# Energy Efficiency # Energy Efficiency Analysis Understanding energy flow in VIC circuits helps optimize performance and evaluate system efficiency. This page covers how to analyze energy storage, transfer, and dissipation in resonant VIC systems. ## Energy in Resonant Circuits In an LC resonant circuit, energy oscillates between the inductor and capacitor: #### Energy Storage: E_L = ½LI² (energy in inductor) E_C = ½CV² (energy in capacitor) #### At Resonance: E_total = E_L,max = E_C,max = ½CV_peak² #### Peak Energy (example):

- C = 10 nF, V_peak = 1000 V - E = ½ × 10×10⁻⁹ × 1000² = 5 mJ

## Energy Flow Diagram ``` Input Power │ ↓ ┌─────────────────────────────────────────────┐ │ VIC CIRCUIT │ │ │ │ ┌──────┐ ┌──────┐ ┌──────┐ │ │ │ L1 │──────│ L2 │──────│ WFC │ │ │ │ DCR │ │ DCR │ │ ESR │ │ │ └──────┘ └──────┘ └──────┘ │ │ │ │ │ │ │ ↓ ↓ ↓ │ │ Heat Loss Heat Loss Heat Loss │ │ (copper) (copper) (solution) │ │ │ │ │ ↓ │ │ Electrochemical │ │ Work (desired) │ └─────────────────────────────────────────────┘ ``` ## Loss Mechanisms

Loss Type	Formula	How to Minimize
Choke DCR Loss	P = I²R_DCR	Use larger wire, copper
Solution Resistance	P = I²R_sol	Optimize water conductivity
Core Loss	P ∝ f^α × B^β	Choose low-loss core material
Skin Effect Loss	Increases R at high f	Use Litz wire at high f
Dielectric Loss	P = ωCV² × tan(δ)	Use low-loss capacitors

## Q Factor and Efficiency Q factor is directly related to energy efficiency per cycle: #### Energy Loss Per Cycle: ΔE_cycle = 2π × E_stored / Q #### Interpretation:

- Q = 10: Lose 63% of energy per cycle - Q = 50: Lose 13% of energy per cycle - Q = 100: Lose 6% of energy per cycle - Q = 200: Lose 3% of energy per cycle

#### Energy Retention: After n cycles: E(n) = E₀ × e^(-2πn/Q) ## Power Flow Analysis ### Input Power P_in = V_in × I_in × cos(φ) For pulsed operation: P_avg = (1/T) × ∫V(t)I(t)dt ### Dissipated Power P_diss = I_rms² × R_total Where R_total = R_DCR1 + R_DCR2 + R_sol + R_other ### Useful Power Power available for electrochemical work: P_useful = P_in - P_diss Or, for the WFC specifically: P_wfc = V_wfc × I_wfc × cos(φ_wfc) ## Efficiency Calculations

Efficiency Type	Formula	Typical Values
Resonant Tank η	η = Q/(Q+1) ≈ 1 - 1/Q	90-99% for high Q
Power Transfer η	η = P_wfc/P_in	50-90%
Voltage Multiplication η	V_out/V_in (at resonance)	10-100× typical

## Energy Balance Verification To verify your analysis is correct, energy must balance: #### Steady State: P_in = P_DCR1 + P_DCR2 + P_sol + P_core + P_other #### Check:

- Sum all loss mechanisms - Compare to measured input power - Large discrepancy indicates missing loss or measurement error

## Loss Breakdown Example

Component	Resistance	Power Loss (at 1A)	% of Total
L1 DCR	2.5 Ω	2.5 W	25%
L2 DCR	3.0 Ω	3.0 W	30%
R_solution	4.0 Ω	4.0 W	40%
Other (core, leads)	0.5 Ω	0.5 W	5%
Total	10 Ω	10 W	100%

## Improving Efficiency #### High-Impact Improvements:

1. **Reduce largest loss first:** In example above, R_sol is 40%—optimize water conductivity 2. **Use larger wire:** Each AWG step down reduces DCR by ~25% 3. **Choose better core:** Low-loss ferrite vs. iron powder 4. **Optimize water conductivity:** Not too high (electrolysis), not too low (resistance loss) 5. **Reduce connection resistance:** Good solder joints, clean contacts

#### Diminishing Returns: Once a loss mechanism is <10% of total, further improvement has limited benefit. Focus on the dominant losses. ## Thermal Considerations All dissipated power becomes heat:

Component	Heat Concern	Mitigation
Choke windings	Wire insulation damage	Adequate wire size, ventilation
Ferrite core	Curie temp, permeability change	Keep below rated temperature
Water/WFC	Boiling, capacitance drift	Monitor temperature, allow cooling
Capacitors	ESR heating, life reduction	Use low-ESR types, derate

**VIC Matrix Calculator:** The simulation module calculates expected power dissipation in each component. Use this to identify thermal hotspots and verify your design won't overheat during operation. *Next: Experimental Validation Methods →*