Average and RMS Values

13.3 Average and RMS Values

Alternating voltages and currents pass through every value between zero and the peak value in one cycle, so some specific values must be defined before waveforms can be compared or measured.

Peak Value

The peak value of an alternating current (or emf) is the maximum value of the current (or emf) in either direction. It is denoted $i_0$ (or $e_0$).

(a) Average Value of AC

The average value of AC is the average of all the values of the current (or voltage) taken over one half cycle.

The average over a full cycle is always zero, because the average value of $\sin\omega t$ over a complete cycle is zero — the positive half exactly cancels the negative half. The mean value of AC over a full cycle therefore has no significance, and the mean value of AC is defined as the average over half a cycle.

The average value of $\sin\theta$ for $\theta$ in the range $0$ to $\pi$ is

$$\dfrac{\displaystyle\int_0^{\pi}\sin\theta\,d\theta}{\displaystyle\int_0^{\pi}d\theta} = \dfrac{\left[-\cos\theta\right]_0^{\pi}}{\pi} = \dfrac{2}{\pi} = 0.637$$

Hence the average value of an alternating current or emf is $0.637$ times its peak value:

$$i_{av} = 0.637\,i_0 = \dfrac{2i_0}{\pi}, \qquad e_{av} = 0.637\,e_0 = \dfrac{2e_0}{\pi}$$

(b) RMS Value

A moving-coil ammeter and voltmeter measure the average value of the current and voltage respectively. It is obvious, therefore, that moving-coil instruments cannot be used to measure alternating current and voltage, since that average over a cycle is zero.

To measure these quantities we must use a property that does not depend on the direction of flow. The heating effect is such a property: heat depends on the square of the current, and a square is always positive.

Consider an alternating current of peak value $i_0$ flowing through a resistance $R$, and let $H$ be the heat produced in time $t$. The same quantity of heat $H$ can be produced in the same resistance $R$ in the same time $t$ by passing a steady current of constant magnitude through it. The value of that steady current is called the effective value, or virtual value, or rms value of the given alternating current, denoted $i_{rms}$.

$$i_{rms} = \sqrt{\dfrac{\displaystyle\int_0^T i_0^2\sin^2\omega t\,dt}{\displaystyle\int_0^T dt}} = \sqrt{\dfrac{i_0^2}{2}} = \dfrac{i_0}{\sqrt2} = 0.707\,i_0$$

using $\displaystyle\int_0^T\sin^2\omega t\,dt = \dfrac{T}{2}$. Similarly, for the emf,

$$e_{rms} = \dfrac{e_0}{\sqrt2} = 0.707\,e_0$$

Heat Produced Over a Complete Cycle

$$H = \int_0^T i^2R\,dt = i_0^2R\int_0^T\sin^2\omega t\,dt = i_0^2R\left(\dfrac{T}{2}\right) = i_{rms}^2\,R\,T$$

This is the same as the heat produced by a direct current of magnitude $i_{rms}$ flowing for the same time — which is exactly what the rms value was defined to mean.

Standard Results

• $i_{av} = 0.637\,i_0$, $e_{av} = 0.637\,e_0$ (half cycle); zero over a full cycle.
• $i_{rms} = 0.707\,i_0$, $e_{rms} = 0.707\,e_0$; conversely $i_0 = \sqrt2\,i_{rms} = 1.414\,i_{rms}$.
• $H = i_{rms}^2Rt$.
• Every AC meter reading, and every quoted supply voltage such as 220 V, is an rms value.

Worked Pattern

For $e = 8\sin 628.4\,t$: comparing with $e = e_0\sin\omega t$ gives $e_0 = 8$ V and $\omega = 628.4$ rad/s, so $f = \omega/2\pi = 628.4/(2\times3.142) = 100$ Hz. At $t = 10$ ms $= 10^{-2}$ s, $e = 8\sin(6.284) = 8\sin(2\pi) = 0$ V.

For $e = 6\sin(100\pi t)$: $e_0 = 6$ V, $f = 100\pi/2\pi = 50$ Hz, $T = 1/f = 0.02$ s, and at $t = 2$ ms, $e = 6\sin(100\pi\times2\times10^{-3}) = 3.527$ V.