An Introduction To
Two – Port Networks
The University of Tennessee
Electrical and Computer Engineering
Knoxville, TN
wlg
Two Port Networks
Generalities:
The standard configuration of a two port:
I1
+
V1
_
Input
Port
I2
The Network
The network ?
The voltage and current convention ?
* notes
Output
Port
+
V2
_
Two Port Networks
Network Equations:
Impedance
Z parameters
Admittance
Y parameters
Transmission
A, B, C, D
parameters
* notes
V1 = z11I1 + z12I2
V2 = b11V1 - b12I1
V2 = z21I1 + z22I2
I2 = b21V1 – b22I1
I1 = y11V1 + y12V2
V1 = h11I1 + h12V2
I2 = y21V1 + y22V2
Hybrid
H parameters
I2 = h21I1 + h22V2
V1 = AV2 - BI2
I1 = g11V1 + g12I2
I1 = CV2 - DI2
V2 = g21V1 + g22I2
Two Port Networks
Z parameters:
V
z  1
11 I
1
V
z  1
12 I
2
I 0
1
V
z  2
21 I
1
I 0
2
V
 2
22 I
2
I 0
1
z
* notes
I 0
2
z11 is the impedance seen looking into port 1
when port 2 is open.
z12 is a transfer impedance. It is the ratio of the
voltage at port 1 to the current at port 2 when
port 1 is open.
z21 is a transfer impedance. It is the ratio of the
voltage at port 2 to the current at port 1 when
port 2 is open.
z22 is the impedance seen looking into port 2
when port 1 is open.
Two Port Networks
Y parameters:
I
y  1
11 V
1
I
y  1
12 V
2
V 0
1
I
y  2
21 V
1
V 0
2
I
 2
22 V
2
V 0
1
y
* notes
V 0
2
y11 is the admittance seen looking into port 1
when port 2 is shorted.
y12 is a transfer admittance. It is the ratio of the
current at port 1 to the voltage at port 2 when
port 1 is shorted.
y21 is a transfer impedance. It is the ratio of the
current at port 2 to the voltage at port 1 when
port 2 is shorted.
y22 is the admittance seen looking into port 2
when port 1 is shorted.
Two Port Networks
Z parameters:
Example 1
Given the following circuit. Determine the Z parameters.
I1
8
I2
10 
+
V1
_
+
20 
20 
V2
_
Find the Z parameters for the above network.
Two Port Networks
Z parameters:
Example 1 (cont 1)
For z11:
For z22:
Z11 = 8 + 20||30 = 20 
Z22 = 20||30 = 12 
I1
For z12:
8
+
+
20 
V1
V
z  1
12 I
2
I2
10 
I 0
1
20 xI 2 x 20
V1 
 8 xI 2
20  30
20 
_
V2
_
Therefore:
z12
8 xI 2

8
I2

=
z 21
Two Port Networks
Z parameters:
Example 1 (cont 2)
The Z parameter equations can be expressed in
matrix form as follows.
V1   z11
V    z
 2   21
z12   I 1 



z 22   I 2 
V1  20 8   I 1 
V   
I 

 2   8 12  2 
Two Port Networks
Z parameters:
Example 2 (problem 18.7 Alexander & Sadiku)
You are given the following circuit. Find the Z parameters.
I1
I2
4
1
+
+
+
V1
_
1
2
Vx
-
V2
2Vx
_
Two Port Networks
Z parameters:
V
z  1
11 I
1
Example 2 (continue p2)
I1
I 0
2
+
;
V1
1
2
Vx
-
V2
2Vx
_
but V x  V1  I 1
Other Answers
Z21 = -0.667 
Substituting gives;
3V1  I 1 
I1 
2
+
+
V
V  2V x
6V x  V x  2V x
I1  x  x

1
6
6
3V x
I1 
2
I2
4
1
V1
5
 z11  
or
I1
3
Z12 = 0.222 
Z22 = 1.111 
_
Two Port Networks
Transmission parameters (A,B,C,D):
The defining equations are:
V1   A B   V2 
I   
  I 
C
D
 2 
 1 
V1
A
V2
I1
C
V2
I2 = 0
V1
B
 I2
V2 = 0
I2 = 0
I1
D
 I2
V2 = 0
Two Port Networks
Transmission parameters (A,B,C,D):
Example
Given the network below with assumed voltage polarities and
Current directions compatible with the A,B,C,D parameters.
I1
+
-I2
+
R1
V1
R2
_
V2
_
We can write the following equations.
V1 = (R1 + R2)I1 + R2I2
V2 = R2I1 + R2I2
It is not always possible to write 2 equations in terms of the V’s and I’s
Of the parameter set.
Two Port Networks
Transmission parameters (A,B,C,D):
Example (cont.)
V1 = (R1 + R2)I1 + R2I2
V2 = R2I1 + R2I2
From these equations we can directly evaluate the A,B,C,D parameters.
V1
A
V2
C
I1
V2
I2 = 0
I2 = 0
=
=
R1  R2
R2
1
R2
V1
B
 I 2 V2 = 0
=
I1
D
 I2
=
V2 = 0
R1
1
Later we will see how to interconnect two of these networks together for a final answer
* notes
Two Port Networks
Hybrid Parameters:
V1   h11
 I   h
 2   21
V1
h11 
I1
I2
h21 
I1
* notes
V2 = 0
V2 = 0
The equations for the hybrid parameters are:
h12   I 1 
h22  V2 
V1
h12 
V2
I2
h22 
V2
I1 = 0
I1 = 0
Two Port Networks
The following is a popular model used to represent
a particular variety of transistors.
Hybrid Parameters:
I1
I2
K1
+
+
+
V1
K2V2
_
K3V1
_
V2
_
We can write the following equations:
V1  AI 1  BV2
V2
I 2  CI 1 
D
* notes
K4
Two Port Networks
Hybrid Parameters:
V1  AI 1  BV2
V2
I 2  CI 1 
D
We want to evaluate the H parameters from the above set of equations.
V1
h11 
I1
I2
h21 
I1
=
K1
V2 = 0
=
V2 = 0
K3
V1
h12 
V2
I2
h22 
V2
=
K2
I1 = 0
=
I1 = 0
1
K
4
Two Port Networks
Another example with hybrid parameters.
Hybrid Parameters:
Given the circuit below.
I1
-I2
+
The equations for the circuit are:
+
R1
V1
R2
V2
_
_
V1 = (R1 + R2)I1 + R2I2
V2 = R2I1 + R2I2
The H parameters are as follows.
h11 
h21 
V1
I1
I2
I1
V2=0
=
=
V2=0
R1
-1
h12 
V1
V2
h22 
I2
V2
I1=0
I1=0
=
1
=
1
R2
Two Port Networks
Modifying the two port network:
Earlier we found the z parameters of the following network.
I1
8
I2
10 
+
V1
+
20 
_
V1   20 8   I 1 
V    8 12  I 
  2
 2 
* notes
20 
V2
_
Two Port Networks
Modifying the two port network:
We modify the network as shown be adding elements outside the two ports
6
+
10 v
_
I1
8
I2
10 
+
V1
+
20 
_
20 
V2
_
We now have:
V1 = 10 - 6I1
V2 = - 4I2
4
Two Port Networks
Modifying the two port network:
We take a look at the original equations and the equations describing
the new port conditions.
V1   20 8   I 1 
V    8 12  I 
  2
 2 
So we have,
10 – 6I1 = 20I1 + 8I2
-4I2 = 8I1 + 12I2
* notes
V1 = 10 - 6I1
V2 = - 4I2
Two Port Networks
Modifying the two port network:
Rearranging the equations gives,
 I 1   26
I   8
 2 


16

8
1
10 
 0 
 I1 
0.4545 
 I   -0.2273 
 
 2
Two Port Networks
Y Parameters and Beyond:
Given the following network.
I1
+
V1
I2
1
+
1
s
s
_
V2
_
1
(a) Find the Y parameters for the network.
(b) From the Y parameters find the z parameters
Two Port Networks
Y Parameter Example
I
y  1
11 V
1
I1 = y11V1 + y12V2
I
y  1
12 V
2
V 0
2
V 0
1
I2 = y21V1 + y22V2
I1
I2
1
+
V1
y
I
 2
21 V
1
y
V 0
2
I
 2
22 V
2
+
1
s
s
_
V2
_
short
1
To find y11
V1  I 1 (
2
s ) I  2 
1
21 s
 2 s  1 
so
We use the above equations to
evaluate the parameters from the
network.
I
y  1
11 V
1
V 0
2
=
s + 0.5
V 0
1
Two Port Networks
Y Parameter Example
I1
y
I
 2
21 V
1
V 0
2
+
V1
I2
1
+
1
s
s
_
1
We see
V1   2I 2
I
y  2
21 V
1
= 0.5 S
V2
_
Two Port Networks
Y Parameter Example
I1
To find y12 and y21 we reverse
things and short V1
I
y  1
12 V
2
+
V1
short
I2
1
+
1
s
s
_
V2
_
1
V 0
1
We have
y
I
 2
22 V
2
V 0
1
We have
V2   2I1
I
y  1
12 V 2
= 0.5 S
2s
V2  I 2
( s  2)
1
y22  0.5 
s
Two Port Networks
Y Parameter Example
Summary:
Y
=
 y11
y
 21
y12   s  0.5
 0.5 


y22    0.5 0.5  1 s 
Now suppose you want the Z parameters for the same network.
Two Port Networks
Going From Y to Z Parameters
For the Y parameters we have:
For the Z parameters we have:
V Z I
I Y V
From above;
1
V Y I  Z I
Therefore
Z Y
1
z 
z
11
12
 
 
z
z
 21 22 
y 
 y
22
12




 Y
Y 
 y

y
11 
 21
 
 
 Y
Y 
where
Y  detY
Two Port Parameter Conversions:
Two Port Parameter Conversions:
To go from one set of parameters to another, locate the set of parameters
you are in, move along the vertical until you are in the row that contains
the parameters you want to convert to – then compare element for element
H
z11 
h22
Interconnection Of Two Port Networks
Three ways that two ports are interconnected:
*
Y parameters
ya
Parallel
 y ya  
yb
*
Z parameters
z  za  
za
Series
yb 
zb
zb 
ABCD parameters
* Cascade
Ta
Tb
T  Ta  Tb 
Interconnection Of Two Port Networks
Consider the following network:
I1
Find
V2
V1
R1
I2
R1
+
+
V1
_
T1
R2
T2
R2
V2
_
Referring to slide 13 we have;
 R1  R2

V1   R
2
I   
 1  1
 R2
 R  R
2
R1   1
  R2
 1
1 
  R2

R1  V
 2 



I

1  2


Interconnection Of Two Port Networks
 R1  R2

V1   R
2
I   
 1  1
 R2
 R  R
2
R1   1
  R2
 1
1 
  R2

R1  V


 2 
I
1   2


Multiply out the first row:
  R  R  2 R 

 R  R 

2   1  V   1
2  R  R  (  I )
V1    1
1
2 
  R

 R
 1
 2
R




2
2
2


 

Set I2 = 0 ( as in the diagram)
V2
V1

R2 2
R1  3 R1 R2 R2 2
2
Can be verified directly
by solving the circuit
Basic Laws of Circuits
End of Lesson
Two-Port Networks