What is RMS value? — Physics

Definition 8.13 University Physics — Year 1 · Chapter 8 — Sinusoidal Steady State and Impedance

The root-mean-square (rms, or effective) value of a periodic signal $x(t)$ is $X_{\mathrm{rms}} = \sqrt{\langle x^2\rangle}$ , the square root of the time average of $x^2$ over a period. For a sinusoid of amplitude $X$ , $X_{\mathrm{rms}} = X/\sqrt2$ . The $230\,\mathrm{V}$ of the mains is an rms value; its amplitude is $325\,\mathrm{V}$ .

Examples

Example 8.15 (Power factor correction)

A workshop motor on $230\,\mathrm{V}$ rms draws $2.0\,\mathrm{kW}$ with $\cos\varphi = 0.70$ (inductive): $I_{\mathrm{rms}} = 2000/(230 \times 0.70) = 12.4\,\mathrm{A}$ , against $8.7\,\mathrm{A}$ if the power factor were $1$ — the extra current heats the utility’s cables for nothing, hence the penalty. A capacitor in parallel supplies the coil’s quarter-period energy swings locally: the reactive power $P\tan\varphi = 2.0\,\mathrm{kvar}$ needs $C = P\tan\varphi/(\omega U_{\mathrm{rms}}^2) \approx 120\,\text{µ}\mathrm{F}$ at $50\,\mathrm{Hz}$ , and the line current drops to $8.7\,\mathrm{A}$ .

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