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8 AC theory: RMS, reactance, impedance & resonance

AC, RMS & peak

RMS values V_{\mathrm{rms}}=\frac{V_{\text{peak}}}{\sqrt2}\qquad I_{\mathrm{rms}}=\frac{I_{\text{peak}}}{\sqrt2} RMS is the “effective” DC-equivalent value. NZ mains 230 V is RMS (≈ 325 V peak). AC power P = I<sub>rms</sub> V<sub>rms</sub>

Reactance & impedance

Capacitive reactance X_C=\frac{1}{2\pi fC} Falls as frequency rises: capacitors “pass” high frequencies. Inductive reactance X<sub>L</sub> = 2π f L Rises with frequency: inductors “block” high frequencies. Impedance (RLC series) Z=\sqrt{R^2+\left(X_L-X_C\right)^2}\qquad V=IZ Combines resistance and reactance. Resonance f_0=\frac{1}{2\pi\sqrt{LC}} X_L = X_C, impedance is minimum (just R), current is maximum.

Phase relationships

  • In a capacitor, current leads voltage by 90° (“ICE”).
  • In an inductor, current lags voltage by 90° (“ELI”).
  • Mnemonic: ELI the ICE man, E leads I in an inductor (L); I leads E in a capacitor (C).
Worked example Capacitive reactance Find the reactance of a 2.2 µF capacitor at 50 Hz, and the RMS current if 230 V RMS is across it. X_C = 1/(2πfC) = 1/(2π × 50 × 2.2×10⁻⁶) = 1/(6.91×10⁻⁴) = 1447 Ω. I_rms = V_rms / X_C = 230 / 1447 = 0.159 A. ✓ X_C ≈ 1.4 kΩ; I_rms ≈ 0.16 A.

Practise just this