---
id: power-optimization
title: "Power Optimization and the Hungry Streamer Principle"
status: established
source_sections: "spark-physics.txt: Part 2 (lines 75-124), Part 9 (lines 666-700), Part 11 (lines 740-744)"
related_topics: [circuit-topology, thevenin-method, coupled-resonance, field-thresholds, energy-and-growth, thermal-physics, streamers-and-leaders, capacitive-divider, branching-physics, empirical-scaling, lumped-model, distributed-model, equations-and-bounds]
key_equations:
  - "R_opt_phase"
  - "R_opt_power"
  - "Power delivered to load P_load"
  - "Impedance phase at R_opt_power"
key_terms:
  - "R_opt_power"
  - "R_opt_phase"
  - "hungry streamer principle"
  - "power transfer"
  - "impedance matching"
  - "self-optimization"
  - "thermal ionization"
  - "conductivity"
  - "causality reversal"
  - "QCW power paradigm"
images:
  - power-vs-resistance-curves.png
  - hungry-streamer-feedback-loop.png
  - impedance-matching-concept.png
examples:
  - calculating-ropt.md
open_questions:
  - "What is the time constant for the plasma to converge to R_opt_power after a step change in drive conditions?"
  - "Under what conditions does the hungry streamer feedback loop become unstable (oscillatory resistance)?"
  - "How does branching affect the effective R seen at the topload -- does each branch independently optimize?"
  - "Is the convergence to R_opt_power monotonic, or can the plasma overshoot and oscillate?"
---

# Power Optimization and the Hungry Streamer Principle

This document derives the two critical resistance values for Tesla coil spark modeling -- R_opt_phase and R_opt_power -- and establishes the physical mechanism by which real spark plasmas self-optimize toward maximum power extraction. The "hungry streamer" principle, credited to Steve Conner, is the conceptual cornerstone linking circuit theory to plasma behavior.

## 1. Two Critical Resistance Values

### 1.1 R_opt_phase: The Most Resistive-Looking Impedance

Starting from the admittance expressions derived in [[circuit-topology]], the impedance phase angle phi_Z depends on the spark resistance R. The value of R that minimizes |phi_Z| (makes the impedance look as resistive as possible) is found by differentiating phi_Z with respect to G = 1/R and setting the result to zero.

**Result:**

```
R_opt_phase = 1 / (omega * sqrt(C_mut * (C_mut + C_sh)))
```

At this resistance, the impedance phase angle equals the fundamental minimum:

```
phi_Z(R_opt_phase) = phi_Z_min = -atan(2 * sqrt(r * (1 + r)))
```

where r = C_mut / C_sh.

**Physical meaning:** R_opt_phase is the resistance at which the spark presents the closest approximation to a purely resistive load. However, due to the [[circuit-topology]] phase constraint, this "closest approximation" is still significantly capacitive (typically -50 to -70 degrees).

**When is R_opt_phase relevant?** In situations where minimizing reactive power flow is more important than maximizing real power -- for example, when the source has limited reactive current capability, or for minimizing circulating currents in the primary tank.

### 1.2 R_opt_power: Maximum Real Power Transfer

The real power delivered to the spark, for a fixed topload voltage magnitude |V_top|, is:

```
P_spark = 0.5 * |V_top|^2 * Re{Y}
        = 0.5 * |V_top|^2 * G * B_2^2 / (G^2 + (B_1 + B_2)^2)
```

Maximizing P_spark with respect to G (equivalently R) by setting dP/dG = 0:

```
d/dG [G * B_2^2 / (G^2 + (B_1 + B_2)^2)] = 0
```

The numerator of the derivative gives:

```
B_2^2 * [(G^2 + (B_1 + B_2)^2) - 2G^2] = 0
B_2^2 * [(B_1 + B_2)^2 - G^2] = 0
```

Since B_2 is nonzero, this requires G^2 = (B_1 + B_2)^2, giving G_opt = B_1 + B_2 = omega*(C_mut + C_sh).

**Result:**

```
R_opt_power = 1 / (omega * (C_mut + C_sh))
```

**Numerical example:** At f = 200 kHz with C_mut + C_sh = 12 pF:

```
omega = 2 * pi * 200e3 = 1.257e6 rad/s
R_opt_power = 1 / (1.257e6 * 12e-12) = 1 / (1.508e-5) = 66.3 kOhm
```

### 1.3 Relationship Between the Two Optima

**R_opt_power is always less than R_opt_phase:**

```
R_opt_power / R_opt_phase = sqrt(C_mut * (C_mut + C_sh)) / (C_mut + C_sh)
                          = sqrt(C_mut / (C_mut + C_sh))
                          = sqrt(r / (1 + r))    where r = C_mut/C_sh
```

Since r/(1+r) < 1 for all positive r, R_opt_power < R_opt_phase always.

For r = 1 (equal capacitances): R_opt_power / R_opt_phase = sqrt(0.5) = 0.707
For r = 0.5: R_opt_power / R_opt_phase = sqrt(1/3) = 0.577
For r = 2: R_opt_power / R_opt_phase = sqrt(2/3) = 0.816

**Impedance phase at R_opt_power:** Substituting G = omega*(C_mut + C_sh) into the phase expression:

```
phi_Z(R_opt_power) is typically -55 to -75 degrees
```

This is more negative (more capacitive) than phi_Z_min, meaning R_opt_power does NOT correspond to the minimum phase point. The maximum power condition accepts a worse phase angle in exchange for delivering more real power.

![Power vs. resistance curves showing both optima](../assets/power-vs-resistance-curves.png)

### 1.4 Power at the Two Optima

At R_opt_power, the maximum power is:

```
P_max = 0.5 * |V_top|^2 * B_2^2 / (2 * (B_1 + B_2))
      = 0.5 * |V_top|^2 * omega * C_sh^2 / (2 * (C_mut + C_sh))
```

At R_opt_phase, the power is lower. The ratio depends on r but is typically 0.7 to 0.9 of P_max. Except in unusual geometries, the difference is modest -- but over a long spark growth event (tens of milliseconds), the accumulated energy difference can be significant.

### 1.5 Causality Reversal: Spark Loading Drives Quench, Not Vice Versa

Richie Burnett (richieburnett.co.uk) identified a critical insight for understanding power delivery to spark loads:

**"It is not early quenching that produces good sparks, but rather good spark loading that leads to an early quench."**

The causality runs: the spark efficiently absorbs energy → secondary voltage drops → gap quenches (SGTC) or primary current drops (DRSSTC). A well-optimized spark near R_opt_power extracts power efficiently, pulling V_top down and naturally terminating the drive. This is the hungry streamer principle viewed from the source side: maximum power transfer produces maximum damping.

**Practical consequence:** Attempts to optimize spark performance by adjusting quench timing (SGTC) or burst duration (DRSSTC) are attacking the symptom, not the cause. The primary lever is optimizing the impedance match and power delivery to the spark itself.

### 1.6 QCW vs Burst: Fundamentally Different Power Paradigms

Community builder data [Phase 6 QCW community survey, 2026-02-10] reveals that QCW and burst mode represent fundamentally different approaches to power delivery:

| Aspect | QCW | Burst DRSSTC |
|--------|-----|-------------|
| Power delivery | Sustained low power over 10-22 ms | Brief high power over 70-150 us |
| Secondary voltage | 40-70 kV | 200-600 kV |
| How growth works | Continuous leader extension through persistent conducting channel | Single-shot streamer reach set by peak voltage |
| Limiting factor | Capacitive voltage division at tip | Streamer reach (voltage-limited) |
| Efficiency metric | Spark:secondary ratio (7-16x) | Bang energy to length scaling |

The most striking data point: davekni measured **~600 kV for 2-3 m burst sparks vs ~40 kV for equivalent QCW sparks** at 450 kHz — a 15:1 voltage ratio for similar spark lengths. This proves that QCW operates via a completely different mechanism: sustained energy delivery through a thermally persistent leader (see [[thermal-physics]]), not high instantaneous voltage.

**Implication for power optimization:** In burst mode, R_opt_power analysis at a single frequency is approximately valid because the entire event occurs within a few hundred microseconds. In QCW mode, R_opt_power shifts continuously during the 10-22 ms ramp as C_sh grows (spark extends). The matching strategy should target 50-70% of final spark length, as described in Section 4.2.

## 2. The Hungry Streamer Principle

### 2.1 Origin

Steve Conner observed that Tesla coil streamers appear to actively seek out conditions that maximize power extraction from the resonant circuit. He termed this the "hungry streamer" principle: the plasma is "hungry" for power and adjusts its properties to consume as much as possible.

### 2.2 Physical Mechanism: The Feedback Loop

The hungry streamer principle is not mystical -- it follows from well-understood plasma physics through a feedback loop:

**Step 1: Power injection drives Joule heating.**
Current I flows through the spark resistance R, depositing power P = I^2 * R in the plasma channel.

**Step 2: Heating increases temperature.**
The deposited energy raises the gas temperature T in the channel. For a thin channel with thermal time constant tau_thermal (see [[thermal-physics]]), the temperature responds on millisecond timescales.

**Step 3: Temperature drives thermal ionization.**
At elevated temperatures (above ~3000-5000 K), thermal ionization of air molecules becomes significant. The electron density n_e increases approximately exponentially with temperature (Saha equation):

```
n_e ~ exp(-E_ion / (2 * k_B * T))
```

where E_ion is the ionization energy (~14.5 eV for N2).

**Step 4: Ionization increases conductivity.**
Electrical conductivity sigma is proportional to electron density and inversely related to collision frequency:

```
sigma = n_e * e^2 / (m_e * nu_collision)
```

Higher n_e directly increases sigma, decreasing R.

**Step 5: Changed R modifies power transfer.**
A lower R changes the admittance and thus the power delivered. If R was above R_opt_power, decreasing R moves toward the optimum and increases power. If R was below R_opt_power, decreasing R moves away from the optimum and decreases power.

**Step 6: Geometry changes modify capacitances.**
As the channel heats and expands, its diameter changes, which weakly affects C_mut and C_sh (logarithmic dependence on diameter). The expanding, lengthening channel also increases C_sh linearly with length. These capacitance changes shift R_opt_power to a new value.

**Step 7: Stable equilibrium at R_actual ~ R_opt_power.**
The negative feedback loop (less power -> cooling -> higher R -> approaching R_opt from above) and positive feedback (more power -> heating -> lower R -> approaching R_opt from below, up to a point) create a stable attractor near R_opt_power. The plasma self-regulates.

![Hungry streamer feedback loop diagram](../assets/hungry-streamer-feedback-loop.png)

### 2.3 Why R_opt_power, Not R_opt_phase?

The feedback loop selects for maximum power, not minimum phase angle. Physical reasoning:

- More power -> more heating -> plasma responds to power, not to phase
- The plasma has no mechanism to "sense" phase angle; it responds to energy deposition (I^2*R)
- R_opt_power maximizes I^2*R for fixed source conditions
- The equilibrium is reached when no perturbation in R can increase I^2*R further

This is analogous to maximum power transfer in classical circuit theory, except the "load" actively adjusts itself.

### 2.4 Stability Analysis

Near R_opt_power, consider a small perturbation delta_R:

- If R = R_opt_power + delta_R (too high): power decreases -> less heating -> temperature drops -> ionization decreases -> R increases further. This is POSITIVE feedback away from optimum! However, as R increases beyond R_opt_power, the spark also cools, which eventually leads to the spark extinguishing or branching to find a better path. In practice, the spark stalls or a new streamer launches.

- If R = R_opt_power - delta_R (too low): power decreases (since we are below optimum on the P vs. R curve) -> less heating -> temperature drops -> ionization decreases -> R increases back toward R_opt_power. This is NEGATIVE feedback, stabilizing.

The equilibrium is thus stable from below but has a "cliff" above R_opt_power. In practice, this asymmetry manifests as the tendency for sparks to either burn brightly at or below R_opt or extinguish rapidly when the resistance drifts too high. The dynamic is further stabilized by the thermal inertia of the channel.

## 3. Constraints on Optimization

### 3.1 Source Limitations

The analysis above assumes fixed |V_top|. In reality, the source (Tesla coil primary circuit) has finite current and voltage capability:

- **Current-limited:** If the primary cannot supply the current demanded by the load at R_opt_power, the topload voltage collapses. The spark operates at a higher effective R (source impedance dominates).

- **Voltage-limited:** If V_top is insufficient to maintain the field threshold at the spark tip (see [[field-thresholds]]), the spark stalls regardless of R optimization.

### 3.2 Inception Threshold

The spark must first form. Inception requires E_tip > E_inception ~ 2-3 MV/m at the topload surface. If the topload voltage never reaches the inception field, no spark forms and the optimization loop never starts.

### 3.3 Physical Conductivity Bounds

The spark resistance cannot be arbitrarily low or high:

```
R_min ~ 1 kOhm     (very hot, thick, fully thermalized leader plasma)
R_max ~ 100 MOhm   (cold, thin, barely ionized streamer)
```

If R_opt_power falls outside [R_min, R_max], the plasma cannot reach the optimum:

```
R_actual = clip(R_opt_power, R_min, R_max)
```

When clipping occurs, the spark is constrained and operates sub-optimally. Check whether the source can still provide adequate power at the clipped resistance.

### 3.4 Thermal Time Constants

The plasma cannot adjust instantaneously. Thermal time constants (see [[thermal-physics]]) set the response speed:

- Thin streamers (d ~ 100 um): tau ~ 0.1-0.2 ms
- Thick leaders (d ~ 5 mm): tau ~ 300-600 ms

If the drive conditions change faster than the plasma can respond (e.g., burst-mode pulses shorter than tau), the plasma cannot track R_opt_power in real time. The effective R will lag behind the instantaneous optimum.

### 3.5 Sub-Optimal Operation

When constraints prevent reaching R_opt_power, several outcomes are possible:

1. **Spark stalls:** Growth stops; the field threshold is not met at the tip.
2. **Spark operates at R_max:** Cold streamer that cannot heat up further. Low power, inefficient.
3. **Spark operates at R_min:** Fully ionized, very hot. May occur in arc-like conditions. Power is high but limited by source.
4. **Spark branches:** Rather than one channel adjusting R, multiple channels form, each seeking its own optimum. Total power may be shared.

## 4. Impedance Matching for Target Spark Length

### 4.1 The Matching Dilemma

During QCW operation, the spark grows from zero to its final length over 5-20 ms. As it grows:
- C_sh increases (more length, more capacitance to ground)
- R_opt_power changes (shifts with capacitance)
- The impedance presented to the source changes continuously

The coil designer must choose a single matching condition (or a tracking strategy). See [[coupled-resonance]] for frequency tracking aspects.

### 4.2 QCW Matching Strategy

**Recommended: Match at 50-70% of target length.**

Reasoning:
- At 0% length: no spark, pure open circuit (infinite impedance). Matching here is meaningless.
- At 100% length: spark is at maximum extent, about to stall. Little time spent here.
- At 50-70%: spark is in its fastest growth phase, consuming the most power. Matching here maximizes energy delivered during the critical growth window.

**Rule of thumb: Match at 60% for first design iteration.**

### 4.3 Formal Optimization

Minimize total energy over the growth trajectory:

```
E_total = integral_0^T [epsilon * L(t) / eta(t)] dt
```

where eta(t) is the power transfer efficiency at time t, and epsilon is the energy per meter (see [[energy-and-growth]]).

**Procedure:**
1. Simulate growth with match points at 0%, 30%, 50%, 70%, 100% of target length.
2. For each match point, compute E_total to reach target length.
3. Choose the match point that minimizes E_total.

### 4.4 Burst Mode Matching

For non-ramping burst operation (fixed drive amplitude):
- Match to final spark length (100%)
- The coil rings up quickly to steady state
- Steady-state impedance matching dominates over transient growth

## 5. Numerical Sensitivity

### 5.1 Sensitivity of R_opt_power to Capacitance Errors

Since R_opt_power = 1/(omega * C_total):

```
dR/R = -dC/C
```

A 20% error in C_total produces a 20% error in R_opt_power. Given that FEMM capacitance extraction is accurate to ~10% and plasma variability is ~50%, this is acceptable.

### 5.2 Sensitivity of Power to R Errors

Near R_opt_power, the power curve is relatively flat. A factor of 2 error in R (R = 0.5*R_opt or R = 2*R_opt) reduces power by only about 20%. This flatness is why the simplified R = R_opt_power approach works well even with significant uncertainties.

### 5.3 Sensitivity to Frequency

Since R_opt_power is inversely proportional to omega:

```
dR/R = -domega/omega = -df/f
```

A 5% frequency shift (common when a spark loads the system; see [[coupled-resonance]]) produces a 5% shift in R_opt_power. This is small compared to other uncertainties.

## 6. Connection to Other Topics

### Key Relationships

- **Derives from:** [[circuit-topology]] (the admittance expressions and phase constraint provide the mathematical foundation)
- **Enables:** [[lumped-model]] (R_opt_power is the default resistance assignment: R = 1/(omega*C_total))
- **Enables:** [[distributed-model]] (each segment's R_opt is computed from its local capacitances using the same principle)
- **Constrains:** [[energy-and-growth]] (the power available for spark growth is bounded by P at R_opt_power)
- **Interacts with:** [[coupled-resonance]] (frequency shift changes R_opt_power; the spark must track)
- **Interacts with:** [[thermal-physics]] (thermal time constants limit how quickly the plasma can adjust to R_opt)
- **Interacts with:** [[streamers-and-leaders]] (streamer vs. leader determines whether R is near R_min or R_max)
- **Measured via:** [[thevenin-method]] (Thevenin extraction allows computing power to any R without re-simulation)

### Summary of Key Results

1. R_opt_power = 1/(omega*(C_mut + C_sh)) maximizes real power to the spark.
2. R_opt_phase = 1/(omega*sqrt(C_mut*(C_mut + C_sh))) minimizes impedance phase magnitude.
3. R_opt_power < R_opt_phase always. R_opt_power gives phi_Z ~ -55 to -75 degrees.
4. The hungry streamer principle: plasma self-optimizes toward R_opt_power via thermal feedback.
5. Constraints (source limits, physical R bounds, thermal lag) can prevent reaching R_opt_power.
6. QCW matching at ~60% of target length is a good first-order design rule.
7. Power is relatively insensitive to R errors near the optimum (flat peak).