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Impedancematching Series Line SectionsAuthor: Edmund A. Laport In highfrequency practice, conditions are often favorable for the insertion of a short length of line in series with a main feeder as an impedancematching device. When this line section has the correct characteristic impedance and length and is located at the correct position in a mismatched feeder, the standing waves can be suppressed for one frequency or a small band of frequencies.
In a twowire balanced feeder, a reduction in Z_{0} can be effected by closer spacing, larger conductors, or both. Whereas an increase in Z_{0} is effected in the opposite manner, a large increase in Z_{0} may not be physically realizable.
For this reason, the quarterwave impedancematching section will usually start at a current maximum so that it can use a characteristic impedance less than that of the main feeder. The matching section may be made with reduced characteristic impedance by using a fourwire cross section, with two spaced wires in parallel on each side, or by using much larger conductors, or by reducing the spacing between wires. For illustrations of these possibilities see Figs. 4.34 and 4.35. With a fourwire balanced line of type XVI, where there are two wires in parallel on each side of the feeder, the characteristic impedance of a quarterwave matching section may be made lower or higher than that of the main feeder by contracting or expanding the spacings of the sides or between the sides, or by changing conductor sizes, or by increasing the number of wires in parallel, or combinations of all these. In such a feeder, this seriessection method of impedance matching is convenient and practical.31,32 Figure 4.35 shows various possibilities. In this figure A and B are sections of reduced characteristic impedance, and C is a section with increased value. Figure 4.89 shows this method in use. The matching of impedances by this method is not restricted to the use of quarterwave matching sections. In the generalized case the computations are more involved.
For this purpose the circle diagram of a transmission line is useful. Such a diagram is shown in Figure 4.58These curves are read as follows: The abscissa of a rectangular system of coordinates is taken as the scale of resistances, and the ordinates, positive and negative, are taken as the scales for positive and negative reactances, using the same scale as for resistance. In order to make the chart as general as possible, a characteristic impedance of 1.0 is used for all computations. This makes all resistances and reactances read directly in proportional parts of the characteristic impedance of any transmission line. The focal point is 1.0 ± j0. This is the input impedance of a perfect line with Z_{0} = 1.0 and terminated in a resistance R_{t} = Z_{0}, whatever the distance from the termination. When the line is terminated in a resistance R_{t} > Z_{0}, the input impedance Z_{in}, as a function of distance from the termination, will describe a circle enclosing Z_{0}. This circle will be centered on the resistance axis but will be eccentric with respect to the Z_{0} focal point. The distances from the terminal end fall on semicircles centered on the reactance axis where R_{in} = 0. The direction of change of Z_{in} with βl in degrees from the termination is clockwise from the resistance axis, starting at the right of Z_{0} when R_{t} > Z_{0}, and sweeps a semicircle in the first 90 degrees and a full circle back to the starting point on the resistance axis in 180 degrees (onehalf wavelength from the termination). To test this method with figures, read from the chart (Fig. 4.58) R_{t}, βl, R_{in} and X_{in}:
When R_{t} < Z_{0}, the starting point is on the resistance axis to the left of the focal point and electrical length is read clockwise, starting from this axis as zero and continuing around the complete circle at 180 degrees. The chart is not marked this way, to avoid confusion, but in use one subtracts 90 degrees from the electrical lengths marked on the radial lines in the upper half of the chart and adds 90 degrees in the lower half of the chart. For example, test the following points on Fig. 4.58:
These relative values of impedance are from the resistive termination or from the current maximum or current minimum nearest to the termination when Z_{t} is complex, or when Z_{t} <> Z_{0}. The ratio R_{t}/Z_{0}Q is the standingwave ratio, marked on the confocal circles. By means of this circle diagram the impedance at any point on a mismatched feeder can be determined if the standingwave ratio can be measured and the point of minimum or maximum current located. From the knowledge of the impedance Z/θ at any chosen point on a mismatched line, one can proceed to determine the electrical length βl_{00} and characteristic impedance Z_{00} of a matching line to terminate the main feeder in its characteristic impedance Z_{0}.


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