Appendices 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. Glossary of Terms Appendix B: Wire Gauge & Material Tables Complete reference tables for wire properties used in VIC choke design. All values at 20°C (68°F) unless noted. AWG Wire Gauge Reference AWG Diameter (mm) Diameter (in) Area (mm²) Area (kcmil) Cu Ω/1000ft Cu Ω/km 10 2.588 0.1019 5.261 10.38 0.9989 3.277 12 2.053 0.0808 3.309 6.530 1.588 5.211 14 1.628 0.0641 2.081 4.107 2.525 8.286 16 1.291 0.0508 1.309 2.583 4.016 13.17 18 1.024 0.0403 0.823 1.624 6.385 20.95 20 0.812 0.0320 0.518 1.022 10.15 33.31 22 0.644 0.0253 0.326 0.642 16.14 52.96 24 0.511 0.0201 0.205 0.404 25.67 84.22 26 0.405 0.0159 0.129 0.254 40.81 133.9 28 0.321 0.0126 0.081 0.160 64.90 212.9 30 0.255 0.0100 0.051 0.101 103.2 338.6 32 0.202 0.0080 0.032 0.063 164.1 538.3 34 0.160 0.0063 0.020 0.040 260.9 856.0 36 0.127 0.0050 0.013 0.025 414.8 1361 38 0.101 0.0040 0.008 0.016 659.6 2164 40 0.080 0.0031 0.005 0.010 1049 3441 Highlighted rows indicate commonly used gauges for VIC chokes. Wire Material Resistivity Material Resistivity ρ (Ω·m) Relative to Cu Temp Coefficient α (/°C) Silver (Ag) 1.59 × 10⁻⁸ 0.95× 0.0038 Copper (Cu) 1.68 × 10⁻⁸ 1.00× (reference) 0.00393 Gold (Au) 2.44 × 10⁻⁸ 1.45× 0.0034 Aluminum (Al) 2.65 × 10⁻⁸ 1.58× 0.00429 Brass 6-9 × 10⁻⁸ 4-5× 0.002 Steel 1.0 × 10⁻⁷ 6× 0.005 Stainless Steel 6.9 × 10⁻⁷ 41× 0.001 Nichrome 1.1 × 10⁻⁶ 65× 0.0004 Temperature Correction Resistance at Temperature T: R(T) = R₂₀ × [1 + α(T - 20)] Example (Copper wire): R₂₀ = 10 Ω at 20°C At 50°C: R = 10 × [1 + 0.00393(50-20)] = 10 × 1.118 = 11.18 Ω At 80°C: R = 10 × [1 + 0.00393(80-20)] = 10 × 1.236 = 12.36 Ω Magnet Wire Specifications Magnet wire has enamel insulation. Overall diameter includes insulation: AWG Bare Dia. (mm) Overall Dia. (mm) Turns/cm Turns/inch 18 1.024 1.09 9.2 23.3 20 0.812 0.87 11.5 29.2 22 0.644 0.70 14.3 36.3 24 0.511 0.56 17.9 45.4 26 0.405 0.45 22.2 56.4 28 0.321 0.36 27.8 70.6 30 0.255 0.29 34.5 87.6 32 0.202 0.24 41.7 106 Current Capacity Guidelines For chassis wiring (in open air): AWG Max Current (A) AWG Max Current (A) 10 15 24 1.4 12 9.3 26 0.9 14 5.9 28 0.55 16 3.7 30 0.35 18 2.3 32 0.22 20 1.8 34 0.14 22 2.1 36 0.09 For coils, derate by 50% due to limited cooling. Magnet wire rated for higher temperature can handle more current. Skin Depth Reference At high frequencies, current flows near the wire surface. Skin depth δ: δ = √(ρ / πfμ₀μᵣ) Skin Depth in Copper: Frequency Skin Depth (mm) Max Useful Wire Dia. 1 kHz 2.1 mm ~4 mm (AWG 6) 10 kHz 0.66 mm ~1.3 mm (AWG 16) 50 kHz 0.30 mm ~0.6 mm (AWG 22) 100 kHz 0.21 mm ~0.4 mm (AWG 26) Use wire diameter ≤ 2×δ for effective use of conductor cross-section. For larger currents at high frequencies, use Litz wire. Quick Reference: DCR Calculation For Copper Wire: DCR (Ω) = Length (m) × Resistance (Ω/km) / 1000 DCR (Ω) = Length (ft) × Resistance (Ω/1000ft) / 1000 For Other Materials: DCR material = DCR Cu × (ρ material /ρ Cu ) Wire Gauge Tables Appendix C: Core Specifications Reference specifications for magnetic cores commonly used in VIC choke design. Includes ferrite toroids, iron powder cores, and E-cores. Core Material Overview Material Type μᵣ Range Frequency Range Best For MnZn Ferrite 800-10,000 1 kHz - 2 MHz High L, moderate f NiZn Ferrite 15-1,500 500 kHz - 100 MHz High frequency Iron Powder 8-100 10 kHz - 10 MHz High current, low cost MPP (Molypermalloy) 14-550 DC - 1 MHz Low loss, stable Kool Mµ 26-125 DC - 500 kHz High current, moderate loss Air Core 1 Any No saturation, linear Common Ferrite Materials MnZn Ferrite Materials Material μᵢ B sat (mT) Frequency Notes Fair-Rite 77 2000 480 <1 MHz General purpose, high μ Fair-Rite 78 2300 480 <500 kHz Very high μ TDK N87 2200 490 <500 kHz Popular, low loss TDK N97 2300 410 <300 kHz Very low loss Ferroxcube 3C90 2300 470 <200 kHz Low loss at high B Ferroxcube 3F3 2000 440 <500 kHz Higher frequency Iron Powder Core Mix Chart Iron powder cores (Micrometals/Amidon) are identified by color code: Mix Color μᵣ Frequency Range Application -26 Yellow/White 75 DC - 1 MHz EMI/RFI filters -2 Red/Clear 10 250 kHz - 10 MHz RF, resonant circuits -6 Yellow/Clear 8.5 3 - 40 MHz Higher frequency -1 Blue/Clear 20 500 kHz - 5 MHz Medium frequency -3 Gray/Clear 35 50 kHz - 500 kHz Medium μ, low f -52 Green/Blue 75 DC - 200 kHz High μ, DC bias Common Toroid Sizes FT (Ferrite Toroid) Series Size OD (mm) ID (mm) H (mm) Aₗ (77 mat) Aₗ (43 mat) FT-37 9.5 4.7 3.2 884 440 FT-50 12.7 7.1 4.8 1140 570 FT-82 21.0 13.0 6.4 2170 557 FT-114 29.0 19.0 7.5 2640 603 FT-140 35.5 23.0 12.7 3170 885 FT-240 61.0 35.5 12.7 4820 1075 Aₗ values in nH/turn². Highlighted sizes are commonly used for VIC chokes. T (Iron Powder Toroid) Series Size OD (mm) ID (mm) H (mm) Aₗ (-2 mix) Aₗ (-26 mix) T-37 9.5 4.9 3.2 4.0 27 T-50 12.7 7.7 4.8 4.9 33 T-68 17.5 9.4 4.8 5.7 38 T-80 20.2 12.6 6.4 8.5 55 T-94 24.0 14.5 7.9 8.4 70 T-106 26.9 14.0 11.1 13.5 90 T-130 33.0 19.7 11.1 11.0 96 T-200 50.8 31.8 14.0 12.0 120 Inductance Calculations Using Aₗ Value: L (nH) = Aₗ × N² N = √(L / Aₗ) Example: Want L = 10 mH = 10,000,000 nH Using FT-240-77 (Aₗ = 4820 nH/turn²) N = √(10,000,000 / 4820) = 45.6 turns Use 46 turns for L ≈ 10.2 mH Saturation Considerations Saturation Flux Density (B sat ): Material Type B sat (mT) MnZn Ferrite 400-500 NiZn Ferrite 250-350 Iron Powder 800-1000 MPP 750 Calculating Peak Flux: B = (V × t) / (N × A e ) Where A e is effective core area. Keep B < 0.5 × B sat for linear operation. Temperature Effects Material Curie Temp (°C) Max Operating (°C) μ vs. Temp MnZn Ferrite 200-250 100-120 Peaks near 80°C, then drops NiZn Ferrite 300-500 150 Relatively stable Iron Powder 770 (iron) 125 (coating limited) Stable Core Selection Guide for VIC For Primary Choke (L1): Moderate L (1-50 mH typical) Moderate current handling Consider: FT-82-77, FT-114-77, T-106-26 For Secondary Choke (L2): May need higher L (10-100 mH) for high Q Lower current typically Consider: FT-140-77, FT-240-77 For High Frequency (>100 kHz): Use lower-μ materials to maintain SRF margin Consider: Iron powder -2 or -6 mix, NiZn ferrite Quick Reference: Turns Calculation Desired L FT-82-77 FT-240-77 T-106-26 1 mH 21 turns 14 turns 105 turns 5 mH 48 turns 32 turns 236 turns 10 mH 68 turns 46 turns 333 turns 25 mH 107 turns 72 turns 527 turns 50 mH 152 turns 102 turns 745 turns Approximate values. Verify with actual Aₗ from manufacturer datasheet. Core Specifications Glossary of Terms A comprehensive glossary of technical terms used throughout the VIC Matrix educational content and calculator. A A L (Inductance Factor) A core specification in nH/turn² that allows quick calculation of inductance: L = A L × N² Alpha (α) - Cole-Cole Distribution parameter (0-1) in the Cole-Cole model. α=0 is ideal Debye relaxation; higher values indicate broader distribution of relaxation times. Alpha (α) - Damping Damping factor in an RLC circuit: α = R/(2L). Determines how quickly oscillations decay. Amplitude The maximum value of an oscillating quantity, such as voltage or current. B Bandwidth (BW) The frequency range over which a resonant circuit responds effectively. BW = f₀/Q for a series RLC circuit. Bifilar Winding A winding technique where two wires are wound together in parallel, creating tight magnetic coupling and significant inter-winding capacitance. Blocking Electrode An electrode where no Faradaic (electrochemical) reactions occur, behaving purely as a capacitor. C Capacitance (C) The ability to store electric charge. Measured in Farads (F). C = Q/V where Q is charge and V is voltage. Characteristic Impedance (Z₀) The ratio √(L/C) for an LC circuit. Represents the impedance level of the resonant system. Charge Transfer Resistance (R ct ) The resistance associated with electron transfer at an electrode surface during electrochemical reactions. Choke An inductor used in a circuit to block or impede certain frequencies while allowing others to pass. In VIC context, the resonating inductors. Cole-Cole Model A mathematical model describing frequency-dependent dielectric behavior with distributed relaxation times. Constant Phase Element (CPE) A circuit element with impedance Z = 1/[Q(jω) n ], used to model non-ideal capacitor behavior in electrochemical systems. Coupling Coefficient (k) A measure of magnetic coupling between inductors (0-1). k = M/√(L₁L₂) where M is mutual inductance. D DCR (DC Resistance) The resistance of an inductor measured with direct current. Primary contributor to inductor losses. Debye Length (λ D ) The characteristic thickness of the diffuse layer in an electrochemical double layer. Decreases with increasing ion concentration. Diffuse Layer The outer region of the electric double layer where ion concentration gradually returns to bulk values. Dielectric An insulating material that can be polarized by an electric field. Water is a dielectric with high permittivity (ε r ≈ 80). Double Layer See Electric Double Layer (EDL). E EDL (Electric Double Layer) The structure formed at an electrode-electrolyte interface, consisting of a compact layer of ions and a diffuse layer extending into solution. EIS (Electrochemical Impedance Spectroscopy) A technique for characterizing electrochemical systems by measuring impedance across a range of frequencies. ESR (Equivalent Series Resistance) The resistive component of a capacitor's impedance, causing power dissipation. F Faradaic Reaction An electrochemical reaction involving electron transfer at an electrode, such as water electrolysis. Ferrite A ceramic magnetic material used for inductor cores, suitable for high-frequency applications. Frequency (f) The number of complete oscillation cycles per second. Measured in Hertz (Hz). G-H Helmholtz Layer The compact inner layer of the EDL, where ions are closest to the electrode surface. Hysteresis Energy loss in magnetic materials due to the lag between applied field and magnetization. I Impedance (Z) The total opposition to alternating current, including both resistance and reactance. Measured in Ohms (Ω). Inductance (L) The property of a conductor that opposes changes in current by storing energy in a magnetic field. Measured in Henries (H). IHP (Inner Helmholtz Plane) The plane passing through the centers of specifically adsorbed ions in the EDL. L-M LC Circuit A circuit containing an inductor and capacitor, capable of oscillating at a resonant frequency. Mutual Inductance (M) The inductance linking two coils, allowing energy transfer between them. N-O Nyquist Plot A plot of imaginary vs. real impedance (-Z'' vs Z') used in EIS analysis. OHP (Outer Helmholtz Plane) The plane of closest approach for solvated (hydrated) ions in the EDL. P Parasitic Capacitance Unintended capacitance in an inductor, arising from turn-to-turn and layer-to-layer effects. Permittivity (ε) A measure of how much electric field is reduced in a material compared to vacuum. ε = ε₀ε r . Permeability (μ) A measure of how well a material supports magnetic field formation. μ = μ₀μ r . PLL (Phase-Locked Loop) A control system that maintains frequency lock with a reference signal, used to track resonance. Q Q Factor (Quality Factor) A dimensionless parameter indicating the "sharpness" of resonance. Q = ωL/R = Z₀/R. Higher Q means narrower bandwidth and higher voltage magnification. R Randles Circuit An equivalent circuit model for electrochemical cells consisting of R s , C dl , R ct , and Z W . Reactance The imaginary part of impedance. Inductive reactance X L = ωL; capacitive reactance X C = 1/(ωC). Resonance The condition where inductive and capacitive reactances are equal, resulting in maximum energy storage and voltage magnification. Ringdown The decay of oscillations after excitation stops, characterized by the time constant τ = 2L/R. S Self-Resonant Frequency (SRF) The frequency at which an inductor's parasitic capacitance resonates with its inductance. Above SRF, the inductor behaves as a capacitor. Skin Effect The tendency of AC current to flow near the surface of a conductor, increasing effective resistance at high frequencies. Solution Resistance (R s ) The ionic resistance of the electrolyte between electrodes. Step Charging A technique using multiple resonant pulses to progressively build voltage on a capacitor. Stern Layer The combined compact and diffuse layer model of the EDL. T Tank Circuit A parallel LC circuit that "tanks" or stores energy, oscillating between magnetic and electric forms. Tau (τ) - Time Constant The characteristic time for decay. For an RLC circuit: τ = 2L/R. Toroidal Core A doughnut-shaped magnetic core providing a closed magnetic path and good field containment. V VIC (Voltage Intensifier Circuit) A resonant circuit configuration using chokes and capacitors to develop high voltage across a water fuel cell. Voltage Magnification The ratio of voltage across a reactive element to the source voltage at resonance. Equals Q for a series RLC circuit. W Warburg Impedance (Z W ) Impedance arising from diffusion of electroactive species, characterized by 45° phase angle and Z ∝ 1/√ω. WFC (Water Fuel Cell) An electrochemical cell where water serves as the medium between electrodes, acting as a capacitive-resistive load in VIC circuits. Z Z₀ (Characteristic Impedance) The natural impedance level of an LC circuit: Z₀ = √(L/C). Also Q × R for a series RLC circuit. Zero-Current Switching (ZCS) A switching technique where transistors turn off when current is zero, minimizing switching losses. Glossary compiled for the VIC Matrix educational series.