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directional antenna gain calculation

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Gain of Directional Antennas: WJ Tech Notes 1976
1
Watkins-Johnson Company Tech-notes Copyright © 1976 Watkins-Johnson Company Vol. 3 No. 4 July/August 1976
Gain of Directional Antennas
John E. Hill Gain is an antenna property dealing with an antenna’s ability to direct its radiated power in a desired direction, or synonymously, to receive energy preferentially from a desired direction. However, gain is not a quantity which can be defined in terms of physical quantities such as the Watt, ohm or joule, but is a dimensionless ratio. As a consequence, antenna gain results from the interaction of all other antenna characteristics. This article will explore these interactions using elementary definitions of antenna properties. Antenna characteristics of gain, beamwidth and efficiency are independent of the antenna’s use for either transmitting or receiving. Generally these characteristics are more simply described for the transmitting antenna; however, the properties described in this article apply to both cases. Gain definitions, and antenna characteristics related to gain, are found in a glossary on page 10, and will appear in italics within text. First, the concept of directive gain will be examined, followed by related antenna factors such as beamwidth and efficiency. Some simple equations are listed at the conclusion which permit approximate computations of directive gain and half-power beamwidth for directional type antennas.
Directive Gain from a Hypothetical Antenna
An antenna does not amplify. It only distributes energy through space in a manner which can best make use of energy available. Directive gain is related to and is a measure of this energy distribution. To visualize the concept of directive gain, assume an elastic sphere is filled with an incompressible medium having a shape as shown in Figure 1a. A dot at the center of the sphere represents a hypothetical
isotropic radiator
which has equal
radiation intensity
in all directions. Let the radius of the sphere be proportional to the power radiated by the isotropic radiator. Next, the sphere is deformed to create a new shape as shown in Figure 1b. As a result of our assumption that the sphere is filled with an incompressible medium, the volume must remain unchanged regardless of the change in shape; the sphere surface must bulge outward somewhere if another area of the surface is depressed. For the surface shown in Figure 1b, the distance from the center dot to all points on the
Gain of Directional Antennas: WJ Tech Notes 1976
2
sphere surface is no longer everywhere equal, although the average distance, which is equal to the srcinal radius (r
o
), remains the same. The distance from the center to a point on the deformed surface is now proportional to the radiation intensity in that direction. The ratio of the distance from the center to any particular point on the surface (r
d
), to the average distance (or srcinal sphere radius, r
o
) is the
directive gain
in that direction. The value of the directive gain in the direction of its maximum value is the
directivity.
a) Symmetric radiation pattern of an isotropic radiator. b) Directive radiation pattern. Figure 1. Directive gain resulting from the shape of the radiation pattern in a certain direction.
To accomplish this power distribution change, the hypothetical antenna at the sphere’s center must be replaced by an antenna with the ability to direct radiated power in a desired direction. It is important to note that directive gain, as just described, is related only to the shape of the antenna’s radiation pattern, and does not include efficiency factors.
Directive Gain and Beamwidth
An antenna’s beamwidth is usually understood to mean the
half-power beamwidth,
that is, the angle between the two directions in which the directive gain of the major
radiation
lobe
is one half the maximum value (one half the directivity), and is shown in Figures 2a, 2b, and 2c. Each curve represents the same antenna
radiation pattern,
but plotted to a different scale: in watts, voltage, and decibels(dB).
Gain of Directional Antennas: WJ Tech Notes 1976
3
Figure 2. Equivalent half-power beamwidth representations of an antenna’s radiation pattern.
For the power plot, the half-power beamwidth is measured at a value which is one half (.5) the peak of the
beam,
and is 30º in the illustrated example. For the voltage plot, the half-power beamwidth is measured at a point which is .707 of the beam maximum .5 = .707
2
), and is 30º. For the decibel plot, the half-power beam-width is 3dB from the beam maximum (10 log
10
0.5 = -3dB), and is 30º. Assuming that a significant amount of radiated power is not diverted into side lobes, then the directive gain is inversely proportional to beamwidth; as the beamwidth decreases, the directive gain increases. A simplified approximation to an antenna’s directive gain may be obtained by considering a convenient spherical-shaped boundary at which the power radiated by a hypothetical directional antenna can be measured. All power radiated from the hypothetical antenna may be imagined to flow outward and through the surface shown in Figure 3a. This surface may be divided into square areas which are independent of radius, each occupying one degree in the vertical plane and one degree in the horizontal plane, and containing a total of 41,253 square degrees.* If all the power radiated by a directional
radiator
could be constrained to flow through one square degree, shown in Figure 3b, the directive gain in that direction would be 41,253 times the average directive gain. The directive gain for this power distribution is; g
d
= 41,253 1
*
where all the power radiated is assumed to flow through an area of one square degree. Usually, directive gain is expressed in decibels, and for the directive gain just calculated, is equal to: G
d
= 10 log
10
g
d
= 46 dB.
*
4
ππ
square radians (steradians) = 4
ππ
X (57.3)
2
square degrees = 41,253 square degrees.
Gain of Directional Antennas: WJ Tech Notes 1976
4
a) Power flow through a convenient spherical boundary b) Power flow through a square area of one square degree c) Power flow through a circular area of
ππ
/4 square degrees Figure 3. Simplified assumptions as to the shape of the radiated power yield approximate calculations of directive gain.
A more accurate approximation of the directive gain from the radiated pattern assumes that all the power radiated by a directional radiator is constrained to flow through an area which is circular in cross section, as shown in Figure 3c. Since the power radiated is constrained to flow through an area which is
π
/4.78% (as large. the resulting directive gain will be greater. and is given by: g
d
= 41,253 ã 4
θ
1
θ
2
π
or g
d
= 52,525
θ
1
θ
2
where
θ
1
and
θ
2
are orthogonal beamwidths, and represent the major and minor axis of the beam. For a circular beam shape,
θ
1
is equal to
θ
2
. In practical antenna applications, the beam is usually circular in cross section with many minor radiation lobes, or side lobes, present. To account for power flow in directions other than the beam’s direction, an assumption is made that approximately 55% of the power radiated flows within the half-power beamwidth. The directive gain is now approximated by: g
d
= 29,000
θ
1
θ
2
where
θ
1
and
θ
2
are the orthogonal half-power beamwidths of an asymmetric beam. Although this last equation is very useful in obtaining an antenna’s directive gain knowing the beamwidth, it must be remembered that it serves only as an approximation. The directive gain which results is based upon a radiation pattern exhibiting low-power losses in the side lobes. This is not always a good assumption. It

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