---
id: phys-04
title: "Empirical ε Values and Calibration"
section: "Spark Growth Physics"
difficulty: "intermediate"
estimated_time: 35
prerequisites: ["phys-03"]
objectives:
  - Learn typical ε values for different operating modes
  - Understand why QCW, DRSSTC, and burst modes have different ε
  - Calibrate ε from experimental measurements
  - Apply thermal accumulation effects to refine ε predictions
tags: ["epsilon", "calibration", "QCW", "DRSSTC", "burst-mode", "thermal-accumulation"]
---

# Empirical ε Values and Calibration

The energy per meter (ε) is not a universal constant - it depends strongly on the operating mode. Understanding typical values and calibration methods is essential for accurate spark growth modeling.

## Typical ε Values by Operating Mode

### QCW (Quasi-Continuous Wave)

**ε ≈ 5-15 J/m**

**Characteristics:**
- Long ramp times: 5-20 ms
- Channel stays hot throughout growth
- Efficient leader formation
- Minimal re-ionization needed
- Each joule efficiently extends length

**Why low ε (efficient)?**
- Continuous power maintains channel ionization
- Thermal ionization kept active
- Leaders form and persist
- Minimal energy wasted on re-starting

**Typical coil parameters:**
- Medium-high power: 10-100 kW
- Moderate duty cycle: 1-10%
- Linear voltage ramp
- Long sparks: 2-5+ m

### Hybrid DRSSTC (Moderate Duty Cycle)

**ε ≈ 20-40 J/m**

**Characteristics:**
- Medium pulse lengths: 1-5 ms
- Mix of streamers and leaders
- Some thermal accumulation between pulses
- Moderate efficiency

**Why moderate ε?**
- Not quite continuous like QCW
- Some cooling between bursts
- Partial re-ionization required
- Both streamer and leader mechanisms active

**Typical coil parameters:**
- High power: 50-200 kW peak
- Moderate duty cycle: 5-15%
- Partial interrupter control
- Good balance: length and brightness

### Burst Mode (Hard-Pulsed)

**ε ≈ 30-100+ J/m**

**Characteristics:**
- Short pulses: <500 μs typical
- Channel cools between pulses
- Mostly streamers, bright but short
- Must re-ionize repeatedly
- Poor length efficiency

**Why high ε (inefficient)?**
- Peak power → intense brightening and branching
- Channel cools between bursts (ms timescale)
- Energy dumped into light and heat, not length
- Must restart from cold each time
- High ionization overhead

**Typical coil parameters:**
- Very high peak power: 100-500+ kW
- Low duty cycle: 0.1-2%
- Bang energy: 10-100+ J per burst
- Short sparks: 0.5-2 m despite high energy

### Single-Shot Impulse

**ε ≈ 50-150+ J/m**

**Characteristics:**
- One-time discharge (capacitor bank)
- No thermal memory from previous events
- All energy must come from single pulse
- Very high ε due to complete inefficiency

**Why very high ε?**
- Starting from completely cold air
- No accumulated ionization
- Transient streamer formation
- Most energy into flash and noise

## Physical Explanation for ε Differences

### QCW Efficiency (Low ε)

**Energy flow:**
```
1. Initial streamers form (t = 0)
2. Current flows → Joule heating (t = 0-1 ms)
3. Channel heats → thermal ionization (t = 1-2 ms)
4. Leader forms from base (t = 2-5 ms)
5. Leader maintained by continuous power (t = 5-20 ms)
6. New growth builds on existing hot ionization
7. Minimal wasted energy
```

**Result:** Each joule goes into extending the channel, not re-creating what already exists.

### Burst Inefficiency (High ε)

**Energy flow:**
```
1. Pulse creates bright streamer (t = 0-100 μs)
2. Pulse ends, no more power (t = 100 μs)
3. Channel begins cooling (t = 0.1-1 ms)
4. Thermal diffusion and convection cool channel
5. Ionization recombines
6. Next pulse must re-ionize cold gas (t = 1-10 ms)
7. Energy wasted heating the same air repeatedly
```

**Result:** Energy into brightening and repeated ionization overhead, not cumulative length.

### Analogy: Boiling Water

**Low ε (QCW):**
- Keep burner on continuously
- Maintain simmer (steady state)
- Efficient: minimal energy to maintain temperature

**High ε (Burst):**
- Pulse burner on/off repeatedly
- Water cools between pulses
- Inefficient: must reheat repeatedly

## Calibration Procedure

To calibrate ε for your specific coil:

### Step 1: Measure Delivered Energy

**From SPICE simulation:**
```
E_delivered = ∫ P_spark(t) dt

where P_spark = instantaneous power to spark
Integration from t = 0 to end of ramp
```

**From measurements (if available):**
```
E_delivered ≈ E_capacitor - E_losses

where E_capacitor = ½ C_primary V_primary²
      E_losses = resistive, core, switching losses
```

### Step 2: Measure Final Spark Length

**Direct measurement:**
- Photograph spark with scale reference
- Measure from topload to tip
- Average over multiple runs (sparks vary!)
- Use median or typical length, not maximum outlier

**Typical measurement uncertainty:**
- ±10-20% due to spark variability
- Branching makes "length" ambiguous
- Use main channel length

### Step 3: Calculate ε

```
ε = E_delivered / L_final  [J/m]

Example:
E_delivered = 45 J (from SPICE)
L_final = 1.8 m (measured)

ε = 45 J / 1.8 m = 25 J/m
```

### Step 4: Verify and Refine

**Repeat for different power levels:**
- Change primary voltage or pulse width
- Measure new E_delivered and L_final
- Calculate ε for each run
- Average to get robust estimate

**Check for consistency:**
- ε should be approximately constant (±30%)
- Large variations indicate:
  - Voltage-limited at some power levels
  - Thermal accumulation effects
  - Operating mode changes

## Thermal Accumulation Effects

For more advanced modeling, ε can decrease during long ramps due to thermal accumulation:

```
ε(t) = ε₀ / (1 + α × ∫P_stream dt)

where:
  ε₀ = initial energy per meter [J/m]
  α = thermal accumulation factor [1/J]
  ∫P_stream dt = accumulated energy [J]
```

**Physical meaning:**
- As channel heats up, ionization becomes easier
- Less energy needed per meter as temperature rises
- ε decreases with accumulated heating

**Typical values:**
- ε₀ ≈ 15 J/m (initial, cold start)
- α ≈ 0.01-0.05 [1/J]
- After 50 J accumulated: ε ≈ 15/(1 + 0.03×50) = 6 J/m

**When to use:**
- Long QCW ramps (>10 ms)
- High accumulated energy (>30 J)
- For short bursts: ε ≈ ε₀ (constant)

**Simplified model:**
Most practitioners use constant ε for simplicity:
- Choose ε representing average over ramp
- Simpler and usually adequate
- Advanced users can implement ε(t) in simulation

---

## WORKED EXAMPLE: Calibration from Data

**Given:**
Three experimental runs on a QCW coil:

| Run | V_primary | E_delivered | L_measured |
|-----|-----------|-------------|------------|
| 1   | 200 V     | 25 J        | 2.2 m      |
| 2   | 250 V     | 38 J        | 3.1 m      |
| 3   | 300 V     | 55 J        | 4.5 m      |

**Find:**
(a) Calculate ε for each run
(b) Average ε for this coil
(c) Assess consistency

### Solution

**Part (a): ε for each run**

```
Run 1: ε₁ = E₁ / L₁ = 25 J / 2.2 m = 11.4 J/m
Run 2: ε₂ = E₂ / L₂ = 38 J / 3.1 m = 12.3 J/m
Run 3: ε₃ = E₃ / L₃ = 55 J / 4.5 m = 12.2 J/m
```

**Part (b): Average ε**

```
ε_avg = (ε₁ + ε₂ + ε₃) / 3
      = (11.4 + 12.3 + 12.2) / 3
      = 12.0 J/m

Recommended value: ε ≈ 12 J/m
```

**Part (c): Consistency assessment**

```
Standard deviation: σ ≈ 0.5 J/m
Coefficient of variation: CV = σ/μ = 0.5/12 = 4.2%

Excellent consistency! (<5% variation)
```

**Interpretation:**
- ε is nearly constant across power range
- Coil is NOT voltage-limited in this range
- Pure power-limited growth (field threshold always met)
- Can confidently use ε = 12 J/m for predictions

**If we saw large variation:**
```
Example: ε₁ = 10 J/m, ε₂ = 15 J/m, ε₃ = 30 J/m

This would indicate:
- Run 3 hitting voltage limit (inefficient growth)
- Possible mode transition (streamers vs leaders)
- Need to reassess model assumptions
```

---

## WORKED EXAMPLE: Predicting Performance Change

**Given:**
- Current coil: Burst mode, ε = 65 J/m, E_bang = 80 J, L_typical = 1.2 m
- Proposed upgrade: Convert to QCW with ε = 12 J/m, same E_total = 80 J

**Find:**
(a) Predicted length after QCW conversion
(b) Percentage improvement
(c) Required power for 10 ms ramp

### Solution

**Part (a): Predicted QCW length**

```
L_QCW = E_total / ε_QCW
      = 80 J / 12 J/m
      = 6.67 m

Predicted length ≈ 6.7 m
```

**Part (b): Improvement**

```
Improvement = (L_QCW - L_burst) / L_burst × 100%
            = (6.67 - 1.2) / 1.2 × 100%
            = 456% increase in length!

Or: 6.67/1.2 = 5.6× longer sparks
```

**Part (c): Required power**

```
For 10 ms ramp:
P_avg = E_total / T_ramp
      = 80 J / 0.010 s
      = 8,000 W
      = 8 kW average

Peak power higher (depends on waveform)
Typical: P_peak ≈ 1.5-2 × P_avg ≈ 12-16 kW
```

**Reality check:**
- 6.7 m prediction assumes NOT voltage-limited
- Actual length limited by topload voltage capability
- Still expect major improvement over burst mode
- Might achieve 3-4 m instead of 6.7 m (voltage limit)

---

## Summary Table: ε by Operating Mode

| Mode | ε Range [J/m] | Characteristics | Best For |
|------|---------------|-----------------|----------|
| **QCW** | 5-15 | Efficient leaders, long ramps | Maximum length |
| **DRSSTC Hybrid** | 20-40 | Mixed streamers/leaders | Balanced length & brightness |
| **Burst Mode** | 30-100+ | Bright streamers, short pulses | Visual spectacle, music |
| **Single-Shot** | 50-150+ | One-time discharge | Impulse testing, demonstrations |

**Choosing operating mode:**
- **Goal: Length** → QCW (low ε)
- **Goal: Brightness** → Burst (high peak power)
- **Goal: Music/modulation** → Burst (rapid on/off)
- **Goal: Efficiency** → QCW (low ε, lower losses)

---

## Key Takeaways

- **QCW: ε ≈ 5-15 J/m** - Most efficient, maintains hot channel
- **Hybrid DRSSTC: ε ≈ 20-40 J/m** - Moderate efficiency, mixed mechanisms
- **Burst mode: ε ≈ 30-100+ J/m** - Least efficient, repeated re-ionization
- **Calibration**: ε = E_delivered / L_measured from experimental runs
- **Consistency check**: ε should be approximately constant if power-limited
- **Thermal accumulation**: Advanced models use ε(t) decreasing with heating
- **Operating mode choice**: Trades off length efficiency vs brightness/aesthetics

## Practice

{exercise:phys-ex-04}

**Problem 1:** A coil delivers 60 J in burst mode and produces 0.9 m sparks. Calculate ε. If converted to QCW with same energy, estimate new length assuming ε = 10 J/m.

**Problem 2:** Calibration runs give: ε₁ = 14 J/m (25 J delivered), ε₂ = 13 J/m (40 J), ε₃ = 28 J/m (90 J). What does the sudden increase in ε₃ suggest?

**Problem 3:** Explain why burst mode has higher ε than QCW despite delivering the same total energy. What happens to the "wasted" energy?

---
**Next Lesson:** [Thermal Memory Effects](05-thermal-memory.md)