Download Study Guide

1
Chapter 16 – Fourier Series
Study Guide
Objectives:
1. Be able to calculate the trigonometric form of the Fourier coefficients for a periodic
waveform using the definition of the coefficients and the simplifications possible if the
waveform exhibits one or more types of symmetry.
2. Know how to analyze a circuit’s response to a periodic waveform using Fourier
coefficients and superposition.
3. Be able to estimate the average power delivered to a resistor using a small number of
Fourier coefficients.
4. Be able to calculate the exponential form of the Fourier coefficients for a periodic
waveform and use them to generate magnitude and phase-spectrum plots for that
waveform.
Mastering the Objectives:
1. Read the Introduction, Section 16.1, and Section 16.2.
a) Define the following terms:
b) i. Periodic function; ii. Fundamental frequency; iii. Harmonic frequency.What is
another name for the Fourier coefficien av? Find av for the waveforms in Figs.
16.3 and 16.4 without integrating.
c) If you have a plotting program, it is instructive to use it to create a periodic
function by adding successive terms in its Fourier series. For example, consider
the periodic function in Fig. 16.5 and its Fourier series in Example 16.1. Let Vm
= 1 V and 0 = 1 rad/s. Begin by plotting the first term in the Fourier series, then
the sum of the first two terms, then the sum of the first three terms, and so on.
How many terms of the Fourier series must be used to create a good replica of the
original periodic function?
d) Solve Chapter Problem 16.1.
2. Read Section 16.3.
a) Make a table with
 The four types of symmetry;
 The definition of each type of symmetry;
 The simplification of the Fourier coefficients that can be made if that
symmetry is present.
b) Using the table you created in (a), identify the symmetry exhibited by the
following waveforms, if any:
i.
P16.1;
ii. P16.3;
iii. P16.12;
iv.
P16.36;
v.
P16.37.
c) Solve Assessment Problem 16.3.
3. Read Sections 16.4 and 16.5.
a) How is the alternative trigonometric form of the Fourier coefficients related to the
original trigonometric form of the Fourier coefficients?
b) How do the four types of symmetry affect the computation of the alternative form
of the Fourier coefficients?
c) What advantage does the alternative trigonometric form of the Fourier
coefficients have over the original form?
2
d) What basic circuit-analysis tool is used in analyzing a circuit whose source is a
periodic function represented as a Fourier series?
e) What type of filter does the RC circuit in Fig. 16.12(a) represent? Can you see the
effects of the filter on the harmonic components of the output of the circuit (look
at Eq. 16.56)?
f) Solve Assessment Problems 16.5 and 16.6.
4. Read Sections 16.6 and 16.7.
a) How can you approximate the average power delivered to a resistor given the
Fourier series representation of the voltage drop across the resistor?
b) How can you approximate the average power delivered to a resistor given the
Fourier series representation of the current through the resistor?
c) How do you know how many terms you should use from the Fourier series
representation of a voltage or current for a resistor to get an accurate calculation
of the power delivered to the resistor?
d) How can you approximate the rms value of a voltage or current in a circuit, given
the Fourier series representation of that voltage or current?
e) How do you know how many terms you should use from the Fourier series
representation of a voltage or current to get an accurate calculation of the rms
value of that voltage or current?
f) Solve Assessment Problem 16.7 and Chapter Problem 16.29.
5. Read Sections 16.8 and 16.9.
a) How is the exponential form of the Fourier series related to the trigonometric
form of the Fourier series?
b) How does the presence of each type of symmetry affect the computation of the
exponential form of the Fourier series?
c) Define “amplitude spectrum” and “phase spectrum.” How do these plots help you
to understand the effect of a circuit when transforming an input periodic function
to an output periodic function?
d) Describe the effect of the following filter circuits on the amplitude spectrum of
the output of the filter:
i.
Low-pass filter;
ii. High-pass filter;
iii. Bandpass filter;
iv.
Bandreject filter.
e) Solve Chapter Problem 16.45.
Assessing Your Mastery:
Review the Objectives for this unit. Once you are satisfied that you have achieved these
Objectives, take the Chapter 16 Test.