By Wai-Kai Chen
Energetic community and suggestions Amplifier concept
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Extra info for Active Network Analysis. Feedback Amplifier Theory
The first two conditions are relatively easy to check. The third constraint is somewhat difficult to verify, because the nonnegative definiteness of the hermitian part must be investigated for all s in the open RHS. In the case where a matrix is rational, each element being the ratio of two polynomials, an equivalent set of conditions can be stated that reduces the test for the points in the vast open RHS to the points on the boundary, the real-frequency axis ( j w-axis). Its justification can be found in Chen (1976).
161) where u1 and u2 are real n-vectors. 162) because u1 Hh (σ0 )u2 = u2 Hh (σ0 )u1 . 159). 160) does not mean that their values are equal. 163) Consider, for example, the case where H(σ0 ) = 12 , the identity matrix of order 2. Choosing u = [ j, 1] shows that the left-hand side is 2 whereas the right-hand side is 0. 5 On the real-frequency axis or the positive σ -axis, a linear timeinvariant n-port network is passive at a point s0 = j ω0 or s0 = σ0 if and only if the hermitian part of its associated general hybrid matrix is nonnegative definite at s0 .
131) will now be employed in characterizing the singlefrequency behavior of a linear, time-invariant n-port network. 132) where Hh (s) and Hs (s) denote the hermitian part and the symmetric part of H(s), respectively, and s0 = σ0 + j ω0 . 132) is defined to be zero for s0 = 0. For s0 = σ0 , Eq. 132) is to be taken for all finite nonzero real n-vectors u0 . The statement of the definition of passivity at a single point s0 is not really meaningful for real networks, because passivity must always occur at the complex conjugate pair of the complex frequencies s0 and s 0 .
Active Network Analysis. Feedback Amplifier Theory by Wai-Kai Chen