# 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](01-introduction.md)** (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](02-basic-circuit-model.md)** (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](03-admittance-analysis.md)** (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](04-phase-angles.md)** (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](05-phase-constraint.md)** (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?](06-why-not-45-degrees.md)** (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](07-measurement-port.md)** (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](08-review-exercises.md)** (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](../02-optimization/README.md)**

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