By Wai-Kai Chen
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Extra info for Active Network Analysis. Feedback Amplifier Theory
Denotes the set of all excitation-response pairs that can be supported by the n-port, there exist n linearly independent excitation vectors uk (t)’s in the set. 1 A linear passive n-port network that can support n linearly independent excitations for all time is also causal. PROOF. Let y(t) and y0 (t) be the responses of the excitations u(t) and u0 (t), respectively. 40) Denote by yˆ (t) the response corresponding to the excitation u ˆ (t). 42) t0 for all t (see Prob. 5). 43) negative. 45) Let uk (t), k = 1, 2, .
See the footnote on page 33. 134) for some finite complex n-vector u0 . We emphasize the difference of the concepts that an n-port is passive and that an n-port is passive at a closed RHS point s0 . 135) for all complex n-vectors u0 . 136) u0∗ Hh (s0 )u0 |s0 | 0 for all complex n-vectors u0 . As demonstrated in Eq. 131), Eq. 136) is a consequence of positive realness of H(s). However, the converse is not necessarily true. 3. 127). The definitions of passivity and activity at a single complex frequency s0 in the closed RHS suggest that for any given n-port, the closed right half of the complex frequency s-plane can be partitioned into regions of passivity and activity.
The right half of the complex s-plane will be abbreviated as RHS. Likewise, the left half of the complex s-plane will be abbreviated as LHS. The open RHS is the region defined by Re s > 0 (Re denotes the real part of), and the closed RHS is the region defined by Re s 0. When we say that A is analytic in a region, we mean that every element of A is analytic in the region. On the other hand, when we say that A has a pole at s0, we mean that at least one element of A has a pole at s0. 6: Positive-real matrix An n × n matrix function A(s) of the complex variable s is said to be a positive-real matrix if it satisfies the following three conditions: 1.
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