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By Wai-Kai Chen

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**Additional info for Active Network Analysis. Feedback Amplifier Theory**

**Example text**

92b) For Hh ( j ω) to be nonnegative definite, it is necessary and sufficient that all of its principal minors are nonnegative (App. I). 93c) By assuming that G 1 , G 2 , C1 , C2 , and gm are all nonnegative, Eqs. 93b) are clearly satisfied, because G 1 C2 − C1 C2 σ0 0. 94) 2 , Eq. 6, if 4G 1 G 2 gm h nonnegative definite. Consequently, the hybrid matrix H(s) is positive real. If 2 , Eq. 94) can be violated for any value of ω on the real-frequency 4G 1 G 2 < gm axis, indicating that Hh ( j ω) is not nonnegative definite and H(s) is not positive real.

We illustrate the above results by the following example. 4 Consider the high-frequency equivalent network of a bipolar transistor as shown in Fig. 17. Let u(s) ˜ = [I1 (s), V2 (s)] and y˜ (s) = V1 (s), I2 (s) , where Vk (s) and Ik (s) are the transforms of vk (t) and i k (t), respectively. 17 A high-frequency equivalent network of a bipolar transistor. September 1, 2016 10:33 Active Network Analysis: Feedback …– 9in x 6in b2428-ch01 page 25 CHARACTERIZATIONS OF NETWORKS 25 where G 1 = 1/R1 . They are obtained from the network of Fig.

Likewise, we compute h 12 (s) and h 22 (s). The resulting hybrid matrix is given by (see Prob. 67) we let u(s) ˜ = [I1 (s), I2 (s)] and y˜ (s) = [V1 (s), V2 (s)] From Eq. 68b) which are obtained from the network of Fig. 17 by open-circuiting the output port. In a similar manner, we can compute z 12 (s) and z 22 (s). 70) we let u(s) ˜ = [V1 (s), V2 (s)] and y˜ (s) = [I1 (s), I2 (s)] . From Eq. 71b) September 1, 2016 10:33 Active Network Analysis: Feedback …– 9in x 6in b2428-ch01 page 26 26 ACTIVE NETWORK ANALYSIS They are obtained by short-circuiting the output port of the two-port network of Fig.