Capacitors & Charge
voltage on a cap, charging v = v₀[1–e^(–t/τ)] v₀ is the battery voltage v is the voltage after time t R is resistance in ohms, C is capacitance in farads t is time in seconds RC = τ = time constant After 1τ, v = 0.63v₀ After 2τ, v = 0.86v₀ After 3τ, v = 0.95v₀ After 5τ, v = 0.993v₀ current into a cap, charging v = v₀[1–e^(–t/τ)] v₀ is the battery voltage i is the current after time t R is resistance in ohms, C is capacitance in farads t is time in seconds RC = τ = time constant i = (v₀/R)[–e^(–t/τ)] voltage on a cap, discharging v = v₀e^(–t/τ) v₀ is the initial voltage on the cap v is the voltage after time t R is resistance in ohms, C is capacitance in farads t is time in seconds RC = τ = time constant After 1τ, v = 0.37v₀ After 2τ, v = 0.14v₀ After 3τ, v = 0.05v₀ After 5τ, v = 0.807v₀ Current in an inductor Iʟ = (V₀/R)(1–e^(-t/τ)) V₀/R = I₀, steady state current Voltage across inductor Vʟ = V₀e^(-t/τ) Vʟ is the voltage after time t V₀ is the battery voltage R is resistance in ohms L is inductance in henries t is time in seconds L/R = τ = time constant Parallel plate cap C = ε₀εᵣ(A/d) in Farads ε₀ is vacuum permittivity, 8.854e-12 F/m εᵣ is dielectric constant or relative permittivity of the material (vacuum = 1) A and d are area of plate in m² and separation in m or C = ε₀εᵣ(A/d) in pF, ε₀ is 8.854 Capacitance/unit length of a long cylinder is C/L = (2πεᵣε₀) / (ln (b/a)) b is radius of outside conductor a is radius if inside conductor εᵣ is dielectric constant (vacuum = 1) ε₀ is 8.8542e-12 F/m Capacitance of a sphere is C = 4πε₀εᵣR εᵣ is dielectric constant (vacuum = 1) ε₀ is 8.8542e-12 F/m R is radius in meters Energy in a Capacitor in Joules E = ½CV² = ½QV = ½Q²/C Q = CV Q is charge in coulombs C is capacitance in Farads V is voltage in volts E is energy in Joules V = Energy / charge Capacitor markings Most capacitors have numbers printed on their bodies to indicate their electrical characteristics. Larger capacitors like electrolytics usually display the actual capacitance together with the unit (for example, 220 μF). Smaller capacitors like ceramics, however, use a shorthand consisting of three numbers and a letter, where the numbers show the capacitance in pF (calculated as XY × 10Z for the numbers XYZ) and the letter indicates the tolerance (J, K or M for ±5%, ±10% and ±20% respectively). Additionally, the capacitor may show its working voltage, temperature and other relevant characteristics. A capacitor with the text 473K 330V on its body has a capacitance of 47 × 103 pF = 47 nF (±10%) with a working voltage of 330 V. relative permittivity or dielectric constant εᵣ Vacuum 1 (by definition) Air 1.000590 PTFE/Teflon 2.1 Polyethylene 2.25 Polyimide 3.4 Polypropylene 2.2–2.36 Polystyrene 2.4–2.7 Carbon disulfide 2.6 Paper 3.85 Electroactive polymers 2–12 Silicon dioxide 3.9 [3] Concrete 4.5 Pyrex (Glass) 4.7 (3.7–10) Rubber 7 Diamond 5.5–10 Salt 3–15 Graphite 10–15 Silicon 11.68 Ammonia 26, 22, 20, 17 (−80, −40, 0, 20 °C) Methanol 30 Ethylene Glycol 37 Furfural 42.0 Glycerol 41.2, 47, 42.5 (0, 20, 25 °C) Water 0º 88 Water 20º 80.1 Water 100º 55.3 Water 200º 34.5 Hydrofluoric acid 83.6 (0 °C) Formamide 84.0 (20 °C) Sulfuric acid 84–100 (20–25 °C) Hydrogen peroxide 128 aq–60 (−30–25 °C) Hydrocyanic acid 158.0–2.3 (0–21 °C) Titanium dioxide 86–173 Strontium titanate 310 Barium strontium titanate 500 Barium titanate 1250–10,000 (20–120 °C) Lead zirconate titanate 500–6000 Conjugated polymers 1.8-6 up to 100,000 Calcium copper titanate >250,000 |
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