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Two-Port Networks
Chapter 19
ch19 Two-Port Networks
19.1 Introduction
• A
two-port network
is an electrical network
with two separate ports for input and output.
ch19 Two-Port Networks
3
19.2 Impedance Parameters,
z
2 22 1 21 2
2 12 1 11 1
I
z
I
z
V
I
z
I
z
V
[ ]
I I
z
I I
z
z
z
z
V V
ch19 Two-Port Networks
Impedance Parameters,
z
z^11
= Open-circuit input impedance
z^12
= Open-circuit transfer impedance from port 1 to port 2
z^21
= Open-circuit transfer impedance from port 2 to port 1
z^22
= Open-circuit output impedance
0 (^22)
22 0 (^21) 21
0 (^12) 12 0 (^11) 11
1
2
1
2
=
=
=
=
I
I
I
I
V I
z
V I
z
V I
z
V I
z
ch19 Two-Port Networks
5
Fig 19.
(^22) 22 (^12) 12
(^21) 21 (^11) 11
V^ I
z
V I
z
V I
z
V I
z
ch19 Two-Port Networks
Fig 19.4 & Fig 19.
ch19 Two-Port Networks
7
Fig 19.
2
1 2 1
I
I
V
V^
n
n^
ch19 Two-Port Networks
Example 19.
[ ]^
= = = =
Thus
2
2
(^22) 22
(^22)
(^12) 12
(^11)
(^21) 21
1
1
(^11) 11
z
I
I
V I
z
I I
V I
z
I I
V I
z
I
I
V I
z
(^0) I (^20) I (^20) I 1 0 I 1
-^
Determine the
z^ parameters for the circuit in Fig. 19.7.
-^
Solution:
ch19 Two-Port Networks
13
Admittance Parameters,
y
y^11
= Short-circuit input admittance
y^12
= Short-circuit transfer admittance from port 1 to port 2
y^21
= Short-circuit transfer admittance from port 2 to port 1
y^22
= Short-circuit output admittance
1
2
1
2
, ,
=
=
=
=
V
V
V
V
I V
y
I V
y
I V
y
I V
y
ch19 Two-Port Networks
Fig 19.
(^22) 22 (^12) 12
(^21) 21 (^11) 11
V^ I
y
V I
y
V I
y
V I
y
ch19 Two-Port Networks
15
Example 19.
S (^5). 0 (^4585)
4 5 (^828)
S 625
. 0 8 5
(^8) ) 5 (^2) || 8 (
S (^5). 0 (^2343)
2 3 (^424)
S (^75). 0 4 3
(^4) ) 3 (^2) || 4 (
0 1 2 21 2 2 1
2 2 0 2 2 22 2
2 2
0 2 12
0 11
1 1
−= − =
= ⇒ = =+ −
= = = ⇒ = =
−= − =
= ⇒ = =+ −
= = = ⇒ = = = = = =
(^11) V V
(^11)
1 V
1 1 2
1 1 1 1 V
1
1 1
I I
I V y I I I
I I I V y I
I V
I I
I V y I I I
I I I V y I
I V
2 2
-^
Obtain the
y^ parameters for the
network shown in
Fig. 19.14.
-^
Solution:
ch19 Two-Port Networks
-^
Obtain the
y^
parameters for the T network shown in
Fig. 19.16.
Practice Problem 19.
ch19 Two-Port Networks
17
Example 19.
-^
Determine the
y^ parameters for the
T^
network shown
in Fig. 19.17.
ch19 Two-Port Networks
Example 19.
S
node At
S
But
node At
2 1 21
2
2 1
1 1 11
1
1
1
1
1 1
1
1
o o
o
o o
o
o o
o o o o o o
o o
o
o o
o V^ V I V y
V V V I I I V
V V
I V
y
V V V I V V V V V
V V V V V I
V
V
I
V
V
ch19 Two-Port Networks
19
Example 19.
.
reciprocalt isn' network the since case, in this
that Notice
S 25
. 0
(^625). 0 5. 2
(^625). 0
2 8 ) (^5). 2 1 ( 4 (^25). 0
0
2 4 2, nodeAt
S 05
. 0
/
(^5). 2
2 2 4
0
4 2 8 0
0 8 But
4 2 2 0 8 1, nodeAt
19.18(b). Fig. using and get we
Similarly,
(^12) 2 21 2 (^222)
2 1 2 1 2 (^21)
2 2
2
1
2
21 (^12) y y
V V
I V y
V
V V
V I
I I V V
V V I V y
V V V V V V V V V V V I V V V I V
y y
1 ≠
=
=
⇒
−= −
−
−→
=
−
−= −= = ⇒
= → −
−= →
−
−= → − =
−
= −
o o
o
o o
o o
o o
o
o o o
o o o
o
o o
o
ch19 Two-Port Networks
Practice Problem 19.
-^
Determine the
y^
parameters for the
T^
network shown
in Fig. 19.19.
ch19 Two-Port Networks
25
Example 19.
-^
Find the hybrid parameters for the two-port networkof Fig. 19.22.
ch19 Two-Port Networks
Example 19.
19.23(a),Fig. 2 (
From
0 2 1 21
1 1
2
0 (^11) 11
1
1 1
(^22)
= V = V
I I
h
I
I
I
V I
h
I
I
V
ch19 Two-Port Networks
27
Example 19.
91 S 9
Also,
19.23(b),
Fig.
From
0 2 2 22
2 2
2
0 1 2 12
2 2
1
= I 1 = I 1
I V
h
I
I
V
V V
h
V
V
V
ch19 Two-Port Networks
Practice Problem 19.
-^
Find the hybrid parameters for the two-port networkof Fig. 19.24.
ch19 Two-Port Networks
29
Example 19.
-^
Determine the Thevenin equivalent at the output portof the circuit in Fig. 19.25.
ch19 Two-Port Networks
Example 19.6^22
1 21 2
11 12
1
12 1 11 1
1
1
2
2 22 1 21 2
2 12 1 11 1
getwe 40
,I
and, 1
But
h
I
h
I
h
h
I
h
I
h
I
V
V
V
h
I
h
I
V
h
I
h
V
ch19 Two-Port Networks
31
Example 19.
× × + + × ×
−
Therefore,
6
6
3
22 12 21 22 11
11
2 (^22) TH
11
22 12 21 22 11
11
12 21 22 2
h
h
h
h
h
h
I
V I
Z
h
h h h h h h
h
h
h
I
ch19 Two-Port Networks
Example 19.
2 12 22 21
11
2 22 21 1 2 22 (^121)
2 12 1 11
2 12 (^111) 1
2
1
1
1 1
and
or
output, At the
intput at the
19.26(b), Fig. From
V
h h h
h
V
h h I V h I h
V
h I
h
V
h Ih I
I
I
V
V
I
⎡^ ⎢⎣
ch19 Two-Port Networks
37
Example 19.
[ ]
⎤ ⎥ ⎥ ⎥ ⎥⎦
⎡ ⎢ ⎢ ⎢ ⎢⎣
++ +
−
=
++ +
=
=
=
⎞⎟⎠
⎛^ ⎜⎝
=
−=
=
−=
=
= ) 1 (
1
1 1
1 1
1 1
Thus,
) 1 (
1
1 1 1
or 1 1
Also,
1 1
or 19.29(b), 1 1 Fig. From
2
2
0 (^22) 22
2 2
0 1 2 12 2
1
s ss s s
s
s
s ss s s s
s s
s
s g
V I g
I V
I I g I
I
1
1
V
V
ch19 Two-Port Networks
Practice Problem 19.
-^
Find the
g^
parameters as functions of s for the circuit
in Fig. 19.30.
ch19 Two-Port Networks
39
19.5 Transmission Parameters,
T
2 2 1
2 2
1
DI
CV
I
BI
AV
V
[^
]^
2 2
2 2
(^11)
V^ I
T
V I
D
C
B
A
V I
ch19 Two-Port Networks
Transmission Parameters,
T
A^
= Open-circuit voltage ratio
B^
= Negative short-circuit transfer impedance
C^
= Open-circuit transfer admittance
D^
= Negative short-circuit current ratio
0 1 2
0 1 2
0 (^12)
0 1 2
2
2
2
2
, ,
=
=
=
=
Ι − =
=
−=
=
V
I
V
I
I
D
I V C
V I
B
V V A
ch19 Two-Port Networks
41
Inverse Transmission Parameters,
t
a^ = Open-circuit voltage gain b^ = Negative short-circuit transfer impedance c^ = Open-circuit transfer admittance d^ = Negative short-circuit current gain
0 2 1
0 2 1
0 (^21)
0 2 1
1
1
1
1
, ,
=
=
=
=
Ι−
=
−=
=
V
I
V
I
I d
I V c
V I
b
V V a
−^
bc
ad
BC
AD
ch19 Two-Port Networks
42
Example 19.
-^
Find the transmission parameters for the two-port networkin Fig. 19.
ch19 Two-Port Networks
43
Example 19.
S
Thus
and
19.33(a),
Fig.
From
1 1 0 1 2
1 1 0 1 2
1 1 1 2 1 1 1
=
=^
I I
I V
C
I I
V V
A
I I I V I I V
2
2
I
I
ch19 Two-Port Networks
Ω 44
=
−
−=
=
−=
−=
⇒
− ⇒
=
= ⇒
−
=
=
− −
=
=
(^29). 15 ) (^20) / (^17) (
13
, (^176). 1 (^2017)
Therefore,
1720 0
3 20
13 , 3
, 10 /)
(
and 3
But
0
20
10
19.33(b), Fig. From
1 1
0 1 2
0 1 2
2 1
2 1 1
1 1 1
1 1 1
2
1
I^ I
V V B
I I D
I I
I I I
I V I V
V V I I V
I V V V
2
2
V
V a
a
a
a a
Example 19.
ch19 Two-Port Networks
49
19.6 Relationships Between
Parameters
, 1
,
,
,
1
1
22 22 (^222)
2 (^222) 2 22
1 2 22 2
2
1 2 22
2 22
(^222)
22
(^212)
22
2 (^222) 1
22
(^212)
22
2 22 1 2 22
22 1 2 2
2 1
z h z z
h z z h z h
I V h h
h h I V z
z z
z z
z
z z z z V I
V z z I
z
z z z z V
V z I z z I I z I z V I z I z V
1 1 1 1
11
1
1 11
1
1
1
11 (^12)
1
1
11 1
1 2 2
1
2 1 11 1
=
−=
=
Δ=
⇒
⎤ ⎥⎦ ⎡⎤ ⎢⎥⎣⎦
⎡=⎢⎣ ⎤ ⎥⎦ ⎤ ⎥⎡⎥⎢⎥⎣ ⎥⎦
⎡ ⎢ ⎢ ⎢ ⎢⎣
− −
⎤=⎥⎦ ⎡ ⎢⎣ →
−
= →
−=
→
=
=
z
(^11) ] [ ][
][ ][
− − = ≠ T t
h g
ch19 Two-Port Networks
ch19 Two-Port Networks
51
Example 19.
S
][
][
Thus,
22
21
12
11
22
21
12
11
g
z
B A
g
A
g
A
g
C A
g
D C
z
C
z
C
z
A C
z
T T
-^
Find [
z ] and [
g ] of a two-port network if
-^
Solution:
⎡^ ⎢⎣
=^
S 2
]
[ T
Table From. 37 3 40
is matrix the oft
determinan the4,
2, 1.5, , 10
If
BC
AD
D
C
B
A T
ch19 Two-Port Networks
Example 19.
-^
Obtain the
y^ parameters of the op amp circuit in Fig. 19.37.
Show that the circuit has no
z^ parameters.
ch19 Two-Port Networks
53
Example 19.
inverse. no has matrix] [ the0,
Since
is matrix] [ the oft
determinan The
Hence,( . /
But
Also,
as and of in terms
expressed be can which
ams, op the of inals
input term
enter the can current no Since
21 12 22 11
3 22 (^231) 1 21
2 3 1 (^231) 1 2
1
2 1 1 (^23) 2
(^11)
2 1 (^23) 2
12
11 2 1 1
2 1
1
yy y yy
y y
y
V V I V I V
V
I
I
I
V
y
y V V I
V
V
I
y o y
o
R
R RR
R
R
R RR
R
R
R
R
R
R
R
R
R
ch19 Two-Port Networks
Practice Problem 19.
-^
Obtain the
z^ parameters of the op amp circuit in Fig. 19.38.
Show that the circuit has no
y^
parameters.
ch19 Two-Port Networks
55
19.7 Interconnection of Networks
] [ ] [ ][
b a^
z z z^
=
-^
The series connection
b a b a
b a b a
22 22 2 2
2 2
22 2
2
22 22
2 2
2 2 2
2 2
z z z z
z z z z z z
z z
I z z I z z
V
V
V
I z z I z z
V
V
V
I I I I I I
Iz Iz
V
Iz Iz V
Iz Iz V
Iz Iz V
1 1
1 1 11 11
1
1 11
2 b a 1 1b 1a
b a
2 b 1 a 1 1 11b 11a
1b 1a 1
2b 2a 2 1b 1a 1
2b 22 1b 21 2b
2b 12 1b 11 1b
2a 22 1a 21 2a
2a 12 1a 11 1a
ch19 Two-Port Networks
Fig 19.
]
[
]
[
][
b a^
y
y
y^
22b 22a 21b 21a
12b 12a 11b 11a 22 21
12 11
2 22b 22a 1 21b 21a
2b 2a 2
2 12b 12a 1 11b 11a
1b 1a 1
2b 2a 2 1b 1a 1
2b 22 1b 21 2b
2b 12 1b 11 1b
2a 22 1a 21 2a
2a 12 1a 11 1a
y y y y
y y y y y y
y y
)V
y (y )V y (y
I
I
I
)V
y (y )V y (y
I
I
I
V
V
V,
V
V
V
V
y V y I
V
y V y I
V
y V y I
V
y V y I
ch19 Two-Port Networks
61
Example 19.
-^
Find the
y^ parameters of the two-port in Fig. 19.44.
ch19 Two-Port Networks
Example 19.
S
4 4 2 6 ] [ ] [ ] [
S
]
[
or
and
S
]
[
or
22
11 21
12
22
11 21
12
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
b a b
b
a a
b
a
a
a a
a
y
y
y y
y
y
y
y
y
y
y
y
y
ch19 Two-Port Networks
63
Practice Problem 19.
-^
Find the
y^ parameters of the two-port in Fig. 19.45.
ch19 Two-Port Networks
Example 19.
⎡^ ⎢⎣
Ω = × + = = + =
⎡^ ⎢⎣
Ω = × + = = + =
S
]
[
S, 1
and
S 1
]
[
S, 1
and
3 2
2
2
3 2 1 3
1 2 b
b
b
b
a b
a
a
a
a
R R
R
R
R
R
R
R
R R
T
D
C
B
T A
D
C
B
A
D
C
B
A
-^
Find the transmission parameters for the circuit in Fig. 19.46.
-^
Solution:
ch19 Two-Port Networks
65
Example 19.
. 1
that Notice
42 S (^5). 5
206 27
4 9 6 1 (^5). 0 9 1 1
4 44 6 5 (^5). 0 44 1 5
4 (^5). 0
6 1 9 1
44 5 ] [] [ ] [
19.46, Fig.in
network total for the, Thud
= Δ= Δ= Δ
⎤ ⎥⎦
⎡^ ⎢⎣
Ω
=
⎤ ⎥⎦
⎡ ⎢⎣
× + × × + × × + × × + × = ⎤ ⎥⎦
⎡⎤ ⎢⎥⎣⎦
⎡=⎢⎣ ⋅ =
T T b T a
b a T T T
ch19 Two-Port Networks
Practice Problem 19.
-^
Find the transmission parameters for the circuit in Fig. 19.48.
ch19 Two-Port Networks
67
19.9 Applications 19.9.1 Transistor Circuits
0 (^22) out
1 1 in
(^2121)
=
V^ s
v i
s
I
s
V
Z
s
I
s
V
Z
s
I
s
I
A
s
V
s
V
A
ch19 Two-Port Networks
Transistor Circuits
admittance
Output
gain
current
collector-
Basic
ratio
feedback
voltage
Reverse
imdepance
input
Basic
,^
22
21
12
= ie re fe oe
o
f
r
i h h h h
h h h h h h h h
ch19 Two-Port Networks
73
Fig 19.
2 22 1 21 2
2 12 1 11 1
V
V
I
V
V
I
y
y
y
y
L^ L
s^
Y
y
Y
y
H^
21 22
ch19 Two-Port Networks
74
Ladder Network Synthesis
and
22
21
o
e o
e
o e
L
o e o
e
o e
L
o
o e
o e
e
e o
e o
o e o
e
e
e o
o
e o
e o
s
s
s s
s
N
D D
N
D D
y Y
N
N D
N
N D
y Y
N
D
D
D
N
N
D
D
D N H N D D
N
N
D
D
N
H
D
D
N
N
N D
H
ch19 Two-Port Networks
75
Example 19.
-^
Design the
LC
ladder network terminated with a 1-
reistor that has the normalized transfer function.
-^
Solution:
s
s s
y
s
s
y
y y
s
Y
R
y y
s
s s
s
s
s
s
s
s
s
s
s
s
L L
L L
when
(^23) 22
3 21 21 22
21 22
3 2 3
2
3
2
3
H
Y Y
D H
ch19 Two-Port Networks
76
Example 19.
H, 5.
2
3
2
3
3 2 22
s
s
Z
L
s
s
s
Z
Z
sL
s
s
s
y
Z
B
A
B
A
H 5
andF
which
from
1 2 1
2
2
L
sL
s
Y
C
Y
sC
s
s
s
s
Z
Y C
C
B B
