Question

Variation of voltage with time is shown in the above figure.
The rms voltage is found to be $N\sqrt{\frac{2}{3}}V$. Find $N$.

Answer 1

$V_{rms}^2=\frac{1}{4}{\int }_0^4{V}^{2}dt=\frac{1}{4}[{\int }_0^1{4}^{2}dt+{\int }_1^2(-4(t-1){)}^{2}dt+{\int }_2^3{4}^{2}dt+{\int }_3^4(-4(t-3){)}^{2}dt]=\frac{1}{4}[{\int }_0^{1}16dt+{\int }_1^{2}16(t^2-2t+1)dt+{\int }_2^{3}16dt+{\int }_3^{4}16(t^2-6t+9)dt]=\frac{1}{4}[16+16(\frac{8-1}{3}-2\times \frac{4-1}{2}+1)+16+16(\frac{64-27}{3}-6\times \frac{16-9}{2}+9\left)\right]=\frac{32}{3}{V}_{rms}=\frac{4\sqrt{6}}{3}=\frac{4\sqrt{2}}{\sqrt{3}}V$
N=4