# Sinusoidal Current Distribution

Author: Edmund A. Laport

The radiation pattern for a vertical radiator with sinusoidal current distribution may be found from the following equation, in which G may be any value, large or small, and a is the angle above the horizon:

When G = 90 degrees (height is one-quarter wavelength), this equation

reduces to

When G = 60 degrees, the following equation may be used for the vertical radiation pattern:

These equations only give the shape of the pattern in relative values of field strength.

Figure 2.6 shows the vertical patterns, in rectangular coordinates, for vertical radiators from 45 to 225 degrees high, based on sinusoidal current distribution. Table 2.5 gives the relative values of the vertical pattern, in more convenient form for computational purposes, for eight different heights corresponding to those of most frequent application for broadcasting.

 FIG. 2.6. Relative vertical radiation patterns for vertical radiators with sinusoidal current distributions for various electrical heights (G).

In this table are included the approximate values of the patterns in the region of a pattern null, which is the result of the fact that in practice the current never is zero at a node. The occurrence of a minimum instead of zero current at a node in the radiator produces an analogous effect on the radiation pattern, in that the pattern will have a minimum instead of a complete null. The phase of the electric field goes through the same kind of transition in passing a minimum as did the phase of the current in passing a node. There is a minimum in the vertical radiation pattern for every current minimum along the radiator.

Broadcasting applications almost never make use of radiators having more than one node, not counting the one that exists at the top of the antenna. The node at the top of the antenna has its radiation counterpart as a null in the pattern directly above the vertical radiator, or where α = 90 degrees.

 TABLE 2.5. VERTICAL RADIATION PATTERNS FOR VERTICAL RADIATORS OF DIFFERENT ELECTRICAL HEIGHTS WITH SINUSOIDAL CURRENT DISTRIBUTIONS α 60° 90° 120° 150° 165° 180° 195° 210° 0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5° 0.981 0.977 10° 0.980 0.976 0.970 0.961 0.954 0.943 0.928 0.909 15° 0.958 0.950 0.937 0.913 0.896 0.873 0.844 0.804 20° 0.929 0.916 0.888 0.851 0.820 0.785 0.734 0.675 25° 0.890 0.869 0.833 0.777 0.736 0.683 0.616 0.524 30° 0.845 0.816 0.768 0.695 0.645 0.578 0.490 (0.510)a 0.375 (0.400)a 35° 0.792 0.756 0.702 0.610 0.548 0.470 0.368 0.234 40° 0.735 0.695 0.629 0.527 0.458 0.370 (0.390)a 0.256 (0.270)a 0.111 (0.150)a 45° 0.673 0.628 0.554 0.446 0.370 0.278 0.159 0.008 50° 0.607 0.559 0.483 0.371 0.293 0.203 (0.225)a 0.081 (0.125)a -0.066 (0.160)a 55° 0.536 0.488 0.413 0.301 0.227 0.136 0.022 -0.115 60° 0.464 0.414 0.344 0.238 0.170 0.087 (0. 110)a -0.017 (0.080)a -0.143 (-0.200)a 65° 0.388 0.345 0.281 0.187 0.125 0.052 -0.040 -0.149 70° 0.312 0.271 0.218 0.140 0.088 0.025 (0.050)a -0.048 (0.075)a -0.138 75° 0.237 0.204 0.162 0.098 0.058 0.010 -0.047 -0.113 80° 0.158 0.138 0.106 0.063 0.036 0.003 -0.036 -0.082 85° 0.079 0.070 0.053 0.031 0.017 0.001 -0.020 -0.043 90° 0 0 0 0 0 0 0 0 b) 190 195 203 216 225 236 250 265 c) 306 314 326 348 362 380 402 426

b) Unattenuated field strength at 1 mile with 1,000 watts radiated, in millivolts per meter.

c) Unattenuated field strength at 1 kilometer with 1,000 watts radiated, in millivolts pei- meter.

a) Figures in parentheses represent very nearly the actual values encountered in practice, taking into account the deviation from sinusoidal current distribution due to radiation losses in the antenna, shown only where the differences are of importance.

Having now the shape of the vertical patterns for simple vertical radiators, it remains to set the actual values of field strength that will result from a given power radiated from a given vertical radiator. Table 2.1 gives the values of unattenuated field strength at the surface of the ground at a distance of 1 mile from the radiator, for various values of G and power radiated.

A radiator of nonuniform cross section has a current distribution that departs from sinusoidal distribution from trivial to considerable amounts, depending on the geometry of the radiator. The distribution then becomes empirical and has to be solved as an individual case. The resulting vertical pattern is also empirical. Solutions for such cases have been published.

It can be seen from the figures in Table 2.1 that the vertical directivity has an important influence on the horizontal field strength at 1 mile. The effect can be seen more clearly in Fig. 2.1, which shows the field strength at unit distance as a function of electrical height G for constant radiated power, using three different radiator heights as 100 percent.

Last Update: 2011-03-19