# RMS Value of the Fourier Series

RMS Value of the Fourier Series Swapnil Sunil Jain December 28, 2006

RMS Value of the Fourier Series

If a function $f(t)$ is given by its Fourier series i.e.

 $\displaystyle f(t)$ $\displaystyle=$ $\displaystyle\frac{a_{0}}{2}+\sum_{k=1}^{\infty}[a_{k}\cos(k\omega t)+b_{k}% \sin(k\omega t)]$

The RMS value $F_{\mathrm{rms}}$ of $f(t)$ is

 $\displaystyle F_{\mathrm{rms}}$ $\displaystyle=$ $\displaystyle\frac{{a_{0}}^{2}}{4}+\frac{1}{2}\sum_{k=1}^{\infty}{a_{k}}^{2}+{% b_{k}}^{2}$

Proof:

The RMS value of a function $f(t)$ is, by definition, given by

 $\displaystyle F_{\mathrm{rms}}$ $\displaystyle=$ $\displaystyle\frac{1}{T}\int_{t_{0}}^{t_{0}+T}[f(t)]^{2}dt$

Then,

 $\displaystyle F_{\mathrm{rms}}$ $\displaystyle=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}[f(t)]^{2}dt$ $\displaystyle=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}f(t)\cdot f(t)dt$ $\displaystyle=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}f(t)\cdot\Big{(}\frac{a_{0}}{2}% +\sum_{k=1}^{\infty}\Big{[}a_{k}\cos(k\omega t)+b_{k}\sin(k\omega t)\Big{]}% \Big{)}dt$ $\displaystyle=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}\Big{(}\frac{a_{0}f(t)}{2}+\sum% _{k=1}^{\infty}\Big{[}a_{k}f(t)\cos(k\omega t)+b_{k}f(t)\sin(k\omega t)\Big{]}% \Big{)}dt$ $\displaystyle=\frac{1}{T}\Big{(}\int_{t_{0}}^{t_{0}+T}\frac{a_{0}f(t)}{2}dt+% \sum_{k=1}^{\infty}\Big{[}\int_{t_{0}}^{t_{0}+T}a_{k}f(t)\cos(k\omega t)dt+% \int_{t_{0}}^{t_{0}+T}b_{k}f(t)\sin(k\omega t)dt\Big{]}\Big{)}$ $\displaystyle=\frac{1}{T}\Big{(}\frac{a_{0}}{2}\int_{t_{0}}^{t_{0}+T}f(t)dt+% \sum_{k=1}^{\infty}\Big{[}a_{k}\int_{t_{0}}^{t_{0}+T}f(t)\cos(k\omega t)dt+b_{% k}\int_{t_{0}}^{t_{0}+T}f(t)\sin(k\omega t)dt\Big{]}\Big{)}$ $\displaystyle=\frac{1}{T}\Big{(}\frac{a_{0}}{2}(a_{0}\frac{T}{2})+\sum_{k=1}^{% \infty}\Big{[}a_{k}(a_{k}\frac{T}{2})+b_{k}(b_{k}\frac{T}{2})\Big{]}\Big{)}$ $\displaystyle=\frac{{a_{0}}^{2}}{4}+\sum_{k=1}^{\infty}\frac{{a_{k}}^{2}}{2}+% \frac{{b_{k}}^{2}}{2}$ $\displaystyle=\frac{{a_{0}}^{2}}{4}+\frac{1}{2}\sum_{k=1}^{\infty}{a_{k}}^{2}+% {b_{k}}^{2}$
Title RMS Value of the Fourier Series RMSValueOfTheFourierSeries1 2013-03-11 19:30:56 2013-03-11 19:30:56 swapnizzle (13346) (0) 1 swapnizzle (0) Definition