Question

An alternating current is given by $ I={{I}_{1}}\cos \omega t+{{I}_{2}}\sin \omega t $ . The root mean square current is given by:

• A. $ \frac{({{I}_{1}}+{{I}_{2}})}{\sqrt{2}} $
• B. $ \frac{{{({{I}_{1}}+{{I}_{2}})}^{2}}}{2} $
• C. $ \sqrt{\frac{I_{1}^{2}+I_{2}^{2}}{2}} $
• D. $ \frac{\sqrt{I_{1}^{2}+I_{2}^{2}}}{2} $

Answer 1

Answer: (C)

The equation of A.C. is
$I=I_{1} \cos \omega t+I_{2} \sin \omega t$
The resultant current is given by
$I_{0}=\sqrt{I_{1}^{2}+I_{2}^{2}}$
Hence, the rms current from relation is
$I_{ rms } =\frac{I_{0}}{\sqrt{2}}$
$=\frac{\sqrt{I_{1}^{2}+I_{2}^{2}}}{\sqrt{2}} \,\,$[ from eq. (1)]
$=\sqrt{\frac{I_{1}^{2}+I_{2}^{2}}{2}}$