Some Network-Theoretic Properties of Nonlinear DC Transistor Networks

Several results are presented in Ref. 1 concerning the equation F(x) + Ax = B (1) (with F(-) a "diagonal" nonlinear mapping of real Euclidean n-space E" into itself, a n d A a r e a l n X n m a t r i x ) w h i c h p l a y s a c e n t r a l r o l e in the dc analysis of transistor networks. In particular, a necessary and sufficient condition on A is given such that the equation possesses a unique solution x for each real n-vector B and each strictly monotone increasing F(-) that maps En onto itself. Several circuit-theoretic implications of the results are also described in Ref. 1; for example, it is shown that the short-circuit admittance matrix of the linear portion of the dc model of an interesting class of switching circuits must violate a certain dominance condition. In Ref. 1 the word transistor was used to refer to the three-terminal device whose dc equivalent circuit is shown in Fig. 1(a). Although this equivalent circuit is frequently used in the design and computer analysis of transistor networks it is, from a physical standpoint, somewhat incomplete. A more exact dc model of a physical transistor is that of Fig. 1(h) in which the presence of scries resistance in each of the transistor's leads has been accounted for. In this paper we report on several extensions of the previous results. The motivation for much of this work was to enable the model of Fig. 1 (b) to be taken into account. In addition, we present here 1293