ANTENNA ARRAY THEORY
2.4 Demonstration of beamscanning using linear arrays
ANTENNA ARRAY THEORY CHAPTER 2
(h) Reduction in handoff rate: When the number of mobiles in a cell exceeds its capacity. cell capacity is used to create new cells, each with its own base-station and new frequency assignment. A consequence of this is an increased handoIT due to reduced cell size. This may be reduced using an array of antennas. Instead of cell splitting, the capacity is increased by creating independent beams using more antennas. Each beam is adapted or adjusted as the mobile locations change.
The beam follows a cluster of mobiles or a single mobile. as the case may be. and no handoffis necessary as long as the mobiles served by different beams using the same frequency do not cross each other.
(i) Reduction in cross talk: Cross talk may be caused by unknown propagation conditions when an array is transmitting multiple cochannel signals to various receivers. Adaptive transmitters based upon the feedback obtained from probing the mobiles could help eliminate this problem. The mechanism works by transmitting a probing signal periodically. The received feedback from mobiles is used to identify the propagation conditions, and this infonnation is then incorporated into the beamfonning mechanism.
ANTENNA ARRAY THEORY CHAPTER 2
computation of radiation patterns. The effect of practical elements with nonisotropic elements will be considered in section (2.4.1.1).
N
r,
r ,
r,
r,
y dcos9
Fig. 2.7: Far-field 2.eometrv of N element array of isotropic point sources positioned along the z-axis
The time delay of a signal between two successive elements is given by:
dcosB
T =
=""'-
cwhere c is the free-space signal propagation velocity.
This time delay corresponds to a phase shift of /If = lVT + fJ
= 27rf d cosB + fJ c
=-dcosB+fJ 27r A.
= kdcosB+ fJ
where f is the frequency and k = 21r is the wave number.
A.
(2.1 )
(2.2)
The array factor (AF) is now introduced. It is basically a mathematical concept that is a function of the time delay (or phase shift) between the array elements. For a uniform array. the array factor is given by
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ANTENNA ARRAY THEORY
AF = I + e+}(kdC058+P) + e+ j2(kdcosO+/J) + .... + e+ ,(N-1Xkdo;os8+/J)
= LN e+)(II-1X-Wo;osO+P)
., '
which can be written as
. ,'
where If = kd cosB +
fJ
CHAPTER 2
(2.3)
(2.4)
Since the total array factor for the uniform array is a summation of exponentials, it can be represented by the vector sum of N phasors each of unit amplitude and progressive phase 't' relative to the previous one. Graphically this is illustrated by the phasor diagram in figure (2.8).
_._---
IL(N - I)'V#N #4
I L341 3'4'
#2
Fig. 2.8: Phasor diagram of N element linear array
It is apparent from the phasor diagram that the amplitude and phase of the AF can be controlled in uniform arrays by properly selecting the relative phase 't' between the elements.
The array factor of equation (2.4) can also be expressed in an alternate, compact and closed form whose functions and their distribution are more recognizable. This is accomplished as follows:
Multiplying both sides of equation (2.4) by ei'l' it can be wrinen as
(AF)eill' = eifll + el2!" + ej3\1' + .... + eJ(N-I)1I' + ejN'(I (2.5)
Subtracting equation (2.4) from equation (2.5) reduces to
(2.6) which can also be written as
ANTENNA ARRAY THEORY CHAPTER 2
AF = e - = eil(N-I)12],- e - e = eil(N-I)12]'- 2 (2.7)
[ jN. I] [ j(Na). -j(NI». ]
[Sin(NVI)]
j" - 1 ej(l/2)1'" _ e-J(1/2)\I" • 1
e sm(- Vl)
2
If the reference point is the physical centre of the aTTay, the aTTay factor of equation (2.7) reduces to
AF= 2
[
Sin(N
VI)]
sin(~VI)
2
(2.8)
The maximum of equation (2.8) occurs when "{l=O but the aTTay factor (AF) is indeterminate since both the numerator and denominator are zero. However, by applying L'Hopital's rule (differentiating numerator and denominator separately) and setting "{l=O results in equation (2.8) having a maximum value ofN. To normalize the array factors so that the maximum value of each is unity, equation (2.8) is written in the normalized form as
(AF). =
~[Sin( tVl)]
sin(- Vl) 2 where 'I' = kd cos B +
P
2.4.1.1 Principle of pattern multiplication
(2.9)
The reasons for using isotropic elements for computation of the array factor becomes apparent in the ensuing discussion. The array factor is a function of the geometry of the array and the excitation phase. By varying the separation d and andlor the phase
p
between the elements, the characteristics of the array factor and of the total field of the array can be controlled. The far~zone field of a uniform array of identical elements is equal to the product of the field of a single element, at a selected reference pOi"t (usually the origin), and the array faclOr of that array [11]. That is
E(total) = [£(singleelemellt at reference point)] x [array faclor ET = EeE(J
(2.10) This is referred to as pattern multiplication for arrays of identical elements. Each array has its own array factor. The array factor, in general, is a function of the number
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ANTENNA ARRAY THEORY CHAPTER 2
of elements, their geometrical arrangement, their relative magnitudes, their relative phases, and their spacing. Since the array factor does not depend on the directional characteristics of the radiating elements themselves, it can be fonnulated by replacing the actual elements with isotropic (point sources). Once the array factor has been derived using the point source array, the total field of the actual array is obtained by use of equation (2.10). Each point source is assumed to have the amplitude, phase and location of the corresponding element it is replacing. Squaring both sides of equation (2.10) converts it into a relationship of power density-patterns (radiation patterns) in which the squares of the electric-intensity factors may be replaced by corresponding power-density pattern factors i.e. p. = pp.
r ' "
in using the principle of pattern multiplication, certain rules must be observed [25]:
(a) The pattern to be used for the array elements is the pattern that a single element would have ifit were isolated, that is, not in the "array envi.ronment".
(b) The array pattern to be used is the one calculated fOT isotropic point-source elements spaced and phased the same as the actual elements. The phasing cannot be calculated on the basis of a transmission-line feed system designed for the input impedances of the elements operating as isolated elements. Their actual input impedances will be affected by mutual coupling with other elements.
(c) All the array elements must be alike; they must have similar patterns, similarly orientated. For example, if the elements are dipoles, they must have their axes parallel to one another. If the elements are horns, they must be of the same fonn and size, and all "aimed" in the same direction.
Equation (2.10) is only an approximation for many problems of array design. It ignores mutual coupling (section (2.5)) and it does not take account of the scattering or diffraction of radiation by the adjacent array elements. These effects cause the element pattern to be different when located within the array in the presence of the other elements than when isolated in free space. In order to obtain an exact computation of the array radiation pattern, the presence of each element must be measured in the presence of all the others. The array pattern may be found by summing the contributions of each element, taking into account the proper amplitude and phase.
ANTENNA ARRAY THEORY CHAPTER 2
2.4.1.2 Phased (scanning) array
In many applications it is desired to orientate/steer the maximum radiation in a particular direction to ronn a scanning array. Assume that the maximum radiation of the array is required to be oriented at an angle So (00 ~ So ~ 18(0). Referring to equation (2.9), the maximum of the array factor occurs when 'f'=0,
\If
=
kd cosB +Ill ... . =
0=>
11
= -kd cosB, The array factor of equation (2.9) now becomes(2.11 )
sin[Nm:1 (COSB-cosBJ]
(AF). =
~
(2.12)NSin[ T (COSB-coSBJ]
Two arrays that are of interest are the broadside and the end fire arrays. In the broadside array, the maximum radiation is directed nonna! to the axis of the array (e = 90<> in figure (2.7». Substituting 9, = 90<> into equation (2.11) yields
p
= O. Thus for a broadside array, the elements have zero progressive phase lead current excitation. However, it is necessary that all the elements have the same phase excitation (in addition to the same amplitude excitation). For an endfire array, the maximum radiation can be directed at either 90 = 00 or 1800 i.e. along the axis of the array. To direct the maximum toward 9
0 = 00, the progressive lead current excitation
(p)
should be -kd while if the maximum is desired towardse ,
= ISO<>, thenp
= kd.2.4.1.3 Antenna element spacing to avoid grating lobes
One of the objectives in many designs is to avoid multiple maxima, in addition to the main maximum, which are referred to as grating lobes. A grating lobe is defined as a lobe, other than the main lobe, produced by an array antenna when the interelement spacing is sufficiently large to pennit the in-phase addition of radiated fields in more than one direction. They are undesirable since, in addition to directing the main lobe in the desired direction, the array also directs grating lobes to undesired directions.
Thus there is no maximization of power in the desired direction. Often it may be required 10 select the largest spacing between the elements but with no grating lobes.
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ANTENNA ARRAY THEORY CHAPTER 2
Consider a broadside array (fJ = 0 and 'I' = kd cosO = 27r d cosO). This value of \jI
A
when substituted in equation (2.9) makes the array factor attain its maximum value when casO = IIA for n = 1,2,3, ... The maximum at cosO = 0 defines the main beam.
d
If the spacing between elements is a half-wavelength (d = 0.5),), the first grating lobe (n = ±l) does not appear in real space since cosO = 2 which is undefined. Grating lobes appear at ()o and 18()O when d = A' For a nonscanning array this condition
(d = ,..) is usually satisfactory for the prevention of grating lobes. Practical antenna elements that are designed to maximize the radiation at
e
= 900, generally have negligible radiation in the directionse
= ()o ande
= 18()O directions. To avoid any grating lobe in a nonscanning array, the largest spacing between the elements should be less than one wavelength (dmax <),J.
Using an argument similar to the nonscanning array described above, for a scanning array, grating lobes appear at an angle 8g when the array factor of equation (2.12) attains its maximum value, or
: (cosO. -cOSOJ=±II7r
d ± 11
or - = ;---:-='---,--;
A IcosB, -cosB,,1
(2.13)
If a grating lobe is pennitted to appear at 8
g = ()o when the main beam is steered to 80 = 18()O, it is found from equation (2.13) that d = '),/2. Thus the element spacing must not be larger than a half-wavelength if the beam is to be steered over a wide angle without having undesirable grating lobes. Therefore a criterion for determining the maximum element spacing for an array scanned to a given angle 8
0 is to set the spacing so that the nearest greatest lobe is at the boundary of the visible region.
Using equation (2.13) this leads to the condition
d I
A ,;; "11---'co'-s-=o.'1 (2.14) Therefore the common rule that half-wave spacing precludes grating lobes is not quite accurate, as part of the grating lobe may be visible for extreme scan angles.
ANTENNA ARRAY THEORY CHAPTER 2
Grating lobes can also be avoided by unequal spacing of the elements i.e. by making the array aperiodic [8]. In an aperiodic array, the element weights are made equal but all the design freedom is employed in the element locations. There is no restriction imposed on the element spacing. However, the elements are usually overspaced to avoid the effect of mutual coupling. This process of overspacing the array is known as thinning [8]. In addition to the hazard of grating lobes, which is avoided by unequal spacing of the elements, a thinned alTay introduces another problem. For an array of fixed length, the number of elements is reduced. While this lowers cost and mutual coupling, both highly desirable changes, the reduction in element numbers reduces the designer's control of the radiation pattern in the sidelobe region. This reduction in design control influences the level of the peak sidelobe, which is of major concern to the alTay designer. Since the number of degrees of freedom is reduced in direct proportion to the number of elements, deviations from the desired radiation pattern must be anticipated. The beam width and approximate shape of the main lobe are not severely altered by thinning (beam width is not affected since the length of the array is fixed). The dominant effect of thinning is observed in the sidelobes. [8]
provides a detailed discussion on aperiodic arrays and thinning.
2.4. t.4 Simulation work
Appendix A.2 contains a MA TLAB program for the analysis of linear arrays using the theory developed thus far. The program plots the array factor pattern (for elements placed along the z-axis) as a function of the number of elements, element spacing and scan angle. An extension is also made to illustrate the pattern multiplication rule (section 2.4.1.1) using the half-wavelength dipole antenna as the element. The concept of grating lobes is also illustrated.
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ANTENNA ARRAY THEORY CHAPTER 2