Resonant Charging

Resonant Charging Principle 

 Resonant charging is a technique where energy is transferred to a capacitive load (the WFC) in a controlled, oscillatory manner. Unlike direct DC charging, resonant charging can achieve higher efficiency and allows voltage magnification beyond the source voltage. 

 Conventional vs. Resonant Charging 

 Aspect 

 DC Charging (R-C) 

 Resonant Charging (L-C) 

 Final voltage 

 = V source 

 Can exceed V source (up to 2× for half-wave) 

 Energy efficiency 

 50% max (half lost in R) 

 Can approach 100% (minimal loss in L) 

 Charging curve 

 Exponential (slow) 

 Sinusoidal (faster) 

 Peak current 

 V/R at start 

 V/Z₀ (controlled by L) 

 Basic Resonant Charging Circuit 

 Switch (S)

 ────○/○────┬───────────────┬────

 │ │

 V_source │ │

 + │ ┌─────┐ ─┴─

 │ │ L │ ─┬─ C (WFC)

 │ └──┬──┘ │

 │ │ │

 ───────────┴───────┴───────┴────

 GND

 When S closes:

 1. Current builds in L (energy stored in magnetic field)

 2. Current flows into C, charging it

 3. Voltage on C rises

 4. At peak voltage, current reverses (or S opens)

 

 Half-Cycle Resonant Charging 

 In half-cycle mode, the switch opens when capacitor voltage reaches maximum: 

 Ideal Half-Cycle Charging (lossless): 

 V C,max = 2 × V source 

 Charging Time: 

 t charge = π√(LC) = π/ω₀ = 1/(2f₀) 

 This is exactly half the resonant period. 

 Why 2× Voltage? 

 Energy Conservation: 

 

 Initially: All energy in source (voltage V s ) 

 Quarter cycle: Energy split between L (current max) and C (V = V s ) 

 Half cycle: All energy in C, current = 0 

 For energy to be conserved: ½CV c ² = C×V s ² (accounting for work done by source) 

 This gives V c = 2V s 

 

 Resonant Charging with Losses 

 Real circuits have losses that reduce the voltage gain: 

 With Resistance (damped case): 

 V C,max = V source × (1 + e -πR/(2√(L/C)) ) 

 V C,max = V source × (1 + e -π/(2Q) ) 

 Approximation for high Q: 

 V C,max ≈ 2V source × (1 - π/(4Q)) 

 Voltage Gain vs. Q Factor 

 Q Factor 

 V C,max /V source 

 Efficiency 

 ∞ (ideal) 

 2.00 

 100% 

 100 

 1.98 

 98.4% 

 50 

 1.97 

 96.9% 

 20 

 1.92 

 92.5% 

 10 

 1.85 

 85.5% 

 5 

 1.73 

 73% 

 Continuous Resonant Excitation 

 In the VIC, instead of single pulses, we drive the circuit continuously at the resonant frequency: 

 Steady-State Resonance: 

 Energy from the source compensates for losses each cycle, maintaining a steady oscillation amplitude. 

 Voltage Magnification: 

 V C = Q × V source 

 This is much greater than the 2× from single-pulse resonant charging when Q > 2. 

 Resonant Charging in VIC Context 

 The VIC uses resonant charging principles in several ways: 

 Primary tank: C1 is resonantly charged through L1 

 Secondary transfer: Energy transfers resonantly to WFC through L2 

 Cumulative effect: Multiple stages multiply the magnification 

 Timing and Switching 

 For optimal resonant charging: 

 Critical Timing Points: 

 

 

 Turn-on: When capacitor voltage is minimum (or at desired starting point) 

 Turn-off: When current through inductor reaches zero (zero-current switching) 

 Period: Should match or be a harmonic of the resonant frequency 

 

 

 Zero-Current Switching (ZCS): 

 Turning off when current is zero minimizes switching losses and eliminates inductive kick. 

 Energy Flow Analysis 

 Time →

 V_C: ────╱╲ ╱╲ ╱╲ ╱╲────

 ╱ ╲ ╱ ╲ ╱ ╲ ╱ ╲

 ╱ ╲╱ ╲╱ ╲╱ ╲

 I_L: ──╱╲ ╱╲ ╱╲ ╱╲────

 ╱ ╲ ╱ ╲ ╱ ╲ ╱ ╲

 ╱ ╲╱ ╲╱ ╲╱ ╲

 Energy in C: High → Low → High → Low

 Energy in L: Low → High → Low → High

 Total energy (minus losses) remains constant in steady state.

 

 Advantages of Resonant Charging for WFC 

 High voltage: Achieves voltages beyond source capability 

 Low current draw: Source only provides loss compensation 

 Controlled energy delivery: Sinusoidal rather than impulsive 

 Efficient: Minimal resistive losses when Q is high 

 Self-limiting: Voltage limited by Q factor, not infinite 

 Key Principle: Resonant charging exploits the energy storage capability of inductors and capacitors. By timing the energy injection to match the natural oscillation, we can build up substantial energy in the circuit with modest input power—the same principle used in pushing a swing at just the right moment. 

 Next: Step-Charging Ladder Effect →