AC Circuits and Three-Phase Power

AC Fundamentals and RMS

Alternating current varies sinusoidally with time: $v(t) = V_m \sin(\omega t + \phi)$ where V_m is the peak (amplitude), ω = 2πf is the angular frequency in rad/s, f is the frequency in hertz (60 Hz in North America, 50 Hz in much of the world), and φ is the phase angle. 707,V_m, \qquad I_{rms} = \frac{I_m}{\sqrt{2}}$$ Every AC voltage rating you see — 120 V outlets, 480 V industrial feeders — is an RMS value; the 120 V outlet actually peaks near 170 V. ** A frequent FE error is plugging peak values into P = VI or treating a quoted line voltage as a peak.

Impedance and Reactance

In AC circuits, capacitors and inductors oppose current in a frequency-dependent, phase-shifting way described by impedance Z, the AC generalization of resistance, expressed as a complex number: $Z = R + jX, \qquad |Z| = \sqrt{R^2 + X^2}, \qquad \theta = \arctan\!\frac{X}{R}$

The mnemonic ELI the ICE man captures the phasing: in an inductor (L), voltage E leads current I (ELI); in a capacitor (C), current I leads voltage E (ICE). Reactance grows with frequency for an inductor (X_L = ωL) but shrinks for a capacitor (X_C = 1/ωC) — at very high frequency an inductor blocks and a capacitor passes current.

Series RLC, Resonance, and the Power Triangle

For a series RLC circuit the reactances of L and C subtract because they are 90° out of phase with each other (opposite signs): $Z = R + j\left(\omega L - \frac{1}{\omega C}\right), \qquad |Z| = \sqrt{R^2 + (X_L - X_C)^2}$ At the resonant frequency the two reactances are equal (X_L = X_C) and cancel, leaving Z = R (purely resistive) and current at its maximum: $f_0 = \frac{1}{2\pi\sqrt{LC}}$

Worked impedance example. A series circuit has R = 30 Ω, X_L = 40 Ω, X_C = 10 Ω. Net reactance = 40 − 10 = 30 Ω (inductive). |Z| = √(30² + 30²) = √1,800 = 42.4 Ω, with phase angle arctan(30/30) = 45° (current lags). At 120 V_rms, I = 120/42.4 = 2.83 A.

The power triangle. In AC, current and voltage may be out of phase, splitting power into three parts:

They relate by S² = P² + Q², i.e. S = √(P² + Q²). Only real power P does useful work or shows as heat; reactive power Q sloshes back and forth into inductors/capacitors each cycle. The power factor is: $PF = \cos\phi = \frac{P}{S} = \frac{R}{|Z|}$ Inductive (lagging) loads — motors, transformers — dominate industry. Utilities penalize low power factor because it raises line current for the same useful power; adding capacitors corrects a lagging PF toward unity, cutting current and losses.

Three-Phase Power

Large power systems use three-phase AC: three voltages 120° apart, delivering constant total power and using conductors efficiently. Loads connect in wye (Y / star) or delta (Δ):

where V_L is line-to-line voltage, V_φ phase voltage, I_L line current, I_φ phase current. For a balanced load, total real and apparent power are: $P_{total} = \sqrt{3}\,V_L I_L \cos\phi, \qquad S_{total} = \sqrt{3}\,V_L I_L$ Using line quantities, the √3 already accounts for all three phases — do not multiply by 3 again. The common 480/277 V building system is wye: 480 V between lines, 277 V (= 480/√3) line-to-neutral.

Worked power-factor example. A load draws P = 5 kW of real power and Q = 4 kVAR of reactive power. Apparent power S = √(P² + Q²) = √(5² + 4²) = √41 = 6.40 kVA, so PF = P/S = 5/6.40 = 0.78 lagging. To raise the PF to unity, a capacitor bank supplying 4 kVAR of leading reactive power is added, canceling the load's 4 kVAR and dropping line current.

Worked three-phase example. A balanced three-phase motor runs at V_L = 480 V, draws I_L = 10 A, PF = 0.85, with 90% efficiency. Input power P_in = √3 × 480 × 10 × 0.85 = 7,064 W. Output power P_out = 0.90 × 7,064 = 6,358 W ≈ 8.5 hp (÷ 745.7 W/hp).

Measuring devices. A voltmeter (high internal resistance) connects in parallel; an ammeter (low resistance) connects in series; a wattmeter uses a parallel voltage coil and a series current coil; an ohmmeter is used only on a de-energized circuit. Connecting an ammeter in parallel (a near-short) is a classic destructive mistake.