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;
3V1  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 
21 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 