Whites, EE 481/581
Lecture 20
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Lecture 20: Transmission (ABCD) Matrix.
Concerning the equivalent port representations of networks
we’ve seen in this course:
1. Z parameters are useful for series connected networks,
2. Y parameters are useful for parallel connected networks,
3. S parameters are useful for describing interactions of
voltage and current waves with a network.
There is another set of network parameters particularly suited
for cascading two-port networks. This set is called the ABCD
matrix or, equivalently, the transmission matrix.
Consider this two-port network (Fig. 4.11a):
I1
+
V1
-
I2
A B
C D 


+
V2
-
Unlike in the definition used for Z and Y parameters, notice that
I 2 is directed away from the port. This is an important point and
we’ll discover the reason for it shortly.
The ABCD matrix is defined as
V1   A B  V2 
 I   C D    I 
  2
 1 
© 2015 Keith W. Whites
(4.69),(1)
Whites, EE 481/581
Lecture 20
Page 2 of 7
It is easy to show that
A
C
V1
V2
,
B
I 2 0
I1
V2
V1
I 2 V 0
2
, D
I2 0
I1
I2
V2  0
Note that not all of these parameters have the same units.
The usefulness of the ABCD matrix is that cascaded two-port
networks can be characterized by simply multiplying their
ABCD matrices. Nice!
To see this, consider the following two-port networks:
I1
V1
I 2
I2
 A1
C
 1
B1 
D1 
V2
V2
I3
 A2
C
 2
B2 
D2 
V3
In matrix form
and
V1   A1
 I   C
 1  1
B1  V2 

D1   I 2 
(4.70a),(2)
V    A
 2  2
 I 2  C2
B2  V3 

D2   I 3 
(3)
When these two-ports are cascaded,
Whites, EE 481/581
Lecture 20
I1
+
V1
-
I 2
+ +
V2 V2
- -
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I3
I2
 A1
C
 1
B1 
D1 
 A2
C
 2
B2 
D2 
+
V3
-
it is apparent that V2  V2 and I 2  I 2 . (The latter is the reason
for assuming I 2 out of the port.) Consequently, substituting (3)
into (2) yields
V1   A1 B1   A2 B2  V3 
(4.71),(4)
 I   C D   C D    I 
1  2
2  3
 1  1
We can consider the matrix-matrix product in this equation as
describing the cascade of the two networks. That is, let
 A3 B3   A1 B1   A2 B2 
(5)
 C D   C D    C D 
3
1  2
2
 1
 3
V1   A3 B3  V3 
(6)
so that
 I   C D    I 
3  3
 1  3
I1
where
+
V1
-
I3
 A3
C
 3
B3 
D3 
+
V3
-
In other words, a cascaded connection of two-port networks is
equivalent to a single two-port network containing a product of
the ABCD matrices.
It is important to note that the order of matrix multiplication
must be the same as the order in which the two ports are
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Lecture 20
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arranged in the circuit from signal input to output. Matrix
multiplication is not commutative, in general. That is,
 A   B    B    A .
Text example 4.6 shows the derivation of the ABCD parameters
for a series (i.e., “floating”) impedance, which is the first entry
in Table 4.1 on p. 190 of the text.
In your homework, you’ll derive the ABCD parameters for the
next three entries in the table. In the following example, we’ll
derive the last entry in this table.
Example N20.1 Derive the ABCD parameters for the T network:
I1
Z1
Z2
+
+
V1
Z3
VA
-
-
I2
+
V2
-
Z in,1
Recall from (1) that by definition
V1  AV2  BI 2 and I1  CV2  DI 2
 To determine A:
A
V1
V2
I2 0
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Lecture 20
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we need to open-circuit port 2 so that I 2  0 . Hence,
Z3
VA 
V1  V2
Z1  Z 3
A
which yields,
 To determine B:
V1
V2
B
 1
I 2 0
Z1
Z3
V1
I 2 V 0
2
we need to short-circuit port 2 so that V2  0 . Then, using
current division:
Z3
I2 
I1
Z 2  Z3
Substituting this into the expression for B above we find
 Z 
V1  Z 2 
 1  
  Z1  Z 2 Z 3  1  2 
B
I
Z 3  V 0
 Z3 
1 
2
 Z in,1
V2  0
 Z2 
Z1Z 2
 Z1 
 Z 2 Z 3 1  
Z3
 Z3 
Z Z Z  Z2
ZZ
 Z1  1 2  2 3 3
Z3
Z 2  Z3 Z3
ZZ
B  Z1  Z 2  1 2
Therefore,
Z3
Whites, EE 481/581
Lecture 20
 To determine C:
C
I1
V2
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I2 0
we need to open-circuit port 2, from which we find
VA  I1Z 3  V2
C
Therefore,
 To determine D:
I1
V2
D
I1
I2

I2 0
1
Z3
V2  0
we need to short-circuit port 2. Using current division, as
above,
Z3
I2 
I1
Z 2  Z3
Therefore,
D
I1
I2
 1
V2  0
Z2
Z3
These ABCD parameters agree with those listed in the last entry
of Table 4.1.
Properties of ABCD parameters
As shown on p. 191 of the text, the ABCD parameters can be
expressed in terms of the Z parameters. (Actually, there are
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Lecture 20
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interrelationships between all the network parameters, which are
conveniently listed in Table 4.2 on p. 192.)
From this relationship, we can show that for a reciprocal
network
A B
Det 
 1 or AD  BC  1

C D 
If the network is lossless, there are no really outstanding features
of the ABCD matrix. Rather, using the relationship to the Z
parameters we can see that if the network is lossless, then
Z
A  11  A real
 From (4.73a):
Z 21
Z Z  Z12 Z 21
 From (4.73b):
 B imaginary
B  11 22
Z 21
1
 From (4.73c):
C
 C imaginary
Z 21
Z
D  22  D real
 From (4.73d):
Z 21
In other words, the diagonal elements are real while the offdiagonal elements are imaginary for an ABCD matrix
representation of a lossless network.