Sinusoidal Steady State, Phasors and AC Power

In sinusoidal steady state, every voltage and current is a sine wave at the source frequency. Phasors turn the differential equations into complex algebra.

RMS and average values

Form factor = RMS/average; crest (peak) factor = peak/RMS (√2 for a sine).

Phasors and impedance

• v(t) = V_m cos(ωt + φ) ↔ phasor V = V∠φ (using RMS magnitudes in power work).
• Impedances: Z_R = R, Z_L = jωL, Z_C = 1/(jωC) = −j/(ωC).
• In an inductor current lags voltage by 90°; in a capacitor it leads by 90° (ELI the ICE man).
• Impedances combine like resistances. Admittance Y = 1/Z = G + jB.

AC power

With RMS phasors V and I and the angle φ between them:

• Real power P = VI cos φ (W).
• Reactive power Q = VI sin φ (VAR); positive for inductive loads.
• Apparent power S = VI (VA), with S² = P² + Q².
• Complex power S = V I* = P + jQ.
• Power factor = cos φ = P/S (lagging for inductive loads).

Power factor correction: a capacitor in parallel supplies reactive power Q_C = P(tan φ₁ − tan φ₂), with C = Q_C/(ωV²).

Maximum power transfer (AC): Z_L = Z_th* (complex conjugate), giving P_max = |V_th|²/(4R_th).

Worked example

A load takes 10 kW at 0.8 pf lagging from 230 V, 50 Hz. Find the capacitor to raise the pf to 1.

• tan φ₁ = 0.75, so Q = 10 × 0.75 = 7.5 kVAR.
• C = 7500/(2π × 50 × 230²) = 451 µF.

Practice