# Derive an expression for the average or mean value of AC for half cycle

In this short piece of article, we will derive an expression for the average or mean value of AC for half cycle, so let’s get started…

## What is the average value of AC?

It is defined as that value of direct current that sends the same charge in a circuit at the same time as is sent by the given alternating current in its half-time period. It is denoted by $l_{a v}$ or $I_{m}$

## Relation between average value and peak value of AC

The value of alternating current at any instant $t$ is given by $$I=I_{0} \sin \omega t$$ This current can be assumed to remain constant for a short time $d t$. Then the amount of charge that flows through the circuit in a short time $d t$ is given by $$d q=I. d t=I_{0} \sin \omega t . d t$$ The total charge that flows through the circuit, say in the first half cycle, i.e., from $t=0$ to $t=T / 2$ is given by \begin{aligned} q &=\int_{0}^{T / 2} d q=\int_{0}^{T / 2} I_{0} \sin \omega t d t\\&=I_{0}\left[-\frac{\cos \omega t}{\omega}\right]_{0}^{T / 2} \\ &=-\frac{I_{0}}{2 \pi / T}\left[\cos \frac{2 \pi}{T} t\right]_{0}^{T / 2} \\ &=-\frac{I_{0} T}{2 \pi}[\cos \pi-\cos 0] \quad\left[\because \omega=\frac{2 \pi}{T}\right.\\ &=-\frac{I_{0} T}{2 \pi}[-1-1]=\frac{I_{0} T}{\pi} \end{aligned} $\therefore$ The average value of a.c. over the first half cycle is $$l_{a v}=\frac{\text { Charge }}{\text { Time }}=\frac{q}{T / 2}=\frac{2 q}{T}$$ $$=\frac{2}{T} \cdot \frac{I_{0} T}{\pi}$$ or $$I_{a v}=\frac{2}{\pi} I_{0}=0.637 I_{0}$$

Thus, the mean or average value of an alternating current is $2/\pi$ or $0.637$ times its peak value. A similar relation can be proved for the alternating emf, which is given as $$\mathcal{E}_{av}=\frac{2}{\pi}\mathcal{E_0}=0.637\mathcal{E}_0$$

## Frequently Asked Questions – FAQs

##### What is the average value of AC during a half cycle?

The average or mean value over the full cycle of a sine wave is always zero. But for the half cycle, its value is $0.637 I_{0}$, Where $I_0$ is the peak value of AC.

##### What is meant by an average value of AC?

Mean value of alternating current: Mean value of a.c. over any half cycle is that steady current that sends the same charge through a circuit in half time period of alternating current as is sent by the alternating current in the same time through the same circuit.

##### How do you find the average value of alternating current?

The average value of AC can be obtained by integrating the instantaneous values of current or voltage over the half cycle i.e. area of the curve over the half cycle and dividing the result by the base length of the half cycle.

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