + jX | - |
|
|
| **Z_th** | Ω | Thévenin output impedance | 100-200 Ω + j(-2000 to -3000 Ω) |
|
|
| **V_th** | V (kV) | Thévenin open-circuit voltage | 200-500 kV |
|
|
| **φ_Z** | ° or rad | Impedance phase angle | -55° to -75° typical |
|
|
| **φ_Z,min** | ° or rad | Minimum achievable phase: -atan(2√(r(1+r))) | More negative than -45° usually |
|
|
| **r** | - | Capacitance ratio: C_mut/C_sh | 0.5-2.0 typical |
|
|
|
|
### Frequency and Quality Factor
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **f** | Hz (kHz) | Operating frequency | 100-400 kHz |
|
|
| **f₀** | Hz | Unloaded resonant frequency | - |
|
|
| **f_L** | Hz | Loaded resonant frequency (with spark) | Lower than f₀ |
|
|
| **ω** | rad/s | Angular frequency: 2πf | 6.28×10⁵ - 2.5×10⁶ |
|
|
| **Q₀** | - | Unloaded quality factor | 50-200 typical |
|
|
| **Q_L** | - | Loaded quality factor (with spark) | 20-80 typical |
|
|
| **τ** | s (ms) | Time constant for decay | τ = 2Q/ω |
|
|
|
|
### Power and Energy
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **P** | W (kW) | Real (average) power | - |
|
|
| **P_spark** | W (kW) | Power dissipated in spark | 10-200 kW |
|
|
| **P_avg** | W (kW) | Average power over time | - |
|
|
| **P_max** | W (kW) | Theoretical maximum (conjugate match) | Usually unachievable |
|
|
| **E** | J | Energy | - |
|
|
| **E_total** | J | Total energy to grow spark | ε × L |
|
|
| **ε** (epsilon) | J/m | Energy per meter for growth | 5-15 (QCW), 30-100 (burst) |
|
|
| **ε₀** | J/m | Initial energy per meter | Before thermal accumulation |
|
|
|
|
### Electric Fields
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **E** | V/m (MV/m) | Electric field strength | - |
|
|
| **E_tip** | V/m (MV/m) | Field at spark tip | κ × V_top/L |
|
|
| **E_average** | V/m (MV/m) | Average field: V_top/L | - |
|
|
| **E_inception** | V/m (MV/m) | Field for initial breakdown | 2-3 MV/m |
|
|
| **E_propagation** | V/m (MV/m) | Field for sustained growth | 0.4-1.0 MV/m |
|
|
| **κ** (kappa) | - | Tip enhancement factor | 2-5 typical |
|
|
|
|
### Geometric Variables
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **L** | m | Spark length | 0.3-6 m typical |
|
|
| **L_target** | m | Target design length | - |
|
|
| **L_segment** | m | Length of one segment (distributed model) | L_total/n |
|
|
| **d** | m (mm) | Spark channel diameter | 0.1-5 mm (streamers-leaders) |
|
|
| **d_nominal** | m (mm) | Assumed diameter for FEMM | 1 mm (burst), 3 mm (QCW) |
|
|
| **n** | - | Number of segments (distributed model) | 5-20, typically 10 |
|
|
| **i** | - | Segment index (1 to n) | 1=base, n=tip |
|
|
|
|
### Thermal Variables
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **T** | K | Temperature | 1000 K (streamer) - 20000 K (leader) |
|
|
| **ΔT** | K | Temperature rise above ambient | - |
|
|
| **τ_thermal** | s (ms) | Thermal diffusion time: d²/(4α) | 0.1 ms (thin) - 300 ms (thick) |
|
|
| **τ_effective** | s (ms) | Observed persistence time | Longer than τ_thermal |
|
|
| **α_thermal** | m²/s | Thermal diffusivity of air | ~2×10⁻⁵ m²/s |
|
|
|
|
### Matrix and Optimization
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **[C]** | F (pF) | Maxwell capacitance matrix (n+1)×(n+1) | - |
|
|
| **C[i,j]** | F (pF) | Matrix element i,j | Diagonal >0, off-diagonal <0 |
|
|
| **R[i]** | Ω (kΩ) | Resistance of segment i | Increases from base to tip |
|
|
| **α_damp** | - | Damping factor for iteration | 0.3-0.5 |
|
|
| **position** | - | Normalized position: (i-1)/(n-1) | 0=base, 1=tip |
|
|
|
|
### Measurement Variables
|
|
|
|
| Variable | Units | Definition | Typical Values |
|
|
|----------|-------|------------|----------------|
|
|
| **V_top** | V (kV) | Voltage at topload (peak) | 200-600 kV |
|
|
| **V_tip** | V (kV) | Voltage at spark tip | V_top × C_mut/(C_mut+C_sh) |
|
|
| **I_spark** | A | Current through spark | 0.5-3 A |
|
|
| **I_base** | A | Current at secondary base (WRONG for spark) | Includes displacement currents |
|
|
| **A₁, A₂** | V, A | Consecutive peak amplitudes in ringdown | - |
|
|
|
|
---
|
|
|
|
## Appendix B: Formula Quick Reference
|
|
|
|
### Basic Circuit Analysis
|
|
|
|
**Admittance of spark circuit:**
|
|
```
|
|
Y = [(G + jB₁) × jB₂] / [G + j(B₁ + B₂)]
|
|
|
|
where: G = 1/R
|
|
B₁ = ωC_mut
|
|
B₂ = ωC_sh
|
|
```
|
|
|
|
**Real and imaginary parts:**
|
|
```
|
|
Re{Y} = GB₂² / [G² + (B₁+B₂)²]
|
|
|
|
Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
|
|
```
|
|
|
|
**Impedance phase:**
|
|
```
|
|
φ_Z = atan(-Im{Y}/Re{Y})
|
|
```
|
|
|
|
**Power calculation:**
|
|
```
|
|
P = 0.5 × Re{V × I*} (with peak phasors)
|
|
P = 0.5 × |V|² × Re{Y}
|
|
P = 0.5 × |I|² × Re{Z}
|
|
P = 0.5 × |V| × |I| × cos(φ_v - φ_i)
|
|
```
|
|
|
|
### Optimal Resistances
|
|
|
|
**Maximum power transfer:**
|
|
```
|
|
R_opt_power = 1 / [ω(C_mut + C_sh)]
|
|
|
|
Example: f=200 kHz, C_total=12 pF
|
|
R_opt_power = 1/(2π×200×10³×12×10⁻¹²) ≈ 66 kΩ
|
|
```
|
|
|
|
**Minimum phase angle magnitude:**
|
|
```
|
|
R_opt_phase = 1 / [ω√(C_mut(C_mut + C_sh))]
|
|
|
|
Always: R_opt_power < R_opt_phase
|
|
```
|
|
|
|
**Minimum phase angle:**
|
|
```
|
|
φ_Z,min = -atan(2√[r(1+r)])
|
|
|
|
where r = C_mut/C_sh
|
|
|
|
Critical value: r = 0.207 gives φ_Z,min = -45°
|
|
If r > 0.207: cannot achieve -45°
|
|
```
|
|
|
|
### Thévenin Equivalent
|
|
|
|
**Measuring Z_th (drive off, test source on):**
|
|
```
|
|
Z_th = V_test / I_test = 1V / I_test
|
|
|
|
Apply 1V AC at topload-to-ground
|
|
Measure current I_test
|
|
```
|
|
|
|
**Measuring V_th (drive on, no load):**
|
|
```
|
|
V_th = V(topload) with spark removed
|
|
```
|
|
|
|
**Power to any load:**
|
|
```
|
|
P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|²
|
|
```
|
|
|
|
**Theoretical maximum (conjugate match):**
|
|
```
|
|
Z_load = Z_th* (complex conjugate)
|
|
P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
|
|
|
|
Usually unachievable due to topological constraints
|
|
```
|
|
|
|
### Ringdown Method
|
|
|
|
**Quality factor from decay:**
|
|
```
|
|
Q = πf × Δt / ln(A₁/A₂)
|
|
|
|
where Δt = time between peaks
|
|
A₁, A₂ = consecutive peak amplitudes
|
|
```
|
|
|
|
**At loaded resonance:**
|
|
```
|
|
Q_L = ω_L C_eq R_p = R_p/(ω_L L)
|
|
|
|
Therefore:
|
|
R_p = Q_L/(ω_L C_eq) = Q_L ω_L L
|
|
G_total = ω_L C_eq/Q_L = 1/(Q_L ω_L L)
|
|
```
|
|
|
|
**Capacitance from frequency shift:**
|
|
```
|
|
C_eq = C₀(f₀/f_L)²
|
|
ΔC = C_eq - C₀
|
|
```
|
|
|
|
**Spark admittance approximation:**
|
|
```
|
|
Y_spark ≈ (G_total - G_0) + jω_L ΔC
|
|
```
|
|
|
|
### Spark Growth Physics
|
|
|
|
**Growth rate equation:**
|
|
```
|
|
dL/dt = P_stream/ε (when E_tip > E_propagation)
|
|
dL/dt = 0 (when E_tip ≤ E_propagation, stalled)
|
|
```
|
|
|
|
**Time to reach target length (constant power):**
|
|
```
|
|
T = ε × L_target / P_stream
|
|
```
|
|
|
|
**Total energy required:**
|
|
```
|
|
E_total = ε × L_target
|
|
```
|
|
|
|
**Energy per meter with thermal accumulation:**
|
|
```
|
|
ε(t) = ε₀ / (1 + α∫P dt)
|
|
|
|
where α has units [1/J]
|
|
```
|
|
|
|
**Field thresholds:**
|
|
```
|
|
E_inception ≈ 2-3 MV/m (initial breakdown)
|
|
E_propagation ≈ 0.4-1.0 MV/m (sustained growth)
|
|
E_tip = κ × E_average = κ × V_top/L
|
|
```
|
|
|
|
### Thermal Time Constants
|
|
|
|
**Pure thermal diffusion:**
|
|
```
|
|
τ_thermal = d² / (4α)
|
|
|
|
where α ≈ 2×10⁻⁵ m²/s for air
|
|
|
|
Examples:
|
|
d = 100 μm → τ ≈ 0.125 ms
|
|
d = 5 mm → τ ≈ 312 ms
|
|
```
|
|
|
|
**Convection velocity (buoyancy):**
|
|
```
|
|
v ≈ √(g × d × ΔT/T_amb)
|
|
|
|
where g = 9.8 m/s²
|
|
```
|
|
|
|
### Capacitive Divider
|
|
|
|
**Open-circuit voltage division:**
|
|
```
|
|
V_tip = V_topload × C_mut/(C_mut + C_sh)
|
|
|
|
As spark grows: C_sh increases → V_tip decreases
|
|
```
|
|
|
|
**With finite resistance (more complex):**
|
|
```
|
|
V_tip = V_topload × Z_mut/(Z_mut + Z_sh)
|
|
|
|
where Z_mut = (1/jωC_mut) || R
|
|
Z_sh = 1/(jωC_sh)
|
|
```
|
|
|
|
### FEMM Capacitance Extraction
|
|
|
|
**For 2-body system (topload + spark):**
|
|
```
|
|
Maxwell matrix:
|
|
[Top] [Spark]
|
|
[Top] C₁₁ C₁₂
|
|
[Spark] C₂₁ C₂₂
|
|
|
|
Extraction:
|
|
C_mut = |C₁₂| = |C₂₁| (absolute value)
|
|
C_sh = C₂₂ - |C₁₂|
|
|
|
|
Validation: C_sh ≈ 2 pF/foot × L_spark
|
|
```
|
|
|
|
### Distributed Model
|
|
|
|
**Simplified resistance calculation:**
|
|
```
|
|
For each segment i:
|
|
C_total[i] = Σⱼ |C[i,j]| (sum of absolute values)
|
|
R[i] = 1/(ω × C_total[i])
|
|
R[i] = clip(R[i], R_min[i], R_max[i])
|
|
```
|
|
|
|
**Position-dependent bounds:**
|
|
```
|
|
position = (i-1)/(n-1) (0 at base, 1 at tip)
|
|
|
|
R_min[i] = 1 kΩ + (10 kΩ - 1 kΩ) × position
|
|
R_max[i] = 100 kΩ + (100 MΩ - 100 kΩ) × position²
|
|
```
|
|
|
|
**Iterative optimization (damped update):**
|
|
```
|
|
R_new[i] = α × R_optimal[i] + (1-α) × R_old[i]
|
|
|
|
where α = 0.3-0.5 (damping factor)
|
|
```
|
|
|
|
---
|
|
|
|
## Appendix C: Physical Constants and Typical Values
|
|
|
|
### Universal Constants
|
|
|
|
| Constant | Symbol | Value | Units |
|
|
|----------|--------|-------|-------|
|
|
| Permittivity of free space | ε₀ | 8.854×10⁻¹² | F/m |
|
|
| Pi | π | 3.14159... | - |
|
|
| Gravitational acceleration | g | 9.81 | m/s² |
|
|
| Electron charge | e | 1.602×10⁻¹⁹ | C |
|
|
|
|
### Air Properties (Sea Level, 20°C)
|
|
|
|
| Property | Symbol | Value | Units |
|
|
|----------|--------|-------|-------|
|
|
| Density | ρ_air | 1.2 | kg/m³ |
|
|
| Thermal diffusivity | α | 2×10⁻⁵ | m²/s |
|
|
| Thermal conductivity | k | 0.026 | W/(m·K) |
|
|
| Specific heat | c_p | 1005 | J/(kg·K) |
|
|
| Molecular density | n | 2.5×10²⁵ | molecules/m³ |
|
|
| Ionization energy | E_ion | ~15 | eV/molecule |
|
|
|
|
### Field Thresholds (Dry Air, Sea Level)
|
|
|
|
| Parameter | Value | Units | Notes |
|
|
|-----------|-------|-------|-------|
|
|
| E_inception | 2-3 | MV/m | Initial breakdown, smooth electrode |
|
|
| E_propagation | 0.4-1.0 | MV/m | Sustained leader growth |
|
|
| Altitude correction | -20 to -30 | %/1000m | Lower air density → lower threshold |
|
|
| Humidity effect | ±10 | % | Variable, depends on conditions |
|
|
|
|
### Energy per Meter by Mode
|
|
|
|
| Operating Mode | ε Range | Units | Characteristics |
|
|
|----------------|---------|-------|-----------------|
|
|
| QCW (5-20 ms ramp) | 5-15 | J/m | Efficient, leader-dominated |
|
|
| Hybrid DRSSTC | 20-40 | J/m | Mixed streamers/leaders |
|
|
| Burst mode (<1 ms) | 30-100+ | J/m | Inefficient, streamer-dominated |
|
|
| Single-shot burst | 50-150 | J/m | Very inefficient, bright but short |
|
|
|
|
### Typical Spark Resistance (@ 200 kHz)
|
|
|
|
| Spark Type | Length | Total R | Notes |
|
|
|------------|--------|---------|-------|
|
|
| Short burst | 0.5-1 m | 100-300 kΩ | Streamer-dominated |
|
|
| Medium burst | 1-2 m | 150-400 kΩ | Mixed |
|
|
| Long burst | 2-3 m | 200-500 kΩ | Difficult, high R |
|
|
| QCW (short) | 0.5-1 m | 20-80 kΩ | Leader-dominated |
|
|
| QCW (medium) | 1-2 m | 30-120 kΩ | Efficient |
|
|
| QCW (long) | 2-4 m | 40-200 kΩ | Best mode for length |
|
|
|
|
### Frequency Dependence
|
|
|
|
| Frequency | R_typical | C_sh (per meter) | Notes |
|
|
|-----------|-----------|------------------|-------|
|
|
| 100 kHz | 5-50 kΩ | ~6 pF | Low frequency, low R |
|
|
| 150 kHz | 10-100 kΩ | ~6 pF | Typical small coils |
|
|
| 200 kHz | 20-200 kΩ | ~6 pF | Common frequency |
|
|
| 300 kHz | 30-300 kΩ | ~6 pF | Higher frequency |
|
|
| 400 kHz | 40-400 kΩ | ~6 pF | Very high, smaller coils |
|
|
|
|
**Note:** R ∝ 1/f approximately, C_sh relatively constant
|
|
|
|
### Thermal Time Constants
|
|
|
|
| Channel Type | Diameter | τ_thermal | Persistence | Notes |
|
|
|--------------|----------|-----------|-------------|-------|
|
|
| Thin streamer | 50-100 μm | 0.05-0.2 ms | 1-5 ms | Convection extends |
|
|
| Medium streamer | 200-500 μm | 0.2-1.5 ms | 2-10 ms | Mixed |
|
|
| Thin leader | 1-2 mm | 6-25 ms | 50-500 ms | Buoyancy significant |
|
|
| Thick leader | 5-10 mm | 150-600 ms | Seconds | Persistent column |
|
|
|
|
### Tesla Coil Typical Parameters
|
|
|
|
| Parameter | Small Coil | Medium Coil | Large Coil | Units |
|
|
|-----------|------------|-------------|------------|-------|
|
|
| Frequency | 300-500 | 150-250 | 80-150 | kHz |
|
|
| Topload C₀ | 15-25 | 25-40 | 40-80 | pF |
|
|
| Secondary Q₀ | 100-200 | 80-150 | 50-120 | - |
|
|
| Spark length | 0.3-1.0 | 1.0-2.5 | 2.0-4.0 | m |
|
|
| Power | 1-10 | 10-100 | 50-300 | kW |
|
|
| Z_th magnitude | 1-3 | 0.5-2 | 0.3-1 | kΩ |
|
|
| Z_th phase | -85 to -88 | -86 to -89 | -87 to -89 | degrees |
|
|
|
|
---
|
|
|
|
## Appendix D: SPICE Component Reference
|
|
|
|
### Basic Elements
|
|
|
|
**Resistor:**
|
|
```
|
|
R<name> node1 node2 <value>
|
|
Example: R1 topload spark 50k
|
|
R2 n1 n2 {R_value} ; parameterized
|
|
```
|
|
|
|
**Capacitor:**
|
|
```
|
|
C<name> node1 node2 <value>
|
|
Example: C_mut topload spark 10p
|
|
C_sh spark 0 6p
|
|
```
|
|
|
|
**Voltage source:**
|
|
```
|
|
V<name> node+ node- <type> <value>
|
|
Example: V1 topload 0 AC 1V
|
|
V2 drive 0 AC 100k ; 100 kV
|
|
```
|
|
|
|
**Current source:**
|
|
```
|
|
I<name> node+ node- <type> <value>
|
|
Example: I1 topload 0 AC 1m
|
|
```
|
|
|
|
### Parameterized Components
|
|
|
|
**Define parameters:**
|
|
```
|
|
.param freq=200k
|
|
.param omega={2*pi*freq}
|
|
.param C_mut=10p
|
|
.param C_sh=6p
|
|
.param R={1/(omega*(C_mut+C_sh))}
|
|
```
|
|
|
|
**Use in components:**
|
|
```
|
|
C1 n1 n2 {C_mut}
|
|
R1 n2 n3 {R}
|
|
```
|
|
|
|
### Controlled Sources (for capacitance matrix)
|
|
|
|
**Voltage-controlled current source:**
|
|
```
|
|
G<name> node+ node- ctrl+ ctrl- <gain>
|
|
Example: G1 n1 0 n2 0 {j*omega*C[1,2]}
|
|
```
|
|
|
|
**Behavioral source:**
|
|
```
|
|
B<name> node+ node- V={expression}
|
|
Example: B1 n1 0 V={j*omega*C_mut*V(n2)}
|
|
```
|
|
|
|
### Analysis Commands
|
|
|
|
**AC analysis:**
|
|
```
|
|
.ac lin <points> <fstart> <fstop>
|
|
Example: .ac lin 1 200k 200k ; single frequency
|
|
.ac lin 100 180k 220k ; sweep 100 points
|
|
```
|
|
|
|
**Transient analysis:**
|
|
```
|
|
.tran <tstep> <tstop>
|
|
Example: .tran 0.1u 10m ; 0.1 μs steps, 10 ms total
|
|
```
|
|
|
|
**Print/plot:**
|
|
```
|
|
.print ac v(topload) i(V1) vp(topload) ip(V1)
|
|
.plot ac vdb(topload) ; dB magnitude
|
|
```
|
|
|
|
### Mutual Inductance (for transformer)
|
|
|
|
**Inductors with coupling:**
|
|
```
|
|
L1 n1 n2 <L1_value>
|
|
L2 n3 n4 <L2_value>
|
|
K1 L1 L2 <coupling_coefficient>
|
|
|
|
Example:
|
|
Lpri drive n1 100u
|
|
Lsec n2 base 10m
|
|
K_couple Lpri Lsec 0.15 ; k=0.15
|
|
```
|
|
|
|
### Subcircuits (for modular models)
|
|
|
|
**Define subcircuit:**
|
|
```
|
|
.subckt spark_model topload ground
|
|
+ params: C_mut=10p C_sh=6p R=50k
|
|
C1 topload n1 {C_mut}
|
|
R1 n1 n2 {R}
|
|
C2 n2 ground {C_sh}
|
|
.ends
|
|
```
|
|
|
|
**Use subcircuit:**
|
|
```
|
|
X1 topload 0 spark_model params: C_mut=12p C_sh=8p R=60k
|
|
```
|
|
|
|
### Example: Complete Lumped Model
|
|
|
|
```
|
|
* Tesla Coil Spark Lumped Model
|
|
* Frequency: 200 kHz
|
|
|
|
.param freq=200k
|
|
.param omega={2*pi*freq}
|
|
|
|
* Spark parameters from FEMM
|
|
.param C_mut=10p
|
|
.param C_sh=6p
|
|
.param R_opt={1/(omega*(C_mut+C_sh))}
|
|
|
|
* Clip to physical bounds
|
|
.param R_min=5k
|
|
.param R_max=500k
|
|
.param R={min(max(R_opt,R_min),R_max)}
|
|
|
|
* Circuit
|
|
V_topload topload 0 AC 1V
|
|
C_mut topload n1 {C_mut}
|
|
R_spark n1 n2 {R}
|
|
C_sh n2 0 {C_sh}
|
|
|
|
* Analysis
|
|
.ac lin 1 {freq} {freq}
|
|
.print ac v(topload) i(V_topload) vp(topload) ip(V_topload)
|
|
|
|
* Calculate admittance in post-processing:
|
|
* Y = I/V, extract real and imaginary parts
|
|
* Power = 0.5 * |V|^2 * Re{Y}
|
|
|
|
.end
|
|
```
|
|
|
|
---
|
|
|
|
## Appendix E: FEMM Quick Start Guide
|
|
|
|
### Installation
|
|
|
|
1. **Download:** Visit www.femm.info
|
|
2. **Install:** Run installer (Windows), or use Wine (Linux/Mac)
|
|
3. **Launch:** Open FEMM 4.2 (main application)
|
|
|
|
### Basic Interface
|
|
|
|
**Main window sections:**
|
|
- **Toolbar:** Problem type, zoom, view controls
|
|
- **Drawing area:** Geometry creation
|
|
- **Status bar:** Coordinates, snap mode
|
|
- **Menus:** File, Edit, View, Problem, Mesh, Analysis
|
|
|
|
### Creating Electrostatic Problem
|
|
|
|
**Step 1: New document**
|
|
```
|
|
File → New
|
|
Select: Electrostatics Problem
|
|
Frequency: 0 (electrostatic)
|
|
Length units: Centimeters (or your preference)
|
|
Problem type: Axisymmetric
|
|
Precision: 1e-8
|
|
```
|
|
|
|
**Step 2: Define materials**
|
|
```
|
|
Problem → Materials Library
|
|
Select: Air (ε_r = 1.0)
|
|
Add to model
|
|
|
|
If needed, define custom materials:
|
|
Problem → Materials → Add Property
|
|
Name: Custom
|
|
Permittivity: (relative value)
|
|
```
|
|
|
|
**Step 3: Draw geometry**
|
|
```
|
|
Use toolbar buttons:
|
|
- Draw nodes (points): Click to place
|
|
- Draw lines: Select two nodes
|
|
- Draw arcs: Select two nodes, define angle
|
|
- Draw circles: Center + radius
|
|
|
|
For axisymmetric:
|
|
- Draw in r-z plane (r ≥ 0)
|
|
- r = 0 is axis of symmetry
|
|
```
|
|
|
|
### Tesla Coil Spark Geometry Example
|
|
|
|
**Toroid (topload):**
|
|
```
|
|
1. Draw circle (minor diameter) at z=0, r=15 cm
|
|
2. Use circular rotation: Operations → Mirror/Rotate
|
|
3. Create toroidal surface
|
|
```
|
|
|
|
**Spark (cylinder):**
|
|
```
|
|
1. Draw vertical line from topload base to tip
|
|
Example: r=0.1 cm, z=-5 to z=-105 cm (1 m spark)
|
|
2. This represents axis of cylinder
|
|
3. For multiple segments: Draw each as separate line
|
|
```
|
|
|
|
**Ground plane:**
|
|
```
|
|
1. Draw large circle or line at z = (below spark)
|
|
2. Large enough to approximate "infinity"
|
|
```
|
|
|
|
**Outer boundary:**
|
|
```
|
|
1. Draw rectangle enclosing entire problem
|
|
2. Far from coil (5-10× max dimension)
|
|
```
|
|
|
|
### Assigning Properties
|
|
|
|
**Step 4: Define conductors**
|
|
```
|
|
Problem → Conductors
|
|
Add conductor groups:
|
|
- Conductor 1: Name "Topload", Voltage = 1V
|
|
- Conductor 2: Name "Spark1", Floating
|
|
- Conductor 3: Name "Spark2", Floating
|
|
...
|
|
- Conductor n+1: Name "Ground", Voltage = 0V
|
|
```
|
|
|
|
**Step 5: Assign to geometry**
|
|
```
|
|
Select line/arc/circle
|
|
Right-click → Set Boundary
|
|
Choose conductor group
|
|
|
|
All segments of spark: Assign to separate conductors
|
|
Topload surface: Assign to topload conductor
|
|
Ground: Assign to ground conductor
|
|
```
|
|
|
|
**Step 6: Assign materials**
|
|
```
|
|
Select region (click inside enclosed area)
|
|
Right-click → Set Block Property
|
|
Material: Air
|
|
Mesh size: Auto or specify
|
|
```
|
|
|
|
**Step 7: Boundary conditions**
|
|
```
|
|
Problem → Boundaries
|
|
- Outer boundary: V=0 (Dirichlet)
|
|
- r=0: Axisymmetric boundary
|
|
- Others: Default (Neumann, E field normal)
|
|
```
|
|
|
|
### Meshing and Solving
|
|
|
|
**Step 8: Create mesh**
|
|
```
|
|
Mesh → Create Mesh
|
|
Wait for triangulation (seconds to minutes)
|
|
Check mesh quality: Zoom in near conductors
|
|
```
|
|
|
|
**Step 9: Solve**
|
|
```
|
|
Analysis → Run
|
|
Wait for solution (seconds to minutes)
|
|
Look for convergence message
|
|
```
|
|
|
|
### Post-Processing
|
|
|
|
**Step 10: View results**
|
|
```
|
|
File → Open Postprocessor
|
|
(or automatically opens after solve)
|
|
|
|
View field:
|
|
- View → Contour Plot → V (voltage)
|
|
- View → Vector Plot → E (field)
|
|
- View → Density Plot → Field magnitude
|
|
```
|
|
|
|
**Step 11: Extract capacitance matrix**
|
|
```
|
|
Circuit Properties window (usually visible)
|
|
If not: View → Circuit Properties
|
|
|
|
Shows capacitance matrix [C]
|
|
Copy values to spreadsheet/text file
|
|
|
|
Format:
|
|
[1] [2] [3] ...
|
|
[1] C₁₁ C₁₂ C₁₃
|
|
[2] C₂₁ C₂₂ C₂₃
|
|
...
|
|
```
|
|
|
|
**Step 12: Calculate electric field at point**
|
|
```
|
|
Click on specific point
|
|
View → Point Values
|
|
Shows: V, E_r, E_z, |E| at that location
|
|
|
|
For tip field: Click at spark tip
|
|
```
|
|
|
|
### Tips and Tricks
|
|
|
|
**Efficient meshing:**
|
|
```
|
|
- Finer mesh near conductors (small triangle size)
|
|
- Coarse mesh far away (large triangles)
|
|
- Specify manually: Set Block Property → Mesh size
|
|
```
|
|
|
|
**Symmetry exploitation:**
|
|
```
|
|
- Use axisymmetric for cylindrical symmetry (2D → 3D)
|
|
- Use planar for 2D problems
|
|
- Reduces element count by 10-100×
|
|
```
|
|
|
|
**Convergence issues:**
|
|
```
|
|
- Increase precision (Problem → Precision: 1e-10)
|
|
- Refine mesh near conductors
|
|
- Enlarge outer boundary
|
|
- Check for geometry errors (gaps, overlaps)
|
|
```
|
|
|
|
**Large matrix extraction:**
|
|
```
|
|
For n=20 segments → 21×21 matrix
|
|
Circuit Properties window may be small
|
|
Resize window or copy values programmatically
|
|
Consider exporting to CSV
|
|
```
|
|
|
|
### Automation with Lua Scripting
|
|
|
|
**FEMM supports Lua scripts for automation:**
|
|
```lua
|
|
-- Example: Create spark segment
|
|
newdocument(0) -- Electrostatics
|
|
for i=1,10 do
|
|
z_start = -i*10
|
|
z_end = -(i+1)*10
|
|
addnode(0.1, z_start)
|
|
addnode(0.1, z_end)
|
|
addsegment(0.1, z_start, 0.1, z_end)
|
|
selectsegment(0.1, (z_start+z_end)/2)
|
|
setconductor("Spark"..i, 0) -- Floating
|
|
end
|
|
```
|
|
|
|
**Useful for:**
|
|
- Parametric sweeps (vary length, diameter)
|
|
- Batch processing multiple geometries
|
|
- Extracting results programmatically
|
|
|
|
---
|
|
|
|
## Appendix F: Troubleshooting Guide
|
|
|
|
### Problem: Negative Phase Angle Too Large (φ_Z < -80°)
|
|
|
|
**Symptoms:**
|
|
- Impedance phase more negative than -80°
|
|
- Very capacitive
|
|
- Low power transfer
|
|
|
|
**Possible causes:**
|
|
1. R too high (not optimized)
|
|
2. Capacitances overestimated
|
|
3. Frequency too high for given R
|
|
|
|
**Solutions:**
|
|
- Run iterative R optimization
|
|
- Verify FEMM capacitance extraction
|
|
- Check R bounds (R_max too high?)
|
|
- Recalculate R_opt_power
|
|
|
|
---
|
|
|
|
### Problem: Power Balance Doesn't Close
|
|
|
|
**Symptoms:**
|
|
- P_spark > P_input (violates conservation)
|
|
- Or P_spark << P_input (most energy missing)
|
|
|
|
**Possible causes:**
|
|
1. Incorrect power calculation (missing 0.5 factor?)
|
|
2. Using RMS instead of peak values inconsistently
|
|
3. Missing loss terms
|
|
4. Measuring wrong current (I_base instead of I_spark)
|
|
|
|
**Solutions:**
|
|
- Verify formula: P = 0.5 × Re{V × I*} with peak
|
|
- Check all quantities are peak (or all RMS, consistently)
|
|
- Account for secondary losses separately
|
|
- Measure I_spark on return path, not I_base
|
|
|
|
---
|
|
|
|
### Problem: FEMM Capacitance Matrix Not Symmetric
|
|
|
|
**Symptoms:**
|
|
- C[i,j] ≠ C[j,i]
|
|
- Non-physical
|
|
|
|
**Possible causes:**
|
|
1. Numerical error (insufficient precision)
|
|
2. Mesh quality poor
|
|
3. Geometry errors (overlaps, gaps)
|
|
|
|
**Solutions:**
|
|
- Increase precision: Problem → Precision: 1e-10
|
|
- Refine mesh near conductors
|
|
- Check geometry for errors (zoom in, look for gaps)
|
|
- Ensure proper boundary conditions
|
|
|
|
---
|
|
|
|
### Problem: Distributed Model Doesn't Converge
|
|
|
|
**Symptoms:**
|
|
- Iterative optimization oscillates
|
|
- R values jumping around
|
|
- No stable solution after many iterations
|
|
|
|
**Possible causes:**
|
|
1. Damping factor α too high
|
|
2. Weakly coupled segments (tip)
|
|
3. R bounds too restrictive
|
|
4. Power curve very flat
|
|
|
|
**Solutions:**
|
|
- Reduce α to 0.2-0.3 (more damping)
|
|
- Accept tip segments not converging (physical)
|
|
- Widen R_max bounds for tip segments
|
|
- Use simplified method if iterative fails
|
|
|
|
---
|
|
|
|
### Problem: Simulation Predicts Too Short Spark
|
|
|
|
**Symptoms:**
|
|
- Predicted length << measured
|
|
- Model underestimates performance
|
|
|
|
**Possible causes:**
|
|
1. ε too high (overestimating energy needed)
|
|
2. E_propagation set too high
|
|
3. Power transfer underestimated (R not optimized)
|
|
4. Capacitances wrong (affects R_opt)
|
|
|
|
**Solutions:**
|
|
- Calibrate ε from measurements
|
|
- Check E_propagation threshold
|
|
- Verify R optimization ran correctly
|
|
- Re-check FEMM extraction
|
|
|
|
---
|
|
|
|
### Problem: Simulation Predicts Too Long Spark
|
|
|
|
**Symptoms:**
|
|
- Predicted length >> measured
|
|
- Model overestimates performance
|
|
|
|
**Possible causes:**
|
|
1. ε too low (underestimating energy needed)
|
|
2. E_propagation set too low
|
|
3. Not accounting for capacitive divider voltage drop
|
|
4. Using burst-mode ε for QCW (or vice versa)
|
|
|
|
**Solutions:**
|
|
- Increase ε (burst needs higher value)
|
|
- Verify field threshold appropriate for conditions
|
|
- Check V_tip calculation (capacitive division)
|
|
- Use correct ε for operating mode
|
|
|
|
---
|
|
|
|
### Problem: R_total Outside Expected Range
|
|
|
|
**Symptoms:**
|
|
- Total resistance 10× too high or too low
|
|
- Doesn't match measurements or expectations
|
|
|
|
**Possible causes:**
|
|
1. Wrong frequency
|
|
2. Capacitance extraction error
|
|
3. Optimization failure
|
|
4. Physical bounds too restrictive
|
|
|
|
**Solutions:**
|
|
- Verify frequency used in R calculation
|
|
- Re-check capacitance matrix from FEMM
|
|
- Try simplified R method as sanity check
|
|
- Compare segment-by-segment to expected profile
|
|
|
|
---
|
|
|
|
### Problem: SPICE Simulation Gives Nonsense Results
|
|
|
|
**Symptoms:**
|
|
- Negative resistance calculated
|
|
- Infinite impedance
|
|
- Convergence errors
|
|
|
|
**Possible causes:**
|
|
1. Capacitance matrix implementation wrong
|
|
2. Negative capacitor values
|
|
3. Ground reference missing
|
|
4. Parameter syntax error
|
|
|
|
**Solutions:**
|
|
- Use partial capacitance transformation (all positive)
|
|
- Verify every capacitor value >0
|
|
- Ensure at least one node grounded
|
|
- Check .param syntax (use {expression} for calculations)
|
|
|
|
---
|
|
|
|
### Problem: Measured vs Simulated Impedance Differs by Factor >2
|
|
|
|
**Symptoms:**
|
|
- Model predicts Z = 200 kΩ
|
|
- Measurement shows Z = 450 kΩ (or 90 kΩ)
|
|
|
|
**Possible causes:**
|
|
1. Measurement method wrong (V_top/I_base)
|
|
2. Spark branching in measurement (not modeled)
|
|
3. Operating mode different (burst vs QCW)
|
|
4. Frequency shift not accounted for
|
|
|
|
**Solutions:**
|
|
- Use correct measurement port (topload-to-ground)
|
|
- Model cannot capture branching (expected discrepancy)
|
|
- Ensure ε appropriate for actual mode
|
|
- Remeasure at loaded resonance frequency
|
|
|
|
---
|
|
|
|
### Problem: Growth Stalls Before Target Length
|
|
|
|
**Symptoms:**
|
|
- Spark stops growing
|
|
- More power doesn't help
|
|
|
|
**Possible causes:**
|
|
1. Voltage-limited (E_tip < E_propagation)
|
|
2. Capacitive divider drops V_tip too much
|
|
3. E_propagation higher than assumed
|
|
4. Topload too small for target length
|
|
|
|
**Solutions:**
|
|
- Check E_tip calculation at stall length
|
|
- Consider ramping voltage higher
|
|
- Increase topload capacitance (less voltage division)
|
|
- Reduce target length (be realistic)
|
|
|
|
---
|
|
|
|
### Problem: QCW Gives Same Length as Burst (Expected Longer)
|
|
|
|
**Symptoms:**
|
|
- QCW and burst same performance
|
|
- Not seeing efficiency advantage
|
|
|
|
**Possible causes:**
|
|
1. Using same ε for both (should be different)
|
|
2. QCW ramp too short (not exploiting thermal memory)
|
|
3. Insufficient power for QCW
|
|
4. Leader formation not occurring
|
|
|
|
**Solutions:**
|
|
- Use ε_QCW = 8-15 J/m, ε_burst = 40-80 J/m
|
|
- Lengthen ramp time (10-20 ms)
|
|
- Increase average power
|
|
- Check current sufficient for leader (>0.5 A)
|
|
|
|
---
|
|
|
|
### Quick Diagnostic Flowchart
|
|
|
|
```
|
|
Problem occurs
|
|
|
|
|
├─ Unreasonable value (negative, infinite, 1000× off)
|
|
| → Check units, formula, syntax
|
|
| → Verify all inputs are correct quantities
|
|
|
|
|
├─ Non-convergence (oscillation, no stable solution)
|
|
| → Reduce damping factor α
|
|
| → Check if problem has solution (bounds?)
|
|
| → Try simpler model first
|
|
|
|
|
├─ Mismatch with measurement (factor 2-5)
|
|
| → Verify measurement method
|
|
| → Check operating mode matches
|
|
| → Calibrate ε, E_propagation from data
|
|
|
|
|
└─ Physical impossibility (violates conservation, etc.)
|
|
→ Review assumptions
|
|
→ Check for double-counting or missing terms
|
|
→ Verify reference frames consistent
|
|
```
|
|
|
|
---
|
|
|
|
## Appendix G: Worked Solutions to Comprehensive Problems
|
|
|
|
### Part 2 Comprehensive Design Exercise (Solution)
|
|
|
|
**Given:**
|
|
- f = 190 kHz
|
|
- C_topload = 30 pF
|
|
- Target spark: 3 feet (estimate C_sh)
|
|
- C_mut = 9 pF (from FEMM)
|
|
- Z_th = 105 - j2100 Ω, V_th = 320 kV
|
|
|
|
---
|
|
|
|
**Task 1: Calculate capacitance ratio and phase constraint**
|
|
|
|
```
|
|
C_sh = 2 pF/ft × 3 ft = 6 pF
|
|
|
|
r = C_mut/C_sh = 9/6 = 1.5
|
|
|
|
φ_Z,min = -atan(2√[r(1+r)])
|
|
= -atan(2√[1.5×2.5])
|
|
= -atan(2√3.75)
|
|
= -atan(2×1.936)
|
|
= -atan(3.872)
|
|
= -75.5°
|
|
|
|
Cannot achieve -45° (r = 1.5 > 0.207) ✓
|
|
```
|
|
|
|
---
|
|
|
|
**Task 2: Determine optimal resistances**
|
|
|
|
```
|
|
ω = 2π × 190×10³ = 1.194×10⁶ rad/s
|
|
|
|
R_opt_power = 1/(ω(C_mut + C_sh))
|
|
= 1/(1.194×10⁶ × 15×10⁻¹²)
|
|
= 1/(17.91×10⁻⁶)
|
|
= 55.8 kΩ
|
|
|
|
R_opt_phase = 1/(ω√(C_mut(C_mut+C_sh)))
|
|
= 1/(1.194×10⁶ × √(9×10⁻¹² × 15×10⁻¹²))
|
|
= 1/(1.194×10⁶ × 11.62×10⁻¹²)
|
|
= 1/(13.87×10⁻⁶)
|
|
= 72.1 kΩ
|
|
|
|
R_opt_power < R_opt_phase ✓ (55.8 < 72.1)
|
|
|
|
At R_opt_power, expect φ_Z ≈ -76° (slightly more capacitive than minimum)
|
|
```
|
|
|
|
---
|
|
|
|
**Task 3: Build lumped spark model**
|
|
|
|
```
|
|
Circuit:
|
|
Topload ---[C_mut=9pF]---+--- [C_sh=6pF]---GND
|
|
|
|
|
[R=55.8kΩ]
|
|
|
|
Calculate Y_spark:
|
|
G = 1/R = 1/55800 = 17.92 μS
|
|
B₁ = ωC_mut = 1.194×10⁶ × 9×10⁻¹² = 10.75 μS
|
|
B₂ = ωC_sh = 1.194×10⁶ × 6×10⁻¹² = 7.16 μS
|
|
|
|
Re{Y} = GB₂²/[G² + (B₁+B₂)²]
|
|
= 17.92 × 51.27 / [321.1 + 319.7]
|
|
= 918.8 / 640.8
|
|
= 1.434 μS
|
|
|
|
Im{Y} = B₂[G² + B₁(B₁+B₂)] / [G² + (B₁+B₂)²]
|
|
= 7.16 × [321.1 + 191.7] / 640.8
|
|
= 7.16 × 512.8 / 640.8
|
|
= 5.73 μS
|
|
|
|
Y_spark = 1.434 + j5.73 μS
|
|
```
|
|
|
|
---
|
|
|
|
**Task 4: Predict performance with Thévenin**
|
|
|
|
```
|
|
Convert Y_spark to Z_spark:
|
|
|Y_spark| = √(1.434² + 5.73²) = 5.91 μS
|
|
|Z_spark| = 1/5.91×10⁻⁶ = 169 kΩ
|
|
|
|
φ_Y = atan(5.73/1.434) = 76.0°
|
|
φ_Z = -76.0°
|
|
|
|
Z_spark = 169 kΩ ∠-76.0°
|
|
= 169 × cos(-76°) + j × 169 × sin(-76°)
|
|
= 41 - j164 kΩ
|
|
|
|
Total impedance:
|
|
Z_total = Z_th + Z_spark
|
|
= (105 - j2100) + (41000 - j164000)
|
|
= (41105 - j166100) Ω
|
|
= 41.1 - j166.1 kΩ
|
|
|
|
|Z_total| = √(41.1² + 166.1²) = 171 kΩ
|
|
|
|
Current:
|
|
I = V_th/Z_total = 320 kV / 171 kΩ = 1.87 A
|
|
|
|
Power to spark:
|
|
P_spark = 0.5 × I² × Re{Z_spark}
|
|
= 0.5 × 1.87² × 41000
|
|
= 0.5 × 3.50 × 41000
|
|
= 71.7 kW
|
|
```
|
|
|
|
---
|
|
|
|
**Task 5: Compare to theoretical maximum**
|
|
|
|
```
|
|
For conjugate match: Z_load = Z_th* = 105 + j2100 Ω
|
|
|
|
P_max = 0.5 × |V_th|² / (4 × Re{Z_th})
|
|
= 0.5 × (320×10³)² / (4 × 105)
|
|
= 0.5 × 1.024×10¹¹ / 420
|
|
= 122 MW
|
|
|
|
Actual percentage:
|
|
71.7 kW / 122000 kW = 0.0588%
|
|
|
|
Spark extracts only 0.06% of theoretical maximum!
|
|
|
|
Why such huge difference?
|
|
- Conjugate match needs Z_load = 105 + j2100 Ω (very low R, inductive)
|
|
- Actual spark: Z_spark = 41000 - j164000 Ω (high R, capacitive)
|
|
- Topological constraints prevent achieving conjugate match
|
|
- This is NORMAL for Tesla coils
|
|
- The 71.7 kW is still significant useful power
|
|
```
|
|
|
|
---
|
|
|
|
### Part 4 Final Comprehensive Problem (Partial Solution)
|
|
|
|
**Given:**
|
|
- f = 195 kHz, 2 m target, QCW 10 ms
|
|
- Topload 35 cm, P_input = 120 kW
|
|
- Z_th = 110 - j2300 Ω, V_th = 340 kV
|
|
|
|
---
|
|
|
|
**Part 1: Distributed model setup**
|
|
|
|
```
|
|
Choose n = 10 (good balance accuracy/speed)
|
|
|
|
FEMM geometry (axisymmetric r-z):
|
|
- Toroid: major R=17.5 cm, minor r=5 cm, center z=0
|
|
- Segments: 10 cylinders, each 20 cm long
|
|
Segment 1: r=0.15 cm, z=-5 to -25 cm
|
|
Segment 2: z=-25 to -45 cm
|
|
...
|
|
Segment 10: z=-185 to -205 cm
|
|
- Ground plane: z=-220 cm, r=0 to 400 cm
|
|
- Outer boundary: r=400 cm, z=±300 cm
|
|
|
|
Validation checks after [C] extraction:
|
|
1. Symmetry: C[i,j] = C[j,i] within 0.1%
|
|
2. All diagonal positive
|
|
3. All off-diagonal negative
|
|
4. C_sh_total ≈ 2 pF/ft × 6.56 ft ≈ 13 pF
|
|
(Sum across segments)
|
|
```
|
|
|
|
---
|
|
|
|
**Part 2: Resistance calculation (simplified method)**
|
|
|
|
```
|
|
ω = 2π × 195×10³ = 1.225×10⁶ rad/s
|
|
|
|
Assume FEMM gives C_total[i] = [14, 11, 9, 7.5, 6.5, 5.5, 4.5, 3.5, 2.8, 2.0] pF
|
|
|
|
R[i] = 1/(ω × C_total[i]):
|
|
|
|
R[1] = 1/(1.225×10⁶ × 14×10⁻¹²) = 58.3 kΩ
|
|
R[2] = 1/(1.225×10⁶ × 11×10⁻¹²) = 74.6 kΩ
|
|
R[3] = 92.1 kΩ
|
|
R[4] = 110 kΩ
|
|
R[5] = 127 kΩ
|
|
R[6] = 150 kΩ
|
|
R[7] = 184 kΩ
|
|
R[8] = 236 kΩ
|
|
R[9] = 294 kΩ
|
|
R[10] = 408 kΩ
|
|
|
|
R_total = 1734 kΩ
|
|
|
|
Expected range at 195 kHz for 2m QCW: 30-120 kΩ
|
|
Actual: 1734 kΩ (high, but long spark distributed can be higher)
|
|
|
|
Bounds check: All R[i] between 5 kΩ and 500 kΩ ✓
|
|
Distribution: Monotonically increasing ✓
|
|
```
|
|
|
|
---
|
|
|
|
**Part 3: Performance prediction (abbreviated)**
|
|
|
|
```
|
|
Build SPICE with [C] matrix and R[i] values
|
|
Run AC analysis at 195 kHz
|
|
|
|
Expected results (estimated):
|
|
Z_spark ≈ 600 kΩ ∠-72°
|
|
I ≈ 0.5 A
|
|
P_spark ≈ 40 kW
|
|
|
|
Percentage of theoretical max: <0.1% (typical)
|
|
```
|
|
|
|
---
|
|
|
|
**Part 4: Growth analysis**
|
|
|
|
```
|
|
Power available: 40 kW (from part 3)
|
|
ε = 12 J/m (QCW calibrated)
|
|
Target: L = 2 m, Time: T = 10 ms
|
|
|
|
Energy needed: E = ε × L = 12 × 2 = 24 J
|
|
|
|
Power needed: P = E/T = 24/0.010 = 2.4 kW
|
|
|
|
Available: 40 kW >> 2.4 kW needed ✓
|
|
Power is MORE than sufficient
|
|
|
|
Voltage check:
|
|
V_top = 340 kV (from V_th, approximately)
|
|
κ = 3.2, E_prop = 0.7 MV/m
|
|
E_tip = κ × V_top/L = 3.2 × 340 kV / 2 m
|
|
= 3.2 × 170 kV/m = 544 kV/m = 0.544 MV/m
|
|
|
|
E_tip = 0.544 MV/m < E_prop = 0.7 MV/m ✗
|
|
|
|
Growth is VOLTAGE-LIMITED!
|
|
Cannot reach 2 m with 340 kV
|
|
|
|
Required voltage:
|
|
V_required = E_prop × L / κ = 0.7×10⁶ × 2 / 3.2
|
|
= 437.5 kV
|
|
|
|
Need to ramp to 438 kV to sustain growth to 2 m
|
|
With 340 kV, maximum length ≈ 340/438 × 2 = 1.55 m
|
|
|
|
Conclusion: Voltage limited, not power limited
|
|
Need higher voltage ramp or accept shorter spark
|
|
```
|
|
|
|
---
|
|
|
|
**Part 5: Validation plan**
|
|
|
|
```
|
|
Measurements to take:
|
|
1. Ringdown: f₀, Q₀ (unloaded); f_L, Q_L (loaded)
|
|
→ Extract Y_spark, compare to model
|
|
2. High-speed video: Growth rate dL/dt
|
|
→ Validate power/ε relationship
|
|
3. V_top with E-field probe (calibrated)
|
|
→ Check voltage predictions
|
|
4. Final spark length with ruler/laser
|
|
→ Validate growth model
|
|
|
|
Refinement process:
|
|
1. If measured length > predicted:
|
|
- Reduce ε (more efficient than assumed)
|
|
- Check E_prop (may be lower)
|
|
2. If measured length < predicted:
|
|
- Increase ε
|
|
- Check for branching (wastes energy)
|
|
3. Adjust R distribution if impedance mismatch
|
|
|
|
Expected accuracy:
|
|
- Length: ±30% (good agreement)
|
|
- Power: ±40% (acceptable)
|
|
- Impedance: ±25% (reasonable)
|
|
|
|
Better than factor of 2 on all parameters = success!
|
|
```
|
|
|
|
---
|
|
|
|
## Appendix H: Further Resources
|
|
|
|
### Online Communities
|
|
|
|
**4hv.org Forums**
|
|
- Active Tesla coil community
|
|
- Design sharing and troubleshooting
|
|
- DRSSTC, QCW, SGTC sections
|
|
- Measurement techniques
|
|
|
|
**High Voltage Forum (highvoltageforum.net)**
|
|
- International community
|
|
- Advanced projects
|
|
- Safety discussions
|
|
|
|
### Software Tools
|
|
|
|
**FEMM (femm.info)**
|
|
- Free 2D electromagnetic FEA
|
|
- This framework's primary tool
|
|
- Active development and support
|
|
|
|
**LTSpice (analog.com/ltspice)**
|
|
- Free SPICE simulator
|
|
- Excellent for circuit analysis
|
|
- Large component library
|
|
|
|
**Python Scientific Stack**
|
|
- NumPy: Matrix operations
|
|
- SciPy: Optimization algorithms
|
|
- Matplotlib: Plotting
|
|
- Free and powerful
|
|
|
|
### Books and Papers
|
|
|
|
**Lightning Physics:**
|
|
- Uman, M.A. "The Lightning Discharge" (comprehensive)
|
|
- Rakov & Uman "Lightning: Physics and Effects" (modern)
|
|
|
|
**Plasma Physics:**
|
|
- Chen, F.F. "Introduction to Plasma Physics" (accessible)
|
|
- Raizer, Y.P. "Gas Discharge Physics" (detailed)
|
|
|
|
**High Voltage Engineering:**
|
|
- Kuffel, Zaengl, Kuffel "High Voltage Engineering Fundamentals"
|
|
- Wadhwa, C.L. "High Voltage Engineering"
|
|
|
|
**Tesla Coil Specific:**
|
|
- "The Spark Gap" magazine archives (historical)
|
|
- Tesla coil design guides (various online)
|
|
|
|
### Academic Resources
|
|
|
|
**IEEE Xplore**
|
|
- Search: "spark discharge modeling"
|
|
- "Tesla transformer"
|
|
- "resonant transformer"
|
|
|
|
**arXiv.org**
|
|
- Physics preprints
|
|
- Some Tesla coil research
|
|
|
|
### Safety Resources
|
|
|
|
**ALWAYS prioritize safety:**
|
|
- High voltage safety guidelines
|
|
- Grounding and bonding practices
|
|
- First aid for electrical injuries
|
|
- Equipment safety ratings
|
|
|
|
**Key principle:** If you're not sure, DON'T DO IT.
|
|
|
|
---
|
|
|
|
## Closing Remarks
|
|
|
|
**You now have:**
|
|
- Complete theoretical framework
|
|
- Practical implementation guide
|
|
- Worked examples throughout
|
|
- Troubleshooting resources
|
|
- Validation methodologies
|
|
|
|
**Next steps:**
|
|
1. Start with lumped model (simple coil)
|
|
2. Calibrate ε from one measurement
|
|
3. Predict new operating point
|
|
4. Progress to distributed model
|
|
5. Share results with community
|
|
|
|
**Remember:**
|
|
- All models are approximations
|
|
- Plasma physics has uncertainties
|
|
- ±20-50% agreement is GOOD
|
|
- Document your assumptions
|
|
- Compare to measurements
|
|
- Iterate and improve
|
|
|
|
**Most importantly:**
|
|
- Stay safe
|
|
- Have fun
|
|
- Learn continuously
|
|
- Contribute back to community
|
|
|
|
**This framework is a starting point, not the final word. As you gain experience, you'll develop intuition and may improve upon these methods. That's the goal!**
|
|
|
|
---
|
|
|
|
**END OF APPENDICES**
|
|
|
|
**END OF COMPLETE TESLA COIL SPARK MODELING LESSON PLAN**
|
|
|
|
---
|
|
|
|
**Total lesson plan:**
|
|
- Part 1: ~18,000 tokens (Foundation)
|
|
- Part 2: ~17,500 tokens (Optimization)
|
|
- Part 3: ~17,800 tokens (Growth Physics & FEMM)
|
|
- Part 4: ~17,900 tokens (Distributed Models)
|
|
- Appendices: ~14,500 tokens (Reference)
|
|
- **Grand Total: ~85,700 tokens**
|
|
|
|
**Ready for teaching Tesla coil spark modeling from beginner to advanced!**
|
|
|
|
|