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Question

Variation of voltage with time is shown in figure.

The rms voltage is found to be ,$N\sqrt{\frac{2}{3}}V$. Find N.

Moderate

Solution

In such cases it is difficult to develop a single equation.

Hence, it is usual to consider two equations, one applicable from 0 to 1 ms and another from 1 to 2 ms.

For t lying between 0 and 1 ms, V_{1} = 4. For t lying between 1 and 2 ms, V_{2 }= -4t+4.

(a) ${V}_{rms}=\sqrt{\frac{1}{2}\left({\int}_{0}^{1}{V}_{1}^{2}dt+{\int}_{1}^{2}{V}_{2}^{2}dt\right)}$

${V}_{rms}^{2}=\frac{1}{2}\left[{\int}_{0}^{1}{4}^{2}dt+{\int}_{1}^{2}{(-4t+4)}^{2}dt\right]$

$=\frac{1}{2}\left[{\left|16t\right|}_{0}^{1}+{\left|\frac{{\displaystyle 16{t}^{3}}}{{\displaystyle 3}}\right|}_{1}^{2}+{\left|16t\right|}_{1}^{2}-{\left|\frac{{\displaystyle 32{t}^{2}}}{{\displaystyle 2}}\right|}_{1}^{2}\right]=\frac{32}{3}$

${V}_{rms}=\sqrt{\frac{32}{3}}=4\sqrt{\frac{2}{3}}V.HenceN=4$

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