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 | Vout = Q × Vin | 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 | fd = √(f₀² - α²/(4π²)) | Hz |
| Ringdown Cycles (to 1%) | N ≈ 0.733 × Q | cycles |
4. Capacitance Formulas
| Formula | Equation | Notes |
|---|---|---|
| Parallel Plate | C = ε₀εrA/d | ε₀ = 8.854×10⁻¹² F/m |
| Concentric Cylinders | C = 2πε₀εrL / ln(ro/ri) | L = length |
| Capacitors in Series | 1/Ctotal = 1/C₁ + 1/C₂ + ... | |
| Capacitors in Parallel | Ctotal = 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 |
| AL Method | L = AL × N² | AL in nH/turn² |
| Inductors in Series | Ltotal = 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 | CH = ε₀εrA/d | d ≈ 0.3 nm |
| Debye Length | λD ≈ 0.304/√c (nm) | c in mol/L |
| Total EDL (series) | 1/C = 1/CStern + 1/Cdiff |
9. Cole-Cole Model
Complex Permittivity:
ε* = ε∞ + (εs - ε∞) / [1 + (jωτ)(1-α)]
Effective Capacitance:
Ceff(ω) = C₀ × [1 + (ωτ)2(1-α)]-1/2
10. Step Charging
| Formula | Equation | Notes |
|---|---|---|
| Ideal N pulses | VC,N = 2N × Vs | Lossless |
| Maximum voltage | Vmax ≈ (4Q/π) × Vs | 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) | εr | ~80 at 20°C |
| Copper resistivity | ρCu | 1.68 × 10⁻⁸ Ω·m |
| Elementary charge | e | 1.602 × 10⁻¹⁹ C |
| Boltzmann constant | kB | 1.381 × 10⁻²³ J/K |
Reference complete. Use with the VIC Matrix Calculator for automated calculations.