This document discusses two-port networks and various parameters used to describe them, including impedance parameters, admittance parameters, and hybrid parameters. Impedance parameters relate terminal voltages to terminal currents, admittance parameters relate terminal currents to terminal voltages, and hybrid parameters relate input voltage to output current and vice versa. Examples are provided to demonstrate calculating the parameters for simple two-port networks.

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  





