plasma-expert/context/coupled-resonance.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
coupled-resonance	Coupled Resonance, Pole Splitting, and Frequency Tracking	established	spark-physics.txt: Part 4 (lines 192-210), Part 9 (lines 666-700)	[circuit-topology power-optimization thevenin-method energy-and-growth qcw-operation lumped-model distributed-model equations-and-bounds]	[Pole frequencies (eigenfrequencies) C_sh increase with spark length Loaded pole shift Power at loaded pole vs fixed frequency]	[coupled resonant system eigenfrequency lower pole upper pole pole splitting frequency tracking PLL DRSSTC detuning loaded Q geometric mean]	[frequency-shift-with-loading.png drsstc-operating-modes.png loaded-pole-analysis.png]	[]	[What is the optimal PLL bandwidth for tracking the loaded pole during QCW ramp? How does the upper pole behave under heavy spark loading -- does it ever become the dominant mode? Can the frequency tracking strategy be adapted in real time based on spark impedance feedback? What is the quantitative power penalty for operating 5% off the loaded pole versus exactly on it? How do higher-order modes (if present in long secondaries) interact with the spark-loaded poles? Does the ~1 MHz breakdown voltage minimum affect inception behavior for high-frequency DRSSTCs (>200 kHz)? How does frequency tracking performance vary between 300-600 kHz QCW builds and 50-100 kHz burst DRSSTCs?]

Coupled Resonance, Pole Splitting, and Frequency Tracking

This document addresses the coupled resonant nature of the Tesla coil system and the critical role of frequency tracking when a spark loads the circuit. The central finding is that comparing spark impedances at a fixed frequency conflates two distinct effects (impedance matching and detuning), and that frequency tracking is THE most important often-missed concept in Tesla coil spark modeling.

1. The Coupled Resonant System

1.1 Unloaded Tesla Coil: Two Eigenfrequencies

A Tesla coil consists of two resonant circuits (primary and secondary) coupled magnetically with coupling coefficient k. Even without a spark, this coupled system does not have a single resonant frequency. Instead, it has two resonant modes (eigenfrequencies or poles):

• Lower pole (f_lower): Below the geometric mean of the uncoupled primary and secondary frequencies. In this mode, the primary and secondary currents are approximately in phase.

• Upper pole (f_upper): Above the geometric mean. In this mode, the primary and secondary currents are approximately in antiphase.

For a coil with uncoupled primary frequency f_p and uncoupled secondary frequency f_s, and coupling coefficient k:

f_lower ~ f_mean * sqrt(1 - k)      (approximate, for tuned case f_p ~ f_s)
f_upper ~ f_mean * sqrt(1 + k)      (approximate, for tuned case f_p ~ f_s)

where f_mean = sqrt(f_p * f_s) ~ f_p ~ f_s  (for tuned coils)

The splitting between the two poles is proportional to k. For typical DRSSTCs with k = 0.15-0.25, the splitting is 15-25% of the center frequency.

1.2 Mode Characteristics

Each pole has its own:

• Frequency: f_lower and f_upper
• Quality factor (Q): Determined by losses in both primary and secondary
• Voltage gain: The ratio of topload voltage to primary voltage differs between modes
• Current distribution: The pattern of currents in primary and secondary differs

For most DRSSTCs, the lower pole provides higher topload voltage because the secondary's distributed capacitance and the topload capacitance are in a favorable configuration.

1.3 Why Poles, Not "Resonant Frequency"

It is incorrect to speak of "the resonant frequency" of a Tesla coil. The system has two distinct resonances. The coil's behavior depends critically on which pole the drive is tuned to. Most DRSSTCs are tuned to operate at or near the lower pole, but this is a design choice, not a physical necessity.

1.4 Frequency Dependence of Air Breakdown

A relevant physical phenomenon for coupled resonance design: the breakdown voltage in air shows a frequency dependence, with a minimum near ~1 MHz. At frequencies well below this, breakdown follows quasi-DC (Paschen) behavior. Near and above 1 MHz, electrons can survive the field reversal between half-cycles, reducing the effective breakdown threshold. The effect becomes significant when the RF half-period approaches the electron attachment time in air (~16 ns at STP). [Becker et al. 2005, Ch 2, p. 30; Kunhardt 2000]

For Tesla coils operating at 50-400 kHz, this effect is relatively minor but not negligible:

Operating Frequency	Estimated Effect on Inception
50 kHz	Essentially DC-like breakdown
100-200 kHz	Possibly 1-5% reduction vs. DC predictions
200-400 kHz	Possibly 5-10% reduction vs. DC predictions
~1 MHz (some small SSTCs)	Approaching minimum, potentially 20-30% reduction

This frequency dependence is an additional factor (beyond the pole-tracking effects in Sections 2-3) that should be considered when comparing spark performance across coils operating at very different frequencies. A coil at 400 kHz may have a slight inherent advantage in spark inception over an otherwise identical coil at 100 kHz, independent of coupling and power considerations.

See field-thresholds Section 4.4 for the broader discussion of frequency effects on breakdown.

1.5 QCW Operating Parameters from Community Survey

A comprehensive survey of QCW builder data [Phase 6 QCW community survey, 2026-02-10] reveals that QCW operation occupies a distinct parameter space compared to burst-mode DRSSTCs:

Parameter	QCW Range	Burst DRSSTC	Implication
Coupling (k)	0.3-0.55+	0.05-0.2	QCW needs tight coupling for adequate power transfer at low peak current
Operating frequency	300-600 kHz	50-110 kHz	Higher frequency enables continuous heating (see thermal-physics)
Tank capacitance	5-15 nF	50-300 nF	Smaller tank for faster ring-up
Ramp duration	10-22 ms	N/A (burst ~70-150 us)	100-200x longer pulse
Peak primary current	50-200 A	200-1000+ A	QCW uses far less peak current
Secondary voltage	40-70 kV	200-600 kV	QCW voltage is 5-15x lower
Spark:secondary ratio	7-16x	2-4x	QCW produces 3-5x more spark per unit secondary
Growth rate	~170 m/s	N/A (single-shot)	Half the speed of sound

Key insight — QCW secondary voltage is LOW: Multiple independent builders (Steve Ward, davekni, Loneoceans) have measured QCW secondary voltages of only 40-70 kV despite producing meter-length sparks. The most dramatic comparison: davekni measured ~600 kV for 2-3 m burst-mode sparks vs ~40 kV for equivalent QCW sparks at 450 kHz — a 15:1 voltage ratio. This proves that QCW growth is driven by sustained energy injection through a persistent leader channel, not by high instantaneous voltage. See streamers-and-leaders for the physical explanation.

Coupling requirement (k >= 0.3): All successful QCW sword-spark builds use k >= 0.3, typically 0.35-0.55. Higher coupling enables sufficient power transfer at QCW's lower peak currents (50-200 A vs 200-1000+ A for burst). It also widens the pole separation, making frequency tracking more robust against the shifting loaded pole during the ramp. However, Loneoceans' SSTC3 (single-resonant, lower coupling) still produces straight sparks at 380-420 kHz, suggesting that the coupling requirement is primarily an engineering constraint (adequate power delivery) rather than a physics constraint (straightness).

2. Spark-Induced Pole Modification

2.1 How the Spark Modifies the System

When a spark forms, it adds the spark circuit (C_mut || R in series with C_sh, per circuit-topology) at the topload node. This modifies the system in two ways:

1. Frequency shift: The additional capacitance (primarily C_sh, which grows with spark length at ~2 pF/foot) increases the total capacitance at the topload, lowering both pole frequencies. The lower pole drops more because it is more sensitive to topload loading.

2. Damping increase: The spark resistance R adds loss to the system, reducing the Q of both poles. This is the desired effect -- power dissipated in R is the power delivered to the spark.

Critical distinction: The spark modifies both frequency AND damping, not just one or the other. Ignoring either effect leads to incorrect power predictions.

2.2 Quantitative Frequency Shift

For a spark of length L (in meters), with C_sh ~ 6.6 pF/m * L:

C_total_new = C_top_original + C_sh(L) + coupling_corrections

Approximate frequency shift:
delta_f / f_0 ~ -C_sh(L) / (2 * C_top_original)    (first order)

For a medium coil with C_top = 30 pF and a 2-meter spark (C_sh ~ 13 pF):

delta_f / f_0 ~ -13 / (2 * 30) ~ -22%

This is a very large frequency shift. A 22% detuning can reduce power transfer by an order of magnitude if the drive frequency is not adjusted.

2.3 Damping Increase

The loaded Q at the lower pole decreases as:

1/Q_loaded = 1/Q_unloaded + 1/Q_spark

where Q_spark ~ omega_L * C_total / G_spark    (for the spark contribution)

For a well-coupled spark near R_opt_power, Q_spark might be 10-30, while Q_unloaded might be 100-300. The spark dominates the loaded Q, which is desirable (most power goes to the spark, not secondary losses).

3. The Frequency Tracking Problem

3.1 The Fundamental Issue

Consider the following common (but flawed) simulation approach:

1. Set drive frequency to f_0 (unloaded resonance).
2. Attach spark load with resistance R_1.
3. Measure power P_1.
4. Change to R_2.
5. Measure power P_2.
6. Compare P_1 and P_2 to determine "which R is better matched."

This is wrong. The comparison is invalid because the loaded pole frequency shifts when R changes (through the change in damping and the coupling between R and the reactive elements). At fixed drive frequency:

• Some R values will happen to place the loaded pole near the drive frequency (accidentally "tuned"), giving misleadingly high power.
• Other R values will shift the loaded pole far from the drive frequency ("detuned"), giving misleadingly low power.

What is actually being measured: The comparison conflates two independent effects:

1. Impedance matching quality (how close R is to R_opt_power)
2. Frequency detuning (how far the drive is from the loaded pole)

These must be separated to draw valid conclusions.

3.2 The Correct Approach

For each R value:

1. Sweep the drive frequency over a band (e.g., +/-15% of f_0).
2. Find the frequency of maximum |V_top| -- this is the loaded pole frequency f_L(R).
3. Measure the power delivered to the spark AT that loaded pole frequency.
4. Record P(R) at the optimally tuned frequency.

This procedure isolates the impedance matching quality from frequency effects. The resulting P(R) curve peaks at R_opt_power, as predicted by power-optimization.

3.3 Quantitative Impact of Ignoring Frequency Tracking

The power penalty for operating at a fixed frequency when the loaded pole has shifted can be estimated. For a system with loaded Q_L:

P(f) / P(f_L) ~ 1 / (1 + Q_L^2 * (f/f_L - f_L/f)^2)

For Q_L = 20 and 5% detuning:

P(f) / P(f_L) ~ 1 / (1 + 400 * (0.05)^2) ~ 1 / 2 = 50%

A 5% frequency error costs half the power. For 10% detuning, the penalty is ~80%. This is why frequency tracking is so important.

Steve Conner and others in the Tesla coil community have identified frequency tracking as the single most important factor that separates high-performance coils from underperformers. A coil with excellent frequency tracking will outperform one with better static impedance matching but poor tracking. The power difference can be a factor of 3-5.

4. DRSSTC Operating Modes

4.1 Fixed Frequency

Description: The drive inverter operates at a pre-set frequency determined by a crystal oscillator, RC timer, or similar fixed reference.

Advantages:

• Simple implementation
• Predictable behavior
• No feedback loop to destabilize

Disadvantages:

• No compensation for spark loading
• As spark grows and poles shift, the coil detunes
• Power delivery drops dramatically during spark growth
• Only competitive for very short pulses (burst mode) where the spark has minimal time to load the system

When appropriate: Short burst-mode operation (<100 us pulses) where frequency shift is minimal, or for initial testing and debugging.

4.2 PLL (Phase-Locked Loop)

Description: A phase comparator measures the phase relationship between the drive signal and a feedback signal (typically the secondary base current or a current transformer on the primary). The PLL adjusts the drive frequency to maintain a target phase relationship, tracking the loaded pole.

Advantages:

• Automatically tracks the loaded pole as spark grows
• Maintains near-optimal power transfer throughout spark growth
• Most common approach in high-performance DRSSTCs

Disadvantages:

• PLL bandwidth must be chosen carefully:
  • Too slow: cannot track rapid impedance changes
  • Too fast: may overshoot or oscillate, especially during spark inception
• Phase detector may lock to wrong pole (upper instead of lower)
• Noise from spark can corrupt the feedback signal
• Complex implementation

Design considerations:

• PLL bandwidth should be fast enough to track the spark growth timescale (~1 ms changes) but slow enough to reject RF noise (>1 MHz)
• Typical PLL bandwidth: 1-10 kHz
• The feedback signal must be filtered to extract the fundamental frequency

4.3 Programmed Frequency

Description: The drive frequency is pre-programmed as a function of time, based on anticipated or pre-measured loading. For QCW operation, the frequency ramp can be designed to match the expected pole shift during voltage ramp-up.

Advantages:

• No feedback loop (stable, predictable)
• Can be optimized for specific operating conditions
• No noise sensitivity

Disadvantages:

• Requires advance knowledge of loading (or iterative calibration)
• Does not adapt to variations (spark length, humidity, proximity to objects)
• Must be re-programmed for different operating conditions

When appropriate: Highly repeatable operating conditions, competition coils optimized for a specific target, or as a supplement to PLL (pre-programmed nominal trajectory with PLL corrections).

4.4 Hybrid Approaches

Modern high-performance DRSSTCs often combine approaches:

• Programmed frequency ramp for the nominal trajectory
• PLL correction for deviations from nominal
• Mode switching: fixed frequency during ring-up, PLL during spark growth
• Adaptive algorithms that learn the pole trajectory over multiple pulses

5. Pole Behavior Under Heavy Loading

5.1 Pole Migration

As spark loading increases (C_sh grows, R decreases toward R_opt):

1. Lower pole: Frequency decreases, Q decreases. This is the primary operating pole for most DRSSTCs. Under very heavy loading, the lower pole can shift by 20-30% from its unloaded position.

2. Upper pole: Frequency also shifts (less dramatically), Q decreases. The upper pole may become so heavily damped that it effectively disappears as a distinct resonance.

3. Pole merging: In extreme cases (very heavy loading or very tight coupling), the two poles can merge into a single, heavily damped resonance. This is unusual in normal operation but can occur during arc strikes to grounded objects.

5.2 Mode Coupling

The spark introduces a coupling between what were previously relatively independent modes. At moderate loading:

• Energy can transfer between modes
• The simple two-pole picture becomes less clean
• Transient analysis may show beating between modes

For simulation purposes, sweeping frequency to find the actual loaded pole (as described in Section 3.2) automatically accounts for these effects.

6. Interaction with Thevenin Analysis

6.1 Frequency-Dependent Thevenin Parameters

The Thevenin equivalent (see thevenin-method) captures the coupled resonance behavior implicitly through the frequency dependence of Z_th(omega) and V_th(omega):

• Near the lower pole: Z_th has a peak in its real part, V_th has a peak in magnitude
• Near the upper pole: smaller secondary peaks
• Between poles: Z_th and V_th vary smoothly

By measuring Z_th and V_th over the full frequency band, the Thevenin approach automatically accounts for pole shifting and mode coupling.

6.2 Practical Implication

When using the Thevenin method for power prediction:

1. Compute Z_load for the spark at the operating frequency.
2. Compute P_load = 0.5 * |V_th(f)|^2 * Re{Z_load(f)} / |Z_th(f) + Z_load(f)|^2.
3. Sweep f to find the frequency that maximizes P_load for each set of spark parameters.
4. That frequency is the loaded pole.

7. Practical Recommendations

7.1 For Simulation

• Always sweep frequency when comparing different spark loads. Never evaluate at a single fixed frequency.
• Report power at the loaded pole, not at the unloaded resonant frequency.
• Track both poles to ensure you are operating on the correct one.
• Include primary tank components in the model; they affect pole locations significantly.

7.2 For Coil Design

• Design PLL bandwidth for the expected spark growth timescale.
• Allow sufficient frequency range in the drive electronics (at least +/-15% of nominal).
• Monitor for pole-hopping: If the PLL locks onto the wrong pole, power delivery can drop dramatically.
• Consider QCW ramp rate: Faster ramps require faster frequency tracking. Typical QCW ramps of 5-20 ms are well within PLL capability if bandwidth is 1-10 kHz.

7.3 For Measurement

• Ringdown measurements (see thevenin-method) give Q and frequency at a single operating point. Multiple measurements at different loading levels map out the pole trajectory.
• Real-time frequency monitoring (e.g., counting zero crossings of the secondary current) provides the loaded pole frequency during operation.

8. Connection to Other Topics

Key Relationships

• Derives from: Coupled oscillator theory (standard physics of two inductively coupled LC circuits)
• Depends on: circuit-topology (the spark load impedance is what modifies the poles)
• Interacts with: power-optimization (R_opt_power changes with frequency; frequency tracking ensures the correct R_opt is used)
• Measured via: thevenin-method (Z_th(omega) captures pole behavior; ringdown gives loaded Q)
• Affects: energy-and-growth (power delivery during spark growth depends on how well the system tracks the loaded pole)
• Affects: lumped-model and distributed-model (simulations must include frequency tracking for accurate power predictions)

Summary of Key Results

1. A Tesla coil has two eigenfrequencies (poles), not one "resonant frequency."
2. Spark loading shifts both poles lower in frequency and increases damping.
3. Comparing spark loads at fixed frequency conflates impedance matching with detuning.
4. The correct procedure: for each load, find the loaded pole, then measure power there.
5. Frequency tracking (PLL or programmed) is the single most impactful design feature.
6. A 5% frequency error can halve the delivered power; 10% can cost 80%.
7. Three DRSSTC operating modes: fixed frequency, PLL, programmed. PLL is most common.
8. Power penalty from poor frequency tracking: factor of 3-5 in real coils.