# 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:

1. **Use the largest wire that fits:** Fill the available winding area 2. **Choose copper:** Unless current limiting is specifically needed 3. **Use higher permeability core:** Fewer turns needed for same L 4. **Optimize core size:** Larger cores have more room for thicker wire 5. **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 →*