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id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
fund-02	The Basic Spark Circuit Model	Fundamentals	beginner	25	[fund-01]	[Understand what capacitance represents physically Distinguish between mutual capacitance (C_mut) and shunt capacitance (C_sh) Learn the empirical 2 pF/foot rule for spark capacitance Draw the correct circuit topology for a Tesla coil spark Identify the topload port as the measurement reference]	[capacitance circuit-topology C_mut C_sh measurement]

The Basic Spark Circuit Model

Introduction

A spark isn't just a resistor - it's a complex structure with multiple electrical properties. Understanding how to model a spark as a circuit with the correct topology is essential for analyzing Tesla coil performance.

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)

Shunt Capacitance and the 2 pF/Foot Rule

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: Estimating C_sh

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.

Why Sparks Have TWO Capacitances

A spark channel is a conductor in space with:

Proximity to the topload → mutual capacitance C_mut
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

Field line visualization:

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: The same spark channel participates in BOTH capacitances! This is why we need a specific circuit topology.

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?

C_mut couples topload voltage to spark
R represents plasma resistance (where power is dissipated)
C_sh provides current return path to ground
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

We'll explore why V_top/I_base is incorrect in a later lesson.

Worked Example: Drawing the Complete Circuit

Given:

Spark is 3 feet long
FEMM analysis gives C_mut = 8 pF (between topload and spark)
Assume R = 100 kΩ
Estimate C_sh using empirical rule

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: Alternative representation showing 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.

Key Takeaways

Capacitance stores energy in electric fields, depends on geometry
C_mut: mutual capacitance between topload and spark
C_sh: shunt capacitance from spark to ground, approximately 2 pF/foot
Both capacitances exist simultaneously on the same conductor
Correct topology: (R || C_mut) in series with C_sh
Topload port: measurement reference between topload and ground
Ground means "remote return path" in this context

Practice

{exercise:fund-ex-02}

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?

Problem 3: 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? (Hint: Consider how C_mut and C_sh combine in the circuit.)

Next Lesson: Admittance Analysis

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