First slide

Alternating current and voltage

Question

A direct current of 5 A is superimposed on an alternating current $I = 10 \sin (ωt)$ flowing through a wire. The effective value of the resulting current will be

Moderate

A

(15/2) A
B

$5 \sqrt{3} A$
C

$5 \sqrt{5} A$
D

15 A

Solution

Given: $I = 5 + 10 sinωt$
$I_{eff} = {[\begin{array}{l} \int_{0}^{T} I^{2} dt \\ \int_{0}^{T} dt \end{array}]}^{1 / 2}$
$= {[\frac{1}{T} \int_{0}^{T} (5 + 10 sinωt)^{2} dt]}^{1 / 2}$
$= [\frac{1}{T} \int_{0}^{T} {(25 + 100 sinωt + 100 \sin^{2} ωt]}^{1 / 2}$
$But as \frac{1}{T} \int_{0}^{T} \sin ωt dt = 0$
$and \frac{1}{T} \int_{0}^{T} \sin^{2} ωt dt = \frac{1}{2}$
$so I_{eff} = {[25 + \frac{1}{2} \times 100]}^{1 / 2}$
$= 5 \sqrt{3} A$

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