THEORETICAL LIMITATIONS ON THE BROADBAND MATCHING OF ARBITRARY IMPEDANCES *

This paper deals with the general problem of matching an arbitrary load impedance to a pure resistance by means of a reactive network. It consists primarily of a systematic study of the origin and nature of the theoretical limitations on the tolerance and bandwidth of match and of their dependence on the characteristics of the given load impedance. Necessary and sufficient conditions are derived for the physical realizability of a function of frequency representing the input reflection coefficient of a matching network terminated in a prescribed load impedance. These conditions of physical realizability are then transformed into a set of integral relations involving the logarithm of the magnitude of the reflection coefficient. Such relations are particularly suitable for the study of the limitations on the bandwidth and tolerance of match. Definite expressions for these quantities are obtained in special cases. The practical problem of approaching the optimum theoretical tolerance by means of a network with a finite number of elements is also considered. Design curves are provided for a particularly simple but very important type of load impedance. In addition, a very convenient method is presented for computing the values of the elements of the resulting matching network. 1. TFiR MATCHING PROBLEM The transfer of power from a generator to a load constitutes one of the fundamental problems in the design of communication systems. A problem of this type involves in every case the design of a lossless coupling network to transform a given load impedance into another specified impedance. One refers to this operation as “impedance matching. ” In most practical cases this problem can be idealized as indicated in Fig. 1. The generator is assumed to consist of an ideal voltage source in series with a pure resistance ; maximum power transfer is then obtained when the impedance presented to the generator is equal to the source resistance. * Technical Report No. 41, Massachusetts Institute of Technology, Research Laboratory of Electronics. The work presented in this paper is part of a thesis with the same title submitted by the author in partial fulfillment of the requirements for the degree of Doctor of Science at the Massachusetts Institute of Technology (June, 1947). A summary was presented to the National Electronics Conference in Chicago on November 3, 1947. The research reported in this document was made possible through support extended the Massachusetts Institute of Technology, Research Laboratory of Electronics, jointly by the Army Signal Corps, the Navy Department (Office of Naval Research), and the Army Air Forces (Air Materiel Command), under the Signal Corps Contract No. W-36-039 ~~-32037. 1 Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 57 58 R. h’l. FASO [J. 17. 1. It will be shown later that it is not possible to match an arbitrary impedance to a pure resistance over the whole frequency spectrum, or even at all frequencies within a finite frequency band. On the other hand, it is evidently possible to obtain a match at any desired number of frequencies, provided the given impedance has a finite resistive component at those frequencies. Such a matching, however, has little practical value because it is incorrect to assume that one can obtain a reasonable match over a frequency band by correctly matching at a sufficiently large number of frequencies within the desired band. It becomes clear at this point that the statement of any matching problem must include the maximum tolerance on the match as well as the minimum bandwidth within which the match is to be obtained. Furthermore, it is reasonable to expect that, for a given load impedance and a given maximum tolerance, there is an upper limit to the bandwidth that can be obtained by means of a physically realizable coupling network. It follows that an investigation of such a limit should be the first step in any systematic study of matching networks. Before this