An Introduction To
Two – Port Networks
Two Port Networks
Generalities: The standard configuration of a two port:
I1 I2
+ +
Input Output
V1
_ Port The Network Port
V2
_
The voltage and current convention
Two Port Networks
Network Equations:
Impedance
V1 = z11I1 + z12I2
Inverse Transmission
V2 = A’V1 – B’I1
Z parameters parameters
V2 = z21I1 + z22I2 I2 = C’V1 – D’I1
I1 = y11V1 + y12V2 V1 = h11I1 + h12V2
Admittance Hybrid
Y parameters H parameters I2 = h21I1 + h22V2
I2 = y21V1 + y22V2
Transmission V1 = AV2 - BI2 Inverse Hybrid I1 = g11V1 + g12I2
A, B, C, D H parameters
parameters I1 = CV2 - DI2 V2 = g21V1 + g22I2
Z parameters/Impedance Parameters/
Open Circuit Parameters:
• In matrix form as:
V1 z11 z12 I1
V = z
2 21 z 22 I 2
• The Z-parameter that we want to determine are z11, z12, z21, z22.
• The value of the parameters can be evaluated by setting:
1. I1= 0 (input port open – circuited)
2. I2= 0 (output port open – circuited)
Z parameters/Impedance Parameters/ V1 = z11I1 + z12I2
Open Circuit Parameters: V2 = z21I1 + z22I2
V z11 is the impedance seen looking into port 1
z = 1
11 I I =0 when port 2 is open.
1 2
V z12 is a transfer impedance. It is the ratio of the
z = 1 voltage at port 1 to the current at port 2 when
12 I I =0
2 1 port 1 is open.
V z21 is a transfer impedance. It is the ratio of the
z = 2
21 I I =0 voltage at port 2 to the current at port 1 when
1 2
port 2 is open.
V
z = 2 z22 is the impedance seen looking into port 2
22 I I =0
2 1 when port 1 is open.
* notes
Z parameters: Example 1
Given the following circuit. Determine the Z parameters.
I1 I2
8 10
+ +
V1 20 20 V2
_ _
Find the Z parameters for the above network.
Z parameters: Example 1 (cont 1)
For z11: For z22:
Z11 = 8 + 20||30 = 20 Z22 = 20||30 = 12
I1 I2
8 10
For z12:
+ +
V1 20 20 V2
V
z = 1 _ _
12 I I =0
2 1
20 xI 2 x 20 Therefore: 8 xI 2
V1 = = 8 xI 2 z12 = =8 = z 21
20 + 30 I2
Z parameters: Example 1 (cont 2)
The Z parameter equations can be expressed in
matrix form as follows.
V1 z11 z12 I 1
V = z
2 21 z 22 I 2
V1 20 8 I 1
V = I
2 8 12 2
Z parameters: Example 2 (problem 18.7 Alexander & Sadiku)
You are given the following circuit. Find the Z parameters.
I1 I2
1 4
+ +
+ 2
1 Vx
V1 V2
- 2Vx
_
_
Z parameters: Example 2 (continue p2)
I1 I2
V 1 4
z = 1
11 I I =0 + +
1 2 + 2
1 Vx
V1 V2
-
V V + 2V x 6V x + V x + 2V x 2Vx
I1 = x + x = _
_
1 6 6
3V x
I1 = ; but V x = V1 − I 1 Other Answers
2
Z21 = -0.667
Substituting gives;
Z12 = 0.222
3(V1 − I 1 ) or
V1 5
= z11 =
I1 =
2 I1 3
Z22 = 1.111
Y parameters/Admittance Parameters/
Short Circuit Parameters:
I1 y11 y12 V1
I = y
2 21 y22 V2
Y parameters/Admittance Parameters/ I1 = y11V1 + y12V2
Short Circuit Parameters: I2 = y21V1 + y22V2
I y11 is the admittance seen looking into port 1
y = 1
11 V V =0 when port 2 is shorted.
1 2
I y12 is a transfer admittance. It is the ratio of the
y = 1 current at port 1 to the voltage at port 2 when
12 V V =0
2 1 port 1 is shorted.
I y21 is a transfer admittance. It is the ratio of the
y = 2
21 V V =0 current at port 2 to the voltage at port 1 when
1 2
port 2 is shorted.
I
y = 2 y22 is the admittance seen looking into port 2
22 V V =0
2 1 when port 1 is shorted.
* notes
Example
Find the Y – parameter of the circuit shown below.
5Ω
I1 I2
+ +
V1 20Ω 15Ω V2
_ _
Solution
i) V2 = 0
5Ω I2
I1
+
V1 20Ω
Ia
_
ii) V1 = 0
I1 5Ω
I2
+
15Ω Ix V2
_
• In matrix form;
1 1
4 −
Y = 1 5 S
4
−
5 15
T (ABCD) PARAMETER
• T – parameter or ABCD – parameter is another set of
parameters relates the variables at the input port to those at
the output port.
• T – parameter also called transmission parameters because
this parameter are useful in the analysis of transmission lines
because they express sending – end variables (V1 and I1) in
terms of the receiving – end variables (V2 and -I2).
• The “black box” that we want to replace with T –
parameter is as shown below.
I1 I2
A11 B12
+ +
V1 V2
- -
C21 D22
• The equation is: V1 = AV2 − BI 2 .......(1)
I1 = CV2 − DI 2 .......(2)
• In matrix form is:
V1 A B V2
I = C D − I
1 2
• The T – parameter that we want determine are A, B, C and D
where A and D are dimensionless, B is in ohm (Ω) and C is in
siemens (S).
• The values can be evaluated by setting
i) I2 = 0 (output port open – circuit)
ii) V2 = 0 (output port short circuit)
• Thus;
V1 V1
A= B=
V2 I 2 =0
I2 V2 = 0
I1 I1
C= D=
V2 I 2 =0
I2 V2 = 0
Example
Find the ABCD – parameter of the circuit
shown below.
I1 2Ω 4Ω I2
+ +
V1 10Ω V2
_ _
Solution
i) I2 = 0,
I1 2Ω
+ +
V1 10Ω V2
_ _
10
ii) V2 = 0, I 2 = − I1
14
I1 2Ω 4Ω I2 I1
D = − = 1 .4
I2
+
V1 = 2 I1 + 10(I1 + I 2 )
V1 10Ω
I1 + I2 V1 = 12 I1 + 10 I 2
_
14
V1 = 12 − I 2 + 10 I 2
10
1.2 6.8
T =
V1
B = − = 6.8
0 .1 1 . 4 I2
CONDITION FOR SYMMETRY
AND RECIPROCITY OF Z PARAMETERS
Symmetry : For the network to be symmetrical, the voltage-to-
current ratio at one port should be the same as the voltage-to-current
ratio at the other port with one of the port open circuited.
Condition for symmetry
23
A network is said to be reciprocal if the the ratio of
excitation at one port to response at other port is
Reciprocity : same if excitation and responses are
interchanged
Condition for Reciprocity
24
CONDITION FOR SYMMETRY
AND RECIPROCITY OF Y PARAMETERS
Symmetry : For the network to be symmetrical, the voltage-to-
current ratio at one port should be the same as the voltage-to-current
ratio at the other port with one of the port short circuited.
Condition for symmetry
25
A network is said to be reciprocal if the the ratio of excitation at
one port to response at other port is same if excitation and
responses are interchanged
Condition for Reciprocity
26
CONDITION FOR SYMMETRY
AND RECIPROCITY OF ABCD PARAMETERS
Symmetry : For the network to be symmetrical, the voltage-to-
current ratio at one port should be the same as the voltage-to-current
ratio at the other port with one of the port short circuited.
Condition for symmetry
27
Reciprocity :
28
Two Port Networks
Hybrid Parameters: The equations for the hybrid parameters are:
V1 h11 h12 I 1
I = h h22 V2
2 21
V1 V1
h11 = h12 =
I1 V2 = 0 V2 I1 = 0
I2 I2
h21 = h22 =
I1 V2 = 0 V2 I1 = 0
* notes
CONDITION FOR SYMMETRY
AND RECIPROCITY OF h PARAMETERS
Condition for
symmetry
30
Reciprocity :
31
Two Port Networks
Hybrid Parameters: Another example with hybrid parameters.
Given the circuit below.
I1 -I2
The equations for the circuit are:
+ +
R1
V1 = (R1 + R2)I1 + R2I2
V1 R2 V2
_ V2 = R2I1 + R2I2
_
The H parameters are as follows.
V1 V1
h11 = = h12 = = 1
I1 V2=0
R1 V2 I1=0
I2 I2 1
h21 = = --11 h22 = =
I1 V2=0 V2 I1=0 R2
SYNOPSIS OF ALL PARAMETERS
33
Inter-relationships between parameters
Inter-relationships between parameters
Given the following network.
I1 1
I2
+ +
1 s
V1 V2
s
_ _
1
(a) Find the Y parameters for the network.
(b) From the Y parameters find the z parameters
Y Parameter Example
I I
y = 1 y = 1
I1 = y11V1 + y12V2 11 V V =0 12 V
1 2 2 V =0
1
I2 = y21V1 + y22V2
I I
I2 y = 2 y = 2
I1 1 21 V V =0 22 V V =0
1 2 2 1
+ +
1 s
V1
s V2
short
_ _ We use the above equations to
1 evaluate the parameters from the
network.
To find y11
2
V1 = I 1 ( s )= I 2 so
I
y = 1 s + 0.5
1
2 s + 1
=
2+1 s 11 V
1 V =0
2
Y Parameter Example
I1 I2
1
I
y = 2 + +
21 V V =0
1 2 1 s
V1
s V2
_ _
1
We see
I
V1 = − 2I 2 y = 2
21 V = 0.5 S
1
Y Parameter Example
I1 I2
1
To find y12 and y21 we reverse short + +
1 s
things and short V1 V1
s V2
_ _
I 1
y = 1
12 V V =0
2 1
I
We have y = 2
22 V V =0
2 1
We have
V2 = − 2I1
2s 1
V2 = I 2 y22 = 0.5 +
I
( s + 2) s
y = 1 = 0.5 S
12 V 2
Y Parameter Example
Summary:
y11 y12 s + 0.5 − 0 .5
y =
y22 − 0.5 0.5 + 1 s
Y =
21
Now suppose you want the Z parameters for the same network.
Going From Y to Z Parameters
For the Y parameters we have: For the Z parameters we have:
I =Y V V =Z I
From above; V = Y −1 I = Z I
Therefore
y −y
22 12
where
z z
Y
−1
= Y Y = det Y
Z =Y 11 12
= − y
21 22
z z
21
y
11
Y Y
Two Port Parameter Conversions:
Interconnection Of Two Port Networks
Interconnection Of Two Port Networks
Three ways that two ports are interconnected:
ya Y parameters
y = ya + yb
* Parallel
yb
Z parameters
z = za + zb
za
* Series
zb
ABCD parameters
* Cascade Ta Tb T = Ta Tb
Interconnection Of Two Port Networks
Consider the following network:
I1 I2
R1 R1
V2
Find + +
V1
V1 R2 R2 V2
T1 T2
_ _
Referring to slide 13 we have;
R1 + R2 R + R
R1 1 2 R1 V
V1 R R2 2
I = 2
1 1 1 − I
1 1 2
R2 R2
Interconnection Of Two Port Networks
R1 + R2 R + R
R1 1 2 R1 V
V1 R R2
I = 2 2
1 −I
1 1 1 1 2
R2 R2
Multiply out the first row:
R + R 2 R R + R
V1 = 1 2 + 1 V + 1 2 R + R ( − I )
R R 2 R 1 1
2
2 2 2
Set I2 = 0 ( as in the diagram)
V2 R2 2
= Can be verified directly
R1 + 3 R1 R2 R2 2 by solving the circuit
2
V1
Example : Two identical sections of the network shown in figure
are connected in series .Obtain Z parameters of over all network.
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