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Y-PARAMETERS
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The terminal characteristics may also be written as in (9), where I1 and I2 are expressed in terms of V1 and V2 . I1 Y11 V1 Y12 V2 I2 Y21 V1 Y22 V2 9
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The coe cients 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
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TWO-PORT NETWORKS
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Y11 Y12 Y21 Y22
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I1 V1 V2 0 I1 V2 V1 0 I 2 V1 V2 0 I2 V
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EXAMPLE 13.4
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Find the Y-parameters of the circuit in Fig. 13-5.
Fig. 13-5
Fig. 13-6 We apply KCL to the input and output nodes (for convenience, we designate the admittances of the three branches of the circuit by Ya , Yb , and Yc as shown in Fig. 13-6). Thus, Ya 1 3 2 5s=3 5s 6 1 2 Yb 3 5s=2 5s 6 1 s Yc 5 6=s 5s 6
11
The node equations are I1 V1 Ya V1 V2 Yc Ya Yc V1 Yc V2 I2 V2 Yb V2 V1 Yc Yc V1 Yb Yc V2 By comparing (9) with (12), we get
12
TWO-PORT NETWORKS
[CHAP. 13
Y11 Ya Yc Y12 Y21 Yc Y22 Yb Yc Substituting Ya , Yb , and Yc in (11) into (13), we nd Y11 s 3 5s 6 s 5s 6 14 13
Y12 Y21 Y22 Since Y12 Y21 , the two-port circuit is reciprocal. s 2 5s 6
PI-EQUIVALENT OF RECIPROCAL NETWORKS
A reciprocal network may be modeled by its Pi-equivalent as shown in Fig. 13-6. The three elements of the Pi-equivalent network can be found by reverse solution. We rst nd the Y-parameters of Fig. 13-6. From (10) we have Y11 Ya Yc Y12 Yc Y21 Yc Y22 Yb Yc from which Ya Y11 Y12 Yb Y22 Y12 Yc Y12 Y21 16 [Fig. 13.7 a [Fig. 13-7 b [Fig. 13-7 a [Fig. 13-7 b
15
The Pi-equivalent network is not necessarily realizable.
Fig. 13-7
APPLICATION OF TERMINAL CHARACTERISTICS
The four terminal variables I1 , I2 , V1 , and V2 in a two-port network are related by the two equations (1) or (9). By connecting the two-port circuit to the outside as shown in Fig. 13-1, two additional equations are obtained. The four equations then can determine I1 , I2 , V1 , and V2 without any knowledge of the inside structure of the circuit.
CHAP. 13]
TWO-PORT NETWORKS
EXAMPLE 13.5
The Z-parameters of a two-port network are given by Z11 2s 1=s Z12 Z21 2s Z22 2s 4 Find I1 , I2 , V1 , and V2 .
The network is connected to a source and a load as shown in Fig. 13-8.
Fig. 13-8 The terminal characteristics are given by V1 2s 1=s I1 2sI2 V2 2sI1 2s 4 I2 17
The phasor representation of voltage vs t is Vs 12 V with s j. From KVL around the input and output loops we obtain the two additional equations (18) Vs 3I1 V1 0 1 s I2 V2 Substituting s j and Vs 12 in (17) and in (18) we get V1 jI1 2jI2 V2 2jI1 4 2j I2 12 3I1 V1 0 1 j I2 V2 from which I1 3:29 10:28 V1 2:88 37:58 I2 1:13 131:28 V2 1:6 93:88 18
CONVERSION BETWEEN Z- AND Y-PARAMETERS
The Y-parameters may be obtained from the Z-parameters by solving (1) for I1 and I2 . Applying Cramer s rule to (1), we get Z22 Z V1 12 V2 DZZ DZZ Z21 Z I2 V 11 V DZZ 1 DZZ 2 I1
19
where DZZ Z11 Z22 Z12 Z21 is the determinant of the coe cients in (1). By comparing (19) with (9) we have
TWO-PORT NETWORKS
[CHAP. 13
Y11 Y12 Y21 Y22
Z22 DZZ Z12 DZZ Z21 DZZ Z11 DZZ
20
Given the Z-parameters, for the Y-parameters to exist, the determinant DZZ must be nonzero. versely, given the Y-parameters, the Z-parameters are Z11 Z12 Z21 Z22 Y22 DYY Y12 DYY Y21 DYY Y11 DYY
Con-
21
where DYY Y11 Y22 Y12 Y21 is the determinant of the coe cients in (9). For the Z-parameters of a two-port circuit to be derived from its Y-parameters, DYY should be nonzero.
EXAMPLE 13.6 Referring to Example 13.4, nd the Z-parameters of the circuit of Fig. 13-5 from its Y-parameters. The Y-parameters of the circuit were found to be [see (14)] Y11 s 3 5s 6 Y12 Y21 s 5s 6 Y22 s 2 5s 6
Substituting into (21), where DYY 1= 5s 6 , we obtain Z11 s 2 Z12 Z21 s Z22 s 3 The Z-parameters in (22) are identical to the Z-parameters of the circuit of Fig. 13-2. The two circuits are equivalent as far as the terminals are concerned. This was by design. Figure 13-2 is the T-equivalent of Fig. 13-5. The equivalence between Fig. 13-2 and Fig. 13-5 may be veri ed directly by applying (6) to the Z-parameters given in (22) to obtain its T-equivalent network. 22
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