Question

If a direct current of value $A$ ampere is superimposed on an alternating current $I=b\sin \omega t$ flowing through a wire, what is the effective (rms) value of the resulting current in the circuit?

Answer 1

Total current,

$i=a+b\sin \left(\omega t\right)$

By definition of rms value,

$i_{rms}=\sqrt{\frac{{\int }_0^T{i}^{2}dt}{T}}$

$i_{rms}=\sqrt{\frac{{\int }_0^T[a^2+b^2{\sin }^2(\omega t)+2ab\sin (\omega t\left)\right]dt}{T}}$

$i_{rms}=\sqrt{\frac{{\int }_0^T[a^2+b^2/2(1-\sin \left(2\omega t\right))+2ab\sin (\omega t\left)\right]dt}{T}}$

Average value of $\sin \left(\omega t\right)$ and $\sin \left(2\omega t\right)$ is $0$ over a time period.

$⟹i_{rms}=\sqrt{a^2+\frac{b^2}{2}}A$