Transcript Document
Guided By: Prof. Naveen Sharma
Two Port Networks
Generalities:
The standard configuration of a two port:
I1
+
V_1
Input
Port
I2
The Network
Output +
V_2
Port
Two Port Networks
Network Equations:
Impedance
Z parameters
V1 = z11I1 + z12I2
Admittance
Y parameters
I1 = y11V1 + y12V2
V2 = z21I1 + z22I2
I2 = y21V1 + y22V2
Two Port Networks
Z parameters:
V
z 1
11 I
1
I 0
2
z11 is the impedance seen looking into port 1
when port 2 is open.
I 0
1
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.
V
z 2
21 I
1
I 0
2
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.
V
2
22 I
2
I 0
1
V
z 1
12 I
2
z
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
V 0
2
y11 is the admittance seen looking into port 1
when port 2 is shorted.
V 0
1
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.
I
y 2
21 V
1
V 0
2
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.
I
y 2
22 V
2
V 0
1
I
y 1
12 V
2
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
+
V1
V
z 1
12 I
2
I2
10
I 0
1
20xI 2 x 20
V1
8 xI 2
20 30
+
20
20
V2
_
_
Therefore:
z12
8 xI 2
8
I2
=
z 21
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
+
Vx
-
2
V2
2Vx
_
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 8 12 I
2
2
Two Port Networks
Z parameters:
V
z 1
11 I
1
Example 2 (continue p2)
I1
I 0
2
+
V
V 2V x
6V x V x 2V x
I1 x x
1
6
6
3V x
I1
2
;
V1
+
1
+
Vx
-
2
V2
2Vx
_
but Vx V1 I 1
Other Answers
Z21 = -0.667
Substituting gives;
3V1 I 1
I1
2
I2
4
1
V1
5
z11
or
I1
3
Z12 = 0.222
Z22 = 1.111
_
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
I2 = y21V1 + y22V2
I1
+
V1
y
I2
1
I
2
21 V
1
I
y 2
22 V
2
V 0
2
+
1
s
s
_
V2
_
1
short We use the above equations to
evaluate the parameters from the
network.
To find y11
2
2
s
V1 I 1 (
) I1
21 s
2 s 1
so
I
y 1
11 V
1
V 0
2
=
s + 0.5
V 0
1
V 0
1
Two Port Networks
Y Parameter Example
y
I
2
21 V
1
V 0
2
I1
+
V1
We see
V1 2I 2
+
1
s
s
_
1
I
y 2
21 V
1
I2
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
_
1
V 0
1
I
y 2
22 V
2
We have
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
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
22
Y
y
21
Y
y
12
Y
y
11
Y
where
Y detY
Two Port Parameter Conversions:
Interconnection Of Two Port Networks
Three ways that two ports are interconnected:
ya
Y parameters
* Parallel
yb
y ya
yb
za
zb
*
Series
Z parameters
z za
zb