plasma-expert/spark-lessons/lessons/01-fundamentals

Part 1: Fundamentals

Overview

This section provides the foundational knowledge for Tesla coil spark modeling. You'll learn the circuit theory, analysis techniques, and key concepts needed to understand and predict spark behavior.

Lessons

1. Introduction to Tesla Coil Spark Modeling (20 min)

   • AC circuit fundamentals review
   • Peak vs RMS values
   • Complex numbers and phasors
   • Power calculations with peak phasors

2. The Basic Spark Circuit Model (25 min)

   • Physical meaning of capacitance
   • Mutual capacitance (C_mut) vs shunt capacitance (C_sh)
   • The 2 pF/foot empirical rule
   • Correct circuit topology: (R || C_mut) in series with C_sh

3. Admittance Analysis (30 min)

   • Why use admittance for parallel circuits
   • Deriving the total admittance formula
   • Calculating Re{Y} and Im{Y}
   • Converting between Y and Z

4. Phase Angles and Their Meaning (20 min)

   • Impedance phase φ_Z vs admittance phase φ_Y
   • Physical interpretation of phase angles
   • The "famous -45°" and why it's special
   • Typical spark phase angles: -55° to -75°

5. The Topological Phase Constraint (25 min)

   • What is a topological constraint?
   • Deriving φ_Z,min = -atan(2√[r(1+r)])
   • The critical ratio r = 0.207
   • Why -45° is usually impossible

6. Why Not -45 Degrees? (15 min)

   • Historical origin of the -45° target
   • Why it's often impossible for Tesla coils
   • R_opt_phase vs R_opt_power
   • What to target instead

7. The Measurement Port (20 min)

   • Understanding displacement current
   • Why V_top/I_base gives wrong impedance
   • Multiple current paths in a Tesla coil
   • Correct measurement methods

8. Review and Integration (45 min)

   • Complete concepts checklist
   • Integration exercise combining all topics
   • Checkpoint quiz
   • Preview of Part 2

Total Time

Approximately 3-4 hours for complete mastery

Learning Outcomes

After completing Part 1, you will be able to:

• Use peak values and phasor notation correctly
• Model a spark with proper circuit topology
• Calculate impedance using admittance formulas
• Understand phase angle constraints and their physical meaning
• Recognize why -45° is rarely achievable
• Measure spark impedance correctly
• Avoid common measurement pitfalls
• Apply integrated circuit analysis to real Tesla coil scenarios

Prerequisites

• Basic algebra and trigonometry
• Familiarity with sine waves and AC circuits (helpful but not required)
• Scientific calculator or Python/MATLAB for calculations

Key Formulas

Admittance:

Re{Y} = GB₂² / [G² + (B₁ + B₂)²]
Im{Y} = B₂[G² + B₁(B₁ + B₂)] / [G² + (B₁ + B₂)²]
where G = 1/R, B₁ = ωC_mut, B₂ = ωC_sh

Topological constraint:

φ_Z,min = -atan(2√[r(1 + r)])
where r = C_mut/C_sh

Empirical rule:

C_sh ≈ 2 pF/foot

Power:

P = 0.5 × Re{V × I*}

Image Placeholders

The following images should be created for the assets folder:

1. field-lines-capacitances.png - C_mut and C_sh field lines
2. geometry-to-circuit.png - 3D geometry to circuit schematic
3. complex-plane-admittance.png - Y and Z on complex planes
4. phase-angle-visualization.png - Phase angles on impedance plane
5. phase-constraint-graph.png - φ_Z,min vs r graph
6. current-paths-diagram.png - Multiple current paths in Tesla coil

Next Steps

After mastering Part 1, proceed to:

Part 2: Optimization and Power Transfer

Topics include:

• R_opt_power and R_opt_phase derivations
• Thévenin equivalent method
• The "hungry streamer" self-optimization
• Q measurements and ringdown analysis