1. Chapter 17 The Fourier Series
EEEB123 Circuit Analysis 2
Chapter 17
The Fourier Series
Materials from Fundamentals of Electric Circuits (4th Edition),
Alexander & Sadiku, McGraw-Hill Companies, Inc.

2. The Fourier Series - Chapter 17
17.1 Introduction
17.2 Trigonometric Fourier Series
17.3 Symmetry Considerations
17.4 Circuit Applications

3. 17.1 Introduction(1)
Previously, have considered analysis of circuits with sinusoidal sources.
The Fourier series provides a means of analyzing circuits with periodic non-sinusoidal excitations.
Fourier is a technique for expressing any practical periodic function as a sum of sinusoids.
Fourier representation + superposition theorem, allows to find response of circuits to arbitrary periodic inputs using phasor techniques.

4. 17.2 Trigonometric Fourier Series (1)
The Fourier series of a periodic function f(t) is a representation that resolves f(t) into a dc component and an ac component comprising an infinite series of harmonic sinusoids.
Given a periodic function f(t)=f(t+nT) where n is an integer and T is the period of the function.
where w0=2p/T is called the fundamental frequency in radians per second.

5. 17.2 Trigonometric Fourier Series (2)
and Fourier coefficients, an and bn , are:
in alternative form of f(t)
where

6. 17.1 Trigometric Fourier Series (3)
Conditions (Dirichlet conditions) on f(t) to yield a convergent Fourier series:
f(t) is single-valued everywhere.
f(t) has a finite number of finite discontinuities in any one period.
f(t) has a finite number of maxima and minima in any one period.
The integral

7. 17.2 Trigometric Fourier Series (4)
Example 17.1
Determine the Fourier series of the waveform shown below. Obtain the amplitude and phase spectra
*Refer to textbook, pg. 760

8. 17.2 Trigonometric Fourier Series (5)
Solution:
a) Amplitude and
b) Phase spectrum
Truncating the series at N=11

9. 17.3 Symmetry Considerations (1)
Three types of symmetry
1. Even Symmetry : a function f(t) if its plot is symmetrical about the vertical axis.
In this case,
Typical examples of even periodic function

10. 17.3 Symmetry Considerations (2)
2. Odd Symmetry : a function f(t) if its plot is anti-symmetrical about the vertical axis.
In this case,
Typical examples of odd periodic function

11. 17.3 Symmetry Considerations (3)
3. Half-wave Symmetry : a function f(t) if
Typical examples of half-wave odd periodic functions

12. 17.3 Symmetry Considerations (4)
Example 17.3
Find the Fourier series expansion of f(t) given below.
Ans:
*Refer to textbook, pg. 771

13. 17.3 Symmetry Considerations (5)
Example 17.4
Determine the Fourier series for the half-wave cosine function as shown below.
Ans:
*Refer to textbook, pg. 772

14. 17.4 Circuit Applications (1)
Steps for Applying Fourier Series
Express the excitation as a Fourier series.
Example, for periodic voltage source:
Transform the circuit from the time domain to the frequency domain.
Find the response of the dc and ac components in the Fourier series.
Add the individual dc and ac response using the superposition principle.

15. 17.4 Circuit Applications (2)
Example 17.6
Find the response v0(t) of the circuit below when the voltage source vs(t) is given by
*Refer to textbook, pg. 775

16. 17.4 Circuit Applications (3)
Solution
Phasor of the circuit
For dc component, (wn=0 or n=0), Vs = ½ => Vo = 0
For nth harmonic,
In time domain,
Amplitude spectrum of the output voltage

17. Ch 17 Useful Formula
Given: