AC Fundamentals and Single-Phase Circuits
This domain moves from DC to alternating quantities: how an EMF is generated, the meaning of RMS and average values, reactance, impedance and power factor in R, L and C circuits.
Generating an alternating EMF
A coil rotating in a uniform field produces e = E_m sin ωt, where ω = 2πf radians per second and f = PN/120 for a machine. In India the supply is 230 V, 50 Hz, single phase.
RMS, average and factors
For a sine wave: RMS = E_m/√2 = 0.707 E_m; average over a half cycle = 2E_m/π = 0.637 E_m. Form factor = RMS/average = 1.11; peak factor = E_m/RMS = 1.414. The average over a complete cycle is zero. Quoted AC voltages and currents are RMS values.
Reactance and impedance
X_L = 2πfL (rises with frequency); X_C = 1/(2πfC) (falls with frequency). For series R–L–C: Z = √(R² + (X_L − X_C)²), and I = V/Z.
Power
Power factor cos φ = R/Z. P = VI cos φ (watt), Q = VI sin φ (var), S = VI (volt-ampere). Only resistance consumes real power; ideal L and C circulate reactive power.
Worked example
R = 6 Ω in series with X_L = 8 Ω across 230 V, 50 Hz. Z = √(36 + 64) = 10 Ω. I = 230/10 = 23 A. cos φ = 6/10 = 0.6. P = 230 × 23 × 0.6 = 3174 W.
Traps
- Treating the peak value as the rated value.
- Adding X_L and X_C arithmetically instead of taking their difference.
- Writing a power factor greater than 1, or applying cos φ to a purely resistive circuit.
- Mixing var and watt in the same answer.