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: C geo = ε₀ε r A/d 

 EDL capacitance: C edl (in series, at each electrode) 

 Effective capacitance: C wfc = f(C geo , C edl , frequency) 

 

 

 At typical VIC frequencies (1-50 kHz), C wfc is dominated by C geo . 

 Secondary Resonant Frequency 

 Secondary Resonance: 

 f₀ secondary = 1 / (2π√(L2 × C wfc )) 

 For Maximum Voltage Transfer: 

 Ideally, f₀ secondary = f₀ primary 

 This means: L1 × C1 = L2 × C wfc 

 Q Factor of Secondary Side 

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

 Secondary Q Factor: 

 Q L2 = (2π × f₀ × L2) / (R2 + R wfc ) 

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

 Total Voltage Magnification: 

 V WFC = Q L1C × Q L2 × V in 

 Example: 

 

 Q L1C = 30, Q L2 = 20, V in = 12V 

 V WFC = 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 

 Q L1C 

 L2-WFC not tuned 

 Only secondary resonance 

 Q L2 

 L1-C1 not tuned 

 Dual resonance 

 Q L1C × Q L2 

 Maximum magnification 

 Harmonic secondary 

 Variable 

 Secondary at 2f₀, 3f₀, etc. 

 Impedance Matching Considerations 

 For efficient energy transfer between primary and secondary: 

 Characteristic Impedance Match: 

 Z₀ primary = √(L1/C1) 

 Z₀ secondary = √(L2/C wfc ) 

 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 C wfc 

 Lower f₀, lower Z₀ 

 Increase L2 or reduce C1 

 Higher R wfc 

 Lower Q L2 

 Use purer water or optimize gap 

 Larger electrode area 

 Higher C wfc 

 Requires larger L2 

 Narrower gap 

 Higher C wfc , lower R wfc 

 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 / (4π²f₀²C wfc ) 

 Example: 

 

 

 f₀ = 10 kHz 

 C wfc = 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: 

 P wfc = I² wfc × R wfc 

 Where: 

 I wfc = V WFC / Z wfc ≈ V WFC × ω × C wfc 

 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 = V WFC / d 

 Where d is the electrode gap. 

 Example: 

 

 

 V WFC = 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 

 Match resonant frequency: L2 should resonate with C wfc at the same frequency as L1-C1 

 Minimize DCR: R2 directly reduces Q L2 and thus voltage magnification 

 Consider coupling: If using transformer-coupled design, mutual inductance matters 

 Account for WFC changes: C wfc varies with temperature, voltage, and bubble formation 

 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. 

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