Tài liệu Fundamentals of circuits and filters

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 CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5887-1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. 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TK7867.F835 2009 621.3815--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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