In this book we attempt to develop the fundamental results of resistive network analysis, based upon a sound mathematical structure. The axioms upon which our development is based are Ohm's Law, Kirchhoff's Voltage Law, and Kirchhoff's Current Law. In order to state these axioms precisely, and use them in the development of our network analysis, an elaborate mathematical structure is introduced, involving concepts of graph theory, linear algebra, and one dimensional algebraic topology. The graph theory and one dimensional algebraic topology used are developed from first principles; the reader needs no background in these subjects. However, we do assume that the reader has some familiarity with elementary linear algebra. It is now stylish to teach elementary linear algebra at the sophomore college level, and we feel that the require ment that the reader should be familiar with elementary linear algebra is no more demanding than the usual requirement in most electrical engineering texts that the reader should be familiar with calculus. In this book, however, no calculus is needed. Although no formal training in circuit theory is needed for an understanding of the book, such experience would certainly help the reader by presenting him with familiar examples relevant to the mathematical abstractions introduced. It is our intention in this book to exhibit the effect of the topological properties of the network upon the branch voltages and branch currents, the objects of interest in network analysis.
1. Connected Networks.- 1.0 Introduction.- 1.1 Set Theory.- 1.2 Sets with Two or Less Elements.- 1.3 Generalized Union.- 1.4 Relations and Functions.- 1.5 Superpositions and Inverses.- 1.6 Restrictions.- 1.7 Cartesian Products.- 1.8 Some Special Symbols.- 1.9 Finite Sequences.- 1.10 Networks.- 1.11 Geometrical Realization of a Network.- 1.12 Subnetworks.- 1.13 Degree of a Vertex.- 1.14 Path in a Network.- 1.15 Proper Path in a Network.- 1.16 Reduction of a Path to a Proper Path.- 1.17 Connected Networks.- 1.18 Isolated Vertices.- 1.19 Connected Sets of Branches.- 1.20 Path Connected Set of Branches.- 1.21 Union of Connected Sets of Branches.- 1.22 Connectedness of Paths.- 1.23 Component of a Set of Branches.- 1.24 Existence of Components.- 1.25 Partition into Components.- 1.26 Removal of a Branch.- 2. Loops, Trees, and Cut Sets.- 2.0 Introduction.- 2.1 Loop in a Network.- 2.2 Loops.- 2.3 Subloops of a Loop.- 2.4 Branches and Vertices of a Loop.- 2.5 Paths in a Loop.- 2.6 Removal of a Branch from a Loop.- 2.7 Tree in a Network.- 2.8 Trees.- 2.9 Connected Subset of a Tree.- 2.10 Branches and Vertices of a Tree.- 2.11 Number of Vertices of a Connected Set of Branches.- 2.12 Addition of a Branch to a Tree.- 2.13 Existence of Maximal Trees.- 2.14 Cut Set in a Network.- 2.15 Existence of Cut Sets.- 2.16 Alternate Characterization of Cut Sets.- 3. Incidence Functions and Incidence Matrices.- 3.0 Introduction.- 3.1 Incidence Functions.- 3.2 Matrices and Arrays.- 3.3 Submatrices.- 3.4 Determinants.- 3.5 Incidence Matrices.- 3.6 Square Submatrices of an Incidence Matrix.- 3.7 Unimodular Matrices.- 3.8 Laplacian Expansion of a Determinant.- 3.9 Reduced Incidence Matrix of a Tree.- 3.10 Incidence Matrix of a Loop.- 4. Linear Algebra Review.- 4.0 Introduction.- 4.1 The Field of Scalars.- 4.2 Addition and Scalar Multiplication of Functions.- 4.3 Linear Space of 0-Chains.- 4.4 Canonical Base of the Space of 0-Chains.- 4.5 Inner Product.- 4.6 Linear Maps.- 4.7 Transpose of a Linear Map.- 4.8 Direct Sum Decomposition.- 4.9 Dimension and Direct Sum Decomposition.- 5. Boundary Operator and Coboundary Operator.- 5.0 Introduction.- 5.1 Assumptions of This Chapter.- 5.2 Chain Spaces.- 5.3 The Boundary Operator.- 5.4 Boundaries and Cycles.- 5.5 Summation Over Finite Sets.- 5.6 The Coboundary Operator.- 5.7 Coboundaries and Cocycles.- 5.8 Boundaries, Coboundaries, and Inner Products.- 5.9 Orthogonality of Cycles and Coboundaries.- 5.10 Orthogonality of Boundaries and Cocycles.- 5.11 Decomposition of ?(K) into Cycles and Coboundaries.- 5.12 Decomposition of ? (V) into Boundaries and Cocycles.- 5.13 Isomorphism of Coboundaries and Boundaries.- 5.14 Dimension of the Space of Cocycles.- 6. Axioms of Network Analysis.- 6.0 Introduction.- 6.1 Assumptions of This Chapter.- 6.2 Resistive Networks.- 6.3 Currents and Voltages.- 6.4 Ohm's Law.- 6.5 Sources.- 6.6 Kirchhoff's Laws for Voltage Sources.- 6.7 Kirchhoff's Laws for Current Sources.- 7. Existence and Uniqueness of Solutions.- 7.0 Introduction.- 7.1 Assumptions of This Chapter.- 7.2 Linearity of L and H.- 7.3 Existence and Uniqueness with Voltage Sourcess.- 7.4 Existence and Uniqueness with Current Sources.- 7.5 Current Variables.- 7.6 Voltage Variables.- 8. Kirchhoff's Third and Fourth Laws.- 8.0 Introduction.- 8.1 Assumptions of This Chapter.- 8.2 The Cycle Map.- 8.3 The Chord Map.- 8.4 The Sum of Tree Chord Products.- 8.5 The Current Chain with Voltage Sources.- 8.6 The Coboundary Map.- 8.7 The Tree Branch Map.- 8.8 The Sum of Tree Branch Products.- 8.9 The Voltage Chain with Current Sources.- 8.10 Invariance Under Change of Incidence.- References.
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