Alternating Current - Quick Revision
AC basics and rms values
- An ac voltage v = vm sin(omega t) drives a sinusoidal current; the average current over a cycle is zero, but the average power is not.
- RMS (effective) value: Irms = I0/sqrt2 = 0.707 I0; Vrms = V0/sqrt2 = 0.707 V0. Mains '220 V' is rms (peak about 311 V).
- The rms current is the equivalent dc current producing the same average Joule heating.
Single elements
- Resistor: V = I R, voltage and current in phase (phi = 0).
- Inductor: XL = omega L = 2 pi f L; current lags voltage by pi/2; XL grows with frequency.
- Capacitor: XC = 1/(omega C); current leads voltage by pi/2; XC falls with frequency.
- Average power over a cycle in a pure L or pure C is zero (wattless current).
Series LCR circuit
- Impedance: Z = sqrt(R^2 + (XL - XC)^2); current amplitude im = vm/Z.
- Phase: tan(phi) = (XC - XL)/R. XC > XL: capacitive (current leads). XL > XC: inductive (current lags).
Resonance
- At resonance XL = XC, so Z = R (minimum) and current is maximum (im = vm/R).
- omega0 = 1/sqrt(LC). Needs both L and C present.
- Q-factor = omega0 L/R = 1/(omega0 C R) measures sharpness of resonance; used in radio/TV tuning.
Power and power factor
- P = Vrms Irms cos(phi) = I^2 R; power is dissipated only in R.
- Power factor cos(phi) = R/Z: 1 for pure R, 0 for pure L or C.
- Low power factor means large current and large I^2 R transmission loss; improved with a parallel capacitor.
Transformer
- Vs/Vp = Ns/Np; for an ideal transformer Is/Ip = Np/Ns (power conserved).
- Ns > Np: step-up (voltage up, current down); Ns < Np: step-down.
- Losses: flux leakage, winding resistance (I^2 R), eddy currents (use laminated core), hysteresis.
Common traps
- Quoted ac values are rms unless stated otherwise.
- Pure L and pure C dissipate no power; only R does.
- Voltages across elements add as phasors, not arithmetically (VR + VC can exceed source V).
- Resonant frequency does NOT depend on R.