FCC guidelines provide approximate models that may be used to calculate power density at AM, FM and television broadcast stations, as well as near aperture antennas.
The power density surrounding an antenna varies as a function of location, and is dependent on distance and orientation. The fields around an antenna may be divided into two principal regions, one near the antenna called the near field and one at a large distance from the antenna called the far field. The boundary between the two is often taken to be at the radius
where L is the maximum dimension of the antenna and λ is the wavelength.
In the far field the shape of the antenna pattern is independent of distance while in the near field, the shape of the field pattern depends on the distance R. Antenna patterns published by manufacturers are typically only applicable in the far field and are therefore only applicable for power density calculations in the far field of the antenna.
For high gain arrays at broadband PCS frequencies, this boundary can be at a significant distance from the antenna. At broadband PCS frequencies (~1900 MHz) the boundary between the near and the far field is 50 meters for a 2 meter high antenna. The majority of potential RF hazard zones at a site occur in the near field, resulting in the necessity to predict both the near and the far field power density of an antenna array.
Rigorous analytical techniques and software methods are available that predict fields surrounding antennas. These techniques typically model antennas as small wire elements and metal plates with some of the elements fed by signal sources. These methods have a practical limitation, however, because they require detailed information on the physical structure of the antenna that is typically not available. Other less rigorous techniques have been developed to obtain adequate estimates of power density near antennas. These techniques make use of available information including the physical dimensions and the published gain patterns of the antennas.
As described by the FCC, in the case of a single radiating antenna, a prediction for power density in the far field of the antenna can be made by using the following general equation
where S is the power density, P is the power input to the antenna, Gi is the gain of the antenna relative to an isotropic radiator, and R is the distance to the center of radiation.
An alternative expression is
where EIRP is the effective isotropically radiated power.
This model can be modified to consider both ground reflection Τ and the gain of the antenna or EIRP in a particular direction as:
where EIRP(θ,φ) is the antenna EIRP at a particular azimuth θ and elevation φ, is found by extrapolating the published horizontal and vertical gain patterns of the antenna to form a three-dimensional antenna gain pattern.
A method of estimating the power density in the near field of a collinear omnidirectional array is described in a technical report prepared for the FCC. This Cylindrical Method describes a technique of predicting power density in the near field of collinear arrays commonly used by wireless operators which is useful only in the main beam of the antenna. To overcome this limitation, a new method has been developed that models collinear antennas as an array of elements. This Collinear Method is useful anywhere in the near field of a collinear array.
A cylindrical radiation model involves computing the average power density on the surface of a cylinder, with a height equal to the antenna’s aperture, and a radius equal to the distance of interest. This is illustrated below.
This model is useful in the near field within the aperture of the antenna. Measurement of collinear arrays show that the power density at a fixed height above the surface falls off exponentially as the antenna’s height is raised above the surface. The cylindrical model does not reflect this exponential decrease in power density.
By modeling collinear antennas as an array of elements with a length of half a wavelength, it is possible to estimate the power density of the array in both the near and far field to within approximately one wavelength of the array. The accuracy of this technique is dependent on how well an array of linear elements fed in phase represents the real antenna. In practice, the elements in an array are not necessarily fed in phase. The error introduced by assuming linear phase is, however, small if the power is averaged spatially over a number of wavelengths. At cellular and broadband PCS frequencies a human body is about 5 and 10 wavelengths long, respectively. Averaging the predicted power density over the height of a human body, at cellular and broadband PCS frequencies, provides a reasonable estimate of exposure.
The near field power density of a collinear array is modeled by treating the vertical collinear antenna as an array of N elements spaced one wavelength apart.
The collinear method estimates the number of elements in the array and the gain pattern of each element. The power density near the antenna is calculated by combining the contributions from each element in the array. This method is described in detail later.
The collinear model has been compared against published measured data. A report prepared for the FCC published measured power density as a function of distance along the main beam of three collinear antennas. Some of the data have been published in an earlier paper. These published measured data are compared against the so-called cylindrical (1/r) model and the collinear method described in the previous section.
Measurements performed on a Swedcom Model ALP-9209 directional collinear antenna, mounted with its center of radiation 1.75 meters above the floor, were taken along the main beam of the antenna up to a distance of 4 meters from the antenna. The ALP-9209 antenna has a physical height of 70 centimeters and a gain of 8.2 dBd. The power input to the antenna was 25 watts. A comparison between measured data in predicted results using the cylindrical and the collinear array model is shown below.
Similar measurements were taken from two other antennas. The relevant test parameters were:
| Antenna | Gain (dBd) | Height of radiation center (m) | Antenna length (m) | Power (W) |
| Decibel Products dB 586 | 6 | 1.02 | 1.05 | 25 |
| Decibel Products dB 809K | 9 | 3.83 | 1.95 | 160 |
The following two graphs show the comparison between the measured data and results predicted using both the cylindrical and collinear models.
These results illustrate that both theoretical methods track the measured results although the predicted results tend, on average, to be conservative. The data are insufficient to draw statistically significant conclusions, but the above results indicate that the average error between predicted and measured values appears to be less than 3 dB for measured data that is not spatially averaged, and less than 1 dB for spatially averaged measured data.
From the above results it is clear that both models are adequate for predicting power density in the near field within the main beam of the antenna. In many cases, predicting the decrease in power density as a function of height below the antenna is also a requirement. This situation is particularly important on rooftop sites, where antennas are elevated to reduce exposure on the rooftop. The paper prepared for the FCC provides some normalized measured data that shows the decrease in power density below antenna at a distance of four feet from the antennas. These data have been compared with the Collinear Method and the results are shown below.
The correlation of the predictions to the measured data varies. For the short Swedcom antenna the predicted results are conservative, while the predictions for the longer Decibel antenna, that is more representative of a collinear array, correlate more closely. Assuming one wavelength spacing between elements, the Swedcom antenna dimensions appear to provide only sufficient spacing for a single element. For a single element “array”, the array pattern is the same as the element pattern. The Swedcom manufacturer’s data sheet describes the antenna as a Log Periodic reflector antenna. This is not a low gain element and the Collinear Method will simply predict a conservative near field approximation using the far-field gain pattern.
The following approximation is used to estimate the gain of an individual element in any direction. Data supplied by antenna manufacturers typically includes the far field horizontal and vertical gain pattern and the physical length of the antenna, L. If one assumes that the collinear array comprises elements with a spacing of one wavelength, λ, then the number of elements in the array, N, may be estimated as
The maximum gain of each element in the array, 
, is
where 
is the maximum gain of the array.
Given the following coordinate system:
The normalized horizontal gain pattern of each element in the vertical collinear array is estimated to be the same as the normalized horizontal gain pattern of the array GA(φ). In contrast, the vertical gain pattern of each element is not readily extracted from the vertical gain pattern of the array because the shape of the array pattern is highly dependent on the phasing and spacing of the array elements. It is however possible to make a reasonable approximation if the gain of each element is less than about 3 dB. The normalized vertical gain pattern of the main lobe of the element is approximated as GEl(φ) = cos3 θ where θ is the elevation angle. This pattern corresponds to a vertical half power beamwidth of 75 degrees.
The gain of each element in any direction is limited to a minimum of 20 dB less than the maximum gain of an element. This has the effect of filling in the nulls of the element pattern and is a conservative approximation used to ensure that the gain of the element is not underestimated in any direction.
The gain of an element in any direction is thus calculated as
Normalized vertical array polar pattern (dB)
As an example, consider a directional collinear array with the following vertical and horizontal gain patterns:
Normalized horizontal array polar pattern (dB)
As described above, the normalized horizontal array pattern is the same as the normalized horizontal element pattern. Note that the vertical array pattern is not used. The vertical pattern is approximated using a cos3 θ function and the element pattern is limited to a minimum gain of 20 dB less than the maximum gain. This is illustrated below.
A three dimensional rendering of the element pattern looks as follows when displayed using the same logarithmic scale shown above.
Using a linear scale, the same antenna appears as follows:
The time rate of energy flow per unit area is the Poynting vector, or power density (watts per square meter) in the far field of an antenna is
where EIRP(θ,φ) is the effective isotropic radiated power.
The effective isotropic radiated power from a single element is
where P is the power fed to the array. The relationship between EIRP, ERP, and P is
EIRP = 1.64ERP = PGA
where GA is the power gain of the array relative to an isotropic source.
The relationship between the Poynting vector and the amplitude of the total electric field intensity, E, at a point in the far field is
where Z is the intrinsic impedance of the medium (Z=377 Ω in free space).
The peak electric field from any element at the location of measurement is,
where R is the distance from the center of antenna to the location of measurement.
The signals from the elements have a different amplitude and a different phase when arriving at the point of measurement. The signals can be added vectorially (or as phasors). Using the center of the array as a phase reference, the relative phase of the ith element is
where RRef is the distance to the center of the array with phase zero and RE is the distance to the ith element.
The contribution from each element is broken into its components,
The X and Y components of each phasor are added and the equivalent root mean square voltage is calculated as
The root mean square electric field is
It may be shown that the average power density is
