Question

In a circuit an alternating current and a direct current are supplied together. The expression of the instantaneous current is given as i = 3+ 6 sin $\omega$t. Then calculate the r.m.s. value of the current.

Answer 1

Rms value of current $I_{avg}^2=\frac{{\int }_0^T{i}^{2}dt}{{\int }_0^{T}dt}$

Or $I_{avg}^2=\frac{{\int }_0^T(3+6\sin wt{)}^{2}dt}{T}$

Or $I_{avg}^2=\frac{{\int }_0^T(9+36{\sin }^{2}wt+36sinwt)dt}{T}$

Or $I_{avg}^2=\frac{1}{T}({\int }^{2}9dt+36{\int }_0^T{\sin }^{2}wtdt+36{\int }_0^T\sin wt)$

$⟹I_{avg}^2=\frac{1}{T}(9T+36\times 2T+36\times 0)$

$⟹I_{avg}^2=\frac{81}{2}$

$⟹I_{rms}=\sqrt{I_{avg}^2}=\frac{9}{\sqrt{2}}A$