![]() ![]() In the next section we will be able to determine analytically the minimum IL required to achieve a perfect match. Stated another way, for certain combinations of ZL and 1, a perfect match cannot be attained for any LA and LB if IL = 0. Depending upon the normalized load impedance ZL = YOZL and the chosen value of 1, a transmission-line section between the location of the first stub and the load may or may not be needed in order to achieve a perfect match. The four parameters involved are as follows: LA = length of first stub LB = length of second stub I = distance between the stubs IL = distance between first stub and load. A load ad- mittance YL is to be matched to a main line with characteristic admittance Y0. ANALYTICAL FORMULATION Consider the double-stub arrangement in Fig. The entire com- puter program takes only about 1 s to run on a DEC-10 computer. Much faster answers with much greater accuracy can be obtained than are possible with a Smith chart. 3) If the problem specifies a longer distance between the first stub and the load than the minimum required distance calculated in 1), compute the appropriate stub lengths. 2) If a match is not possible, calculate the minimum required distance between the first stub and the load as well as the stub lengths. 1) Test whether a match is possible if the first stub is connected directly across the load, and if a match is possible, calculate the required stub lengths. Assuming a given load impedance (normalized with respect to the characteristic impedance of the transmission line) and a chosen distance between the stubs, the computer program does the follow- ing. A Fortran computer program is written which enables the complete solution of the double- stub impedance-matching problem. The authors are with the Department of Electrical and Computer Engineering, Syracuse University, Syracuse, NY 13210. When the chosen distance between the stubs is such that a match is not possible for a given load impedance if the first stub is connected directly across the load, formulas for the minimum required distance between the first stub and the load and for the appropriate Manuscript received Marevised August 9, 1982. In this paper, we make use of the ABCD-matrix (the chain- parameter) representation for a transmission-line network to obtain an analytical solution for the general double-stub impedance-matching problem. The minimum value of this distance is not easily found on a Smith chart, nor are analytical formulas available for the determination of this minimum distance. In that case, a distance must be allowed between the first stub and the load. ![]() However, de- pending upon the chosen distance between the stubs and upon the load impedance, a match may not be possible if the first stub is connected across the load. For simplicity, it is desirable to connect one of the stubs (the first stub) directly across the load. It involves four param- eters namely, the lengths of the two stubs, the distance between the stubs, and the distance between the first stub and the load. The general problem of impedance matching using double-stub tuners, on the other hand, is more complicated. In the case of single- stub matching, the required stub length and its distance from a given load can be found relatively easily on a Smith chart. The techniques for solving impedance-matching problems on a Smith chart are well known. The best known and most widely used graphical transmission-line chart is the Smith chart, which is a graphical plot of normalized resistance and reactance loci in the reflection-coefficient plane. This tedium can be alleviated by the use of a graphi- cal solution. INTRODUCTION CALCULATIONS concerning transmission lines involve \_manipulations of complex numbers and often are very tedious. A computer program is attached and examples are given. The formulas are easily programmed on a digital com- puter and numerical answers can be obtained much more quickly and accurately than is possible with a Smith chart. Different combinations of the load impedance and the distance between the stubs are considered. CHENG, FELLOW, IEEE, AND CHANG-HONG LIANG Abstract-An analytical solution is formulated for the general double- stub impedance-matching problem for transmission lines. 4, NOVEMBER 1982 Computer Solution of Double-Stub Impedance-Matching Problems DAVID K.
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