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13.12 Find (a) the Z-parameters of the circuit of Fig. 13-23(a) and (b) an equivalent model which uses three positive-valued resistors and one dependent voltage source.
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Fig. 13-23 (a) From application of KVL around the input and output loops we nd, respectively, V1 2I1 2I2 2 I1 I2 4I1 V2 3I2 2 I1 I2 2I1 5I2 The Z-parameters are Z11 4, Z12 0, Z21 2, and Z22 5. (b) The circuit of Fig. 13-23(b), with two resistors and a voltage source, has the same Z-parameters as the circuit of Fig. 13-23(a). This can be veri ed by applying KVL to its input and output loops.
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13.13 (a) Obtain the Y-parameters of the circuit in Fig. 13-23(a) from its Z-parameters. (b) Find an equivalent model which uses two positive-valued resistors and one dependent current source.
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(a) From Problem 13.12, Z11 4, Z12 0, Z21 2; Z22 5, and so DZZ Z11 Z22 Z12 Z21 20. Hence, Y11 Z22 5 1 DZZ 20 4 Y12 Z12 0 DZZ Y21 Z21 2 1 20 10 DZZ Y22 Z11 4 1 DZZ 20 5
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(b) Figure 13-24, with two resistors and a current source, has the same Y-parameters as the circuit in Fig. 13-23(a). This can be veri ed by applying KCL to the input and output nodes.
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13.14 Referring to the network of Fig. 13-23(b), convert the voltage source and its series resistor to its Norton equivalent and show that the resulting network is identical with that in Fig. 13-24.
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The Norton equivalent current source is IN 2I1 =5 0:4I1 . But I1 V1 =4. Therefore, IN 0:4I1 0:1V1 . The 5-
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resistor is then placed in parallel with IN . The circuit is shown in Fig. 13-25 which is the same as the circuit in Fig. 13-24.
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Fig. 13-24
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Fig. 13-25
TWO-PORT NETWORKS
[CHAP. 13
13.15 The h-parameters of a two-port network are given. Show that the network may be modeled by the network in Fig. 13-26 where h11 is an impedance, h12 is a voltage gain, h21 is a current gain, and h22 is an admittance.
Fig. 13-26 Apply KVL around the input loop to get V1 h11 I1 h12 V2 Apply KCL at the output node to get I2 h21 I1 h22 V2 These results agree with the de nition of h-parameters given in (23).
13.16 Find the h-parameters of the circuit in Fig. 13-25.
By comparing the circuit in Fig. 13-25 with that in Fig. 13-26, we nd h11 4
; h12 0; h21 0:4; h22 1=5 0:2
13.17 Find the h-parameters of the circuit in Fig. 13-25 from its Z-parameters and compare with results of Problem 13.16.
Refer to Problem 13.13 for the values of the Z-parameters and DZZ . Use Table 13-1 for the conversion of the Z-parameters to the h-parameters of the circuit. Thus, h11 DZZ 20 4 5 Z22 h12 Z12 0 Z22 h21 Z21 2 0:4 5 Z22 h22 1 1 0:2 Z22 5
The above results agree with the results of Problem 13.16.
13.18 The simpli ed model of a bipolar junction transistor for small signals is shown in the circuit of Fig. 13-27. Find its h-parameters.
Fig. 13-27 The terminal equations are V1 0 and I2 I1 . By comparing these equations with (23), we conclude that h11 h12 h22 0 and h21 .
CHAP. 13]
TWO-PORT NETWORKS
13.19 h-parameters of a two-port device H are given by h11 500
h12 10 4 h21 100 h22 2 10 6
Draw a circuit model of the device made of two resistors and two dependent sources including the values of each element.
From comparison with Fig. 13-26, we draw the model of Fig. 13-28.
Fig. 13-28
13.20 The device H of Problem 13-19 is placed in the circuit of Fig. 13-29(a). Replace H by its model of Fig. 13-28 and nd V2 =Vs .
Fig. 13-29
TWO-PORT NETWORKS
[CHAP. 13
The circuit of Fig. 13-29(b) contains the model. 13-29(c) from which I1 Vs =2000 Thus, V2 =Vs 50.
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