ENE 104
Electric Circuit Theory
Lecture 12:
Two-Port Networks
Week #13 : Dejwoot KHAWPARISUTH
http://webstaff.kmutt.ac.th/~dejwoot.kha/
Objectives : Ch17
Page 2
Objectives include:
• Learning the distinction between one-port and
two-port network
• Techniques for characterizing networks by 𝐲,
𝐳, 𝐡, and 𝐭 parameters
• Transformation methods between 𝐲, 𝐳, 𝐡, and
𝐭 parameters
• Performing circuit analysis using network
parameters, including cascaded networks
ENE 104
Two-Port Networks
Week #13
Introduction:
Page 3
The circuit is linear, and the ability to measure
voltages and currents:
It is possible to characterize such a
network with a set of parameters that allow us to
predict how the network will interact with other
networks.
ENE 104
Two-Port Networks
Week #13
A multiport Network:
Page 4
A one-port network.
ia must be equal ib
A two-port network.
ia  ib
ENE 104
ic  id
Two-Port Networks
Week #13
A multiport Network:
Page 5
Z11I1  Z12I 2  Z13I 3  ...  Z1N I N  V1
Z 21I1  Z 22I 2  Z 23I 3  ...  Z 2 N I N  V2
Z 31I1  Z 32I 2  Z33I 3  ...  Z3 N I N  V3
.......................................................
Z N 1I1  Z N 2I 2  Z N 3I 3  ...  Z NN I N  VN
ENE 104
Two-Port Networks
Week #13
A multiport Network:
Page 6
The circuit determinant:
Z11
Z12
Z13
... Z1N
Z 21
Z 22
Z 23
... Z 2 N
 Z  Z31
Z32
Z 33
... Z 3 N
...
...
...
Z N1 Z N 2
ENE 104
...
...
Z N 3 ... Z NN
Two-Port Networks
Week #13
A multiport Network:
Page 7
The Cramer’s rule:
V1
0
0
...
0
I1 
Z11
Z 21
Z 31
...
Z N1
ENE 104
Z12
Z 22
Z 32
...
ZN 2
Z12
Z 22
Z 32
...
ZN 2
Z13
Z 23
Z 33
...
ZN3
Z13
Z 23
Z 33
...
ZN3
...
...
...
...
...
...
...
...
...
...
Z1N
Z2N
Z3 N
...
Z NN
Z1N
Z2 N
Z3 N
...
Z NN
Two-Port Networks
V111
I1 
Z
minor
V1  Z
Zin 

I1 11
Week #13
Example:
Page 8
Calculate the input impedance:
ENE 104
Two-Port Networks
Week #13
Example:
Page 9
By inspection:
V1 
10I1  10I 2
0   10I1  17I 2
0
 2I 2
0
 5I 2
ENE 104
Two-Port Networks
 2I 3
 7I 3
 I3
 5I 4
 I4
 26I 4
Week #13
Example:
Page 10
By inspection:
V1 
10I1  10I 2
0   10I1  17I 2
0
 2I 2
0
 5I 2
 2I 3
 7I 3
 I3
 5I 4
 I4
 26I 4
10  10
0
0
 10
17  2  5
Z 
0 2
7 1
0  5  1 26
ENE 104
Two-Port Networks
= 9680 Ω4
Week #13
Example:
Page 11
By inspection:
V1 
10I1  10I 2
0   10I1  17I 2
0
 2I 2
0
 5I 2
 2I 3
 7I 3
 I3
 5I 4
 I4
 26I 4
17  2  5
11   2
7  1  2778
5
1
26
Zin  9680
2778  3.485 Ω
ENE 104
Two-Port Networks
Week #13
Practice: 17.1
Page 12
Find the input impedance of the network shown in
Figure below if it is formed into a one-port network
by breaking it at terminals: (a) a and a’; (b) b and
b’; (c) c and c’.
ENE 104
Two-Port Networks
Week #13
Practice: 17.1
ENE 104
Two-Port Networks
Page 13
Week #13
Practice: 17.1
ENE 104
Two-Port Networks
Page 14
Week #13
Practice: 17.1
ENE 104
Two-Port Networks
Page 15
Week #13
Example:
Page 16
Calculate the input impedance:
ENE 104
Two-Port Networks
Week #13
Example:
Page 17
10I1  10I 2
 10I1  17I 2
 2I 3
 5I 4
 2I 2
 7I 3
 I4
 V1
 0

0
and I 4  0.5I a  0.5I 4  I3 
or
ENE 104
 0.5I3  1.5I 4  0
Two-Port Networks
Week #13
Example:
Page 18
10I1  10I 2
 10I1  17I 2
 2I 3
 5I 4
 2I 2
 7I 3
 I4
 V1
 0

0
 0.5I3  1.5I 4  0
10  10
0
0
 10
17
2 5
Z 
 590 Ω
0 2
7 1
0
0  0.5 1.5
ENE 104
Two-Port Networks
Week #13
Example:
Page 19
10I1  10I 2
 10I1  17I 2
 2I 3
 5I 4
 2I 2
 7I 3
 I4
 V1
 0

0
 0.5I3  1.5I 4  0
17
11   2
2 5
7  1  159 Ω
0  0.5 1.5
590
Zin  159
 3.711 Ω
ENE 104
Two-Port Networks
Week #13
Practice: 17.2
Page 20
Write a set of nodal equations for the circuit of
figure below, calculate Δ𝑌 , and then find the input
admittance seen between: (a) node 1 and the
reference node; (b) node 2 and the reference.
ENE 104
Two-Port Networks
Week #13
Practice: 17.2
ENE 104
Two-Port Networks
Page 21
Week #13
Practice: 17.2
ENE 104
Two-Port Networks
Page 22
Week #13
Admittance Parameters:
Page 23
A general two-port with terminal voltages and currents specified.
The two-port is composed of:
- linear elements, no energy stored within the circuits
- possibly including dependent sources,
- but not containing any independent sources.
- all external connections must be made to either the input port
or the output port.
ENE 104
Two-Port Networks
Week #13
Admittance Parameters:
Page 24
We may begin with
the set of equations:
I1  y11V1  y12V2
I 2  y 21V1  y 22V2
The matrix equation:
 I1   y11 y12   V1 
I   y



 2   21 y 22  V2 
ENE 104
Two-Port Networks
Week #13
Admittance Parameters:
Page 25
We may begin with
the set of equations:
I1  y11V1  y12V2
I 2  y 21V1  y 22V2
The short-circuit admittance parameters
The short-circuit input admittance:
The short-circuit output admittance:
ENE 104
Two-Port Networks
I1
y11 
V1
V2 0
I2
y 22 
V2
V1 0
Week #13
Admittance Parameters:
Page 26
We may begin with
the set of equations:
I1  y11V1  y12V2
I 2  y 21V1  y 22V2
The short-circuit admittance parameters
The short-circuit transfer admittance:
I1
y12 
V2
ENE 104
I2
y 21 
V1 V 0
V1 0
2
Two-Port Networks
Week #13
Example:
Page 27
Find the short-circuit admittance parameters:
I1
y11 
V1
I2
y 22 
V2
ENE 104
V2 0
I1
y12 
V2
V1 0
I2
y 21 
V1 V 0
V1 0
2
Two-Port Networks
Week #13
Example:
Page 28
Find the short-circuit admittance parameters:
I1
y11 
V1
I2
y 22 
V2
ENE 104
V2 0
V1 0
15

 0.3 S.
5 10
30

 0.15 S.
10  20
Two-Port Networks
Week #13
Example:
Page 29
Find the short-circuit admittance parameters:
-V2
- V2
When V1  0, I1 
 y12  10
 0.1 S.
10
V2 V 0
1
-V1
- V1
When V2  0, I 2 
 y 21  10
 0.1 S.
10
V1 V 0
2
ENE 104
Two-Port Networks
Week #13
Example:
Page 30
Find the short-circuit admittance parameters:
I1  y11V1  y12V2
I 2  y 21V1  y 22V2
Write the expressions:
V1 V1  V2
I1 

 0.3V1  0.1V2
 0.3  0.1
5
10
y

V2  V1 V2
 0.1 0.15 
I2 

 0.1V1  0.15V2
10
20
ENE 104
Two-Port Networks
Week #13
Practice: 17.3
Page 31
By applying the appropriate 1-V sources and short
circuits to the circuit shown in figure below, find (a)
𝐲11 ; (b) 𝐲21 ; (c) 𝐲22 ; (d) 𝐲12
ENE 104
Two-Port Networks
Week #13
Practice: 17.3
ENE 104
Two-Port Networks
Page 32
Week #13
Practice: 17.3
ENE 104
Two-Port Networks
Page 33
Week #13
The Terminal Equations:
Page 34
There are six different
ways in which to
combine the four
variables:
V1  z11I1  z12I 2 V1  a11V2  a12I 2 V1  h11I1  h12V2
V2  z 21I1  z 22I 2 I1  a 21V2  a 22I 2 I 2  h 21I1  h 22V2
I1  y11V1  y12V2 V2  b11V1  b12I1 I1  g11V1  g12I 2
I 2  y 21V1  y 22V2 I 2  b 21V1  b 22I1 V2  g 21V1  g 22I 2
ENE 104
Two-Port Networks
Week #13
Admittance parameters:
Page 35
I1  y11V1  y12V2
I 2  y 21V1  y 22V2
I1
y11 
V1
I2
y 22 
V2
ENE 104
V2 0
I1
y12 
V2
V1 0
I2
y 21 
V1 V 0
V1 0
2
Two-Port Networks
Week #13
impedance parameters:
Page 36
V1  z11I1  z12I 2
V2  z 21I1  z 22I 2
V1
z11 
I1
V2
z 22 
I2
ENE 104
I 2 0
V1
z12 
I2
I1  0
I1  0
V2
z 21 
I1
I 2 0
Two-Port Networks
Week #13
transmission parameters:
Page 37
V1  a11V2  a12I 2
I1  a 21V2  a 22I 2
V1
a11 
V2
 I1
a 22 
I2
I 2 0
V2  0
 V1
a12 
I2
I1
a 21 
V2

V2 0
S.
I 2 0
Note: Hayt’s use “t” instead of “a”
ENE 104
Two-Port Networks
Week #13
transmission parameters:
Page 38
V2  b11V1  b12I1
I 2  b 21V1  b 22I1
V2
b11 
V1
 I2
b 22 
I1
ENE 104
I1  0
V1  0
 V2
b12 
I1
I2
b 21 
V1
Two-Port Networks

V1 0
S.
I1  0
Week #13
hybrid parameters:
Page 39
V1  h11I1  h12V2
I 2  h 21I1  h 22V2
V1
h11 
I1
I2
h 22 
V2
ENE 104
V2 0
V1
h12 
V2
I1  0
I1  0
I2
h 21 
I1
V2  0

S.
Two-Port Networks
Week #13
hybrid parameters:
Page 40
I1  g11V1  g12I 2
V2  g 21V1  g 22I 2
I1
g11 
V1
V2
g 22 
I2
ENE 104
I 2 0
I1
g12 
I2
V1 0
V2
g 21 
V1
S.

Two-Port Networks
V1  0
I 2 0
Week #13
Relationships Among …:
Page 41
To find the z-parameters as function of y-parameters,
I1  y11V1  y12V2
V1  z11I1  z12I 2
I 2  y 21V1  y 22V2
V2  z 21I1  z 22I 2
 I1   y11 y12   V1 
I   y



 2   21 y 22  V2 
I1
I2
V1 
y11
y 21
ENE 104
y12
y 22
y 22
y12

I1 
I2
y12 y
y
y 22
1
 V1   y11 y12   I1 
V   y



 2   21 y 22  I 2 
V2 
Two-Port Networks
I1
I2
y12
y 22 y 22
y12

I1 
I2
y
y
y
Week #13
Transformation between …:
ENE 104
Two-Port Networks
Page 42
Week #13
Reciprocal Two-Port Circuits:
Page 43
The following relationship exist,
z12

z 21,
y12

y 21,
a11a 22  a12a 21
 a  1,
b11b 22  b12b 21  b  1,
ENE 104
h12

 h 21,
g12

 g 21
Two-Port Networks
Week #13
Reciprocal Two-Port Circuits:
Page 44
A reciprocal two-port circuit:
A reciprocal twoport circuit is
symmetric if its
ports can be
interchanged
without
disturbing the
values of the
terminal currents
and voltages.
ENE 104
Two-Port Networks
Week #13
Some Equivalent Networks:
Page 45
(a, b) Two-ports which are equivalent to any general linear two-port.
The dependent source in part a depends on V1, and that in part b
depends on V2. (c) An equivalent for a bilateral network.
ENE 104
Two-Port Networks
Week #13
Analysis of the Terminated …:
Page 46
Six characteristics of the terminated two-port
circuits define its terminal behavior:
ENE 104
Two-Port Networks
Week #13
Analysis of the Terminated …:
Page 47
Six characteristics of the terminated two-port
circuits define its terminal behavior:
1. The input impedance
2. the output current
3. the Thevenin voltage and impedance
4. The current gain
5. The voltage gain
6. The voltage gain
ENE 104
V2
V1
V2
Vg
Two-Port Networks
Week #13
Analysis of the Terminated …:
Page 48
Six characteristics in terms of the z-parameters:
The input impedance:
(the impedance seen
looking into port 1)
V1  z11I1  z12I 2
V2  z 21I1  z 22I 2
V1  Vg  I1Z g
V2  I 2 Z L
ENE 104
... *
... * *
... * * *
... * * * *
Two-Port Networks
V1
Z in 
I1
V2  I 2 Z L  z 21I1  z 22I 2
 z 21I1
 I2 
Z L  z 22
sub I 2 in *
Week #13
Analysis of the Terminated …:
ENE 104
Two-Port Networks
Page 49
Week #13
Example:
ENE 104
Page 50
Two-Port Networks
Week #13
Example:
ENE 104
Page 51
Two-Port Networks
Week #13
Example:
ENE 104
Page 52
Two-Port Networks
Week #13
Example:
ENE 104
Page 53
Two-Port Networks
Week #13
Example:
ENE 104
Page 54
Two-Port Networks
Week #13
Interconnected Two-port Circuits:
Page 55
The five basic interconnections of two-port circuit:
(a) Cascade
ENE 104
Two-Port Networks
Week #13
Interconnected Two-port Circuits:
Page 56
(a) Cascade
ENE 104
Two-Port Networks
Week #13
Example:
ENE 104
Page 57
Two-Port Networks
Week #13
Example:
ENE 104
Page 58
Two-Port Networks
Week #13
Example:
ENE 104
Page 59
Two-Port Networks
Week #13
Interconnected Two-port Circuits:
Page 60
(b) Series
ENE 104
(c) Parallel
Two-Port Networks
Week #13
Interconnected Two-port Circuits:
Page 61
(d) Series- Parallel
ENE 104
(e) Parallel- Series
Two-Port Networks
Week #13
Example: Final 2/47
ENE 104
Two-Port Networks
Page 62
Week #13
Example: Final 2/47
ENE 104
Two-Port Networks
Page 63
Week #13
Example: Final 2/47
ENE 104
Two-Port Networks
Page 64
Week #13
Example: Final 2/46
ENE 104
Two-Port Networks
Page 65
Week #13
Example: Final 2/46
ENE 104
Two-Port Networks
Page 66
Week #13
Example: Final 2/46
ENE 104
Two-Port Networks
Page 67
Week #13
Example: Final 2/46
ENE 104
Two-Port Networks
Page 68
Week #13
Reference:
•W.H. Hayt, Jr., J.E. Kemmerly, S.M. Durbin, Engineering Circuit Analysis, Sixth Edition.
Copyright ©2002 McGraw-Hill. All rights reserved.
•James W. Nilsson, Susan A. Riedel, “Electric Circuits” Sixth edition, Addison Wesley, 2001
ENE 104
Circuit Analysis and Electrical Engineering
Page 69