Question

An alternating current is given by the equation $i=i_1\sin \omega t+i_2\cos \omega t$. The rms current will be:

• A. $\frac{1}{2}{({i_1}^2+{i_2}^2)}^{1/2}$
• B. $\frac{1}{\sqrt{2}}{({i_1}^2+{i_2}^2)}^{1/2}$
• C. $\frac{1}{\sqrt{2}}{(i_1+i_2)}^2$
• D. $\frac{1}{\sqrt{2}}(i_1+i_2)$

Answer 1

The correct option is B

$\frac{1}{\sqrt{2}}{({i_1}^2+{i_2}^2)}^{1/2}$

Step 1. Given data

Current equation, $i=i_1\sin \omega t+i_2\cos \omega t$

Step 2. Finding the RMS value of the current, $I_{rms}$

By using the formula,

$$\begin{gathered} I_0=\sqrt{{i_1}^2+{i_2}^2+2i_1{i}_{\mathbf{2}}\cos \left(\theta \right)} \\ =\sqrt{{i_1}^2+{i_2}^2+2i_1{i}_{\mathbf{2}}\cos \left(90°\right)} \\ =\sqrt{{i_1}^2+{i_2}^2+2i_1{i}_{\mathbf{2}}\left(0\right)} \\ =\sqrt{{i_1}^2+{i_2}^2} \end{gathered}$$ [$I_0$ is maximum value of current]

We know that,

$I_{rms}=\frac{I_0}{\sqrt{2}}$

Thus, $I_{rms}=\frac{\sqrt{{i_1}^2+{i_2}^2}}{\sqrt{2}}$

Hence, option $\left(B\right)$ is correct.