Question

Find the root mean square (rms) voltage for the given variation of voltage.

• A. $V_0\sqrt{\frac{5}{24}}$
• B. $V_0\sqrt{\frac{1}{2}}$
• C. $V_0\sqrt{\frac{7}{24}}$
• D. $V_0$

Answer 1

The correct option is C $V_0\sqrt{\frac{7}{24}}$

The rms voltage is found by parameterizing the voltage.
$V=\{{}_{V_0/21<t<2sec}^{V_{0}t0<t<1sec}$
$V_{rms}={\left\{\frac{1}{2}\left[{\int }_0^2{V}^2\right(t\left)dt\right]\right\}}^{1/2}$
$={\left\{\frac{1}{2}[V_0^2{\int }_0^1{t}^{2}dt+\frac{V_0^2}{4}{\int }_1^{2}dt]\right\}}^{1/2}$
$={\left\{\frac{1}{2}[V_0^2{\left(\frac{t^3}{3}\right)}_0^1+\frac{V_0^2}{4}(t{)}_1^2]\right\}}^{1/2}$
$={\left\{\frac{1}{2}[\left(\frac{V_0^2}{3}\right)+\frac{V_0^2}{4}(1\left)\right]\right\}}^{1/2}$
$=V_0{\{\frac{1}{6}+\frac{1}{8}\}}^{1/2}$
$=V_0\sqrt{\frac{7}{24}}$