If a direct current of ‘a’ units is superimposed with an alternating current I=b sin $ω t,$ then the effective value of resulting current is

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$I_{r m s} = {(a^{2} + b^{2})}^{\frac{1}{2}}$

$I_{r m s} = {(\frac{a^{2}}{2} + b^{2})}^{\frac{1}{2}}$

$I_{r m s} = {(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2}}$

$I_{r m s} = {(a^{2} + \frac{b^{2}}{2})}^{\frac{1}{2}}$

answer is D.

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Detailed Solution

$\begin{array}{l} i = i_{1} + i_{2} = a + b \sin w t \\ T {I_{r m s}}^{2} = \int_{0}^{T} {(a + b \sin w t)}^{2} d t = \int_{0}^{T} (a^{2} + b^{2} \sin^{2} w t + 2 a b \sin w t) d t \end{array}$

$\begin{array}{l} T I_{r m g^{2}} = \int_{0}^{T} a^{2} d t + \int_{0}^{T} b^{2} \sin^{2} w t + \int_{0}^{T} 2 a b \sin w t d t \\ T {I_{r m s}}^{2} = a^{2} T + b^{2} \int_{0}^{T} \frac{1 - \cos 2 w t}{2} d t + \int_{0}^{T} 2 a b \sin w t d t \\ T {I_{r m s}}^{2} = a^{2} T + b^{2} [\frac{T}{2} - \frac{1}{2} {[\frac{\sin w t}{w}]}_{0}^{T} + 2 a b {[\frac{- \cos w t}{w}]}_{0}^{T}] \\ {T I_{r m s}}^{2} = a^{2} T + b^{2} (\frac{T}{2} + 0) + 0 \\ {I_{r m s}}^{2} = a^{2} + \frac{b^{2}}{2} \\ I_{r m s} = {[a^{2} + \frac{b^{2}}{2}]}^{1 / 2} \end{array}$