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1. Fundamentals of Electric Circuits
AC Circuits
Chapter 16. Two-port networks
16.1. Introduction
16.2. Impedance parameters
16.3. Admittance parameters
16.4. Hybrid parameters
16.5. Transmission parameters
16.6. Interconnection of networks

2. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.1. Introduction
+ Port: A pair of terminals through which a current may enter or leave a
network  is an access to the network and consists of a pair of terminals
+ One-port networks: two-terminal devices or elements (R, L, C)
+ Two-port networks: four-terminal devices (op amps, transistors, transformers)
A two-port network is an electrical network with two separate
ports for input and output
+ Study of two-port networks:
 Useful in communications, control systems, power systems,…
 Treat circuit as a “black box” when embedded within a larger network

3. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.2. Impedance parameters
+ Impedance & admittance parameters are commonly used in the
synthesis of filters
+ A two-port network may be voltage-driven or current driven  the
terminal voltage can be related to the terminal currents as:







2
22
1
21
2
2
12
1
11
1
I
Z
I
Z
V
I
Z
I
Z
V






  

























2
1
2
1
22
21
12
11
2
1
I
I
I
I
Z
Z
Z
Z
V
V






Z
Z: impedance parameters

4. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.2. Impedance parameters
+ The value of the parameters: open circuit impedance
Open circuit input impedance: 0
2
1
1
11 
 I
I
V
Z 


Open circuit transfer impedance
from port 2 to port 1:
0
1
2
1
12 
 I
I
V
Z 


Open circuit transfer impedance
from port 1 to port 2:
0
2
1
2
21 
 I
I
V
Z 


Open circuit ouput impedance: 0
1
2
2
22 
 I
I
V
Z 



5. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.2. Impedance parameters
+ Characteristics of impedance parameters
 two-port network is said to be symmetrical when Z11 = Z22
 two-port network is said to be reciprocal when Z12 = Z21 (a linear two- port network and no
dependent sources
 The T-equivalent circuit:

6. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.2. Impedance parameters
R1 R2
R3
R3
R1 R2
R1 R2
R
3
. .
I V
2
.
I1
.
V1 2
+ Example 1: Determine the z-parameters for the given circuit
Solution
Method 1: Using definition equation
 
3
1
1
1
3
1
0
1
1
11 2
R
R
I
I
R
R
I
V
Z I




  



 3
1
1
3
0
1
2
21 2
R
I
I
R
I
V
Z I


  




 Open the output port: I2 = 0
 Open the intput port: I1 = 0
 
3
2
2
2
3
2
0
2
2
22 1
R
R
I
I
R
R
I
V
Z I




  



 3
2
2
3
0
2
1
12 1
R
I
I
R
I
V
Z I


  




Method 2: Using mesh current method
 
 









2
3
2
1
3
2
2
3
1
3
1
1
I
R
R
I
R
V
I
R
I
R
R
V







7. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.3. Admittance parameters
+ There are some cases (i.e. ideal transformer) that the impedance
parameters may not exist for a two-port network  need an alternative
means of describing
+ Express the terminal currents in terms of the terminal voltages 
admittance parameters







2
22
1
21
2
2
12
1
11
1
V
Y
V
Y
I
V
Y
V
Y
I






  

























2
1
2
1
22
21
12
11
2
1
V
V
V
V
Y
Y
Y
Y
I
I






Y
Y: admittance parameters

8. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.3. Admittance parameters
+ The value of the parameters: short circuit admittance
Short circuit input admittance:
0
1
1
11 2 
 V
V
I
Y 


Short circuit transfer admittance
from port 2 to port 1: 0
1
2
21 2 
 V
V
I
Y 


Short circuit transfer admittance
from port 1 to port 2: 0
2
1
12 1 
 V
V
I
Y 


Short circuit ouput admittance : 0
2
2
22 1 
 V
V
I
Y 



9. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.3. Admittance parameters
+ Characteristics of admittance parameters
 two-port network is said to be symmetrical when Y11 = Y22
 two-port network is said to be reciprocal when Y12 = Y21 (a linear two- port network and no
dependent sources
 The Π-equivalent circuit:

10. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.3. Admittance parameters
g1 g2
g3
g1 g2
g3
g1 g2
g3
.
V1
.
V2
.
I1
.
I2
+ Example 2: Determine the Y-parameters for the given circuit
Solution
Method 1: Using definition equation
 Shorten the output port: V2 = 0
 Shorten the intput port: V1 = 0
Method 2: Using node voltage method
 
 










2
3
2
1
3
2
2
3
1
3
1
1
V
g
g
V
g
I
V
g
V
g
g
I






 
3
1
1
1
3
1
0
1
1
11 2
g
g
V
V
g
g
V
I
Y V




  



 3
1
1
3
0
1
2
21 2
g
V
V
g
V
I
Y V




  




 
3
2
2
2
3
2
0
2
2
22 1
g
g
V
V
g
g
V
I
Y V




  



 3
2
2
3
0
2
1
12 1
g
V
V
g
V
I
Y V




  





11. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.3. Admittance parameters
+ Example 3: Obtain the Y parameters for the given circuit
Solution
Shorten the output port
0
1
0
0
0
1
0
0
1
0
1
5
4
2
8
2
4
2
2
8
V
V
V
V
V
V
V
V
I
V
V 





















At node 1:
S
V
I
Y
V
V
V
V
V
I 15
.
0
75
.
0
8
5
8 1
1
11
0
0
0
0
1
1 

















At node 2:
S
V
V
V
I
Y
V
I
I
I
V
25
.
0
5
25
.
1
2
25
.
1
0
2
4
0
0
0
1
2
21
0
2
2
1
1





















12. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.3. Admittance parameters
+ Example 3: Obtain the Y parameters for the given circuit
Solution
Shorten the input port
0
2
2
0
0
0
2
0
0
1
0
5
.
2
4
2
8
2
4
2
2
8
0
V
V
V
V
V
V
V
V
V
I
V 























At node 1:
S
V
V
V
I
Y 05
.
0
5
.
2
1
8 0
0
2
1
12 








At node 2:
S
V
V
V
I
Y
V
I
I
I
V
V
25
.
0
5
.
2
625
.
0
625
.
0
0
2
4 0
0
2
2
22
0
2
2
1
2
0




















 network is not reciprocal

13. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.4. Hybrid parameters
+ Input voltage and output current as function of input current and output voltage of a two-port network
  

























2
1
2
1
22
21
12
11
2
1
V
I
V
I
H
H
H
H
I
V






H
+ Value of the parameters:
Short circuit input impedance 0
1
1
11 2 
 V
I
V
H 


Short circuit forward current gain 0
1
2
21 2 
 V
I
I
H 

 Open circuit output admittance 0
2
2
22 1 
 I
V
I
H 


Open circuit reverse voltage gain 0
2
1
12 1 
 I
V
V
H 



14. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.4. Hybrid parameters
+ Input current and output voltage of a two-port network as function of input voltage and output current
 G parameters
  

























2
1
2
1
22
21
12
11
2
1
I
V
I
V
G
G
G
G
V
I






G
+ Value of the parameters:
Open circuit input admittance 0
1
1
11 2 
 I
V
I
G 


Open circuit forward voltage gain 0
1
2
21 2 
 I
V
V
G 

 Short circuit output impedance 0
2
2
22 1 
 V
I
V
G 


Short circuit reverse current gain 0
1
1
12 1 
 V
V
I
G 



15. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.5. Transmission parameters
+ Transmission parameters:  relates the variables at the input port to
those at the output port
  

























2
2
2
2
22
21
12
11
1
1
I
V
I
V
A
A
A
A
I
V






A
+ Transimission parameters  useful in the analysis of transmission
lines (cable, fiber) and in the design of telephone system, microwave
network,…
+ A - parameters:
+ Value of the A parameters:
Open circuit voltage ratio 0
2
1
11 2 
 I
V
V
A 


Open circuit transfer admittance 0
2
1
21 2 
 I
V
I
A 


Short circuit transfer impedance 0
2
1
12 2 
 V
I
V
A 


Short circuit current ratio 0
2
1
22 2 
 V
I
I
A 



16. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.5. Transmission parameters
  

























1
1
1
1
22
21
12
11
2
2
I
V
I
V
B
B
B
B
I
V






B
+ B - parameters:
+ Inverse transmission parameters  express the variables at the output port in term of the variables at the
input port
+ Value of the B - parameters:
Open circuit voltage gain 0
1
2
11 1 
 I
V
V
B 


Open circuit transfer admittance 0
1
2
21 1 
 I
V
I
B 


Short circuit transfer impedance 0
1
2
12 1 
 V
I
V
B 


Short circuit current gain 0
1
2
22 1 
 V
I
I
B 


+ Reciprocal network: 1
21
12
22
11 
 A
A
A
A
1
21
12
22
11 
 B
B
B
B

17. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.5. Transmission parameters
. .
+ Example 4: Find the transmission parameters for the given circuit
 Open the output port: I2 = 0
Solution
 
1
1
1
2
1
1
1
17
3
20
30
20
10
I
I
I
V
I
I
V













765
.
1
17
30
0
2
1
11 2


 
I
V
V
A 


059
.
0
17 1
1
0
2
1
21 2


 
I
I
V
I
A I 




 Shorten the output port: V2 = 0
1
2
1
3
0
20
10
I
V
I
V
V
V
a
a
a











At node A:
10
1
1
a
V
V
I


 

 





 
29
.
15
20
/
17
13
1
1
0
2
1
12 2
I
I
I
V
A V 




176
.
1
20
3
10
3
13 1
1
1
1
0
2
1
22 2






 
I
I
I
I
I
I
A V 







18. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.6. Interconnection of networks
+ Large, complex network  divided into sub-networks (2 port network) for the purposes of analysis and
design
+ Two-port networks - as building blocks - that can be interconnected (in series, in parallel, or in cascade) to
form a complex network
+ The value of parameters of the complex network:  calculated from the value of each parameters of
each building block

19. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.6. Interconnection of networks
Series connection
[Z] = [Za] + [Zb]
Parallel connection
[Y] = [Ya] + [Yb]
Cascade connection
[T] = [Ta][Tb]

20. FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits
16.6. Interconnection of networks
Zn2
Zd
Zn1
Tn2
Td
Tn1
T  Tn1.Td .Tn2
+ Example 5: Find the transmission parameters of the given Pi circuit.

I 2 
V2

V1

I1 Zd V1
  V2

 I1   I2 
1 Z
.
1   
0
d 

 
 
    
 

V2

V1 Zn1

I 2

I1
V1
  V2

 I1    I2 
 Zn1
1
1
0  
.
1  
 
 


 
 
 

T  Tn1.Td .Tn2   1
 Zn1   Zn2 
1 0
1 Z
1 0

1  1
10 1
d 
 

 
 
 
Zn2

 Zn1 Zn2 Zn1.Zn2 Zn1 
1
Zd
 1

1
 Zd
1

d
Zd
Z
 T  





