Peak and rms values

Describing an alternating supply

• An alternating voltage or current varies sinusoidally with time, reversing direction every half cycle:

$$V = V_{\text{MAX}}\sin\omega t \qquad \omega = 2\pi f$$

• $V_{\text{MAX}}$ — the peak voltage (V)

• $\omega$ — the angular frequency (rad s⁻¹)

• $f$ — the frequency (Hz); NZ mains is $50$ Hz

• $t$ — time (s)

• The average value of a sine wave over a whole cycle is zero, because it spends equal time positive and negative. So the average is useless as a description of an AC supply, and a different measure is needed.

Change the frequency and amplitude below and watch the waveform, with the rms level marked:

Why rms values are used

• The useful measure is the one that describes the energy delivered.
• Power in a resistor is $P = I^2R$, and because the current is squared, the power is positive in both half cycles — a negative current heats a resistor exactly as much as a positive one.
• The root mean square (rms) value is defined so that:

An AC supply of a given rms value delivers the same average power to a resistor as a DC supply of that same value.

• This is why a $230$ V rms mains supply is directly comparable with a $230$ V battery: both would light the same lamp to the same brightness.

$$V_{\text{rms}} = \frac{V_{\text{MAX}}}{\sqrt{2}} \qquad I_{\text{rms}} = \frac{I_{\text{MAX}}}{\sqrt{2}}$$

• Equivalently, $V_{\text{MAX}} = \sqrt{2}\,V_{\text{rms}} \approx 1.414\,V_{\text{rms}}$.
• The factor $\sqrt{2}$ comes from averaging $\sin^2$ over a cycle, which gives $\tfrac{1}{2}$ — so the mean square is half the peak squared, and the root of that is the peak over $\sqrt{2}$.

Which value is quoted

• Quoted AC voltages and currents are always rms unless stated otherwise. The "$230$ V" mains supply is $230$ V rms; its peak is $\sqrt{2} \times 230 = 325$ V.
• Meters (multimeters, ammeters) read rms.
• An oscilloscope displays the waveform, so you read the peak (or peak-to-peak) from it and convert.
• Peak-to-peak is $2V_{\text{MAX}}$ — the full vertical extent of the trace, and a common oscilloscope reading.

Power in AC circuits

$$P_{\text{average}} = I_{\text{rms}}V_{\text{rms}} = I_{\text{rms}}^2R = \frac{V_{\text{rms}}^2}{R}$$

• The familiar DC power relationships work unchanged with rms values — which is the entire point of defining them that way.
• Using peak values in these expressions gives the peak power, which is twice the average power. Questions frequently test this.