Introduction to Arrays

Transcription

1 Intoduction to Aays Page 1 Intoduction to Aays The antennas we have studied so fa have vey low diectivity / gain. While this is good fo boadcast applications (whee we want unifom coveage), thee ae cases whee we want a moe focused antenna patten to pevent wasting powe illuminating aeas/diection whee we do not need coveage. Fo example, if we ae tying to send a signal to a teminal on the hoizon (θ = 90 ), a dipole is quite wasteful because even ±45 fom the hoizon we ae still boadcasting half the adiation intensity (the HPBW points fo an ideal dipole.) Although we could design moe diective antenna elements, one staightfowad way to incease the diectivity of a single antenna is to assemble it with othe antennas to fom an antenna aay. Then, using intefeence between the fields ceated by the individual aay elements, it is possible to synthesize a vaiety of diective beam pattens. 1 Two-Element Antenna Aays Let s conside a simple case of two ideal dipoles spaced a distance S apat along the z-axis. Since the elements themselves ae oiented along the z-axis, we call this a collinea aay. This analysis looks vey simila to that which we caied out fo the λ/2 dipole. Recall that we divided the dipole into many sections of ideal dipoles and used supeposition (the summation of all the elements esponses) to detemine the esulting electic and magnetic fields. Hee we will use the same appoach except that we only have to woy about the contibution of two segments. Recall the θ-component of the electic field adiated in the fa field by a Hetzian dipole is E θ = jkηi z 4π } {{ } E s e jkr sin θ R. (1) Let s call the fist faction E s since it was peviously defined as the stength facto of the dipole and did not depend on the geomety of the situation. The total E θ -field poduced by the two

2 Intoduction to Aays Page 2 dipoles, by supeposition, is E T = E s sin θ 1 e jkr 1 R 1 + E s sin θ 2 e jkr2 R 2. (2) In the fa field, vey fa fom the aay, we can make the following appoximations: θ 1 = θ 2 = θ; (3) 1 R 1 = 1 R 2 = 1. (4) Recall that we cannot simply say that R 1 = R 2 = fom the phase tem (the complex exponential tem) because even if R 1 R 2, exp( jkr 1 ) exp( jkr 2 ) exp( jk). (5) But, making the paallel ay appoximation, we can say R 1 S 2 R 2 + S 2 cos θ (6) cos θ. (7) Theefoe, E T = E s sin θ [ e jk( s 2 cos θ) + e jk(+ s 2 cos θ)] (8) e jk ] = E s sin θ [e j ks 2 cos θ + e j ks 2 cos θ (9) e jk = 2E s sin θ cos (k S2 ) cos θ (10)

3 Intoduction to Aays Page 3 e The E jk s sin θ tem is exactly the patten of a single dipole, if placed at the oigin (the cente of the aay.) So what has happened is that the oiginal field of the dipole has been doubled (which we expect, because we have two dipoles diven with the same amplitude as the single dipole peviously), and multiplication by a facto 2 cos (k S2 ) cos θ, (11) which we call the aay patten o moe commonly the aay facto. The oiginal element patten is modified by multiplying by this new facto. The aay facto esults puely fom summing the phase tems coesponding to the diffeent distances involved in the aay. Hee, fo this specific example, AF = e j ks 2 cos θ + e j ks 2 cos θ. (12) Notice that the aay facto is only a function of wavelength (k), element spacing (S) and obsevation angle (θ.) We also notice that it epesents the esponse of the aay if the elements used had been puely isotopic; that is, if E e jk (13) (a facto which is found using the fa-field E and H of the dipole and dopping angula dependence.) Notice that the AF has no dependence on the sin θ patten facto associated with the constituent elements: the AF tem is sepaable fom the total field expession. The total patten is the multiplication of the aay facto and the field poduced by the constituent element. This popety is called patten multiplication. Notice also that fo the degeneate case of a one-element aay, egadless of the element type, AF = 1. (14) Example: 2-element dipole aay with an element spacing of half a wavelength (S = λ/2). AF = 2 cos(k S 2 cos θ) = 2 cos(2π λ λ cos θ) (15) ( 4 π ) = 2 cos 2 cos θ (16) The total patten is the patten facto multiplied by the aay facto. Gaphically, this is achieved as follows [1]: We see that the esulting patten is slightly moe diective than that of the individual elements composing the aay. In geneal, all sots of beam possibilities can be achieved by changing the wavelength, element spacing, and as we will soon see, the numbe of elements as well as the amplitude and phase of the element excitations. Hee we have only consideed two elements diven with cuents of identical amplitude and phase. Howeve, the analysis technique is identical: the pinciple of supeposition is always used.

4 Intoduction to Aays Page 4 2 Intepetation of Aay Facto fo the Two-Element Case We have developed a fomula fo aay facto fo the two-element case. It is instuctive to see physically was is happening fo a few examples. Remembe, the AF epesents the patten of an aay of isotopic elements. Example: S = λ/2 (gaphics fom [1]) ( π ) AF = 2 cos 2 cos θ (17) Example: S = λ (gaphics fom [1]) The aay facto is will have nulls wheeve AF = 2 cos (π cos θ) (18) cos(π cos θ) = 0. (19) Nulls occu between the additive points in the patten that is, wheeve the contibutions fom both souces ae 180 out of phase. Evaluating, π cos θ = ± π 2, ±3π 2 (20) θ = ±60, ±120. (21) Refeences [1] W. L. Stutzman and G. A. Theile, Antenna theoy and design. John Wiley and Sons, Inc., 1998.

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