Determination of the Minimum Antenna Mast Height with Nonzero Path Inclination: Method II

In this paper, a second method for the determination of the minimum antenna mast height for line of site wireless communication link with nonzero path inclination and with known height of one of the antennas is presented. In the first method,(not presented here), none of the antenna height is known. In this second paper, the height of one of the antenna is known, particularly, the antenna that is above the maximum obstruction height. This places further constraint in the determination of the minimum antenna mast height for the lower antenna. In this paper, both the mathematical models and the algorithm are presented along with sample numerical example using path profile data for a 3 GHz microwave communication link with path length of 38.8876 km. The know antenna height is 20 m above the maximum height of the tip of the obstruction which is found to be 146.62 m at a distance of 14306.98 m from the transmitter. From the result, the receiver antenna height is 166.6 m and transmitter antenna height is 135.35 m whereas, the transmitter antenna mast height is 45.51 m while the receiver antenna mast height is 117.1 m the path inclination is 0.804. The ideas presented in this paper are particularly useful when a line of sight link is to be extended from an existing transmitting point.


Introduction
In wireless communication systems, Line Of Site (LOS) communication is a form of communication used when the signal, such as microwave, can travel in a straight line [1][2][3][4][5]. In that case, the transmitter and receiver antennas are raised and aligned to each other above the surrounding obstructions in the signal path. In order to determine the minimum antenna height for clear line of sight certain terrain and network parameters are considered; namely, the terrain elevation profile, the earth bulge, the obstruction height, the signal frequency, radius of the Fresnel zone, among others [6][7][8][9][10][11].
Basically, determination of the minimum antenna mast height is to minimize cost of construction, installation and maintenance of the mast. The higher the mast, the higher the cost. When installing a fresh LOS link, the minimum height of the transmitter and the receiver antennas need to be determined from the available link and terrain parameters. In that case, none of the antenna height is known. The approach to determining the minimum antenna mast heights in such case is presented in method I. In this paper, the method II is presented for determination of the antenna mast height when the path inclination is not equal to zero and the height of one of the antennas is known, particularly, the antenna that is above the maximum obstruction height. This places further constraint in the determination of the minimum antenna mast height for the lower antenna.
In any case, the methods consists of mathematical expressions and algorithm for determination of the various requisite parameters along with the minimum the transmitter and the receiver antenna mast heights.

Determination of the Minimum Antenna Mast Height When the Path Inclination Is not Equal to Zero, ( ) and the Higher Antenna Height ( ) Is Known
In this analysis, a communication link (figure 1) with the transmitter (T) and the receiver (R) at distance d apart is considered. it is assumed that the higher antenna is the receiver antenna and it is known. As such , H H . If however, H H , then the notation r for transmitter and t for receiver will have to be swapped, whereby the transmitter becomes the receiver and vice versa. Let be defined as the difference between the given receiver antenna height ( ) and the obstruction height at location x which is a distance from the transmitter. That means, Where is the actual line of sight (LOS) clearance height (in m) at point x.
is the obstruction height (in m) at point x measured from the ground level whereas is the obstruction height (in m) at point x measured from the sea level, where; ! is the height (in m) of the earth bulge at location x between the transmitter and the receiver; it is given as [8,-10]; At the transmitter and the receiver, , = 0, hence, 0.
is the elevation (in m) at point x, which is a distance of , from the transmitter and a distance of from the receiver.
Let -./ be defined as the expected line of sight height at point x. Basically, -./ gives the equation for line of sight in terms of . At the transmitter, x = 0, 0, and is the distance from the transmitter to point x. In addition, let be the distance from the receiver to point x. Then, = . Consider two points on the line of sight at location x1 and x2 (where location x1 is at a distance & from the transmitter and a distance & from the receiver, Also, location x2 is at a distance ' from the transmitter and a distance ' from the receiver). Then, by using similar triangle (in Figure 1 ) on triangles ARD and BRD, -./ & (which is the expected line of sight clearance height at point x1) and -./ ' (which is the expected line of sight clearance height at point x2) are related as follows: Hence, In order to satisfy the line of sight clearance requirement at point x the following condition must be met: Where 5678 is the clearance of the Fresnel zone, n where [12][13][14][15]; Where ; for n =1, 2, 3, … and ,( ) >> λ in metres is given as; The height (in meters) of the transmitter antenna mast measured from the ground is given as h (LMN ) where; Where , is the elevation at the transmitter. The height (in meters) of the receiver antenna mast measured from the ground is denoted as h (LMN ) ;

The Procedure for Determining the Minimum Antenna Mast Height When the Path Inclination Is Not Equal to Zero and the Higher Antenna Height Is Known
The following algorithm states the procedure for determining the minimum transmitter and receiver antenna mast height when the path inclination is not equal to zero and the higher antenna height is known.

Results and Discussions
The LOS link parameters are used in the computation are; path length = 38887.6 m, frequency = 3 GHz, effective earth radius factor (k-factor) = 1.33333 and obstruction height (hob) = 10 m. The receiver antenna is assumed to be 20 m above the maximum height of the tip of the obstruction and the specified minimum LOS percentage clearance with respect to Fresnel zone 1 is 60%. Some of the elevation profile is given is Table 1. The maximum elevation of the obstruction from sea level (that is, maximum ( ( ) ) 146.62 m and it occurred at a the distance of 14306.98 m from the transmitter. The receiver antenna is assumed to be 20 m above maximum ( ( ) ). So, the receiver antenna is 166 m while from the results the transmitter is obtained as 135.35 m high.
In table 1 the minimum percentage clearance of 60% with respect to Fresnel zone 1 occurred at the location of the maximum height of the tip of the obstruction which is a distance of 14306.98 m from the transmitter. The radius of the first Fresnel zone at that point is 30.07 m and the LOS clearance height at that point is -18 m which gives the percentage clearance of 60% at that point. The 60% percentage clearance tallies with the 60% clearance specified at the link design stage.

Conclusion
In this paper, a second method for determination of the minimum antenna mast height when the path inclination is not equal to zero and the higher antenna height is known is presented. In the first method,(not presented here), none of the antenna height is known. In this second paper, the height of one of the antenna is known, particularly, the antenna that is above the maximum obstruction height. This places further constraint in the determination of the minimum antenna mast height for the lower antenna. In this paper, both the mathematical models and the algorithm are presented along with sample numerical example using path profile data for a line of site microwave communication link.