Name: Purely Inductive Circuit
Uploaded: 2019-07-19
Duration: 5 min 40 s
Description: How does an inductor behave in a circuit? This video will explain every parameter involved in the derivation of a pure inductor working with AC supply.

Question

Derive an expression for average or mean value of alternating current for half cycle.

Accepted Answer

Expression for mean value of a.c. for half cycle:

Suppose an a.c. current: $I = I_{\circ} \sin (ω t)$

The charge sent by a.c. current in $d t$ time is: $d q = I d t = I_{\circ} \sin (ω t) d t$

For an a.c. half cycle, $t$ is from $0$ to $\frac{T}{2}$ .

Thus, $\int d q = \int_{0}^{\frac{T}{2}} I_{\circ} \sin (ω t) d t$

$\Rightarrow q = I_{\circ} \int_{0}^{\frac{T}{2}} \sin (ω t) d t = I_{\circ} {[- \frac{\cos (ω t)}{ω}]}_{0}^{\frac{T}{2}}$

$\Rightarrow q = - \frac{I_{\circ}}{ω} {[\cos (ω t)]}_{0}^{\frac{T}{2}} = - \frac{I_{\circ}}{\frac{2 π}{T}} {[\cos (\frac{2 π}{T} t)]}_{0}^{\frac{T}{2}} = - \frac{I_{\circ} T}{2 π} [\cos (\frac{2 π}{T} \frac{T}{2}) - \cos 0]$

$q = - \frac{I_{\circ} T}{2 π} (- 2) = \frac{I_{\circ} T}{π}$ ............................... $(1)$

Let $I_{m}$ be the mean value of a.c. over positive half cycle, then charge sent by it in the $\frac{T}{2}$ is given by,

$q = I_{m} \frac{T}{2}$ ............................... $(2)$

Since, $(1)$ and $(2)$ then, $\frac{I_{\circ} T}{π} = I_{m} \frac{T}{2}$

$\Rightarrow I_{m} = \frac{2 I_{\circ}}{π} = 0.631 I_{\circ}$

Hence, the expression for mean value of alternating current for half cycle is $0.631 I_{\circ}$ .

Derive an expression for average or mean value of alternating current for half cycle.

Important Questions on Alternating Current

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