Question

What is meant by mean value of A.C? Derive an expression for mean value of alternating current and emf.

Answer 1

Hint: Mean value of an alternating current is the total charge flown for one complete cycle divided by the time taken to complete the cycle i.e. time period T. The value of an alternating current changes with time. Also, its direction changes after every half cycle. Hence, for a full cycle, the average value is zero. We find the average value for half cycle.

Formula used:
$i_{av}={\displaystyle \frac{\underset{0}{\overset{T/2}{\int }}idt}{{\displaystyle \frac{T}{2}}}}$

Complete answer:
When we have a DC circuit, the current of the circuit is constant. Hence, there is no problem in specifying the magnitude of the current.
In an AC circuit, the current in the circuit is an alternating current. Therefore, its value continuously changes with respect to time. Therefore, we have the difficulty of specifying the magnitude of the current as it is not a constant value. Hence, we calculate the average value of the alternating current.
Mean value of an alternating current is the total charge flown for one complete cycle divided by the time taken to complete the cycle i.e. time period T.
The magnitude of an alternating current changes from time to time and its direction also reverses after every half cycle. This means that the current is positive in one-half cycle and it is negative in the other half cycle. Because of this, the net charge flown is zero.
Therefore, the mean value of an alternating current for a complete cycle is zero. But, of zero value as no meaning and not useful for calculations. As a result, we measure the mean value of an alternating current only for the positive half cycle.
Let derive an expression for the average value of sinusoidal alternating current over a positive half cycle. A sinusoidal alternating current is given as $i=i_m\sin \omega t$,
$i_m$ is the amplitude of the sine wave (maximum current), $\omega$ is the angular frequency of the function and t is time.
Average current for half cycle is given as $i_{av}={\displaystyle \frac{\underset{0}{\overset{T/2}{\int }}idt}{{\displaystyle \frac{T}{2}}}}$ , where T is the time period of the function.
And $T={\displaystyle \frac{2\pi }{\omega }}$$\Rightarrow {\displaystyle \frac{T}{2}}={\displaystyle \frac{\pi }{\omega }}$
$i_{av}={\displaystyle \frac{\underset{0}{\overset{\pi /\omega }{\int }}{i}_m\sin \omega tdt}{{\displaystyle \frac{\pi }{\omega }}}}$
$={\displaystyle \frac{-{\displaystyle \frac{i_m}{\omega }}{\cos \omega t|}_0^{\pi /\omega }}{{\displaystyle \frac{\pi }{\omega }}}}$
$={\displaystyle \frac{-2i_m{\cos \omega t|}_0^{\pi /\omega }}{\pi }}$
$={\displaystyle \frac{-2i_m(\cos \omega \left({\displaystyle \frac{\pi }{\omega }}\right)-\cos (0))}{\pi }}$
$={\displaystyle \frac{-2i_m(0-1)}{\pi }}={\displaystyle \frac{2i_m}{\pi }}$.
Therefore, the mean value of the alternating current is ${\displaystyle \frac{2i_m}{\pi }}$.

Note:
Other than the mean value, we use another value of the alternating current called root mean square value.
The root mean square value is the root of the mean of the square of the alternating current over one full cycle.
The root mean square (rms) value of a sinusoidal alternating current is equal to ${\displaystyle \frac{i_m}{\sqrt{2}}}$.