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52) This completes the proof of the theorem. The ideal one-port short circuit is noncausal under voltage excitation-current response because it admits an infinitude of responses for the sole excitation September 1, 2016 10:33 Active Network Analysis: Feedback …– 9in x 6in b2428-ch01 page 21 CHARACTERIZATIONS OF NETWORKS 21 v(t) = 0. It fails the sufficient condition of the above theorem in that the oneport does not support any independent excitation. However, it is causal under current excitation-voltage response because at each instant of time all the current excitations are related by a multiplicative constant, meaning that the ideal oneport short circuit supports one independent excitation, thus fulfilling the sufficient requirement of the theorem.

It is clear that if any one of them is singular, the conversion is not possible, and the corresponding set of parameters does not exist. For example, the ideal transformer of Fig. 15 does not possess the impedance, admittance, or transmission representation except the hybrid representation of Eq. 35) or its inverse. In addition to the port characterizations of a network, it is sometimes advantageous to consider a network with n accessible terminals. Terminal characterization of a network and its applications will be considered in the following chapter.

4 Let y(t) be the response of an excitation u(t) of an n-port network N. 97) t0 for all initial time t0 and for all excitations u(t). The condition is clearly necessary, because Eq. 10) must be true for all time t and, in particular, for t = ∞, and because u (t)y(t) = v (t)i(t). To show that Eq. 97) also implies Eq. 10) is negative for some t = t1 . 97) will also be negative, contradicting our hypothesis. 10) must be nonnegative for all t t0 . Without loss of generality, we assume that the n-port under consideration is initially relaxed.

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