Two Port Networks
The standard configuration of a two port:
I1
+
V1
_
Input
Port
I2
The Network
Output
Port
+
V2
_
Two Port Networks
Network Equations:
Impedance Z parameters
Admittance Y parameters
V1 = z11I1 + z12I2
I1 = y11V1 + y12V2
V2 = z21I1 + z22I2
I2 = y21V1 + y22V2
Hybrid H parameters
V1 = h11I1 + h12V2
V2 = b11V1 - b12I1
I2 = h21I1 + h22V2
I2 = b21V1 – b22I1
Transmission A, B, C, D parameters
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
V
z = 2
21 I
1
V
z = 2
22 I
2
I =0
2
I =0
1
I =0
2
I =0
1
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
I
y = 2
21 V
1
I
y = 2
22 V
2
V =0
2
V =0
1
V =0
2
V =0
1
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
8Ω
I2
10 Ω
+
V1
_
+
20 Ω
20 Ω
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 Ω
Z22 = 20||30 = 12 Ω
I1
For z12:
8Ω
+
+
20 Ω
V1
V
z = 1
12 I
2
I2
10 Ω
I =0
1
20 xI 2 x 20
V1 =
= 8 xI 2
20 + 30
20 Ω
_
V2
_
Therefore:
z 12
8 xI 2
=
=8
I2
Ω=
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:
You are given the following circuit. Find the Z parameters.
I1
1Ω
Ω
I2
4Ω
+
V1
_
+
1Ω
Ω
+
Vx
-
2Ω
V2
2Vx
_
Two Port Networks
Z parameters:
V
z = 1
11 I
1
Example 2 (continue p2)
I1
I =0
2
;
I2
4Ω
+
V
V + 2V x
6V x + V x + 2V x
=
I1 = x + x
1
6
6
3V x
I1 =
2
1Ω
Ω
V1
+
1Ω
Ω
+
Vx
-
2Ω
V2
2Vx
_
but V x = V1 − I 1
Answers
Substituting gives;
3(V1 − I 1 )
I1 =
2
V1
5
or
= z 11 = Ω
I1
3
Z21 = -0.667 Ω
Z12 = 0.222 Ω
Z22 = 1.111 Ω
_
Two Port Networks
Transmission parameters (A,B,C,D):
The defining equations are:
V1   A B   V 2 
I  = 



 1  C D   − I 2 
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
I1
C=
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
Two Port Networks
Hybrid Parameters:
V1   h11
 I  = h
 2   21
V1
h11 =
I1
I2
h21 =
I1
The equations for the hybrid parameters are:
h12   I 1 
h22  V2 
V2 = 0
V1
h12 =
V2
V2 = 0
I2
h22 =
V2
I1 = 0
I1 = 0
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
8Ω
I2
10 Ω
+
V1
+
20 Ω
_
V1   20 8   I 1 
V  =  8 12  I 
  2
 2 
20 Ω
V2
_
Two Port Networks
Modifying the two port network:
We modify the network as shown be adding elements outside the two ports
6Ω
Ω
+
10 v
_
I1
8Ω
I2
10 Ω
+
V1
+
20 Ω
_
20 Ω
V2
_
We now have:
V1 = 10 - 6I1
V2 = - 4I2
4Ω
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
V1 = 10 - 6I1
V2 = - 4I2
Two Port Networks
Modifying the two port network:
Rearranging the equations gives,
 I 1   26
I  =  8
 2 
 I
 I

1
2

16

8

=


−1
 10
 0
 0.4545 
 -0.2273 


Two Port Networks
Y Parameters and Beyond:
Given the following network.
I1
+
V1
I2
1Ω
+
1
s
s
_
V2
_
1Ω
(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
+
V1
I
y = 2
21 V
1
I2
1Ω
I
y = 2
22 V
2
V =0
2
+
1
s
s
_
1Ω
V2
_
short
To find y11
2
s )= I  2 
V1 = I 1 (
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
I
y = 2
21 V
1
V =0
2
+
V1
I2
1Ω
+
1
s
s
_
1Ω
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
+
1
s
s
_
1Ω
V =0
1
I
y = 2
22 V
2
We have
V2 = − 2I1
I
y = 1
12 V 2
short
V =0
1
2s
We have V2 = I 2
( s + 2)
= 0.5 S
I2
1Ω
1
y22 = 0.5 +
s
V2
_
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;
V =Y
−1
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
= Det Y
Two Port Parameter Conversions:
To
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 
− I2 


1


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
2  R + R  ( − I )
 1

V1 =   1
+
1
2 
  R

 1
 R
 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