Tower Height Design

Introduction

This applet is designed to permit antenna heights and thus, tower heights to be calculated for a Line of Sight (LOS) Microwave Radio Link. The path profile is entered into the applet from a simple text file (profile.txt) comprising the point number, distance (km) and the height of the terrain (ft or m) at a specific distance from the "A" site. The applet, scales, calculates and plots the flat Earth path profile and after the average tree height and the k-factor have been entered, plots the modified path profile. The radio button on the top left hand side of the screen permits calculations to be performed according to whether the profile contour height data, entering the programme from the profile.txt file, is in feet or metres. Once the frequency (GHz) has been entered, and the CCIR recommendation for the fraction of the First Fresnel Zone radius, the Fresnel zone is drawn. The antenna heights at both "A" and "B" sites can be adjusted to ensure that the Fresnel zone is not entered by any obstruction in the path profile. A print out on the screen indicates the antenna heights (m), the site heights above mean sea level (a.m.s.l.) (m), the chosen tree heights (m), the radius of the First Fresnel zone at mid path, the maximum and minimum heights of the path profile, the path length (km), and the mid-path Earth Bulge. Below is a typical applet plot.

Theory

Central to tower height calculations, in radio relay link design, are the mechanisms of refraction and diffraction. The first of these two mechanisms give rise to the k-factor which is instrumental in causing the Earth to bulge up and cause obstruction fading. The second mechanism gives rise to a cigar-shaped region centred around the line connecting the antennas at either ends of the link (boresight) whose edges give rise to a received signal which is half a wavelength different from the received signal travelling along the boresight. The rim of the cigar shaped region is known as the first Fresnel zone and carries 96% of the power between the two antennas.

Refraction

All electromagnetic waves are refracted when they pass from a material of one refractive index to a material of another refractive index. In the atmosphere, the changes in refractive index are at most times gradual, since the density of the air decreases with height, theoretically, at a uniform rate. Above 30 MHz, the water content of the air plays a predominant part in refractivity changes, for the dielectric constant of water is approximately eight times that of air. In a normal atmosphere, the specific humidity of the mass of water vapour per unit mass of air, is constant. This means that the dielectric constant and the refractive index both decrease continuously with increasing height. The general vertical changes in refractive index of the atmosphere produce a curving of the waves in the vertical plane, as they travel from the transmitter to the receiver. The amount of path curvature varies with time due to changes in temperature, pressure and humidity. Under normal propagation conditions, the path curves away from the true Earth's surface so that the radio horizon is effectively extended. When the vertical refractivity gradient increases, however, the path of the radio wave is bent towards the true Earth's surface, reducing the clearance over the underlying terrain. This situation may produce an effect where the Earth itself, or the vegetation, causes an obstruction. Under conditions such as these, obstruction fading may occur. Good radio link design prevents this from occurring, by insuring that the line of sight from the transmitter and receiver is rarely lost under worst refractivity conditions. To do this, the design determines the height at which the antennas should be placed. However, the higher the antennas are to be placed, the higher the towers must be. As increasing the tower height of a self-supporting tower increases almost exponentially the cost, some consideration must be given to keeping the tower heights to a minimum.

The changes in the refractive index n, of the air of only a few parts per million can have an effect on the propagation of the radio wave. The values of n are very close to unity (typically 1.00035) and so it is more usual to deal with the refractivity N, in the dimensionless N-units, where:

One of the most significant factors in the influence of radio wave propagation is the large-scale variation of refractive index with height, and the extent to which this changes with time. In practice, the measured median of the refractivity gradient in the first kilometer above ground in most temperate regions is about -40 N-units/km. Under the assumption of a constant refractivity gradient, the radio wave is an arc of a circle r, related to the refractive index n, by:

where h is the height above the Earth's surface in the same units as r. The effective Earth's radius a_e, due to the change in refractive index is given by:

where a is the Earth's radius (6.37 x 10³ km). If the effective Earth's radius a_e, is given by:

then the k-factor, or the effective Earth's radius factor is given by:

The effects of the k-factor become evident when considering the Earth Bulge at any given point along the radio path, as now discussed.

Earth Bulge

Given the length of the path d (km), at any distance d₁ (km), from a site (A or B), the distance to the other site (B or A) d₂(km), can be calculated as (d - d₁) (km). For a given k-factor k, the Earth Bulge h (m), at this distance d₁ (km) is given by:

The path profile, which is obtained from contour maps by drawing a line between the two sites A and B and plotting the contour height (m) against distance along the path (km), is usually for flat Earth. That is, without the effects of refractivity taken into account. By taking this flat Earth profile (k ® ¥ ) and adding the mean tree heights (m) to it and then adding the distance h, to the tops of the trees for each kilometer along the path, the resulting path profile is that which would exist for the value of k-factor chosen. The smaller the k-factor is less than unity, the more pronounced is the Earth bulge, and as a consequence of this, the higher towers must be. In most temperate climates throughout the world, 99% of the time will see a k-factor of 4/3. The remaining 1% of the time may see a k-factor reducing to 2/3. It is these probability factors which are taken into account in the radio link outage calculations, which depend on the worst month experience in a location over a 20-year period. For example, if it had been recorded that the k-factor dropped to 1/3 once in 20 years, and it was for a period of only one hour and that the link had been designed so that the path was only obstructed for a k-factor of 1/3, then the radio link outage time would be 1 hour in the number of hours in 20 years. That is, 5.7 x 10^-6, or 0.00057% of the time. It should be noted that the Earth bulge is at a maximum in the centre of the path, and only becomes a problem if there is a high point or there are high points in this region. At site A and site B the Earth bulge reduces to zero. The applet permits the k-factor to be entered in one of the scroll bars, in the tower height design. Negative k-factors can occur. Negative values of k-factor indicate that "ducting" in this region will occur, where the radio beam is trapped as if in a duct or waveguide. The above equation for Earth bulge does not take into account negative values of k-factor.

Diffraction

Diffraction of radio waves is, in practice, the bending of the waves around objects. The amount of bending increases as the object thickness is reduced and also as the wavelength of the wave increases. The amount of bending or diffraction by radio waves is therefore much greater than that of light around the same sized object. The reason why diffraction concerns the radio link design engineer is that obstructions that are not necessarily in the direct path (boresight) of the radio wave may cause the wave to become attenuated. This attenuation is due to the diffraction losses and is called diffraction attenuation. This attenuation arises from the concept of Fresnel zones which are ellipsoids surrounding the direct path or boresight of the radio wave from the transmitter antenna to the receiver antenna. The first ellipsoid has the locus of its outer shell such that any signal reaching the receiver antenna via this path (the indirect path) will travel half the carrier wavelength, l /2, more than a signal travelling along boresight (the direct path). This indirect path signal when arriving at the receiver antenna will be 180° out of phase with the direct path signal. Because the indirect path is longer than the direct path the received signal will be more attenuated, and only partial cancellation of the received signal will occur. The amount of partial cancellation, of course depends on how much longer the indirect path is over the direct path. The region inside this first ellipsoid is called the first Fresnel zone. The first Fresnel zone ellipsoid contains approximately 96% of the power that reaches the receiver. In practice, propagation is assumed to be pure line of sight, that is, without any diffraction attenuation occurring, if there is no obstacle within the first Fresnel zone. Besides the first Fresnel zone, there is a family of ellipsoids surrounding this first shell. These are the second, third, fourth, etc. Fresnel zones. They have little effect in producing noticeable diffraction attenuation due to the small signal power contained within them. The radius of the family of ellipsoids about the boresight varies along the propagation path and is given by:

metres

where; n is the Fresnel zone number (1,2,3,...), l is the carrier wavelength, d₁ is the distance from one terminal or site to the point where the Fresnel zone radius is being calculated, d₂ is the distance from the other terminal or site to the point where the Fresnel zone radius is being calculated, and d = d₁ + d_2. All quantities are in the same units. The radius of the first Fresnel zone F₁, is given by:

where; d_1,d₂ and d are in km, and the carrier frequency f is in GHz.

Diffraction losses differ for different types of terrain that may obstruct the first Fresnel zone. These types of terrain are modelled as smooth sphere, plane Earth and knife-edge. All of these models show that at F/F₁ = 1/Ö 3 = 0.577, the obstruction produces no attenuation of the received signal level over that of a signal in free-space. For this reason, the CCIR has recommended for a k-factor of 2/3, that an obstruction may enter the first Fresnel zone up to 0.6F₁, that is, the CCIR Fresnel factor scrollbar in the applet should take a value of 0.6. This recommendation permits the tower height to be reduced. However, when a k-factor of 4/3 is being used, it is recommended that no obstruction enters the first Fresnel zone, that is the CCIR Fresnel factor scrollbar in the applet should take a value of unity.

Reflection

Not taken account of in this applet, is reflection. The variations in transmission loss for many paths that have been studied arise from two relatively simple propagation mechanisms. These are, the refraction associated with the time-varying vertical gradient of refractive index, as discussed above, and the phase interference patterns due to diffraction and reflection by the Earth's surface and atmospheric discontinuities in the refractive index. Under certain conditions, the wave travelling along the direct path will be interfered with by a ground reflected ray or other multipath rays. The most severe fading occurs when the reflected or multipath ray is of the same amplitude as, and in opposite phase to the direct path wave. The strength of a reflected ground wave, that is a wave reflected off of a lake, marsh, building, etc., that reaches the receiving antenna depends to a large extent on the directivity of the antennas, the height of the antennas above the Earth and the nature of the surface and the length of the path. Reflection point calculations can be completed to permit the point of reflection on a path given the antenna heights at either end of the path and the k-factor. If the reflection point on a path occurs on water, or a good reflecting surface, then the height of one of the antennas must be reconsidered.

The Applet

The profile.txt file

This file contains the data for the path profile of the radio link under consideration. On writing the file yourself, based on information obtained from a contour map, the data should be entered as given in the example below:

Point number	Distance (km)	Height (ft or m)
1	0	4123.7
2	1	4136.8
3	2	4143.5
…	additional points inserted here	additional heights inserted here
n	last distance point	site B height

The applet will self scale the profile. The distance axis will print on the screen, every third point in the profile.txt file. The vertical scale will be automatically sized according to the maximum and minimum values of height in your profile and a horizontal grid line will be produced for each height shown on the screen. The maximum number of points in your profile is limited to 150 (which is more than normally required). An example profile is provided with this applet as can be seen from the above diagram. The name of this example file is "Profile.txt". Your profile must replace this file and be called "Profile.txt". Save the example "Profile.txt" as "Profile0.txt" if you do not want to lose it.

The Web Browser

This applet works best with Microsoft's Internet Explorer. The Netscape browser tends to take to much rewriting time before the applet stabilizes on a scrollbar value.

The radio button

This button is a toggle button. It will toggle between feet and metres. If the height data from the profile.txt file is given in metres, then the button must be toggled for metres. The height axis will change to metres and the fm indicator, above the height axis label, will indicate feet or metres. It is important to make sure this toggle is set to the data coming into the programme because of the calculation required for the Fresnel zone and Earth Bulge (there being a factor of 3.28 between feet and metres).

Entering Data into the scrollbars

The top three scrollbars are for entering the trial height of the antennas at site A and site B and for entering the average height of the vegetation, or trees. Each value is entered in the height scale being used. The bottom three scrollbars are for entering the k-factor (0 to 2), the CCIR recommended clearance factor as a decimal fraction of the first Fresnel zone radius, and the carrier frequency in GHz. As the k-factor is reduced it can be seen that a second profile starts to separate itself from the main (or flat Earth) profile. The addition of tree heights to the applet makes the Earth bulge profile stand out more clearly. This is shown in the above diagram. To have the first Fresnel zone drawn on the screen, the CCIR Fres fact. and the Frequency scrollbars require an entry. Note, the first Fresnel zone will only be shown if the CCIR Fres factor is 1. For a value less than 1, that is a clearance factor less than 1, only the fraction of the first Fresnel zone is drawn. The object of the excercise is to increase or decrease the antenna height at site A or site B until the Fresnel zone is just slightly above the obstruction. It can be seen that as the antenna heights are varied, the Fresnel zone will follow the boresight, or the line between the tops of the towers at both sites. In this applet, it is assumed that the antenna at both sites is placed at the top of the tower. The tower heights required, in practice, must be higher than the required antenna heights. By increasing or decreasing the frequency, it can be seen that the first Fresnel zone or portion of it, decreases or increases in size.

Screen printout

The Earth Bulge at mid-path is printed out on the top part of the left-hand side of the screen. Below this are the antenna heights at both A and B sites, above the level of the site heights. Below this is the value of the selected tree or foliage height. The site heights from the profile.txt file are printed out on the right-hand side of the screen, together with the value of the first Fresnel zone radius at mid-path. All units again are in the chosen height units. The centre of the screen shows the link distance in kilometres and the maximum peak and minimum valley of the path profile in the chosen height units.

The source code (version Rev.1 98/08/07) is available according to the GNU Public License.

Tony Townsend, tonyart@ieee.org

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