Revisiting Matrix Theory and Electric Circuit Analysis

Matrix Theory has long been applied to many branches of engineering. However, numerical difficulties have limited access to students, especially at the elementary level. This limitation has partially been removed with the development of canned software and modern scientific calculators which operate matrices. However, the approach with matrices ignores this and still follows traditional ways. This work presents an alternative view of matrix applications to electric circuit analysis, linking matrices and equations with circuit solutions, with heavy emphasis on appropriate setting of equations and interpretation of results. As a result, a more effective use is made to obtain applications in areas that usually require more work or symbolic analysis, such as when dealing with symbolic sources, equivalent port representations, two ports, etc. Index Terms – Matrix analysis, Circuit Theory, Systems education INTRODUCTION Matrices have been used in circuit analysis for a long time [1-3]. Yet, matrices have been more useful for “advanced” courses or for computed aided oriented books [4, 5]. Popular introductory circuit books usually make limited use of them when coming to the mathematical analysis of the circuits. The main emphasis is either for the solution of a set of equations using Cramer’s or determinants [6, 7] or mainly for preparing the equations for use in Matlab® [8]. Gottling [9] focuses more on the use of matrices, but emphasis is definitely on Matlab programming. Almost all teachers would agree that the use of matrices in the introductory courses, without more than a precalculus level treatment, would focus in the solution of the set of linear equations, and this is in fact how matrices are used at this level and will certainly be most of the time. Properties of matrices are also linked with some properties of circuits, although this relationship is seldom considered. The main problem with matrices in introductory courses is of numerical nature since hand and pencil analysis becomes usually more complicated, except for simple cases; Cramer’s rule is impractical for more than three equations. Software tools such as Matlab®, Mathematica® or MathCad® alleviate, but do not necessarily solve the problem, since there are still practical problems associated for home study such as the program availability, the computer, and so on. On the other side, modern graphical calculators such as the TI-89TM, HP49TM and many others have matrix operations included, so students count now with a computation capacity allowing them to focus on the solution of the equations as easily as with the use of expensive software and laptops. However, the power of such tools can only be appreciated at elementary courses if we reconsider the way matrices are used in circuit analysis. True, the main use of matrices at this level will still be the solution of equations. Yet, the equations have more power than usually thought when properly considered and the solutions adequately interpreted. Most texts use matrices as an intermediate tool in a solution process which becomes invisible at the moment the circuit includes elements such as symbolic sources or multiple sources as functions of time in resistive circuits, etc. In these situations, the student without access to calculators or programs with symbolic manipulation power becomes lost for circuits with a little more complexity. The purpose of this work is to look at elementary matrix theory, or better, at the use of matrices to solve equations, under a renewed perspective, which puts at the reach of early classes in circuits the tools of matrices in a formal and useful way. For all practical purposes, the approach given here is extendable to any linear system, mechanical, electrical, etc. No new theorem or theory is introduced, only an approach from a new perspective. Among features of this approach attractive for using matrices early we can mention: • There is no need to consider complicated notations such as sub matrices and operations with them. [3,4,10] • Only numerical operations are involved, so there is no need for programs or calculators with symbolic capabilities. • There is a direct and immediate connection between theoretical concepts in linear circuits and how equations are set up and solutions interpreted. The theory and examples presented in this paper are limited to resistive circuits due to space limitations. The method and numerical procedures are of course extendable to the phasor domain. Other topics that stem from an extension of this approach are left outside this paper. The setting of equations directly using characteristics of equations is not considered, since the rules are already popular and can be looked elsewhere, like in [7, p. 90, p. 105]