Resonance, power and power factor
Resonance in a series LCR circuit: X_L and X_C change with frequency in opposite ways. At one frequency they become equal: ωL = 1/(ωC). This is the resonant frequency: ω₀ = 1/√(LC), or f₀ = 1/(2π√(LC)). At resonance, Z = R (minimum), the current is maximum and V and I are in phase. Radio and TV tuning circuits use resonance to pick one station from many. The sharpness of resonance is described by the quality factor Q = ω₀L/R; a smaller R gives a sharper peak.
Worked example: L = 0.1 H, C = 10 μF: √(LC) = √(10⁻⁶) = 10⁻³ s, so f₀ = 1/(2π × 10⁻³) ≈ 159 Hz.
Average power in AC: P = V_rms I_rms cos φ, where cos φ is the power factor and cos φ = R/Z.
- Pure resistor: φ = 0, cos φ = 1, P = V_rms I_rms.
- Pure inductor or pure capacitor: φ = 90°, cos φ = 0, P = 0.
- At resonance in LCR: cos φ = 1.
Wattless current: in a circuit with only L or only C, current flows but no average power is used. The component I_rms sin φ of current, which is 90° out of phase with the voltage, is called the wattless current. A choke coil (an inductor with small resistance) is used to limit current in AC circuits such as tube lights with very little wasted power.
Worked example: V_rms = 200 V, Z = 50 Ω, R = 30 Ω. I_rms = 4 A, cos φ = 0.6, P = 200 × 4 × 0.6 = 480 W.
Key points
- f₀ = 1/(2π√LC); at resonance Z = R and I is maximum.
- P = V_rms I_rms cos φ; cos φ = R/Z.
- Power factor of pure L or pure C is zero.
- Q factor = ω₀L/R.
Exam tips
- Many questions give L and C in mH and μF; convert to H and F first.
- If asked why a choke is preferred to a resistor, the answer is that a choke wastes almost no power.
Common mistakes
- Thinking impedance is zero at resonance (it equals R).
- Forgetting cos φ when finding average power.
- Confusing ω₀ with f₀ (differ by 2π).