Circle Diagram of a Transmission Line

Author: Edmund A. Laport

In general, for any short low-loss line,

If this equation is computed for a range of values of βl from 0 to 180 degrees when Z_t = R_t varies from 0 (short circuit) to infinity (open circuit), these values can be plotted in rectangular coordinates to give the well-known circle diagram for a transmission line of characteristic impedance Z₀. However, this diagram has a more general application if the values are computed for Z₀ = 1, because then it can be used for a line of any characteristic impedance by using its value as a proportionality factor applied to all values read from the circle diagram. A chart of this kind is of great utility in making practical adjustments and measurements of transmission lines.

In plotting such a set of computations in rectangular¹ coordinates it is found that the locus of Z_in = jX_in lies on a semicircle centered on the imaginary axis when βl is constant and R_t is varied. Also, it is found that when R_t is constant and βl is varied, the locus of Z_in is a circle centered on the real axis. The full range of values for R_t yields a set of confocal circles approaching the focal point 1-j0 when R_t = Z₀. Since the ratio R_t/Z₀ is the standing-wave ratio Q, an easily measurable quantity, the chart yields all necessary information concerning the impedance at any point in a transmission line having a known standing-wave pattern both in magnitude and in position on the line. It provides the general solution for Eq. (8) for its complete field of values and permits any unknown factor to be read directly when all the other factors are known. Such a chart is shown in Fig. 4.58.

In practice, Z₀ is a resistive quantity that can always be computed with adequate precision for any ordinary engineering uses. The electrical length βl is known from linear measurements, based on the propagation velocity. The position and magnitude of a standing wave can be measured by simple means, in the laboratory and in the field. A current minimum (voltage maximum) point is one where the impedance of the line is resistive and equal to QZ₀. A current maximum (voltage minimum) point is one where the line impedance is resistive and is equal to Z₀/Q. This information also provides a handy method of determining the value of an unknown complex impedance at the load end of the line.

FIG. 4.58. Transmission-line circle diagram in rectangular coordinates.

The same chart may be read and used in terms of admittance, conductance, and susceptance for those applications where the parallel components of impedance are more convenient than the series components. All quantities then are reciprocals except Q and βl.

1)	The circle diagram can also be presented in polar form.10 We use the rectangular form here for convenience in constructing such charts for field use whenever required.