1.8 Combination of Two Port Networks

5/23/2018 1.8 Combination of Two Port Networks

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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

Copyright 2003, 1997, 1986, 1965 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.


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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


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.


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


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



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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)


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


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):


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


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)


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


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.


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.


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


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


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).


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.



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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.


Fig. 13-23

Fig. 13-24 Fig. 13-25


19/24

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 .


Fig. 13-26

Fig. 13-27


20/24

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.


Fig. 13-28

Fig. 13-29


21/24

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.


Fig. 13-30


22/24

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


Fig. 13-31

Fig. 13-32


23/24

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.


Fig. 13-33

Fig. 13-34

Fig. 13-35


24/24

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


Fig. 13-37

Fig. 13-38

Fig. 13-36

1.8 Combination of Two Port Networks

Documents

port circuit

circuit of

terminal network

port networks13

v1 2i1 i2 si1 i2

terminal voltage

si1 si2v2 3i2 si1 i2

terminal current