Question

The r.m.s. value of current for a variable current $i=i_1$ cos $\omega t+i_2$:

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

Answer 1

The correct option is C $\frac{1}{\sqrt{2}}(i_1^2+i_2^2{)}^{1/2}$

current in the circuit

$i=i_1\cos \omega t+i_2$

the rms current $\left(i\right)$ is given as:

$i^2=\frac{\int i^{2}dt}{\int dt}=\frac{{\int }_0^T(i_1\cos \omega t+i_2{)}^{2}dt}{{\int }_0^{T}dt}$ $\left[{\int }_0^{T}dt=T,timeperiod\right]$

$=\frac{{\int }_0^T(i_2^2+2i_1{i}_2\cos \omega t+i_1^2{\cos }^2\omega t)dt}{T}$

in a full cycle ${\int }_0^T\cos \omega t=0$ & ${\int }_0^T{\cos }^2\omega t=\frac{1}{2}$

so, $\overline{i_2}=(i_2^2+\frac{i_1^2}{2})$

$\Rightarrow \overline{i}={(i_2^2+\frac{i_1^2}{2})}^{1/2}$