plasma-expert/spark-lessons/lessons/03-spark-physics/08-freau-relationship.md

id	title	section	difficulty	estimated_time	prerequisites	objectives	tags
phys-08	Freau's Empirical Relationship	Spark Growth Physics	advanced	35	[phys-03 phys-04 phys-07]	[Understand Freau's empirical L ∝ √E scaling for burst mode Derive the physical explanation from capacitive divider effects Recognize differences between burst mode and QCW scaling Apply scaling laws to predict performance changes]	[freau scaling-laws sub-linear burst-mode QCW empirical]

Freau's Empirical Relationship

Tesla coil community observations have revealed consistent patterns in how spark length scales with energy. Understanding these scaling laws helps predict performance and set realistic expectations.

The Empirical Observations

Daniel Freau and others in the Tesla coil community documented:

Single-Shot Burst Mode

L ∝ √E

where:
  L = spark length [m]
  E = bang energy (capacitor energy per pulse) [J]

Example measurements:

• 25 J → 0.8 m
• 100 J → 1.6 m (4× energy → 2× length)
• 400 J → 3.2 m (16× energy → 4× length)

Sub-linear scaling: Doubling energy does NOT double length; only increases by √2 ≈ 1.41×.

Repetitive Burst Operation

L ∝ P_avg^n

where:
  P_avg = average power [W]
  n ≈ 0.3 to 0.5 (empirical exponent)

Example:

• 10 kW → 1.2 m
• 40 kW → 2.0 m (4× power → 1.67× length, n ≈ 0.4)

Still sub-linear: More power helps, but with diminishing returns.

QCW Mode

L ∝ E^m

where:
  m ≈ 0.6 to 0.8 (closer to linear than burst)

Example:

• 50 J → 3.5 m
• 200 J → 9.0 m (4× energy → 2.6× length, m ≈ 0.7)

Less sub-linear: QCW shows better scaling than burst mode.

Physical Explanation: Voltage-Limited Burst Mode

The L ∝ √E relationship for burst mode comes from the interplay of capacitive divider effects and voltage limitations.

Derivation from First Principles

Step 1: Growth stops when E_tip = E_propagation

E_tip = κ × V_tip / L

At stall:
κ × V_tip / L_max = E_propagation

Solving for L_max:
L_max = κ × V_tip / E_propagation

Step 2: Voltage division affects V_tip

From capacitive divider (Lesson 07):

V_tip ≈ V_topload × C_mut / (C_mut + C_sh)

For long sparks (C_sh >> C_mut):
C_sh ≈ αL  (α ≈ 6.6 pF/m)

V_tip ≈ V_topload × C_mut / (αL)
     ∝ V_topload / L

Step 3: Substitute into stall condition

L_max = κ × V_tip / E_propagation
      = κ × (V_topload/L_max) / E_propagation

Multiply both sides by L_max:
L_max² = κ × V_topload / E_propagation

Solving for L_max:
L_max = √(κ × V_topload / E_propagation)
      ∝ √V_topload

Step 4: Connect to energy

For a capacitor discharge (burst mode):

E_bang = ½ C_primary V_primary²

If transformer ratio is fixed:
V_topload ∝ V_primary ∝ √E_bang

Therefore:
L_max ∝ √V_topload ∝ √(√E_bang) ∝ E_bang^(1/4) to E_bang^(1/2)

The exact exponent depends on:

• Whether topload voltage saturates (breakdown limit)
• Impedance matching (affects voltage transfer)
• Spark loading (changes transformer ratio during pulse)

Empirically observed: The exponent clusters around 0.5, giving L ∝ √E.

Simplified Intuition

The vicious cycle:

Longer spark → Higher C_sh → Lower V_tip → Lower E_tip → Harder to grow

E_tip ∝ V_tip/L ∝ (V_top/L)/L ∝ V_top/L²

Growth requires: V_top/L² ≥ E_propagation/κ
                V_top ≥ (E_propagation/κ) × L²

For fixed V_top:
L_max² ≤ κ × V_top/E_propagation
L_max ∝ √V_top ∝ √E

Physical meaning: The capacitive divider creates a quadratic penalty (E_tip ∝ 1/L²), resulting in square-root scaling with energy/voltage.

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WORKED EXAMPLE: Burst Mode Scaling

Given:

• Coil operates in burst mode
• Test 1: E_bang = 40 J → L = 1.1 m
• Test 2: E_bang = 160 J → L = ?

Find: Predicted length for Test 2 using L ∝ √E

Solution

L₂/L₁ = √(E₂/E₁)

L₂ = L₁ × √(E₂/E₁)
   = 1.1 m × √(160/40)
   = 1.1 m × √4
   = 1.1 m × 2
   = 2.2 m

Predicted: 2.2 m for 160 J

Verification:

• 4× energy (40 J → 160 J)
• 2× length (1.1 m → 2.2 m)
• Consistent with √E scaling ✓

If scaling were linear (wrong):

L₂ = 1.1 m × (160/40) = 4.4 m (incorrect!)

Key insight: Quadrupling energy only doubles length in voltage-limited burst mode.

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Why QCW Shows Different Scaling

QCW mode shows less sub-linear scaling (L ∝ E^0.6 to E^0.8) because of active mitigation:

QCW Advantages

1. Voltage ramping:

V_topload(t) increases during ramp
Actively compensates for capacitive divider
Can maintain E_tip > E_propagation longer

2. Leader formation:

Lower ε (5-15 J/m vs 30-100 J/m for burst)
Same energy produces longer spark
Better inherent efficiency

3. Thermal accumulation:

Channel stays hot (no cooling between pulses)
Effective ε decreases during ramp
Later growth more efficient than early growth

Modified Scaling

Effective relationship:

L_max ∝ (V_top(t_final) / ε_effective)

Both numerator and denominator improve during QCW ramp:
- V_top(t) increases (ramping)
- ε_effective decreases (thermal accumulation)

Result: L ∝ E^m where m ≈ 0.6-0.8

Still sub-linear, but better than burst mode:

• Burst: L ∝ E^0.5
• QCW: L ∝ E^0.7 (typical)

Ratio improvement:

For 4× energy increase:
Burst: 4^0.5 = 2.0× longer
QCW: 4^0.7 = 2.64× longer

QCW gains 32% more length for same energy increase!

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WORKED EXAMPLE: Comparing Modes

Given:

• Burst mode coil: 100 J → 1.5 m (baseline)
• QCW conversion: Same 100 J total energy
• Burst scaling: L ∝ E^0.5
• QCW scaling: L ∝ E^0.7

Find: (a) Predicted QCW length at 100 J (b) Energy needed for 3 m in each mode (c) Which mode is more "scalable"?

Solution

Part (a): QCW length at 100 J

Need calibration point for QCW. Assume QCW has lower ε:

From ε perspective:
Burst: ε_burst = 100 J / 1.5 m = 67 J/m
QCW: ε_QCW ≈ 12 J/m (typical)

Linear estimate:
L_QCW = 100 J / 12 J/m = 8.3 m

But voltage limit will reduce this.
Realistic with same topload: ~4-5 m

We'll use 4.5 m as calibration point.

Part (b): Energy for 3 m in each mode

Burst mode:

L ∝ E^0.5
L₁ = 1.5 m at E₁ = 100 J
L₂ = 3 m at E₂ = ?

(L₂/L₁)² = E₂/E₁
(3/1.5)² = E₂/100
4 = E₂/100
E₂ = 400 J needed for 3 m

QCW mode:

L ∝ E^0.7
L₁ = 4.5 m at E₁ = 100 J
L₂ = 3 m at E₂ = ?

(L₂/L₁)^(1/0.7) = E₂/E₁
(3/4.5)^1.43 = E₂/100
0.667^1.43 = E₂/100
0.568 = E₂/100
E₂ = 56.8 J needed for 3 m

Actually, 3 m < 4.5 m, so less energy needed.
Correct calculation:
(3/4.5)^1.43 = E₂/100
E₂ ≈ 56.8 J

Wait, let me recalculate for going DOWN in length:

If QCW produces 4.5 m at 100 J, then for 3 m:
(E₂/E₁) = (L₂/L₁)^(1/0.7)
E₂/100 = (3/4.5)^1.43
E₂ = 100 × 0.568 ≈ 57 J

QCW needs only 57 J for 3 m
Burst needs 400 J for 3 m

QCW is 7× more energy-efficient!

Part (c): Which is more scalable?

Scalability = how much length increases per energy increase

Burst: L ∝ E^0.5
  Doubling energy: 2^0.5 = 1.41× length gain

QCW: L ∝ E^0.7
  Doubling energy: 2^0.7 = 1.62× length gain

QCW is more scalable: 15% better length gain per energy doubling

Practical implication: QCW benefits more from increased energy/power than burst mode.

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Repetitive Operation Scaling

For repetitive burst mode (many pulses per second):

L ∝ P_avg^n  where n ≈ 0.3-0.5

Physical explanation:

Thermal memory between pulses:

• If repetition rate is fast enough (~100+ Hz)
• Some ionization/thermal memory carries over
• Effective ε decreases slightly
• Better scaling than single-shot (n > 0.5)

Power vs energy:

P_avg = E_bang × f  (f = pulse rate)

For fixed E_bang:
L ∝ P^n ∝ (E × f)^n ∝ f^n

More frequent pulses help, but sub-linearly

Example:

100 Hz, 40 J per pulse: P_avg = 4 kW → L₁
200 Hz, 40 J per pulse: P_avg = 8 kW → L₂

L₂/L₁ = (8/4)^0.4 = 2^0.4 = 1.32

Only 32% longer despite doubling pulse rate

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Practical Implications

Design Decisions

For maximum length:

• Use QCW mode (better scaling, lower ε)
• Large topload (fight capacitive divider)
• Modest energy with long ramp (exploit thermal accumulation)

For visual spectacle:

• Use burst mode (bright, branched)
• High peak power (dramatic but short sparks)
• Accept poor energy efficiency

Performance Predictions

When upgrading primary capacitance:

C_primary doubles → E_bang doubles (same V_primary)

Burst mode: L increases by √2 = 1.41×
QCW mode: L increases by 2^0.7 = 1.62×

QCW benefits more from the upgrade

When adding more power:

QCW mode: More sensitive to power increases
  Can ramp voltage higher/faster
  Better return on investment

Burst mode: Less sensitive
  Voltage-limited earlier
  Diminishing returns

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Key Takeaways

• Burst mode scaling: L ∝ √E (square root of energy)
• Physical origin: Capacitive divider creates E_tip ∝ 1/L² penalty
• QCW scaling: L ∝ E^0.7 (less sub-linear, better than burst)
• QCW advantages: Voltage ramping + lower ε + thermal accumulation
• Repetitive burst: L ∝ P^0.3-0.5, slight improvement over single-shot
• Design implication: QCW is more "scalable" - better returns on energy/power increases
• Realistic expectations: Quadrupling energy only doubles burst-mode length

Practice

{exercise:phys-ex-08}

Problem 1: A burst coil produces 1.4 m sparks with 60 J per pulse. Using L ∝ √E, predict: (a) Length with 135 J per pulse (b) Energy needed for 2.1 m sparks

Problem 2: Compare two upgrade paths for a QCW coil currently at 80 J, 3.2 m (assume L ∝ E^0.7):

• Option A: Upgrade to 160 J
• Option B: Upgrade to 240 J Calculate expected length for each option.

Problem 3: Explain why QCW shows L ∝ E^0.7 instead of L ∝ √E. What three mechanisms contribute to better-than-square-root scaling?

Problem 4: A repetitive burst coil runs at 150 Hz with 30 J/pulse (4.5 kW average) and produces 1.0 m sparks. If pulse rate increases to 300 Hz (9 kW, same energy/pulse) and L ∝ P^0.4, predict new length.

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Next Lesson: Part 3 Review & Exercises