CHAPTER
Two-Port
Networks
THE LEARNING GOALS FOR THIS CHAPTER
ARE THAT STUDENTS SHOULD BE ABLE TO:
■ Calculate the admittance, impedance, hybrid, and
transmission parameter for two-port networks.
■ Convert between admittance, impedance, hybrid, and
transmission parameters.
■ Describe the interconnection of two-port networks to
form more complicated networks.
16
AN EXPERIMENT THAT HELPS STUDENTS
DEVELOP AN UNDERSTANDING OF THE
MODELS USED FOR TWO-PORT NETWORKS IS:
■ Impedance Transmission Parameters: Impedance
parameters for two-port networks for several circuits are
determined analytically and from results obtained from
PSpice. Measurements of the impedance parameters are
performed on real circuits.
We say that the linear network in Fig. 16.1a has a single port—that is, a single pair of terminals. The pair of terminals A-B that constitute this port could represent a single element
(e.g., R, L, or C), or it could be some interconnection of these elements. The linear network
in Fig. 16.1b is called a two-port. As a general rule, the terminals A-B represent the input
port and the terminals C-D represent the output port.
In the two-port network shown in Fig. 16.2, it is customary to label the voltages and currents as shown; that is, the upper terminals are positive with respect to the lower terminals,
the currents are into the two-port at the upper terminals and, because KCL must be satisfied
at each port, the current is out of the two-port at the lower terminals. Since the network is
linear and contains no independent sources, the principle of superposition can be applied to
determine the current I1, which can be written as the sum of two components, one due to
V1 and one due to V2. Using this principle, we can write
16.1
Admittance
Parameters
I1 = y11V1 + y12V2
where y11 and y12 are essentially constants of proportionality with units of siemens. In a similar manner I2 can be written as
I2 = y21V1 + y22V2
16-1
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06/04/15 12:45 pm
16-2
CHAPTER 16
TWO-PORT NETWORKS
Figure 16.1
A
A
C
(a) Single-port network;
(b) two-port network.
Linear
network
Linear
network
B
B
D
(a)
(b)
I1
Figure 16.2
I2
+
Generalized two-port network.
+
Linear
network
V1
−
V2
−
Therefore, the two equations that describe the two-port network are
I1 = y11V1 + y12V2
I2 = y21V1 + y22V2
16.1
or in matrix form,
[I ] = [y
I1
y11
2
21
y12
y22
] [V ]
V1
2
Note that subscript 1 refers to the input port and subscript 2 refers to the output port, and
the equations describe what we will call the Y parameters for a network. If these parameters
y11, y12, y21, and y22 are known, the input/output operation of the two-port is completely
defined.
From Eq. (16.1), we can determine the Y parameters in the following manner. Note from
the equations that y11 is equal to I1 divided by V1 with the output short-circuited (i.e., V2 = 0).
∣
I1
y11 = —
16.2
V1 V2 = 0
Since y11 is an admittance at the input measured in siemens with the output short-circuited,
it is called the short-circuit input admittance. The equations indicate that the other Y parameters can be determined in a similar manner:
∣
I
=
V ∣
I
=
V ∣
I1
y12 = ___
V2
y21
y22
V1 = 0
2
___
1 V2 = 0
16.3
2
___
2 V1 = 0
y12 and y21 are called the short-circuit transfer admittances and y22 is called the shortcircuit output admittance. As a group, the Y parameters are referred to as the short-circuit
admittance parameters. Note that by applying the preceding definitions, these parameters
could be determined experimentally for a two-port network whose actual configuration is
unknown.
c16Two-PortNetworks.indd 2
06/04/15 12:45 pm
16-3
SECTION 16.1 ADMITTANCE PARAMETERS
We wish to determine the Y parameters for the two-port network shown in Fig. 16.3a. Once
these parameters are known, we will determine the current in a 4-Ω load, which is connected
to the output port when a 2-A current source is applied at the input port.
EXAMPLE
From Fig. 16.3b, we note that
SOLUTION
(
1 1
I1 = V1 — + —
1 2
16.1
)
Therefore,
3
y11 = — S
2
As shown in Fig. 16.3c,
V
I1 = − —2
2
2Ω
I1
+
V1
1Ω
3Ω
−
2Ω
I1
I2
+
+
V2
V1
−
−
1Ω
(a)
I2
1Ω
3Ω
2Ω
I1
V2 = 0
V2
−
(c)
I2
+
+
V1 = 0
3Ω
(b)
2Ω
I1
I2
2A
V1
+
1Ω
3Ω
−
V2
4Ω
−
(d)
Figure 16.3
Networks employed in Example 16.1.
and hence,
1
y12 = − — S
2
Also, y21 is computed from Fig. 16.3b using the equation
V
I2 = − —1
2
and therefore,
1
y21 = − — S
2
c16Two-PortNetworks.indd 3
06/04/15 12:45 pm
16-4
CHAPTER 16
TWO-PORT NETWORKS
Finally, y22 can be derived from Fig. 16.3c using
(
1 1
I2 = V2 — + —
3 2
)
and
5
y22 = — S
6
Therefore, the equations that describe the two-port itself are
3
1
I1 = — V1− — V2
2
2
5
1
I2 = − — V1 + — V2
6
2
These equations can now be employed to determine the operation of the two-port for some
given set of terminal conditions. The terminal conditions we will examine are shown in
Fig. 16.3d. From this figure, we note that
I1 = 2 A
and
V2 = −4I2
Combining these with the preceding two-port equations yields
3
1
2 = — V1 − — V2
2
2
1
13
0 = − — V1 + — V2
2
12
or in matrix form
[
3
2
1
−—
2
—
1
−—
2
13
—
12
]
[ VV ] = [ 20 ]
1
2
Note carefully that these equations are simply the nodal equations for the network in
Fig. 16.3d. Solving the equations, we obtain V2 = 811 V and therefore I2 = −211 A.
LEARNING ASSESSMENTS
E16.1 Find the Y parameters for the two-port network shown in Fig. E16.1.
ANSWER: 1
y11 = — S;
14
21 Ω
1
1
y12 = y21 = − — S; y22 = — S.
21
7
42 Ω
10.5 Ω
Figure E16.1
E16.2 If a 10-A source is connected to the input of the two-port network in
Fig. E16.1, find the current in a 5-Ω resistor connected to the output port.
c16Two-PortNetworks.indd 4
ANSWER: I2 = − 4.29 A.
06/04/15 12:45 pm
SECTION 16.2 IMPEDANCE PARAMETERS
E16.3 Find the Y parameters for the two-port network shown in Fig. E16.3.
ANSWER: y11 = 282.2 μS; y12 = −704 nS;
60I1
I1
16-5
y21 = 16.9 mS; y22 = 7.74 μS.
500 Ω
20 kΩ
50 Ω
Figure E16.3
Once again, if we assume that the two-port network is a linear network that contains no independent sources, then by means of superposition we can write the input and output voltages
as the sum of two components, one due to I1 and one due to I2:
V1 = z11I1 + z12I2
V2 = z21I1 + z22I2
16.4
16.2
Impedance
Parameters
These equations, which describe the two-port network, can also be written in matrix form as
[ VV ] = [ z
1
z11
2
21
z12
z22
] [ II ]
1
16.5
2
Like the Y parameters, these Z parameters can be derived as follows:
∣
V
=
I ∣
V
=
I ∣
V
=
I ∣
V
z11 = ___1
I1
z12
z21
z22
I2 = 0
___1
2 I1 = 0
16.6
___2
1 I2 = 0
___2
2 I1 = 0
In the preceding equations, setting I1 or I2 = 0 is equivalent to open-circuiting the input or
output port. Therefore, the Z parameters are called the open-circuit impedance parameters.
z11 is called the open-circuit input impedance, z22 is called the open-circuit output impedance, and z12 and z21 are termed open-circuit transfer impedances.
We wish to find the Z parameters for the network in Fig. 16.4a. Once the parameters are known,
we will use them to find the current in a 4-Ω resistor that is connected to the output terminals
when a 12 0° −V source with an internal impedance of 1 + j0 Ω is connected to the input.
EXAMPLE
From Fig. 16.4a, we note that
SOLUTION
16.2
z11 = 2 − j4 Ω
z12 = −j4 Ω
z21 = −j4 Ω
z22 = −j4 + j2 = −j2 Ω
c16Two-PortNetworks.indd 5
06/04/15 12:45 pm
16-6
CHAPTER 16
TWO-PORT NETWORKS
2Ω
I1
j2 Ω
1Ω
I2
+
+
−j4 Ω
V1
V2
2Ω
j2 Ω
I2
+
+
12 0° V
+
−
−
−
I1
−j4 Ω
V1
4Ω
−
−
(a)
V2
(b)
Figure 16.4
Circuits employed in Example 16.2.
The equations for the two-port network are, therefore,
V1 = (2 − j4)I1 − j4I2
V2 = −j4I1 − j2I2
The terminal conditions for the network shown in Fig. 16.4b are
V1 = 12 0° − (1)I1
V2 = −4I2
Combining these with the two-port equations yields
12 0° = (3 − j4)I1 − j4I2
0 = −j4I1 + (4 − j2)I2
It is interesting to note that these equations are the mesh equations for the network. If we solve
the equations for I2, we obtain I2 = 1.61 137.73° A, which is the current in the 4-Ω load.
LEARNING ASSESSMENTS
E16.4 Find the Z parameters for the network in Fig. E16.4. Then compute the current in
a 4-Ω load if a 12 0° −V source is connected at the input port.
12 Ω
ANSWER: I2 =−0.73 0° A.
3Ω
6Ω
Figure E16.4
E16.5 Determine the Z parameters for the two-port network shown in Fig. E16.5.
2I1
I1
ANSWER: z11 = 15 Ω; z12 = 5 Ω;
z21 = 5 − j8 Ω;
z22 = 5 − j4 Ω.
10 Ω
−j4 Ω
5Ω
Figure E16.5
c16Two-PortNetworks.indd 6
06/04/15 12:45 pm
SECTION 16.3 HYBRID PARAMETERS
Under the assumptions used to develop the Y and Z parameters, we can obtain what are commonly
called the hybrid parameters. In the pair of equations that define these parameters, V1 and I2 are
the independent variables. Therefore, the two-port equations in terms of the hybrid parameters are
V1 = h11I1 + h12V2
or in matrix form,
[ VI ] = [ h
h11 h12
21 h22
1
2
16.3
Hybrid
Parameters
16.7
I2 = h21I1 + h22V2
] [ VI ]
1
16-7
16.8
2
These parameters are especially important in transistor circuit analysis. The parameters are
determined via the following equations:
∣
V
= ∣
V
I
=
I ∣
I
= ∣
V
V
h11 = ___1
I1
h12
h21
h22
V2 = 0
___1
2 I1 = 0
16.9
__2
1 V2 = 0
2
___
2 I1 = 0
The parameters h11, h12, h21, and h22 represent the short-circuit input impedance, the opencircuit reverse voltage gain, the short-circuit forward current gain, and the open-circuit
output admittance, respectively. Because of this mix of parameters, they are called hybrid
parameters. In transistor circuit analysis, the parameters h11, h12, h21, and h22 are normally
labeled hi, hr, hf, and ho.
An equivalent circuit for the op-amp in Fig. 16.5a is shown in Fig. 16.5b. We will determine the hybrid parameters for this network.
EXAMPLE
Parameter h11 is derived from Fig. 16.5c. With the output shorted, h11 is a function of only
Ri, R1, and R2 and
R1R2
h11 = Ri + _______
R1 + R2
SOLUTION
+
−
+
+
+
R2
V1
I1 +
R2
−
−
R1
+
V1
Vi
V2
+
−
−
(b)
R2
−
AVi
−
(a)
I1 +
+
Ro
V1
−
I2
Ri
V2
R1
Vi
16.3
Ri
I1 = 0 +
I2
+
Ro
R1
+ AVi
−
−
(c)
V2 = 0
V1
Vi
R2
−
I2
Ri
Ro
R1
AVi
−
+
−
+
V2
−
(d)
Figure 16.5
Circuits employed in Example 16.3.
c16Two-PortNetworks.indd 7
06/04/15 12:45 pm
16-8
CHAPTER 16
TWO-PORT NETWORKS
Fig. 16.5d is used to derive h12. Since I1 = 0, Vi = 0 and the relationship between V1 and
V2 is a simple voltage divider:
V2R1
V1 = _______
R +R
1
2
Therefore,
R1
h12 = _______
R +R
1
2
KVL and KCL can be applied to Fig. 16.5c to determine h21. The two equations that relate
I2 to I1 are
Vi = I1Ri
−AVi _______
I1R1
I2 = _____
R −R + R
o
1
2
Therefore,
(
ARi _______
R1
h21 = − ___
Ro + R1 + R2
)
Finally, the relationship between I2 and V2 in Fig. 16.5d is
Ro( R1 + R2 )
V2 ____________
___
=
I2
Ro + R1 + R2
and therefore,
Ro + R1 + R2
h22 = ____________
Ro( R1 + R2 )
The network equations are, therefore,
(
)
R1R2
R1
_______
V1 = Ri + _______
R + R I1 + R + R V2
1
2
1
2
R1
AR
Ro + R1 + R2
____________
I2 = − ___i + _______
R1 + R2 I1 + Ro ( R1 + R2 ) V2
Ro
(
)
LEARNING ASSESSMENTS
E16.6 Find the hybrid parameters for the network shown in Fig. E16.4.
ANSWER: E16.7 If a 4-Ω load is connected to the output port of the network examined in Learning
Assessment E16.6, determine the input impedance of the two-port with the load connected.
ANSWER: E16.8 Find the hybrid parameters for the two-port network shown in Fig. E16.3.
ANSWER: c16Two-PortNetworks.indd 8
2
2
h11 = 14 Ω; h12 = —; h21 = − —;
3
3
1
h22 = — S.
9
Zi = 15.23 Ω.
h11 = 3543.6 Ω;
h12 = 2.49 × 10−3; h21 = 59.85;
h22 = 49.88 μS.
06/04/15 12:45 pm
SECTION 16.4 TRANSMISSION PARAMETERS
The final parameters we will discuss are called the transmission parameters. They are defined
by the equations
V1 = AV2 − BI2
I1 = CV2 − DI2
16.10
16-9
16.4
Transmission
Parameters
or in matrix form,
V
[ I ] = [ CA DB ] [ −I ]
1
V2
1
2
16.11
These parameters are very useful in the analysis of circuits connected in cascade, as we will
demonstrate later. The parameters are determined via the following equations:
V
A = ___1
V2
∣
I2=0
V1
B = ____
−I2
I1
C = ___
V2
∣
∣
V2=0
16.12
I2=0
I1
D = ____
−I2
∣
V2=0
A, B, C, and D represent the open-circuit voltage ratio, the negative short-circuit transfer
impedance, the open-circuit transfer admittance, and the negative short-circuit current ratio,
respectively. For obvious reasons, the transmission parameters are commonly referred to as
the ABCD parameters.
We will now determine the transmission parameters for the network in Fig. 16.6.
EXAMPLE
Let us consider the relationship between the variables under the conditions stated in the
parameters in Eq. (16.12). For example, with I2 = 0, V2 can be written as
SOLUTION
16.4
( )
V1
1
—
V2 = ________
1 + 1jω jω
or
V
A = ___1
V2
∣
I2=0
= 1 + jω
Similarly, with V2 = 0, the relationship between I2 and V1 is
V1
−I2 = ____________
1jω
1+—
1 + 1jω
(
1jω
________
1 + 1jω
)
or
V1
B = ____
= 2 + jω
−I2
In a similar manner, we can show that C = jω and D = 1 + jω.
I1
1Ω
1Ω
+
V1
−
c16Two-PortNetworks.indd 9
I2
+
1F
Figure 16.6
Circuit used in
Example 16.4.
V2
−
06/04/15 12:45 pm
16-10
CHAPTER 16
TWO-PORT NETWORKS
LEARNING ASSESSMENTS
ANSWER: E16.9 Compute the transmission parameters for the two-port network in Fig. E16.1.
A = 3; B = 21 Ω;
1
3
C = — S; D = — .
6
2
E16.10 Find the transmission parameters for the two-port network shown in Fig. E16.5.
ANSWER: A = 0.843 + j1.348; B = 4.61 + j3.37 Ω;
C = 0.056 + j0.09 S; D = 0.64 + j0.225.
ANSWER: Vs = 1015.9 −137.63° V rms.
E16.11 Find Vs if V2 = 220 0° V rms in the network shown in Fig. E16.11.
1Ω
Vs
j2 Ω
I1
I2
+
+
5:1
+
−
Two–port
network
V1
−
Figure E16.11
−(1.333 + j6)
−(0.333 + j0.333)
0.333 + j0.333
V1
=
I1
j0.1667
Parameter
Conversions
−
Ideal
[ ] [
16.5
10 kVA
0.8 lag
V2
][ ]
V2
I2
If all the two-port parameters for a network exist, it is possible to relate one set of parameters
to another since the parameters interrelate the variables V1, I1, V2, and I2.
Table 16.1 lists all the conversion formulas that relate one set of two-port parameters to
another. Note that ∆Z, ∆Y, ∆H, and ∆T refer to the determinants of the matrices for the Z, Y,
hybrid, and ABCD parameters, respectively. Therefore, given one set of parameters for a
network, we can use Table 16.1 to find others.
TABLE 16.1 Two-port parameter conversion formulas
[
[
[
[
c16Two-PortNetworks.indd 10
z11 z12
z21 z22
z22
___
∆Z
−z21
_____
∆Z
[
]
−z12
_____
∆Z
z11
___
∆Z
z11 ___
∆z
___
z21 z21
z22
1
___
___
z21 z21
∆Z
___
z12
___
z22 z22
−z21 ___
1
_____
z22 z22
]
]
−y12
y22 _____
___
∆Y
−y21
_____
∆Y
]
[
[
∆Y
y11
___
∆Y
y11 y12
y21 y22
−y
−1
____
y21 y21
−y11
−∆y _____
—
y21
y21
[
∆H
___
∆
C
D
—
C
h12
___
h22 h22
−h21 ___
1
_____
h22 h22
T
—
[ ]
D
B
1
−—
B
]
−y12
1 _____
___
[ ]
[ ]
[ ]
—
22
—
y11 y11
y21 ∆Y
___
___
y11 y11
]
A
C
1
—
C
—
]
]
[
A B
C D
]
[ ]
B
D
1
−—
D
—
−h12
1 _____
___
−∆
B
A
—
B
T
—
∆
D
C
—
D
T
—
h11 h11
h21 ___
∆H
___
h11 h11
[
−∆H _____
−h11
_____
h21
−h22
_____
h21
[ hh
11
21
h21
−1
____
h21
h12
h22
]
]
06/04/15 12:45 pm
16-11
SECTION 16.6 INTERCONNECTION OF TWO-PORTS
LEARNING ASSESSMENT
ANSWER: E16.12 Determine the Y parameters for a two-port if the Z parameters are
[
18
Z=
6
6
9
1
1
1
y11 = — S; y12 = y21 = −— S; y22 = — S..
21
14
7
]
Interconnected two-port circuits are important because when designing complex systems it is
generally much easier to design a number of simpler subsystems that can then be interconnected to form the complete system. If each subsystem is treated as a two-port network, the
interconnection techniques described in this section provide some insight into the manner in
which a total system may be analyzed and/or designed. Thus, we will now illustrate techniques
for treating a network as a combination of subnetworks. We will, therefore, analyze a two-port
network as an interconnection of simpler two-ports. Although two-ports can be interconnected
in a variety of ways, we will treat only three types of connections: parallel, series, and cascade.
For the two-port interconnections to be valid, they must satisfy certain specific requirements that are outlined in the book Network Analysis and Synthesis by L. Weinberg (McGrawHill, 1962). The following examples will serve to illustrate the interconnection techniques.
In the parallel interconnection case, a two-port N is composed of two-ports Na and Nb
connected as shown in Fig. 16.7. Provided that the terminal characteristics of the two
networks Na and Nb are not altered by the interconnection illustrated in the figure, then the
Y parameters for the total network are
y11a + y11b y12a + y12b
y11 y12
=
y21a + y21b y22a + y22b
21 y22
[y
] [
]
16.6
Interconnection
of Two-Ports
16.13
and hence to determine the Y parameters for the total network, we simply add the Y parameters of the two networks Na and Nb.
Likewise, if the two-port N is composed of the series connection of Na and N b, as
shown in Fig. 16.8, then once again, as long as the terminal characteristics of the two
I1a
I2a
Figure 16.7
Na
y11a y12a
y21a y22a
V1a
I1
Parallel interconnection
of two-ports.
V2a
I2
V1
V2
I1b
I2b
Nb
y11b y12b
y21b y22b
V1b
I1a
V1a
V1
Figure 16.8
V2a
Na
I1b
V2b
c16Two-PortNetworks.indd 11
I2a
Series interconnection
of two-ports.
V2
I2b
Nb
V2b
V2b
06/04/15 12:45 pm
16-12
CHAPTER 16
TWO-PORT NETWORKS
I1
Figure 16.9
I1a
+
V1
−
Cascade interconnection
of networks.
+
V1a
−
Na
I2a
I1b
I2a
+
V2a
−
+
V1b
−
Nb
+
V2b
−
I2
+
V2
−
networks Na and N b are not altered by the series interconnection, the Z parameters for
the total network are
z11a + z11b z12a + z12b
z12
z22 = z21a + z21b z22a + z22b
] [
z
[ z1121
]
16.14
Therefore, the Z parameters for the total network are equal to the sum of the Z parameters for
the networks Na and Nb.
Finally, if a two-port N is composed of a cascade interconnection of Na and Nb, as shown
in Fig. 16.9, the equations for the total network are
[ VI ] = [ C
1
Aa
1
a
Ba
Da
] [ AC
b
b
Bb
Db
] [ −IV ]
2
16.15
2
Hence, the transmission parameters for the total network are derived by matrix multiplication
as indicated previously. The order of the matrix multiplication is important and is performed
in the order in which the networks are interconnected.
The cascade interconnection is very useful. Many large systems can be conveniently
modeled as the cascade interconnection of a number of stages. For example, the very
weak signal picked up by a radio antenna is passed through a number of successive
stages of amplification—each of which can be modeled as a two-port subnetwork. In
addition, in contrast to the other interconnection schemes, no restrictions are placed on
the parameters of Na and Nb in obtaining the parameters of the two-port resulting from
their interconnection.
EXAMPLE
16.5
SOLUTION
We wish to determine the Y parameters for the network shown in Fig. 16.10a by considering it to be a parallel combination of two networks as shown in Fig. 16.10b. The capacitive
network will be referred to as Na, and the resistive network will be referred to as Nb .
The Y parameters for Na are
1
y11a = j — S
2
1
y21a = −j — S
2
1
y12a = −j — S
2
1
y22a = j — S
2
and the Y parameters for Nb are
3
y11b = — S
5
1
y21b = −— S
5
1
y12b = −— S
5
2
y22b = — S
5
Hence, the Y parameters for the network in Fig. 16.10 are
)
3
1
1
1
y11 = — + j — S
y12 = − — + j — S
5
2
5
2
2
1
1
1
y21 = − — + j — S y22 = — + j — S
5
5
2
2
(
(
)
To gain an appreciation for the simplicity of this approach, you need only try to find the
Y parameters for the network in Fig. 16.10a directly.
c16Two-PortNetworks.indd 12
06/04/15 12:45 pm
SECTION 16.6 INTERCONNECTION OF TWO-PORTS
16-13
−j2 Ω
−j2 Ω
1Ω
2Ω
1Ω
2Ω
1Ω
1Ω
(a)
(b)
Figure 16.10
Network composed of the parallel combination of two subnetworks.
Let us determine the Z parameters for the network shown in Fig. 16.10a. The circuit is
redrawn in Fig. 16.11, illustrating a series interconnection. The upper network will be
referred to as Na and the lower network as Nb .
EXAMPLE
The Z parameters for Na are
SOLUTION
2 − 2j
z11a = — Ω
3 − 2j
2
z21a = — Ω
3 − 2j
16.6
2
z12a = — Ω
3 − 2j
2 − 4j
z22a = — Ω
3 − 2j
and the Z parameters for Nb are
z11b = z12b = z21b = z22b = 1 Ω
Hence, the Z parameters for the total network are
5 − 4j
z11 = — Ω
3 − 2j
5 − 2j
z12 = — Ω
3 − 2j
5 − 2j
z21 = — Ω
3 − 2j
5 − 6j
z22 = — Ω
3 − 2j
We could easily check these results against those obtained in Example 16.5 by applying the
conversion formulas in Table 16.1.
−j2 Ω
1Ω
Figure 16.11
2Ω
Network in Fig. 16.10a
redrawn as a series
interconnection of
two networks.
1Ω
c16Two-PortNetworks.indd 13
06/04/15 12:45 pm
16-14
CHAPTER 16
EXAMPLE
TWO-PORT NETWORKS
16.7
SOLUTION
Let us derive the two-port parameters of the network in Fig. 16.12 by considering it to be a
cascade connection of two networks as shown in Fig. 16.6.
The ABCD parameters for the identical T networks were calculated in Example 16.4 to be
A = 1 + jω
C = jω
B = 2 + jω
D = 1 + jω
Therefore, the transmission parameters for the total network are
[ CA DB ] = [ 1 +jωjω
2 + jω
1 + jω
] [ 1 +jωjω
2 + jω
1 + jω
]
Performing the matrix multiplication, we obtain
[ CA DB ] = [ 1 +2jω4jω−−2ω2ω
2
2
1Ω
Figure 16.12
4 + 6jω − 2ω2
1 + 4jω − 2ω2
2Ω
]
1Ω
Circuit used in
Example 16.7.
1F
EXAMPLE
16.8
1F
Fig. 16.13 is a per-phase model used in the analysis of three-phase high-voltage transmission lines. As a general rule in these systems, the voltage and current at the receiving end are known, and it is the conditions at the sending end that we wish to find. The
transmission parameters perfectly fit this scenario. Thus, we will find the transmission
parameters for a reasonable transmission line model and, then, given the receiving-end
voltages, power, and power factor, we will find the receiving-end current, sending-end
voltage and current, and the transmission efficiency. Finally, we will plot the efficiency
versus the power factor.
150-mile-long
transmission line
L
R
Highvoltage
substation
(sending
end)
C
C
Lowervoltage
substation
(receiving
end)
VL = 300 kV
P = 600 MW
pf = 0.95 lagging
Transmission line model
Figure 16.13
A π-circuit model for power transmission lines.
c16Two-PortNetworks.indd 14
06/04/15 12:45 pm
SECTION 16.6 INTERCONNECTION OF TWO-PORTS
R
ZL
V
V1
∣
I2=0
ZC
=—
ZC + ZL + R
V
A = —1
V2
∣
ZC + ZL + R
=—
= 0.9590 0.27°
ZC
—2
+
V1
+
−
ZC
ZC
V2
−
R
V1
+
−
−I
V1
∣
V
B = —1
−I2
∣
ZL
2
—
ZC
I2
R
ZL
V
I1
∣
V
C = —2
I1
∣
—2
+
I1
ZC
ZC
V2
−
R
I2=0
ZC
= ZL + R = 100.00 84.84° Ω
2ZC + ZL + R
= ——
= 975.10 90.13° μS
Z2C
I2=0
ZL
I2
V2=0
1
=—
ZL + R
Z2C
= ___________
2ZC + ZL + R
I2=0
−I
I1
∣
I
D = —1
−I2
∣
—2
I1
V2=0
16-15
V2=0
ZC
=—
ZC + ZL + R
ZC + ZL + R
=—
= 0.950 0.27°
ZC
V2=0
Figure 16.14
Equivalent circuits used to determine the transmission parameters.
Given a 150-mile-long transmission line, reasonable values for the π-circuit elements of the
transmission line model are C = 1.326 μF, R = 9.0 Ω, and L = 264.18 mH. The transmission
parameters can be easily found using the circuits in Fig. 16.14. At 60 Hz, the transmission
parameters are
A = 0.9590 0.27°
B = 100.00 84.84° Ω
SOLUTION
C = 975.10 90.13° μS
D = 0.9590 0.27°
To use the transmission parameters, we must know the receiving-end current, I2. Using
standard three-phase circuit analysis outlined in Chapter 11, we find the line current to be
−1
600 cos (pf)
= −1.215 −18.19° kA
I2 = − ____________
—
√3 (300)(pf)
where the line-to-neutral (i.e., phase) voltage at the receiving end, V2 , is assumed to have
zero phase. Now we can use the transmission parameters to determine the sending-end volt—
age and power. Since the line-to-neutral voltage at the receiving end is 300√ 3 = 173.21 kV,
the results are
V1 = AV2 − BI2 = (0.9590 0.27°)(173.21 0°)
+ (100.00 84.84°)(1.215 −18.19°) = 241.92 27.67° kV
c16Two-PortNetworks.indd 15
06/04/15 12:45 pm
16-16
CHAPTER 16
TWO-PORT NETWORKS
I1 = CV2 − DI2 = (975.10 × 10−6 90.13°)(173.21) 0°)
+ (0.9590 0.27°)(1.215 −18.19°) = 1.12 −9.71° kA
At the sending end, the power factor and power are
pf = cos (27.67 − (−9.71)) = cos (37.38) = 0.80 lagging
P1 = 3V1I1(pf) = (3)(241.92)(1.12)(0.80) = 650.28 MW
Finally, the transmission efficiency is
P2
600
—
η = ____
P1 = 650.28 = 92.3%
This entire analysis can be easily programmed into an Excel spreadsheet. A plot of the transmission efficiency versus power factor at the receiving end is shown in Fig. 16.15. We see
that as the power factor decreases, the transmission efficiency drops, which increases the cost
of production for the power utility. This is precisely why utilities encourage industrial customers to operate as close to unity power factor as possible.
Figure 16.15
100
Transmission efficiency
The results of an Excel
simulation showing the
effect of the receivingend power factor on the
transmission efficiency.
Because the Excel
simulation used more
significant digits, slight
differences exist between
the values in the plot and
those in the text.
90
80
70
60
50
0.65
0.75
0.85
0.95
1.05
Receiving-end power factor
SUMMARY
■ Four of the most common parameters used to describe a
two-port network are the admittance, impedance, hybrid, and
transmission parameters.
■ If all the two-port parameters for a network exist, a set of
■ When interconnecting two-ports, the Y parameters are added
for a parallel connection, the Z parameters are added for a
series connection, and the transmission parameters in matrix
form are multiplied together for a cascade connection.
conversion formulas can be used to relate one set of two-port
parameters to another.
PROBLEMS
16.1 Given the two networks in Fig. P16.1, find the Y parameters for the circuit in Fig. P16.1 (a) and the Z parameters for the circuit in
Fig. P16.1 (b). Calculate (a) y11, (b) y12, (c) y21, (d) y22, (e) z11, (f) z12, (g) z21, (h) z22 if ZL = 30 Ω.
I1
+
ZL
I2
+
–
–
(a)
c16Two-PortNetworks.indd 16
I2
+
V2
V1
Figure P16.1
I1
V1
+
ZL
–
V2
–
(b)
06/04/15 12:45 pm
PROBLEMS
16.2 Find the Y parameters for the two-port network shown in
16.7 Find the Z parameters for the two-port network in Fig. P16.7.
Fig. P16.2.
I1
12 Ω
I2
21 Ω
+
12 Ω
16-17
+
42 Ω
V1
12 Ω
10.5 Ω
V2
–
–
Figure P16.7
Figure P16.2
16.3 Find the Y parameters ((a) y11, (b) y12, (c) y21, (d) y22) for the
16.8 Find the Z parameters for the two-port network shown in
Fig. P16.8 and determine the voltage gain of the entire
circuit with a 4-kΩ load attached to the output.
two-port network in Fig. P16.3.
3Ω
16 Ω
60I1
9Ω
I1 500 Ω
+
20 kΩ
+
–
Figure P16.3
50 Ω
V1
4 kΩ
Vo
16.4 Determine the Y parameters for the two-port network shown
–
in Fig. P16.4.
Figure P16.8
2i
16.9 Determine a two-port network that is represented by the
following Z parameters.
i
8Ω
4Ω
[Z] =
I2
Z11
in Fig. P16.5.
C2
C3
R
Z21
Figure P16.10
–
16.11 Find the voltage gain of the two-port network in Fig. Figure P16.5
P16.10 if a 12-kΩ load is connected to the output port.
16.6 Find the Y parameters for the two-port network in Fig. P16.6.
I1
Z2
+
–
Figure P16.6
Z22
V2
–
Z1
Z
V2
+
gV1
Z12
V1
I2
+
V1
82j
I1
16.5 Determine the admittance parameters for the network shown
C1
]
5 − j2
in Fig. P16.10 in terms of the Z parameters and the load
impedance Z.
Figure P16.4
V1
6 + j3
16.10 Determine the input impedance of the network shown
2Ω
I1
[ 5 − j2
–
+
16.12 Find the input impedance of the network in Fig. P16.10.
16.13 Find the Y parameters for the network in Fig. P16.13.
I2
+
γV1
+
C1
V2
–
+
V1
C2
R
–
+
AV1
C3
L
V2
–
–
Figure P16.13
c16Two-PortNetworks.indd 17
06/04/15 12:45 pm
16-18
CHAPTER 16
TWO-PORT NETWORKS
16.14 Find the Z parameters for the network in Fig. P16.14.
R1
I1
γV1
R3
–+
+
V1
16.19 Find the Z parameters for the two-port network shown in
Fig. P16.19.
I2
I1
+
+
V2
R2
–
V1
–
I2
+
1 kΩ
+
–
(3)10–4 V2
40I1
20 μS
–
V2
–
Figure P16.14
Figure P16.19
16.15 Find the Z parameters of the two-port network in
16.20 Find [Z] of a two-port network if
Fig. P16.15.
jωM
I1
[Z] =
+
+
jωL1
V1
jωL2
–
[
]
10 1.5
2
4
16.21 Determine the Z parameters for the two-port network in
Fig. P16.21.
I2
V2
R2
–
Figure P16.15
1:n
R1
16.16 Given the network in Fig. P16.16, (a) find the Z parameters for
the transformer, (b) write the terminal equation at each end of
the two-port, and (c) use the information obtained to find V2.
Figure P16.21
j1 Ω
2Ω
+
+
12 0° V
j4 Ω
Ideal
j3 Ω
4Ω
–
V2
16.22 Determine (a) R1, (b) R2, (c) R3 in the circuit diagram shown
in Fig. P16.22 that has the following Y parameters.
–
[ ]
15
56
[Y] =
1
−—
8
—
Figure P16.16
16.17 Find the Z parameters for the network in Fig. P16.17.
1
−—
8
18
—
80
R2
50 kΩ
+
+
VA
–
R1
+
2 kΩ
R3
0.04 VA
1 kΩ
V1
V2
Figure P16.22
–
–
16.23 Determine Z1 ((a) real part and (b) imaginary part), Z2
((c) real part and (d) imaginary part), Z3 ((e) real part and
(f) imaginary part) in the circuit diagram shown in
Fig. P16.23 that has the following Z parameters (all in
ohms):
6 2 j1 4 − j7
[Z] =
4 − j7 7 + j4
Figure P16.17
16.18 Calculate I1 and I2 in the circuit.
I1
+
100 0° V
+
–
V1
–
I2
Z11 = 40 Ω
Z12 = j20 Ω
Z21 = j30 Ω
Z22 = 50 Ω
[
+
V2
10 Ω
Z1
]
Z3
–
Z2
Figure P16.18
Figure P16.23
c16Two-PortNetworks.indd 18
06/04/15 12:45 pm
PROBLEMS
16.24 Compute the hybrid parameters for the network in
16-19
16.29 Determine the hybrid parameters for the network in
Fig. P16.29. R1 = 30 Ω, R2 = 21 Ω, C = 170 mF,
a = 0.8, v = 100 radians per second.
Fig. E16.21.
16.25 Compute the hybrid parameters ((a) h11, (b) h12, (c) h21,
αI1
(d) h22) for the network in Fig. P16.25.
27 Ω
54 Ω
I1
10.5 Ω
R1
I2
1
––––
jωC
+
R2
V1
Figure P16.25
+
V2
–
–
16.26 Find the hybrid parameters for the network in Fig. P16.23.
16.27 Consider the network in Fig. P16.27. The two-port network
is a hybrid model for a basic transistor. Determine the voltage gain of the entire network, V2/Vs, if a source Vs with
internal resistance R1 is applied at the input to the two-port
network and a load RL is connected at the output port.
I1
R1
+
–
VS
+
V1
h11
Figure P16.29
16.30 Find the ABCD parameters for the networks in Fig. P16.1.
16.31 Find the transmission parameters for the network in
Fig. P16.31.
I2
–j1 Ω
+
1 kΩ
+
–
h12V2
h21I1
–
1
–––
h
V2
22
1Ω
RL
–
Figure P16.31
Figure P16.27
16.28 Find the hybrid parameters of the two-port network.
2Ω
3Ω
6Ω
Figure P16.28
16.32 Given the network in Fig. P16.32, find the transmission parameters for the two-port network and then find Io using the
terminal conditions.
j17 Ω
–j34 Ω
68 Ω
Io
34 Ω
12 0° V
+
–
j34 Ω
j68 Ω
+
–
6 30° V
Figure P16.32
Enter arguments from 0 to 2π.
c16Two-PortNetworks.indd 19
06/04/15 12:45 pm
16-20
CHAPTER 16
TWO-PORT NETWORKS
16.33 Find the voltage gain V2/V1 for the network in Fig. P16.33
16.36 Draw the circuit diagram (with all passive elements in
ohms) for a network that has the following
Y parameters:
using the ABCD parameters.
I1
I2
+
ABCD
parameters
known
V1
–
+
[ ]
5
11
[Y] =
2
−—
11
—
ZL
V2
–
2
−—
11
3
—
11
16.37 Draw the circuit diagram for a network that has the
following Z parameters:
Figure P16.33
16.34 Find the input admittance of the two-port in Fig. P16.34 if
[Z] =
the Y parameters are y11 = 17 S, y12 = –16 S,
y21 = –16 S, y22 = 17 S and the load YL is 8 S.
[
6 + j4
4 + j6
4 + j6
10 + j6
]
16.38 Following are the hybrid parameters for a network:
Two–port
Y
parameters
known
Yin
[
YL
h11
h12
h21
h22
]
[ ]
11
5
=
2
−—
5
—
2
5
1
—
5
—
Determine the Y parameters for the network.
Figure P16.34
16.39 If the Y parameters for a network are known to be
16.35 Find the voltage gain V2/V1 for the network in Fig. P16.35
using the Z parameters.
I1
+
V1
[
I2
Z
parameters
known
–
y11
y12
y21
y22
+
[ ]
3
−—
11
3
—
11
find the Z parameters ((a) z11, (b) z12, (c) z21, (d) z22).
ZL
V2
]
2
11
=
3
−—
11
—
16.40 Find the Z parameters ((a) z11, (b) z12, (c) z21, (d) z22)
if a = 3, b = 5, c = 4 S, d = 9.
–
16.41 Find the hybrid parameters in terms of the Z parameters.
Figure P16.35
16.42 Find the transmission parameters for the two-port in Fig. P16.42.
1Ω
j1 Ω
1Ω
–j 1 Ω
j4 Ω
–j 1 Ω
j3 Ω
Figure P16.42
16.43 Find the transmission parameters for the two-port in Fig. P16.43 and then use the terminal conditions to compute Io.
Io
1:2
1Ω
6 0° V
+
–
–j 1 Ω
2Ω
–j 1 Ω
+
–
(4–j 4)V
Figure P16.43
c16Two-PortNetworks.indd 20
06/04/15 12:45 pm
PROBLEMS
16.44 Find the Y parameters for the network in Fig. P16.44.
16.48 The Y parameters of a network are
Z1
[Y] =
Z2
16-21
[
–0.5
–0.2
–0.2
0.4
]
Determine the Z parameters of the network.
16.49 Find the transmission parameters of the network in
Z2
Fig. E16.4 by considering the circuit to be a cascade
interconnection of three two-port networks as shown
in Fig. P16.49.
Z1
24 Ω
Figure P16.44
3Ω
16.45 Find the Y parameters of the two port in Fig. P16.45.
j4 S
6Ω
2S
3S
Nb
Na
Nc
Figure P16.49
4S
–j 2 S
16.50 Find the transmission parameters of the circuit.
–j 6 S
4Ω
8Ω
6Ω
Figure P16.45
1Ω
2Ω
16.46 Determine the Y parameters ((a) y11, (b) y12, (c) y21,
(d) y22) for the network shown in Fig. P16.46.
Z1
Figure P16.50
Z2
16.51 Find the transmission parameters for the two-port network.
Z4
Z5
Z6
I1
3I1
10 Ω
I2
+–
Z3
20 Ω
Figure P16.46
16.47 Find the Y parameters of the two-port network
Figure P16.51
in Fig. P16.47. Find the input admittance of the network
when the capacitor is connected to the output port.
j2 Ω
16.52 Find the ABCD parameters for the circuit in Fig. P16.52.
1H
1Ω
1F
2Ω
Yin
2Ω
Figure P16.52
2Ω
−j2 Ω
Figure P16.47
c16Two-PortNetworks.indd 21
06/04/15 12:45 pm
16-22
CHAPTER 16
TWO-PORT NETWORKS
16.53 Determine the output voltage Vo in the network in Fig. P16.53 if the Z parameters for the two-port are
Z=
2
3
3
2
12 30° V
VA
VB
+–
j1 Ω
2 60° A
[ ]
+
–j2 Ω
Two–port
1Ω
Vo
–
Figure P16.53
16.54 Find the Z parameters for the two-port network in Fig. P16.54 and then determine Io for the specified terminal conditions.
j1 Ω
8Ω
j2 Ω
57 0° V
–j2 Ω
3Ω
–j4 Ω
j3 Ω
+
–
Io
2Ω
Figure P16.54
Enter arguments from 0 to 2π.
16.55 Determine the output voltage Vo in the network in Fig. P16.55 if the Z parameters for the two-port are
Z=
VA
j1 Ω
2 60° A
[
120 80
80 120
12 30° V
+–
]
VB
–j2 Ω
+
Two–port
1Ω
Vo
–
Figure P16.55
Enter arguments from 0 to 2π.
16.56 Find the transmission parameters of the two-port in Fig. 16.56 and then use the terminal conditions to compute Io.
Io
1:2
10 Ω
9Ω
–j6 Ω
–j3 Ω
24
+ –––
– √2
–45° V
Ideal
Figure P16.56
16.57 The h parameters of a circuit are given as
h11 = 600 Ω, h12 = 10–3, h21 = 120, h22 = 2 × 10–6 S
Draw a circuit model of the device including the value of each element.
c16Two-PortNetworks.indd 22
21/04/15 4:55 pm
16-23
TYPICAL PROBLEMS FOUND ON THE FE EXAM
TYPICAL PROBLEMS FOUND ON THE FE EXAM
16PFE-1 A two-port network is known to have the following
parameters:
1
1
1
y11 = — S y12 = y21 = − — S y22 = — S
21
14
7
If a 2-A current source is connected to the input terminals as shown in Fig. 16PFE-1, find the voltage across
this current source.
16PFE-4 Find the Z parameters of the network shown in
Fig. 16PFE-4.
+
I1
I2
6Ω
V1
12 Ω
3Ω
–
+
V2
–
Figure 16PFE-4
2A
Two–port
Figure 16PFE-1
a. 36 V
c. 24 V
b. 12 V
d. 6 V
16PFE-2 Find the Thévenin equivalent resistance at the output
terminals of the network in Fig. 16PFE-1.
a. 3 Ω
c. 12 Ω
b. 9 Ω
d. 6 Ω
16PFE-3 Find the Y parameters for the two-port network shown
5
19
7
a. z11 = — Ω, z12 = — Ω, z22 = — Ω
3
3
3
7
22
9
b. z11 = — Ω, z12 = — Ω, z22 = — Ω
5
5
5
12
36
18
c. z11 = — Ω, z12 = — Ω, z22 = — Ω
7
7
7
7
27
13
d. z11 = — Ω, z12 = — Ω, z22 = — Ω
6
6
6
16PFE-5 Calculate the hybrid parameters of the network in
Fig. 16PFE-5.
8Ω
2Ω
4Ω
in Fig. 16PFE-3.
16 Ω
4Ω
8Ω
Figure 16PFE-3
5
5
9
a. y11 = — S, y21 = y12 = −— S, y22 = — S
14
32
14
3
7
7
b. y11 = — S, y21 = y12 = − — S, y22 = — S
16
48
16
1
3
4
c. y11 = — S, y21 = y12 = − — S, y22 = — S
15
25
15
1
3
3
d. y11 = — S, y21 = y12 = − — S, y22 = — S
28
56
28
c16Two-PortNetworks.indd 23
Figure 16PFE-5
2
28
2
1
a. h11 = — Ω, h21 = − —, h12 = —, h22 = — S
3
3
3
6
1
16
1
3
b. h11 = — Ω, h21 = − —, h12 = —, h22 = — S
5
5
5
10
3
19
3
5
c. h11 = — Ω, h21 = − —, h12 = —, h22 = — S
4
4
4
8
2
32
2
1
d. h11 = — Ω, h21 = − —, h12 = —, h22 = — S
9
9
9
18
06/04/15 12:45 pm