plasma-expert/spark-lessons/lessons/02-optimization/06-frequency-tracking.md

id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
opt-06	Frequency Tracking and Loaded Poles	Optimization & Simulation	advanced	45	[opt-05 opt-01 fund-08]	[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]	[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:

# 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

* 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.

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Next Lesson: Part 2 Review and Comprehensive Exercises