AC Circuits: Phasors, Resonance and Power For students.

An AC circuit is one where voltage and current reverse direction repeatedly, following a sine wave — the kind of electricity that comes from wall sockets and power grids. Because everything keeps changing, AC analysis uses special tools: phasors, impedance and the idea of resonance.

Almost all generated and distributed power is AC. Motors, transformers, radios and chargers all depend on AC behaviour — phasors, resonance and power factor are daily vocabulary for electrical engineers.

Key Concepts

• Sine wave: v(t) = Vm × sin(ωt), with peak Vm and angular frequency ω = 2πf.
• RMS value: the heating-equivalent value: Vrms = Vm / √2 ≈ 0.707 × Vm. Mains "230 V" is RMS; its peak is about 325 V.
• Phasor: a rotating arrow representing a sine wave by magnitude and phase angle, like 230 ∠ 30° V.
• Impedance (Z): the AC version of resistance, measured in ohms: Z = R + jX, combining resistance R with reactance X.
• Reactance: XL = 2πfL grows with frequency; XC = 1/(2πfC) shrinks with frequency.
• Resonance: the frequency where XL equals XC and they cancel.
• Power: real P = Vrms × Irms × cos φ (watts), reactive Q = Vrms × Irms × sin φ (VAR), apparent S = Vrms × Irms (VA). Power factor = cos φ.

Formulas

• Inductive reactance: XL = 2πfL ohms, with f in hertz and L in henries. Capacitive reactance: XC = 1 / (2πfC) ohms, with C in farads.
• Series impedance: Z = √(R² + (XL − XC)²) ohms, and phase angle φ = arctan((XL − XC) / R). Assumes a series R-L-C circuit with sinusoidal steady state.
• Resonant frequency: f₀ = 1 / (2π√(LC)) hertz — where inductive and capacitive reactances are equal.
• Real power: P = Vrms × Irms × cos φ watts. Utilities penalise a low power factor because the extra current heats their cables without doing useful work.

Worked Example

A series circuit has R = 30 Ω and a coil with XL = 40 Ω at 50 Hz, fed by 100 V RMS. Find the current, phase angle and real power.

1. Impedance: Z = √(30² + 40²) = √(900 + 1600) = √2500 = 50 Ω.
2. Current: I = V / Z = 100 / 50 = 2 A RMS.
3. Phase angle: φ = arctan(40 / 30) = arctan(1.333) = 53.13 degrees — the current lags the voltage because the circuit is inductive.
4. Power factor: cos φ = R / Z = 30 / 50 = 0.6.
5. Real power: P = 100 × 2 × 0.6 = 120 W. Apparent power is 200 VA, so 80 VAR circulates uselessly — a capacitor could correct this.

Resonance check: with L = 0.1 H and C = 100 µF, f₀ = 1/(2π√(0.1 × 100 × 10⁻⁶)) = 50.33 Hz — at this frequency the circuit draws maximum current in phase with the voltage.

Common Misconceptions

• "230 V mains peaks at 230 V." 230 V is RMS; the peak is 230 × √2 ≈ 325 V, and insulation must withstand the peak, not the RMS value.
• "Resonance always means maximum current." Only in a series circuit. In a parallel R-L-C circuit, resonance gives minimum total current (maximum impedance) — the dual behaviour.
• "Reactive power is wasted power." It is not consumed, but it still flows through wires and transformers, loading them — which is exactly why industries install capacitor banks to cancel it.

Real-World Applications

• Power grids: entire transmission systems run at 50 or 60 Hz AC, with power-factor correction at substations.
• Radio tuning: a resonant L-C circuit selects one station's frequency.
• Induction motors: the workhorses of industry run on AC, and their starting current is an impedance problem.

AC circuits use phasors and impedance; resonance at f₀ = 1/(2π√(LC)) cancels reactances; only P = VI cos φ does real work — the ideas behind power grids and radio tuning alike.