Formula sheet: Alternating Current
Alternating Current14 min readLesson 1 of 6
1.1 Core formulas
| Quantity | Expression | Notes |
|---|---|---|
| AC voltage | v = v_m sin omega t | v_m is the peak value |
| RMS voltage | V_rms = v_m / square root of 2 | about 0.707 v_m |
| RMS current | I_rms = i_m / square root of 2 | used in power formulas |
| Inductive reactance | X_L = omega L | rises with frequency |
| Capacitive reactance | X_C = 1 / (omega C) | falls with frequency |
| Series LCR impedance | Z = square root of (R squared + (X_L - X_C) squared) | ohms |
| Phase angle | tan phi = (X_L - X_C) / R | phi positive when the circuit is inductive |
| Current amplitude | i_m = v_m / Z | |
| Resonance frequency | omega_0 = 1 / square root of (L C) | when X_L = X_C |
| Quality factor | Q = omega_0 L / R = (1/R) times square root of (L/C) | sharpness of resonance |
| Average power | P = V_rms I_rms cos phi | cos phi is the power factor |
| Wattless current | I_rms sin phi | zero average power from this part |
| LC oscillation frequency | omega = 1 / square root of (L C) | charge and current oscillate |
| Transformer voltage ratio | V_s / V_p = N_s / N_p | ideal transformer |
| Transformer current ratio | I_s / I_p = N_p / N_s | ideal transformer |
| Transformer efficiency | eta = output power / input power |
1.2 Numbers worth remembering
The mains supply in India is 220 V at 50 Hz, and the 220 V is the rms value, so the peak value is about 311 V. The angular frequency of the 50 Hz supply is 2 pi times 50 = 314 rad per second. In a pure resistor the current and voltage are in phase and the power factor is one. In a pure inductor the current lags the voltage by 90 degrees, and in a pure capacitor it leads by 90 degrees, so the power factor is zero and the average power is zero in both cases.
1.3 The three special circuits
For a pure resistor, v and i are in phase, X = R and the power is V I. For a pure inductor, the current lags by pi by 2, the reactance is omega L and the average power is zero. For a pure capacitor, the current leads by pi by 2, the reactance is 1/(omega C) and the average power is again zero. The phrase in phase, lags or leads is the first thing an examiner looks for in an AC answer.