10 KiB
| id | title | section | difficulty | estimated_time | prerequisites | objectives | tags |
|---|---|---|---|---|---|---|---|
| phys-07 | The Capacitive Divider Problem | Spark Growth Physics | advanced | 45 | [fund-04 fund-05 phys-01 phys-02] | [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] | [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