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id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
phys-04	Empirical ε Values and Calibration	Spark Growth Physics	intermediate	35	[phys-03]	[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]	[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

11 KiB Raw Blame History

Empirical ε Values and Calibration

Typical ε Values by Operating Mode

QCW (Quasi-Continuous Wave)

Hybrid DRSSTC (Moderate Duty Cycle)

Burst Mode (Hard-Pulsed)

Single-Shot Impulse

Physical Explanation for ε Differences

QCW Efficiency (Low ε)

Burst Inefficiency (High ε)

Analogy: Boiling Water

Calibration Procedure

Step 1: Measure Delivered Energy

Step 2: Measure Final Spark Length

Step 3: Calculate ε

Step 4: Verify and Refine

Thermal Accumulation Effects

WORKED EXAMPLE: Calibration from Data

Solution

WORKED EXAMPLE: Predicting Performance Change

Solution

Summary Table: ε by Operating Mode

Key Takeaways

Practice

11 KiB

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