---
id: opt-05
title: "Direct Power Measurement Method"
section: "Optimization & Simulation"
difficulty: "intermediate"
estimated_time: 25
prerequisites: ["opt-04", "opt-01"]
objectives:
  - Understand the direct measurement alternative to Thévenin
  - Set up simulations for direct power measurement
  - Extract spark resistance through power optimization
  - Compare advantages and disadvantages of each method
tags: ["power-measurement", "simulation", "optimization", "methodology"]
---

# Direct Power Measurement Method

While the Thévenin equivalent method is powerful and elegant, there's an alternative approach: directly measure power delivered to the spark in a full simulation. Each method has advantages and trade-offs.

## The Direct Measurement Approach

### Concept

Instead of extracting a simplified equivalent circuit, keep the **full coupled model** with the spark load present and directly measure power flow.

**Setup:**
1. Build complete simulation (primary, secondary, coupling, spark load)
2. Drive primary at operating frequency and amplitude
3. Run AC analysis (or transient with post-processing)
4. Measure power dissipated in spark resistance
5. Repeat for different spark resistance values

**Goal:** Find the spark resistance R that maximizes measured power

### Procedure

**Step 1: Build Full Model**
- Primary tank circuit (L_primary, C_MMC)
- Secondary coil (distributed or lumped model)
- Topload capacitance
- Magnetic coupling k
- **Spark load** modeled as R||C_mut in series with C_sh

**Step 2: Set Operating Point**
- Drive frequency: f_drive (initially at unloaded resonance)
- Drive amplitude: V_drive or I_drive
- Spark parameters: Choose initial R, C_mut, C_sh

**Step 3: Run AC Analysis**
- Solve circuit at drive frequency
- Extract voltage and current at spark resistor
- Calculate power: P = 0.5 × Re{V_spark × I_spark*}

Or more directly:
```
P = 0.5 × |I_R|² × R

where I_R is current through the resistance R
```

**Step 4: Sweep R Values**
- Vary R from 10 kΩ to 200 kΩ (typical range)
- For each R, measure P
- Plot P vs R
- Find R that gives maximum P → this is R_opt_power

**Step 5: Validate**
- Compare numerical R_opt_power to analytical formula
- Check that it matches: R_opt = 1/[ω(C_mut + C_sh)]

## Power Measurement in SPICE

### Method 1: Using Current Through Resistor

```
.param Rspark = 50k
Rspark topload node2 {Rspark}
Cmut node2 0 8p
Csh topload 0 6p

.ac lin 1 185k 185k
.step param Rspark list 10k 30k 50k 70k 100k 150k

.meas ac Ispark_mag find mag(I(Rspark))
.meas ac Pspark param '0.5 * Ispark_mag^2 * Rspark'
```

This sweeps Rspark and calculates power for each value.

### Method 2: Direct Power Function

Some SPICE variants support direct power measurement:

```
.meas ac Pspark_real find Re(V(topload)*conj(I(Rspark)))
```

This directly computes complex power and extracts the real part.

### Method 3: Voltage and Current

```
.meas ac Vtop_mag find mag(V(topload))
.meas ac Ispark_mag find mag(I(Rspark))
.meas ac phase_diff param 'ph(V(topload)) - ph(I(Rspark))'
.meas ac Pspark param '0.5 * Vtop_mag * Ispark_mag * cos(phase_diff)'
```

This accounts for phase difference in power calculation.

## Worked Example: Direct Optimization

**Given:**
- DRSSTC simulation at f = 185 kHz
- Primary drive: V_drive produces V_top ≈ 350 kV (unloaded)
- Spark model: C_mut = 8 pF, C_sh = 6 pF, R = variable

**Goal:** Find R_opt_power

### Analytical Prediction

First, predict what we should find:

```
C_total = C_mut + C_sh = 8 + 6 = 14 pF
ω = 2π × 185×10³ = 1.162×10⁶ rad/s

R_opt_power = 1/(ωC_total)
            = 1/(1.162×10⁶ × 14×10⁻¹²)
            = 61.5 kΩ
```

We expect maximum power near 61.5 kΩ.

### Simulation Sweep

**Run AC analysis with R values:**
- R = 20 kΩ → P = 85 kW
- R = 40 kΩ → P = 115 kW
- R = 60 kΩ → P = 125 kW ← **Maximum**
- R = 80 kΩ → P = 118 kW
- R = 100 kΩ → P = 105 kW

**Result:** Maximum power at R ≈ 60 kΩ

**Validation:** Simulation (60 kΩ) matches theory (61.5 kΩ) within rounding!

## Advantages of Direct Measurement

### 1. No Approximations

- Full coupled model captures all interactions
- No linearization assumptions
- Includes all nonlinear effects (if using transient analysis)

### 2. Intuitive

- Directly see what you care about: power to spark
- No intermediate steps
- Easy to visualize results

### 3. Flexibility

- Can use any circuit simulator
- Works with complex topologies
- Easy to add additional elements (damping, protection, etc.)

### 4. Transient Capability

- Can extend to time-domain (transient) analysis
- Capture burst mode, ramping, dynamics
- See energy transfer over time

## Disadvantages of Direct Measurement

### 1. Computational Cost

- Must re-run full simulation for each R value
- Sweep of 20 points = 20 full simulations
- Slow for large parameter spaces

### 2. Limited Insight

- Doesn't reveal underlying equivalent circuit
- Harder to understand why maximum occurs where it does
- Less portable to different load types

### 3. Frequency Coupling

- Operating frequency may need adjustment for each R (see next lesson!)
- Fixed-frequency comparison can be misleading
- Must account for resonance shift

### 4. Sensitivity to Setup

- Results depend on drive amplitude, frequency, damping
- Harder to isolate spark effects from system effects

## Comparison: Thévenin vs Direct

| Aspect | Thévenin Method | Direct Method |
|--------|----------------|---------------|
| **Speed** | Fast (single extraction + algebra) | Slow (simulation per R value) |
| **Insight** | High (reveals equivalent circuit) | Moderate |
| **Accuracy** | Excellent (if linear) | Excellent (includes nonlinearities) |
| **Flexibility** | Any load instantly | One load per simulation |
| **Complexity** | Requires understanding of method | Straightforward |
| **Best for** | Sweeps, optimization, understanding | Validation, nonlinear cases |

## When to Use Each Method

### Use Thévenin When:
- Exploring many different load configurations
- Optimizing spark parameters
- Building intuition about matching
- Preparing design curves
- Speed is important

### Use Direct Measurement When:
- Validating Thévenin results
- Dealing with significant nonlinearities
- Need transient/time-domain behavior
- Checking specific operating points
- Learning circuit behavior

### Best Practice: Use Both

1. **Start with Thévenin:** Fast exploration, find optimal regions
2. **Validate with Direct:** Confirm key points, check assumptions
3. **Iterate:** If discrepancies exist, understand why

## Accounting for Displacement Currents

Both methods can fall victim to the "I_base error" discussed in Module 2.4.

### The Problem

**Wrong:** Measuring total current returning through secondary base

**Right:** Measuring current specifically through spark resistance

### Why It Matters

Total base current includes:
- Spark current (what we want)
- Displacement currents from secondary to ground
- Coupling currents to primary
- Environmental coupling

**In SPICE:** This isn't usually a problem because you can measure specific branch currents. Use I(Rspark) not I(V_secondary_base).

**In physical measurements:** You must use current probes on the spark return path, not the coil base.

## Implementation Tips

### Tip 1: Automate Sweeps

Use SPICE .STEP or scripting:

```
.step param Rspark 10k 200k 5k
```

This automatically sweeps from 10 kΩ to 200 kΩ in 5 kΩ steps.

### Tip 2: Log Scale for Wide Ranges

Spark resistance varies over decades (10 kΩ to 1 MΩ). Use logarithmic stepping:

```
.step param Rspark list 10k 20k 50k 100k 200k 500k
```

### Tip 3: Extract Peak Directly

Use .MEAS to find maximum automatically:

```
.meas ac Pmax MAX Pspark
.meas ac Ropt WHEN Pspark=Pmax
```

### Tip 4: Verify Power Components

Separately measure real and reactive power:

```
P_real = Re{V × I*}
Q_reactive = Im{V × I*}
S_apparent = |V × I*|
```

Check that Q >> P (highly reactive, as expected).

## Key Takeaways

- **Direct measurement:** Keep full model, measure power in spark, sweep R
- **Advantages:** Intuitive, no approximations, handles nonlinearity
- **Disadvantages:** Slow, less insight, multiple simulations required
- **Power formula:** P = 0.5 × |I_R|² × R or P = 0.5 × Re{V × I*}
- **Find R_opt:** Sweep R, plot P vs R, identify maximum
- **Validation:** Should match analytical R_opt = 1/[ω(C_mut + C_sh)]
- **Best practice:** Use Thévenin for exploration, direct measurement for validation
- **Beware:** Measure spark current, not base current (displacement current issue)

## Practice

{exercise:opt-ex-05}

**Problem 1:** You run simulations with the following results:

| R (kΩ) | P (kW) |
|--------|--------|
| 30 | 92 |
| 50 | 118 |
| 70 | 128 |
| 90 | 125 |
| 110 | 115 |

(a) Estimate R_opt_power from this data
(b) If C_total = 12 pF and f = 200 kHz, what does theory predict?
(c) Do they match?

**Problem 2:** A simulation reports I_R = 2.1 A (peak) through R = 55 kΩ. Calculate the power dissipated.

**Problem 3:** You measure V_topload = 340 kV ∠0° and I_spark = 1.8 A ∠-72°.
(a) Calculate apparent power S = V × I*
(b) Extract real power P = Re{S}
(c) Extract reactive power Q = Im{S}
(d) Is the spark more resistive or reactive?

**Problem 4:** List two scenarios where direct measurement would be preferred over Thévenin extraction.

**Problem 5:** Why is it important to measure I(Rspark) rather than I(V_secondary_base) when calculating power? Sketch the circuit showing both current paths.

---
**Next Lesson:** [Frequency Tracking and Loaded Poles](06-frequency-tracking.md)