--- id: thevenin-method title: "Thevenin Equivalent Extraction and Impedance Measurement" status: established source_sections: "spark-physics.txt: Part 3 (lines 128-524), Part 11 (lines 753-803)" related_topics: [circuit-topology, power-optimization, coupled-resonance, lumped-model, distributed-model, femm-workflow, equations-and-bounds] key_equations: - "Thevenin impedance Z_th" - "Thevenin voltage V_th" - "Power to load P_load" - "Theoretical maximum power P_max" - "Ringdown Q and conductance extraction" - "Equivalent capacitance from frequency shift" key_terms: - "Thevenin equivalent" - "output impedance" - "open-circuit voltage" - "conjugate match" - "measurement port" - "ringdown method" - "loaded Q" - "Rogowski coil" - "E-field probe" - "VNA" images: - thevenin-measurement-setup.png - impedance-matching-concept.png examples: - thevenin-extraction.md open_questions: - "How much does primary coupling coefficient uncertainty affect Z_th extraction accuracy?" - "Can V_th be measured in situ during spark operation using E-field probes, or only in simulation?" - "What is the best frequency resolution for Z_th(omega) sweeps to capture pole behavior?" - "How does the Thevenin approach extend to time-varying loads (transient spark growth)?" --- # Thevenin Equivalent Extraction and Impedance Measurement This document describes the correct method for characterizing a Tesla coil as a source (Thevenin equivalent) and evaluating its power delivery to arbitrary spark loads. The central message is that naive impedance measurements (V_top/I_base) are fundamentally flawed, and the Thevenin port method provides a rigorous alternative. Three measurement approaches are presented: Thevenin extraction (recommended), direct power measurement, and ringdown analysis. ## 1. Why V_top / I_base Is Wrong ### 1.1 The Common Mistake A tempting approach to measuring spark impedance is to divide the topload voltage by the base current: Z_apparent = V_top / I_base. This is incorrect and produces misleading results. ### 1.2 Physical Reason The base current I_base is the total current flowing into the bottom of the secondary winding. This current includes ALL displacement currents returning to ground from the secondary: 1. **Distributed secondary capacitance to ground:** Every turn of the secondary coil has capacitance to the ground plane, strike ring, and nearby objects. These displacement currents flow through the base. 2. **Strike ring coupling:** If a strike ring is present, capacitive coupling between the secondary and strike ring contributes additional current. 3. **Primary-to-secondary capacitance:** The inter-winding capacitance between primary and secondary contributes displacement current. 4. **Spark current:** The actual current flowing through the spark load (the quantity of interest) is only one component of I_base. Computing V_top/I_base therefore mixes the spark load impedance with all parasitic return paths. The result has no clear physical interpretation and cannot be used for impedance matching analysis. ### 1.3 Quantitative Impact In a typical DRSSTC: - Total I_base might be 2 A (peak) at the operating frequency - Of this, perhaps 0.5-1.0 A is spark current - The remainder is parasitic displacement currents Using I_base overestimates the current through the spark by a factor of 2-4, which underestimates the spark impedance by the same factor. This leads to incorrect R_opt calculations and misleading efficiency estimates. ## 2. The Correct Measurement Port ### 2.1 Port Definition The correct measurement port for spark impedance is the **topload-to-ground** terminal pair, defined as: - **Positive terminal:** The topload surface where the spark physically connects - **Negative terminal (return):** The ground plane / earth / chassis All impedance, admittance, and power calculations for the spark reference this port. ### 2.2 Why This Port The topload is the node where the spark load physically attaches to the Tesla coil circuit. The Thevenin theorem states that any linear circuit, viewed from a single port, can be replaced by a voltage source V_th in series with an impedance Z_th. By defining the port at the topload, we cleanly separate: - **The source:** Everything behind the topload (primary circuit, coupling, secondary winding, parasitic capacitances) is characterized by V_th and Z_th. - **The load:** The spark circuit (C_mut, R, C_sh as described in [[circuit-topology]]) connects at this port. ## 3. Thevenin Equivalent Extraction (Recommended Method) ### 3.1 Overview The Thevenin method characterizes the Tesla coil as a two-terminal source, then evaluates power delivery to any load by simple circuit calculation. This completely decouples coil characterization from load analysis. ### 3.2 Step 1: Measure Z_th (Output Impedance, Drive Off) **Setup:** - Set the primary drive source to AC 0V (effectively short-circuit the voltage source in the primary tank). This is critical: the voltage source is replaced by a short circuit, NOT removed. All tank components (MMC capacitor, primary inductance, damping resistors) remain in the circuit. - Apply a 1V AC test source at the topload-to-ground port. - Measure the resulting current I_test (complex: magnitude and phase). **Calculation:** ``` Z_th = V_test / I_test = 1V / I_test = R_th + j*X_th ``` **Physical meaning:** Z_th is the impedance the spark "sees" looking back into the Tesla coil. It includes the reflected impedance of the entire primary tank circuit through the magnetic coupling, plus the secondary's own impedance (distributed capacitance, winding resistance, etc.). **Practical notes:** - In SPICE, this is straightforward: replace the primary voltage source with a short, add a 1V AC source at the topload node. - Z_th is complex and frequency-dependent. At the operating frequency, it is typically dominated by the reflected primary tank impedance. - R_th (real part) represents all losses in the coil plus the reflected primary resistance. - X_th (imaginary part) is typically capacitive near resonance. ### 3.3 Step 2: Measure V_th (Open-Circuit Voltage, Drive On) **Setup:** - Remove the test source from Step 1. - Restore the primary drive source to its normal operating conditions (full voltage, operating frequency). - Remove the spark load (open-circuit the topload; no spark, no load impedance). - Measure V_th = V(topload), both magnitude and phase. **Calculation:** ``` V_th = V(topload)|_{open circuit, drive on} ``` **Physical meaning:** V_th is the voltage the Tesla coil would produce at the topload if no spark were present. It represents the "driving force" available for spark power. **Practical notes:** - In SPICE, simply run the normal coil simulation without any spark load attached. - V_th depends on drive conditions (bus voltage, pulse width, coupling) and is typically 50-500 kV peak for medium-to-large DRSSTCs. - V_th is the voltage that determines whether the inception field threshold (see [[field-thresholds]]) is met. ### 3.4 Step 3: Calculate Power to Any Load Given Z_th and V_th, the power delivered to any candidate load impedance Z_load is: ``` P_load = 0.5 * |V_th|^2 * Re{Z_load} / |Z_th + Z_load|^2 ``` This is the standard Thevenin power transfer formula with the peak-value convention (factor of 0.5). **For the spark circuit specifically:** Z_load is the impedance of the spark network (C_mut || R in series with C_sh), as derived in [[circuit-topology]]: ``` Z_load = 1/Y_spark = [G + j*(B_1 + B_2)] / [(G + j*B_1) * j*B_2] ``` **Theoretical maximum power (conjugate match sanity check):** If a perfect conjugate match were achievable (Z_load = Z_th*): ``` P_max_conjugate = 0.5 * |V_th|^2 / (4 * R_th) ``` This is an upper bound. The actual spark power will be less because: 1. The spark topology constrains the achievable phase angle (see [[circuit-topology]]). 2. Z_load cannot be freely chosen to equal Z_th* -- it is constrained by the (C_mut, R, C_sh) topology. ### 3.5 Advantages of the Thevenin Method 1. **One-time characterization:** Measure Z_th and V_th once for a given coil geometry and drive setup. Then evaluate any number of spark loads by plugging Z_load into the power formula. 2. **No re-simulation:** Changing spark parameters (R, C_mut, C_sh, spark length) does not require re-simulating the coil. Just recalculate Z_load and use the power formula. 3. **Clean separation:** "Coil behavior" (Z_th, determined by winding geometry, coupling, tank circuit) is separated from "drive conditions" (V_th, determined by bus voltage, pulse timing) and from "load behavior" (Z_load, determined by spark physics). 4. **Design insight:** Z_th reveals the coil's output characteristics independent of any particular spark. A coil with lower R_th can deliver more power; a coil with different X_th may require different spark impedance for optimal matching. ### 3.6 Enhancement: Frequency-Dependent Characterization For the most accurate analysis, measure Z_th(omega) and V_th(omega) over a frequency band of +/-10% around the nominal operating frequency. **Why:** When a spark loads the Tesla coil, the resonant frequency shifts (see [[coupled-resonance]]). The coil may not operate at its nominal frequency. Having Z_th and V_th as functions of frequency allows evaluating power delivery at the actual loaded frequency, not just the design frequency. **Procedure:** - Sweep the AC analysis frequency over the band [0.9*f_0, 1.1*f_0]. - Record Z_th(f) and V_th(f) at each frequency point. - For a given spark load at a given frequency, use the appropriate Z_th(f) and V_th(f). ## 4. Direct Power Measurement (Alternative Method) ### 4.1 Approach Instead of extracting the Thevenin equivalent, directly measure power delivered to the spark in a full coupled simulation: 1. Build the complete SPICE model: primary tank, magnetic coupling, secondary, topload, AND spark load (C_mut || R in series with C_sh). 2. Drive the primary at the operating frequency and amplitude. 3. Run AC analysis. 4. Measure spark power: P = 0.5 * Re{V(topload) * conj(I(spark))}. 5. Step R through a range and record P(R). 6. Find the R that maximizes P. ### 4.2 Critical Detail: Retune for Each R **This is the most commonly overlooked step.** When R changes, the loaded pole frequency shifts. If you measure P at a fixed frequency for each R, you are measuring the combined effect of impedance matching AND detuning. These two effects are conflated and the result is misleading. **Correct procedure:** For each R value: 1. Sweep frequency to find the loaded pole (frequency of maximum |V_top|). 2. Measure P at that loaded pole frequency. 3. Record P(R) at the matched frequency. This gives the true power transfer capability as a function of R, independent of frequency tracking. ### 4.3 Comparison with Thevenin Method | Aspect | Thevenin | Direct | |--------|----------|--------| | Number of simulations | 2 (Z_th + V_th) | Many (one per R value) | | Frequency tracking | Requires separate Z_th(omega) sweep | Naturally included if done correctly | | Physical insight | Separates source from load | Shows only total result | | Re-usability | Characterize once, evaluate many loads | Must re-simulate for each new scenario | | Accuracy | Exact (same circuit equations) | Exact (same circuit equations) | | Complexity | Lower (once setup is understood) | Higher (must retune for each R) | **Recommendation:** Use Thevenin for design and parameter sweeps. Use direct measurement for validation of specific operating points. ## 5. Ringdown Method ### 5.1 Principle An alternative experimental (not just simulation) technique. When a Tesla coil rings down after the drive is removed, the decay rate reveals the total system Q, from which the total conductance (and hence spark conductance) can be extracted. ### 5.2 Parallel RLC Equivalent At the loaded resonance omega_L, the system near the topload looks like a parallel RLC: ``` Q_L = omega_L * C_eq * R_p = R_p / (omega_L * L) ``` where R_p is the equivalent parallel resistance (representing all losses including the spark), C_eq is the equivalent capacitance, and L is the equivalent inductance. **Solving for R_p:** ``` R_p = Q_L / (omega_L * C_eq) [using Q = omega*C*R_p form] R_p = Q_L * omega_L * L [using Q = R_p/(omega*L) form] ``` **Total conductance:** ``` G_total = 1/R_p = omega_L * C_eq / Q_L = 1 / (Q_L * omega_L * L) ``` ### 5.3 Measurement Procedure 1. **Unloaded measurement:** Measure the resonant frequency f_0 and quality factor Q_0 without spark. From geometry or separate measurement, determine C_0 (topload + secondary distributed capacitance). 2. **Loaded measurement:** With spark present, measure the loaded resonant frequency f_L and loaded quality factor Q_L. 3. **Equivalent capacitance:** ``` C_eq = C_0 * (f_0 / f_L)^2 ``` This accounts for the frequency shift caused by the additional spark capacitance. 4. **Capacitance change:** ``` delta_C = C_eq - C_0 ``` 5. **Total conductance (loaded):** ``` G_total = omega_L * C_eq / Q_L ``` 6. **Unloaded conductance:** ``` G_0 = omega_0 * C_0 / Q_0 ``` 7. **Spark admittance:** ``` Y_spark ~ (G_total - G_0) + j * omega_L * delta_C ``` The real part gives the spark's conductance (1/R); the imaginary part gives the net reactive change, which should be consistent with C_mut and C_sh. ### 5.4 Limitations of the Ringdown Method - **Sensitivity to primary coupling:** The primary tank circuit affects the ringdown behavior. If coupling is not well characterized, errors propagate into the extracted Q and hence into Y_spark. - **Transient vs. steady-state:** The ringdown captures the impedance at the moment the drive is removed. If the spark is evolving (growing, cooling), this is a snapshot, not the steady-state value. - **Mode identification:** The Tesla coil has two coupled modes. The ringdown may excite both, and careful analysis is needed to separate them. **The Thevenin port method is more robust** because it operates in the frequency domain and does not require separating coupled mode contributions. ## 6. Direct Physical Measurement ### 6.1 Voltage Measurement: E-Field Probe The topload voltage V_top can be measured using a calibrated E-field probe: - Capacitive divider probe placed near (but not touching) the topload - Must be calibrated for frequency response and geometry - Provides V_top(t) in the time domain; FFT gives V_top(omega) ### 6.2 Current Measurement: Rogowski Coil or CT The spark return current (NOT I_base) can be measured using: - **Rogowski coil** around the spark ground return conductor - **Current transformer (CT)** on the ground return path - Must measure the current flowing through the spark circuit specifically, not the total secondary base current **Critical:** The current sensor must be placed to capture only the spark-associated current, not all displacement currents. This typically means placing it on a dedicated ground return wire from the spark target or strike object. ### 6.3 VNA (Vector Network Analyzer) For low-level characterization without spark: - Capacitive pickup at topload - VNA drives through a coupling network - Measures Z_th(omega) across a frequency band - Cannot measure V_th directly (requires active drive) - Useful for validating SPICE models before spark testing ### 6.4 Calculating Impedance from Measurements With V_top and I_spark measured: ``` Y_measured = I_spark / V_top Z_measured = V_top / I_spark ``` From Y_measured, extract R by fitting to the circuit model (see [[circuit-topology]]): - Known: omega, C_mut (from FEMM), C_sh (from FEMM or estimated from length) - Unknown: R - Solve: Y(R) = Y_measured for R ## 7. Practical Workflow ### 7.1 Recommended Sequence 1. **Build SPICE model** of complete Tesla coil (primary tank, coupling, secondary, topload). 2. **Extract Z_th** (Step 1: short drive, apply test source at topload). 3. **Extract V_th** (Step 2: normal drive, open topload). 4. **Compute power curves:** For a range of spark lengths (and corresponding C_mut, C_sh from [[femm-workflow]]), calculate P_load(R) for each length. 5. **Identify R_opt_power** for each length (should match [[power-optimization]] formula). 6. **Validate:** Check that P_max is consistent with known coil performance. 7. **Frequency sweep (optional):** Repeat Steps 1-2 across +/-10% band for frequency tracking analysis. ### 7.2 Common Pitfalls - **Forgetting to short the drive source** in Step 1 (leaving it open gives wrong Z_th). - **Removing tank components** in Step 1 (they must remain; they are part of the source impedance). - **Using I_base instead of I_spark** in direct measurements. - **Comparing R values at fixed frequency** without retuning (see [[coupled-resonance]]). - **Ignoring the 0.5 factor** in power (peak-value convention). ## 8. Connection to Other Topics ### Key Relationships - **Derives from:** Linear circuit theory (Thevenin's theorem) applied to the Tesla coil system - **Requires:** [[circuit-topology]] (defines the spark load Z_load that connects to the Thevenin port) - **Validates:** [[power-optimization]] (P_load(R) curve from Thevenin analysis should peak at R_opt_power) - **Interacts with:** [[coupled-resonance]] (frequency-dependent Z_th captures pole splitting and detuning) - **Feeds into:** [[lumped-model]] and [[distributed-model]] (Thevenin source drives the spark circuit model) - **Complements:** [[femm-workflow]] (FEMM provides C_mut, C_sh; Thevenin provides source characterization) ### Summary of Key Results 1. V_top/I_base is wrong because I_base includes parasitic displacement currents. 2. The correct measurement port is topload-to-ground. 3. Thevenin extraction: Z_th from drive-off test, V_th from drive-on open-circuit. 4. P_load = 0.5 * |V_th|^2 * Re{Z_load} / |Z_th + Z_load|^2. 5. P_max (conjugate match) = 0.5 * |V_th|^2 / (4*R_th) is an upper bound. 6. Ringdown method extracts Y_spark from Q and frequency measurements but is sensitive to coupling. 7. Direct measurement requires E-field probe (voltage) and Rogowski/CT on spark return (current). 8. The Thevenin method is the most robust and reusable approach.