AC Circuit Formulas: Reactance and Impedance
AC circuit formulas: RMS and peak, frequency and period, inductive and capacitive reactance, series RLC impedance and resonance, with a worked example.
The formulas
| To find | Formula |
|---|---|
| RMS from peak (sine) | Vrms = Vpeak / √2 |
| Peak-to-peak | Vpp = 2 × Vpeak |
| Period and angular frequency | T = 1 / f, ω = 2π f |
| Inductive reactance | XL = 2π f L |
| Capacitive reactance | XC = 1 / (2π f C) |
| Series RLC impedance | |Z| = √(R² + (XL − XC)²) |
| Phase angle | φ = arctan((XL − XC) / R) |
| Resonant frequency | f0 = 1 / (2π √(L C)) |
f in hertz, L in henries, C in farads, R, XL, XC and Z in ohms. A positive angle means the current lags the voltage (inductive).
Worked example: R = 10 Ω, L = 100 mH, C = 100 µF at 50 Hz
- XL = 2π × 50 × 0.1 = 31.42 Ω
- XC = 1 / (2π × 50 × 0.0001) = 31.83 Ω
- |Z| = √(10² + (31.42 − 31.83)²) = 10.01 Ω
- f0 = 1 / (2π × √(0.1 × 0.0001)) = 50.33 Hz, close to 50 Hz, so the circuit is near resonance
Reactance of 100 mH and 100 µF
| Frequency | XL (100 mH) | XC (100 µF) |
|---|---|---|
| 50 Hz | 31.42 Ω | 31.83 Ω |
| 60 Hz | 37.7 Ω | 26.53 Ω |
| 400 Hz | 251.3 Ω | 3.979 Ω |
Frequently asked questions
Why does a capacitor pass more current at a higher frequency?
Its reactance is 1 / (2π f C), which falls as the frequency rises. An inductor does the opposite: XL rises with frequency.
Is the 230 V on a mains supply the RMS or the peak value?
It is the RMS value. The peak of a 230 V sine wave is 230 × √2 = 325 V.
What happens at resonance?
In a series circuit XL equals XC, they cancel, and the impedance falls to R alone, so the current is at its largest.
Do these formulas work for non-sinusoidal waveforms?
Only for each frequency component separately. The RMS-to-peak ratio of √2 applies to a pure sine wave only.
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