1.8 combination of two port networks
DESCRIPTION
niceTRANSCRIPT
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Two-Port Networks
13.1 TERMINALS AND PORTS
In a two-terminal network, the terminal voltage is related to the terminal current by the impedance
Z 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 variablesi1,i2,v1, andv2 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
Z11
I1
Z12
I2
V2 Z21I1 Z22I21
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 V1
I1
I20
Z12 V1
I2
I10
Z21 V2
I1
I20
Z22 V2
I2
I10
2
Fig. 13-1
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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 2I1 sI1 I2 2 sI1 sI2
V2 3I2 sI1 I2 sI1 3 sI23
By comparing (1) and (3), the Z-parameters of the circuit are found to be
Z11s 2
Z12Z21s
Z22s 3
4
Note that in this example Z12Z21.
Reciprocal and Nonreciprocal Networks
A two-port network is called reciprocal if the open-circuit transfer impedances are equal;
Z12 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 Z12I 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 2I1 I2 sI1 I2 2 sI1 s 1I2
V2 3I2 sI1 I2 sI1 3 sI2
CHAP. 13] TWO-PORT NETWORKS 311
Fig. 13-2
Fig. 13-3
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The Z-parameters are
Z11s 2
Z12s 1
Z21s
Z22s 3
5
With the dependent source in the circuit, Z126Z21 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 Z11 Z12
Zb Z22 Z21
Zc Z12 Z21
6
The T-equivalent network is not necessarily realizable.
EXAMPLE 13.3 Find theZ-parameters of Fig. 13-4.
Again we apply KVL to obtain
V1 ZaI1 ZcI1 I2 Za ZcI1 ZcI2
V2 ZbI2 ZcI1 I2 ZcI1 Zb ZcI27
By comparing (1) and (7), the Z-parameters are found to be
Z11 Za Zc
Z12 Z21Zc
Z22 Zb Zc
8
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 Y11V1 Y12V2
I2 Y21V1 Y22V29
The coefficients Yijhave the dimension of admittance and are called the Y-parameters or short-circuit
admittance parametersbecause 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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Y11 I1
V1
V20
Y12 I1
V2
V10
Y21 I2
V1
V20
Y22 I2
V2
V10
10
EXAMPLE 13.4 Find the Y-parameters of the circuit in Fig. 13-5.
We apply KCL to the input and output nodes (for convenience, we designate the admittances of the threebranches of the circuit by Ya, Yb, and Yc as shown in Fig. 13-6). Thus,
Ya 1
2 5s=3
3
5s 6
Yb 1
3 5s=2
2
5s 6
Yc 1
5 6=s
s
5s 6
11
The node equations are
I1 V1Ya V1 V2Yc Ya YcV1 YcV2
I2 V2Yb V2 V1Yc YcV1 Yb YcV212
By comparing (9) with (12), we get
CHAP. 13] TWO-PORT NETWORKS 313
Fig. 13-5
Fig. 13-6
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Y11 Ya Yc
Y12 Y21 Yc
Y22 Yb Yc
13
SubstitutingYa, Yb, and Yc in (11) into (13), we find
Y11 s 3
5s 6
Y12 Y21 s
5s 6
Y22 s 2
5s 6
14
SinceY12 Y21, the two-port circuit is reciprocal.
13.5 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 first find theY-parameters
of Fig. 13-6. From (10) we have
Y11 Ya Yc [Fig. 13.7a
Y12 Yc [Fig. 13-7b
Y21 Yc [Fig. 13-7a
Y22 Yb Yc [Fig. 13-7b
15
from which
Ya Y11 Y12 Yb Y22 Y12 Yc Y12 Y21 16
The Pi-equivalent network is not necessarily realizable.
13.6 APPLICATION OF TERMINAL CHARACTERISTICS
The four terminal variablesI1,I2,V1, andV2 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 knowl-
edge of the inside structure of the circuit.
314 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-7
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EXAMPLE 13.5 The Z-parameters of a two-port network are given by
Z112s 1=s Z12 Z212s Z22 2s 4
The network is connected to a source and a load as shown in Fig. 13-8. Find I1, I2, V1, and V2.
The terminal characteristics are given by
V1 2s 1=sI1 2sI2
V2 2sI1 2s 4I217
The phasor representation of voltage vstis 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 sI2 V218
Substituting s j and Vs 12 in (17) and in (18) we get
V1 jI1 2jI2
V2 2jI1 4 2jI2
12 3I1 V1
0 1 jI2 V2
from which
I1 3:29 10:28 I2 1:13 131:28
V1 2:88 37:58 V2 1:6 93:88
13.7 CONVERSION BETWEEN Z- AND Y-PARAMETERS
The Y-parameters may be obtained from the Z-parameters by solving (1) for I1 and I2. Applying
Cramers rule to (1), we get
I1
Z22
DZZ V1
Z12
DZZ V2
I2 Z21
DZZV1
Z11
DZZV2
19
where DZZ Z11Z22 Z12Z21 is the determinant of the coefficients in (1). By comparing (19) with (9)
we have
CHAP. 13] TWO-PORT NETWORKS 315
Fig. 13-8
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Y11 Z22
DZZ
Y12 Z12
DZZ
Y21 Z21
DZZ
Y22 Z11
DZZ
20
Given the Z-parameters, for the Y-parameters to exist, the determinant DZZ must be nonzero. Con-
versely, given the Y-parameters, the Z-parameters are
Z11 Y22
DYY
Z12 Y12
DYY
Z21 Y21
DYY
Z22 Y11
DYY
21
where DYY Y11Y22 Y12Y21 is the determinant of the coefficients in (9). For theZ-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, find 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 Z21s
Z22 s 3
22
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 verified directly by applying ( 6) to theZ-parameters given in(22) to obtain its T-equivalent network.
13.8 h-PARAMETERS
Some two-port circuits or electronic devices are best characterized by the following terminal
equations:
V1 h11I1 h12V2
I2 h21I1 h22V223
where the hijcoefficients are called the hybrid parameters, or h-parameters.
EXAMPLE 13.7 Find theh-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
316 TWO-PORT NETWORKS [CHAP. 13
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By comparing (24) and (23) we get
h1150 h120 h21300 h22 0 25
13.9 g-PARAMETERS
The terminal characteristics of a two-port circuit may also be described by still another set of hybrid
parameters given in (26).
I1 g11V1 g12I2
V2 g21V1 g22I226
where the coefficients gijare called inverse hybridor g-parameters.
EXAMPLE 13.8 Find the g-parameters in the circuit shown in Fig. 13-10.
This is the simple model of a field effect transistor in its linear region of operation. To find theg-parameters,
we first derive the terminal equations by applying Kirchhoffs laws at the terminals:
V1 109I1At the input terminal:
V2 10I2 103V1At the output terminal:
or I1 109V1 and V2 10I2 10
2V1 (28)
By comparing (27) and (26) we get
g11 109 g120 g21 10
2 g2210 28
13.10 TRANSMISSION PARAMETERS
The transmission parametersA,B,C, andD express the required source variables V1and I1 in terms
of the existing destination variablesV2 andI2. They are calledABCDor T-parameters and are defined
by
CHAP. 13] TWO-PORT NETWORKS 317
Fig. 13-9
Fig. 13-10
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V1 AV2 BI2
I1 CV2 DI229
EXAMPLE 13.9 Find the T-parameters of Fig. 13-11 where Za and Zb are nonzero.
This is the simple lumped model of an incremental segment of a transmission line. From (29) we have
AV1
V2
I2 0
Za Zb
Zb1 ZaYb
B V1
I2
V2 0
Za
C I1
V2
I2 0
Yb
D I1
I2
V2 0
1
30
13.11 INTERCONNECTING TWO-PORT NETWORKS
Two-port networks may be interconnected in various configurations, such as series, parallel, or
cascade connection, resulting in new two-port networks. For each configuration, 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 configuration, we use the Z-parameters since
they are combined as a series connection of two impedances. TheZ-parameters of the series connection
are (see Problem 13.10):
318 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-11
Fig. 13-12
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Z11 Z11;a Z11;b
Z12 Z12;a Z12;b
Z21 Z21;a Z21;b
Z22 Z22;a Z22;b
31a
or, in the matrix form,
Z Za Zb 31b
Parallel Connection
Figure 13-13 shows a parallel connection of two-port networks a andb with short-circuit admittance
parametersYa andYb. In this case, theY-parameters are convenient to work with. TheY-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
32a
or, in the matrix form
Y Ya Yb 32b
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 AaAb BaCb
B AaBb BaDb
C CaAb DaCb
D CaBb DaDb
33a
or, in the matrix form,
T TaTb 33b
CHAP. 13] TWO-PORT NETWORKS 319
Fig. 13-13
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13.12 CHOICE OF PARAMETER TYPE
What types of parameters are appropriate to and can best describe a given two-port network or
device? Several factors influence the choice of parameters. (1) It is possible that some types of
parameters do not exist as they may not be defined 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 theZ- and Y-parameters of Fig. 13-15.
We apply KVL to the input and output loops. Thus,
V1 3I1 3I1 I2Input loop:
V2 7I1 2I2 3I1 I2Output loop:
or V1 6I1 3I2 and V2 10I1 5I2 (34)
By comparing (34) and (2) we get
Z116 Z123 Z2110 Z22 5
The Y-parameters are, however, not defined, since the application of the direct method of ( 10) or the conversion
from Z-parameters (19) produces DZZ65 310 0.
13.13 SUMMARY OF TERMINAL PARAMETERS AND CONVERSION
Terminal parameters are defined by the following equations
Z-parameters h-parameters T-parameters
V1 Z11I1 Z12I2 V1 h11I1 h12V2 V1 AV2 BI2V2 Z21I1 Z22I2 I2 h21I1 h22V2 I1 CV2 DI2
V ZI
Y-parameters g-parameters
I1 Y11V1 Y12V2 I1 g11V1 g12I2I2 Y21V1 Y22V2 V2 g21V1 g22I2I YV
320 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-14
Fig. 13-15
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Table 13-1 summarizes the conversion between the Z-, Y-, h-, g-, and T-parameters. For the
conversion to be possible, the determinant of the source parameters must be nonzero.
Solved Problems
13.1 Find the Z-parameters of the circuit in Fig. 13-16(a).
Z11 and Z21 are obtained by connecting a source to port #1 and leaving port #2 open [Fig. 13-16(b)].
The parallel and series combination of resistors produces
Z11V1
I1
I2 0
8 and Z21V2
I1
I2 0
1
3
Similarly, Z22
and Z12
are obtained by connecting a source to port #2 and leaving port #1 open [Fig.
13-16(c)].
Z22V2
I2
I10
8
9 Z12
V1
I2
I1 0
1
3
The circuit is reciprocal, since Z12Z21.
CHAP. 13] TWO-PORT NETWORKS 321
Table 13-1
Z Y h g T
Z
Z11 Z12 Y22
DYY
Y12DYY
Dhh
h22
h12
h22
1
g11
g12g11
A
C
DTT
C
Z21 Z22 Y21DYY
Y11
DYY
h21h22
1
h22
g21g11
Dgg
g11
1
C
D
C
Y
Z22
Dzz
Z12Dzz
Y11 Y12 1
h11
h12h11
Dgg
g22
g12g22
D
B
DTTB
Z21
Dzz
Z11
Dzz
Y21 Y22 h21
h11
Dnn
h11
g21
g22
1
g22
1
B
A
B
h
Dzz
Z22
Z12
Z22
1
Y11
Y12Y11
h11 h12 g22
Dgg
g12Dgg
B
D
DTT
D
Z21Z22
1
Z22
Y21
Y11
Dyy
Y11
h21 h22 g21
Dgg
g11Dgg
1
D
C
D
g
1
Z11
Z12Z11
DYY
Y22
Y12
Y22
h22
Dhh
h12Dhh
g11 g12 C
A
DTTA
Z21
Z11
DZZ
Z11
Y21Y22
1
Y22
h21Dhh
h11
Dhh
g21 g22 1
A
B
A
T
Z11Z21
DZZZ21
Y22Y21
1Y21
Dhhh21
h11h21
1g21
g22g21
A B
1
Z21
Z22
Z21
DYYY21
Y11Y21
h22h21
1
h21
g11g21
Dgg
g21
C D
DPP P11P22 P12P21 is the determinant ofZ; Y; h; g; or T-parameters.
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13.2 The Z-parameters of a two-port network N are given by
Z11 2s 1=s Z12 Z21 2s Z22 2s 4
(a) Find the T-equivalent ofN. (b) The networkNis connected to a source and a load as shown
in the circuit of Fig. 13-8. Replace N by its T-equivalent and then solve for i1, i2, v1, and v2.
(a) The three branches of the T-equivalent network (Fig. 13-4) are
Za Z11 Z122s 1
s 2s
1
s
Zb Z22 Z122s 4 2s 4
Zc Z12 Z212s
(b) The T-equivalent ofN, along with its input and output connections, is shown in phasor domain in Fig.
13-17.
322 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-16
Fig. 13-17
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By applying the familiar analysis techniques, including element reduction and current division, to
Fig. 13-17, we find i1, i2, v1, and v2.
In phasor domain In the time domain:
I1 3:29 10:28 i1 3:29cos t 10:28
I2 1:13 131:28 i2 1:13cos t 131:28V1 2:88 37:58 v1 2:88cos t 37:58
V2 1:6 93:88 v2 1:6cos t 93:88
13.3 Find the Z-parameters of the two-port network in Fig. 13-18.
KVL applied to the input and output ports obtains the following:
V1 4I1 3I2 I1 I2 5I1 2I2Input port:
V2 I2 I1 I2 I1 2I2Output port:
By applying (2) to the above, Z11
5, Z12
2, Z21
1, and Z22
2:
13.4 Find theZ-parameters of the two-port network in Fig. 13-19 and compare the results with those
of Problem 13.3.
KVL gives
V1 5I1 2I2 and V2 I1 2I2
The above equations are identical with the terminal characteristics obtained for the network of Fig.
13-18. Thus, the two networks are equivalent.
13.5 Find the Y-parameters of Fig. 13-19 using its Z-parameters.
From Problem 13.4,
Z115; Z12 2; Z211; Z222
CHAP. 13] TWO-PORT NETWORKS 323
Fig. 13-18
Fig. 13-19
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Since DZZ Z11Z22 Z12Z21 52 21 12,
Y11 Z22
DZZ
2
12
1
6 Y12
Z12DZZ
2
12
1
6 Y21
Z21DZZ
1
12 Y22
Z11
DZZ
5
12
13.6 Find theY-parameters of the two-port network in Fig. 13-20 and thus show that the networks of
Figs. 13-19 and 13-20 are equivalent.
Apply KCL at the ports to obtain the terminal characteristics and Y-parameters. Thus,
I1 V1
6
V2
6Input port:
I2 V2
2:4
V1
12Output port:
Y11 1
6 Y12
1
6 Y21
1
12 Y22
1
2:4
5
12and
which are identical with the Y-parameters obtained in Problem 3.5 for Fig. 13-19. Thus, the two networks
are equivalent.
13.7 Apply the short-circuit equations (10) to find the Y-parameters of the two-port network in Fig.
13-21.
I1 Y11V1jV20 1
12
1
12
V1 or Y11
1
6
I1 Y12V2jV10 V2
4 V2
12 1
4 1
12
V2 or Y12 1
6
I2 Y21V1jV20 V1
12 or Y21
1
12
I2 Y22V2jV10 V2
3
V2
12
1
3
1
12
V2 or Y22
5
12
324 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-20
Fig. 13-21
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13.8 Apply KCL at the nodes of the network in Fig. 13-21 to obtain its terminal characteristics andY-
parameters. Show that two-port networks of Figs. 13-18 to 13-21 are all equivalent.
I1 V1
12
V1 V212
V2
4Input node:
I2
V2
3
V2 V112Output node:
I1 1
6V1
1
6V2 I2
1
12V1
5
12V2
TheY-parameters observed from the above characteristic equations are identical with theY-parameters of
the circuits in Figs. 13-18, 13-19, and 13-20. Therefore, the four circuits are equivalent.
13.9 Z-parameters of the two-port network N in Fig. 13-22(a) are Z11 4s, Z12 Z21 3s, and
Z22 9s. (a) Replace N by its T-equivalent. (b) Use part (a) to find input current i1 for
vs cos 1000t(V).
(a) The network is reciprocal. Therefore, its T-equivalent exists. Its elements are found from (6) and
shown in the circuit of Fig. 13-22(b).
CHAP. 13] TWO-PORT NETWORKS 325
Fig. 13-22
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Za Z11 Z124s 3s s
Zb Z22 Z219s 3s 6s
Zc Z12Z21 3s
(b) We repeatedly combine the series and parallel elements of Fig. 13-22(b), with resistors being in k ands
in krad/s, to find Zin in k
as shown in the following.
Zins Vs=I1 s 3s 66s 12
9s 18 3s 4 or Zinj 3j 4 5 36:98 k
and i1 0:2cos 1000t 36:98 (mA).
13.10 Two two-port networksa and b, with open-circuit impedancesZa andZb, are connected in series
(see Fig. 13-12). Derive theZ-parameters equations (31a).
From network a we have
V1a Z11;aI1a Z12;aI2aV2a Z21;aI1a Z22;aI2a
From network b we have
V1b Z11;bI1b Z12;bI2b
V2b Z21;bI1b Z22;bI2b
From connection between a and b we have
I1 I1a I1b V1 V1a V1b
I2 I2a I2b V2 V2a V2b
Therefore,
V1 Z11;a Z11;bI1 Z12;a Z12;bI2
V2 Z21;a Z21;bI1 Z22;a Z22;bI2
from which the Z-parameters (31a) are derived.
13.11 Two two-port networks a and b, with short-circuit admittances Ya and Yb, are connected in
parallel (see Fig. 13-13). Derive the Y-parameters equations (32a).
From network a we have
I1a Y11;aV1a Y12;aV2a
I2a Y21;aV1a Y22;aV2a
and from network b we have
I1b Y11;bV1b Y12;bV2b
I2b Y21;bV1b Y22;bV2b
From connection between a and b we have
V1 V1a V1b I1 I1a I1b
V2 V2a V2b I2 I2a I2b
Therefore,
I1 Y11;a Y11;bV1 Y12;a Y12;bV2
I2 Y21;a Y21;bV1 Y22;a Y22;bV2
from which the Y-parameters of (32a) result.
326 TWO-PORT NETWORKS [CHAP. 13
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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.
(a) From application of KVL around the input and output loops we find, respectively,
V1 2I1 2I2 2I1 I2 4I1
V2 3I2 2I1 I2 2I1 5I2
The Z-parameters are Z114, Z120, Z212, and Z225.
(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 verified by applying KVL to its input and output loops.
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.
(a) From Problem 13.12, Z114, Z120, Z212; Z225, and so DZZ Z11Z22 Z12Z2120.
Hence,
Y11 Z22
DZZ
5
20
1
4 Y12
Z12DZZ
0 Y21Z21
DZZ
2
20
1
10 Y22
Z11
DZZ
4
20
1
5
(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 verified by applying KCL to the input and output nodes.
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.
The Norton equivalent current source is IN 2I1=5 0:4I1. But I1 V1=4. Therefore,
IN0:4I1 0:1V1. The 5- 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.
CHAP. 13] TWO-PORT NETWORKS 327
Fig. 13-23
Fig. 13-24 Fig. 13-25
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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.
Apply KVL around the input loop to get
V1
h11
I1
h12
V2
Apply KCL at the output node to get
I2 h21I1 h22V2
These results agree with the definition ofh-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 find
h11 4 ; h120; h21 0:4; h221=5 0:2 1
13.17 Find the h-parameters of the circuit in Fig. 13-25 from itsZ-parameters and compare with results
of Problem 13.16.
Refer to Problem 13.13 for the values of the Z-parameters andDZZ. Use Table 13-1 for the conversion
of the Z-parameters to the h-parameters of the circuit. Thus,
h11DZZ
Z22
20
5 4 h12
Z12
Z220 h21
Z21Z22
2
5 0:4 h22
1
Z22
1
5 0:2
The above results agree with the results of Problem 13.16.
13.18 The simplified model of a bipolar junction transistor for small signals is shown in the circuit of
Fig. 13-27. Find its h-parameters.
The terminal equations are V1 0 andI2 I1. By comparing these equations with (23), we conclude
that h11h12h220 and h21 .
328 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-26
Fig. 13-27
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13.19 h-parameters of a two-port device H are given by
h11 500 h12 104 h21 100 h22 210
6 1
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.
13.20 The device Hof Problem 13-19 is placed in the circuit of Fig. 13-29(a). ReplaceHby its model
of Fig. 13-28 and find V2=Vs.
CHAP. 13] TWO-PORT NETWORKS 329
Fig. 13-28
Fig. 13-29
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The circuit of Fig. 13-29(b) contains the model. With good approximation, we can reduce it to Fig.
13-29(c) from which
I1 Vs=2000 V2 1000100I1 1000100Vs=2000 50Vs
Thus,V2=Vs 50.
13.21 A load ZL is connected to the output of a two-port device N (Fig. 13-30) whose terminal
characteristics are given by V1 1=NV2 and I1 NI2. Find (a) the T-parameters ofN
and (b) the input impedance Zin V1=I1.
(a) The T-parameters are defined by [see (29)]V1 AV2 BI2
I1 CV2 DI2
The terminal characteristics of the device are
V1 1=NV2
I1 NI2By comparing the two pairs of equations we get A 1=N, B 0, C 0, and D N.
(b) Three equations relating V1, I1, V2, and I2 are available: two equations are given by the terminal
characteristics of the device and the third equation comes from the connection to the load,
V2 ZLI2
By eliminating V2 and I2 in these three equations, we get
V1 ZLI1=N2 from which Zin V1=I1 ZL=N
2
Supplementary Problems
13.22 The Z-parameters of the two-port network N in Fig. 13-22(a) are Z114s, Z12 Z21 3s, and Z229s.
Find the input current i1 forvs cos 1000t(V) by using the open circuit impedance terminal characteristic
equations ofN, together with KCL equations at nodes A, B, and C.
Ans: i1 0:2cos 1000t 36:98(A)
13.23 Express the reciprocity criteria in terms ofh-, g-, and T-parameters.
Ans: h12 h210, g12
g21
0, and AD BC 1
13.24 Find the T-parameters of a two-port device whose Z-parameters are Z11s, Z12Z2110s, and
Z22100s. Ans: A 0:1; B 0; C 101
=s, and D 10
13.25 Find the T-parameters of a two-port device whose Z-parameters are Z11106s, Z12Z2110
7s, and
Z22108s. Compare with the results of Problem 13.21.
330 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-30
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Ans: A 0:1; B 0; C 107=s andD 10. For high frequencies, the device is similar to the device of
Problem 13.21, with N10.
13.26 TheZ-parameters of a two-port deviceNareZ11ks,Z12 Z2110ks, andZ22 100ks. A 1- resistor
is connected across the output port (Fig. 13-30). (a) Find the input impedance Z in V1=I1 and construct
its equivalent circuit. (b) Give the values of the elements for k 1 and 10
6
.
Ans: a Zin ks
1 100ks
1
100 1=ks
The equivalent circuit is a parallel RL circuit with R 102 and L 1 kH:
b For k 1;R 1
100 andL 1 H. For k 106;R
1
100 and L 106 H
13.27 The device N in Fig. 13-30 is specified by its following Z-parameters: Z22 N2Z11 and
Z12Z21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z11Z22p
NZ11. Find Zin V1=I1 when a load ZL is connected to the output terminal.
Show that ifZ11ZL=N2 we have impedance scaling such that Zin ZL=N
2.
Ans: Zin ZLN2 ZL=Z11
. For ZL N2Z11; Zin ZL=N2
13.28 Find theZ-parameters in the circuit of Fig. 13-31. Hint: Use the series connection rule.
Ans: Z11 Z22s 3 1=s; Z12Z21 s 1
13.29 Find theY-parameters in the circuit of Fig. 13-32. Hint: Use the parallel connection rule.Ans: Y11 Y229s 2=16; Y12Y21 3s 2=16
CHAP. 13] TWO-PORT NETWORKS 331
Fig. 13-31
Fig. 13-32
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13.30 Two two-port networksa and b with transmission parameters Ta and Tb are connected in cascade (Fig. 13-
14). Given I2a I1b and V2a V1b, find the T-parameters of the resulting two-port network.
Ans: A AaAb BaCb, B AaBb BaDb, C CaAb DaCb, D CaBb DaDb
13.31 Find the T- andZ-parameters of the network in Fig. 13-33. The impedances of capacitors are given. Hint:Use the cascade connection rule.
Ans: A 5j 4, B 4j 2, C 2j 4, and D j3, Z111:3 0:6j, Z220:3 0:6j,
Z12 Z21 0:2 0:1j
13.32 Find theZ-parameters of the two-port circuit of Fig. 13-34.
Ans: Z11 Z2212
Zb Za; Z12Z2112
Zb Za
13.33 Find theZ-parameters of the two-port circuit of Fig. 13-35.
Ans: Z11 Z221
2
Zb2Za Zb
Za Zb; Z12 Z21
1
2
Z2b
Za Zb
13.34 Referring to the two-port circuit of Fig. 13-36, find the T-parameters as a function of! and specify their
values at ! 1, 103, and 106 rad/s.
332 TWO-PORT NETWORKS [CHAP. 13
Fig. 13-33
Fig. 13-34
Fig. 13-35
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Ans: A 1 109!2 j109!, B 1031 j!, C 106j!, and D 1. At ! 1 rad/s, A 1,
B 1031 j, C 106j, and D 1. At ! 103 rad/s, A 1, B j, C 103j, and D 1.
At ! 106 rad/s, A 103, B 103j, C j, and D 1
13.35 A two-port network contains resistors, capacitors, and inductors only. With port #2 open [Fig. 13-37(a)], a
unit step voltage v1 ut produces i1 etut mA and v2 1 e
tut(V). With port #2 short-
circuited [Fig. 13-37(b)], a unit step voltage v1 ut delivers a current i1 0:51 e2tut mA. Find
i2 and the T-equivalent network. Ans: i2 0:51 e2tut [see Fig. 13-37(c)]
13.36 The two-port networkNin Fig. 13-38 is specified byZ112, Z12Z21 1, andZ224. FindI1,I2, and
I3. Ans: I1 24 A; I2 1:5 A; andI3 6:5 A
CHAP. 13] TWO-PORT NETWORKS 333
Fig. 13-37
Fig. 13-38
Fig. 13-36