Mathematics for Circuits

The reader probably met simultaneous equations at first in examples of this type :

‘If 5x +3y = 11 and 9x − 2y = 5 determine the values of x and y.’

Ask a student of calculus in the early stages of his study how this branch of mathematics may be used, and it is likely that he will think first of the finding of maximum or minimum values of variable quantities. Our previous work will immediately afford a simple example of this.

One of the difficulties about the analysis of circuits is that it is not at all easy to visualise what is going on in the system under consideration. This is not quite so much the case with mechanical devices, where the ‘if you push here, this much will happen there’approach helps one to form a mental picture of the problem.

Reference has been made, in dealing with oscillatory systems, to quantities which vary sinusoidally with time at a certain frequency. This form of variation is extremely common; the vast majority of domestic and industrial supply voltages, for example, have a waveform which can be expressed in mathematical terms as where ω is a constant known as the angular frequency of the voltage wave. (Figure 4.1.) From the diagram it can be seen that the voltage reaches its maximum value when the time t ₁ is such that or when .

A brief reference was made (Section 4.5) to the fact that there is a special technique for manipulating quantities which vary sinusoidally with time. It makes use of the concept that such a quantity may be completely represented by a vector rotating at a constant speed. The diagram (Figure 5.1) shows a line of length V _m rotating at an angular speed ω radians per second. If time t is measured from the datum X′X the projection of the line on to the vertical axis Y′Y at any instant will be V _m sin ωt. If V _m represents the maximum amplitude of a voltage wave it follows that the value of voltage at any instant will be completely defined by the position of the line. The whole wave-shape may be developed from a sequence of lines spaced by equal time intervals (Figure 5.2). Developing this idea further, let us consider two sine waves displaced in time

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The operator i = √(−l) is a useful mathematical concept which finds application in alternating current circuit theory. However, in the study of networks the symbol i is already much in demand and usually denotes instantaneous current. In order to avoid confusion, therefore, the notation for √( −1) will henceforth be j and not i.

A serious study of any of the specialist subjects in the field of electrical engineering requires a thorough knowledge of circuit theory and the associated mathematics. These specialist studies are concerned with the design and performance of electrical machines, the interconnexion of power supply systems, the design and use of electronic devices, telecommunication systems, special methods of measurement, and the design of control systems which meet a certain requirement.