---
id: phys-07
title: "The Capacitive Divider Problem"
section: "Spark Growth Physics"
difficulty: "advanced"
estimated_time: 45
prerequisites: ["fund-04", "fund-05", "phys-01", "phys-02"]
objectives:
  - Understand how voltage divides between C_mut and C_sh
  - Calculate V_tip as a function of spark length
  - Recognize why tip voltage drops as spark grows
  - Apply capacitive division to predict sub-linear scaling
tags: ["capacitive-divider", "voltage-division", "C_mut", "C_sh", "V_tip", "sub-linear"]
---

# The Capacitive Divider Problem

A critical limitation affects all Tesla coils: as the spark grows longer, the voltage at the tip **decreases** even if topload voltage is maintained. This "capacitive divider effect" creates progressively harder conditions for continued growth.

## Review: Spark Circuit Topology

From Fundamentals, recall the spark circuit:

```
       [C_mut]
Topload ----||---- Node_spark (spark base)
                      |
                     [R]
                      |
                   [C_sh]
                      |
                     GND
```

**Components:**
- **C_mut**: Mutual capacitance between topload and spark
- **C_sh**: Shunt capacitance from spark to ground
- **R**: Spark resistance (varies with ionization)

**Key insight:** The spark sees a **voltage divider** between topload and ground!

## Voltage Division Equation

The general voltage divider with complex impedances:

```
V_tip = V_topload × Z_mut / (Z_mut + Z_sh)

where:
  Z_mut = (1/jωC_mut) || R  (parallel combination of capacitance and resistance)
  Z_sh = 1/(jωC_sh)  (capacitive reactance)
```

**In complex form:**

```
Y_mut = jωC_mut + 1/R  (admittance of parallel combination)
Z_mut = 1/Y_mut

Y_sh = jωC_sh
Z_sh = 1/Y_sh

V_tip = V_topload × Z_mut / (Z_mut + Z_sh)
```

This is complex-valued (magnitude and phase).

## Open-Circuit Limit (No Current Flow)

**Simplified case:** When R → ∞ (no conduction, purely capacitive):

```
V_tip = V_topload × C_mut / (C_mut + C_sh)
```

This is the **capacitive voltage divider** formula.

**Physical interpretation:**
- Charges distribute between two capacitors in series
- Voltage splits proportionally to inverse capacitances
- As C_sh increases, V_tip decreases

### The Problem: C_sh Grows with Length

**Empirical relationship:**

```
C_sh ≈ 2 pF/foot × L_feet

Or in SI units:
C_sh ≈ 6.6 pF/m × L_meters
```

**As spark grows:**
- Length L increases
- C_sh increases (proportional to length)
- Denominator (C_mut + C_sh) increases
- V_tip decreases!

**This is self-limiting:** Longer sparks make it harder to grow even longer.

---

## WORKED EXAMPLE: Open-Circuit Voltage Division

**Given:**
- V_topload = 400 kV (constant, maintained by primary)
- C_mut = 8 pF (approximately constant)
- Spark grows from 1 ft to 6 ft

**Find:** V_tip at L = 1, 2, 3, 4, 5, 6 feet

### Solution

**At L = 1 ft:**

```
C_sh = 2 pF/ft × 1 ft = 2 pF

V_tip = 400 kV × 8/(8+2)
      = 400 kV × 8/10
      = 320 kV (80% of V_topload)
```

**At L = 2 ft:**

```
C_sh = 4 pF

V_tip = 400 × 8/12
      = 267 kV (67%)
```

**At L = 3 ft:**

```
C_sh = 6 pF

V_tip = 400 × 8/14
      = 229 kV (57%)
```

**At L = 4 ft:**

```
C_sh = 8 pF

V_tip = 400 × 8/16
      = 200 kV (50%)
```

**At L = 5 ft:**

```
C_sh = 10 pF

V_tip = 400 × 8/18
      = 178 kV (44%)
```

**At L = 6 ft:**

```
C_sh = 12 pF

V_tip = 400 × 8/20
      = 160 kV (40%)
```

### Summary Table

| Length | C_sh | V_tip | % of V_top | E_avg (MV/m) |
|--------|------|-------|------------|--------------|
| 1 ft (0.3 m) | 2 pF | 320 kV | 80% | 1.07 |
| 2 ft (0.6 m) | 4 pF | 267 kV | 67% | 0.89 |
| 3 ft (0.9 m) | 6 pF | 229 kV | 57% | 0.76 |
| 4 ft (1.2 m) | 8 pF | 200 kV | 50% | 0.67 |
| 5 ft (1.5 m) | 10 pF | 178 kV | 44% | 0.59 |
| 6 ft (1.8 m) | 12 pF | 160 kV | 40% | 0.53 |

**Observations:**
- V_tip drops to 40% of V_topload by 6 ft
- E_avg = V_tip/L decreases even faster
- Growth becomes progressively harder

{image:voltage-division-vs-length-plot}

---

## With Finite Resistance

Real sparks have finite resistance R ≈ R_opt_power (from optimization):

```
R_opt_power ≈ 1/(ω(C_mut + C_sh))
```

**Effect of finite R:**

```
Z_mut = R || (1/jωC_mut)

For R ≈ R_opt:
Z_mut ≈ (1-j)/(2ωC_mut)  (complex, 45° phase lag)

V_tip magnitude is LOWER than open-circuit case
V_tip has phase shift relative to V_topload
```

**Result:** Voltage division is **worse** than the open-circuit case!

### Detailed Calculation (Advanced)

For R = R_opt_power = 1/(ω(C_mut + C_sh)):

```
Y_mut = jωC_mut + 1/R
      = jωC_mut + ω(C_mut + C_sh)
      = ω(C_mut + C_sh) + jωC_mut

Z_mut = 1/Y_mut
      = 1 / [ω(C_mut + C_sh)(1 + jC_mut/(C_mut + C_sh))]

Z_sh = 1/(jωC_sh)

Ratio:
V_tip/V_top = Z_mut/(Z_mut + Z_sh)

After algebra (details omitted):
|V_tip/V_top| ≈ C_mut/(C_mut + C_sh) × (1/√2)

Approximately 0.707× the open-circuit value!
```

**Practical conclusion:** With conduction current, voltage division is ~30% worse than capacitive-only case.

## Impact on E_tip and Growth

Recall the tip field:

```
E_tip = κ × V_tip / L
```

**As L increases:**

**Numerator effect (voltage division):**
```
V_tip ∝ C_mut / (C_mut + C_sh)
     ≈ C_mut / (C_mut + αL)  (where α = 6.6 pF/m)
     ≈ 1 / (1 + αL/C_mut)

For large L: V_tip ∝ 1/L
```

**Denominator effect (geometry):**
```
Division by L
```

**Combined:**
```
E_tip ∝ V_tip / L
     ∝ (1/L) / L
     ∝ 1/L²

E_tip decreases as L²!
```

**This is devastating for long spark growth.**

## Sub-Linear Scaling Prediction

From the capacitive divider effect, we can predict scaling:

**Growth stops when:**
```
E_tip(L_max) = E_propagation

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

**Substituting voltage division:**
```
κ × [V_topload × C_mut/(C_mut + αL_max)] / L_max = E_propagation

Rearranging:
V_topload × C_mut / (C_mut + αL_max) = E_propagation × L_max / κ

V_topload × C_mut = E_propagation × L_max × (C_mut + αL_max) / κ
```

**For large L (C_sh >> C_mut):**
```
V_topload × C_mut ≈ E_propagation × L_max × αL_max / κ

V_topload × C_mut ≈ (E_propagation × α / κ) × L_max²

Solving for L_max:
L_max ∝ √(V_topload × C_mut)
      ∝ √(V_topload)  (if C_mut approximately constant)
```

**Connection to energy:**

If topload voltage is limited by breakdown, V_top ∝ √E (from capacitor energy):
```
E_cap = ½ C_top V_top²
V_top ∝ √E

Therefore:
L_max ∝ √V_top ∝ √(√E) ∝ E^(1/4) to E^(1/2)

Approximately: L ∝ √E
```

**This explains Freau's empirical observation:** For burst mode (voltage-limited), spark length scales as square root of energy!

---

## WORKED EXAMPLE: Scaling Prediction

**Given:**
- Coil A: V_top = 300 kV, produces L = 1.2 m spark
- Coil B: Same design, but V_top = 450 kV (1.5× voltage)

**Find:** Predicted length for Coil B using:
(a) Linear scaling (naive)
(b) Sub-linear scaling (capacitive divider)

### Solution

**Part (a): Linear scaling (incorrect)**

```
If L ∝ V:
L_B = L_A × (V_B/V_A)
    = 1.2 m × (450/300)
    = 1.2 m × 1.5
    = 1.8 m
```

**Part (b): Sub-linear scaling (more realistic)**

```
If L ∝ √V (from capacitive divider):
L_B = L_A × √(V_B/V_A)
    = 1.2 m × √(450/300)
    = 1.2 m × √1.5
    = 1.2 m × 1.225
    = 1.47 m

Only 1.47 m instead of 1.8 m!
```

**Actual measurements typically show:** L_B ≈ 1.4-1.5 m, confirming sub-linear scaling.

**Percentage improvement:**
- Linear prediction: 50% longer (wrong)
- Sub-linear prediction: 23% longer (correct)
- Capacitive divider limits gains from higher voltage

---

## Mitigation Strategies

How can we fight the capacitive divider effect?

### 1. Increase C_mut

**Larger topload:**
```
C_top increases → C_mut increases
→ C_mut/(C_mut + C_sh) ratio improves
→ Better V_tip retention
```

**Effect:**
- Diminishes relative impact of C_sh
- Requires larger topload (practical limits)

### 2. Active Voltage Ramping (QCW)

**Strategy:**
```
Ramp V_topload upward as spark grows
Compensate for voltage division
Maintain E_tip above threshold longer
```

**This is the QCW advantage:**
- Not fighting capacitive divider directly
- But actively increasing numerator (V_topload)
- Allows longer sparks than fixed voltage

### 3. Reduce C_sh (Limited Options)

**Physical constraints:**
- C_sh ∝ L (fundamental geometry)
- Cannot eliminate
- Thin spark slightly better (smaller cross-section)
- But thermal/ionization requirements limit how thin

### 4. Accept the Limitation

**Reality:**
- Capacitive divider is fundamental
- Cannot be eliminated
- Design around it (optimize topload, use QCW ramping)
- Accept sub-linear scaling

---

## Comparison: QCW vs Burst Mode

### Burst Mode (Fixed Voltage)

```
V_topload = constant (capacitor discharge)

As spark grows:
- V_tip decreases (capacitive divider)
- E_tip decreases rapidly
- Growth stalls at voltage limit
- L ∝ √E scaling dominates
```

### QCW Mode (Ramped Voltage)

```
V_topload(t) increases with time

As spark grows:
- V_tip still affected by divider
- But V_topload increasing compensates partially
- Can maintain E_tip > E_propagation longer
- Better scaling: L ∝ E^0.6 to E^0.8
```

**QCW doesn't eliminate the divider, but actively fights it!**

---

## Key Takeaways

- **Voltage divider**: V_tip = V_topload × C_mut/(C_mut + C_sh)
- **C_sh grows with length**: C_sh ≈ 6.6 pF/m × L, making growth self-limiting
- **V_tip drops dramatically**: Can reach 40% of V_topload by 6 ft
- **E_tip ∝ 1/L²**: Combined effect of voltage division and geometric scaling
- **Sub-linear scaling**: L ∝ √E for voltage-limited burst mode (Freau's observation)
- **Finite R worsens effect**: Conduction current creates additional voltage drop
- **QCW mitigation**: Active voltage ramping compensates for divider effect
- **Fundamental limit**: Cannot be eliminated, only managed through design

## Practice

{exercise:phys-ex-07}

**Problem 1:** V_top = 350 kV, C_mut = 10 pF. Calculate V_tip for:
(a) L = 1 ft (C_sh = 2 pF)
(b) L = 5 ft (C_sh = 10 pF)
What percentage of voltage is lost?

**Problem 2:** A spark needs E_propagation = 0.6 MV/m and κ = 3 to grow. For a 2 m spark, calculate the required V_tip. Then, if C_mut = 8 pF and C_sh = 13 pF (for 2 m), what V_topload is needed?

**Problem 3:** Explain why spark length scales as L ∝ √E for voltage-limited burst mode. Connect this to the capacitive divider effect and the E_tip ∝ 1/L² relationship.

**Problem 4:** Two coils: Coil A has C_mut = 6 pF, Coil B has C_mut = 12 pF (larger topload). Both operate at V_top = 400 kV and grow 1.5 m sparks. Calculate V_tip for each. Which suffers less from voltage division?

---
**Next Lesson:** [Freau's Empirical Relationship](08-freau-relationship.md)