DCR Effects DC Resistance and Q Factor The DC resistance (DCR) of an inductor is the primary factor limiting its Q factor and thus the voltage magnification achievable in a VIC circuit. Understanding and minimizing DCR is essential for high-performance designs. What is DCR? DCR is simply the resistance of the wire used to wind the inductor, measured with direct current: R dc = ρ × l wire / A wire Where: ρ = resistivity of wire material (Ω·m) l wire = total wire length (m) A wire = wire cross-sectional area (m²) DCR and Inductor Design For a given inductance, DCR depends on the design choices: Design Change Effect on L Effect on DCR Net Q Effect More turns L ∝ N² R ∝ N Q ∝ N (improves) Larger wire gauge No change R decreases Q improves Higher μ core L increases Fewer turns needed Variable* Larger core L increases Longer mean turn Often improves Copper vs. SS wire No change R × 40-60 Q ÷ 40-60 *Core losses may offset wire resistance reduction at high frequencies Q Factor Calculation Q Factor at Operating Frequency: Q = 2πfL / R total Total Resistance includes: R total = R dc + R skin + R proximity + R core At low frequencies, R dc dominates. At high frequencies, skin effect and core losses become significant. Voltage Magnification Impact Since voltage magnification equals Q at resonance: Example Comparison: Scenario L DCR Q @ 10kHz V out (12V in) 22 AWG Copper 10 mH 5 Ω 126 1,508 V 26 AWG Copper 10 mH 13 Ω 48 580 V 22 AWG SS316 10 mH 220 Ω 2.9 34 V 22 AWG Nichrome 10 mH 320 Ω 2.0 24 V Measuring DCR Method 1: Multimeter Simple and quick Set meter to lowest resistance range Subtract lead resistance Accuracy: ±1-5% Method 2: 4-Wire (Kelvin) Measurement Eliminates lead resistance error Required for low DCR (<1 Ω) Uses separate sense and current leads Accuracy: ±0.1% Method 3: LCR Meter Measures L and DCR together Can measure at different frequencies Shows equivalent series resistance (ESR) Best for complete characterization Optimizing DCR Design Strategies: Use the largest wire that fits: Fill the available winding area Choose copper: Unless current limiting is specifically needed Use higher permeability core: Fewer turns needed for same L Optimize core size: Larger cores have more room for thicker wire Consider parallel windings: Two parallel wires = half the DCR Practical Limits: Wire must fit on the core with proper insulation Multiple layers increase parasitic capacitance Very thick wire is hard to wind neatly Cost and availability of materials Temperature Effects Wire resistance increases with temperature: R(T) = R 20°C × [1 + α(T - 20)] Where α ≈ 0.00393 /°C for copper Example: At 80°C: R = R 20°C × 1.24 (+24% increase) This means Q drops by ~20% when the choke heats up! DCR in the VIC System The total resistance in a VIC circuit includes: Source Typical Range Mitigation L1 DCR 1-50 Ω Optimize winding L2 DCR 1-50 Ω Optimize winding Capacitor ESR 0.01-1 Ω Use low-ESR caps WFC solution resistance 10-10000 Ω Electrode design, electrolyte Connection resistance 0.01-1 Ω Solid connections Driver output resistance 0.1-10 Ω Low R ds(on) MOSFETs Practical Example Target: 10 mH inductor at 10 kHz with Q > 50 Required R max : Q = 2πfL/R → R = 2πfL/Q = 2π × 10000 × 0.01 / 50 = 12.6 Ω Wire selection (100 turns on 25mm toroid): Mean turn length ≈ 80mm, total wire = 8m 22 AWG copper: 8m × 0.053 Ω/m = 0.42 Ω ✓ 26 AWG copper: 8m × 0.134 Ω/m = 1.07 Ω ✓ 30 AWG copper: 8m × 0.339 Ω/m = 2.71 Ω ✓ 22 AWG SS316: 8m × 2.3 Ω/m = 18.4 Ω ✗ (Q = 34) Result: 22-30 AWG copper all meet the requirement. 22 AWG gives highest Q but may be harder to wind. VIC Matrix Calculator: Enter your wire gauge and material in the Choke Design tool. It calculates DCR automatically and shows how it affects Q factor and voltage magnification. The calculator warns if your DCR is too high for effective resonance. Chapter 5 Complete. Next: Water Fuel Cell Design →