Harmonic Analysis

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Harmonics formed on waves are generated as component frequencies of a fundamental frequency of the wave.

The fundamental & the higher frequencies(harmonics) generate periodic signals from the original wave. And every periodic signal can be written as a sum of the various harmonics using the Fourier series. Read this article to learn more about the harmonic analysis.

harmonic analysis 1

Hence, to find the various harmonics using the fourier series, we can use…

nth harmonic : (ancosx+bnsinx)

where,

harmonic analysis 2

&

harmonic analysis 3

where p is the number of unique values of the function y. The following example will make things a bit more clear.

Example : y is a function of x periodic with period 2pi. Some experimental values of y are given below calculated for certain values of x. Expand y to 2 harmonics.

harmonic analysis 4

Solution :

Clearly, in the above, p=6,

& We simply need to find:

1st harmonic + 2nd harmonic = (a1cosx+b1sinx) + (a2cos2x+b2sin2x)

So, all we need is a1, b1, a2 &
b
2

for which we use the formula mentioned above:

harmonic analysis 5

harmonic analysis 6

&

harmonic analysis 7

harmonic analysis 8

where xi=0, 60, 120… & so on.

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Founder at Durofy. Blogs on technology & startups, runs a full service digital agency, teaches AP Calculus.

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