Impedance: resistance plus reactance
On AC an inductor and a capacitor oppose current with reactance, which depends on frequency. Inductive reactance rises with f; capacitive reactance falls. In a series circuit they add to the resistance as a complex impedance:
X_L = 2πf·L, X_C = 1 / (2πf·C) Z = R + j(X_L − X_C) |Z| = √(R² + (X_L − X_C)²), θ = atan((X_L − X_C)/R)
Lead and lag
The angle θ is how far the voltage leads the current. An inductor resists changes in current, so the current peaks a quarter cycle after the voltage: inductive loads lag. A capacitor must charge before its voltage rises, so its current peaks first: capacitive loads lead. A pure resistor keeps them in step.
- R only: θ = 0°, in phase
- L only: current lags by 90°
- C only: current leads by 90°
- RL: lags by 0–90°; RC: leads by 0–90°
Real, reactive and apparent power
Only the part of the current in phase with the voltage delivers energy. The rest flows back and forth each cycle. The three powers form a right-angled triangle, and the power factor is the cosine of θ.
S = V·I (VA) P = V·I·cos θ (W) Q = V·I·sin θ (var) PF = cos θ = P / S
Power factor correction and resonance
Motors are inductive, so factories draw lagging reactive power that the utility must still carry. A capacitor across the load supplies leading Q that cancels it, raising the power factor toward 1 and cutting the line current.
In a series RLC circuit, X_L and X_C cancel at one frequency, the resonant frequency. There Z = R, the current is largest and the power factor is exactly 1.
f0 = 1 / (2π·√(L·C))
Worked example (the simulator’s default RL load)
- RL load: R = 10 Ω, L = 50 mH on 220 V, 50 Hz.
- X_L = 2π·50·0.050 = 15.71 Ω; |Z| = √(10² + 15.71²) = 18.62 Ω; θ = atan(15.71/10) = 57.5°.
- I = 220/18.62 = 11.8 A, lagging by 57.5°; PF = cos 57.5° = 0.537 lagging.
- P = I²R = 1.40 kW; Q = I²X_L = 2.19 kvar; S = V·I = 2.60 kVA.
- To correct to PF = 1, add C = Q/(ωV²) = 2193/(314.2·220²) ≈ 144 µF in parallel.
Common mistakes
- Adding R and X directly. They are at right angles: |Z| = √(R² + X²), not R + X.
- Mixing up the sign: inductive loads lag (θ > 0); capacitive loads lead (θ < 0).
- Thinking reactive power is wasted energy. It does no work on average, but it still flows in the wires and loads them.
Common questions
What is power factor?
The ratio of real power to apparent power, PF = P/S = cos θ, where θ is the phase angle between voltage and current. It runs from 0 (purely reactive) to 1 (purely resistive).
Why does current lag in an inductor?
An inductor’s voltage is L·di/dt, so it is largest where the current changes fastest: as the current crosses zero. That puts the current’s peak a quarter cycle (90°) after the voltage’s.
How do you find the resonant frequency of an RLC circuit?
f0 = 1/(2π√(LC)), where X_L = X_C. With L = 50 mH and C = 100 µF, f0 ≈ 71 Hz.