For a sinusoidal voltage:
is called the crest factor, which varies for different waveforms.
is therefore 
, which is about 325 V. The peak-to-peak value 
of the 230 V AC is double that, at about 650 V.
![\begin{align}
V_\mathrm{rms} &=\sqrt{\frac{1}{T} \int_0^{T}[{V_{pk}\sin( \omega t+\phi)]^2 dt}}\\
&=V_{pk}\sqrt{\frac{1}{2T} \int_0^{T}[{1-\cos(2\omega t+2\phi)] dt}}\\
&=V_{pk}\sqrt{\frac{1}{2T} \int_0^{T}{ dt}}\\
&=\frac{V_{pk}}{\sqrt {2}}
\end{align}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sxvKcnq6Bk6rtbbEmU5SI6Ur_ascxrNJjC07IHKKLbPoB-xs00zEPorkgvtQjOb-q7F_3M9oQRqn7XJvVm7BRTQD9SkVxm7L89qXhqVYMUamlS1Vhh5evnmN8yBQ3f8yZzMim8GbO_PLT-98t3lA=s0-d)
is called the crest factor, which varies for different waveforms.- For a triangle waveform centered about zero
- For a square waveform centered about zero
- For an arbitrary periodic waveform
of period
:
![V_\mathrm{rms}=\sqrt{\frac{1}{T} \int_0^{T}{[v(t)]^2 dt}}.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vcKODke1kdPEGttAWxnrcmgBqEaGWxRLKaNV1qVOHOeJnmHLqO3ViNjnqAPMzuR18LqqM87003y9OaLZoOFWEFD1vJQuW0L2tILzfL0lFgFTomjEDhdSCsAgCnOi6XIr8ZH6DMfQz7siIGj_3zfg=s0-d)
Example
To illustrate these concepts, consider a 230 V AC mains supply used in many countries around the world. It is so called because its root mean square value is 230 V. This means that the time-averaged power delivered is equivalent to the power delivered by a DC voltage of 230 V. To determine the peak voltage (amplitude), we can rearrange the above equation to:
is therefore , which is about 325 V. The peak-to-peak value of the 230 V AC is double that, at about 650 V.
No comments:
Post a Comment