AC Circuits and Three-Phase Power

AC Fundamentals

Sinusoidal Waveform

$v(t) = V_m \sin(\omega t + \phi)$

where:

• Vm = peak (maximum) voltage
• ω = 2πf = angular frequency (rad/s)
• f = frequency (Hz); standard in US: 60 Hz
• φ = phase angle

RMS Values

$V_{rms} = \frac{V_m}{\sqrt{2}} \approx 0.707 V_m$

RMS values are used for power calculations. When someone says "120 V" for household power, they mean 120 V rms (peak is about 170 V).

Impedance

Impedance Z is the AC equivalent of resistance: $Z = R + jX$

where:

• R = resistance (Ω) — real part
• X = reactance (Ω) — imaginary part
• j = √(-1)

Mnemonic: ELI the ICE man — In an inductor (L), voltage (E) leads current (I). In a capacitor (C), current (I) leads voltage (E).

Series RLC Circuit

$Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$

$|Z| = \sqrt{R^2 + (X_L - X_C)^2}$

Resonance (XL = XC): $f_0 = \frac{1}{2\pi\sqrt{LC}}$

At resonance: Z = R (purely resistive), current is maximum.

AC Power

Power Triangle

$S = P + jQ$ $|S| = \sqrt{P^2 + Q^2}$

Power Factor

$PF = \cos(\phi) = \frac{P}{S} = \frac{R}{|Z|}$

Most industrial loads are inductive (motors, transformers) with lagging power factor. Capacitors are added to improve (correct) power factor toward unity, reducing utility penalties.

Three-Phase Power

Balanced Three-Phase Systems

where:

• VL = line-to-line voltage
• Vφ = phase voltage
• IL = line current
• Iφ = phase current

Three-Phase Power (Balanced Load)

$P_{total} = \sqrt{3} V_L I_L \cos(\phi)$ $S_{total} = \sqrt{3} V_L I_L$

Motor Efficiency

$\eta = \frac{P_{out}}{P_{in}} = \frac{P_{mechanical}}{P_{electrical}}$

Example: A 3-phase motor draws 10 A at 480 V with PF = 0.85 and efficiency = 90%. Output power?

Pin = √3 × 480 × 10 × 0.85 = 7,064 W Pout = 0.90 × 7,064 = 6,358 W ≈ 8.5 hp (÷ 745.7)

Measuring Devices