Question

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

Accepted 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 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 sinusodial alternating current is given as $i = i_{m} S i n ω t$ , $i_{m}$ is the amplitude of the sine wave (maximum current), $ω$ is the angular frequency of the function and $t$ is time. Average current for half cycle is given as $i_{a v} = \frac{\int_{0}^{\frac{T}{2}} i d t}{\frac{T}{2}}$ , where $T$ is the time period of the function. And $T = \frac{2 π}{ω} \Rightarrow \frac{T}{2} = \frac{π}{ω}$

$i_{a v} = \frac{\int_{0}^{\frac{π}{ω}} i_{m} S i n ω t d t}{\frac{π}{ω}}$

$= \frac{- \frac{i_{m}}{ω} C o s ω t_{0}^{\frac{π}{ω}}}{\frac{π}{ω}}$

$\frac{- 2 i_{m} C o s ω t_{0}^{\frac{π}{ω}}}{π}$

$= \frac{- 2 i_{m} (C o s ω (\frac{π}{ω}) - C o s (0))}{π}$

$= \frac{- 2 i_{m} (0 - 1)}{π} = \frac{2 i_{m}}{π}$ .

Therefore, the mean value of the alternating current is $\frac{2 i_{m}}{π}$ .

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

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