**APD PROGRAM**

**1. Introduction**

The Antenna Power Density (APD) program provides simplified procedures
for estimating the near field power density of a number of common types
of antennas and graphically checking the compliance of systems with
different emission exposure standards or user defined limits. The
emission exposure standards include EPA, ANSI/IEEE C95.1-1991, FCC
1.1310, and National Council on Radiation Protection (NCRP). The outputs
of the program are distance and power density, either in the form of a
report or in the form of a graph of power density vs. distance. It also
provides the free-space power density at one far-field point.

The APD program was developed at NTIA and the analysis methods have been
outlined in a series of Technical Memoranda. The original APD document
is NTIA TM-87-129 \[1\], in which the procedure for calculating power
density of a number of frequently used antennas was given. The antennas
included the circular and rectangular aperture antennas and six of the
most frequently occurring wire antenna types on the Government Master
File (GMF), i.e., Dipole, Monopole, Whip, Log Periodic, Yagi and
Collinear. The second document NTIA TM-88-135 expanded this work to
include corner reflector aperture antennas plus additional antenna
types, i.e., Folded and Coaxial Dipole - single and stacked,
Omnidirectional: Stub (blade), Discone and Inverted Discone, Ground
plane, Dipole and Stacked Arrays. NTIA TM-90-145 \[3\] is the
engineering document for Version 3, which added an off-axis angle
capability to the circular aperture antenna option. In subsequent
versions, minor changes were made to add or modify the output report or
graph. In the original program, only the emission exposure standard ANSI
C95.1-1982 was used. Later, it was changed to ANSI/IEEE C95.1-1991
standard. In the current version, more emission exposure standards have
been added and in many cases, two different numbers are used, one for
controlled environments (occupational) and the other for uncontrolled
environments (general population).

The procedure used for the calculations is discussed in Section 2
followed by Sections on Input Data and Output.

**2. Analysis**

The APD program is based on simple analytical procedures for calculating
the power densities of aperture and some linear antennas. The procedures
are simple, easy to use and require no additional database other than
what is generally made available by the federal agencies prior to a
system review. Available computer and analytical models such as
Numerical Electromagnetic Code (NEC), NEC-REF (Reflector), NEC-BSC
(Basic Scattering Code), and Mini Numerical Electromagnetic Code
(MININEC) were examined and used in the selection and development of the
procedures used in this program. The APD program can handle most of the
antennas listed in the Government Master File (GMF). The procedure is
different for three main types of antenna: wire, aperture and corner
reflector.

**2.1 Methods for Obtaining Near-Field Intensities**

The behavior of electric and magnetic vectors in the near and far fields
is very different. For linear antennas the far field vectors are linear
and orthogonal, whereas, the near field vectors are generally ellipses
and non-orthogonal. Consequently, the intrinsic impedance varies
spatially and is not a constant 377 ohms as often assumed in the far
field. Also, the **E** and **H** vectors have radial components near the
antenna, which are negligible in the far field.

The MININEC code was run to obtain near field data for all wire antenna
types except the corner reflector. Both NEC and MININEC use method of
moments to solve integral equations for E and H fields generated by wire
currents. MININEC is faster and simpler, but has some restrictions. An
important restriction is that the wire radius must be very small
compared to segment length by about two orders of magnitude. Also, only
free space and perfectly conducting half-space can be simulated for near
field calculations with MININEC.

The NEC-BSC code is used for aperture antennas and corner reflector
antennas. The NEC-BSC code is not a combination code, but is based on
the Geometrical Theory of Diffraction (GTD) and treats structures
composed of flat plates by asymptotic techniques.

2.  **Derivation of Near Field Power Densities**

The near field intensity and distance from the antenna's electrical
center are normalized to their values at a distance considered just
within the far field. Normalization enables an antenna type to be
represented by a single curve and to be independent of antenna power
input and frequency. The far field power density at the normalized
distance (d) is multiplied by a correction factor taken from the
correction curve for a particular antenna type. The E-field intensity is
used for all calculations because it yields a higher, therefore, more
conservative correction than the H-field intensity in all the antennas
under consideration.

For **wire** antennas, the normalized distance corresponding to the
actual distance considered is found by the proportional relationship:

> d~w~ = R /(1.5 λ) (2.1)

where: d~w~ is the normalized distance for wire antennas

R is the distance under consideration, in meters

> λ is the wavelength, in meters

Similarly, for **aperture** antennas:

d~A~ = Rλ/(2D^2^) (2.2)

where, d~A~ is the normalized distance for aperture antennas

D is the largest aperture length, in meters

For **corner reflectors:**

d~C~ = R/(5.0 λ) (2.3)

where: d~C~ is the normalized distance for corner reflectors

The near field power density equations for wire, aperture and corner
reflectors are given below. The inputs to the program are wavelength,
main lobe on-axis distance from the antenna, antenna input power and
antenna gain.

The far field power density for any antenna is:

P~F~ = p~t~ g~t~ / 4πR^2^ (2.4)

where: P~F~ is the far-field power density, in watts/m^2^

p~t~ is the antenna input power, in meters

g~t~ is the antenna gain referenced to an isotropic source

(gain in dBi must be converted to a power ratio)

The near field and far field intensities are related by

E~N~ = E~F~ (CF)~E~ (2.5)

where: E~N~ is the near field electric intensity in volts/meter

E~F~ is the far field electric intensity in volts/meter

(CF)~E~ is the E-field correction factor at d

By definition,

P~F~ = (E~F~)^2^/(120π) (2.6)

where: 120π is the intrinsic impedance in free space, in ohms

Assuming: P~N~ = (E~N~)^2^/(120π)

where: P~N~ is near-field power density, in mW/cm^2^

Combining Equations 2.4, 2.5 and 2.6,

P~N~ = 0.00796 p~t~ g~t~ \[(CF)~E~/R\]^2^ (2.7)

From Equation 2.1, for wire or linear antennas,

R = 1.5 λ (2.8)

Combining Equations 2.7 and 2.8 for wire antennas

P~N~ = 0.00354 p~t~ g~t~ \[(CF)~E~/λ\]^2^ (2.9)

From Equation 2.2, for aperture antennas,

R = 2 D^2^/λ (2.10)

Combining Equations 2.7 and 2.10, for **aperture antennas,**

P~N~ = 0.00199 p~t~ g~t~ \[(CF)~E~λ/D^2^\]^2^ (2.11)

From Equation 2.3, for corner reflectors,

> R = 5.0λ (2.12)

Combining Equation 2.7 with 2.11, for **corner reflectors,**

P~N~ = 0.00318 p~t~ g~t~ \[(CF)~E~/λ\]^2^ (2.13)

In summary, the steps for calculating the near field on-axis power
density for any type of antenna are as follows:

1.  Select a normalized correction curve for a specified antenna type.

```{=html}
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1.  Calculate the normalized distance by Equation 2.1 or 2.2 or 2.3 for
    wire antennas, aperture antennas, and corner reflectors
    respectively.

2.  Find the correction factor (CF)~E~ corresponding to d.

3.  Calculate the near field power density P~N~ at R by Equation 2.9, or
    2.11 or 2.13 for wire antennas, aperture antennas and corner
    reflectors respectively.

The normalized correction curves and the Tables needed for calculating
near field power density for various kinds of antennas are included in
the APD program.

2.  **Determination of Normalized Correction Curves**

The methods for calculating correction curves for different types of
antennas are given in the following sections for wire, corner reflector
and aperture antennas.

1.  **Wire Antennas**

Dipole antennas are used in a large number of systems in the GMF because
of simplicity, low cost and desirable electrical characteristics. A
half-wave dipole may be used as a building block for most wire antennas.
Therefore, field intensities for most antennas can be found by using an
appropriate correction factor as a multiplier for the field of an
electric dipole. If a wire antenna is less than a tenth of a wavelength
long, it can be treated as a current element, where the current along
the element is constant. Closed form expressions for both near and far
field intensities for a current element may be found in textbooks on
electromagnetic theory. For systems in which such electrically short
antennas are placed over a perfectly conducting ground, a good
approximation may be achieved by multiplying the free space calculation
results by a factor of 2.

As the length of the wire antenna becomes larger, the current is no
longer constant. For a center-fed linear antenna, the current
distribution on the antenna is very close to sinusoidal provided that
the antenna cross-section is small compared to a wavelength. Jordan
\[4\] has derived the following closed form expressions for the fields
produced by a dipole assuming sinusoidal current distribution

$E_{z}$=
-*j3oI~m~*$(\frac{e^{- {j\beta R}_{1}}}{R_{1}} + \frac{e^{- {j\beta R}_{2}}}{R_{2}} - 2\text{cos}\beta H\frac{e^{- j\beta r}}{r})$
(2.14)

*E~y~ = j30I~m~*
$(\frac{z - H}{y}\frac{e^{- {j\beta R}_{1}}}{R_{1}} + \frac{z + H}{y}\frac{e^{- {j\beta R}_{2}}}{R_{2}} - \frac{2z\text{cos}\beta H}{y}\frac{e^{- j\beta r}}{r})$
(2.15)

*H~φ\ =~*
$\frac{j\text{30}I_{m}}{\eta y}(e^{- {j\beta R}_{1}} + e^{- {j\beta R}_{2}} - 2\text{cos}\beta\text{He}^{- j\beta r})$
(2.16)

Where I~m~ is the maximum amplitude of current along the length of the
antenna. Figure 1 shows some of the notations used in the expressions
(2.14) to (2.16). The results obtained using the above equations were
found to be in good agreement with those obtained from the NEC model as
shown in Figure 2. As discussed earlier, for antennas located over a
ground plane, the results obtained using the above equations may be
multiplied by a factor of two.

![](media/image1.jpeg){width="5.0in" height="4.104166666666667in"}

The correction factors for off-axis points on a sphere surrounding a
half-wave and a full-wave dipole in free space are given in Figures 3
and 4 respectively for θ = 22.5^o^, 45^o^, 67.5^o^, and 90^o^. The data
for H-field is also included. The data shown in Figures 2, 3 and 4 are
also used to estimate the radiation hazards of other linear antennas as
discussed below.

2.3.1.1 [Monopole]{.underline}

A monopole antenna consists of a straight wire perpendicular to a ground
plane. Field of a monopole antenna is the same as that of a center-fed
dipole antenna in the upper half space.

# 2.3.1.2 Whip

Whip antennas constitute a large category of vehicular antennas, which
are commercially available and are used in many government
telecommunication systems. They are generally low gain antennas and may
not have their feed point at the ground plane. They are used with both
high and low power transmitters. Their far fields are similar to those
of a dipole. The near fields may be approximated by that obtained from a
dipole, as found by comparison using the NEC code.

# 2.3.1.3 Log Periodic

A log periodic antenna is a frequency independent antenna, which may be
an array of a number of dipoles of different length. In the HF range,
the physical size of the antenna is quite large and is supported from a
tower with guy wires. The VHF log periodic antennas are mounted on a
mast above ground. The antenna elements may be vertical or horizontal
depending on the desired polarization. Gain values may be as high as 16
dBi and output power may be in kilowatts. To a first order
approximation, the field produced by a log periodic antenna is the sum
total of the fields produced by each of the dipoles of the array,
neglecting the coupling effects. The NEC model, which includes the
coupling effects, was used to calculate the near electric and magnetic
field intensity produced by an eight-element and a twelve-element dipole
array. The comparison of the normalized near electrical and magnetic
fields of the log periodic dipole arrays with fields produced by a
single electric dipole shows that the correction factors for the single
dipole is higher than those for the log periodic dipole arrays \[1\].
Therefore, the correction factor for a dipole constitutes a conservative
multiplying factor for a log periodic antenna.

# 2.3.1.4 Yagi Antenna

A Yagi antenna is a directional array, in which there is a driven
element and several parasitic elements in which the fields of the driven
element induce currents. A parasitic element is usually located behind
the driven element and is called reflector. The other parasitic elements
are located in front of the driven element and are called directors. The
elements in a Yagi antenna are nearly equal in length while the spacing
between them varies. Yagi antennas are used in many government
telecommunication systems in the

![](media/image3.jpeg){width="4.294444444444444in"
height="4.607638888888889in"}

VHF range. The gain of these antennas may be as high as 12 dBi and
operate with a transmitter power of up to 600 watts. The normalized near
electric and magnetic field intensity of a Yagi antenna were compared to
the near fields of a single dipole with identical input power. The
correction factors for a single dipole are higher than those for a Yagi
antenna \[1\]. Therefore, the correction factor for a dipole constitutes
a conservative multiplying factor for a Yagi antenna.

2.3.1.5 [Collinear Antennas]{.underline}

A collinear antenna is an array of collinear half-wave dipoles. High
gain collinear antennas are used in base stations, whereas low gain
collinear antennas with two or three elements are used as vehicular
antennas. A collinear antenna has omnidirectional pattern with main beam
pointed at low elevation angle. The half wave dipoles are fed in phase
and a quarter wave stub is used for phase correction. Collinear antennas
are extensively used in Government telecommunication systems. The NEC
model was used to compare the near fields of a collinear antenna to
those of a full-wave dipole and it was found that the correction factors
are very nearly equal \[1\]. Therefore, the correction factors for a
full-wave dipole are used to estimate the near fields of a collinear
antenna.

# 2.3.1.6 Single Folded Dipole

The ideal folded dipole is basically a half-wave antenna formed by
shaping a loop into a thin rectangle as shown in the inset of Figure 5.
This type of antenna is widely used for Government systems in the
VHF-UHF bands. It is more mechanically sturdier than the thin wire
dipole; its characteristic impedance is increased fourfold and uses a
balanced feedline. The plot of the correction factor vs. normalized
distance for an ideal folded dipole as well as the practical dipole is
given in Figure 5. It is seen from the plot that that ideal curve is
higher than the practical curve for most distances. Therefore, the
correction curve for the ideal folded dipole will be chosen as a
conservative estimate.

# 2.3.1.7 Stacked Folded Dipole Array

For high power VHF-UHF transmission, folded dipoles equally spaced along
a vertical supporting pole is used. For non-interacting dipoles,
changing the alternate arrangements of dipoles to an offset of one half
wavelength from the supporting pole, there will be an elliptical
radiation pattern with a 6 dB gain. It was found that the supporting
pole has little effect on the nearby dipole, as confirmed by a numerical
model of dipoles spaced half a wavelength from one and two wavelengths
long supporting pole using MININEC \[2\]. The gain of the stacked folded
dipole array referred to an isotropic antenna, P~s~/P~i~, is given
bellow.

P~s~/P~i~ = (P~f~/P~i~) (P~s~/P~f~)

where, P~f~/P~i~ is the folded dipole antenna gain referred to an
isotropic antenna, 1.6 (2.1 dBi),

and, P~s~/P~f~ is the stacked folded dipole array gain referred to a
folded dipole antenna max. 4.0 (6.0 dBi)

Therefore, P~s~/P~i~ will vary from 1.6 (2.1 dBi) to 6.4 (8.1 dBi).

# 2.3.1.8 Coaxial Dipole: Single and Stacked

The coaxial dipole is a half-wave dipole with one half of the antenna
inside a coaxial cable that also acts as the transmission line. Since
the radiation pattern is similar to that of a half-wave dipole, the
near-field correction is same as for the folded dipole antenna, as given
in Figure 5. The stacked coaxial dipole array radiation characteristics
would also apply for widely separated non-interacting dipoles.

# 2.3.1.9 Omnidirectional Antennas

Omnidirectional antennas include stub and blade, ground plane, coaxial,
discone and inverted discone antennas. They are generally low-frequency
low-gain linear radiators, except the stub antennas, which work in the
upper UHF range. A radially symmetric signal is radiated from an antenna
vertical to a conducting flat or sloping ground plane. The radiating
element varies in configuration from the narrowband quarter-wave whip of
a ground plane aerial to the broadband cone of a discone antenna. The
ground plane ranges from stiff radial wires to sheet metal discs and
cones. MININEC near-field intensities were plotted for all
omnidirectional types and compared against more basic antenna correction
curves \[2\].

Most stub and blade antennas are found as self-supporting VHF-UHF
quarter-wave radiators using an aircraft's metal surface as a ground
plane. For the stub or blade, numerical calculations of a basic
base-loaded quarter-wave vertical wire model above a perfectly
conducting plane, as shown in Figure 6, would lead to conservative
radiation hazard estimates.

The basic configuration for ground plane antenna is the vertical
half-wave dipole with the lower half replaced by a conducting plane
termed the ground-plane as shown in the inset of Figure 6. There are
other variations, as shown in Reference 2. Ideally, a large perfectly
conducting ground plane will reflect all the radiation from the
quarter-wave dipole, thus doubling the total radiated power to yield a 3
dB gain over a half-wave dipole. This gain is used as an upper bound.
The ground plane varies from radial wires at MF to UHF to flat metal
plates at UHF and SHF and is often angled to offset a raised main lobe
pattern. MININEC calculations for three common ground plane antennas
show that their correction curves are very close the curve in Figure 6
for normalized distances greater than 0.15.

The discone antenna was derived from the biconical dipole by replacing a
quarter-wave section with a conducting disc or a ground plane. The
discone antenna has segmented radial construction and a simple antenna
feed, and it can cover several octaves in frequency with little change
in input impedance or the omnidirectional radiation pattern.

![](media/image5.jpeg){width="5.142361111111111in"
height="4.602083333333334in"}

When a disc is not practical, the radiating cone is inverted and placed
over ground radials as shown in the inset of Figure 7. The near field
intensity of the discone and inverted discone antennas as shown in
Figure 7 are greater than the values shown in Figure 6. Therefore Figure
7 is used for discone and inverted discone antennas.

# 2.3.1.10 Dipole and Stacked Arrays

Stacked arrays usually consist of dipoles arranged vertically. Uniformly
spaced half-wave dipoles, which interact in a linear array to radiate a
bilaterally symmetrical azimuthal pattern, are considered here.
Different phase and magnitude of excitation at antenna inputs and
variable antenna spacing will give rise to an infinite number of
radiation patterns. For the same excitation vector applied to each
element of a uniformly spaced antenna array, the broadside and endfire
designs concentrate radiation best in the main lobes and therefore, used
for radiation hazard assessment.

If the dipoles in a linear array are spaced a half wave apart and fed in
phase, a far-field broadside radiation pattern will result with nulls
along the array axis. This is seen in Figure 8a, where the main lobes
are along the x-axis and the nulls are on the y-axis. The near-field
radiation has a similar pattern. MININEC results for 1, 4, and 8 element
broadside arrays are plotted as normalized curves in Figure 8a. Since
the 1 element array or a single half-wave dipole has E-field intensities
higher than others, it is used as a conservative estimate for the near
field correction of broadside dipole arrays.

If the dipoles in a linear array are fed with a phase shift of 180
degrees between adjacent elements, the endfire pattern will result with
main lobe along the y-axis and nulls in the x-direction, as shown in
Figure 8b. Since the elements are now on the axis of the main lobe, it
is important to choose the starting point for measuring the distance.
The midpoint of the outermost element of the array is chosen as the
starting point. MININEC endfire results are plotted in Figure 8b for 4
and 8 element arrays along with that of a single dipole. Again, the
single half-wave dipole has higher correction factors than any of the
endfire arrays; its correction curve is used for endfire arrays.

**2.3.2 Corner Reflectors**

Corner reflector antennas consist of two parts: a metallic reflector
consisting of two plane surfaces meeting at an inclined angle of 180
degrees or less, and a radiator, usually a thin wire or bowtie half-wave
dipole, placed equidistant from the plane surfaces, as shown in the
inset of Figure 9. Corner reflectors can increase forward gain by 12 dB
or more with high aperture illumination.

Geometric optics analysis using the Method of Images can be done for
corner reflector antennas, provided the included angle Φ~CR~ of the
corner is an integral submultiple of π, i.e.,

Φ~CR~ = π/n n = 1,2,3,4,

![](media/image7.jpeg){width="5.625in" height="5.299305555555556in"}

![](media/image8.jpeg){width="4.354166666666667in"
height="4.272222222222222in"}

![](media/image9.jpeg){width="4.725in" height="4.75in"}

In practice, the aperture must be at least several square wavelengths in
size and therefore, these antennas are used most often at UHF or higher
frequencies. Nearly all corner reflector antennas have 60- or 90-degree
reflector, an aperture area of at least 3 square wavelengths, and a
half-wave dipole oriented parallel to the reflector's vertex of the
reflector.

NEC-BSC was used for 60- and 90- degree corner reflectors with both
small and large apertures (1.5 λ to 4.5 λ) and the calculated field
intensities are normalized to 5λ. The results for small and large
apertures are shown in Figure 9a and 9b respectively. A typical large
aperture will give rise to a correction curve with a slightly higher
intensity than for a smaller one. Therefore, the correction curves for
larger aperture corner reflectors as shown in Figure 9b are used for
conservative radiation hazard assessment.

3.  **Aperture Antennas**

The procedure for calculation of near field of aperture antennas
involves calculation of power density at a distance of 2D^2^/λ, and then
multiplying the far-field power density by a correction factor. The
correction factor for each aperture is a function of the aperture
distribution, the ratio D/λ and the shape of the aperture, i.e.,
rectangular or circular.

1.  [Circular Apertures]{.underline}

Circular apertures may use a variety of distribution functions. A simple
distribution factor \[5\], which gives a conservative estimate and is
used by many of the government agencies, is given below.

f (ρ) = (1 -- ρ^2^) ^n^ 0 \< ρ \<1 (2.14)

where, ρ is a variable normalized to the aperture radius R.

There are a number of other distribution functions, which have been used
for far field calculations, generally do not yield closed form solutions
for the near field calculations. The exponent in Equation 2.14 is
generally assumed to be an integer greater than zero. In practice, n is
a number greater than zero and less than 5. The side lobe levels and the
beamwidth of the antenna depend on the value of n. Hence, knowing the
beamwidth and the first side lobe level, n can be determined.

For n = 0, the distribution is uniform and the expression for on-axis
near field will be

P (x) = $\left| 1 - \text{Exp}(j\alpha) \right|^{2}$ (2.15)

![](media/image10.jpeg){width="4.607638888888889in" height="4.625in"}

where

$\alpha = 4{\pi D}_{\lambda}^{2}\left\lbrack x - \sqrt{x^{2} + \frac{1}{(}\text{16}D_{\lambda}^{2})} \right\rbrack$

*D~λ~ =*2R/λ, where R is the radius and D is the diameter of the
aperture

x = zλ/2D^2^ = normalized on-axis distance

Using the aperture distribution in Equation 2.14, the expression for the
on-axis near field may be derived as

P (x) = $\left| 1 - \frac{\text{nI}}{\beta^{n}}e^{j} \right|^{2}$ n \> 0
(2.16)

where,

I = $\int_{}^{}{Y^{n - 1}e^{\text{jY}}}\text{dY}$

β = π / (2 $\sqrt{\text{16}x^{2} + \frac{1}{D_{\lambda^{2}}}})$

The near field correction factor may be written as

Correction Factor = Normalized on-axis power density = P (x) / P
(2D^2^/λ) (2.17)

For circular apertures, the dependency of the side lobe levels and
beamwidth of an antenna on the parameter n may be determined by using
the far field expression developed by Silver \[4\]. The far field
expression using the distribution function of Equation 2.14 is given by

g~n~ (u) = 2πR^2^
$\int_{0}^{1}{(1 - \rho^{2})^{n}J_{0}(u\rho)\rho d\rho}$ (2.18)

where, u = πD sin θ/λ and J~0~ is Bessel function of zero order.

For integer values of n, the closed form solution is given by

g~n~ (u) = πR^2^ Λ~n+1~ (u)/n+1 (2.19)

where the functions Λ~n~ are available in tabular form. However, when n
is not an integer, the integration may be performed by expanding the
Bessel function in an infinite series and then integrating the series
term-by-term.

The result may be written as

g~n~ (u) = πR^2^
$\sum_{0}^{\infty}{( - 1)^{k}\frac{(\frac{u}{2})^{2k}}{(k!)^{2}}\int_{0}^{1}{Y^{n}(1 - Y)^{k}\text{dY}}}$
(2.20)

A computer program was prepared to carry out the calculation of Equation
2.20. The major characteristics of the patterns are given in Table 2-1

**TABLE 1**

**FAR FIELD PATTERN CHARACTERISTICS PRODUCED BY A**

**DISTRIBUTION (1-ρ^2^)^n^ OVER A CIRCULAR APERTURE**

**n Half-Power Beamwidth Position of 1^st^ zero First Sidelobe in dB**

**(radian) (radian) Below Peak Intensity**

0.0 1.02 λ/D arc sin (1.21 λ/D) 17.6

0.2 1.08 λ/D arc sin (1.27 λ/D) 19.1

0.5 1.15 λ/D arc sin (1.43 λ/D) 21.3

0.75 1.21 λ/D arc sin (1.53 λ/D) 23.0

1.0 1.27 λ/D arc sin (1.62 λ/D) 24.6

1.2 1.34 λ/D arc sin (1.72 λ/D) 26.0

1.5 1.40 λ/D arc sin (1.85 λ/D) 27.7

1.75 1.43 λ/D arc sin (1.94 λ/D) 29.1

2.0 1.46 λ/D arc sin (2.04 λ/D) 30.6

2.2 1.53 λ/D arc sin (2.10 λ/D) 31.7

2.5 1.59 λ/D arc sin (2.23 λ/D) 33.3

2.75 1.62 λ/D arc sin (2.32 λ/D) 34.6

3.0 1.66 λ/D arc sin (2.42) λ/D 35.9

One can determine the parameter n uniquely from the information given in
Table 2.1, given the half-power beamwidth, position of first zero, or
the level of first sidelobe in dB for a circular antenna. Generally, the
information on the antenna sidelobe level or half-power beamwidth may be
obtained from the NTIA Form, prepared by the agency seeking spectrum
support.

The program CIRCAPE is used to compute and plot the normalized on-axis
power density for different aperture sizes. The power density at the
distance z along the aperture axis is normalized to the value of the
power density at 2D^2^/λ. The normalized power density referred to as
the correction factor for an aperture is plotted as function of
normalized distance along the antenna axis. The distance along the axis
is normalized to 2D^2^/λ. The input parameters are RLD (R/λ) and EN (n
of Equation 2.14).

# Off-Axis Calculations

As discussed above, the on-axis calculations depend on knowing the
normalized near-field power density variation as a function of distance
with n and D/λ as parameters. An off-axis procedure includes the on-axis
calculation plus the effect of going off-axis in the near field. Because
of rotational symmetry, the average power density at a radial distance
is completely expressed for a given pair of n and D/λ by a single
function f (θ), where θ is the off-axis angle ranging from 0 to 90
degrees.

Sets of relative power density vs. off-axis angle curves were derived
from the procedure given in existing literature \[3\]. The parameters of
the power density curves are shown in Table 2-2.

## TABLE 2

**PARAMETERS OF THE POWER DENSITY CURVES**

> **Parameter Categories**
>
> D/λ ratio D/λ \< 30
>
> D/λ ≥ 30
>
> Aperture taper n n \< 0.5
>
> 0.5 ≤ n \<1.5
>
> 1.5 ≤ n \<3.5
>
> Normalized distance p 0.07 \< p ≤ 0.1

1.  \< p ≤ 0.2

2.  \< p ≤ 0.5

> p \>0.5

If four distances are grouped together on one chart, we obtain the six
charts (2 x 3) of the procedural relative power curves \[3\]. Very close
distances are omitted because of difficulty in assessing the rapid
changes in the sidelobe levels including deep bifurcation of the
mainlobe. On the other hand, for p greater than 0.5, the sidelobes are
well behaved and tend toward far-field conditions.

To obtain the off-axis power density, the on-axis power density is
multiplied by the off-angle correction.

The relative power density curves can be replaced by mathematical
algorithms in the AFI program. A typical relative power density curve is
shown in Figure 10. It is made up of linear log segments, each of which
can be represented by a general equation:

![](media/image12.jpeg){width="6.0in" height="7.490277777777778in"}

G = A + B log θ θ~1~ \< θ \<θ~2~ (2.21)

Where A and B are constants, and 1 and 2 denote the range of θ.

For the line segment between points 1 and 2 in Figure 10, the following
proportional equivalent can be set up between slopes:

> $$
> $\frac{G_{i} - G_{1}}{\text{log}\left( \frac{\theta_{i}}{\theta_{1}} \right)} = \frac{G_{2} - G_{1}}{\text{log}\left( \frac{\theta_{2}}{\theta_{1}} \right)}$

where, G~i~ is the density for a given angle θi.

Solving for G~i~ = G~1~ + (G~2~ -- G~1~)
$\frac{\text{log}\left( \frac{\theta_{i}}{\theta_{1}} \right)}{\text{log}\left( \frac{\theta_{2}}{\theta_{1}} \right)}$
(2.22)

Equation 2.22 can be used with any two pairs of (q, G) to get a value of
G at an intermediate point. Validity of the equation can be checked with
the curves in Reference 3.

In the program, the sequence of steps of obtaining the off-axis power
density would be:

1.  Determine the on-axis power density.

2.  Reach a table of stored (θ, G) for a curve for the parameters D/λ, n
    and p shown in Table 2.

3.  Substitute the off-axis angle θ and a pair of stored values in
    Equation 2.22 to get G~i~.

4.  Multiply the on-axis power density of step 1 by G~i~.

    1.  [Rectangular Aperture]{.underline}

Rectangular apertures are used in a large number of the Government
telecommunication systems. Square apertures are treated as a special
case of the rectangular apertures. Examples of rectangular apertures are
horn antennas and rectangular reflectors.

The field intensities in front of a rectangular aperture with length 2a
and width 2b are derived using the distribution functions:

f (x', y' ) = cos^n^ (πx/2a) -a ≤ x' ≤ a (2.23)

where, n = 0, 1, 2, 3, and

f (x', y') = \[cos (πx/2a) cos (πy/2b)\] ^n^ -b ≤ y' ≤ b (2.24)

where, n = 1,2.

For n = 0, the distribution is uniform. In Equation 2.23, the
distribution is cosinusoidal along the x-axis and uniform along the
y-axis. In Equation 2.24, the distribution is cosinusoidal along both x
and y axis. The parameter n affects the far field characteristics
directly. Therefore, n may be determined from these characteristics. The
six distribution characteristics given in Equations 2.23 and 2.24 are
used to derive on-axis near-field relationships given in Table 3.

## TABLE 3

## RELATIONSHIPS FOR ON-AXIS POWER DENSITY CALCULATIONS

## FOR RECTANGULAR APERTURES

### On-Axis Power Density Expression Aperture Distributions

P~co~ = P^2^ Uniform

P~c1~ = P~1~^2^ cos (πx'/2a)

P~c2~ = P + P~2a~^2^ cos^2^ (πx'/2a)

P~c3~ = P~3~ + 3P~1~^2^ cos^3^ (πx'/2a)

P~cc1~ = P~1c~^2^ cos (πx'/2a) cos (πy'/2b)

P~cc2~ = P + P~2a~ + P~2b~ + P~2cc~^2^ cos^2^ (πx'/2a) cos^2^ (πy'/2b)

The Helmholtz potential function for rectangular aperture shown in
Figure 11 is given by:

P =
$\frac{1}{4\pi}\int_{- a}^{a}{\int_{- b}^{b}{J(r')\frac{e^{- \text{jk}|r - r'|}}{|r - r'|}}}\text{dx}'\text{dy}'$
(2.25)

Where J (r') is the aperture distribution and the parameters a, b, r,
and r' are defined in Figure 11. The current distribution on the
aperture is either uniform or cosinusoidal as given by Equations 2.23
and 2.24. Substituting these distribution functions in Equation 2.25,
the potential function becomes:

P =
$\frac{1}{4\pi}\int_{- a}^{a}{\int_{- b}^{b}\text{cos}^{n}}\left( \frac{\pi x'}{2a} \right)\text{cos}^{m}\left( \frac{\pi y'}{2b} \right)\frac{e^{- \text{jk}|r - r'|}}{r - r'}\text{dx}'\text{dy}'$
(2.26)

The integral in Equation 2.26 was evaluated for six different
distribution functions listed in Table 3 and the expressions for the
symbols in the left hand column of the Table are

![](media/image13.jpeg){width="6.0in" height="6.352777777777778in"}

derived in Appendix A of Reference 1. The results of these calculations
are also given in Reference 1.

In order to relate the far field calculations to the parameter n, one
needs to calculate the radiation pattern. The simplified expressions for
the Fraunhofer region derived by Silver \[5\] are used in the
calculation of the pattern. The far field pattern factor is given by:

g~n~ (θ, φ) =
$\int_{- a}^{a}{\int_{- b}^{b}{f(\xi,\eta)e^{\text{jk}\text{sin}\theta(\xi\text{cos}\varphi + \eta\text{sin}\varphi)}}}d\xi d\eta$
(2.27)

where, f (ξ, η) is the distribution function and k is the wave number
(=2π/λ).

Substituting Equations 2.23 and 2.24 into Equation 2.27, one obtains the
far field pattern for each distribution. The characteristics of the far
field patterns produced by a rectangular aperture for different
distribution functions are summarized in Tables 4 and 5.

## TABLE 4

**FAR FIELD CHARACTERISTICS OF RECTANGULAR APERTURES WITH**

**cos**${}^{n}\frac{\pi x'}{2a}$ **DISTRIBUTION FUNCTION**

**n Full Width at First Sidelobe dB**

**Half Power (Radian) Below Peak Intensity**

**Horizontal Vertical Horizontal Vertical**

**Plane Plane Plane Plane**

0 0.88 λ/2a 0.88 λ/2b 13.2 13.2

1 1.20 λ/2a 0.88 λ/2b 23 13.2

2 1.45 λ/2a 0.88 λ/2b 32 13.2

3 1.66 λ/2a 0.88 λ/2b 40 13.2

## 

##  TABLE 5

**FAR FIELD CHARACTERISTICS OF RECTANGULAR APERTURES WITH**

**cos^n^**$\frac{\pi x'}{2a}$ **cos^n^** $\frac{\pi y'}{2b}$
**DISTRIBUTION FUNCTION**

### n Full Width at First Sidelobe dB

**Half Power (Radian) Below Peak Intensity**

**Horizontal Vertical Horizontal Vertical**

**Plane Plane Plane Plane**

1 1.20 λ/2a 1.20 λ/2b 23 23

2 1.45 λ/2a 1.45 λ/2b 32 32

The data in Tables 4 and 5 show the correspondence between the parameter
n and the far field parameters such as sidelobe levels. For example,
using the sidelobe level for the antenna given in NTIA Form 35 for a
system, one can determine n and then find the near field correction
factor outlined earlier.

**3. EMISSION EXPOSURE STANDARDS**

The near and far field power density of a given antenna is compared to
one or more of the emission standards included in the program, one of
which can be a user defined exposure limit. At present, the emission
exposure standards included in the APD program are EPA 10 mW/cm^2^
\[29CFR1910.97\], ANSI C95.1-1991 controlled (occupational) and
uncontrolled (general population) environments, FCC 1.1310
occupational/controlled and general population/uncontrolled exposures
and National Council on Radiation Protection (NCRP) uncontrolled/general
population environment. Distance at which the power density is exceeded
for a standard is listed in the output report.

**4. INPUT PARAMETERS**

The first input parameter to be chosen is the type of antenna. Frequency
of operation in MHz, average transmitter power in watts and antenna gain
in dBi are needed for all antenna types. The antenna type choices are
wire, circular, rectangular and corner reflector.

**4.1 Wire Antenna Input Parameters**

The wire antenna types included in the program are simple dipole,
monopole, coaxial dipole, folded dipole, collinear and monopole whip,
collinear, ground plane, Omnidirectional, stub and blade, discone and
inverted Discone, log periodic, Yagi, broadside and endfire dipole
arrays. For collinear, whip (collinear), log periodic and Yagi antennas,
there are provisions for calculating near field power densities for off
axis angles θ = 22.5, 45 and 67.5 degrees, if gain is known at those
angles.

**4.2 Circular Aperture Input Parameters**

For circular aperture antennas, the half-power beamwidth and the first
sidelobe gain relative to mainbeam are also required in addition to the
frequency, average transmitter power and antenna gain. The program also
has an option of providing power density at a user specified off-axis
angle.

**4.3 Rectangular Aperture Input Parameters**

For rectangular apertures, horizontal and vertical aperture dimensions
and beamwidths are required in addition to the frequency, average
transmitter power and antenna gain. The aperture dimensions may be
expressed in metric or English units (inches). Intensity of the first
sidelobes in both vertical and horizontal directions may be used as
inputs for crosschecks, though they are not required.

**4.4 Corner Reflectors**

For the corner reflectors, the option of using corner angle of 60 or 90
degrees is available with a default of 90 degrees.

**5. OUTPUT**

The program outputs are reports and graphs, if desired. The report
includes the input parameters, the distance to the far field and all
emission exposure standard options chosen by the user with the distances
at which those limits were exceeded for a given antenna type. The graph
plots the power density in mW/cm^2^ vs. distance in meters along with
lines indicating the emission exposure standards chosen by the user. The
points are also displayed and saved to file, if necessary.

**6. REFERENCES**

1\. Farrar A. and Chang E., *Procedure for Calculating Field Intensities
for Antennas,* NTIA Report TM-87-129, U.S. Department of Commerce,
September 1987.

2\. Kobayashi, H. K., *Procedures for Calculating Field Intensities of
Antennas: Phase II,* NTIA Report TM-88-135, U. S. Department of
Commerce, September 1988.

3\. Kobayashi, H. K., *Procedure for Calculating the Power Density of a
Parabolic Circular Reflector Antenna,* NTIA Report TM-90-145, February
1990.

4\. Jordan, E. C. and Balmain, K. G., *Electromagnetic Waves and
Radiating Systems,* Prentice Hall Inc., N. J., 1968.

5\. Silver, S., *Microwave Antenna Theory and design,* Dover
Publications, Inc., New York, 1965.