Ing. Jiri Otypka, CSc

Calculating the Focal Point of an Offset Dish Antenna

VHF Communications 1/1995

Download the full article in pdf format (contains formulae and BASIC program)

It is assumed that readers know how to calculate the focal point of a symmetrical parabolic antenna. Offset antennae are a different matter. The popular belief here is that the focal point can not be established by simple means. However, this is not correct.

To establish the focal point of an offset antenna, we need to measure three parameters instead of two: the large and small elliptical axes and the maximum antenna depth. The bulk of an offset antennae is formed by a three-dimensional space (Fig.1a). The similarity of offset antennae is determined by two parameters, but no explicit expression is available. The principle of offset antennae is described below, and the parameters and marginal conditions required to establish the focal point are given. A listing of a small BASIC program is also included, together with some checking calculations. Fig.2 shows an offset antenna with an even lateral face.

It emerges as a section of a cylindrical paraboloid which is divided by a cut. With a little effort, we can prove that: These important findings make it possible to establish the correction angle for the edge of the offset antenna:

w = arctg u

An equation can also be derived for the co-ordinate, x1: For the focal point, the position is as follows: The correction angle can be derived from equations (1) and (2) if it has not already been given by the manufacturer. There is a simple method by which it can be calculated from the reflector width. It is more difficult to solve equation (3). It can be solved, for example, using the small BASIC program, or with a programmable pocket calculator.

First we must delineate the areas on the x axis which lead to unambiguous solutions for equation (3) for f > f together with z2 > z1. Fig.3 shows the interval selection for the solution of the equation by means of interval halving. We begin in the range x < f, which emerges as the penetration of the two intervals referred to. If no solution is found, the search is continued in the range x > f. It is advantageous to select the interval dx - b. It can be shown that in (3) a (+) in front of the root for x > f never leads to a solution.

If no solution has been found, the areas outside the brackets, which have so far been established as the standard accuracy, G = 0.001 (in the program), dx = accuracy, b = DX (in the program) or N = 1 (dx - Nb), are expanded. We now select a G < 0.001 or N > 1. The co-ordinate x1 is established with an accuracy of ± dx.

In order to obtain a narrower major lobe in the directional diagram of the offset antenna, antennae are manufactured which do not have an even surface (Fig.4). These antennae also have a plane of symmetry y = f, 2a and t can be measured in this plane. The interval 2b can be measured in a plane which runs vertically to the plane y = diameter and which contains the points (x1, z1) and (x2, z2). To put it another way, for any even section selected through the surface of a rotation paraboloid, the points (x1, z1) and (x2, z2) can be obtained, which make it possible to measure the lengths 2a, 2b and t. All the parameters can indeed be measured on a punched-out offset antenna.

There remains only the problem of the correct classfication of points (x1, z1) and (x2, z2), which are in the plane of symmetry. In theory the classification is clear, but in practice there are always problems, so appropriate manufacturers' markings would be of assistance. The point (x1, z1) lies at a position on the edge where the curvature of the section in the plane y = f is greater than the curvature at point (x2, z2). In many cases, only after precise antenna measurements can it be determined that the two points have been transposed.

Fig.5 shows some examples for offset antennae. Here a focal length of 35 mm. is selected, and then some calculations are carried out. The relevant parameters and results are shown in table form. If we use the BASIC program, we must initially increase the intervals after entering the standard parameters. The standard accuracy has been retained, N has been defined as N = 2, and finally N = 3 has been selected, which led to a solution. In the second case, it was not necessary to alter the standard parameters. In the third case, in which the section ran through the impermissible area x1 = f, the intervals had to be increased again. So G = 0.00001 was selected and x1 calculated. The area around x1 = f can be avoided if the input parameter is altered "infinitesimally". In practice, this case presents no difficulties, and so no special researches or calculations were needed. In the fourth case, a higher accuracy was selected for x1, from x = 0.1 mm. to x = 0.01 mm.. If we lay down that 2a = 2b, we are pre-setting the values of a symmetrical parabolic antenna, which is shown in Fig.5.

In accordance with the principles set out above, here are some explicit examples of the application of the program:

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