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Step Charging

Step-Charging Ladder Effect

Step-charging, also known as the "staircase" or "ladder" effect, refers to the progressive buildup of voltage across a capacitor through successive resonant pulses. This technique can achieve voltage levels far beyond what single-pulse resonant charging provides.

The Concept

Instead of maintaining continuous oscillation, step-charging applies discrete pulses that each add a quantum of energy to the capacitor. The voltage builds up incrementally:

    Voltage
       ↑
       │                                    ┌───
       │                               ┌───┘
       │                          ┌───┘
       │                     ┌───┘
       │                ┌───┘
       │           ┌───┘
       │      ┌───┘
       │ ┌───┘
       │─┘
       └─────────────────────────────────────→ Time
         ↑   ↑   ↑   ↑   ↑   ↑   ↑   ↑   ↑
        Pulse Pulse Pulse ...
        1     2     3

    Each pulse adds approximately 2×V_source to capacitor voltage
    (in ideal lossless case with unidirectional diode)

How Step-Charging Works

Step-by-Step Process:

  1. Pulse 1: Capacitor charges from 0 to 2Vs (resonant half-cycle)
  2. Hold: Diode prevents discharge back through inductor
  3. Pulse 2: Starting from 2Vs, capacitor charges to ~4Vs
  4. Hold: Energy stored, waiting for next pulse
  5. Continue: Each pulse adds ~2Vs (minus losses)

Circuit for Step-Charging

         Switch
    V_s ──○/○───┬───────────────┬────▶│────┬────
               │               │      D     │
               │    ┌─────┐    │           ─┴─
               │    │  L  │   ─┴─          ─┬─ C (WFC)
               │    └──┬──┘   ─┬─           │
               │       │       │            │
    ───────────┴───────┴───────┴────────────┴────

    D = Diode prevents reverse current
    C charges in discrete steps

Voltage After N Pulses

Ideal Case (no losses):

VC,N = 2N × Vsource

With Losses (exponential decay factor):

VC,N = 2Vs × Σ(e-π/(2Q))k for k=0 to N-1

Converges to Maximum:

VC,max = 2Vs / (1 - e-π/(2Q))

For high Q: VC,max ≈ (4Q/π) × Vsource

Maximum Voltage vs. Q Factor

Q Factor Vmax/Vsource Pulses to 90%
10 ~12.7 ~6
20 ~25.5 ~12
50 ~63.7 ~30
100 ~127 ~60

Comparison: Continuous vs. Step Charging

Aspect Continuous Resonance Step Charging
Max voltage Q × Vs (AC peak) (4Q/π) × Vs (DC)
Waveform Sinusoidal Staircase
Power delivery Constant Pulsed
Complexity Simpler Needs diode/timing

Step-Charging in VIC Systems

Meyer's designs allegedly used step-charging principles:

  • Unidirectional charging: Diode prevents energy return to source
  • Pulse timing: Gated pulses at resonant frequency
  • Voltage accumulation: Progressive buildup across WFC
  • Controlled discharge: Occasional reset or bleed-off of accumulated voltage

Pulse Train Design

Optimal Pulse Parameters:

  • Pulse duration: π√(LC) = half resonant period
  • Pulse frequency: fpulse < fresonant/2
  • Duty cycle: Typically 10-50%
  • Gap between pulses: Allow ring-down and settling

Energy Considerations

Energy Stored After N Pulses:

EC,N = ½C(VC,N)² = ½C(2NVs)² = 2CN²Vs²

Energy Delivered per Pulse:

ΔE = EC,N - EC,N-1 = 2CVs²(2N-1)

Each successive pulse adds more energy because it's working against a higher voltage!

Practical Implementation

Driver Circuit Requirements:

  1. High-speed switching: MOSFET or IGBT driver
  2. Precise timing: Microcontroller or pulse generator
  3. High-voltage diode: Fast recovery, rated for expected voltages
  4. Voltage monitoring: Feedback to prevent over-voltage

Safety Considerations:

  • Voltages can reach dangerous levels quickly
  • Energy stored in capacitor can be lethal
  • Include bleed resistor for safe discharge
  • Implement hardware over-voltage protection

VIC Matrix Simulation

The VIC Matrix Calculator can simulate step-charging behavior:

  • Step-charge simulation: Predicts voltage after N pulses
  • Loss modeling: Accounts for resistance and dielectric losses
  • Time to saturation: How many pulses to reach maximum voltage
  • Energy efficiency: Tracks energy delivered vs. stored

Key Insight: Step-charging combines the voltage doubling of resonant charging with the cumulative effect of multiple pulses. With sufficient Q factor, extremely high voltages can be developed across the WFC—voltages that would be impossible to achieve directly from the source.

Chapter 4 Complete. Next: Choke Design & Construction →

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