Distance to Obstruction and Obstruction Height

Value to Enter:

If a building, tree line, rolling hillside, or other obstruction is present between the installation location for Radio #1 or Radio #2, you may enter the distance to the obstruction and its height to incorporate the obstruction into the Antenna System Designer calculations. Note that the obstruction for Radio #1 must be be between Radio #1 and the horizon, with the same requirement applying to Radio #2. The "horizon" is based on the half-way point when Distance Between Locations is specified. When one installation height, or both heights are specified, the "horizon" is calculated based on the Refractive Index selected.

Significance of This Value:

An antenna must be high enough to get above the tallest obstruction between it and the horizon by a factor equal to at least 60% of the first Fresnel Zone radius. The Antenna System Designer calculates all these values. The diagram below depicts the various values used, and calculated, that relate to the distance to the horizon (LOS Distance #1 and LOS Distance #2 in the diagram.) The height of the towers (or other installation locations) determines the respective LOS distance to the horizon. The Refractive Index adjusts the LOS to be either visual, radio, or other ratios that all reach further around the sphere of the earth than the flat, geometric calculations would conclude.

Diagram Depicting Various Antenna System Designer Values

In the diagram above it's interesting to consider the 'tilt' of the antenna tower relative to the distance, annotated on the right-hand side of the diagram as 'a' through 'e'. Distance 'a' is the length, along the ground, between the bases of the two towers. It should be evident that a curved path between two points in longer than the straight line path that could be though of as the 'Distance Between Towers'. Because the towers 'tilt away from each other' (they're both extensions of radii of the circle) the tops of the towers will be further apart than the bases. In fact, this discrepancy is the distance 'e', the 'tilt' distance. On the one hand the curved path on the earth is longer than the path from top-to-top, but on the other hand, the tops of the towers tilt away from each other by distance 'e'. The results essentially cancel each other for all practical purposes. Hence, it's assumed that the Distance Between Towers is actually the same as distance 'a', even though they are slightly different.

Here's another aspect to consider. The required height of a tower consists of the height necessary to 'get over the horizon'. This is height 'b' shown above. The tower must be high enough to provide a visual line-of-sight to the horizon. However, the tower must also be high enough to provide 60% Fresnel Zone clearance at the horizon. Height 'd' represents the Fresnel Zone clearance height, with the tower's overall height being b+d. The 'true' height attributable to line-of-sight requirements is actually an extension of the radius of the circle (the 'LOS Height #2' shown in the diagram.) Because the tower 'tilts away' (by a distance equal to 'c') there is a slight error in this part of the calculation as well. As before, this discrepancy is small enough to be ignore for practical calculations. If you're taking the final exam in a math class, however, this type of 'hand waving' won't get you a passing grade!

The Visual Line-of-Sight

The calculation of Line of Sight begins with an approximation of the Pythagorean theorem for right triangles. Consider the diagram shown to the right. The Line of Sight distance (d) is a leg of a right triangle. The other leg is the radius (R) of the earth. The length of the hypotenuse is the sum of the earth's radius (R) and the height of the antenna tower above the surface of the earth (ah). It can then be shown that d=sort(R^2-(R+ah)^2). Squaring (R+ah) yields R^2+2Rh+ah^2 and the R^2 term cancels under the root resulting in d=sort(2Rh+ah^2). When considering antenna towers and installation heights, the value ah is generally very small compared with 2R. The 40 feet of height for an antenna tower is very close to being inconsequential compared to the roughly 3952 miles for the earth's radius. As a result, the distance equation can be approximated by removing the +ah^2 term. Removing this changes the solution for the 40-foot tower height from d=7 miles, 2303.2859 feet to d=7 miles, 2300.6459 feet. The approximation changes the result by roughly 3 feet over roughly 7 1/2 miles. Typically the distance-to-the-horizon formula appear in literature only as the 1st approximation. The Connect802 Antenna Designer, too, uses this form.

There is another simplification of the general formula that is common in literature. Often, the constants under the root are solved and a constant of proportionality is created. When the equation is solved with R and ah in miles, the result is in miles. So to if kilometers are used, or feet, or any other unit of measurement. Doing this, however, is not often convenient. One convenient way to represent the formula is with ah represented in feet and d in miles. Hence, under the root, ah is replaced by ah/5280 and then the value of ah is in feet, with d being in miles. In this case the value of R is in the numerator under the root, and 5280 is in the denominator. The square root of this fraction is brought out from under the root (and adjusted for the Index of Refraction, discussed below) and you see equations like d=1.22*sort(ah) or some other similar variant based on the conversion between units and the Index of Refraction (discussed below.) In the example, the value 1.22 is the constant of proportionality. Of course, all of these 'simplifications' of the Pythagorean 1st approximation must establish a value to use for R, the radius of the earth.

It is well known that the earth bulges slightly at the equator and is slightly elongated at the poles. The planet is not a geometric sphere but, rather, an ellipsoid. Consequently, a great circle path, covering some predetermined number of degrees of arc, will be longer in some directions than in others. A distance measured on a great circle is called a geodetic line. The United States Department of Defense has adopted a reference system that is considered the best fit for the whole earth. This reference system is called the World Geodetic System, and the most recent base measurements were made a standard in 1984, giving rise to the WGS84 standard.

The WGS1984 ellipsoid uses an equatorial radius of 3963.19 miles and a polar radius of 3940.90 miles. The Antenna Designer uses the average of these two values (3952 miles) as the value for the radius of the earth. This is slightly smaller than the average (based on inhabited land mass) used for GPS navigation (3963 miles) but the difference is, at worst, a more conservative value for the calculation of antenna system design.