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![EXAMPLE 13.1 Find the Z-parameters of the two-port circuit in Fig. 13-2.
Apply KVL around the two loops in Fig. 13-2 with loop currents I1 and I2 to obtain
V1 ¼ 2I1 þ sðI1 þ I2Þ ¼ ð2 þ sÞI1 þ sI2
V2 ¼ 3I2 þ sðI1 þ I2Þ ¼ sI1 þ ð3 þ sÞI2
ð3Þ
By comparing (1) and (3), the Z-parameters of the circuit are found to be
Z11 ¼ s þ 2
Z12 ¼ Z21 ¼ s
Z22 ¼ s þ 3
ð4Þ
Note that in this example Z12 ¼ Z21.
Reciprocal and Nonreciprocal Networks
A two-port network is called reciprocal if the open-circuit transfer impedances are equal;
Z12 ¼ Z21. Consequently, in a reciprocal two-port network with current I feeding one port, the
open-circuit voltage measured at the other port is the same, irrespective of the ports. The voltage is
equal to V ¼ Z12I ¼ Z21I. Networks containing resistors, inductors, and capacitors are generally
reciprocal. Networks that additionally have dependent sources are generally nonreciprocal (see
Example 13.2).
EXAMPLE 13.2 The two-port circuit shown in Fig. 13-3 contains a current-dependent voltage source. Find its
Z-parameters.
As in Example 13.1, we apply KVL around the two loops:
V1 ¼ 2I1 À I2 þ sðI1 þ I2Þ ¼ ð2 þ sÞI1 þ ðs À 1ÞI2
V2 ¼ 3I2 þ sðI1 þ I2Þ ¼ sI1 þ ð3 þ sÞI2
CHAP. 13] TWO-PORT NETWORKS 311
Fig. 13-2
Fig. 13-3](https://image.slidesharecdn.com/twoports-140718022754-phpapp01/85/Two-ports-18-320.jpg)
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Two ports
This document discusses different parameter representations for two-port networks: - Z-parameters (open-circuit impedance parameters) relate terminal voltages to currents. - Y-parameters (short-circuit admittance parameters) relate terminal currents to voltages. - T-equivalent and Pi-equivalent models provide lumped-element representations of reciprocal networks. - h-parameters and g-parameters describe the terminal characteristics of some electronic devices. - Transmission (ABCD) parameters express source variables in terms of destination variables. Conversion formulas are provided between the different parameter sets. Examples are included to illustrate determining the parameters for simple circuits.
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Two ports
- 1. 310 Two-Port Networks 13.1 TERMINALSAND PORTS In a two-terminal network, the terminal voltage is related to the terminal current by the impedance Z ¼ V=I. In a four-terminal network, if each terminal pair (or port) is connected separately to another circuit as in Fig. 13-1, the four variables i1, i2, v1, and v2 are related by two equations called the terminal characteristics. These two equations, plus the terminal characteristics of the connected circuits, provide the necessary and sufficient number of equations to solve for the four variables. 13.2 Z-PARAMETERS The terminal characteristics of a two-port network, having linear elements and dependent sources, may be written in the s-domain as V1 ¼ Z11I1 þ Z12I2 V2 ¼ Z21I1 þ Z22I2 ð1Þ The coefficients Zij have the dimension of impedance and are called the Z-parameters of the network. The Z-parameters are also called open-circuit impedance parameters since they may be measured at one terminal while the other terminal is open. They are Z11 ¼ V1 I1
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- 17. I1¼0 ð2Þ Fig. 13-1 Copyright 2003,1997, 1986, 1965 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
- 18. EXAMPLE 13.1 Findthe Z-parameters of the two-port circuit in Fig. 13-2. Apply KVL around the two loops in Fig. 13-2 with loop currents I1 and I2 to obtain V1 ¼ 2I1 þ sðI1 þ I2Þ ¼ ð2 þ sÞI1 þ sI2 V2 ¼ 3I2 þ sðI1 þ I2Þ ¼ sI1 þ ð3 þ sÞI2 ð3Þ By comparing (1) and (3), the Z-parameters of the circuit are found to be Z11 ¼ s þ 2 Z12 ¼ Z21 ¼ s Z22 ¼ s þ 3 ð4Þ Note that in this example Z12 ¼ Z21. Reciprocal and Nonreciprocal Networks A two-port network is called reciprocal if the open-circuit transfer impedances are equal; Z12 ¼ Z21. Consequently, in a reciprocal two-port network with current I feeding one port, the open-circuit voltage measured at the other port is the same, irrespective of the ports. The voltage is equal to V ¼ Z12I ¼ Z21I. Networks containing resistors, inductors, and capacitors are generally reciprocal. Networks that additionally have dependent sources are generally nonreciprocal (see Example 13.2). EXAMPLE 13.2 The two-port circuit shown in Fig. 13-3 contains a current-dependent voltage source. Find its Z-parameters. As in Example 13.1, we apply KVL around the two loops: V1 ¼ 2I1 À I2 þ sðI1 þ I2Þ ¼ ð2 þ sÞI1 þ ðs À 1ÞI2 V2 ¼ 3I2 þ sðI1 þ I2Þ ¼ sI1 þ ð3 þ sÞI2 CHAP. 13] TWO-PORT NETWORKS 311 Fig. 13-2 Fig. 13-3
- 19. The Z-parameters are Z11¼ s þ 2 Z12 ¼ s À 1 Z21 ¼ s Z22 ¼ s þ 3 ð5Þ With the dependent source in the circuit, Z12 6¼ Z21 and so the two-port circuit is nonreciprocal. 13.3 T-EQUIVALENT OF RECIPROCAL NETWORKS A reciprocal network may be modeled by its T-equivalent as shown in the circuit of Fig. 13-4. Za, Zb, and Zc are obtained from the Z-parameters as follows. Za ¼ Z11 À Z12 Zb ¼ Z22 À Z21 Zc ¼ Z12 ¼ Z21 ð6Þ The T-equivalent network is not necessarily realizable. EXAMPLE 13.3 Find the Z-parameters of Fig. 13-4. Again we apply KVL to obtain V1 ¼ ZaI1 þ ZcðI1 þ I2Þ ¼ ðZa þ ZcÞI1 þ ZcI2 V2 ¼ ZbI2 þ ZcðI1 þ I2Þ ¼ ZcI1 þ ðZb þ ZcÞI2 ð7Þ By comparing (1) and (7), the Z-parameters are found to be Z11 ¼ Za þ Zc Z12 ¼ Z21 ¼ Zc Z22 ¼ Zb þ Zc ð8Þ 13.4 Y-PARAMETERS The terminal characteristics may also be written as in (9), where I1 and I2 are expressed in terms of V1 and V2. I1 ¼ Y11V1 þ Y12V2 I2 ¼ Y21V1 þ Y22V2 ð9Þ The coefficients Yij have the dimension of admittance and are called the Y-parameters or short-circuit admittance parameters because they may be measured at one port while the other port is short-circuited. The Y-parameters are 312 TWO-PORT NETWORKS [CHAP. 13 Fig. 13-4
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