# Graphical Synthesis of Impedance-matching Networks

Author: Edmund A. Laport

The design of impedance-matching networks is an important part of antenna engineering. One of the quickest and easiest methods for designing impedance-matching networks is the graphical method described in this chapter. The accuracy of this method is adequate for engineering-design purposes, giving slide-rule accuracy with moderate care in drawing. At radio frequencies this is usually equal to or better than the accuracy of measurements. This type of solution shows at a glance the physical principles of operation that most engineers want to know, as well as the effects of variations in the circuit elements. The method will be demonstrated by a series of typical engineering problems.

There are many ways in which rotating vectors may be employed for circuit analysis, but not all lend themselves to circuit synthesis. The technique best adapted to synthesis is that where only current and potential vectors are used. The results then come out directly in resistance and reactance by using quantitative vector ratios.

The following are general suggestions for obtaining maximum benefit from the graphical method of impedance-matching network synthesis:

1. Reasonable care with drawing will produce 10-inch slide-rule accuracy on 8 1/2- by 11-inch paper. It is a great convenience to use polar coordinate paper for vector calculations, using the printed decimal divisions thereon for scales. Radial vectors and angles are thus directly revealed, while nonradial vectors can be measured with dividers and referred to the basic scale of the paper. This avoids the use of protractor and rule. In using plain paper, the L scale of a slide rule, in combination with dividers, provides a handy decimal-dimension base. Diagrams should be constructed carefully and with a sharp pencil.

2. An impedance is represented by the ratio of a potential vector of ; assigned value and a current vector of assigned value, related by some phase angle.

3. To avoid confusion, the closed arrowhead A should be used for current, and the open arrowhead V for potential.

4. One scale should be used for all potentials and another scale for all currents, in any single vector diagram. There need be no relation between the scales for potentials and currents, except that the indicated vector ratios be correct for the resistances, reactances, or impedances involved.

5. The current and potential vectors representing the load impedance should be drawn as the reference vectors and the network worked backward to the input. This gives vector addition throughout for currents and potentials.

6. The triangle method of addition is ordinarily used instead of the parallelogram, because it makes a simpler, clearer diagram by conservation of lines.

7. Where a network includes two or more stages, each stage is solved separately as an individual problem. The stage that includes the ultimate load is solved first. Then its input impedance is used as the terminal impedance for the next preceding stage, etc. By this method, quite complicated networks can be handled easily.

8. A rotating vector diagram represents steady-state conditions at a single frequency. To analyze the performance of a network at several frequencies, a new vector diagram is required for each frequency.

9. Following generally accepted conventions, advance in time is counterclockwise. A potential across an inductive reactance, leads the current through the element 90 degrees. When the voltage is lagging 90 degrees, it is a capacitive reactance. A current in phase with a potential is a positive resistance, and a current 180 degrees with respect to a potential is a negative resistance.

Last Update: 2011-03-19