An alternating current is given by the equation

Question:

An alternating current is given by the equation $\mathrm{i}=\mathrm{i}_{1} \sin \omega \mathrm{t}+\mathrm{i}_{2} \operatorname{coscot}$.

The rms current will be :

  1. (1) $\frac{1}{2}\left(\mathrm{i}_{1}^{2}+\mathrm{i}_{2}^{2}\right)^{\frac{1}{2}}$

  2. (2) $\frac{1}{\sqrt{2}}\left(\mathrm{i}_{1}^{2}+\mathrm{i}_{2}^{2}\right)^{\frac{1}{2}}$

  3. (3) $\frac{1}{\sqrt{2}}\left(\mathrm{i}_{1}+\mathrm{i}_{2}\right)^{2}$

  4. (4) $\frac{1}{\sqrt{2}}\left(\mathrm{i}_{1}+\mathrm{i}_{2}\right)$


Correct Option: , 2

Solution:

(2)

$I_{0}=\sqrt{I_{1}^{2}+I_{2}^{2}+2 I_{1} I_{2} \cos \theta}$

$I_{0}=\sqrt{I_{1}^{2}+I_{2}^{2}+2 I_{1} I_{2} \cos 90^{\circ}}$

$I_{0}=\sqrt{I_{1}^{2}+I_{2}^{2}+2 I_{1} I_{2}(0)} \Rightarrow \sqrt{I_{1}^{2}+I_{2}^{2}}$

We, know that

So,

$I_{r m s}=\frac{I_{0}}{\sqrt{2}}$

$I_{r m s}=\frac{\sqrt{I_{1}^{2}+I_{2}^{2}}}{\sqrt{2}}$

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