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Some two-port circuits or electronic devices are best characterized by the following terminal equations: V1 h11 I1 h12 V2 I2 h21 I1 h22 V2 where the hij coe cients are called the hybrid parameters, or h-parameters.
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EXAMPLE 13.7 Find the h-parameters of Fig. 13-9. This is the simple model of a bipolar junction transistor in its linear region of operation. By inspection, the terminal characteristics of Fig. 13-9 are V1 50I1 and I2 300I1 24
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CHAP. 13]
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Fig. 13-9 By comparing (24) and (23) we get h11 50 h12 0 h21 300 h22 0 25
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The terminal characteristics of a two-port circuit may also be described by still another set of hybrid parameters given in (26). I1 g11 V1 g12 I2 V2 g21 V1 g22 I2 where the coe cients gij are called inverse hybrid or g-parameters.
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EXAMPLE 13.8 Find the g-parameters in the circuit shown in Fig. 13-10.
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Fig. 13-10 This is the simple model of a eld e ect transistor in its linear region of operation. To nd the g-parameters, we rst derive the terminal equations by applying Kirchho s laws at the terminals: At the input terminal: At the output terminal: or I1 10 9 V1 V1 109 I1 V2 10 I2 10 3 V1 and V2 10I2 10 2 V1 (28)
By comparing (27) and (26) we get g11 10 9 g12 0 g21 10 2 g22 10 28
TRANSMISSION PARAMETERS
The transmission parameters A, B, C, and D express the required source variables V1 and I1 in terms of the existing destination variables V2 and I2 . They are called ABCD or T-parameters and are de ned by
TWO-PORT NETWORKS
[CHAP. 13
V1 AV2 BI2 I1 CV2 DI2
29
EXAMPLE 13.9 Find the T-parameters of Fig. 13-11 where Za and Zb are nonzero.
Fig. 13-11 This is the simple lumped model of an incremental segment of a transmission line. V1 Z Zb a 1 Za Yb V2 I2 0 Zb V B 1 Za I2 V2 0 I C 1 Yb V2 I2 0 I D 1 1 I A
2 V2 0
From (29) we have
30
INTERCONNECTING TWO-PORT NETWORKS
Two-port networks may be interconnected in various con gurations, such as series, parallel, or cascade connection, resulting in new two-port networks. For each con guration, certain set of parameters may be more useful than others to describe the network.
Series Connection Figure 13-12 shows a series connection of two two-port networks a and b with open-circuit impedance parameters Za and Zb , respectively. In this con guration, we use the Z-parameters since they are combined as a series connection of two impedances. The Z-parameters of the series connection are (see Problem 13.10):
Fig. 13-12
CHAP. 13]
TWO-PORT NETWORKS
Z11 Z11;a Z11;b Z12 Z12;a Z12;b Z21 Z21;a Z21;b Z22 Z22;a Z22;b or, in the matrix form, Z Za Zb
31a
31b
Parallel Connection Figure 13-13 shows a parallel connection of two-port networks a and b with short-circuit admittance parameters Ya and Yb . In this case, the Y-parameters are convenient to work with. The Y-parameters of the parallel connection are (see Problem 13.11): Y11 Y11;a Y11;b Y12 Y12;a Y12;b Y21 Y21;a Y21;b Y22 Y22;a Y22;b or, in the matrix form Y Ya Yb 32b
32a
Fig. 13-13
Cascade Connection The cascade connection of two-port networks a and b is shown in Fig. 13-14. In this case the T-parameters are particularly convenient. The T-parameters of the cascade combination are
A Aa Ab Ba Cb B Aa Bb Ba Db C Ca Ab Da Cb D Ca Bb Da Db 33a
or, in the matrix form,
T Ta Tb
33b
TWO-PORT NETWORKS
[CHAP. 13
Fig. 13-14
CHOICE OF PARAMETER TYPE
What types of parameters are appropriate to and can best describe a given two-port network or device Several factors in uence the choice of parameters. (1) It is possible that some types of parameters do not exist as they may not be de ned at all (see Example 13.10). (2) Some parameters are more convenient to work with when the network is connected to other networks, as shown in Section 13.11. In this regard, by converting the two-port network to its T- and Pi-equivalent and then applying the familiar analysis techniques, such as element reduction and current division, we can greatly reduce and simplify the overall circuit. (3) For some networks or devices, a certain type of parameter produces better computational accuracy and better sensitivity when used within the interconnected circuit.
EXAMPLE 13.10 Find the Z- and Y-parameters of Fig. 13-15.
Fig. 13-15 We apply KVL to the input and output loops. Thus, Input loop: Output loop: or By comparing (34) and (2) we get Z11 6 Z12 3 Z21 10 Z22 5 V1 3I1 3 I1 I2 V2 7I1 2I2 3 I1 I2 V1 6I1 3I2 and V2 10I1 5I2 (34)
The Y-parameters are, however, not de ned, since the application of the direct method of (10) or the conversion from Z-parameters (19) produces DZZ 6 5 3 10 0.
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