Contains reports on one research project.Computation Center, M.I.T

Wilde, D. U.

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XIX. ADVANCED COMPUTATION RESEARCHProf. H. M. Teager U. Shimony R. L. WardE. L. Ivie D. U. WildeA. NETWORK ANALYSIS BY DIGITAL COMPUTERThe amount of work required to analyze an electrical network grows rapidly as afunction of the number of network nodes, N. The digital computer is a tool that canreduce this growth rate if the analysis is done in a proper manner. A study of analy-sis procedures that are applicable to digital-computer automation can be divided intotwo parts: topological and matrix. Such a study shows that the computation timerequired by topological methods grows as N!, while that of matrix methods can be madeto grow more slowly. The use of matrix methods requires a set of independent Kirchhoffvoltage or current equations. Further evaluation shows that node-to-datum voltageequations require a minimal amount of equation setup time. These equations are linearin the node-to-datum voltage variables but have polynomial coefficients of the form:Y.. = C..D+ + G..D + K..Dwhere the C's, G's, and K's are either numerical or symbolic literals, and the D's areeither time-domain operators or frequency functions. Because of this generality ofcoefficient and because most practical electrical networks have few interconnected nodepairs, Cramer's Rule was chosen as the procedure to solve these equations. Cramer'sRule requires the expansion of two determinants.The work for expanding a determinant by the straightforward application of the clas-sical Laplace method grows factorially. The topology of the determinant expansionreveals a treelike structure, referred to as the determinant tree.S= d e f a -b cg h ie -f d -f d -ea(ei-fh) - b(di-fg) + c(dh-eg) i h i g h gThe Determinant The Determinant TreeIf the tree is pruned from bottom to top, the order of the simplification is the same asthat of evaluating the determinantal equation, starting on the inside of the parenthesesand working outward. Thus, the determinant tree indicates the minimum expansion*This work is supported in part by the Computation Center, M. I. T.QPR No. 70 307(XIX. ADVANCED COMPUTATION RESEARCH)work procedure. The structure of the tree can be pruned as the expansion progresses,and the memory space required to store the tree grows only as N 3 . The work forexpanding the determinant of a general network by the tree algorithm grows as (N-1)!.However, the construction rules of the determinant tree allow the computer to greatlyreduce the expansion work for networks having few interconnected node pairs. Forexample, experimental results show that the work required for analyzing a ladder net-work grows geometrically at approximately 1.5 per node instead of factorially.A program has been written in the MAD language and tested on sample networks.The input is a list of network elements and their terminal nodes, the desired unknownvoltage, and the specified datum node. The present state of the loading routine allowsfor linear, passive nonmutual elements and for independent, current-controlled, andvoltage-controlled current sources. Also, one symbolic circuit element may beincluded to allow the study of the circuit behavior as a function of this element. Theoutput of the program is the desired unknown voltage as a ratio of polynomials. Byusing a unity current input, the input admittance or any transfer admittance may befound. The loading routine can easily be modified to handle any other linear circuitelements and more symbolic elements. Also, the expansion subroutine can be gen-eralized for use with determinants with general polynomial entries.D. U. WildeQPR No. 70 308

Advanced Computation Research

http://dspace.mit.edu/bitstream/1721.1/53889/1/RLE_QPR_070_XIX.pdf

Advanced Computation Research

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