# Cấu trúc sóng chức năng trong điện lý thuyết P6

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## Cấu trúc sóng chức năng trong điện lý thuyết P6

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In this chapter, a solution of EM radiation from a prolate spheroidal antenna, excited by a voltage across an infinitesimally narrow gap somewhere around the antenna center, is obtained. Three specific cases are considered: an uncoated antenna, a dielectric-coated antenna, and an antenna enclosed in a confocal radome. The method used is that of separating the scalar wave equation in prolate spheroidal coordinates and then representing the solution in terms of prolate spheroidal wave functions. A simplified version of the solution, after taking account of the fact that the antenna is symmetrical in the &direction, can then be used...

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1. Spheroidal Wave Functions in Electromagnetic Theory Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic) 6 Spheroidal Antennas 6.1 INTRODUCTION In this chapter, a solution of EM radiation from a prolate spheroidal an- tenna, excited by a voltage across an infinitesimally narrow gap somewhere around the antenna center, is obtained. Three specific cases are considered: an uncoated antenna, a dielectric-coated antenna, and an antenna enclosed in a confocal radome. The method used is that of separating the scalar wave equation in prolate spheroidal coordinates and then representing the solution in terms of prolate spheroidal wave functions. A simplified version of the so- lution, after taking account of the fact that the antenna is symmetrical in the &direction, can then be used to obtain the electric and magnetic fields. The Mathematics code written allows a user to model each of the three types of antenna radiation problem discussed in this chapter. The type of antenna used in this chapter is a prolate spheroid excited by a slot cut through the spheroid. Axial symmetry prevails regardless of the location of the slot. In aircraft applications, the antenna used is generally mounted on the nose of the aircraft, and the type of antenna used is often a slot antenna. Therefore, it is possible to model this antenna configuration as a slot antenna mounted on a spheroid. The effects of a protecting coating layer or radome on such a configuration can also be investigated and considered in the optimized operation. 145
2. 146 SPHEROIDAL ANTENNAS 6.2 PROLATE SPHEROIDAL ANTENNA 6.2.1 Antenna Geometry A perfectly conducting prolate spheroidal antenna excited by a specified field over an aperture on its surface and immersed in a homogeneous, isotropic medium is a convenient introductory problem. It is also assumed that the surrounding medium is nonconducting and nonmagnetic. To simplify the situation, it is assumedthat symmetry about the axial direction prevails. The geometry of the prolate antenna is shown in Fig. 6.1. The semimajor and semiminor axes of the spheroid are designated a and b, respectively, and the interfocal distance d, as indicated in previous chapters. In general, the surface of the spheroid is < = 51, and the excitation gap can be located anywhere, say at 7 = ~0. 6.2.2 Maxwell’s Equations for the Spheroidal Antenna For the symmetrical situation (i.e., a/&$= 0), Maxwell’s equations in free space, relating E and H, take the form (6.la) 1 a(P%J (6.lb) hrE77 = -GT’ (6.1~) (6.ld) 1 a(Pw (6.le) hrH,, = -jClowpdSy achvH,) w-QH< -icowhth,E4, - > (6.lf) at drl = where p = dJ(1 - q2)(S2 - 1)/2, ht and h, have been defined in Chapter 2. Also, ~0 and ~0 are the permittivity and permeability of free space, respec- t ively. From Eqs. (6.la) to (6.lf) it is observed that the problem may be split into two parts. If the applied field on the aperture has only an Eq component, the excited magnetic field has only the H$ component and E+ = 0. On the other hand, if the applied field has only an E4 component, the excited electric field has only an E+ component and H& = 0.
3. PROLATE SPHEROlDAL ANTENNA 147 iv=+1 I : b I : I i T -- 1 Fig. 6.1 Prolate spheroid model of an antenna.
4. 148 SPHEROIDAL ANTENNAS Here we consider only the former case; that is, the applied field has only an I$, component and is considered in the subsequent analysis. Then, following Schelkunoff [35], we set A H4=p’ (6 .2) where A is an auxiliary scalar wave function to be defined later. Therefore, from Eqs. (6.la) and (6.lb), we obtain 4 1 dA Et = - (6.3a) &icowd2 Jcrz - l)( 5. PROLATE SPHEROIDAL ANTENNA 149 The orthogonality condition for m = 1 is provided as follows: (6.8) In general, in the exterior region (< > = -@&)I~=& and ~1 has been replaced by ~1 = ~~1~0,to allow for any dielectric medium. ~1 is the relative dielectric constant of the medium. To obtain a,, the orthogonality condition of Eq. (6.8) can be applied to Eq. (6.11). This is done by first multiplying both sidesof Eq. (6.11) bY c2 - v2 and then integrating r) from -1 to +l. The outcome of this manipulation is: jqwd2 an = - 1’ E~({l,~)/ff-J$Vn(V) dV- (6.12) 4Nl,nu~(tl) -1 Equation (6.12) can also be written in the form (6.13) where (6.14) and 770 a fixed angular coordinate where the excitation gap is located. The is V is a generalized voltage. In the case of a delta gap, E: is zero
6. 150 SPHEROIDAL ANTENNAS everywhere on the spheroid surface except at some infinitesimally thin ring at q = 70. Then, in this limiting case, II r7o+&lo Vn = - - -[i V? ??o--nr)o q5 - Wh, d77 bo-+O the voltage across the slot. (6.15) 6.2.5 Far-Field Expressions In the far field (5 ---) oo), by substituting Eq. (6.9) into Eq. (6.2), the following approximate expressions of the magnetic and electric fields can be obtained: 2e-jklr O” H4 x x &(+)ntZ~Vn(?j), (6.16) IclrdJr--;r;i n=l2 9 where ICI = ,/Gko = w,/m, and (6.17) where q = cos8. 6.2.6 Numerical Computations and Mathematics Code Using Eqs. (6.13) and (6.15), the coefficients an can be obtained. The sub script n can range from 1 to 00, but it is possible to obtain truncated coeffi- cients up to only 30. In the Mathematics code, the number of coefficients used in the actual computation of the magnetic field can be even smaller. These values can then be substituted into Eq. (6.16) to compute the magnetic field. The entire code is created as a Mathematics package called SpheroidAn- tenna.nb. In this package there are three main modules, one for each of the three problems: the uncoated antenna, the dielectric-coated antenna, and the antenna enclosed in a radome. The second and third modules are discussed in the following sections, respectively. The first main module, for the uncoated spheroidal antenna, is a Math- ematica module called Uncoated[ul-,erl-,1-,vO-,Vapplied-1, which com- putes the radiation pattern. There are five arguments and they are specified as follows: l ul: the radial coordinate representing the boundary between the spher- oidal antenna and the outer region l erl: the dielectric constant of the outer region l 2: the semi-interfocal distance
7. PROLATE SPHEROIDAL ANTENNA 151 l ~0: the angular coordinate on which the excitation gap lies l Vapplied: the applied voltage The arguments ~1 and I together define the actual size and shape of the antenna. To gain brief insight into the workings of the module, a summary is provided below: l Based on the input arguments, the parameter cl is calculated. l Using the value of cl, the expansion coefficients dp=(cl) and the eigen- values X,, (cl) are computed using the support modules Getdmn and EigenCommon, respectively. l The spheroidal radial functions of the first, second, and fourth kinds and their derivatives are evaluated next. This is achieved through the sup port modules PSpheroidRl, PSpheroidRZ, PSpheroidRZforsmallc, and PSpheroidR2forlargec. l Next, the spheroidal angular function of the first kind is computed via the PSpheroidS module. l With the radial and angular functions computed and tabulated, it is then possible to compute the expansion coefficients using a submodule called [a- n]. In this submodule, the support modules V[m-,n,,c-,v-1, NormFactor[m-,n-,c-1, and UPrime[m-,n-,c-,u,] are called.to com- pute the angular function, the normalization factor N,,(c), and the derivative of the radial function of the fourth kind, respectively. l Prior to computing the &, pattern, the expansion coefficients are first checked to remove those coefficients with magnitudes that are lessthan lo? The reason for this is that these coefficients are too insignificant to affect the computation of the H+ pattern and are removed to reduce computation time. l A table for the &, pattern is then computed by calling the support module Hphi[c-,coeff-,theta]. 6.2.7 Results and Discussion In this section we present and discuss results obtained for prolate spheroidal antennas with different parameters. Here the semi-interfocal distance is indi- cated as 1 = d/2. To verify the Mathematics code written for the uncoated case, results were obtained and compared to those obtained by Weeks [135]. Figure 6.2 shows the radiation patterns for spherical antennas of radii 0.81X0 and 2.1x0 with different excitation gaps. The dotted points stand for results from Weeks [135]. The results obtained in this chapter are in good
8. 152 SPHEROIDAL ANTENNAS agreement with existing results. Therefore, the accuracy of the code has been verified. The results also suggest that the number of lobes increases with ra- dius. Furthermore, for the asymmetrically excited spheres, it can be seen that increasing radius sharpens the main lobe located near the excitation gap. Figure 6.3 shows the radiations patterns from prolate spheroidal antennas excited at 8 = 30’ (Q = 0.866) and 0 = 90° (7 = 0), where semi-interfocal dis- tances are Z/X0 = 5/27r and 47r, respectively. Again, the results are compared with those of Weeks to show the agreement between results. One similar ob- servation as in Fig. 6.2 is that the number of lobes increases with increasing dimensions of the spheroid, indicated by the semi-interfocal distances. Figure 6.4 shows the variation in radiation patterns when the shape of the prolate spheroidal antennas is changed. From the first set of patterns it can be observed that the effect of varying the radial coordinate of the spheroid from 1.005 to 1.10 is insignificant. On the other hand, from the second set of patterns, when larger variations were made, the number of lobes in the patterns is seen to increase with the radial coordinate of the antenna. This is expected because as
9. DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 153 ,+ R=0.81 __zff_ R=2.41 0.8 0 R=0.81 (Weeks) A R=2.41 (Weeks) 0.6 s- z 0.4 0.2 0.0 0 90 180 @ to) (a) q~ = 0, and R = r/X0 0.8 0.6 0.4 0 90 180 @ to) (b) q) = -0.940, and R = r/X0 Fig. 6.2 Radiation patterns of I&,1 versus 7, due to spherical antennas with different conducting sphere radii excited by slots at (a) ~0 = 0 and (b) ~0 = -0.940.
10. 154 SPHEROIDAL ANTENNAS 1.0 0.8 0.6 I 0 ~0=0.866 (Weeks) A TJO=O (Weeks) 0 90 180 @ to) (a) t1 = 1.077, and Z/X0 = 5/2n 1.0 0.8 0.6 0.4 0.2 0.0 I I I 0 90 180 @ co) 04 51 = 1.077, and Z/AC) = 4n Fig. 6.3 Radiation patterns of prolate spheroidal antennas excited by slots at 70 = 0.866 and 0 with different semi-interfocal distances are 1/X0 = (a) 5/27~ and (b) 47~
11. DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 155 1.0 0.8 0.6 0.4 0.2 0.0 0 90 180 6, to) (a) t1 = 1.005, 1.077, and 1.1; and Z/X0 = 0.25 1.0 0.8 0.6 0.4 0.2 0.0 0 90 180 0 to) u-4 45 = 1.1, 2, and 5; and ~/XC) = 0.25 Fig. 6.4 Radiation patterns of prolate spheroidal antennas excited by slots at q = ( with different radial coordinates of (a)
12. 156 SPHEROIDAL ANTENNAS 1.0 0.8 0.6 0.4 0.2 0.0 0 so 180 0 co) (a)
13. DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 157 1 .o 0.8 0.6 0.4 0.2 0.0 0 90 180 e (0) (4 51 = 1.077, and 1/X0 = 0.25 0.8 0.6 0.0 0 90 180 e (0) (b) 51 = 2, and I/& = 0.25 Fig. 6.6 Radiation patterns of prolate spheroidal antennas with different excitation slots locations and radial coordinates (a)
14. 158 SPHEROlDAL ANTENNAS surroundings, the result of which may be corrosion of the antenna. To get around this problem, the antenna is often coated with a protective layer. The geometry of the coated prolate spheroidal antenna is similar to that in Fig. 6.1, but with a confocal dielectric layer. The thickness of the dielectric layer along the semiminor axis is indicated as t. A detailed illustration of the antenna and coating geometry is provided in Fig. 6.7. The first region (region I) contains a coating with a dielectric constant of ~~1. It is bounded by 5 = ci and { = &. The second region (region II), defined by c > c2, has a dielectric constant of ~2, which is actually unity in most cases, to represent free space. There are two regions with different characteristics. Therefore, it is expected that the auxiliary wave function in each region will be different, too. 6.3.2 Obtaining the Auxiliary Wave Functions 6.3.2.1 Region I ((1 < c < 62) We shall first look at the region imme- diately adjacent to theantenna surface (i.e., the region from & to
15. DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 159 jq=+l I I I : : I I -- 1 i v- -- v- Fig. 6.7 Prolate spheroid model of a coated antenna.
16. 160 SPHEROIDAL ANTENNAS The coefficients in Eq. (6.21) can then be calculated from jqwd2 J d ’ Ea rt - q2 Pn = - -Vn (Cl 3 rl) drl7 (6.22) 4Nl,n(Cl) -1 ’ IL - V2 which in the case of an infinitesimally thin slot is given by jcowd V(Cl, rlo)K (6.23) pn z zNl,n(Cl) where Vs is the voltage applied acrossthe slot. Next, we substitute Eq. (6.20) into Eq. (6.18a), to obtain 4 (6.24) Comparing Eq. (6.24) with Eq. (6.20), it can be seen that 1 t pn = M~~~(cl~t) + NnTn(clYC)* (6.25) Equation (6.25) can also be rewritten as - Pn - M~u~(cl~tl) N1 n- (6.26) Ty&l,&) l Substituting Eq. (6.26) into Eq. (6.21), Eq. (6.20) becomes Al = 2 K(Cl7rl) n=1,2 (6.27) 6.3.2.2 Region /I (e >, 62) On the other hand, for the region from 52 to 00, the appropriate representation for A2 is given by A2 = 2 MzUn(C2,
17. DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 161 6.3.3 Imposing the Boundary Conditions The boundary conditions, which require continuity of the tangential fields, can be obtained by comparing Eq. (6.18a) with Eq. (6.29a) and Eq. (6.18b) with Eq. (6.2913)at < = 52. Aft er simplification, the boundary conditions can be stated as follows: Al = A2)5=62 7 (6.30a) (6.30b) It is observed that the left- and right-hand sidesof Eqs. (6.30a) and (6.30b) cannot be matched term by term due to the different Vn’s on both sides. However, Vn can be expressed as follows: (6.31) where the coefficients d.)n(cl) are nonzero when T is even (odd) and 72is odd (even). P!+,(q) is the usual associated Legendre function. Since the same P:+,(q) exists on both sides of the equations now, it is possible to make use of the orthogonality of Legendre polynomials or to compare both sides of the equations using the coefficients of P:+,.(v). Therefore, Eqs. (6.27) and (6.28) can now be written in the respective forms US(c1 9 ~l)Tn(% 0 + pnTn(c1,t) Un(Cl,
18. 162 SPHEROIDAL ANTENNAS from those of even order. This implies that the problem can be broken via decomposition into two parts: solving for the even-ordered coefficients solving for the odd-*ordered coefficients. In principle, the problem of the coated spheroidal antenna is solved once the Mz coefficients are obtained. These coefficients can then be substituted into (6.34) to calculate the magnetic field. 6.3.4 Numerical Computations To solve for unknown coefficients Mi and M#, it is possible to cast the equa- tions in matrix form. To do so, it is convenient to define some intermediate terms as follows: In U#A 9 tl)Ta(cl 7 t2) A(r,n) = dr* (~1) K&I&) - ? (6.35a) q(cl,cl) > In &h2Tn(clk2J B(r, n) = dr9 (~1) (6.3513) q!Jcl,tl) ’ In c(r, n, = d,) (c2)u-(c2Y C2)9 (6.35~) In u;(cly &)q!&(c19 cd D(r, n) = dr9 (Cl) us - 9 (6.35d) T;(c1, In pnTA(c13 t2) E(r, n) = dry (cl) (6.35e) T;(cl,tl) ’ F(r, n) = 3yc2)U~(c2,
19. DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 163 x03 x23 x05 x25 .. .. ... , n odd x43 x45 ... 1 x12 x16 x32 x36 , n even x52 x56 . l . . . . with X being either A, C, D, or F; cy 2n cy 47-h . . . T > , n odd T > , n even 3n 5n . . . cy cy with Y being either B or E; and - 22 24 26 l * )‘,n even with 2 being either M1 or M 2. The matrix equation (6.37) can then be solved twice, for the even and odd cases. In this section, the truncation number is taken to be 30. This means that there are a total of 60 coefficients to be computed (30 each for MA and M..). However, since the odd and even casesare decoupled from each other due to the nature of the coefficients diy”, each case will have 30 coefficients to be solved for. For the odd case, there have to be 30 equations in order to have 30 unique solutions. Since there are two equations for every value of r, r will have to range from 0 to 28 in steps of 2. For the even case, T will range from 1 to 29 in steps of 2. To test for convergence of the results, computations were performed for truncation numbers of 30 and 34. Upon comparing the results, it is observed that the two sets of results match at least to three significant figures, and the smallest value in the solution can be considered to be negligible compared to the largest value in the same solution. For the largest value in the solution set, the results match up to 13 significant figures. Therefore, a truncation number of 30 is sufficient to obtain highly accurate results. 6.3.5 Mathematics Code The module for the dielectric-coated spheroidal antenna is called Coated[ul-, u2-, erl-, er2-, l,, vO,, Vapplied-1. There are seven arguments and they are specified as follows:
20. 164 SPHEROIDAL ANTENNAS l ul: the radial coordinate representing the boundary between the spher- oidal antenna and region I l u2: the radial coordinate representing the boundary between the regions I and II l erl: the relative dielectric constant of region I l er2: the relative dielectric constant of region II l 1: the semi-interfocal distance l ~0: the angular coordinate on which the excitation gap lies l Vupplied: the applied voltage This module developed for computation can be summarized as follows: Based on the input arguments, the parameters cl and c2 are calculated. Using the values of cl and ~2, the expansion coefficients drmn(c) and the eigenvalues A,, (c) are computed for each c using the supporting modules Getdmn and EigenCommon, respectively. The spheroidal radial functions of the first, second, third, and fourth kinds and their derivatives are evaluated subsequently. This is achieved through the supporting modules PSpheroidRl, PSpheroidR2, PSph- eroidR2forsmallc, and PSpheroidR2forlargec. These are computed for all combinations of c and

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