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.
f(t) | = | a02+∞∑k=1[akcos(kωt)+bksin(kωt)] |
The RMS value Frms of f(t) is
Frms | = | a024+12∞∑k=1ak2+bk2 |
Proof:
The RMS value of a function f(t) is, by definition, given by
Frms | = | 1T∫t0+Tt0[f(t)]2𝑑t |
Then,
Frms | =1T∫t0+Tt0[f(t)]2𝑑t | ||
=1T∫t0+Tt0f(t)⋅f(t)𝑑t | |||
=1T∫t0+Tt0f(t)⋅(a02+∞∑k=1[akcos(kωt)+bksin(kωt)])𝑑t | |||
=1T∫t0+Tt0(a0f(t)2+∞∑k=1[akf(t)cos(kωt)+bkf(t)sin(kωt)])𝑑t | |||
=1T(∫t0+Tt0a0f(t)2𝑑t+∞∑k=1[∫t0+Tt0akf(t)cos(kωt)𝑑t+∫t0+Tt0bkf(t)sin(kωt)𝑑t]) | |||
=1T(a02∫t0+Tt0f(t)𝑑t+∞∑k=1[ak∫t0+Tt0f(t)cos(kωt)𝑑t+bk∫t0+Tt0f(t)sin(kωt)𝑑t]) | |||
=1T(a02(a0T2)+∞∑k=1[ak(akT2)+bk(bkT2)]) | |||
=a024+∞∑k=1ak22+bk22 | |||
=a024+12∞∑k=1ak2+bk2 |
Title | RMS Value of the Fourier Series |
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Canonical name | RMSValueOfTheFourierSeries1 |
Date of creation | 2013-03-11 19:30:56 |
Last modified on | 2013-03-11 19:30:56 |
Owner | swapnizzle (13346) |
Last modified by | (0) |
Numerical id | 1 |
Author | swapnizzle (0) |
Entry type | Definition |