CC BY
7
621.372
Mutual Coupling Coefficients of Rotated Rectangular Dielectric Resonators in Open Space
Trubin A. A.
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine
E-mail: alrubin'J&gmail. com
The coefficients of mutual coupling of rectangular dielectric resonators in open space are calculated under the condition of their rotation relative to one of the axes of a given rectangular coordinate system. Analytical formulas for complex coupling coefficients are obtained. The expressions found give complete information about the frequencies and Q-fact.or of coupled oscillations of dielectric resonators. The dependences of the coupling coefficients on the angles of rotation and spatial coordinates of resonators in the case of excitation of the main magnetic types of natural oscillations in them are considered. The concept of pseudo-rotation of resonators is introduced. Cases are noted when the pseudo-rotation of the resonators does not lead to a change in the coupling coefficients. The dependences of the coupling coefficients for different types of resonator pseudo-rotations are investigated. New integral representations are derived for the mutual coupling coefficients of rectangular dielectric resonators provided that their axes rotate in open space. In particular cases of parallelism of the resonator axes of one of the coordinate axes, the analytical expressions found in the work coincide wit.li those obtained earlier. For each case of rotation, approximate analytical formulas are found for the integral representations obtained in this work, expressed in terms of the spherical Hankel functions of the second kind. Comparison of calculations of coupling coefficients by integral formulas and approximate expressions is carried out. It is shown that the approximate expressions have acceptable accuracy for all the considered cases of rotations. The dependences of the coupling coefficients on the coordinates of the resonators are investigated. The regions are marked in which the found integral representations make it possible to correctly describe the coupling coefficients of rectangular resonators. In contrast to integral representations, approximate formulas are correct in the entire spatial region of resonator interaction. The results obtained make it possible to construct analytical models of antennas, multi-element arrays and devices of infrared and optical wavelength ranges, made with the use of rectangular dielectric resonators: significantly reduce computation timecompared to numerical methods and optimize complex multi-cavity structures of microwave and optical communication systems.
Keywords: coupling coefficient: mutual coupling coefficient: rotation: rectangular dielectric resonator: open space
DOI: 10.20535/RADAP. 2022.88.60-68
Introduction
Currently, rotatable dielectric resonators (DRs) have fonnd application in various lattices for the implementation of optical metasnrfaces [2, 4, 6 9], arrays fl, 5, 13], metamaterials [11, 15] and another frequency-selective structures, for the formation and detection of twisted waves, Airy rays [9], as well as antennas [3, 10, 12, 14] and filters [16, 18]. The main advantage of DR lattices in comparison with metal metasnrfaces is the low level of dissipative losses. For calculation and optimization the scattering parameters, it is necessary to build adequate electromagnetic models describing the properties of such lattices. The most convenient way to calculate the physical properties of the lattices is the development of the scattering theory based on the nse of the coupling coefficients of resonators in various structures [17,18].
Knowledge of the coupling coefficients makes it possible to calculate and optimize the basic scattering parameters of the lattices with the least amount of computing power.
1 Statement of the problem
The purpose of this article is calculation and analysis of coupling coefficients of the rectangular dielectric resonators in open space in the rotation of the resonator axes.
Attempts to describe the rotation of dielectric resonators relative to each other lead, usually, to complex and cumbersome analytical expressions. To minimize the size of the resulting formulas, in this work the simplest rotations about only one axis, the allocated rectangular coordinate system, are consi-
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61
dered. For simplicity, wo considered only the basic dipole oscillations of rectangular magnetic resonators H111.
In the general case, the complex coefficient of mutual coupling of two resonators in a waveguide with a cross section a xb, can be represented as an expansion:
2 Coupling coefficients of
X-rotation Rectangular DRs in the Open Space
The main magnetic type of natural oscillations H111, the field of a rectangular dielectric resonator in the local coordinate system (x', y', z') (Fig. ) is represented as:
h\i w^q
k\ -3\
k2 -32 0;
hi
ki -32
2 ¡3y cos(/3xx') sin(3yy') cos(3zz'); 3x sin(3xX) cos(3yy') cos(3zz')
(1)
hy' = k2 - 32 3y3z
hz
3x3z sin(3xxX) cos(3yy') sin{¡3zz' 3y3z cos(3xX) sin(3yy') sin(3zz'
Fig. 1. Rectangular dielectric resonator in the local coordinate system (x',y',z')
ki2 = ^ E (Ci1±)o(^
r|z2-zi|
(2)
where cst± is the expansion coefficients of the DR field in terms of the waveguide field:
c¡± = i¡2w(e 1 -e0) (?, (E±) )dv,
y
resonator:
waveguide mode (E±,H±), ct± = (c't±)0etLz% (c't±)0 -expansion coefficient without taking into account the dependence on the longitudinal coordinate zs-, w2 -energy stored in the resonator material.
To calculate the coupling coefficients in open space, we assumed that the walls of the "virtual" waveguide tend to infinity:
hi cos(/3Xx') cos(3yy') cos(pzz').
Here h1 is the amplitude; w is the circular frequency of natural oscillations; is the magnetic permeability; (/.3x,py) ^e the wave numbers: @X + fiy + 3y = ky; ki = w^QQëi; ko = w^JIQQQeQQ-, £q, ei is the dielectric permittivity of the external space and resonator, respectively.
«12:
: lim kl 2
(3)
as a result, sum (2) was transformed into the integral.
In the case of rotation of the resonator axes relative to the x-axis of the selected rectangular coordinate system, we will assume that the initial orientation of
(Fig. 2, a). Let us also assume that the coordinates of the center of the first resonator are equal (x\,y\,z 1), and the coordinates of the second resonator are equal (x2, y2, z2) and the main type of magnetic oscillations is excited in the resonators Hm, the field of which is described by expression (1).
In the case of rotation of 1 resonator relative to the x-axis (Fig. ,a) by an angle 3i and the second resonator relative to the x-axis by an angle ft2, the mutual coupling coefficient takes the form:
«1 2 = «0
cos[£k0 A.x]
—i -ykoAz
o j0 i(e+v2)
{n
U=1
x< TT iv3yWx(0^y(vcos/3s +-/sin3&) + Ç/3x cos3sWx(C)Wy(ricos 3S + 7sin3s)]wz(vsin3s - jcos/3s)e lr}koAy +
+ n l1l3yWx(0^y (v cos 3s -7 sin 3s) + £3x cos 3swx(i)wy (r] cos 3s - 7sin3s)]wz(v sin 3s +7 cos 3s)e%v 0 y+
s=1
2
+ n K 73y Wx(O^y (v cos 3s + 7 sin 3s) - V73x cos3sWx(£)wy (v cos 3s +7 sin 3s) +
x
where 6s
y
h
x
X
+ (£2 + sin pswx(£)uy (rç cos + 7 sin &)] (rç sin & - 7 cos ¡3s)e ivk°Ay+ 2
+ n [^
yWï(vcos& - 7 sinps) - rp/Px cos(rçcos - 7 sin&) +
S = 1
+ (e2 + sin&rnx(C)uy(Vcos& - 7sin&)] (rçsin& + 7cos^e^0^} d£dV;
r
x
I
a,
i a.
X
X
Aal
y
x
Re[K]
Re[K]
Im[K]
Re[K]
0 2 4 a!
Re[K]
024a,
J
Fig. 2. Rotation of Rectangular DR relatively the x-axis (a): y-axis (b): z-axis (c. d) of the coordinate system in the open space. Dependence of the coupling coefficients on the rotation angle relatively the x-axis (e, i). Dependence of the DR coupling coefficients on the rotation angle a2 relatively the y-axis (f, j). Dependence of the coupling coefficients on the rotation angle a2 relatively the z-axis (g, k) for the initial DR position z' y z, x' y x. Dependence of the coupling coefficients on the angle of rotation a1 for the initial DR position z' || x (h, 1).
(k0Ax = 1; k0Ay = 1; k0Az = 2; a1, ^ = 0).
here £2 + r)2 + 72 = 1;
32^0/ -, s2 PxPyPzqxqyqz
«0 = —2T-(£ir - 1) -TJ-;
n2 vH
oscillations (1)
2
x [pv sinpv cos(gv£) - qv£ cospv sin(gv£)];
vH = £irk0)[p2y+ I32x-Kx
wç = pv ± sinpv cospv; v = (x,y,z); £ir = ei/e.
0
p° - (qviY
x [pv cospv sin(gv£) - qv£sinpv cos(gv£)].
px = ¡3xa0/2-, py = ¡3yb0/2; Pz = L/2 as well Integral ( ) is not very convenient for calculations, as qx = koao/2; qy = k0b0/2] qz = k0L/2 are the it is characterized by poor convergence, therefore, to characteristic parameters of the resonators. For Hm calculate it, we will use the approximation.
1
x
w
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63
The essential region of integration (4) is in the region of small values of the arguments of the functions wv (z), -rnv (z). Therefore we use here an approximate representation of the functions wv(z), (z), v = (x, y, z), typical for small values of variables (£, rj):
wv(z) » Av, (z) » Cv
kzz = k1x j Ax2 + Ay2
h(02)( ko Ar) -
1-3
Ar2
h[2)(koAr) k0Ar
AVAZ ,(2)n A N
kyz = -Ki Ar2 h2 '(koAr);
(8)
(5)
where Av, Cv are constants. Applying ( ) and Sommer fold's integrals [18]. we obtain a simpler formula, expressed in terms of the angles of rotation of the resonators:
K12 ~ KZZ cos ß1 cos ß2 + KYY sin ß1 sin ß2 +
+ KYZ Sin(ßl +ß2), (6)
where
ï-
(7)
(xs, ys, zs) is the coordinates of the center of the s-th
=
1,2); Ax = xi - x2; Ay = yi - y^ Az = |z1 -
Ar = \J(xi - x2)2 + (y i - y2)2 + (zi - Z2)2; hn\z) is the Hankol spherical functions of the second kind.
Under the conditions: Ax = i^d @xCx = PyCy, which occur, for example, for resonators with a square cross-section (a0 = b0), the coefficient k1 = i^^o x IAx@xCxAz |2. The easiest way to choose a numerical value of k1 is based on the mean theorem, directly for the given parameters of the resonator from (4) for a particular orientation, for example, for fti = @2 = 0 f ].
3 Coupling coefficients
of Y-rotation Rectangular DRs
In the case of rotation of each of the resonators, respectively, at an angle a1,a2 (Fig. ,b) relative to the y-axis, we obtain similarly to (4):
/•CO /*CO
Kl2 = VKo
cos[q k0Ay]
-i-ykoAz
o Jo 7(t2 + v2)
{n
XS TT ivßy cosasWy (r))ux(Çcosas + 7 sinas) + £ßxuy (r))mx(Çcosas + 7 sinas)]^z (Çsinas -7 cosas)e ^k°Ax +
^TTtv^y cosasWy (r)ux(€cosas -7 sinas) + £ßx^y (r)^x(^cosas -7 sinas)]^z (£sinas +7 cosas )e <k°Ax+
2
+ TT K7ßy cosasWy (r)iüx(^cosas +7 sin as) - mßx^y (q)'x(^cosas +7 sin as) -
i
- ( C2 + V2) ßy sinas'y (r/)wx(£ cos as + 7sinas)] uz (£ sinas - 7 cosas) e-l^k°Ax+ 2
+ TT K7ßy cos a s'a y (v)cüx(^cosas -7 sinas) - V7ßx^y (q)'x(^cosas -7 sinas) +
=i
+ (£2 + V2) ßy sinas'y(v)wx(£cos as - 7 sinas)] wz(£sinas + 7cosas)el€k°Ax} d^dq. (9)
Using again approximation (5), we find:
K12 » kzz cos ai cos a2 + kxx sin ai sin a2 + kxz sin(ai + a2).
(10)
Here
(2 1
kxx = Ki ■ \ -h^ (koAr) + -
1 Ay2 - Ax2 + Az2
3 A2
h<22)(ko Ar)
} -
(H)
Ax A z (2),, kxz = -Ki Ar2 h2 (koAr).
x
4 Coupling coefficients of the Rectangular DRs under Zl-rotation
In the case of rotation of 1 and 2 resonators relative to z-axis (Fig. ,c), respectively, at an angle a1, a2, relative to z-axis, the mutual coupling coefficient takes the form:
K12 = tK0Jo Jo ^T^) (7)|x
{n
x i TT ( cos as + £sin as)iox(£cos as — V sin as)wy (£sin as + r¡ cos as) —
i
— Px(r¡sinas — £ cos as)rox(£ cos as — r/sinas)wy(£sinas + ?ycosas)] cos (^k0Ax + r/k0Ay) + 2
T IT [@y (^ cos as - Î sin as )lox (£ cos as + r/ sin as )wy (£ sin as - -q cos as ) -
S=1
- px('qsinas + £cosas)inx(£cosas + -qsinas)wy(£sinas - ?ycosas)] cos (^k0Ax - rjk0Ay) +
2
T IT as - r/sin as)ic>x(£cos as - 'qsin as)wy(^sin as + 'qcos as)-
S=1
- Pxl(£, sin as + r/cos as)inx(£cos as - r/sin as)xuy(^sin as + r/cos as)] cos (£k0Ax + r/k0Ay) +
2
T IT cos as + Vsin as)ic>x(£cos as + 'qsin aa)wy(^sin as - qcos as)-
S=1
- Pxl(£, sinas - 'qcosas)xux(£cosas + 'qsinas)xuy(£sinas - ?ycosas)] cos (^k0Ax - r/k0Ay)} d^d-q. (13) Using again approximation (5), we obtain from (13):
«12 « kZZ . (14)
5 Coupling coefficients of the Rectangular DRs under Z2-rotation
Let us consider another important type of resonator rotation about the z-axis, in which the initial position of the axes of both resonators is oriented parallel to the x-axis (Fig. 2, d). In this case
!■ to !■ to g — i-yk0Az
«12 = ÎK0
00
7(e2+v2)
x I TT \Px(v sin as — £ cos as)wx(^)^y (£ sin as + -q cos us)lvz(£ cos as — -q sin as)] cos (ÇkoAx + -qkoAy) +
{n
s=i
i 2
+ TT I^x(i7 sinas + ^cosas)TUx(^)^y(£sinas — r]cosas)(^z(£cosas + ?ysinas)] cos (£koAx — -qkoAy) +
s = 1 2
+ n [Px7(£ sin as + -q cos Us)wx(7)^y(£ sin as + tjcosas) + fty (£2 + r]2)ux(lWy (£ sin as + cos «s^ x
s=1
x wz(£cosas — r/sinas) cos (£&oAx + r/k0Ay) +
2
+ n [fixl(£, sin as —-qcosas)wxXl)tOy(£sin«s — -qcos as) + fty(£2 +^2)^x(7)wy(£sin «s —-qcosas^ x
s=1
x wz(£cosas + rqsinas) cos (£k0Ax —'qk0Ay)} di^d'q. (15) Taking into account (5), expression (15) takes the form:
«12 ~ Kxx cos ai cos a2 + kyy sin ai sin a2 — kxy sin(ai + a2), (16)
where
(2 1
^^^^ = k.i < — h^2^ (kn Ar) + — ^
Í2 (2)r, . , 1 r 1 Ax2 — Ay2 +Az2] l(2),1 . J «FF =«i|-h\0'(koAr) + 1 3--- 42)(^oAr)j ,
h22)(koAr)}, (17)
^xy =«1 AA ^22)(fc0 Ar). (18)
Relations (4.9.13.15) as well as (6-8,10-12,16-18) make it possible to calculate the mutual coupling coefficients of the rectangnlar DRs in the cases of the considered rotations of their axes. Relations (4,9,13,15) are valid under the condition A z > dmax, where dmax is the maximum size of the DR is in the direction of the straight line connecting their centers. In contrast to integral representations, the found approximate formulas (6-8,10-12,16-18) are finite in the entire spatial region of resonator interaction.
6 Analysis of mutual coupling coefficients
In particular cases of parallelism of the resonator axes of one coordinate axes, the found analytical expressions (4-18) coincide with those obtained earlier [17].
The results of calculating the dependences of the mutual coupling coefficients on the coordinates and the angle of rotation of the resonator are shown in Fig. 2-5 for e1r = 36; a0 = 50; L/2a0 = 0, 4; f0 =8 GHz.
Fig. 2 compares the results of the numerical calculation of integral representations and the calculation of the mutual coupling coefficients of the
resonators nsing approximate formulas. Solid curves show the dependences calculated by the formulas (4,9,13,15): the dotted curves show the dependences obtained by approximate formulas (6,10,14,16). As can be seen from the data presented, the approximate relations are in satisfactory agreement with the integral representation of the coupling coefficients in the region of resonator coordinates of interest to us.
Fig. 3 shows the dependences of the coupling coefficients in the case of psendo-rotation of resonators: a2,fî2 = a1,p1 + a, where a = const, according to formulas (6,10,14,16). By psendo-rotation of resonators we mean rotation in which the relative direction of the axes of each resonator remains fixed. A change in the mutual coupling coefficients during psendo-rotation is associated with a change in the relative coordinates of the DR. In the latter case (Fig. 3, d, li), if the centers of the resonators are located on the axis of rotation, then the coefficients of mutual coupling remain constant.
Figures 4,5 show the dependences of the coupling coefficients upon variation of the coordinates of the second resonator, for different orientations of its axis. The reason for the appearance of asymmetry in the dependence in the case of y-rotation (Fig. 5, b, f ) is duo to the different spatial distribution of the vector fields of natural oscillations relative to the plane of symmetry x = 0 of the first resonator.
Fig. 3. Pseudo-rotations (a2, = a1, + 1) of rectangular DRs relatively the x-axis (a, e); y (b,f) and z (c, g); (d,h); (k0Ax = 0 k0Ay = 0 k0Az = 2) - green curves; (k0Ax = — k0Ay = 0 k0Az = 2) - blue curves;
(k0Ax = — k0Ay = 0 k0Az = 2) - red curves.
Fig. 4. Dependence of the DRs rotated about the x-axis (Fig. , a) on the distance ko Az (a, e); rotated relatively the y-axis (Fig. 2,b) (b.f); rotated relatively z-axis (Fig. 2.c) (c. g); rotated relatively z-axis (Fig. 2,d) (d.h): k0Ax = 1; k0Ay = 0; for ai,^l =0 a2,fi2 = 0 - green curves; a2,ft2 =0, 5 - blue curves;
®-2,fi2 = 1 - red curves
Fig. 5. Dependence of the coupling coefficients on the distance k0 Ax (a, d) for rotated DRs about the x-axis (Fig. 2.a) (a.e); rotated DRs relatively the y-axis (Fig. 2,b) (b.f); rotated DRs relatively z-axis (Fig. 2.c) (c.g); rotated DRs relatively z-axis (Fig. ,d) (d, h), (ko ( y1 — y2) = 0 ko Az = 2; a1,^1 = —n/4; a2,@2 = 0 - green curves; a2, = n /4 - blue curves; a2,fi2 = 3n/4 - red curves)
Discussion and Conclusion open space in the case of their rotations about one of
the axes of given coordinate system.
We obtained simple analytical expressions for the mutual coupling coefficients of rectangular DRs in the
The dependences of the coupling coefficients on the angles of rotation and spatial coordinates of resonators are considered.
The concept of psendo-rotation of resonators is introduced. The cases are investigated when the psendo-rotation of the resonators does not lead to a change in the values of the mutual coupling coefficients.
The formulas found have good accuracy and make it possible to calculate and optimize the electromagnetic parameters of complex mnlti-element filters, antennas and different metasnrfaces containing a large number of dielectric resonators much faster compared to numerical simulation methods.
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Коефщ1енти взаемного зв'язку оберто-вих прямокутних д!електричних резо-натор!в у вщкритому простор!
Трубгн О. О.
Розраховап! коеф!ц!епти взаемпого зв'язку д!елек-тричпих резопатор!в прямокутпо! форми у в!дкритому простор! при i'x o6epTaiuii щодо одше! з осей задано! прямокутпо! системи координат. Зпайдеп! вирази дають повпу шформацпо про частоти та добротпост! зв'язапих коливапь д!електричпих резопатор!в. Отримаш апал!ти-nni формули для комплекспих коеф!ц!епт!в взаемпого зв'язку. Розглядаються залежпост! коеф!ц!епт!в зв'язку в!д кут!в обертаппя i просторових координат резопато-р!в в раз! збуджеппя в них осповпих магштпих тип!в власпих коливапь. Введено попяття псевдооберташш ре-зопатор!в. Досл!джепо залежпост! коеф!ц!епт!в зв'язку при р!зпих видах псевдооберташш резопатор!в. В1д-зпачепо вршадки. коли псевдооберташш резопатор!в по призводить до зм!ш1 коефщ1ептв зв'язку. Вгшедепо нов! 1птегральш сшвв1дпошешш для коефщ!епт1в взаемпого зв'язку д1електричпих резопатор!в прямокутпо! форми при обертапш !х осей у в!дкритому простор!. В окре-мих вгшадках паралелыюст осей резопатор!в одп!й з коордипатпих осей, зпайдеп! в робот! апал!тичп! вирази зб!гаються з отримапими paninie. Для кожпого випад-ку обертаппя зпайдеш паближеш апал1тичп1 формули для отримапих в робот! штегральпих ствв!дпошепь. що
виражаються через сферпчш функци Ханкеля другого роду. Проведено пор!вняння обчислень коефцценлв зв'язку за штегральнпмп формулами \ наближеними ви-разами. Показано, що наближеш вирази мають ирийня-тну точшсть для вс1х розглянутих випадшв обертань. Розглянуто залежноста коефщ!ентав зв'язку в!д координат резонатор!в. В1дзначено облает!, в яких знайдеш штегральш вирази дозволяють коректно описувати ко-ефщ!енти зв'язку ирямокутних резонатор!в. На в!дмшу в!д штегральних сшвв1дпошепь, наближеш формули ко-ректш у веш просторовш облает! взаемоди резонатор!в. Отримаш результати дозволяють будувати апалиичш модел! антен, багатоелементних реппток \ пристрош ш-фрачервоного \ оитичного д!апазошв довжин хвиль, виконаних з заетоеуванням д!електричних резонатор!в ирямокутно! форми; значно скорочувати час обчислень в иор!внянш з чисельними методами \ оптим!зувати складш багаторезонаторш структури мшрохвпльових \ оптичних систем зв'язку.
Ключовг слова: коефкцепт зв'язку; коефщ!епт вза-емного зв'язку; обертання; прямокутний д!електричний резонатор; в!дкритий простар
Коэффициенты взаимной связи вращаемых прямоугольных диэлектрических резонаторов в открытом пространстве
Трубин А. А.
Рассчитаны коэффициенты взаимной связи диэлектрических резонаторов прямоугольной формы в открытом пространстве при условии их вращения относительно одной из осей заданной прямоугольной системы координат. Найденные выражения дают полную информацию о частотах и добротности связанных колебаний диэлектрических резонаторов. Выводятся аналитические формулы для комплексных коэффициентов взаимной связи. Рассматриваются зависимости коэффициентов связи от углов вращения и пространственных координат резонаторов в случае возбуждения в
них основных магнитных типов собственных колебаний. Введено понятие псевдовращения резонаторов. Отмечены случаи, когда псевдовращение резонаторов не приводит к изменению значений коэффициентов связи. Исследованы зависимости коэффициентов связи при разных видах псевдовращений резонаторов. Выведены новые интегральные представления для коэффициентов взаимной связи диэлектрических резонаторов прямоугольной формы при условии вращения их осей в открытом пространстве. В частных случаях параллельности осей резонаторов одной из координатных осей, найденные в работе аналитические выражения совпадают с полученными ранее. Для каждого случая вращения получены приближенные аналитические формулы для найденных в работе интегральных представлений, выражаемые через сферические функции Ханкеля второго рода. Проведено сравнение расчетов коэффициентов связи по интегральным формулам и приближенными выражениями. Показано, что приближенные выражения обладают приемлемой точностью для всех рассмотренных случаев вращений. Рассмотрены зависимости коэффициентов связи от координат резонаторов. Отмечены области, в которых найденные интегральные представления позволяют корректно описывать коэффициенты связи прямоугольных резонаторов. В отличие от интегральных представлений, найденные приближенные формулы корректны во всей пространственной области взаимодействия резонаторов. Полученные результаты позволяют строить аналитические модели антенн, многоэлементных решеток и устройств инфракрасного и оптического диапазонов длин волн, выполненных с применением диэлектрических резонаторов прямоугольной формы; значительно сокращать время вычислений по сравнению с численными методами и оптимизировать сложные многорезонаторные структуры микроволновых и оптических систем связи.
Ключевые слова : коэффициент связи; коэффициент взаимной связи; вращение; прямоугольный диэлектрический резонатор; открытое пространство
