# He ain’t heavy, he’s my parabolic

The growth of wireless communications systems means existing support structures, such as towers, are being saddled with an increasing and varied load of antennas and cables. This often necessitates strengthening of the tower, and perhaps its foundation, to support a load that was unplanned when the structure was initially built.

Solid microwave parabolic antennas present more wind resistance than any other antenna type. The following guidelines for calculating wind-exposed areas include a spreadsheet to predict load, based on number and diameters of antennas, and the desired wind-dragging factor coefficient.

A spreadsheet can be prepared showing the combined tower loading of a specific cell site including RF panel antennas as well as parabolic antennas. Every standard diameter of parabolic antenna (solid and grid) can be considered, ranging from one foot to 15 ft. in diameter. The calculations here do not account for satellite antennas or helical, yagi, horn, quad, log periodic, corner reflector or stacked-dipole array antennas.

**Defining spreadsheet fields**

*Site ID — The spreadsheet, as seen in Figure 1 begins with an alphanumeric text field to identify the SITE by NAME and NUMBER. Cellular-system RF sites always have a double nomenclature (name and number). Sometimes a third identifier, such as a mnemonic name, is also used (e.g., Repeater Sumareh—rj 999—RSUM).

*Diameter in feet — This column has numeric cells for the nominal diameters in feet of the solid parabolic antennas.

*Diameter in meters — This column of numeric cells is calculated by multiplying the feet figures to the left by a factor of 0.3048.

*# — This numeric column is for entering the number of microwave solid parabolic antennas for each specified diameter.

*Area without WDF — This numeric column gives the total area in square meters for a given diameter of solid parabolic antenna. If the number of antennas equals “1,” then we will have the parameter named Unitary Area Without WDF. In case we do not have antennas of a given diameter, it is enough to put a zero in the corresponding cell of the column, and the values of the respective areas will be annulled. The values for these cells is the number of antennas in the cells to the left multiplied by the Unitary Area per Antenna:

¼πd^{2}

which is the classical formula from plane Euclidean geometry for the calculation of the area of a circle (where we employ the diameter of the circle instead of its radius). For the transcendent and irrational number pi, you can use the equivalent 3.1415927. Next, we substitute the diameter of the circle by its radius (where R = ½d). So the value for the cell is:

number of antennas * 3.1415927/4 * diameter in meters^{2}

Strictly speaking, the unitary area per antenna is the projected area once the real physical area is the area of a paraboloidal reflector. However, for the considered effects, one must really use the projected area instead of the real physical area. This projected area generates the geometric shape of a circle. The area without WDF does not consider the WDF coefficient. It is, therefore, the projected geometric area.

*Area with WDF — This column is the total fictitious area for a given diameter of a solid parabolic antenna. If the number of antennas equals “1,” then we will have the Unitary Area with WDF. This entity considers the WDF coefficient:

Area with WDF = # of Antennas x Unitary Area per Antenna x WDF or, considering formula [2],

Area With WDF = [Area Without WDF] x WDF

Like the WDF, the coefficient is a number always greater than one (WDF >1 because it is a multiplicative coefficient). We will have:

Area with WDF > Area without WDF

For microwaves, it is common to use a WDF equal to 1.6 (or, in other words, the area with WDF will be 60% greater than the area without WDF).

*Total projected area without WDF — This cell is a summation of all areas (considering all the involved diameters) that do not use the WDF coefficient.

*Total projected area with WDF — This cell is a summation of all areas (considering all the involved diameters) that use the WDF coefficient.

**Using WDF**

Wind dragging factor is the dragging coefficient due to the wind. The WDF coefficient is a pure number, or adimensional (that is, without units). Consider how much more secure the WDF coefficient structure will be for a given antenna loading (on the other hand, the structure will be more expensive).

For parabolic microwave antennas, WDF is usually equal to 1.6. For RF panel antennas WDF is usually equal to 1.2.

An alternative WDF coefficient exists that is additive rather than multiplicative. Its use, however, is much less common than the use of the multiplicative WDF coefficient. When we add the additive WDF coefficient to an area without WDF, we will have the corresponding area with WDF.

Area with WDF = Area without WDF + WDF_{add}

The following illustration shows the mathematical formula for obtaining the additive WDF coefficient in function of the multiplicative WDF coefficient.

WDF_{add} = A_{without}[WDF_{mult} – 1]

or,

WDF_{mult} = 1 + [WDF_{add}/A_{without}]

Where

WDF_{add} = the additive WDF coefficient.

WDF_{mult} = the multiplicative WDF coefficient.

A_{without} = the projected area without WDF.

We can see that the additive WDF coefficient is a function not only of the multiplicative WDF coefficient but also of the projected area without WDF. In other words, the additive WDF coefficient varies with the diameter of the parabolic antenna (on the other side, the multiplicative WDF coefficient does not vary with the projected area). This makes the use of the additive WDF coefficient inefficient in practice.

Figure 2 shows the values of additive WDF coefficients calculated for each diameter of solid parabolic antennas. Notice that, obviously, the area final result will be the same whether one uses additive WDF coefficients or multiplicative WDF coefficients.

Another glaring difference between the coefficients is that the multiplicative WDF coefficient is an adimensional entity while the additive WDF coefficient is a dimensional entity (in this case, the measuring unit is the unit of area used in the problem, typically square meters or square feet).

The program presupposes that all the involved antennas are microwave, solid parabolic antennas. Grid parabolic antennas (such as those used in lower frequencies of the microwave spectrum, for example, in the ISM band) have smaller projected areas than the projected areas of solid parabolic antennas of the same diameter. Thus, the program has a conservative approach when applied to grid antennas, inserting a loading greater than the real loading.

That conservative behavior can be analyzed with a simple, real example. Take the grid antenna model KP4F-23 manufactured by Andrew for use in 2.4GHz ISM band. Based on the model, we infer it is an antenna with a diameter of 4 ft. Its real wind exposition frontal area is 0.5m^{2} (based on the manufacturer’s data). A solid parabolic antenna with the same nominal diameter (in other words, 4 ft.) would have a wind exposition frontal area of 1.17m^{2} (without WDF). The difference, in percentile in this case, is:

Δ% = 100 x [1.17 – 0.50] / [0.50]

Δ% = 134 %

which is certainly a big value.

The calculation of the wind exposition frontal area of a grid antenna is more complex than in the case of a solid antenna. Several types of grid antennas are on the market (such as tubular bars and screens). Screen-type antennas are used in satellite-reception domestic systems in the C band (such systems are technically known as TVRO systems – TV receive only). Grid antennas with a tubular-bar structure are used in spread-spectrum systems at 900MHz and 2.4GHz, as well as in 900MHz point-to-point radio systems.

Note that doubling the diameter of a solid parabolic antenna (from 1.2m to 2.4m, for example) does not double the wind exposition area. Actually, we are going to quadruplicate this area. Remember that the area of a circle varies with the square of the diameter of this circle. If we double the diameter, the area quadruplicates. That is true for any pair of diameters where one is twice the other. Let’s see, therefore, the simple mathematical proof of this affirmation where the subscript “new” assigns the new area or the new diameter and the subscript “old “assigns the old area or the old diameter.

D_{new} = 2 x D_{old}

A_{new} = ¼ x π x [D_{new}]^{2}

A_{new} = ¼ x π x [2 x D_{old}]^{2}

A_{new} = ¼ x π x 4 x [D_{old}]^{2}

but,

A_{old} = ¼ x π x [D_{old}]^{2}

therefore,

A_{new} = 4 x A_{old} Q.E.D.

In a general way, we can say that it is always possible to make loading changes. A loading change occurs when we replace one given antenna for other smaller antennas, with such a form that the final loading, after the change, is less or equal to the loading before the change.

For example: A structure scaled to support a single 8-ft. solid parabolic antenna can easily support the following loading change: three antennas of 4 ft. each, one antenna of 3 ft., one antenna of 2 ft., as well as three antennas of 1 ft.

The effect of supporting-structures loading (towers, monopoles, frames) also varies with the height in which antennas are installed. A tower of height 60m AGL with a 10-ft., solid parabolic antenna located in a platform at 50m will behave differently from an identical tower where a similar antenna is installed in a platform at 15m.

Figure 3 shows the example of the previous loading change (where all new microwave solid parabolic antennas are installed at the same height). Note that the final parameters are equal.

Figure 4 shows the spreadsheet for the RF Panel Antenna Area Calculator (where all physical dimensions, heights and widths are in millimeters).

Praça is a senior telecommunications engineer (transmission coordinator) for Nextel Telecomunicações Ltda., Rio de Janeiro, Brazil. His email address is [email protected]