Complete Formula Reference

Complete Formula Reference 

 This appendix provides a comprehensive reference of all formulas used in VIC circuit design and analysis. Formulas are organized by category for easy lookup. 

 1. Resonance Formulas 

 Formula 

 Equation 

 Units 

 Resonant Frequency 

 f₀ = 1 / (2π√(LC)) 

 Hz 

 Angular Frequency 

 ω₀ = 2πf₀ = 1/√(LC) 

 rad/s 

 Period 

 T = 1/f₀ = 2π√(LC) 

 seconds 

 Inductance (given f₀, C) 

 L = 1 / (4π²f₀²C) 

 Henries 

 Capacitance (given f₀, L) 

 C = 1 / (4π²f₀²L) 

 Farads 

 2. Q Factor and Magnification 

 Formula 

 Equation 

 Notes 

 Q Factor (inductive) 

 Q = 2πfL / R = ωL/R 

 At frequency f 

 Q Factor (capacitive) 

 Q = 1 / (2πfCR) = 1/(ωCR) 

 At frequency f 

 Q from Z₀ 

 Q = Z₀/R = (1/R)√(L/C) 

 Series RLC 

 Voltage Magnification 

 V out = Q × V in 

 At resonance 

 Characteristic Impedance 

 Z₀ = √(L/C) 

 Ohms 

 3. Bandwidth and Damping 

 Formula 

 Equation 

 Notes 

 Bandwidth (-3dB) 

 BW = f₀/Q = R/(2πL) 

 Hz 

 Decay Time Constant 

 τ = 2L/R 

 seconds 

 Damping Factor 

 α = R/(2L) 

 rad/s 

 Damped Frequency 

 f d = √(f₀² - α²/(4π²)) 

 Hz 

 Ringdown Cycles (to 1%) 

 N ≈ 0.733 × Q 

 cycles 

 4. Capacitance Formulas 

 Formula 

 Equation 

 Notes 

 Parallel Plate 

 C = ε₀ε r A/d 

 ε₀ = 8.854×10⁻¹² F/m 

 Concentric Cylinders 

 C = 2πε₀ε r L / ln(r o /r i ) 

 L = length 

 Capacitors in Series 

 1/C total = 1/C₁ + 1/C₂ + ... 

 

 Capacitors in Parallel 

 C total = C₁ + C₂ + ... 

 

 Energy in Capacitor 

 E = ½CV² 

 Joules 

 5. Inductance Formulas 

 Formula 

 Equation 

 Notes 

 Solenoid (air core) 

 L = μ₀N²A/l 

 μ₀ = 4π×10⁻⁷ H/m 

 Wheeler's Formula 

 L(µH) = N²r² / (9r + 10l) 

 r, l in inches 

 A L Method 

 L = A L × N² 

 A L in nH/turn² 

 Inductors in Series 

 L total = L₁ + L₂ (no coupling) 

 

 Mutual Inductance 

 M = k√(L₁L₂) 

 k = coupling coefficient 

 Energy in Inductor 

 E = ½LI² 

 Joules 

 6. Resistance and Wire 

 Formula 

 Equation 

 Notes 

 Wire Resistance 

 R = ρL/A 

 ρ = resistivity 

 Wire Area (AWG) 

 A = π(d/2)² 

 d from wire tables 

 Skin Depth 

 δ = √(ρ/(πfμ)) 

 meters 

 Copper Skin Depth 

 δ(mm) ≈ 66/√f(Hz) 

 Quick approximation 

 Power Dissipation 

 P = I²R = V²/R 

 Watts 

 7. Impedance Formulas 

 Element 

 Impedance 

 Phase 

 Resistor 

 Z = R 

 0° 

 Capacitor 

 Z = 1/(jωC) = -j/(2πfC) 

 -90° 

 Inductor 

 Z = jωL = j2πfL 

 +90° 

 CPE 

 Z = 1/(Q(jω) n ) 

 -n×90° 

 Warburg 

 Z = σ/√ω × (1-j) 

 -45° 

 8. Electric Double Layer 

 Formula 

 Equation 

 Notes 

 Helmholtz Capacitance 

 C H = ε₀ε r A/d 

 d ≈ 0.3 nm 

 Debye Length 

 λ D ≈ 0.304/√c (nm) 

 c in mol/L 

 Total EDL (series) 

 1/C = 1/C Stern + 1/C diff 

 

 9. Cole-Cole Model 

 Complex Permittivity: 

 ε* = ε ∞ + (ε s - ε ∞ ) / [1 + (jωτ) (1-α) ] 

 Effective Capacitance: 

 C eff (ω) = C₀ × [1 + (ωτ) 2(1-α) ] -1/2 

 10. Step Charging 

 Formula 

 Equation 

 Notes 

 Ideal N pulses 

 V C,N = 2N × V s 

 Lossless 

 Maximum voltage 

 V max ≈ (4Q/π) × V s 

 With losses 

 Half-cycle time 

 t = π√(LC) 

 For single pulse 

 Physical Constants 

 Constant 

 Symbol 

 Value 

 Permittivity of free space 

 ε₀ 

 8.854 × 10⁻¹² F/m 

 Permeability of free space 

 μ₀ 

 4π × 10⁻⁷ H/m 

 Relative permittivity (water) 

 ε r 

 ~80 at 20°C 

 Copper resistivity 

 ρ Cu 

 1.68 × 10⁻⁸ Ω·m 

 Elementary charge 

 e 

 1.602 × 10⁻¹⁹ C 

 Boltzmann constant 

 k B 

 1.381 × 10⁻²³ J/K 

 Reference complete. Use with the VIC Matrix Calculator for automated calculations.