---
id: opt-06
title: "Frequency Tracking and Loaded Poles"
section: "Optimization & Simulation"
difficulty: "advanced"
estimated_time: 45
prerequisites: ["opt-05", "opt-01", "fund-08"]
objectives:
  - Understand coupled system poles and eigenfrequencies
  - Recognize frequency shift with loading
  - Implement proper frequency tracking in measurements
  - Avoid common detuning errors in optimization
  - Apply frequency tracking to DRSSTC operating modes
tags: ["frequency-tracking", "coupled-resonators", "detuning", "poles", "DRSSTC"]
---

# Frequency Tracking and Loaded Poles

**This is one of the most commonly overlooked aspects of Tesla coil optimization.** Failing to account for frequency tracking leads to misleading power measurements and incorrect conclusions about optimal operating points.

## The Critical Problem: Fixed-Frequency Comparison

### Common Mistake

**Scenario:** You want to find R_opt_power by measuring power delivered to different spark resistances.

**Wrong approach:**
1. Set drive frequency to f = 200 kHz (unloaded resonance)
2. Measure power with R = 30 kΩ → P₁ = 95 kW
3. Measure power with R = 60 kΩ → P₂ = 110 kW
4. Measure power with R = 90 kΩ → P₃ = 105 kW
5. Conclude: R_opt ≈ 60 kΩ

**What's wrong?** Each different R value changes the system's resonant frequency. By staying at fixed f = 200 kHz, you're comparing:
- R = 30 kΩ at **Δf = +8 kHz detuned**
- R = 60 kΩ at **Δf = +3 kHz detuned**
- R = 90 kΩ at **Δf = -2 kHz detuned**

**You're not measuring inherent matching quality - you're measuring a combination of matching AND detuning!**

### Right Approach

**Correct procedure:**
1. Set R = 30 kΩ
2. **Sweep frequency to find loaded resonance** → f₁ = 192 kHz
3. Measure power at f₁ → P₁ = 108 kW
4. Set R = 60 kΩ
5. **Sweep frequency to find new loaded resonance** → f₂ = 188 kHz
6. Measure power at f₂ → P₂ = 125 kW
7. Set R = 90 kΩ
8. **Sweep frequency to find new loaded resonance** → f₃ = 185 kHz
9. Measure power at f₃ → P₃ = 118 kW
10. Conclude: R_opt ≈ 60 kΩ **(and each was measured at its optimal frequency)**

**Key principle: For each R value, retune to the loaded pole frequency.**

## Why Does Loading Change Frequency?

### Capacitance Changes Resonance

When you change the spark, you change its sheath capacitance C_sh:

**Unloaded:**
```
C_total,0 = C_topload + C_secondary_stray ≈ 28 pF
f₀ = 1/(2π√(L_sec × C_total,0)) = 200 kHz
```

**With spark (R = 60 kΩ, 3-foot leader):**
```
C_sh ≈ 2 pF/foot × 3 feet = 6 pF
C_total,1 = C_total,0 + C_sh = 28 + 6 = 34 pF

f₁ = f₀ × √(C_total,0 / C_total,1)
   = 200 × √(28/34)
   = 200 × 0.907
   = 181 kHz
```

**Frequency dropped by 19 kHz!** This is not a small shift.

### Different Sparks → Different Frequencies

| Spark Length | C_sh | C_total | f_loaded | Δf |
|--------------|------|---------|----------|-----|
| No spark | 0 pF | 28 pF | 200 kHz | 0 |
| 2 feet | 4 pF | 32 pF | 187 kHz | -13 kHz |
| 4 feet | 8 pF | 36 pF | 176 kHz | -24 kHz |
| 6 feet | 12 pF | 40 pF | 167 kHz | -33 kHz |

**Even for the same length, changing R changes the effective loading!**

## Coupled System Poles

Tesla coils are **coupled resonant systems**. Even without a spark, the primary-secondary coupling creates two resonant modes.

### The Two Poles

For coupled resonators with coupling coefficient k:

**Lower pole (f₁):**
```
f₁ = f₀ / √(1 + k) < f₀
```

**Upper pole (f₂):**
```
f₂ = f₀ / √(1 - k) > f₀
```

where f₀ = √(f_primary × f_secondary) is the geometric mean.

**Example with k = 0.15:**
```
f₀ = 200 kHz (geometric mean)
f₁ = 200 / √(1.15) = 186.5 kHz (lower pole)
f₂ = 200 / √(0.85) = 217.0 kHz (upper pole)
```

### Loading Modifies Both Poles

When a spark loads the secondary:
- **Both pole frequencies shift** (usually downward)
- **Both pole damping increases** (Q decreases)
- **Pole separation changes**

The spark doesn't just add capacitance - it adds a complex load that couples into both modes.

### Which Pole Should You Use?

**For DRSSTC operation:**
- Most coils operate on the **lower pole** (more stable)
- Some operate between poles (dual-resonance mode)
- Upper pole is rarely used (harder to control)

**The loaded pole frequency is where voltage gain is maximized.**

## DRSSTC Operating Modes

Different DRSSTC drive strategies interact with frequency tracking differently.

### Mode 1: Fixed Frequency (No Tracking)

**Strategy:** Drive at fixed frequency (e.g., 200 kHz) regardless of loading

**Advantages:**
- Simple control electronics
- No frequency sensing required
- Predictable timing

**Disadvantages:**
- **Detunes as spark grows**
- Voltage gain drops with larger sparks
- Suboptimal power transfer
- Risk of operating off-resonance

**When acceptable:**
- Very short bursts (spark doesn't grow much)
- Controlled environments with consistent sparks
- Systems designed with wide bandwidth

### Mode 2: Frequency Tracking (PLL or Feedback)

**Strategy:** Continuously adjust drive frequency to match loaded pole

**Implementation:**
- Phase-locked loop (PLL) tracks zero-crossing
- Feedback from antenna or current sensor
- Drive frequency follows resonance in real-time

**Advantages:**
- **Always at optimal frequency**
- Maximum voltage gain throughout growth
- Efficient power transfer
- Adapts to varying sparks

**Disadvantages:**
- More complex electronics
- Requires feedback sensing
- Can be unstable if poorly tuned
- Frequency limits needed for safety

**This is the gold standard for QCW and high-performance DRSSTCs.**

### Mode 3: Pre-Programmed Sweep

**Strategy:** Drive frequency ramps down over time (anticipating C_sh increase)

**Implementation:**
- Start at f₀ (unloaded resonance)
- Linearly or exponentially decrease frequency
- End at f_target (expected loaded resonance)

**Advantages:**
- Simpler than PLL
- No feedback required
- Can be optimized per coil

**Disadvantages:**
- Not adaptive (doesn't match actual spark)
- Requires characterization/tuning
- Mismatch if spark growth differs from expectation

**When useful:**
- QCW with consistent spark growth patterns
- Transition from no-spark to steady spark
- Combined with current limiting

## Frequency Response and Bandwidth

### Quality Factor Limits Bandwidth

The resonance has finite width determined by Q:

```
Δf_3dB = f₀ / Q  (3 dB bandwidth)

For Q = 100 at f₀ = 200 kHz:
Δf_3dB = 200 kHz / 100 = 2 kHz
```

**Within ±1 kHz:** Still >70% of peak voltage (acceptable detuning)
**Beyond ±5 kHz:** Down to ~30% of peak voltage (severe detuning)

### High Q vs Low Q

**High Q (narrow bandwidth):**
- Sharper resonance peak
- More sensitive to detuning
- **Frequency tracking more critical**
- Better efficiency when matched

**Low Q (wide bandwidth):**
- Broader resonance peak
- More forgiving of detuning
- Frequency tracking less critical
- Lower peak voltage gain

### Loaded Q vs Unloaded Q

**Unloaded Q₀:**
- No spark, only coil losses
- Typically Q₀ = 100-300

**Loaded Q_L:**
- With spark, additional damping
- Spark resistance adds loss
- Typically Q_L = 20-80

**Effect on bandwidth:**
```
Unloaded: Δf₀ = 200 kHz / 200 = 1 kHz (narrow!)
Loaded:   Δf_L = 185 kHz / 50 = 3.7 kHz (wider)
```

**Ironically, the spark broadens the resonance, making detuning slightly less critical. But the frequency shift is still large enough that you must track it.**

## Implementing Frequency Tracking in Measurements

### Simulation Approach

**For each R value:**

```python
# Pseudocode for proper frequency tracking
for R in [10k, 20k, 30k, ..., 200k]:
    set_spark_resistance(R)

    # Sweep frequency to find loaded pole
    for f in range(150k, 220k, 1k):
        run_AC_analysis(frequency=f)
        V_top[f] = measure_topload_voltage()

    # Find frequency with maximum voltage
    f_loaded = frequency_at_max(V_top)

    # Measure power at loaded frequency
    run_AC_analysis(frequency=f_loaded)
    P[R] = measure_spark_power()

    # Store results
    results[R] = {
        'f_loaded': f_loaded,
        'V_top': V_top[f_loaded],
        'P': P[R]
    }

# Now P[R] represents true matching, not detuning!
R_opt = R_at_max(P)
```

### SPICE Implementation

```spice
* Sweep R and frequency together
.param Rspark = 60k

* First find loaded frequency for this R
.ac dec 100 150k 220k
.meas ac f_loaded WHEN mag(V(topload))=MAX(mag(V(topload)))

* Then measure power at that frequency
.ac lin 1 {f_loaded} {f_loaded}
.meas ac Pspark param '0.5 * mag(I(Rspark))^2 * Rspark'

* Repeat for each R value
.step param Rspark list 10k 30k 50k 70k 90k 110k 150k 200k
```

**Challenge:** SPICE doesn't easily allow nested sweeps where inner result affects outer analysis. You may need to:
- Run multiple simulations
- Use scripting (Python + PySpice, MATLAB, etc.)
- Manually extract f_loaded for key R values

## Worked Example: Impact of Tracking vs Not Tracking

**System:**
- Unloaded: f₀ = 200 kHz, Q₀ = 150
- V_th = 350 kV (at resonance)
- Z_th = 110 - j2400 Ω (at 200 kHz)

**Spark configurations:**

| R | C_sh | C_total | f_loaded | Shift |
|---|------|---------|----------|-------|
| 40k | 5 pF | 33 pF | 188 kHz | -12 kHz |
| 60k | 6 pF | 34 pF | 185 kHz | -15 kHz |
| 80k | 7 pF | 35 pF | 183 kHz | -17 kHz |

### Without Tracking (Fixed f = 200 kHz)

**R = 40 kΩ:**
```
Detuning: Δf = +12 kHz
Voltage penalty: V_actual / V_max ≈ 0.65
Z_spark = 40k - j140k → |Z| = 146 kΩ
I ≈ 350 kV × 0.65 / 146 kΩ = 1.56 A
P = 0.5 × 1.56² × 40k = 48.6 kW
```

**R = 60 kΩ:**
```
Detuning: Δf = +15 kHz
Voltage penalty: ≈ 0.55
Z_spark = 60k - j160k → |Z| = 171 kΩ
I ≈ 350 kV × 0.55 / 171 kΩ = 1.13 A
P = 0.5 × 1.13² × 60k = 38.3 kW (WORSE despite higher R!)
```

**R = 80 kΩ:**
```
Detuning: Δf = +17 kHz
Voltage penalty: ≈ 0.48
Z_spark = 80k - j180k → |Z| = 197 kΩ
I ≈ 350 kV × 0.48 / 197 kΩ = 0.85 A
P = 0.5 × 0.85² × 80k = 28.9 kW
```

**Conclusion from fixed-frequency:** R_opt ≈ 40 kΩ (WRONG!)

### With Tracking (Tune to f_loaded for each R)

**R = 40 kΩ at f = 188 kHz:**
```
Detuning: 0 (by definition - we tuned to loaded pole)
Voltage penalty: 1.0 (at resonance)
I ≈ 350 kV / 146 kΩ = 2.40 A
P = 0.5 × 2.40² × 40k = 115 kW
```

**R = 60 kΩ at f = 185 kHz:**
```
Detuning: 0
Voltage penalty: 1.0
I ≈ 350 kV / 171 kΩ = 2.05 A
P = 0.5 × 2.05² × 60k = 126 kW (MAXIMUM!)
```

**R = 80 kΩ at f = 183 kHz:**
```
Detuning: 0
Voltage penalty: 1.0
I ≈ 350 kV / 197 kΩ = 1.78 A
P = 0.5 × 1.78² × 80k = 127 kW (close!)
```

**Conclusion with tracking:** R_opt ≈ 60 kΩ (CORRECT!)

**Power improvement with tracking:**
- At R = 60 kΩ: 126 kW vs 38 kW = **3.3× more power!**
- At R = 80 kΩ: 127 kW vs 29 kW = **4.4× more power!**

**This is not a small effect. Frequency tracking is critical.**

## Practical Implications

### For Simulation Studies

**Always:**
- Report frequency used for each measurement
- Either track frequency or clearly state fixed-frequency limitations
- Specify whether results assume optimal tuning

**When comparing:**
- Ensure fair comparison (same tracking strategy)
- Document detuning if fixed-frequency is used

### For Physical Coils

**DRSSTC with PLL:**
- Tracks automatically - excellent
- Monitor actual operating frequency
- Check frequency stays within safe limits

**DRSSTC with fixed frequency:**
- Accept voltage/power reduction as spark grows
- Consider pre-tuning to expected loaded frequency
- Wider-bandwidth design helps (lower Q)

**SGTC (Spark Gap):**
- Frequency self-adjusts with loading (inherent tracking)
- Spark gap firing adapts to LC resonance
- Less of an issue for spark gap coils

### For Optimization

**When finding R_opt_power:**
1. Use frequency tracking (simulation or actual)
2. Report f_loaded for each R tested
3. Verify analytical formula matches

**When designing:**
1. Choose f₀ based on unloaded resonance
2. Expect f_operating ≈ f₀ - 10 to 30 kHz with sparks
3. Ensure drive can operate over this range

## Key Takeaways

- **Critical principle:** For each R value, retune to loaded pole frequency
- **Why it matters:** Loading changes C_sh, which shifts resonance by 10-30+ kHz
- **Fixed-frequency comparison is misleading:** Measures detuning, not matching quality
- **Coupled system has two poles:** Lower and upper, both shift with loading
- **DRSSTC modes:** Fixed frequency (simple), PLL tracking (optimal), programmed sweep (compromise)
- **Q affects sensitivity:** Higher Q = narrower bandwidth = more critical tracking
- **Power difference:** Can be 3-5× between tracked and non-tracked measurements
- **Simulation best practice:** Sweep frequency for each load to find f_loaded
- **Physical coils:** PLL tracking gives best performance, fixed frequency is acceptable for short bursts

## Practice

{exercise:opt-ex-06}

**Problem 1:** A coil has f₀ = 195 kHz unloaded with C_total,0 = 30 pF. A 4-foot spark adds C_sh = 8 pF.
(a) Calculate the loaded capacitance
(b) Calculate the loaded frequency
(c) What is Δf (frequency shift)?

**Problem 2:** You measure power at fixed f = 200 kHz:
- R = 50 kΩ, f_loaded = 188 kHz → P₁ = 85 kW
- R = 70 kΩ, f_loaded = 185 kHz → P₂ = 95 kW

If Q = 80, estimate the voltage penalty factor for each case and calculate what power would be measured if you had tracked frequency.

**Problem 3:** Explain why frequency tracking is MORE critical for high-Q coils than low-Q coils.

**Problem 4:** A DRSSTC operates with fixed frequency drive. As the spark grows from 2 feet to 5 feet, what happens to:
(a) Loaded resonant frequency
(b) Detuning (if drive frequency is fixed)
(c) Voltage gain
(d) Power delivered

**Problem 5:** For coupled resonators with k = 0.18 and f₀ = 210 kHz:
(a) Calculate the lower pole frequency
(b) Calculate the upper pole frequency
(c) Which pole is typically used for DRSSTC operation?

**Problem 6:** Sketch V_top vs frequency for three cases:
(a) No spark (unloaded)
(b) R = 60 kΩ spark (lightly loaded)
(c) R = 30 kΩ spark (heavily loaded)

Label the peak frequencies and relative peak heights. Explain how tracking helps maintain peak operation.

---
**Next Lesson:** [Part 2 Review and Comprehensive Exercises](07-review-exercises.md)