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.

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

nth harmonic : (ancosx+bnsinx)

where,

$\120dpi \large a_{n}=\frac{2}{p}\sum_{i=1}^{p}y_{i}cos(nx_{i})$

&

$\120dpi \large b_{n}=\frac{2}{p}\sum_{i=1}^{p}y_{i}sin(nx_{i})$

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.

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:

$\120dpi \large a_{1}=\frac{2}{p}\sum_{i=1}^{p}y_{i}cos(x_{i})$

$\120dpi \large b_{1}=\frac{2}{p}\sum_{i=1}^{p}y_{i}sin(x_{i})$

&

$\120dpi \large a_{2}=\frac{2}{p}\sum_{i=1}^{p}y_{i}cos(2x_{i})$

$\120dpi \large b_{2}=\frac{2}{p}\sum_{i=1}^{p}y_{i}sin(2x_{i})$

where xi=0, 60, 120… & so on.

Share.