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Secondary Side

Secondary Side (L2-WFC) Analysis

The secondary side of the VIC consists of the second inductor (L2) and the water fuel cell (WFC) acting as a capacitor. This stage receives the amplified signal from the primary and delivers the final voltage to the water. Proper design of this stage is critical for efficient energy transfer to the WFC.

Secondary Tank Circuit

L2 and the WFC capacitance form the secondary resonant tank:

    From               R2 (DCR of L2)
    Primary      ┌────────┴────────┐
        ○────────┤                 ├────────┬────────○
    (V_C1)       │      L2         │        │       (+)
                 │                 │       ─┴─
                 └─────────────────┘       │ │  WFC
                                           │ │ (C_wfc)
                                           ─┬─
                                            │
        ○───────────────────────────────────┴────────○
                                                    (−)

    V_C1 ────▶ [ L2 + R2 ] ────▶ [ WFC ] ────▶ V_WFC

    At secondary resonance: V_WFC = Q_L2 × V_C1 = Q_L2 × Q_L1C × V_in

The WFC as a Capacitor

The water fuel cell presents a complex impedance, but at VIC frequencies, it behaves predominantly as a capacitor:

WFC Capacitance Components:

  • Geometric capacitance: Cgeo = ε₀εrA/d
  • EDL capacitance: Cedl (in series, at each electrode)
  • Effective capacitance: Cwfc = f(Cgeo, Cedl, frequency)

At typical VIC frequencies (1-50 kHz), Cwfc is dominated by Cgeo.

Secondary Resonant Frequency

Secondary Resonance:

f₀secondary = 1 / (2π√(L2 × Cwfc))

For Maximum Voltage Transfer:

Ideally, f₀secondary = f₀primary

This means: L1 × C1 = L2 × Cwfc

Q Factor of Secondary Side

The secondary Q factor determines the second stage of voltage magnification:

Secondary Q Factor:

QL2 = (2π × f₀ × L2) / (R2 + Rwfc)

Where Rwfc is the effective resistance of the WFC (solution resistance + losses).

Total Voltage Magnification:

VWFC = QL1C × QL2 × Vin

Example:

  • QL1C = 30, QL2 = 20, Vin = 12V
  • VWFC = 30 × 20 × 12 = 7,200V theoretical

Cascaded Resonance Effects

When both stages resonate at the same frequency, the effects multiply:

Configuration Total Magnification Notes
Only primary resonance QL1C L2-WFC not tuned
Only secondary resonance QL2 L1-C1 not tuned
Dual resonance QL1C × QL2 Maximum magnification
Harmonic secondary Variable Secondary at 2f₀, 3f₀, etc.

Impedance Matching Considerations

For efficient energy transfer between primary and secondary:

Characteristic Impedance Match:

Z₀primary = √(L1/C1)

Z₀secondary = √(L2/Cwfc)

Matching these impedances can improve energy transfer, though it's not always achievable or necessary.

Effect of WFC Properties on Secondary

WFC Parameter Effect on Secondary Design Response
Higher Cwfc Lower f₀, lower Z₀ Increase L2 or reduce C1
Higher Rwfc Lower QL2 Use purer water or optimize gap
Larger electrode area Higher Cwfc Requires larger L2
Narrower gap Higher Cwfc, lower Rwfc Trade-off between C and R

Bifilar Choke Considerations

When L2 is bifilar wound (or when L1 and L2 are wound together as a bifilar pair):

  • Inherent capacitance: The bifilar winding has capacitance between turns
  • Magnetic coupling: Energy transfers inductively between windings
  • Lower SRF: The inter-winding capacitance lowers self-resonant frequency
  • Complex tuning: The system becomes a coupled resonator

Calculating L2 for Given WFC

Given: Target frequency and WFC capacitance

L2 = 1 / (4π²f₀²Cwfc)

Example:

  • f₀ = 10 kHz
  • Cwfc = 5 nF (typical small WFC)
  • L2 = 1 / (4π² × 10⁴² × 5×10⁻⁹) = 50.7 mH

Sanity check: This is a reasonable inductance, achievable with ~500-1000 turns on a ferrite core.

Power Delivery to WFC

The actual power delivered to the WFC depends on its resistive component:

Power in WFC Resistance:

Pwfc = I²wfc × Rwfc

Where:

Iwfc = VWFC / Zwfc ≈ VWFC × ω × Cwfc

This power heats the water and drives electrochemical reactions.

Voltage Distribution Across WFC

The high voltage across the WFC creates an electric field:

Electric Field in WFC:

E = VWFC / d

Where d is the electrode gap.

Example:

  • VWFC = 1000V, d = 1mm
  • E = 1000V / 0.001m = 1 MV/m = 10 kV/cm

This is a substantial electric field that can influence molecular behavior in water.

Design Guidelines for L2

  1. Match resonant frequency: L2 should resonate with Cwfc at the same frequency as L1-C1
  2. Minimize DCR: R2 directly reduces QL2 and thus voltage magnification
  3. Consider coupling: If using transformer-coupled design, mutual inductance matters
  4. Account for WFC changes: Cwfc varies with temperature, voltage, and bubble formation
  5. Leave tuning margin: Design L2 slightly higher, fine-tune with small series capacitor if needed

Key Insight: The secondary side is where VIC theory meets reality. The WFC is not an ideal capacitor—it has losses, frequency-dependent behavior, and changes during operation. Successful VIC design must account for these real-world effects.

Next: Resonant Charging Principle →

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