Harmonic Formulas for Filtering Applications

Trigonometric Series and Harmonic Analysis

The history of the trigonometric series, for all practical purposes, came of age in 1822 with Joseph De Fourier’s book that developed the concept. The basic idea of trigonometric series is to represent periodic functions by series of particular (trigonometric) period functions.

Series of functions in which the general term is , with constant coefficients of a_n and b_n, are called trigonometric series.

Trigonometric series will converge in an interval of length, 2, since this is true the functions are periodic and will converge for all x and represents a periodic function . This function is not necessary continuous however. We only consider series that converges uniformly, and then its sum is continuous. This is what we have for harmonic analysis in electrical systems.

In our case a connection can be established between coefficients a_n and b_n and the sum function of . The multiplication of the series looks something like this

Factors bounded by cos(px) or sin(px), where p is a positive integer allows us to calculate

and

Integration of the series or for the integrals over the interval .

When p these integrals have a value of 0. For p=n they are

for n>0

for n=0

Because n=0 behaves like it does it is now conventional to write the trigonometric series as

The Euler-Fourier formulae for n0 looks like this

Harmonic Analysis and Formulas

Let look back at basic circuits for a moment. We learned from Ohms law V = IR and P=VI and P=I²R. So the power delivered by an a-c circuit at any given time is

Now we need the trigonometric identity we learned years ago

But the average power delivered by a a-c source is the first term only, since the average value of a cosine wave is zero. It may have twice the frequency of the original input current waveform. Equating the average power of an a-c generator to that delivered by a d-c source:

P_av(a-c) = P_d-c

So the effective values give us

The effective value of any quantity plotted as a function of time can be found by using the following equation:

This finally takes us to where we always wanted to go. By applying this equation to the following Fourier series we get

Then form (5) and after performing the indicated operations we have the following:

but, since

Similarly, for

 Looking what is called in mathematics as odd functions, this means x, x³, x⁵, x⁷ and the like for which we have

Understand that even functions have nothing to do with even harmonics, nor odd functions with odd harmonics. An odd function is simply a function with odd powers of x and an odd harmonic has an odd multiple of the fundamental frequency.

This being state it is obvious that the sine wave is odd and the cosine wave is even. Therefore, the sum of sine waves will be odd, and a Fourier series containing only sine components represents an odd function.

Just an aside when even and odd functions are added the sum will be neither odd nor even. We will not get into that in this paper.

Plotting three sine waves, 60 hertz as fundamental, and the 5^th and 7^th harmonic of 60 hertz. Using 360 degrees for the fundamental, we get 1800 degrees for the 5^th and 2520 degrees for the 7^th. The individual sine waves look like this.

We allowed the amplitude to 20 for the fundamental, 10 for the 5^th harmonic and 5 for the 7^th harmonic. Adding these simultaneously we get the following distorted wave.

Assuming these are currents from (8), we get

= 22.913 amps

This is known as the Total Harmonic Distortion Current or THDI.