RADIATION OF ANTENNA ARRAYS WITH GENERALLY ORIENTED DIPOLES

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Cody Barker
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1 Jounal of ELECTRICAL ENGINEERING, VOL. 53, NO. 7-8, 22, RADIATION OF ANTENNA ARRAYS WITH GENERALLY ORIENTED DIOLES Štefan Beník ete Hajach The aim of this aticle is to show the possibilities of shaping the adiation patten in two-element antenna aas which ae geneall oiented in space. esentl solved and used antenna aas contain elements (dipoles which ae oiented in the same wa. The authos pesent the solution of a adiation patten oiented in ais diection. It is clea that the solution of the adiation patten of the dipole can not be used fo the solution of a antenna aa with non-specificall oiented dipoles. In this case thee ae pesented dipole solutions with a tilt angle situated in plane and plane. K e w o d s: antenna aas, adiation patten, dipole 1 INTRODUCTION The adiation patten of an antenna aa might be shaped b appopiate distances between elements in one ais in linea antenna aas o in two aes in plana antenna aas. In geneal we can chaacteise 5 was which can be used fo shaping the whole adiation patten of the antenna aa with identical elements [1]: geometical configuation of the whole aa (linea, cicula, ectangula, etc, distance between elements, amplitude ecitation of single elements, phase ecitation of single elements, adiation patten of individual elements. If the elements in a linea antenna aa, fo eample dipoles, ae oiented in ais diection of the diection is maimal. Geneall, the elements in an antenna aa can be oiented in an diection in the space. In this aticle we have focused on the wa how to solve the adiation patten of a two element antenna aa with geneall oiented elements. Fom the pactical point of view it is possible to use the esults of this solution to stud the attibutes of adiation patten of man element antenna aas in the case of damage in pedefined configuation, fo eample b the influence of weathe. We can evaluate not onl the oveall change of the adiation patten but also the change side lobes, which unfavouabl influence the opeation of ada o communication sstems. 2 GENERALLY ORIENTED DIOLE In man publications the solution is pesented of a dipole oiented in ais. In the fist pat of this aticle the aim is to show the solution of the dipole geneall oiented in plane and the solution of a dipole geneall oiented in plane. I n a Fig. 1. Geneall oiented dipole in plane 2.1 Tilt angle θ n (φ = 9, plane An elementa dipole, o also called a Hetian dipole [5], is a thin, linea conducto whose length L is ve shot compaed with the wavelength λ, this coesponds to a unifom-cuent distibution. The wie is oiented accoding to Fig. 1. We assume dipole displacement in the cente of a Catesian coodinate sstem. The customa appoach fo finding the electic and magnetic fields at point in space due to adiation b a cuent souce is though the etaded vecto potential A [1]. Vecto a is the unit vecto of position vecto, the tilt angle dipole is θ n, and a = sin θ cos φa + sin θ sin φa + cos θa. (1 Depatment of Radioelectonics, Slovak Technical Univesit, Ilkovičova 3, Batislava, Slovakia RE-RINT ISSN c 22 FEI STU

2 Jounal of ELECTRICAL ENGINEERING, VOL. 53, NO. 7-8, whee U = [sin θ cos θ n cos θ sin φ sin θ n ]. n = 4 o = o n = o 27 o n a a a a n = 18 o n = 2 o n = 3 o = o n = o I Fig. 3. Geneall oiented dipole in plane 27 o b n = 18 o n = 6 o Fig. 2. Radiation pattens of the dipole a H plane b E plane We have all components of the vecto potential µi le jk A = a cos θ n, 4π A = a µi le jk 4π sin θ n. (2 Now we tansfom the components of the vecto potential to the spheical coodinate sstem A = sin θ sin φa + cos θa A φ = cos φa A θ = cos θ sin θa + sin θa. The components of the magnetic field (H ae obtained b solving the Mawell equations [1], [2]: H = ot A. (4 In most antenna applications, we ae pimail inteested in the adiation patten of the antenna at geat distances fom the souce. Fo the electical dipole, this coesponds to distances such that λ o, equivalentl, k = 2π/λ 1. This condition allows us to neglect the tems vaing as (k 2, in favo of the tems vaing as (k 1, which ields the fa- field epessions [3], [5]. The components of electic field ae Ė = Ė φ = j k2 I le jk cos φ cos θ n ωε 4π Ė θ = j k2 I le jk U ωε 4π (3 (5 The fa-field due to adiation b the entie antenna is obtained b integating the fields fom all the Hetian dipoles which ceate the antenna [1], [4]. We solve all components of the electomagnetic field H θ, E φ, E θ. The components of the electic field ae: Ė θ = j 6I e jk U D θn (6 Ė φ = j 6I e jk cos φ sin θ n D θn (7 cos(kl cos ψ cos(kl whee D θn = 1 (sin θ sin φ sin θ n + cos θ cos θ n. Figue 2a and 2b efe to the adiation patten when angle θ n is changed. The adiation patten depends on the change of angle θ n, o the adiation patten is dependent on the position the dipole. The magnitude of vecto E can be detemined fom [1]: E = 6I [ (cos 2+ φ sin θn ( ] 2 1/2 sin θ cos θn cos θ sin φ sin θ n Dθn. (8 Assuming θ n =, we can suppess the tilt between the dipole and ais. In this case the dipole is situated along ais. B changing the dipole position we contol the vecto cuent flowing though the wie. The vecto potential is in the diection of the induced cuent. In consequence a change of the dipole position (a souce is ealied b a change of the vecto potential and then shape of adiation patten is also changed. The change of adiation function is in change of maima and minima. The mutual position between maima and minima is constant. E = 6I o F (θ (9 cos ( kl(cos θ cos(kl whee F (θ = sin θ. Equation (9 is the solution of the electomagnetic field of the dipole situated along the ais.

3 24 Š. Beník. Hajach:RADIATION OF ANTENNA ARRAYS WITH GENERALLY ORIENTED DIOLES n = 27 o n = 6 o = o 18 o n = o n = 3 o Fig. 4. Radiation pattens of dipole in the H plane n = - a n = 3 o - b = o 18 o = o 18 o n = o n = 6 o Fig. 5. Radiation pattens of dipole in the E plane a when tilt angle θ n = and 9, b when tilt angle θ n = 3 and Tilt angle θ n (φ =, plane Let us have an infinitesimal (elementa dipole [1], [5] placed at the oigin of a Catesian coodinate sstem. The tilt angle is ealied in the plane. We assume φ =, θ n = ( 18. The situation is shown in Fig. 3. Then µi le jk A = a 4π µi le jk A = a 4π sin θ n cos θ n (1 The components of the vecto potential in a spheical coodinate sstem ae A = A sin θ cos φ + A cos θ A φ = A sin φ (11 A θ = A cos θ cos φ A sin θ. We use the solution b the etaded vecto potentials fo electic and magnetic components of the field. These components of the fa-field in the Faunhofe one ae E = E θ = jk2 I le jk ( cos θn sin θ cos θ cos φ sin θ n ωε4π (12 E φ = jk2 I le jk sin θ n sin φ. ωε4π Equations (12 ae the components of the electic field due to an elementa dipole in the Faunhofe one. The integation ove the length of the dipole antenna gives the following epessions fo electic and magnetic field. The obsevation point is and a is the unit vecto: a = a sin θ cos φ + a sin θ sin φ + a cos θ (13 a a = a sin θ n + a cos θ n Solution of integal ields D θ 2 ( cos(kl cos ψ cos(kl = k ( 1 (sin θ cos φ sin θ n + cos θ cos θ n 2. (14 Othe components have the epessions E θ = j6i e jk ( cos θn sin θ cos θ cos φ sin θ n Dθ (15 E φ = j6i e jk sin θ n sin φ D θ (16 cos(kl cos ψ cos(kl whee D θ = 1 ( 2. sin θ cos φ sin θ n + cos θ cos θ n If the field components ae known, the magnitude of the electic field is given as E = 6I ( (cos θ n sin θ sin θ n cos θ cos φ 2 + (sin θ n sin φ 2 1/2 Dθ (φ=9. (17 Equation (17 is the component of the electomagnetic field adiated b the dipole which changes the position in plane. With kl/2 = π/2 and θ = 9 Eq. (17 eads E = 6I (cos θ n 2 + (sin θ sin φ 2 D θ (θ=9. (18 Equation (18 is the adiation function in plane. B consideing φ = E = 6I (cos θ n sin θ sin θ n cos θ 2 D θ (φ=, (19 then Eq. (19 is the adiation function in the plane. B consideing φ = 9 E = 6I (cos θ n sin θ 2 + (sin θ n 2 D θ (φ=9. (2 Equation (2 is the adiation function in the plane. Figues 4, 5, 6 show the change of the adiation patten when angle θ n is changed. Figue 4 shows the case when θ n =, 3, 5, 9, Fig. 5a when θ n =, 9, and Fig. 5b when θ n = 3, 6.

4 Jounal of ELECTRICAL ENGINEERING, VOL. 53, NO. 7-8, n1 1 I d 1 d n2 2 Fig. 6. Two dipoles oiented in ais Fig. 7. Geneall oiented dipoles on the ais The adiation patten is changed in dependence of the change of angle θ n. on position of the dipole on ais. The position change contols the change of the vecto cuent flowing though wie. The vecto potential is in the diection of induced cuent. In consequence of change position dipole is ealied b change of the vecto potential and then the change shape function aa facto. The change of adiation function is in change of maima and minima. The mutual position between maima and minima is constant. This pat of the pape pesents the solution of a dipole when the dipole is geneall oiented. The adiation pattens ae show in in figues. The shapes of adiation pattens ae identical with the adiation patten of the dipole when the dipole is oiented in ais. 3 TWO GENERALLY ORIENTED DIOLES Man authos pesented linea antenna aas with dipoles oiented in ais (dipoles ae oiented paallel with ais see Fig. 6. The configuation of two halfwave-length dipoles at the distance d is the elementa antenna aa. We assume that the antenna aa is located in a linea, homogeneous, and isotopic field. The total field in the Faunhofe one is defined as a supeposition of electic field components fom both dipoles. If the dipoles ae oiented along ais, the field is with assuming θ θ 1 θ 2 E θ = E θ1 + E θ2 (21 1 d 2 cos θ 2 + d 2 cos θ 1 2. (22 Then the total field in the Faunhofe one is Ė θ = j6i e jk [ ] 1 F (θ 2 cos kd cos θ. (23 2 The aim of this aticle is the solution of linea antenna aas with geneall oiented dipoles. The fist pat is about dipoles geneall oiented on the ais and the second pat is about dipoles geneall oiented on the ais. The electic component adiated field b dipole situated along ais is (Fig. 6 whee Ė θ = j6i e jk F (θ [1 + ep(jkd cos θ] (24 F (θ = cos(kl(cos θ cos(kl sin θ is the adiation function of one dipole. 3.1 Geneall oiented dipoles on the -ais (25 We assume a two element antenna aa with dipoles aange along the ais as shown in Fig. 7. The dipoles change the tilt angle in the plane. All components of the magnetic H and electic field E ae solved using the vecto potentials. We assume the tilt angle θ n1 fo dipole 1 and θ n2 fo dipole 2. Then we use Eq. (24 and obtain the component of the electic field fo this stuctue of antenna aa Ė θ = j6i e jk Ė φ = j6i e jk [F θ (θ, φ 1 +F θ (θ, φ 2 ep(jkd cos θ] [F φ (θ, φ 1 +F φ (θ, φ 2 ep(jkd cos θ] (26 whee F θ (θ, φ 1, F θ (θ, φ 2, F φ (θ, φ 1, F φ (θ, φ 2 ae the adiation functions of dipole 1 and 2. The evaluation of adiation function is F θ (θ, φ 1,2 = ( sin θ cos θ n1,2 cos θ sin φ sin θ n1,2 cos ( kl(sin θ sin φ sin θ n1,2 + cos θ cos θ n1,2 cos(kl 1 (sin θ sin φ sin θ n1,2 + cos θ cos θ n1,2 2. (27 F φ (θ, φ 1,2 = ( cos φ sin θ n1,2 cos ( kl(sin θ sin φ sin θ n1,2 + cos θ cos θ n1,2 cos(kl 1 (sin θ sin φ sin θ n1,2 + cos θ cos θ n1,2 2. The total field is E = a θ Ė θ + a φ Ė φ. (28

5 26 Š. Beník. Hajach:RADIATION OF ANTENNA ARRAYS WITH GENERALLY ORIENTED DIOLES ABS (Fs Cut in - plane q n1,2 = o n1,2 =2 o n1,2 =4 o n1,2 =6 o o n1,2 = 18 o 27 o 36 o n1 1 d 2 1 n2 Fig. 8. Radiation pattens of two dipoles in the H plane Fig. 9. Geneall oiented dipoles on the -ais ABS (Fs 2.5 n = o Cut in - plane n =6 o n =4 o n =2 o n = o 18 o 27 o 36 o Fig. 1. Radiation pattens of two dipole in the E plane If θ n1 = θ n2 =, we obtain Eq. (36. A coss-section in the plane (θ = 9 is shown in Fig. 8. In this case we assume θ n2 = and θ n1 = ( 9. A shot fomula fo E θ and E φ is Ė θ,φ = j6i e jk ( Fθ,φ (θ, φ 1(θ=9 + F θ,φ (θ, φ 2(θ=9 ep(jkd cos θ. (29 When θ n1,2 =, the intefeence function is cicula (omnidiectional, with an incease of θ n1,2 omnidiectional adiation function deceases and the antenna aa is diectional (Fig. 8. The space between adiatos is d =.5λ and the length of dipole is L =.5λ. 3.2 Geneall oiented dipoles on the -ais Now conside a dipole positioned accoding to Fig. 9. The adiatos ae aanged along the ais. The space between dipoles is d. The following equations define the electic component of the electomagnetic field o Ė θ = j6i e jk ( Fθ (θ, φ 1 + F θ (θ, φ 2 ep(jkd sin φ sin θ Ė φ = j6i e jk ( Fφ (θ, φ 1 2π + F φ (θ, φ 2 ep(jkd sin φ sin θ Ė θ = j6i e jk (F θ (θ, φ 1 ep ( d2 jk sin φ sin θ + F θ (θ, φ 2 ep ( jk d 2 sin φ sin θ Ė φ = j6i e jk (F φ (θ, φ 1 ep ( d2 jk sin φ sin θ + F φ (θ, φ 2 ep ( jk d 2 sin φ sin θ (3 (31 The fist equation defines the electic component accoding to Fig. 1, the second equation defines the case, when the dipoles is situated in the coodinate sstem smmeticall with espect to the oigin, o Ė θ,φ = j6i e jk ( Fθ,φ (θ, φ 1 + F θ,φ (θ, φ 2 ep(jkd sin φ sin θ (32 The adiation patten in the plane (φ = 9 is Ė θ,φ = j6i e jk ( Fθ,φ (θ, φ 1(φ=9 + F θ,φ (θ, φ 2(φ=9 ep(jkd sin θ (33 whee F θ(φ=9 1,2 = ( sin θ cos θ n1,2 cos θ sin θ n1,2 cos ( kl(sin θ sin θ n1,2 + cos θ cos θ n1,2 cos(kl 1 (sin θ sin θ n1,2 + cos θ cos θ n1,2 2. (34 F φ(φ=9 1,2 = sin θ n1,2 cos ( kl(cos θ cos θ n1,2 cos(kl 1 (cos θ cos θ n1,2 2.

6 Jounal of ELECTRICAL ENGINEERING, VOL. 53, NO. 7-8, Emploing pesumption θ n1 = θ n2 = we solve the case when the dipole is aanged along the ais and dipoles ae oiented in the ais. Then o Ė θ = j6i e jk (35F (θ ( 1 + ep(jkd sin φ sin θ (35 Ė θ = j6i e jk (35F (θ cos [ 1 2 kd sin θ sin φ] (36 With θ n1 = θ n2, the tilt angle θ n1 = 9, θ n2 = 9 is in the Fig CONCLUSION The pesented aticle solves the adiation of twoelement antenna aas with geneall oiented dipoles. In the fist case the dipoles ae located in ais, and in the second one in the diection of ais. The adiation of the dipole and the adiation of the antenna aas wee solved unde assumption of a sinusoidal cuent distibution along dipoles. Depicted adiation pattens of numeical solutions show the change of the diectional attibutes of the antenna aas with the change of oientation of single dipoles. We can evaluate not onl the whole change of the adiation patten but also the change of sidelobes, which unfavouabl influence the opeation of ada o communication sstems. The solution of the adiation antenna aa, which consists of two elements abitail oiented in the othogonal coodinates, shows that the antenna aa is not diectional. Additional possible contibution is the simulation of antenna aas with an omnidiectional shape of the adiation patten (in an plane, which ae suitable fo the aea of adaptive antenna aas and also fo the aea of passive ada sstems. Refeences [1] BALANIS, C. A. : Antenna Theo, at I. and II., Wile & Sons, New Yok, [2] COLLIN, R. E. ZUCKER, F. J. : Antenna Theo, at 1, McGaw-Hill Co., New Yok, [3] BENÍK, Š. HAJACH,. : Adaptive Antenna Aas-Solving Radiation attens of Two Concuentl Oiented Elements, In: Conf. oceedings Radioelektonika 2, pp , Batislava, Sept. 2. [4] VAVRA, Š. TURÁN, J. : Antennas and opagation of Electomagnetic Waves (Antén a šíenie elektomagnetických vĺn, ALFA, Batislava, (in Slovak [5] ULABY, R. : An Applied Electomagnetic Eneg, New Yok, Received 5 Novembe 21 Štefan Beník (Ing was bon in Batislava, Slovak Republic, on Apil He eceived the Ing degee in electical engineeing fom the VVTŠ of Liptovský Mikuláš, in His eseach inteests include adaptive antennas. ete Hajach (doc, Ing, hd was bon in Batislava, Slovak Republic, on June He eceived the Ing degee in electical engineeing fom the Slovak Technical Univesit of Batislava, in 1969, and the hd degee in adioelectonics fom the STU Batislava, in He is cuentl associate pofesso at the Radioelectonic Depatment of the STU Batislava. His eseach inteests include micowave antennas.

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