--- id: opt-04 title: "Using the Thévenin Equivalent - Power Calculations" section: "Optimization & Simulation" difficulty: "intermediate" estimated_time: 45 prerequisites: ["opt-03", "opt-01"] objectives: - Calculate load voltage and current using Thévenin equivalent - Compute power delivered to arbitrary loads - Determine maximum theoretical power (conjugate match) - Understand why conjugate match is usually unachievable tags: ["thevenin", "power-calculation", "impedance-matching", "circuit-analysis"] --- # Using the Thévenin Equivalent - Power Calculations Now that we've extracted the Thévenin equivalent (V_th and Z_th), we can use it to predict coil performance with any load without re-running full simulations. This lesson shows how to perform these calculations and interpret the results. ## Predicting Behavior with Any Load Once you have V_th and Z_th, the Tesla coil looks like this simple circuit: ``` ┌────┐ │V_th├───[Z_th]───┬─── Output └────┘ │ [Z_load] │ GND ``` This is just a voltage divider! We can apply basic circuit analysis. ### Voltage Across Load Using voltage divider rule: ``` V_load = V_th × [Z_load / (Z_th + Z_load)] ``` **Complex arithmetic:** Both numerator and denominator are complex numbers, so you need to handle magnitude and phase carefully. ### Current Through Load Using Ohm's law on the series circuit: ``` I = V_th / (Z_th + Z_load) ``` This current flows through both Z_th and Z_load since they're in series. ### Power Delivered to Load Power dissipated in the load (real power only): ``` P_load = 0.5 × |I|² × Re{Z_load} ``` Or equivalently: ``` P_load = 0.5 × Re{V_load × I*} ``` where I* is the complex conjugate of I. **Direct formula combining everything:** ``` P_load = 0.5 × |V_th|² × Re{Z_load} / |Z_th + Z_load|² ``` This formula is gold! It lets you sweep different Z_load values and calculate power without any additional simulation. ## Step-by-Step Calculation Process ### Given Information - V_th (complex voltage phasor) - Z_th = R_th + jX_th (complex impedance) - Z_load = R_load + jX_load (spark impedance from model) ### Step 1: Calculate Total Impedance ``` Z_total = Z_th + Z_load = (R_th + R_load) + j(X_th + X_load) R_total = R_th + R_load X_total = X_th + X_load |Z_total| = √(R_total² + X_total²) ``` ### Step 2: Calculate Current ``` I = V_th / Z_total |I| = |V_th| / |Z_total| φ_I = φ_V_th - φ_Z_total ``` where φ_Z_total = atan(X_total / R_total) ### Step 3: Calculate Load Voltage ``` V_load = I × Z_load |V_load| = |I| × |Z_load| φ_V_load = φ_I + φ_Z_load ``` Or use voltage divider directly (often simpler): ``` |V_load| = |V_th| × |Z_load| / |Z_total| ``` ### Step 4: Calculate Power in Load ``` P_load = 0.5 × |I|² × R_load P_load = 0.5 × |I|² × Re{Z_load} ``` The factor of 0.5 accounts for peak phasor to RMS conversion in AC power. ## Worked Example: Complete Thévenin Analysis **Given:** - Z_th = 114 - j2424 Ω (from previous lesson) - V_th = 350 kV ∠0° (measured with drive on, no load) - Spark load: Z_spark = 60 kΩ - j160 kΩ (from lumped model) **Find:** (a) Current through spark (b) Voltage across spark (c) Power dissipated in spark (d) Theoretical maximum power (conjugate match) ### Part (a): Current Through Spark **Calculate total impedance:** ``` Z_total = Z_th + Z_spark = (114 - j2424) + (60000 - j160000) = (60114 - j162424) Ω R_total = 60114 Ω X_total = -162424 Ω |Z_total| = √(60114² + 162424²) = √(3.614×10⁹ + 2.638×10¹⁰) = √(3.000×10¹⁰) = 173.2 kΩ ``` **Calculate current:** ``` I = V_th / Z_total |I| = 350 kV / 173.2 kΩ = 2.02 A peak ``` ### Part (b): Voltage Across Spark **Method 1: Voltage divider** ``` |Z_spark| = √(60000² + 160000²) = √(3.6×10⁹ + 2.56×10¹⁰) = √(2.92×10¹⁰) = 171 kΩ |V_spark| = |V_th| × |Z_spark| / |Z_total| = 350 kV × (171 kΩ / 173.2 kΩ) = 350 kV × 0.987 = 345 kV ``` **Method 2: Using current** ``` |V_spark| = |I| × |Z_spark| = 2.02 A × 171 kΩ = 345 kV ``` **Observation:** Most voltage appears across the spark! This makes sense because Z_spark >> Z_th. ### Part (c): Power in Spark ``` P_spark = 0.5 × |I|² × Re{Z_spark} = 0.5 × (2.02)² × 60000 = 0.5 × 4.08 × 60000 = 122 kW ``` This is the real power dissipated in heating, ionization, radiation, and sound in the spark. ### Part (d): Theoretical Maximum Power The maximum power transfer theorem states that power is maximized when the load impedance is the **complex conjugate** of the source impedance. **Conjugate match condition:** ``` Z_load = Z_th* (complex conjugate) If Z_th = R_th + jX_th Then Z_load = R_th - jX_th For our case: Z_th = 114 - j2424 Ω Z_load_optimal = 114 + j2424 Ω ``` **Why this maximizes power:** - Reactive components cancel: Z_total = Z_th + Z_th* = 2R_th (purely real) - No reactive power circulation - All delivered power is real **Maximum power formula:** ``` P_max = |V_th|² / (8 × R_th) ``` **Calculate:** ``` P_max = (350×10³)² / (8 × 114) = 1.225×10¹¹ / 912 = 134.3 MW ``` **Wait, this seems enormous!** Let's double-check: ``` With Z_load = 114 + j2424 Ω: Z_total = (114 - j2424) + (114 + j2424) = 228 Ω (purely resistive!) I = 350 kV / 228 Ω = 1535 A P = 0.5 × (1535)² × 114 = 134.3 MW ✓ ``` ### Part (e): Reality Check - Why Such a Huge Difference? **Actual spark power:** 122 kW **Theoretical maximum:** 134.3 MW **Efficiency:** 122 / 134,300 = 0.09% of theoretical maximum **Why such a huge discrepancy?** 1. **Conjugate match requires Z_load = 114 + j2424 Ω** - This means R_load = 114 Ω (extremely low!) - This means X_load = +2424 Ω (inductive, not capacitive) 2. **Actual spark: Z_spark = 60 kΩ - j160 kΩ** - R_spark = 60 kΩ (525× too high!) - X_spark = -160 kΩ (capacitive, wrong sign, 66× too large) 3. **Topological constraints prevent achieving conjugate match:** - Spark structure (R||C_mut in series with C_sh) is inherently capacitive - Cannot produce positive (inductive) reactance - Cannot achieve R_load as low as 114 Ω with realistic plasma **This is normal for Tesla coils!** The impedance mismatch is fundamental to the physics of spark discharges. We cannot achieve conjugate match in practice. ## Understanding Efficiency ### What Does 0.09% Mean? It does NOT mean the coil is "inefficient" in the usual sense. Rather: - The coil has very low output impedance (114 Ω) - The spark has very high impedance (171 kΩ) - This is a 1500:1 impedance mismatch - The voltage divider heavily favors the spark (good!) - Most voltage appears at the spark, but current is limited ### Voltage Transfer Efficiency ``` Voltage across spark / Total voltage: 345 kV / 350 kV = 98.6% ``` We achieve excellent voltage transfer! This is what matters for spark length (field at tip). ### Why Not Match Impedances? **In conventional circuits:** Match impedances for maximum power transfer **In Tesla coils:** We WANT high spark impedance because: - High voltage at spark tip drives field - High resistance means controlled current (safety) - Mismatch is unavoidable due to plasma physics - Optimization focuses on maximizing power given the constraints ## Practical Use: Sweeping Spark Parameters The real power of Thévenin analysis is rapid parameter sweeps: **Given:** V_th = 350 kV, Z_th = 114 - j2424 Ω **Sweep:** Spark resistance R from 10 kΩ to 200 kΩ **For each R value:** 1. Construct Z_spark from R and capacitances (using lumped model) 2. Calculate Z_total = Z_th + Z_spark 3. Calculate I = V_th / Z_total 4. Calculate P = 0.5 × |I|² × R 5. Plot P vs R **Result:** You find P_max at R ≈ R_opt_power without any new simulations! ## When Thévenin Analysis Fails ### Nonlinearity **Assumption:** Coil behaves linearly (impedances don't change with voltage/current) **Breaks down when:** - Magnetic cores saturate - Component heating changes parameters - Very large sparks significantly load the coil **Solution:** Iterate - use results to update model, re-extract Thévenin ### Frequency Dependence **Assumption:** Operating at a single frequency **Breaks down when:** - Spark loading shifts resonant frequency - Comparing different loads at fixed frequency (detuning varies) **Solution:** Extract Z_th(ω) and V_th(ω), account for frequency shift (next lessons) ### Coupled Modes **Assumption:** Single-mode operation **Breaks down when:** - Operating between two coupled poles - Mode hopping as spark changes loading **Solution:** Full coupled-mode analysis or stay clearly in one mode ## Key Takeaways - **Thévenin circuit:** Simple series combination of V_th and Z_th - **Load voltage:** V_load = V_th × Z_load/(Z_th + Z_load) - **Load current:** I = V_th / (Z_th + Z_load) - **Load power:** P = 0.5 × |I|² × Re{Z_load} or P = 0.5 × |V_th|² × Re{Z_load}/|Z_th + Z_load|² - **Maximum power:** Requires conjugate match Z_load = Z_th* - **P_max = |V_th|²/(8R_th)** but usually unachievable - **Tesla coils operate far from conjugate match** due to physics constraints - **High voltage transfer efficiency** matters more than impedance matching - **Parameter sweeps** become trivial with Thévenin equivalent ## Practice {exercise:opt-ex-04} **Problem 1:** Given Z_th = 95 - j1850 Ω, V_th = 280 kV, and a spark model with Z_spark = 50 kΩ - j140 kΩ: (a) Calculate total impedance (b) Calculate current through spark (c) Calculate power delivered to spark (d) Calculate theoretical maximum power (conjugate match) (e) What percentage of theoretical maximum is achieved? **Problem 2:** A load Z_load = 200 + j200 Ω is connected to a coil with Z_th = 100 - j2000 Ω and V_th = 300 kV. (a) Calculate the load voltage (b) Calculate power delivered (c) Is this load inductive or capacitive? (d) Is this load closer to conjugate match than a typical spark? **Problem 3:** For Z_th = 120 - j2200 Ω: (a) What load impedance gives conjugate match? (b) Calculate P_max if V_th = 400 kV (c) If actual spark has R = 70 kΩ, X = -180 kΩ, calculate actual power (d) Calculate the power transfer efficiency ratio **Problem 4:** A coil has V_th = 350 kV and Z_th = 110 - j2500 Ω. You want to deliver 100 kW to a purely resistive load. What resistance value is required? (Hint: Set P = 100 kW in power formula and solve for R) **Problem 5:** Explain physically why Tesla coils operate so far from conjugate match. Why can't we just add inductance to the spark to cancel its capacitive reactance? --- **Next Lesson:** [Direct Power Measurement Method](05-direct-measurement.md)