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Chapter 6

Fourier, Taylor, and Maclaurin Series

T his chapter is an introduction to Fourier and power series. We begin with the definition of

sinusoids that are harmonically related and the procedure for determining the coefficients of
the trigonometric form of the series. Then, we discuss the different types of symmetry and
how they can be used to predict the terms that may be present. Several examples are presented to
illustrate the approach. The alternate trigonometric and the exponential forms are also presented.
We conclude with a discussion on power series expansion with the Taylor and Maclaurin series.

6.1 Wave Analysis

The French mathematician Fourier found that any periodic waveform, that is, a waveform that
repeats itself after some time, can be expressed as a series of harmonically related sinusoids, i.e.,
sinusoids whose frequencies are multiples of a fundamental frequency (or first harmonic). For
,
,
example, a series of sinusoids with frequencies 1 MHz 2 MHz 3 MHz , and so on, contains the
fundamental frequency of 1 MHz , a second harmonic of 2 MHz , a third harmonic of 3 MHz ,
and so on. In general, any periodic waveform ft() can be expressed as

1

ft() = --a + a cos ωt + a cos 2ωt + a cos 3ωt + a cos 4ωt + …
-
2 0 1 2 3 4 (6.1)
+ b sin ωt + b sin 2ωt + b sin 3ωt + b sin 4ωt + …
3
2
1
4
or
1 ∞
-
ft() = --a + ∑ ( a cos nωt + b sin nωt ) (6.2)
2 0 n n
n = 1
where the first term a ⁄ 2 is a constant, and represents the DC (average) component of f t() .
0
Thus, if ft() represents some voltage vt() , or current it() , the term a ⁄ 2 is the average value of
0
vt() or it() .
The terms with the coefficients and b 1 together, represent the fundamental frequency compo-
a
1
nent ω * . Likewise, the terms with the coefficients a 2 and b 2 together, represent the second har-
monic component 2ω , and so on.

θ
* We recall that k cos ωt + k sin ωt = kcos ( ωt + θ ) where is a constant.
1
2
Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−1
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