Radiation Resistance

Author: Edmund A. Laport

The radiation resistance is essentially due to the distribution of current on the vertical portions of the antenna. Typically the current in all vertical parts is in phase but varies in amplitude with distance along the vertical, with the maximum value at the base, near ground. If the vertical part has a uniform cross section throughout its length, the current distribution will be some portion of a sine wave. If the antenna consists solely of a vertical part, with no top loading, the current at the top will be zero and the distribution will be sinusoidal as measured from the top. Since a sine wave is practically linear from zero to 30 degrees, a uniform vertical radiator of height 30 degrees or less will have a linear current distribution. Thus current distribution plotted against electrical height forms a right triangle as shown in Fig. 1.1.

When there is capacitive top loading attached to such a vertical antenna, its electrical length is effectively increased. The current I_t at the top of the vertical is no longer zero but has a value that is proportional to the capacitance of the top loading. In almost every practical case I_t is less than the base current I_b. With top loading, the plot of relative current against position along the vertical now becomes a right-angled trapezoid, which as a practical limit may ultimately approach a rectangle. This is realizable only with an electrically short vertical section having a very large amount of top loading.

The radiation resistance of an antenna less than an electrical quarter wavelength long is increased by top loading and by increasing height. A straight vertical radiator of height 30 degrees or less has a radiation resistance Rτ following the equation where G₀ is the electrical height in radians.

(One radian is 57.3 degrees.) Referring again to the geometric form of current distribution with height (Fig. 1.1), this relation holds for the triangular distribution. If the base current I_b is always taken as 1.0 for comparative purposes, we can say that radiation resistance varies as the square of the area of the degree-ampere plot. We can transform this relation into a very useful form that shows the radiation resistance as a function of the degree-ampere area A of the plot of current distribution. This is

The utility of this relation is that it holds for any shape of inphase current distribution. It permits direct computation of the radiation resistance from a known current distribution for an electrically short antenna. A top-loaded antenna having a trapezoidal current distribution has a degree-ampere area where Gν is in degrees of vertical height and I_t and I_b the currents at the top and at the base of the vertical, respectively.

These relations can be used for any arbitrary system where the current distribution in degree-amperes is known, the value of A can be computed, and the base current is taken as unity.

A chart of the values of radiation resistance as a function of physical height Gν in electrical degrees and the ratio I_t/I_b is shown in Fig. 1.2 over the domain where the vertical current distributions are linear.