# Mathematics for Circuits by W. Chellingsworth

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### Introduction

A member of the Royal Society once wrote

Besides the great delight and pleasure there is in these mathematical and philosophical inquiries there is also much real benefit to be learned, particularly for such gentlemen as employing their estates in those chargeable adventures of draining mines, coal-pits, etc..

This is enough to make any mathematician throw up his hands in horror, but it is a hard fact that the elegant language of mathematics has become the medium through which the scientist and engineer express their problem in a quantitative fashion. Mining engineering may have been uppermost in the mind of our seventeenth-century philosopher but the sentiments expressed apply to every branch of applied science.

Electrical engineering is one branch in which the subject matter may be broadly divided under two main headings: the study of field theory, and the study of circuit theory. Each facet requires a sound knowledge of mathematical principles: each requires the same basic approach. First, an appraisal of the nature of the problem, second, a knowledge of the physical laws relating to the particular study in hand, third, the setting out in the mathematical form of the relationships which describe the mode of behavior of the system, and finally, the extraction of an intelligent solution.

It is in respect of this final point that the broad training in engineering is of such vital importance. In the analysis of a circuit problem, for example, it may very well be that two or more solutions will be found in calculating the value of a particular component: it may equally well be that the use of one of the possible values will result in the circuit becoming a charred mass on being put into service. In other words, at least one function of the engineer is to accept the advice offered by the mathematical solution and reject that which is quite irrelevant.

In the following text, there is an introduction to the ways in which certain well-known mathematical processes may be applied to circuit theory. An attempt is made to show how simultaneous equations, differential and integral calculus, certain forms of the differential equations, and complex numbers all find application in a study of networks. There is an indication that mechanical systems may have the same form of response as certain networks simply because the equations which describe their respective modes of behavior are identical.

There can be no great depth of study in such a short introductory text; many volumes have already been written and will continue to be written about the response of networks to certain forms of disturbance. This is simply a brief indication of the kind of tools that are necessary to any student of engineering before a deep and detailed study of the subject can be attempted.

I should like to express my gratitude to Mr. A. J. Moakes and Mr. E. H. Williams for the extremely helpful and constructive suggestions which they have made, and to Miss E. Dewdney for preparing the final draft.