Unified Matrix Theory of Lumped and Distributed Directional Couplers

The directional coupler is an important device in many transmission systems. The theory for the electrical design of certain types of distributed-parameter directional couplers is well established through the contributions of a number of researchers.1-7 The purpose of this paper is to present the theory of transmission-line symmetric directional couplers in matrix form and then extend that theory to arbitrary lossless reciprocal circuits. There are many notational and conceptual advantages in using a matrix formulation which give further insight for the synthesis of different types of directional couplers. 39 40 T H E BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1968 The starting point is to examine the coupled-line directional coupler as a particular application of multiple coupled transmission line theory and then to use the concepts of characteristic impedance matrix and propagation matrix to examine the whole problem, rather than one line at a time or one mode at a time. By using a matrix approach the fundamental properties of the modes become more evident and thus physical intuition and mathematical reasoning blend to give a clearer picture of the situation. For instance, when the lines are no longer identical it is the eigenvectors of the matrices which provide the information of how to extend the mode concept.