Question

An alternating current is given by \(i=i_{1}\sin \omega t-i_{2}\cos \omega t\). The rms current is given by

• A. \(\frac{i_{1}+i_{2}}{\sqrt{2}}\)
• B. \(\frac{\sqrt{i_{1}^{2}+i_{2}^{2}}}{2}\)
• C. \(\sqrt{\frac{i_{1}^{2}+i_{2}^{2}}{2}}\)
• D. \(\sqrt{i_{1}^{2}+i_{2}^{2}}\)

Hint:

Combine the sine and cosine into a single sinusoid and divide its amplitude by the square root of 2.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The rms value is the square root of the mean of \(i^{2}\) over a cycle.

Step 2: Mean of the square
\(i^{2}=i_1^{2}\sin^{2}\omega t+i_2^{2}\cos^{2}\omega t-2i_1i_2\sin\omega t\cos\omega t\).
Over a full cycle, \(\langle\sin^{2}\rangle=\langle\cos^{2}\rangle=\dfrac{1}{2}\) and \(\langle\sin\omega t\cos\omega t\rangle=0\).

Step 3: Result
\(\langle i^{2}\rangle=\dfrac{i_1^{2}+i_2^{2}}{2}\), so \(i_{rms}=\sqrt{\dfrac{i_1^{2}+i_2^{2}}{2}}\).

Final Answer:
The rms current is \(\sqrt{\dfrac{i_1^{2}+i_2^{2}}{2}}\). \[ \boxed{\text{(C) }\sqrt{\dfrac{i_{1}^{2}+i_{2}^{2}}{2}}} \]