Cascade ABCD two-port networks
Telephone subscriber lines have become a topic of intense interest for organizations attempting to transmit Internet signals on telephone lines. Basic loop standards exist, and tables of twisted-pair primary constants extending to 20 MHz are in the literature. Asymmetric digital-subscriber lines (ADSLs) are now available for high-speed Internet service. The telephone industry has always used two-port networks in the form of ABCD matrices to cascade sections of telephone cable. These configurations allow you to join the sections by matrix multiplication. However, because the elements in the matrices are complex numbers and several sections make up a chain, you need to organize the process of matrix multiplication. A short computer program performs this task (Listing 1). You enter the number of matrices in the chain and then enter the real and imaginary parts for each A, B, C, and D element. The product then appears on the screen.
The two-port matrix equation with the ABDC parameters has the following format:
A represents the open-circuit transfer function, B represents the short-circuit transfer impedance, C represents the open-circuit transfer admittance, and D represents the short-circuit current ratio. You can find the ABCD matrix for a ladder network composed of RLC elements by slicing the network into series and shunt matrices. You then multiply the product of the first two by the third, and the progression continues with additional subscripts identifying the components. In the final product, the elements A, B, C, and D—with all their component symbols—may become quite cumbersome. To avoid these complicated expressions in the matrix for the ladder network, don't use them. Rather, enter the numerical values for the series and shunt components along with the necessary 0,0s and 1,0s. You can use this concept for any network comprising passive components that you can separate into isolated two-port sections. You can also use the method to predict the degradation a bridged tap produces in a telephone cable.
The progression of entering values for cascaded networks is normally from left to right, or from source to load with the current pointing toward the load. If you reverse the progression, the direction of the current is usually reversed, and the location of elements A and D is reversed to conform to the reciprocity theorem. The matrices for a single series component and a single shunt component are as follows:
You can use these simple matrices to represent RLC components, transformers, and bridged taps in telephone cables. However, for a telephone cable, you must consider the reflected wave. Thus, you must express the A, B, C, D elements by the following hyperbolic functions:
C=(1/ Z0)sinh(), and
where the propagation constant , =+, d is the electrical length of the cable, and Z0 is the characteristic impedance. In the United States, the nominal value for Z0 is 100V, which is also the specified value for the source and load impedance. Because is a complex number, A, B, C, and D are also complex numbers, or the polar equivalent thereof. These numbers are not difficult to calculate; however, the parameters depend on the application (References 1 and 2). The following example shows how to enter the data and read the results:
Within the solid bars, N1 through N4 show the entered data for the components. The bottom line shows the matrix product as it appears on the screen. For this network, the matrix elements are A=43+j0, B=0–j25, C=0+j12, and D=7+j0. The open-circuit voltage ratio is V1/V2=A= 43. The open-circuit transfer admittanceI1/V2=C= 0+j12. The input impedance isZN=A/C=0–j(43/12). Listing 1 uses easy-to-read and-compile QuickBasic. (DI #2455)
1.Rauschmayer, Dennis, DSL/VDSL Principles, MacMillan Technical Publishing, 1999.
2. Chen, Walter, DLS: Simulation Techniques and Standards Development of Digital Subscriber Line Systems, MacMillan Technical Publishing, 1998.