International Journal of Electrical Engineering Education 40/3 ABCD matrix: a unique tool for linear two-wire transmission line modelling Pedro L. D. Peres, Carlos R. de Souza and Ivanil S. Bonatti School of Electrical and Computer Engineering, University of Campinas, Campinas, Brazil E-mail: peres@dt.fee.unicamp.br Abstract The aim of this note is to show that all the behaviour of a two-wire transmission line can be directly derived from the application of ABCD matrix mathematical concepts, avoiding the explicit use of differential equations. An important advantage of this approach is that the transmission line modelling arises naturally in the frequency domain. Therefore the consideration of frequency- dependent parameters can be carried out in a simple way compared with the time-domain. Some standard examples of transmission lines are analysed through the use of ABCD matrices and a case study of a balun network is presented. Keywords ABCD matrix; electric circuit analysis; two-port networks; two-wire transmission lines Partial differential equations for voltage and current along the line are traditionally derived from an elementary line section of length Dx using the distributed param- eter model for a two-wire transmission line. Several physical interpretations can be obtained from the solution of these equations. 1,3,4,8 The concepts of image impedance and ABCD matrix can be used as an alterna- tive approach to the modelling of the two-wire transmission line, although the oper- ational advantages and capabilities of the ABCD matrix model as a convenient tool for solving almost all problems related to the transmission line are not fully stressed. Indeed, the authors claim that most of the transmission line results which are usually obtained by other means can more easily be produced when nothing more than the ABCD matrix model is used. The importance of ABCD matrix modelling in trans- mission line theory and its indisputable advantages over other tools are presented and discussed in this paper. ABCD matrix fundamentals The approach presented in this section for obtaining the ABCD matrix of a two-wire transmission line is based on image parameters, i.e. image impedance and image transfer constant. 7 Consider an electrical network having two pairs of terminals, one labelled the input (sending) terminals and the other the output (receiving) terminals. A pair of terminals at which the network can be accessed so that the currents in the two ter- minals are the same is called a port. This condition is assured when each port of a network is connected to a similar port of another network. 6 A two-port network with terminal voltages and currents as specified in Fig. 1 can be described by an ABCD matrix if it is composed only of linear elements (zero initial conditions), possibly including dependent sources, but containing no inde- pendent sources.