3.2 AC Steady-State & Phasors

Phasors turn AC into algebra

A sinusoid v(t) = V_m cos(omegat + phi) is represented by a phasor V = V_m angle phi, where omega = 2pi*f. Once every element is an impedance, KVL, KCL, voltage dividers, Thevenin, and node/mesh methods all apply exactly as in DC, but with complex arithmetic. The FE on-screen calculator handles rectangular-to-polar conversion, so keep numbers in whichever form simplifies the step.

Note that FE problems often quote root-mean-square (RMS) values. For a sinusoid, V_rms = V_m / sqrt(2). Power calculations use RMS values.

Impedance and reactance

Each passive element has an impedance Z (units: ohms):

• Resistor: Z_R = R (no phase shift).
• Inductor: Z_L = jX_L where X_L = omegaL = 2pif*L. Voltage leads current by 90 degrees.
• Capacitor: Z_C = -jX_C where X_C = 1/(omegaC) = 1/(2pif*C). Current leads voltage by 90 degrees.

The mnemonic ELI the ICE man captures the phase: in an inductor (L) voltage E leads current I (ELI); in a capacitor (C) current I leads voltage E (ICE).

Total series impedance is Z = R + j(X_L - X_C). Its magnitude is |Z| = sqrt(R^2 + (X_L - X_C)^2) and its angle is theta = arctan((X_L - X_C)/R). The current phasor is I = V / Z.

Complex power: P, Q, and S

The complex power S = V_rms * I_rms* (using the conjugate of current) resolves into:

• Real (average) power P = V_rms I_rms cos(theta), units watts (W). This is the power actually converted to work/heat.
• Reactive power Q = V_rms I_rms sin(theta), units volt-amperes reactive (VAR). It oscillates between source and reactive elements.
• Apparent power S = V_rms I_rms = |P + jQ|, units volt-amperes (VA).

The power triangle relates them: S^2 = P^2 + Q^2, and theta is the angle between voltage and current.

Inductive loads absorb positive Q (lagging pf); capacitive loads supply Q (leading pf).

Power factor correction

Low (lagging) power factor means high current for a given real power, increasing line losses. Adding shunt capacitance supplies reactive power locally and raises pf toward unity. The capacitor reactive power needed is Q_C = P*(tan(theta_1) - tan(theta_2)), moving from initial angle theta_1 to target theta_2.

Resonance

In a series RLC circuit, resonance occurs when X_L = X_C, so 2pif_0L = 1/(2pif_0C), giving

f_0 = 1 / (2pisqrt(L*C)).

At resonance the reactances cancel, impedance is purely resistive and minimum (Z = R), and current is maximum. The sharpness is the quality factor Q = (1/R)sqrt(L/C) = omega_0L/R, and bandwidth BW = f_0 / Q. A parallel RLC circuit resonates at the same f_0 but presents maximum impedance (a tank circuit), so line current is minimum at resonance.