Reference Designer Calculators

Vmax to Vrms formula derivation

The average of the square of a sine wave of Amplitude A in a time period 0 to 2π is give by

$$v_{rms}^{2} = \frac{A^{2} \int_{0}^{2\pi }sin^{^{2}}x dx }{2\pi}$$
simplification leads to
$$v_{rms}^{2} = \frac{A^{2}}{2\pi} \int_{0}^{2\pi}\frac{1-cos2x}{2}dx$$
Integrating the cosine term gives
$$v_{rms}^{2} =\frac{ A^{2}}{2\pi}\left [ \frac{1}{2}x-\frac{1}{4}sin2x \right ]^ {2\pi}_0$$
The sine term becomes 0 so we get
$$v_{rms}^{2} =\frac{A^{2}}{2\pi}\left ( \frac{2\pi}{2} - 0 \right )$$
$$v_{rms}^{2} =\frac{ A^{2}}{2}$$
$$v_{rms}^{2} =\frac{ A}{\sqrt[]2}$$