ABCD Parameters of Transmission Line
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
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
It is easy to show that
A=(V_{1}/V_{2})|_{I2=0}
B=(V_{1}/I_{2})|_{V2=0}
C=(I_{1}/V_{2})|_{I2=0}
D=(I_{1}/I_{2})|_{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:
In matrix form
And
When these two-ports are cascaded,
it is apparent that V_{2}^{’}=V_{2}^{’} and I_{2}^{’} = I_{2}^{’} . (The latter is the reason for assuming I_{2} out of the port.) Consequently, substituting
We can consider the matrix-matrix product in this equation as describing the cascade of the two networks. That is, let
So that
Where
In other words, a cascade 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 arranged in the circuit. Matrix multiplication is not commutative, in general. That is,[A].[B] ¹ [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. 185 of the text. In your homework, you’ll derive the ABCD parameters for the next three entries in the table.
Example N20.1 Derive the ABCD parameters for the T network:
Recall from (1) that by definition
V_{1} = AV_{2} + BI_{2} and I_{1} = CV_{2} + DI_{2}
. To determine A:
A=(V_{1}/V_{2})|_{I2=0}
we need to open-circuit port 2 so that I_{2} = 0. Hence,
V_{A}=(Z_{3}/Z_{1}+Z_{3})V_{1}=V_{2}
which yields,
A=(V_{1}/V_{2})|_{I2=0} = 1+(Z_{1}/Z_{3}
. To determine B:
B=(V_{1}/I_{2})|_{V2=0}
we need to short-circuit port 2 so that V_{2} = 0. Then, using current division:
I_{2}=(Z_{3}/Z_{2}+Z_{3})I_{1}
Substituting this into the expression for B above we find
=Z_{1}+(Z_{1}Z_{2}/Z_{3})+Z_{2}||Z_{3}{1+(Z_{2}/Z_{3})}
= Z_{1} +(Z_{1}Z_{2}/Z_{3})+{(Z_{2}Z_{3})/(Z_{2}+Z_{3)})}(Z_{3}+Z_{2}/Z_{3})
There for
B=Z_{1}+Z_{2}+(Z_{1}Z_{2}/Z_{3})
. To determine C:
C=(I_{1}/I_{2})|_{V2=0}
we need to open-circuit port 2, from which we find
V_{A} = I_{1} Z_{3} =V_{2}
There for
C=(I_{1}/T_{2})|_{V2=0} = 1/Z_{3}
. To determine D:
D=(I_{1}/I_{2})|_{V2=0}
we need to short-circuit port 2. Using current division, as above,
I_{2}=(Z_{3}/Z_{2}+Z_{3})I_{1}
There fore
D=(I_{1}/I_{2})|_{V2=0} = 1+(Z_{2}/Z_{3})
These ABCD parameters agree with those listed in the last entry of Table
Properties of ABCD parameters
As shown on pp. 185-186 of the text, the ABCD parameters can be expressed in terms of the Z parameters. (Actually, there are interrelationships between all the network parameters, which are conveniently listed in Table on p. 187.)
From this relationship, we can show that for a reciprocal Network
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
. From (a): A= Z_{11}/Z_{21} ÞA real
. From (b): B=( Z_{11} Z_{22} -Z_{12} Z_{21} )/Z_{21}Þ B imaginary
. From (c): C=1/Z_{21}Þ C imaginary
. From (d): D =Z_{22}/Z_{21}ÞD real
In other words, the diagonal elements are real while the off diagonal elements are imaginary for an ABCD matrix representation of a lossless network.