By N. Balabanian; T.A. Bickart; S. Seshu
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This is done b y establishing a reference for each voltage and current. The functions i(t) and υ(t) are real functions of time that can take on negative as well as positive values in the course of time. The system of references adopted in this book is shown in Fig. 1. An arrow Fig. 1. Current and voltage references. indicates the reference for t h e current in a branch. This arrow does not mean that t h e current is always in the arrow direction. It means that, whenever the current is in the arrow direction, i(t) will be positive.
1 Any network that does not satisfy this condition is called an active c network; that is, v(x) i(x) dx < 0 for some time t. l ^ - 00 If the network has more than one pair of terminals through which energy can be supplied from the outside, let the terminal voltage and current matrices be with standard references. The instantaneous power supplied to the net work from the outside will then be (65) The network will be passive if, for all t, (66) RECIPROCITY Some networks have the property that the response produced at one point of the network by an excitation at another point is invariant if the positions of excitation and response are interchanged (excitation and response being properly interpreted).
2] ELEMENTARY MATRIX ALGEBRA 25 For the special case of homogeneous equations, namely the case where y = 0 , it should be observed from (45) that a nontrivial solution exists only if the nullity is nonzero. , when A is a square matrix), the nullity is nonzero and a nontrivial solu tion exists only if A is singular. To illustrate the preceding, consider the following set of equations: W e observe that the first four rows and columns 2, 4, 6, and 8 of A form a unit matrix (which is nonsingular) and so rank A > 4.