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id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
phys-09	Part 3 Review: Spark Growth Physics	Spark Growth Physics	intermediate	60	[phys-01 phys-02 phys-03 phys-04 phys-05 phys-06 phys-07 phys-08]	[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]	[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:

High E-field creates streamers
Current flows → Joule heating
Thermal ionization begins
Leader forms from base
Leader tip launches streamers
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:

Sufficient current
Sufficient time
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:

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

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.

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Part 3 Review: Spark Growth Physics

Concepts Summary

Electric Field Thresholds (Lesson phys-01)

Maximum Voltage-Limited Length (Lesson phys-02)

Energy Per Meter (Lesson phys-03)

Empirical ε Values (Lesson phys-04)

Thermal Memory (Lesson phys-05)

Streamers vs Leaders (Lesson phys-06)

Capacitive Divider (Lesson phys-07)

Freau's Scaling Laws (Lesson phys-08)

Comprehensive Practice Problems

Problem 1: Integrated Design Analysis

Problem 2: Mode Comparison

Problem 3: Thermal Physics Analysis

Problem 4: Scaling and Optimization

Problem 5: Capacitive Divider Deep Dive

Problem 6: Troubleshooting Scenario

Conceptual Questions

Question 1: Synthesis

Question 2: Design Trade-offs

Question 3: Physical Mechanisms

Question 4: Limitations

Part 3 Mastery Checklist

Electric Fields

Energy and Power

Operating Modes

Thermal Physics

Discharge Mechanisms

Capacitive Divider

Scaling Laws

Advanced Challenge Problem

Solutions Provided Separately

14 KiB

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