EE 205 Dr. A. Zidouri
Electric Circuits II
Two-Port Circuits
Two-Port Parameters
Lecture #42
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EE 205 Dr. A. Zidouri
The material to be covered in this lecture is as follows:
o
o
o
o
Introduction to two-port circuits
The Terminal Equations
The Two-Port z-parameters
The Two-Port y-parameters
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EE 205 Dr. A. Zidouri
After finishing this lecture you should be able to:
¾
¾
¾
¾
Understand the Importance of Two-Port Circuits
Relate the Current and Voltage at One Port to the Current and Voltage at the Other Port.
Determine the Two-Port z-parameters
Determine the Two-Port y-parameters
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EE 205 Dr. A. Zidouri
Introduction to Two-Port Circuits
In analyzing some electrical systems, focusing on two pairs of terminal is convenient.
Often, a signal is fed into one pair of terminals and then after being processed, is extracted at a
second pair of terminals.
The terminal pairs represent the points where signals are either fed in or extracted. They are
referred to as ports of the system.
Fig. 42-1 illustrates the basic two-port building block.
Use of this building block is subject to several restrictions:
o There can be no energy stored within the circuit
o There can be no independent sources within the circuit
o The current into the port must equal the current out of the port
o All external connections must be made to either the input port or output port, no
connections are allowed between the ports.
The fundamental principle underlying two-port modeling of a system is that only the terminal
variables (i1, v1, i2, and v2) are of interest.
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EE 205 Dr. A. Zidouri
The Terminal Equations
In two-port network we are interested in relating the current and voltage at one port to the current
and voltage at the other port. Fig. 42-1 shows the reference polarities of the terminal voltages and
the reference directions of the terminal currents.
Most general description is carried out in the s domain.
We write all equations in the s domain, resistive networks and sinusoidal steady state solutions
become special cases.
Fig. 42-2 shows the basic building block in terms of the s-domain variables I1, V1, I2, and V2.
I2
I1
V1
V2
Fig. 42-2 The s-domain Two-Port
Basic Building Block
Out these four terminal variables, only two are independent. Thus we can describe a two-port
network with just two simultaneous equations. However there are six ways in which to combine
the four variables:
Impedance Parameters (z-parameters):
V1 = z11 I1 + z12 I 2
V2 = z21 I1 + z22 I 2
(42-1)
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EE 205 Dr. A. Zidouri
Admittance Parameters (y-parameters):
Hybrid Parameters (h-parameters):
Inverse Hybrid Parameters (g-parameters):
Transmission Parameters (a-parameters):
Inverse Transmission Parameters (b-parameters):
I1 = y11V1 + y12V2
I 2 = y21V1 + y22V2
V1 = h11 I1 + h12V2
I 2 = h21 I1 + h22V2
I1 = g11V1 + g12 I 2
V2 = g 21V1 + g 22 I 2
V1 = a11V2 − a12 I 2
I1 = a21V2 − a22 I 2
V2 = b11V1 − b12 I1
I 2 = b21V1 − b22 I1
(42-2)
(42-3)
(42-4)
(42-5)
(42-6)
These six sets of equations may also be considered as three pairs of mutually inverse relations.
The coefficients of the variables are called the parameters of the two-port circuit. We refer to the
z-parameters, y-parameters, a-parameters, b-parameters, h-parameters and g-parameters of the
network.
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EE 205 Dr. A. Zidouri
The Two-Port Parameters
z-parameters:
V1 = z11 I1 + z12 I 2
V2 = z21 I1 + z22 I 2
or in matrix form:
⎡V1 ⎤ ⎡ z11
⎢V ⎥ = ⎢ z
⎣ 2 ⎦ ⎣ 21
z12 ⎤ ⎡ I1 ⎤
⎡ I1 ⎤
=
z
[ ] ⎢I ⎥
z22 ⎥⎦ ⎢⎣ I 2 ⎥⎦
⎣ 2⎦
(42-7)
The values of the parameters can be evaluated by setting I1=0 (input port open-circuited) or I2=0 (output
port open-circuited). Thus,
z11 =
z21 =
V1
V
, z12 = 1
I1 I =0
I 2 I =0
2
1
(42-8)
V2
V
, z22 = 2
I1 I =0
I 2 I =0
2
1
The z-parameters are also called the open-circuit impedance parameters:
¾ z11= Open-circuit input impedance
¾ z12= Open-circuit transfer impedance from port 1 to port 2
¾ z21= Open-circuit transfer impedance from port 2 to port 1
¾ z22= Open-circuit output impedance
Example 42-1 illustrates the determination of the z-parameters for a resistive circuit.
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EE 205 Dr. A. Zidouri
Example 42-1
Find the z-parameters for a resistive circuit shown in Fig. 42-3
Solution:
To obtain z11 and z21 we connect a voltage V1 (or a current source I1)
to port 1 with port 2 open circuited as in Fig. 42-4a.
i1
i2
v1
v2
Fig. 42-3 The Circuit for Example 42-1
z11 =
20 × 20
V1
=
= 10Ω ,
40
I1 I =0
2
When I2 is zero, V2
=
V1
V
0.75V1 × 20
15 = 0.75V1 therefore z21 = 2
=
= 7.5Ω
15 + 5
I1 I =0
V1 10
2
To obtain z12 and z22 we connect a voltage V2 (or a current source I2) to port 2 with port 1 open circuited
as in Fig. 42-4b.
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EE 205 Dr. A. Zidouri
I2
I1=0
V2
v1
Fig. 42-4b Circuit for finding z12 an z22
z22 =
15 × 25
V2
=
= 9.375Ω ,
40
I 2 I =0
1
When I1 is zero, V1
=
V2
V
0.8V2
V
=
= 7.5Ω
( 20 ) = 0.8V2 and I 2 = 2 hence z12 = 1
5 + 20
I 2 I =0 V2 9.375
9.375
1
Note that each of these parameters is the ratio of a voltage to a current and therefore is an
impedance with the dimension of ohms; this is why they are called z-parameters.
When z11 = z22 , the two-port network is said to be symmetrical.
When the two-port network is linear and has no dependent sources, the transfer impedances are equal
(z12=z21), and the two-port network is said to be reciprocal.
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EE 205 Dr. A. Zidouri
I1 = y11V1 + y12V2
y-parameters:
I 2 = y21V1 + y22V2
or in matrix form:
⎡ I1 ⎤ ⎡ y11
⎢I ⎥ = ⎢ y
⎣ 2 ⎦ ⎣ 21
y12 ⎤ ⎡V1 ⎤
⎡V1 ⎤
=
y
[ ] ⎢V ⎥
y22 ⎥⎦ ⎢⎣V2 ⎥⎦
⎣ 2⎦
(42-9)
The values of the parameters can be evaluated by setting V1=0 (input port short-circuited) or V2=0
(output port short-circuited). Thus,
y11 =
y21 =
I1
I
, y12 = 1
V1 V =0
V2 V =0
2
1
(42-10)
I2
I
, y22 = 2
V1 V =0
V2 V =0
2
1
The y-parameters are also called the short-circuit admittance parameters:
¾ y11= Short-circuit input admittance
¾ y12= Short -circuit transfer admittance from port 2 to port 1
¾ y21= Short -circuit transfer admittance from port 1 to port 2
¾ y22= Short -circuit output admittance
Example 42-2 illustrates the determination of the y-parameters for a resistive circuit.
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EE 205 Dr. A. Zidouri
Example 42-2
Obtain the y-parameters for the resistive circuit shown in Fig. 42-5
Solution:
To obtain y11 and y21 we connect a current I1 (or a voltage source
V1) to input port 1 with output port 2 short circuited as in Fig. 426a.
y11 =
I1
I1
I
=
= 1
= 0.75S ,
4 I
V1 V =0 I1 ( 4 2 )
V2 =0
3 1 V2 =0
2
( )
( )
− 2 I1
4
2
4
I2
3
When V2 is zero, − I 2 =
I1 = I1 and V1 = I1 hence y21 =
=
= −0.5S
4 I
V1 I =0
4+2
3
3
2
3 1 I2 =0
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EE 205 Dr. A. Zidouri
To obtain y12 and y22 we connect a current source I2 (or a voltage source V2) to port 2 with port 1 short
circuited as in Fig. 42-6b.
y22 =
I2
I2
=
V2 V =0 I 2 ( 8 2 )
1
When V1 is zero,
− I1 =
=
V2 =0
I2
8 I
5 2 V1=0
= 0.625S
−0.8 I 2
I
I2
8I
=
= −0.5S
(8 ) = 0.8I 2 and V2 = 2 hence y12 = 1
V2 V =0 1.6 I 2 V =0
8+ 2
5
1
1
Note that each of these parameters is the ratio of a current to a voltage and therefore is an
admittance with the dimension of siemens; this is why they are called y-parameters.
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EE 205 Dr. A. Zidouri
Self Test 42:
a) Determine the z-parameters for the circuit in Fig. 42-7
b) Determine the y-parameters for the circuit in Fig. 42-8
Fig. 42-7 Circuit for self test 42a
Fig. 42-8 Circuit for self test 42b
Answer:
a) z11 = 60Ω
z12 = 40Ω z22 = 70Ω z21 = 40Ω
b) y11 = 0.2273S y12 = y21 = −0.0909 S
y22 = 0.1364S
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