Directional Radiators
6.2 CALCULATING DIRECTIVITY OF ANTENNA ARRAY ON THE BASIS OF THE MAIN DIRECTIONAL PATTERNS
As is well known, directivity D is one from basic electrical performances of any antenna.
An antenna gain G is equal to
G = Dh, (6.19)
where h is an efficiency. Knowledge of these magnitudes allows planning the improvement of the antenna characteristics. The value of G can be defined by direct measurements.
As regards magnitudes D and h, it is very difficult to measure them or to interpret the measurements’ results. For example, for an evaluation D it is necessary to know the three-dimensional antenna patterns. But as a rule, these patterns are measured only in two main planes: horizontal and vertical. The calculation difficulties increase with decreasing the cross-section of the main lobe, i.e. with increasing directivity, caused for example by increased numbers of radiators in the antenna array.
Calculating the directivity of the narrow-beam antenna is described in [54]. It is based on the method of calculating the intermediate values in the directional pattern by means of using the measurements’ results, which was proposed in [55].
Sometimes it is regarded that the magnitude of the antenna pattern in arbitrary direction is equal to
F(q, j) = F1(q)F2(q), (6.20)
where F1(q) is a pattern in a vertical plane xOz, F2(j) is a pattern in a horizontal plane xOy, q and j are the angles in the spherical coordinates system, and x, y and z form a rectangular coordinates system (Figure 6.6). The calculations show, that the expression (6.20) is true only in a narrow area, limited by a main lobe of a directional pattern.
Figure 6.6 The coordinates system and a directional pattern.
The method proposed in [55] is based on revealing a curve, which is a locus of points with an identical signal. Here, the angle d = p/2 – q is used instead of an angle q.
Respectively magnitudes of a directional pattern are equal to
f(d, j) = f(d1, 0) = f(0, j1), (6.21) where d1 and j1 are values of coordinates d and j in the points of intersections of a mentioned curve with planes xOz (j = 0) and xOy (d = 0) respectively (see Figure 6.6).
Assume that the direction of the maximum radiation coincides with x-axis, and the pattern is symmetric about the planes xOz and xOy (Figure 6.7). For example, in-phase array, located in a plane yOz, has such directional pattern. In this case curves with identical directivity in the first approximation will be have the form of circles or ellipses:
f(d1, 0) = f1(d1), f(0, j1) = f2(j1). (6.22) If to introduce for convenience a new angular coordinate b, measured from x-axis (see Figure 6.7), then as it is easy to show the new and the old coordinates will be connected among themselves by relation:
b = cos–1(cos d cos j). (6.23)
Figure 6.7 Symmetric directional pattern.
If the curves with identical directivity represent circles, i.e. if the main lobe of the three-dimensional pattern has circular symmetry, then
d1 = j1 = b = cos–1(cosd cosj). (6.24) In more common case these curves are ellipses. Let a1 be length of its vertical axis, i.e. the arc length between the upper and the lower points of the pattern, corresponding to the given magnitude of a signal (to the given magnitude of the directional pattern).
In the same way a2 is the length of its horizontal axis, i.e. the arc length between the left and right points of the pattern corresponding to the given magnitude of a signal.
Relation of lengths of vertical a1 and horizontal a2 axes is equal to a = a1/a2. Then at a > 1
d1 1 d j j1 1 1 d j
=cos (cos cos- a ), = cos (cos cos- )
a a , (6.25)
and at a > 1
d d
j j d
1 1 j
1 1
= Ê
ËÁ ˆ
¯˜ = Ê
ËÁ ˆ
- - ¯˜
acos cos cosa , cos cos cosa . (6.26)
As is known, maximal directivity of antenna with the pattern, symmetrical about planes xOz and xOy, is equal to
D
f d d
=
Ú Ú
p
d j d d j
p p
( , )cos
/
0 2
0
, (6.27)
whence
D
f d d f d d
=
Ú
+Ú Ú Ú
p
d j d d j d y d d y
p
p p p
( , )cos ( , )cos
/
/ / /
0 2
0 2
0 2
0
2 . (6.28)
The first addend of a denominator corresponds to a forward half-space, the second addend—to a back half-space. Here in the second integral the change of variable j = p – y is performed. At a < 1 in accordance with (6.21), (6.22) and (6.25)
f( , )d j = f1( )d1 = ÍÎf1 cos (cos cos-1 d aj)˙˚. (6.29) At a > 1 according to (6.21), (6.22) and (6.26) one can obtain a similar expression.
The expression (6.28) subject to (6.29) allows calculating the antenna directivity, if its directional patterns are given in the main planes.
In Figure 6.8 the experimental directional patterns of a planar uniform antenna array with in-phase excitation are given at the frequency 3.4 MHz. As one can see from the figure, the factor a is equal to 1 in intervals 160–180° and 135–145°, to 0,63 in an interval 145–160° and to 0,69 in an interval from 90° to 120° along an azimuthal angle.
It means that the main lobe of the three-dimensional pattern has circular symmetry, i.e.
the locus of points with identical signal, located on the main lobe, is a circumference.
Such circular symmetry on some side lobes is absent, and that should be taken into account for increasing calculation accuracy. In an interval 120–135° the factor a is greater than 1. Calculation according to the described method at a = 1 gives a directivity value, equal to 18,6 dB. Calculation with allowance for a > 1 in the interval 120–135°gives an outcome 18.2 dB. The measured antenna gain is equal to 18 dB. Thus, one must admit a good conformance of calculated and experimental results.
Figure 6.8 Experimental directional patterns of antenna.
For antennas with one narrow major lobe and small minor lobes, the theory (see, for example, [13]) recommends the expression
D = 41253/(q1q2), (6.30)
where q1 and q2 are half-power beam widths of the pattern (in degrees) in two mutually perpendicular planes. For planar arrays a better approximation is (see in [13])
D = 32400/(q1q2), (6.31)
The calculation in accordance with these formulas for the patterns, presented on Figure 6.8, gives accordingly 20.2 and 19.2 dB, i.e. a much greater error.
The program of directivity calculation used two procedures: the procedure of antenna pattern estimation at intermediate angles and the procedure of integrals calculation by summation of numerical masses. These methods can be used for the solution of other problems too, for example, for an estimation of increasing antenna directivity at the expense of decreasing side lobes.
If it is required to calculate, as far as the directivity will be changed due to diminution of a side lobes to level f0, one must first determine the initial value of the directivity in accordance with (6.27) and next to reduce the side lobes, which exceed f0 (for example in a vertical plane in the range of angles from j11to j11and in a horizontal plane in the range of angles from d11to d11) to level f0 and to calculate the new directivity (at a < 1)
The program of the magnitude D calculation was performed in Matlab and presented in [54].