---
id: phys-09
title: "Part 3 Review: Spark Growth Physics"
section: "Spark Growth Physics"
difficulty: "intermediate"
estimated_time: 60
prerequisites: ["phys-01", "phys-02", "phys-03", "phys-04", "phys-05", "phys-06", "phys-07", "phys-08"]
objectives:
  - Synthesize understanding of spark growth physics
  - Apply multiple concepts to realistic design problems
  - Troubleshoot common performance issues
  - Make informed design decisions based on physics principles
tags: ["review", "synthesis", "design", "troubleshooting", "comprehensive"]
---

# Part 3 Review: Spark Growth Physics

This lesson synthesizes the spark growth physics concepts from Part 3 and provides comprehensive practice problems integrating multiple topics.

## Concepts Summary

### Electric Field Thresholds (Lesson phys-01)

**Key equations:**
```
E_inception ≈ 2-3 MV/m (initial breakdown)
E_propagation ≈ 0.4-1.0 MV/m (sustained growth)
E_tip = κ × E_average = κ × V/L
Growth criterion: E_tip > E_propagation
```

**Key concepts:**
- Tip enhancement factor κ ≈ 2-5
- Altitude/humidity effects: ±20-30%
- Voltage-limited when E_tip < E_propagation

### Maximum Voltage-Limited Length (Lesson phys-02)

**Key equations:**
```
L_max ≈ κ × V_top / E_propagation

FEMM provides: E_tip(V_top, L, geometry)
```

**Key concepts:**
- Both voltage AND power are necessary
- FEMM computes realistic field distributions
- Environmental effects reduce E_propagation at altitude

### Energy Per Meter (Lesson phys-03)

**Key equations:**
```
ΔE ≈ ε × ΔL
dL/dt = P_stream / ε  (when E_tip > E_propagation)
T = ε × L / P_stream  (time to grow)
```

**Key concepts:**
- ε [J/m] is energy per meter of growth
- Includes ionization, heating, radiation, branching
- Theoretical minimum ε ≈ 0.3-0.5 J/m
- Practical values 20-300× higher

### Empirical ε Values (Lesson phys-04)

**Typical ranges:**
```
QCW: ε ≈ 5-15 J/m (efficient leaders)
Hybrid DRSSTC: ε ≈ 20-40 J/m (mixed)
Burst mode: ε ≈ 30-100+ J/m (inefficient streamers)
```

**Key concepts:**
- Calibration: ε = E_delivered / L_measured
- Thermal accumulation: ε(t) = ε₀/(1 + α∫P dt)
- Operating mode choice trades efficiency vs aesthetics

### Thermal Memory (Lesson phys-05)

**Key equations:**
```
τ_thermal = d² / (4α)  where α ≈ 2×10⁻⁵ m²/s
v_convection ≈ √(g × d × ΔT/T_amb)
```

**Typical times:**
```
Thin streamers (d ~ 100 μm): τ ~ 0.1-0.2 ms
Thick leaders (d ~ 3 mm): τ ~ 50-300 ms
Effective persistence: 1-5 ms (streamers), seconds (leaders)
```

**Key concepts:**
- Convection extends persistence beyond pure diffusion
- QCW ramp time << leader thermal time (stays hot)
- Burst gap >> streamer thermal time (cools completely)

### Streamers vs Leaders (Lesson phys-06)

**Comparison:**
```
                Streamers        Leaders
Diameter:       10-100 μm        1-10 mm
Velocity:       ~10⁶ m/s         ~10³ m/s
Temperature:    1000-3000 K      5000-20,000 K
Mechanism:      Photoionization  Thermal ionization
ε:              50-150+ J/m      5-20 J/m
```

**6-step transition:**
1. High E-field creates streamers
2. Current flows → Joule heating
3. Thermal ionization begins
4. Leader forms from base
5. Leader tip launches streamers
6. Fed streamers convert to leader

### Capacitive Divider (Lesson phys-07)

**Key equations:**
```
V_tip = V_topload × C_mut/(C_mut + C_sh)
C_sh ≈ 6.6 pF/m × L
E_tip ∝ V_tip/L ∝ 1/L²  (combined effect)
```

**Key concepts:**
- Voltage division worsens as spark grows
- Self-limiting: longer sparks harder to extend
- Causes sub-linear scaling
- QCW mitigation: active voltage ramping

### Freau's Scaling Laws (Lesson phys-08)

**Empirical relationships:**
```
Burst mode: L ∝ √E  (sub-linear)
QCW mode: L ∝ E^0.7  (less sub-linear)
Repetitive burst: L ∝ P^0.4  (moderate)
```

**Key concepts:**
- Physical origin: capacitive divider + voltage limitation
- QCW advantages: ramping + low ε + thermal accumulation
- Realistic expectations: 4× energy → 2× length (burst)

---

## Comprehensive Practice Problems

### Problem 1: Integrated Design Analysis

**Scenario:**
You are designing a QCW Tesla coil with the following targets:
- Target spark length: L = 2.5 m
- Ramp time: T = 15 ms
- Operating frequency: f = 150 kHz

**Measurements from FEMM:**
- At L = 2.5 m, V_top = 550 kV: E_tip = 0.65 MV/m
- C_mut ≈ 9 pF
- C_sh ≈ 16.5 pF (for 2.5 m spark)

**Questions:**

**(a)** If E_propagation = 0.6 MV/m at your altitude, can the spark reach 2.5 m with 550 kV? Calculate the margin.

**(b)** Assuming ε = 11 J/m for your QCW mode, calculate:
- Total energy required
- Average power required

**(c)** Calculate V_tip using the capacitive divider formula. Compare to the voltage needed if there were no division (C_sh = 0). What percentage is lost?

**(d)** If thermal accumulation reduces ε by 20% during the ramp (ε_effective = 8.8 J/m), recalculate the required power. How much benefit does thermal accumulation provide?

---

### Problem 2: Mode Comparison

**Scenario:**
You have a coil that can operate in either burst mode or QCW mode with the same primary energy E = 120 J.

**Burst mode characteristics:**
- ε_burst = 55 J/m
- No thermal accumulation
- Voltage-limited to L_max = 2.0 m

**QCW mode characteristics:**
- ε_QCW = 13 J/m (initial)
- With thermal accumulation: ε_effective ≈ 10 J/m (average)
- Can ramp voltage to overcome divider partially
- Voltage-limited to L_max = 4.5 m

**Questions:**

**(a)** Calculate predicted spark length for each mode using the power-limited formula L = E/ε. Which limit (power or voltage) dominates in each case?

**(b)** For burst mode at 200 Hz repetition (P_avg = 24 kW), estimate whether thermal memory between pulses affects performance. Use τ_thermal ≈ 0.15 ms for thin streamers.

**(c)** If you want 3 m sparks, which mode should you use? If neither reaches 3 m, what design changes would help?

---

### Problem 3: Thermal Physics Analysis

**Scenario:**
High-speed photography of your QCW coil shows:
- t = 0-0.5 ms: Purple streamers, d ≈ 80 μm
- t = 2-15 ms: White core at base, d ≈ 3 mm
- t > 15 ms (after ramp): Glowing channel rises for ~2 seconds

**Questions:**

**(a)** Calculate thermal diffusion time for:
- Thin streamers (d = 80 μm)
- Thick leaders (d = 3 mm)

**(b)** The observation of leader persistence suggests thermal time constants alone don't explain the 2-second glow. Calculate convection velocity for the 3 mm leader with ΔT = 12,000 K. How does this explain the extended visibility?

**(c)** Your ramp time is 15 ms. Compare this to the leader thermal time constant. Does the leader cool significantly during the ramp? (Use exponential cooling: T(t) ≈ T₀ × exp(-t/τ))

**(d)** Estimate at what time during the ramp the streamer-to-leader transition occurs, given that thermal ionization requires ~5000 K and Joule heating provides ~20 kW to a 1.5 m channel. Use:
- Channel mass: m ≈ d² × L × ρ_air ≈ (3×10⁻³)² × 1.5 × 1.2 ≈ 1.6×10⁻⁵ kg
- Heat capacity: c_p ≈ 1000 J/(kg·K)

---

### Problem 4: Scaling and Optimization

**Scenario:**
You have experimental data from three runs:

| Run | V_primary | E_bang | L_measured | Notes |
|-----|-----------|--------|------------|-------|
| 1   | 300 V     | 45 J   | 1.3 m      | Burst mode |
| 2   | 400 V     | 80 J   | 1.65 m     | Burst mode |
| 3   | 400 V     | 80 J   | 4.2 m      | QCW mode, 12 ms ramp |

**Questions:**

**(a)** Calculate ε for each run. What do the values tell you about the operating modes?

**(b)** Check if Runs 1 and 2 follow L ∝ √E scaling (burst mode). Calculate the predicted L for Run 2 based on Run 1 data.

**(c)** The QCW mode (Run 3) uses the same energy but produces 4.2 m vs 1.65 m for burst. Calculate the efficiency ratio. Where does the "extra length" come from physically?

**(d)** You want to reach 2.5 m in burst mode. Using the L ∝ √E relationship from Runs 1-2, estimate the required energy. Is this upgrade worth it compared to just using QCW mode?

---

### Problem 5: Capacitive Divider Deep Dive

**Scenario:**
Your coil has C_mut = 8.5 pF and operates at V_topload = 480 kV. You want to analyze voltage division effects.

**Questions:**

**(a)** Create a table showing L, C_sh, V_tip, and E_tip (with κ = 3.2) for spark lengths: 0.5 m, 1.0 m, 1.5 m, 2.0 m, 2.5 m, 3.0 m. Use C_sh ≈ 6.6 pF/m × L.

**(b)** If E_propagation = 0.55 MV/m, at what length does growth stall (E_tip = E_propagation)? Use your table and interpolate if needed.

**(c)** Calculate what V_topload would be required to reach 3.0 m if E_propagation = 0.55 MV/m and κ = 3.2. Compare to your current 480 kV capability.

**(d)** Propose two design changes to improve maximum length without increasing V_topload. For each, explain the physical mechanism and estimate the improvement.

---

### Problem 6: Troubleshooting Scenario

**Scenario:**
A coiler reports the following symptoms:
- Coil produces bright, purple, highly-branched 0.8 m sparks
- Primary energy: E_bang = 95 J
- Topload voltage measured: V_top ≈ 420 kV (from FEMM calibration)
- Expected much longer sparks based on energy

**Your analysis:**
- FEMM shows E_tip ≈ 1.1 MV/m at 0.8 m length with 420 kV
- C_mut ≈ 7 pF, C_sh ≈ 5.3 pF (for 0.8 m)
- Operating mode: Hard-pulsed burst, 150 μs pulse width, 200 Hz

**Questions:**

**(a)** Calculate ε from the observed performance. Compare to expected values for burst mode. What does this indicate?

**(b)** The E_tip = 1.1 MV/m is well above typical E_propagation ≈ 0.6 MV/m. Is the coil voltage-limited? What other limit explains the short sparks?

**(c)** The symptom "bright, purple, highly-branched" suggests what type of discharge mechanism? Explain using the streamer vs leader concepts.

**(d)** Calculate thermal diffusion time for a 100 μm streamer. Compare to the 150 μs pulse width and 5 ms gap between pulses. Does thermal memory persist between pulses?

**(e)** Recommend three specific changes to improve spark length. For each, explain the physical principle and estimate the potential improvement.

---

## Conceptual Questions

### Question 1: Synthesis
Explain the complete chain of physics that causes burst mode to scale as L ∝ √E:
- Start with capacitive divider effect
- Connect to E_tip ∝ 1/L²
- Relate to voltage-limited stall condition
- Conclude with scaling relationship

### Question 2: Design Trade-offs
Compare QCW and burst mode for:
- Energy efficiency (ε values)
- Thermal memory utilization
- Voltage division mitigation
- Practical applications
Conclude: when would you choose each mode?

### Question 3: Physical Mechanisms
The streamer-to-leader transition requires three things:
1. Sufficient current
2. Sufficient time
3. Sufficient voltage maintenance

Explain WHY each is necessary using the physics of:
- Joule heating
- Thermal ionization threshold
- Positive feedback mechanisms

### Question 4: Limitations
A coiler claims: "I have 200 kW available, so I should easily get 10 m sparks!"

Identify the flaws in this reasoning. Discuss:
- Voltage vs power limitations
- Energy per meter constraints
- Capacitive divider effects
- Realistic expectations

---

## Part 3 Mastery Checklist

Before proceeding to Part 4, ensure you can:

### Electric Fields
- [ ] Calculate E_average and E_tip from V and L
- [ ] Apply tip enhancement factor κ
- [ ] Determine growth criterion (E_tip vs E_propagation)
- [ ] Account for altitude/environmental effects

### Energy and Power
- [ ] Calculate total energy from ε and L
- [ ] Apply growth rate equation dL/dt = P/ε
- [ ] Predict growth time for target length
- [ ] Distinguish voltage-limited from power-limited

### Operating Modes
- [ ] Explain ε differences between QCW, hybrid, burst
- [ ] Calculate expected length from energy and ε
- [ ] Recognize mode from observed spark characteristics
- [ ] Choose appropriate mode for design goals

### Thermal Physics
- [ ] Calculate thermal diffusion times for different diameters
- [ ] Estimate convection velocity from temperature excess
- [ ] Explain QCW advantage via thermal memory
- [ ] Predict streamer vs leader formation based on timescales

### Discharge Mechanisms
- [ ] Distinguish streamers from leaders (6 key properties)
- [ ] Describe the 6-step transition sequence
- [ ] Explain photoionization vs thermal ionization
- [ ] Predict which mechanism dominates in a given mode

### Capacitive Divider
- [ ] Calculate V_tip from C_mut, C_sh, V_topload
- [ ] Explain how C_sh increases with length
- [ ] Derive E_tip ∝ 1/L² relationship
- [ ] Identify mitigation strategies

### Scaling Laws
- [ ] Apply L ∝ √E for burst mode predictions
- [ ] Explain physical origin of sub-linear scaling
- [ ] Recognize QCW shows better scaling (L ∝ E^0.7)
- [ ] Set realistic expectations for energy/power increases

---

## Advanced Challenge Problem

**Scenario:** Design a QCW coil from scratch to achieve 3.5 m sparks.

**Given constraints:**
- Budget allows C_primary up to 1.0 μF
- V_primary limited to 600 V (safety)
- Topload options: 20 cm toroid (C_top ≈ 25 pF) or 35 cm toroid (C_top ≈ 45 pF)
- Target ramp time: 10-15 ms
- Sea level operation (E_propagation = 0.6 MV/m)

**Your task:**

1. **Energy calculation:**
   - Choose ε for QCW mode
   - Calculate total energy required for 3.5 m
   - Verify this is achievable with C_primary and V_primary

2. **Voltage requirement:**
   - Estimate C_mut for each topload option (use C_mut ≈ 0.7 × C_top as approximation)
   - Calculate C_sh for 3.5 m spark
   - For each topload, calculate V_topload needed to achieve E_tip = 0.7 MV/m at 3.5 m (assume κ = 3.0)
   - Include capacitive division effects

3. **Power analysis:**
   - For T_ramp = 12 ms, calculate required average power
   - Estimate peak power (assume 1.5× average for QCW)
   - Check if this is reasonable for DRSSTC primary

4. **Thermal verification:**
   - Estimate leader diameter (2-4 mm typical)
   - Calculate thermal time constant
   - Verify ramp time << thermal time (QCW condition satisfied)

5. **Final recommendation:**
   - Which topload should be used? Why?
   - Is the 3.5 m target achievable with given constraints?
   - If not, what would you change and why?

---

**Next Section:** [Part 4: Advanced Modeling](../04-advanced-modeling/01-introduction.md)

---

## Solutions Provided Separately

{exercise:phys-ex-comprehensive}

Detailed solutions to all practice problems are available in the solutions guide to allow self-assessment and learning.