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id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
fund-07	The Measurement Port and Why V_top/I_base is Wrong	Fundamentals	intermediate	20	[fund-01 fund-02]	[Understand what displacement current is and why it matters Recognize why V_top/I_base gives incorrect impedance Identify all current paths in a Tesla coil system Learn the correct measurement port definition Calculate power using the correct method]	[measurement displacement-current power troubleshooting]

The Measurement Port and Why V_top/I_base is Wrong

Introduction

One of the most common mistakes in Tesla coil analysis is using V_top/I_base to calculate spark impedance. This seems logical - measure the voltage at the top and the current at the base - but it gives completely wrong results. This lesson explains why and shows the correct approach.

The Displacement Current Problem

What is Displacement Current?

Displacement current flows through capacitances, not through physical conductors. It's given by:

I_displacement = jωC × V

Key insight: At AC, capacitors conduct current even though no charge physically crosses the dielectric!

For Tesla coils:

Every turn of the secondary has capacitance to ground
Higher frequency → larger displacement current (proportional to ω)
These currents return to ground through the secondary base

Multiple Current Paths in a Tesla Coil

A Tesla coil has many current paths returning to ground:

1. Spark current (what we want to measure)

I_spark: From topload → through spark → remote ground → back to secondary base

2. Displacement currents along secondary

I_displacement: From each turn → through C_turn_to_ground → to ground → base
Sum of all displacement currents: I_displacement = Σ(jωC_turn × V_turn)

3. Primary-secondary coupling

I_coupling: Displacement current through C_ps (primary-to-secondary capacitance)
Part of transformer action

4. Environmental coupling

I_environment: Displacement currents to nearby objects, walls, strike ring
Any grounded conductor near the secondary

Total current at secondary base:

I_base = I_spark + I_displacement + I_coupling + I_environment

The problem: Only I_spark goes through the spark! The other currents are parasitic paths that don't tell us about spark behavior.

Why V_top/I_base is Wrong

Z_apparent = V_top / I_base

But I_base >> I_spark (often 3-5× larger!)

Therefore: Z_apparent << Z_spark (impedance appears much lower than actual)

Consequences:

Underestimate impedance: Think load is more resistive than it is
Overestimate power: Calculate far too much power to spark
Wrong optimization: Make decisions based on incorrect data
Model mismatch: Can't reconcile measurements with theory

Diagram description:

RED path: Spark current (I_spark) - the one we want
BLUE paths: Displacement currents along secondary (I_displacement)
GREEN path: Primary-secondary coupling current (I_coupling)
YELLOW paths: Environmental coupling currents (I_environment)
At base: All paths converge: I_base = sum of all currents

Key insight box: "I_base ≠ I_spark! Cannot use V_top/I_base for spark impedance!"

The Correct Measurement Port

Definition: The topload port is the two-terminal reference between topload and ground.

Port definition:
  Terminal 1: Topload (high voltage)
  Terminal 2: Ground reference (0V)

Correct impedance:

Z_spark = V_top / I_spark

where I_spark is the current ONLY through the spark path

Correct power:

P = 0.5 × Re{V_top × I_spark*}
P = 0.5 × |V_top| × |I_spark| × cos(φ_Z)

Methods to Measure I_spark Correctly

Method 1: Separate return path measurement

Run spark ground return through isolated conductor
Measure current with Rogowski coil or current transformer
Only captures I_spark, excludes parasitic currents

Method 2: Circuit modeling

Know V_top (measure with voltage probe/antenna)
Calculate I_spark from circuit model using component values
Use admittance formulas from Lesson 3

Method 3: Thévenin extraction

Characterize coil as Thévenin equivalent (covered in Part 2)
Predict load current from Z_th and V_th
Most accurate for design work

Worked Example: 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%!

Impedance error:

Z_apparent = 60 kΩ (wrong)
Z_spark = 200 kΩ (correct)

Ratio: 200/60 = 3.3× (impedance underestimated)

Why the same ratio? Because I_base/I_spark = 5/1.5 = 3.3× - the displacement currents are 3.3× larger than the spark current in this example!

Why Displacement Current Increases with Frequency

From the capacitor current equation:

I_C = jωC × V

|I_C| = ω × C × |V| = 2πf × C × |V|

Implication: If frequency doubles, displacement current doubles!

For Tesla coils:

Higher frequency operation → larger displacement currents
I_base becomes increasingly dominated by parasitics
V_top/I_base becomes even more wrong at high frequency
200 kHz vs 400 kHz: displacement current 2× larger at 400 kHz

This is why measurement port definition is critical for comparison across different coils.

Common Symptoms of Using I_base

If you're using I_base incorrectly, you'll see:

Impedance too low: Calculate 30-60 kΩ when should be 150-250 kΩ
Power too high: Predict hundreds of kW when actual is tens of kW
Can't match models: Circuit simulations disagree with "measurements"
Phase angle confusion: Measured phase doesn't match expected
Efficiency paradox: Calculate >100% efficiency (impossible!)

If you see these symptoms, check your measurement method!

Key Takeaways

I_base includes multiple current paths: spark + displacement + coupling + environment
Displacement current: I = jωC×V, proportional to frequency
V_top/I_base is wrong: Gives impedance too low, power too high
Correct port: Topload-to-ground with I_spark only
Typical error: 3-5× underestimate of impedance
Frequency dependence: Displacement current ∝ ω, problem worse at high frequency

Practice

{exercise:fund-ex-07}

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 (assume φ_Z = -72°).

Problem 2: Explain why displacement current is proportional to frequency (ω). If frequency doubles from 200 kHz to 400 kHz, 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? What fraction of I_base is parasitic displacement current?

Problem 4: A coil operates at 300 kV with Z_spark = 200 kΩ, φ_Z = -68°. Calculate the correct spark power. If someone incorrectly uses I_base = 4 A instead of the correct I_spark, what power would they calculate? What is the percentage error?

Next Lesson: Review and Exercises

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