Lecture 20: Transmission (ABCD) Matrix.

Transcription

1 Whites, EE 48/58 Lecture 0 Page of 7 Lecture 0: Transmission (ABC) Matrix. Concerning the equivalent port representations of networks we ve seen in this course:. Z parameters are useful for series connected networks,. Y parameters are useful for parallel connected networks,. 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 twoport networks. This set is called the ABC matrix or, equivalently, the transmission matrix. Consider this twoport network (Fig. 4.a): A C B Unlike in the definition used for Z and Y parameters, notice that is directed away from the port. This is an important point and we ll discover the reason for it shortly. The ABC matrix is defined as A B C (4.69),() 05 Keith W. Whites

2 Whites, EE 48/58 Lecture 0 Page of 7 t is easy to show that, A 0 C 0, B 0 0 Note that not all of these parameters have the same units. The usefulness of the ABC matrix is that cascaded twoport networks can be characterized by simply multiplying their ABC matrices. Nice! To see this, consider the following twoport networks: A C B A C B n matrix form and A B C A B C (4.70a),() () When these twoports are cascaded,

3 Whites, EE 48/58 Lecture 0 Page of 7 A C B A C B it is apparent that and. (The latter is the reason for assuming out of the port.) Consequently, substituting () into () yields A B A B C C (4.7),(4) We can consider the matrixmatrix product in this equation as describing the cascade of the two networks. That is, let A B A B A B C C C (5) so that A B C (6) where A C B n other words, a cascaded connection of twoport networks is equivalent to a single twoport network containing a product of the ABC matrices. t is important to note that the order of matrix multiplication must be the same as the order in which the two ports are

4 Whites, EE 48/58 Lecture 0 Page 4 of 7 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 ABC parameters for a series (i.e., floating ) impedance, which is the first entry in Table 4. on p. 90 of the text. n your homework, you ll derive the ABC parameters for the next three entries in the table. n the following example, we ll derive the last entry in this table. Example N0. erive the ABC parameters for the T network: Z Z Z A Z in, Recall from () that by definition A B and C To determine A: A 0

5 Whites, EE 48/58 Lecture 0 Page 5 of 7 we need to opencircuit port so that 0. Hence, Z A Z Z which yields, Z A Z 0 To determine B: B 0 we need to shortcircuit port so that 0. Then, using current division: Z Z Z Substituting this into the expression for B above we find Z Z B ZZ Z Z Z 0 Z in, 0 ZZ Z Z Z Z Z Z ZZ ZZ Z Z Z Z Z Z Z ZZ Therefore, BZZ Z

6 Whites, EE 48/58 Lecture 0 Page 6 of 7 To determine C: C 0 we need to opencircuit port, from which we find Z Therefore, A C 0 Z To determine : 0 we need to shortcircuit port. Using current division, as above, Z Z Z Therefore, Z Z 0 These ABC parameters agree with those listed in the last entry of Table 4.. Properties of ABC parameters As shown on p. 9 of the text, the ABC parameters can be expressed in terms of the Z parameters. (Actually, there are

7 Whites, EE 48/58 Lecture 0 Page 7 of 7 interrelationships between all the network parameters, which are conveniently listed in Table 4. on p. 9.) From this relationship, we can show that for a reciprocal network A B et C or A BC f the network is lossless, there are no really outstanding features of the ABC matrix. Rather, using the relationship to the Z parameters we can see that if the network is lossless, then From (4.7a): Z A A real Z From (4.7b): ZZ ZZ B B imaginary Z From (4.7c): C Z C imaginary From (4.7d): Z real Z n other words, the diagonal elements are real while the offdiagonal elements are imaginary for an ABC matrix representation of a lossless network.