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id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
opt-03	Thévenin Equivalent Method - Extraction	Optimization & Simulation	intermediate	40	[opt-01 fund-08]	[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]	[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

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

10 KiB Raw Blame History

Thévenin Equivalent Method - Extraction

What is a Thévenin Equivalent?

Thévenin's Theorem

Application to Tesla Coils

Step 1: Measuring Z_th (Output Impedance)

Procedure

Physical Meaning of Components

Quality Factor from Z_th

Step 2: Measuring V_th (Open-Circuit Voltage)

Procedure

Important Notes

Worked Example: Extracting Z_th from Simulation

Calculate Z_th

Interpretation

Visual Aid: Thévenin Measurement Setup

Common Pitfalls

Pitfall 1: Removing Tank Components

Pitfall 2: Wrong Frequency

Pitfall 3: Ignoring Phase

Pitfall 4: Using I_base Instead of Port Current

Key Takeaways

Practice

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