Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1968 Resistive n-port network synthesis utilizing lagrangian tree structures Richard Ray Gallagher Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Electrical and Electronics Commons Recommended Citation Gallagher, Richard Ray, "Resistive n-port network synthesis utilizing lagrangian tree structures " (1968). Retrospective Theses and Dissertations. 3549. https://lib.dr.iastate.edu/rtd/3549 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microiihned exactly as received 69-9860 GALLAGHER, Richard Ray, 1942- RESISTIVE N-PORT NETWORK SYNTHESIS UTILIZING LAGRANGIAN TREE STRUCTURES. Iowa State University, Ph.D., 1968 Engineering, electrical University Microfilms, Inc., Ann Arbor, Michigan RESISTIVE N-PORT NETWORK SYNTHESIS UTILIZING LAGRANGIAN TREE STRUCTURES by Richard Ray Gallagher A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Electrical Engineering Approved : Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. Head of Major Department Signature was redacted for privacy. Iowa State University Ames, Iowa 1968 ii TABLE OF CONTENTS Page I. INTRODUCTION 1 II. TERMS AND DEFINITIONS ' 3 III. LITERATURE SEARCH 11 A. Concentration on (n+l)-Node Resistive Networks 11 B. Considerations on Augmented Admittance Matrices 20 IV. REALIZATIONS OF RESISTIVE NETWORKS FROM PORT- ADMITTANCE MATRICES 24 A. Two-Tree Synthesis Process with One Tree Containing Only One Port 24 B. Example One Illustrating Qualified Port Removals from Lagrangian Tree Structures 45 C. Two-Tree Synthesis Process Utilizing Complete Connecting Graph 52 D. Example Two Illustrating (nf2)-Node Synthesis of a Port-Admittance Matrix Which is Nonrealizable by an (nfl)-Node Lagrangian Tree 64 E. Example Three Illustrating (nf2)-Node Synthesis of a Port-Admittance Matrix Which is Nonrealizable by an (n+l)-Node Network 68 F. K-Tree Synthesis Process Utilizing Lagrangian Subtrees 71 G. Example Four Utilizing K-Tree Synthesis of a Port- Admittance Matrix Which is Nonrealizable by an (n+l)-Node Lagrangian Tree 85 H. Example Five Utilizing K-Tree Synthesis of a Port-Admittance Matrix Which is Nonrealizable by an (n+l)-Node Network 91 V. SUMMARY AND SUGGESTED RESEARCH CONSIDERATIONS 98 VI. BIBLIOGRAPHY 102 VII. ACKNOWLEDGMENTS 105 ill LIST OF FIGURES Page Figure 1. Voltage and current orientation for circuit representation and graph representation 7 Figure 2. Basic form for the Lagrangian tree (or subtree) with L ports of interest 10 Figure 3. Basic form for the linear tree (or subtree) with S ports of interest all oriented in the same direction 10 Figure 4. A general port tree structure composed of two subtrees connected by an additional port, which may connect any node of one subtree to any node of the other subtree 25 Figure 5. Basic form for port structure with one isolated port 30 Figure 6. 2n-node realization of the n— row and column of matrix Y 31 Figure 7. Realization of the n— row and column of matrix Y with (n-1) nodes coalesced into one common node ... 32 Figure 8. Parasitic realization of n— row and column of matrix Y • 36b Figure 9. Realizati on c matrix 42 Figure 10. A • "'nation of the port-admittance matrix, Y, Zy nodes 43b Figure 11. A Lagrangian tree showing conductance values for an (n+l)-node realization of Equation 47 46 Figure 12. Network with conductance values illustrating the 4,th row and column realization of Equation 47 on (n+2) nodes 47 Figure 13. Network with conductance values illustrating realization of the 4^ row and coUrain of Equation 49 50 Figure 14. Total realization with (n+-2) nodes of Equation 47 or Equation 49 51 iv Page Figure 15. Basic structure for Lagrangian subtrees corresponding to Equation 53 54 Figure 15. Basic structure for Lagrangian subtrees including connecting port, 54 Figure 17. Graph illustrating the two Lagrangian subtrees and their interconnecting set of unknown conductances . 56 Figure 18. (n+2)-node network with unidentified edges as unknowns based on two Lagrangian subtree structures for a realizable port-admittance matrix 55 Figure 19. Graph illustrating the two Lagrangian subtrees with their set of six unknown connecting conductances 65 Figure 20. Complete realization on (n+-2) nodes for Equation 79 with conductance values given 69 Figure 21. Complete realization on (n+2) nodes for Equation 87 with conductance values given 72 Figure 22. Graph of k Lagrangian subtrees and (k-1) connecting ports 75 Figure 23. Graph of one Lagrangian tree utilizing all (n+k) nodes of the network 78 Figure 24. Two different trees whose combination forms a new graph covering the same set of nodes and providing for the formation of the transformation matrix, Q 81 Figure 25. Graph of two Lagrangian subtrees with one connecting port 87 Figure 26. Graph of one Lagrangian tree utilizing all five nodes of the network 87 Figure 27. Superposition of Figure 26 on Figure 25 providing for the formation of the transformation matrix, Q 88 V Page Figure 28. Complete realization for Equation 115 with k = 2 utilizing k-tree synthesis method with conductance values given 92 Figure 29. Complete realization for Equation 131 with k = 2 utilizing k-tree synthesis method with conductance values given 97 1 I. INTRODUCTION The problem of synthesis of a resistive n-port network from its port-admittance matrix, Y, is certainly not a new one. Much investiga­ tion has been carried on as cited in the Literature Search section; but, the general problem seems far from being solved. Even though many of the known resistive synthesis procedures and their implications can be applied to networks with R, L, and C elements the author restricts the developments to resistive elements. Since the derivations are carried out utilizing nodal analysis terminology the element values of the realized network are referred to as conductances. Linear graph theory plays an important role in analysis and synthesis of networks. After defining many of the basic terms involved with linear graph theory and its applications much of the investigation relies upon the augmentation of the original port-admittance matrix with reference to linear transformations and cut-set notation. Throughout the development and discussion of the realization techniques the n ports of interest corresponding to the (nxn) port- admittance matrix are arranged in a basic Lagrangian tree or in k (2^k^n) Lagrangian subtree structures. This form of port structure lends itself nicely to nodal methods. Two forms of (n+2)-node synthesis methods are developed. One form relies entirely upon the element values of the given port-admittance matrix and requires no solving for unknown quantities. The second method depends upon the solving of a set of independent equalities and inequalities. Compared to the first method this method is a much more flexible; but, involved technique. 2 A third synthesis procedure, which concerns itself with the general case, is developed. Like the second method it requires satisfying a set of independent inequalities. This independent set of inequalities is derived from the augmented port-admittance matrix. The network synthe­ sized by this procedure or by the second procedure is certainly not unique since the solution for the unknown quantities in general will be in the form of a bounded solution. Also, the initial step of each method restricts the topology of the network to basic Lagrangian tree s tructures. 3 II. TERMS AND DEFINITIONS The language associated with linear graph theory and its applica­ tions does not always follow a precise standard. Thus, many of the terms possess various meanings depending upon the presentation or publica­ tion where they are found. In this section the author wishes to clarify the meaning of the terms or operations that are pertinent for the under­ standing of the material found in the following sections. The definitions that the author gives are those of the references cited and of personal preference according to the context of the following sections. They are thought to be the most standard representations. 1. (14) A matrix is a rectangular array containing m rows and n columns of elements of a scalar field F. It is called an (mxn) matrix over F. 2. (14) The transpose of matrix, A = [a_j], denoted by A % is defined as A' = [b. .] where b.. = a... 3. (14) An (nxn) matrix A is non-singular if and only if a matrix B, its inverse. exists such that AB = I. Otherwise, -1 A is said to be singular. We denote the inverse as B = A 4. (14) A linear transformation. T, from a vector space, V, to a vector space, W, both over the scalar field, F, is a mapping of V into W such that for all 0!,PeV and for all a,beF, (aa + bp)T = a(aT) + b(pT). (1) The linear transformation that is used in Section IVF is described as Yg = SY^S' where S' is the non-singular matrix -s • K, that gives the set of voltages, V^, associated with the admittance matrix, Y^, in terms of the set of voltages, V2, associated with the admittance matrix, Yg. 5. A dominant (nxn) matrix, D, satisfies the condition that n d.. ^ 2 |d I, i = 1, . n. (2) i^j j=l 6. (26) Further restricting the matrix D, a hyperdominant matrix also must satisfy the condition, d 3 0; i ^ j; i = 1, " • •, n; j = 1, • • •, n.