---
id: opt-03
title: "Thévenin Equivalent Method - Extraction"
section: "Optimization & Simulation"
difficulty: "intermediate"
estimated_time: 40
prerequisites: ["opt-01", "fund-08"]
objectives:
  - Understand Thévenin's theorem applied to Tesla coils
  - Extract output impedance Z_th through test measurements
  - Extract open-circuit voltage V_th
  - Interpret Z_th components physically
tags: ["thevenin", "impedance-measurement", "circuit-analysis", "simulation"]
---

# Thévenin Equivalent Method - Extraction

The Thévenin equivalent method is a powerful technique that allows us to characterize a Tesla coil **once** and then predict its behavior with **any load** without re-running full simulations. This dramatically simplifies optimization and design work.

## What is a Thévenin Equivalent?

### Thévenin's Theorem

**Statement:** Any linear two-terminal network can be replaced by:
- A voltage source **V_th** (the open-circuit voltage)
- In series with an impedance **Z_th** (the output impedance)

```
┌─────────────┐              ┌────┐
│   Complex   │              │V_th├───[Z_th]───o Output
│   Network   │──o Output ≡  └────┘            |
│             │  |                             GND
└─────────────┘ GND
```

**Key advantage:** The Thévenin equivalent completely characterizes the network's behavior at the output terminals. Once extracted, you can predict performance with any load by simple circuit analysis.

### Application to Tesla Coils

For a Tesla coil, the "complex network" includes:
- Primary tank circuit (L_primary, C_MMC)
- Primary drive (inverter or spark gap)
- Magnetic coupling
- Secondary coil with all its distributed properties
- Topload capacitance
- All parasitic elements

The **output port** is the topload-to-ground connection, where we connect the spark load.

**Thévenin parameters:**
- **V_th:** The voltage that appears at the topload with no spark (open circuit)
- **Z_th:** The impedance "looking into" the topload terminal with the drive turned off

## Step 1: Measuring Z_th (Output Impedance)

The output impedance tells us how the coil "pushes back" against a load. It represents all the losses and reactive elements as seen from the topload.

### Procedure

**Step 1.1: Turn OFF primary drive**
- Set drive voltage to 0V (AC short circuit)
- Keep all tank components in place (MMC, L_primary, damping resistors)
- The tank circuit is still present, just not energized
- This "deactivates" all voltage sources in the network

**Step 1.2: Apply test source**
- Apply 1V AC at operating frequency to topload-to-ground port
- Use small-signal AC source (in simulation or actual test equipment)
- Frequency should match your intended operating frequency

**Step 1.3: Measure current**
```
I_test = current flowing into topload port with 1V applied
```

In SPICE/simulation:
- Place 1V AC source between topload and ground
- Run AC analysis at operating frequency
- Read current magnitude and phase

**Step 1.4: Calculate Z_th**
```
Z_th = V_test / I_test = 1V / I_test

Z_th = R_th + jX_th (complex impedance)
```

### Physical Meaning of Components

**R_th (Resistance):**
- Secondary winding resistance (copper losses)
- Dielectric losses in the coil form
- Damping resistors in primary circuit
- Core losses (if any)
- Typical: 10-100 Ω for medium coils at RF frequencies

**X_th (Reactance):**
- Usually negative (capacitive) due to topload
- Includes reflected impedances from coupling
- May include inductive component from coil
- Typical: -500 to -3000 Ω (strongly capacitive)

**Magnitude |Z_th|:**
- Total opposition to current
- Typical: 500-3000 Ω for Tesla coils at 100-400 kHz

**Phase φ_Z_th:**
- Usually -85° to -88° (nearly pure capacitive)
- Small R_th compared to |X_th| gives phase close to -90°

### Quality Factor from Z_th

The quality factor Q represents how "lossy" the coil is:

```
Q = |X_th| / R_th

Higher Q → lower losses → more efficient
```

Typical values:
- Small coils: Q = 50-150
- Medium coils: Q = 100-300
- Large coils: Q = 200-500

## Step 2: Measuring V_th (Open-Circuit Voltage)

The open-circuit voltage tells us what voltage the coil produces with no load attached.

### Procedure

**Step 2.1: Remove load**
- Disconnect spark (or ensure spark won't break out)
- Topload is in open-circuit condition
- No current flows to external loads

**Step 2.2: Turn ON primary drive**
- Normal operating frequency and amplitude
- Drive the coil exactly as you would for spark operation
- Primary current flows, secondary is excited

**Step 2.3: Measure topload voltage**
```
V_th = V(topload) with no load

Record both magnitude and phase (complex phasor)
```

In simulation:
- Run AC analysis with drive on
- Read voltage at topload node
- This is your V_th

In practice:
- Use high-impedance voltage probe
- Capacitive divider for high voltages
- Or measure primary current and use coupling theory

**Typical values:**
- Small coils (few hundred watts): V_th = 100-300 kV
- Medium coils (1-3 kW): V_th = 200-500 kV
- Large coils (5-10+ kW): V_th = 500 kV - 1 MV+

### Important Notes

**Frequency dependence:**
- Both Z_th and V_th depend on frequency
- Extract at your operating frequency
- Near resonance, small frequency changes cause large V_th changes

**Linearity assumption:**
- Thévenin theorem assumes linear network
- Valid for small-signal analysis
- For large sparks, nonlinear effects may require iterative refinement

**Enhancement for frequency tracking:**
- Measure Z_th(ω) and V_th(ω) over frequency band (±10%)
- Accounts for resonance shift when spark loads the coil
- Enables accurate predictions with different loads

## Worked Example: Extracting Z_th from Simulation

**Simulation setup:**
- DRSSTC at f = 185 kHz
- Primary drive set to 0V (AC short)
- 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 form**
```
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:**
- Represents all resistive losses in the system
- Includes secondary winding resistance
- Includes reflected primary losses
- This is the "cost" of extracting power from the coil

**X_th = -2424 Ω:**
- Strongly capacitive (negative reactance)
- Topload capacitance dominates
- At 185 kHz: C_equivalent ≈ 1/(ω|X_th|) ≈ 35 pF

**Phase ≈ -87°:**
- Nearly pure capacitor (ideal would be -90°)
- Small resistive component (R_th << |X_th|)
- Typical for well-designed Tesla coils

**Quality factor:**
```
Q = |X_th| / R_th = 2424 / 114 ≈ 21
```

This Q is relatively low, likely because:
- Measurement includes all system damping
- Primary circuit losses are reflected
- This is the "loaded" Q of the coupled system

## Visual Aid: Thévenin Measurement Setup

![Thévenin Extraction Setup](assets/thevenin-extraction.png)

*Image shows comparison between:*
- *Left: Full Tesla coil circuit (complex, many components)*
- *Right: Thévenin equivalent (simple: V_th in series with Z_th)*
- *Bottom: Measurement configuration for Z_th extraction*

**Key elements:**
- Primary drive: OFF (0V) for Z_th measurement
- Test source: 1V AC at topload for Z_th
- All tank components remain in circuit
- Ammeter measures test current I_test
- Calculation: Z_th = 1V / I_test

## Common Pitfalls

### Pitfall 1: Removing Tank Components

**Wrong:** Disconnecting C_MMC or shorting L_primary

**Right:** Keep all components, just set drive to 0V

**Why:** The tank circuit affects the output impedance. Removing components gives incorrect Z_th.

### Pitfall 2: Wrong Frequency

**Wrong:** Extracting Z_th at one frequency, using at another

**Right:** Extract at operating frequency, or measure Z_th(ω) over range

**Why:** Impedance is highly frequency-dependent near resonance

### Pitfall 3: Ignoring Phase

**Wrong:** Using only |Z_th| without phase information

**Right:** Keep full complex impedance Z_th = R_th + jX_th

**Why:** Phase affects power calculations and matching

### Pitfall 4: Using I_base Instead of Port Current

**Wrong:** Measuring current at secondary base for Z_th test

**Right:** Measure current through test source at topload port

**Why:** Base current includes displacement currents (see Module 2.4)

## Key Takeaways

- **Thévenin equivalent** reduces complex coil to simple V_th and Z_th
- **Z_th extraction:** Drive OFF, apply 1V test, measure current, Z_th = 1V/I_test
- **V_th extraction:** Drive ON, no load, measure topload voltage
- **Z_th components:** R_th (losses), X_th (reactance, usually capacitive)
- **Typical values:** R_th = 10-100 Ω, X_th = -500 to -3000 Ω, |Z_th| = 500-3000 Ω
- **Quality factor:** Q = |X_th|/R_th indicates coil efficiency
- **Frequency matters:** Extract at operating frequency or measure Z_th(ω)

## Practice

{exercise:opt-ex-03}

**Problem 1:** A test measurement gives I_test = 0.00035 ∠82° A for V_test = 1 ∠0° V at f = 200 kHz. Calculate:
(a) Z_th in polar form
(b) Z_th in rectangular form (R_th + jX_th)
(c) Quality factor Q

**Problem 2:** If Z_th = 85 - j1800 Ω, what is the equivalent capacitance at f = 180 kHz?

**Problem 3:** A coil has Z_th = 120 - j2100 Ω. Calculate:
(a) Impedance magnitude and phase
(b) Quality factor
(c) Would you describe this as "high Q" or "low Q"?

**Problem 4:** Explain why we short the drive voltage source (set to 0V) when measuring Z_th, but keep all passive components in place.

**Problem 5:** Two coils have the same |Z_th| = 2000 Ω but different phases: Coil A has φ = -88°, Coil B has φ = -75°. Which coil has lower losses (higher Q)? Calculate Q for both.

---
**Next Lesson:** [Thévenin Calculations - Using the Equivalent](04-thevenin-calculations.md)