---
id: phys-08
title: "Freau's Empirical Relationship"
section: "Spark Growth Physics"
difficulty: "advanced"
estimated_time: 35
prerequisites: ["phys-03", "phys-04", "phys-07"]
objectives:
  - 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
tags: ["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.

---

## 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.

---

## 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!
```

---

## 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.

---

## 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
```

---

## 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
```

---

## 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.

---
**Next Lesson:** [Part 3 Review & Exercises](09-review-exercises.md)