logistic_guy
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Show that the \(\displaystyle \text{rms}\) output of an ac generator is \(\displaystyle V_{\text{rms}} = NAB\omega/\sqrt{2}\).
ac generator
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logistic_guy
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The induced emf is given by:
\(\displaystyle \mathscr{E} = NBA\omega\sin\omega t\)
The relation between \(\displaystyle V_{\text{rms}}\) and \(\displaystyle \mathscr{E}\) is:
\(\displaystyle V^2_{\text{rms}} = \mathscr{E}^2_{\text{average}} = N^2B^2A^2\omega^2\sin^2_{\text{average}}\omega t\)
\(\displaystyle 0 \leq \sin^2\omega t \leq 1\)
Then, it has an average value of \(\displaystyle \frac{1}{2}\). This gives:
\(\displaystyle V^2_{\text{rms}} = \frac{1}{2}N^2B^2A^2\omega^2\)
Then,
\(\displaystyle V_{\text{rms}} = \sqrt{\frac{N^2B^2A^2\omega^2}{2}} = \textcolor{blue}{\frac{NBA\omega}{\sqrt{2}}}\)
\(\displaystyle \mathscr{E} = NBA\omega\sin\omega t\)
The relation between \(\displaystyle V_{\text{rms}}\) and \(\displaystyle \mathscr{E}\) is:
\(\displaystyle V^2_{\text{rms}} = \mathscr{E}^2_{\text{average}} = N^2B^2A^2\omega^2\sin^2_{\text{average}}\omega t\)
\(\displaystyle 0 \leq \sin^2\omega t \leq 1\)
Then, it has an average value of \(\displaystyle \frac{1}{2}\). This gives:
\(\displaystyle V^2_{\text{rms}} = \frac{1}{2}N^2B^2A^2\omega^2\)
Then,
\(\displaystyle V_{\text{rms}} = \sqrt{\frac{N^2B^2A^2\omega^2}{2}} = \textcolor{blue}{\frac{NBA\omega}{\sqrt{2}}}\)