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Fundamentals of
Circuits and Filters
The Circuits and Filters
Handbook
Third Edition
Edited by
Wai-Kai Chen
Fundamentals of Circuits and Filters
Feedback, Nonlinear, and Distributed Circuits
Analog and VLSI Circuits
Computer Aided Design and Design Automation
Passive, Active, and Digital Filters
The Circuits and Filters Handbook
Third Edition
Fundamentals of
Circuits and Filters
Edited by
Wai-Kai Chen
University of Illinois
Chicago, U. S. A.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2009 by Taylor & Francis Group, LLC
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International Standard Book Number-13: 978-1-4200-5887-1 (Hardcover)
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Library of Congress Cataloging-in-Publication Data
Fundamentals of circuits and filters / edited by Wai-Kai Chen.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-1-4200-5887-1
ISBN-10: 1-4200-5887-8
1. Electronic circuits. 2. Electric filters. I. Chen, Wai-Kai, 1936- II. Title.
TK7867.F835 2009
621.3815--dc22
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
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2008048126
Contents
Preface .................................................................................................................................................. vii
Editor-in-Chief .................................................................................................................................... ix
Contributors ........................................................................................................................................ xi
SECTION I Mathematics
1
Linear Operators and Matrices .......................................................................................... 1-1
Cheryl B. Schrader and Michael K. Sain
2
Bilinear Operators and Matrices ........................................................................................ 2-1
Michael K. Sain and Cheryl B. Schrader
3
Laplace Transformation ....................................................................................................... 3-1
John R. Deller, Jr.
4
Fourier Methods for Signal Analysis and Processing ................................................... 4-1
W. Kenneth Jenkins
5
z-Transform ............................................................................................................................ 5-1
Jelena Kovačevic
6
Wavelet Transforms ............................................................................................................. 6-1
P. P. Vaidyanathan and Igor Djokovic
7
Graph Theory ......................................................................................................................... 7-1
Krishnaiyan Thulasiraman
8
Signal Flow Graphs ............................................................................................................... 8-1
Krishnaiyan Thulasiraman
9
Theory of Two-Dimensional Hurwitz Polynomials ...................................................... 9-1
Hari C. Reddy
10
Application of Symmetry: Two-Dimensional Polynomials,
Fourier Transforms, and Filter Design .......................................................................... 10-1
Hari C. Reddy, I-Hung Khoo, and P. K. Rajan
v
Contents
vi
SECTION II
11
Circuit Elements, Devices, and Their Models
Passive Circuit Elements.................................................................................................... 11-1
Stanisław Nowak, Tomasz W. Postupolski, Gordon E. Carlson,
and Bogdan M. Wilamowski
12
RF Passive IC Components .............................................................................................. 12-1
Tomas H. Lee, Maria del Mar Hershenson, Sunderarajan S. Mohan,
Hirad Samavati, and C. Patrick Yue
13
Circuit Elements, Modeling, and Equation Formulation ........................................... 13-1
Josef A. Nossek
14
Controlled Circuit Elements ............................................................................................. 14-1
Edwin W. Greeneich and James F. Delansky
15
Bipolar Junction Transistor Ampliﬁers .......................................................................... 15-1
David J. Comer and Donald T. Comer
16
Operational Ampliﬁers ...................................................................................................... 16-1
David G. Nairn and Sergio B. Franco
17
High-Frequency Ampliﬁers .............................................................................................. 17-1
Chris Toumazou and Alison Payne
SECTION III
18
Linear Circuit Analysis
Fundamental Circuit Concepts ........................................................................................ 18-1
John Choma, Jr.
19
Network Laws and Theorems .......................................................................................... 19-1
Ray R. Chen, Artice M. Davis, and Marwan A. Simaan
20
Terminal and Port Representations ................................................................................ 20-1
James A. Svoboda
21
Signal Flow Graphs in Filter Analysis and Synthesis ................................................. 21-1
Pen-Min Lin
22
Analysis in the Frequency Domain................................................................................. 22-1
Jiri Vlach and John Choma, Jr.
23
Tableau and Modiﬁed Nodal Formulations .................................................................. 23-1
Jiri Vlach
24
Frequency-Domain Methods ............................................................................................ 24-1
Peter B. Aronhime
25
Symbolic Analysis ............................................................................................................... 25-1
Benedykt S. Rodanski and Marwan M. Hassoun
26
Analysis in the Time Domain .......................................................................................... 26-1
Robert W. Newcomb
27
State-Variable Techniques ................................................................................................. 27-1
Kwong S. Chao
Index ................................................................................................................................................ IN-1
Preface
The purpose of this book is to provide in a single volume a comprehensive reference work covering the
broad spectrum of mathematics for circuits and ﬁlters; circuits elements, devices, and their models; and
linear circuit analysis. This book is written and developed for the practicing electrical engineers in
industry, government, and academia. The goal is to provide the most up-to-date information in the ﬁeld.
Over the years, the fundamentals of the ﬁeld have evolved to include a wide range of topics and a broad
range of practice. To encompass such a wide range of knowledge, this book focuses on the key concepts,
models, and equations that enable the design engineer to analyze, design, and predict the behavior of
large-scale circuits. While design formulas and tables are listed, emphasis is placed on the key concepts
and theories underlying the processes.
This book stresses fundamental theories behind professional applications and uses several examples to
reinforce this point. Extensive development of theory and details of proofs have been omitted. The reader
is assumed to have a certain degree of sophistication and experience. However, brief reviews of theories,
principles, and mathematics of some subject areas are given. These reviews have been done concisely with
perception.
The compilation of this book would not have been possible without the dedication and efforts of
Professors Yih-Fang Huang and John Choma, Jr., and most of all the contributing authors. I wish to
thank them all.
Wai-Kai Chen
vii
Editor-in-Chief
Wai-Kai Chen is a professor and head emeritus of the Department
of Electrical Engineering and Computer Science at the University of
Illinois at Chicago. He received his BS and MS in electrical engineering at Ohio University, where he was later recognized as a
distinguished professor. He earned his PhD in electrical engineering
at the University of Illinois at Urbana–Champaign.
Professor Chen has extensive experience in education and industry and is very active professionally in the ﬁelds of circuits and
systems. He has served as a visiting professor at Purdue University,
the University of Hawaii at Manoa, and Chuo University in Tokyo,
Japan. He was the editor-in-chief of the IEEE Transactions on
Circuits and Systems, Series I and II, the president of the IEEE
Circuits and Systems Society, and is the founding editor and the
editor-in-chief of the Journal of Circuits, Systems and Computers.
He received the Lester R. Ford Award from the Mathematical
Association of America; the Alexander von Humboldt Award from Germany; the JSPS Fellowship
Award from the Japan Society for the Promotion of Science; the National Taipei University of Science
and Technology Distinguished Alumnus Award; the Ohio University Alumni Medal of Merit for
Distinguished Achievement in Engineering Education; the Senior University Scholar Award and the
2000 Faculty Research Award from the University of Illinois at Chicago; and the Distinguished Alumnus
Award from the University of Illinois at Urbana–Champaign. He is the recipient of the Golden Jubilee
Medal, the Education Award, and the Meritorious Service Award from the IEEE Circuits and Systems
Society, and the Third Millennium Medal from the IEEE. He has also received more than a dozen
honorary professorship awards from major institutions in Taiwan and China.
A fellow of the Institute of Electrical and Electronics Engineers (IEEE) and the American Association
for the Advancement of Science (AAAS), Professor Chen is widely known in the profession for the
following works: Applied Graph Theory (North-Holland), Theory and Design of Broadband Matching
Networks (Pergamon Press), Active Network and Feedback Ampliﬁer Theory (McGraw-Hill), Linear
Networks and Systems (Brooks=Cole), Passive and Active Filters: Theory and Implements (John Wiley),
Theory of Nets: Flows in Networks (Wiley-Interscience), The Electrical Engineering Handbook (Academic
Press), and The VLSI Handbook (CRC Press).
ix
Contributors
Peter B. Aronhime
Electrical and Computer
Engineering Department
University of Louisville
Louisville, Kentucky
David J. Comer
Department of Electrical and
Computer Engineering
Brigham Young University
Provo, Utah
Edwin W. Greeneich
Department of Electrical
Engineering
Arizona State University
Tempe, Arizona
Gordon E. Carlson
Department of Electrical and
Computer Engineering
University of Missouri–Rolla
Rolla, Missouri
Donald T. Comer
Department of Electrical and
Computer Engineering
Brigham Young University
Provo, Utah
Marwan M. Hassoun
Department of Electrical and
Computer Engineering
Iowa State University
Ames, Iowa
Artice M. Davis
Department of Electrical
Engineering
San Jose State University
San Jose, California
Maria del Mar Hershenson
Center for Integrated Systems
Stanford University
Stanford, California
Kwong S. Chao
Department of Electrical and
Computer Engineering
Texas Tech University
Lubbock, Texas
Ray R. Chen
Department of Electrical
Engineering
San Jose State University
San Jose, California
Wai-Kai Chen
Department of Electrical and
Computer Engineering
University of Illinois at Chicago
Chicago, Illinois
John Choma, Jr.
Ming Hsieh Department of
Electrical Engineering
University of Southern
California
Los Angeles, California
James F. Delansky
Department of Electrical
Engineering
Pennsylvania State University
University Park, Pennsylvania
John R. Deller, Jr.
Department of Electrical and
Computer Engineering
Michigan State University
East Lansing, Michigan
Igor Djokovic
PairGain Technologies
Tustin, California
Sergio B. Franco
Division of Engineering
San Francisco State University
San Francisco, California
Yih-Fang Huang
Department of Electrical
Engineering
University of Notre Dame
Notre Dame, Indiana
W. Kenneth Jenkins
Department of Electrical
Engineering
Pennsylvania State University
University Park, Pennsylvania
I-Hung Khoo
Department of Electrical
Engineering
California State University
Long Beach, California
Jelena Kovačevic
AT&T Bell Laboratories
Murray Hill, New Jersey
xi
Contributors
xii
Tomas H. Lee
Center for Integrated Systems
Stanford University
Stanford, California
Pen-Min Lin
School of Electrical Engineering
Purdue University
West Lafayette, Indiana
Sunderarajan S. Mohan
Center for Integrated Systems
Stanford University
Stanford, California
David G. Nairn
Department of Electrical
Engineering
Queen’s University
Kingston, Canada
Robert W. Newcomb
Electrical Engineering
Department
University of Maryland
College Park, Maryland
Josef A. Nossek
Institute for Circuit Theory and
Signal Processing
Technical University of Munich
Munich, Germany
Stanisław Nowak
Institute of Electronics
University of Mining and
Metallurgy
Krakow, Poland
Alison Payne
Institute of Biomedical
Engineering
Imperial College of Science,
Technology and Medicine
London, England
Tomasz W. Postupolski
Institute of Electronic Materials
Technology
Warsaw, Poland
P. K. Rajan
Department of Electrical and
Computer Engineering
Tennessee Tech University
Cookeville, Tennessee
James A. Svoboda
Department of Electrical
Engineering
Clarkson University
Potsdam, New York
Hari C. Reddy
Department of Electrical
Engineering
California State University
Long Beach, California
Krishnaiyan Thulasiraman
School of Computer Science
University of Oklahoma
Norman, Oklahoma
and
Department of Computer
Science=Electrical and
Control Engineering
National Chiao-Tung University,
Taiwan
Benedykt S. Rodanski
Faculty of Engineering
University of Technology,
Sydney
Sydney, New South Wales,
Australia
Michael K. Sain
Department of Electrical
Engineering
University of Notre Dame
Notre Dame, Indiana
Hirad Samavati
Center for Integrated Systems
Stanford University
Stanford, California
Cheryl B. Schrader
College of Engineering
Boise State University
Boise, Idaho
Marwan A. Simaan
Department of Electrical and
Computer Engineering
University of Pittsburgh
Pittsburgh, Pennsylvania
Chris Toumazou
Institute of Biomedical
Engineering
Imperial College of Science,
Technology and Medicine
London, England
P. P. Vaidyanathan
Department of Electrical
Engineering
California Institute of
Technology
Pasadena, California
Jiri Vlach
Department of Electrical and
Computer Engineering
University of Waterloo
Waterloo, Ontario, Canada
Bogdan M. Wilamowski
Alabama Nano=Micro Science
and Technology Center
Department of Electrical and
Computer Engineering
Auburn University
Auburn, Alabama
C. Patrick Yue
Center for Integrated Systems
Stanford University
Stanford, California
1
Linear Operators and
Matrices
Cheryl B. Schrader
Boise State University
Michael K. Sain
University of Notre Dame
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Introduction ................................................................................ 1-1
Vector Spaces over Fields ........................................................ 1-2
Linear Operators and Matrix Representations.................... 1-4
Matrix Operations ..................................................................... 1-6
Determinant, Inverse, and Rank ............................................ 1-8
Basis Transformations ............................................................ 1-12
Characteristics: Eigenvalues, Eigenvectors,
and Singular Values................................................................. 1-15
1.8 On Linear Systems................................................................... 1-18
References ............................................................................................ 1-20
1.1 Introduction
It is only after the engineer masters’ linear concepts—linear models and circuit and ﬁlter theory—that the
possibility of tackling nonlinear ideas becomes achievable. Students frequently encounter linear methodologies, and bits and pieces of mathematics that aid in problem solution are stored away. Unfortunately, in memorizing the process of ﬁnding the inverse of a matrix or of solving a system of equations,
the essence of the problem or associated knowledge may be lost. For example, most engineers are fairly
comfortable with the concept of a vector space, but have difﬁculty in generalizing these ideas to the
module level. Therefore, the intention of this section is to provide a uniﬁed view of key concepts in the
theory of linear circuits and ﬁlters, to emphasize interrelated concepts, to provide a mathematical
reference to the handbook itself, and to illustrate methodologies through the use of many and varied
examples.
This chapter begins with a basic examination of vector spaces over ﬁelds. In relating vector spaces, the
key ideas of linear operators and matrix representations come to the fore. Standard matrix operations
are examined as are the pivotal notions of determinant, inverse, and rank. Next, transformations are
shown to determine similar representations, and matrix characteristics such as singular values and
eigenvalues are deﬁned. Finally, solutions to algebraic equations are presented in the context of matrices
and are related to this introductory chapter on mathematics as a whole.
Standard algebraic notation is introduced ﬁrst. To denote an element s in a set S, use s 2 S. Consider
two sets S and T. The set of all ordered pairs (s, t) where s 2 S and t 2 T is deﬁned as the Cartesian
product set S 3 T. A function f from S into T, denoted by f : S ! T, is a subset U of ordered pairs (s, t) 2
S 3 T such that for every s 2 S, one and only one t 2 T exists such that (s, t) 2 U. The function evaluated
at the element s gives t as a solution ( f(s) ¼ t), and each s 2 S as a ﬁrst element in U appears exactly once.
1-1
Fundamentals of Circuits and Filters
1-2
A binary operation is a function acting on a Cartesian product set S 3 T. When T ¼ S, one speaks of a
binary operation on S.
1.2 Vector Spaces over Fields
A ﬁeld F is a nonempty set F and two binary operations, sum (þ) and product, such that the following
properties are satisﬁed for all a, b, c 2 F:
Associativity: (a þ b) þ c ¼ a þ (b þ c); (ab)c ¼ a(bc)
Commutativity: a þ b ¼ b þ a ; ab ¼ ba
Distributivity: a(b þ c) ¼ (ab) þ (ac)
Identities: (Additive) 0 2 F exists such that a þ 0 ¼ a
(Multiplicative) 1 2 F exists such that a1 ¼ a
5. Inverses: (Additive) For every a 2 F, b 2 F exists such that a þ b ¼ 0
(Multiplicative) For every nonzero a 2 F, b 2 F exists such that ab ¼ 1
1.
2.
3.
4.
Examples
.
.
.
.
Field
Field
Field
Field
of real numbers R
of complex numbers C
of rational functions with real coefﬁcients R(s)
of binary numbers
The set of integers Z with the standard notions of addition and multiplication does not form a ﬁeld
because a multiplicative inverse in Z exists only for 1. The integers form a commutative ring. Likewise,
polynomials in the indeterminate s with coefﬁcients from F form a commutative ring F[s]. If ﬁeld
property 2 also is not available, then one speaks simply of a ring. An additive group is a nonempty set G
and one binary operation þ satisfying ﬁeld properties 1, 4, and 5 for addition, that is, associativity and the
existence of additive identity and inverse. Moreover, if the binary operation þ is commutative (ﬁeld
property 2), then the additive group is said to be abelian. Common notation regarding inverses is that the
additive inverse for a 2 F is b ¼ a 2 F. In the multiplicative case b ¼ a1 2 F.
An F-vector space V is a nonempty set V and a ﬁeld F together with binary operations þ: V 3 V ! V
and *: F 3 V ! V subject to the following axioms for all elements v, w 2 V and a, b 2 F:
1.
2.
3.
4.
5.
V and þ form an additive abelian group
a * (vþw) ¼ (a * v)þ(a * w)
(aþb) * v ¼ (a * v)þ(b * v)
(ab) * v ¼ a * (b * v)
1*v¼v
Examples
.
.
Set of all n-tuples (v1, v2, . . . , vn) for n > 0 and vi 2 F
Set of polynomials of degree less than n with real coefﬁcients (F ¼ R)
Elements of V are referred to as vectors, whereas elements of F are scalars. Note that the terminology
vector space V over the ﬁeld F is used often. A module differs from a vector space in only one aspect; the
underlying ﬁeld in a vector space is replaced by a ring. Thus, a module is a direct generalization of a
vector space.
When considering vector spaces of n-tuples, þ is vector addition deﬁned element by element using the
scalar addition associated with F. Multiplication (*), which is termed scalar multiplication, also is deﬁned
Linear Operators and Matrices
1-3
element by element using multiplication in F. The additive identity in this case is the zero vector (n-tuple
of zeros) or null vector, and Fn denotes the set of n-tuples with elements in F, a vector space over F.
e V is called a subspace of V if for each v, w 2 V
e and every a 2 F, v þ w 2 V,
e
A nonempty subset V
e When the context makes things clear, it is customary to suppress the , and write av in
and a * v 2 V.
*
place of a * v.
A set of vectors {v1, v2, . . . , vm} belonging to an F-vector space V is said to span the vector space if
any element v 2 V can be represented by a linear combination of the vectors vi. That is, scalars
a1, a2, . . . , am 2 F are such that
v ¼ a 1 v1 þ a 2 v2 þ þ a m vm
(1:1)
A set of vectors {v1, v2, . . . , vp} belonging to an F-vector space V is said to be linearly dependent over F if
scalars a1, a2, . . . , ap 2 F, not all zero, exist such that
a1 v1 þ a2 v2 þ þ ap vp ¼ 0
(1:2)
If the only solution for Equation 1.2 is that all ai ¼ 0 2 F, then the set of vectors is said to be linearly
independent.
Examples
.
.
.
(1, 0) and (0, 1) are linearly independent.
(1, 0, 0), (0, 1, 0), and (1, 1, 0) are linearly dependent over R. To see this, simply choose a1 ¼ a2 ¼ 1
and a3 ¼ 1.
s2 þ 2s and 2s þ 4 are linearly independent over R, but are linearly dependent over R(s) by
choosing a1 ¼ 2 and a2 ¼ s.
A set of vectors {v1, v2, . . . , vn} belonging to an F-vector space V is said to form a basis for V if it both
spans V and is linearly independent over F. The number of vectors in a basis is called the dimension of
the vector space, and is denoted as dim(V). If this number is not ﬁnite, then the vector space is said to be
inﬁnite dimensional.
Examples
.
.
In an n-dimensional vector space, any n linearly independent vectors form a basis.
Natural (standard) basis
2 3
1
607
6 7
607
7
e1 ¼ 6
6 ... 7,
6 7
405
0
2 3
0
617
6 7
607
7
e2 ¼ 6
6 ... 7,
6 7
405
0
2 3
2 3
0
0
607
607
6 7
6 7
607
617
7, . . . , en1 ¼ 6 . 7,
e3 ¼ 6
.
6 .. 7
6 .. 7
6 7
6 7
415
405
0
0
2 3
0
607
6 7
607
7
en ¼ 6
6 ... 7
6 7
405
1
both spans Fn and is linearly independent over F.
Consider any basis {v1, v2, . . . , vn} in an n-dimensional vector space. Every v 2 V can be represented
uniquely by scalars a1, a2, . . . , an 2 F as
Fundamentals of Circuits and Filters
1-4
v ¼ a 1 v1 þ a 2 v2 þ þ a n vn
2 3
a1
6a 7
6 27
7
¼ [v1 v2 vn ]6
6 .. 7
4 . 5
(1:3)
(1:4)
an
¼ [v1 v2 vn ]a
(1:5)
Here, a 2 Fn is a coordinate representation of v 2 V with respect to the chosen basis. The reader will be able
to discern that each choice of basis will result in another representation of the vector under consideration.
Of course, in the applications, some representations are more popular and useful than others.
1.3 Linear Operators and Matrix Representations
First, recall the deﬁnition of a function f: S ! T. Alternate terminology for a function is mapping,
operator, or transformation. The set S is called the domain of f, denoted by D( f ). The range of f, R( f ), is
the set of all t 2 T such that (s, t) 2 U ( f(s) ¼ t) for some s 2 D( f ).
Examples
Use S ¼ {1, 2, 3, 4} and T ¼ {5, 6, 7, 8}.
.
.
.
e ¼ {(1, 5), (2, 5), (3, 7), (4, 8)} is a function. The domain is {1, 2, 3, 4} and the range is {5, 7, 8}.
U
Û ¼ {(1, 5), (1, 6), (2, 5), (3, 7), (4, 8)} is not a function.
U ¼ {(1, 5), (2, 6), (3, 7), (4, 8)} is a function. The domain is {1, 2, 3, 4} and the range is {5, 6, 7, 8}.
If R( f ) ¼ T, then f is said to be surjective (onto). Loosely speaking, all elements in T are used up. If
f : S ! T has the property that f(s1) ¼ f(s2) implies s1 ¼ s2, then f is said to be injective (one-to-one). This
means that any element in R( f ) comes from a unique element in D( f ) under the action of f. If a function
is both injective and surjective, then it is said to be bijective (one-to-one and onto).
Examples
.
.
e is not onto because 6 2 T is not in R( f). Also U
e is not one-to-one because f(1) ¼ 5 ¼ f(2),
U
but 1 6¼ 2.
U is bijective.
Now consider an operator L: V ! W, where V and W are vector spaces over the same ﬁeld F. L is said
to be a linear operator if the following two properties are satisﬁed for all v, w 2 V and for all a 2 F:
L(av) ¼ aL(v)
(1:6)
L(v þ w) ¼ L(v) þ L(w)
(1:7)
Equation 1.6 is the property of homogeneity and Equation 1.7 is the property of additivity. Together they
imply the principle of superposition, which may be written as
L(a1 v1 þ a2 v2 ) ¼ a1 L(v1 ) þ a2 L(v2 )
(1:8)
for all v1, v2 2 V and a1, a2 2 F. If Equation 1.8 is not satisﬁed, then L is called a nonlinear operator.
Linear Operators and Matrices
1-5
Examples
.
Consider V ¼ C and F ¼ C. Let L: V ! V be the operator that takes the complex conjugate: L(v) ¼ v
for v, v 2 V. Certainly
L(v1 þ v2 ) ¼ v1 þ v2 ¼ v1 þ v2 ¼ L(v1 ) þ L(v2 )
However,
L(a1 v1 ) ¼ a1 v1 ¼ a1 L(v1 ) 6¼ a1 L(v1 )
Then L is a nonlinear operator because homogeneity fails.
.
For F-vector spaces V and W, let V be Fn and W be Fn1. Examine L: V ! W, the operator that
truncates the last element of the n-tuple in V, that is,
L((v1 , v2 , . . . , vn1 , vn )) ¼ (v1 , v2 , . . . , vn1 )
Such an operator is linear.
The null space (kernel) of a linear operator L: V ! W is the set
ker L ¼ {v 2 V such that L(v) ¼ 0}
(1:9)
Equation 1.9 deﬁnes a vector space. In fact, ker L is a subspace of V. The mapping L is injective if and
only if ker L ¼ 0; that is, the only solution in the right member of Equation 1.9 is the trivial solution. In
this case, L is also called monic.
The image of a linear operator L: V ! W is the set
im L ¼ {w 2 W such that L(v) ¼ w for some v 2 V}
(1:10)
Clearly, im L is a subspace of W, and L is surjective if and only if im L is all of W. In this case, L is also
called epic.
A method of relating speciﬁc properties of linear mappings is the exact sequence. Consider a sequence
of linear mappings
~
L
L
V ! W ! U !
(1:11)
~ A sequence is called exact if it is exact at each vector
This sequence is said to be exact at W if im L ¼ ker L.
space in the sequence. Examine the following special cases:
L
0!V !W
~
L
W !U!0
(1:12)
(1:13)
~
Sequence (Equation 1.12) is exact if and only if L is monic, whereas Equation 1.13 is exact if and only if L
is epic.
Further, let L: V ! W be a linear mapping between ﬁnite-dimensional vector spaces. The rank of L,
r(L), is the dimension of the image of L. In such a case
Fundamentals of Circuits and Filters
1-6
r(L) þ dim ( ker L) ¼ dim V
(1:14)
Linear operators commonly are represented by matrices. It is quite natural to interchange these two
ideas, because a matrix with respect to the standard bases is indistinguishable from the linear operator it
represents. However, insight may be gained by examining these ideas separately. For V and W, n- and
m-dimensional vector spaces over F, respectively, consider a linear operator L: V ! W. Moreover, let
{v1, v2, . . . , vn} and {w1, w2, . . . , wm} be respective bases for V and W. Then L: V ! W can be represented
uniquely by the matrix M 2 Fm3n where
2
m11
6 m21
6
M ¼ 6 ..
4 .
m12
m22
..
.
mm1
mm2
...
...
..
.
m1n
m2n
..
.
3
7
7
7
5
(1:15)
. . . mmn
The ith column of M is the representation of L(vi) with respect to {w1, w2, . . . , wm}. Element mij 2 F of
Equation 1.15 occurs in row i and column j.
Matrices have a number of properties. A matrix is said to be square if m ¼ n. The main diagonal of a
square matrix consists of the elements mii. If mij ¼ 0 for all i > j (i < j), a square matrix is said to be upper
(lower) triangular. A square matrix with mij ¼ 0 for all i 6¼ j is diagonal. Additionally, if all mii ¼ 1, a
diagonal M is an identity matrix. A row vector (column vector) is a special case in which m ¼ 1 (n ¼ 1).
Also, m ¼ n ¼ 1 results essentially in a scalar.
Matrices arise naturally as a means to represent sets of simultaneous linear equations. For example, in
the case of Kirchhoff equations, Chapter 7 shows how incidence, circuit, and cut matrices arise. Or
consider a p network having node voltages vi, i ¼ 1, 2 and current sources ii, i ¼ 1, 2 connected across the
resistors Ri, i ¼ 1, 2 in the two legs of the p. The bridge resistor is R3. Thus, the unknown node voltages
can be expressed in terms of the known source currents in the manner
(R1 þ R3 )
1
v1 v2 ¼ i1
R 1 R3
R3
(1:16)
(R2 þ R3 )
1
v2 v1 ¼ i2
R 2 R3
R3
(1:17)
If the voltages, vi, and the currents, ii, are placed into a voltage vector v 2 R2 and current vector i 2 R2,
respectively, then Equations 1.16 and 1.17 may be rewritten in matrix form as
" #
i1
i2
" (R1 þR3 )
¼
R1 R3
R13
R13
(R2 þR3 )
R2 R3
#"
v1
#
v2
(1:18)
A conductance matrix G may then be deﬁned so that i ¼ Gv, a concise representation of the original pair
of circuit equations.
1.4 Matrix Operations
Vector addition in Fn was deﬁned previously as an element-wise scalar addition. Similarly, two matrices
both M and N in Fm3n can be added (subtracted) to form the resultant matrix P 2 Fm3n by
mij nij ¼ pij , i ¼ 1, 2, . . . , m,
j ¼ 1, 2, . . . , n
(1:19)
Linear Operators and Matrices
1-7
Matrix addition, thus, is deﬁned using addition in the ﬁeld over which the matrix lies. Accordingly, the
matrix, each of whose entries is 0 2 F, is an additive identity for the family. One can set up additive
inverses along similar lines, which, of course, turn out to be the matrices each of whose elements is the
negative of that of the original matrix.
Recall how scalar multiplication was deﬁned in the example of the vector space of n-tuples. Scalar
multiplication can also be deﬁned between a ﬁeld element a 2 F and a matrix M 2 Fm3n in such a way
that the product aM is calculated element-wise:
aM ¼ P , amij ¼ pij , i ¼ 1, 2, . . . , m, j ¼ 1, 2, . . . , n
(1:20)
Examples
(F ¼ R): M ¼
.
.
N¼
2
1
3
,
6
a ¼ 0:5
6 0
33
¼
3 7
1þ6
42 3þ3
2 6
M N ¼ ~P ¼
¼
21 16
1 5
(0:5)4 (0:5)3
2 1:5
aM ¼ ^P ¼
¼
(0:5)2 (0:5)1
1 0:5
.
3
,
1
4
2
MþN ¼P ¼
4þ2
2þ1
To multiply two matrices M and N to form the product MN requires that the number of columns of M
equal the number of rows of N. In this case the matrices are said to be conformable. Although vector
multiplication cannot be deﬁned here because of this constraint, Chapter 2 examines this operation in
detail using the tensor product. The focus here is on matrix multiplication. The resulting matrix will have
its number of rows equal to the number of rows in M and its number of columns equal to the number of
columns of N. Thus, for M 2 Fm3n and N 2 Fn3p, MN ¼ P 2 Fm3p. Elements in the resulting matrix P
may be determined by
pij ¼
n
X
mik nkj
(1:21)
k¼l
Matrix multiplication involves one row and one column at a time. To compute the pij term in P, choose
the ith row of M and the jth column of N. Multiply each element in the row vector by the corresponding
element in the column vector and sum the result. Notice that in general, matrix multiplication is not
commutative, and the matrices in the reverse order may not even be conformable. Matrix multiplication
is, however, associative and distributive with respect to matrix addition. Under certain conditions, the
ﬁeld F of scalars, the set of matrices over F, and these three operations combine to form an algebra.
Chapter 2 examines algebras in greater detail.
Examples
(F ¼ R): M ¼
4
2
1 3
3
, N¼
2 4
1
5
6

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