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УДК 621.372
COUPLING COEFFICIENTS OF DIFFERENT CYLINDRICAL DIELECTRIC RESONATORS IN THE OPEN SPACE1
Trubin A. A., Doctor of Engineering, Professor
National Technical University of Ukraine "Kyiv Polytechnic Institute ", Kyiv, Ukraine [email protected]
КОЕФЩДеНТИ ЗВ'ЯЗКУ Р1ЗНИХ ЦИЛ1НДРИЧНИХ Д1ЕЛЕКТРИЧНИХ РЕЗОНАТОР1В У В1ДКРИТОМУ ПРОСТОР1
Tpyôin О. О., д.т.н., професор
Нацюнальний техтчний утверситет Украгни "Кшвський полтехтчний iнститут ",
м. Кшв, Украгна,
Introduction
Cylindrical DRs are applying today in various microwave devices [1 - 6]. For calculation and optimization of such devices is more convenient to use electrody-namic modeling with sufficient accuracy [7, 8]. Calculation of the systems, containing multiple DRs in the open space, is based on computation of mutual coupling coefficients. The coupling coefficients of the Cylindrical DRs did not studied in full measure even in case of the lowest modes. The goal of the present work is the calculation and analysis of the Cylindrical DR coupling coefficients with main modes, located in the open space, in the general case of its arbitrary spatial orientation as well as for cases of various shapes and materials.
Coupling coefficient calculating
Allocation microresonators side by side with each other leads to the coupling oscillations appearance. The fields and frequencies of the DR oscillations are defined by values of the mutual coupling coefficients. In the common case, the coupling coefficient can be determined as a surface integral [9]:
Ksn
2co0wn(l + ÔJ
^{[ës,h;]+[ë;,hs]}nds, (i)
expressed via the eigenmode field (es,hs) of one (s-th) DR on the surface of another (n -th) DR. Here s,n = 1,2; and n - is the normal to the surface sn of n -th DR, w0 - is the resonance frequency of the resonators; wn - is the energy, stored in the dielectric.
s
n
1 http://radap .kpi.ua/radiotechnique/article/view/1223
Fig. 1. A - sketch of two different Cylindrical DRs in the AA position in Open space. The real part of the coupling coefficients K12 (solid lines); K21 (dotted lines) (b, d - f) of two Cylindrical DRs with H+01 modes; (m = 0): slr = 36; s2r = 81; \ = L !2rx = 0,4;
A2 = L / 2r2 = 0,8. Imaginary part of the coupling coefficients versus coordinates of the DR centers (c, g - i). B, c, f, i: k0Ax = k0Ay = 0; d, g: k0Ay = 0; k0Az = 1,1k0(r + r2) ; e, h:
koAx = 0; kAz = 1,1k0(r + r2) .
As follow from (1), in the general case of unequal DRs: ku ^ k22 ; k12 ^ Ki. A direct calculation of the integral (1) in most cases is not possible, so we will use the well-known expressions for the mutual coupling coefficients found for the DR in a rectangular waveguide [10]. Then the same coupling coefficients for the open space can be simply obtained using the integral transformation of the known analytical expression based on assumption that the transmission line metal walls
have been "removed" to the infinity.
Using necessary expressions, for example, 5.2 of the [10], as well as integrals [9, 11], after simplifications obtain:
For the AA position (see fig. 1, a) in the case of two different Cylindrical DRs, with mode И+0Д, the mutual coupling coefficients can be obtained in the form: in the area: Az > r + r2:
ku = к0 • • ф2(л/Т^Г)*И2)(k0Ap^/Г-^2)cos(k0; (2)
( Ap = ^Ay2 +Az2 ) For the AB position (fig. 2, a) in the area: Az > r + r2:
K1,2 = K0 * (3)
. ^bplX sin(k0 Ax^)sin(k0 Ayл) • 5л • d№; j0 J0 У л/Л2 +Y2 V^2 + У2
For the AC position (see fig. 3, a) in the area: Az > r + L2 / 2:
K1,2 = K0 * (4)
q>i(Vл2 + У2), Ф2(У52 + л2)* /л2 + У2 v^2 +Л2
(52 +л2 +у2 = 1);
For the CC position (see fig. 4, a) in the area: Az > Ц / 2 + L2/2:
e-iyk0Az
K1 2 =K0 * L -Ф1(к) -Ф2(к)* J0(k0APK)KdK , (5)
J0 у
( к2 +у2 = 1; Ap -^Ax2 +Ay2 );
where; Ax - xl - x2; Ay - y - y2; Az - |zx - z2\; (xs, ys,zs) - are the rectangular
coordinates of the DR' centers;
8ir P2 P2z (eu - 1)(s2r -1) . K0 - TT~, ; (6)
V2 r2 P1 k0 e2r
v2 = [J12(P2±) - J0(P2±)J2(Pl±)](2P2z + sin2Plz) .
•ХГХГe-iykoAz Ф7 ^ +у J • Ф275 + л J sin(k0 Ax^)cos(k0 Ayл) * 5 * d^;
j0 j0 л/л2 +у2 V52 + л2
Fig. 2. AB position of two Cylindrical DRs in Open space. Mutual coupling coefficients K12 (solid lines); K2i (dotted lines) (b - e) as a function coordinates of the DR centers with
elr= 36 ; e2r= 81 ; A1 = Ll/2rl = 0,4 ; A2 = L2/2r2 = 0,8. B, d: k0Ay = 2 ; k0 Az = 1,1k0(r + r ) ; c, e: k0 Ax = k0 Ay = 2.
Here
[— J0(p s! )J1(q s! s!
)J0(q
s!
qs!
[(^)2 -^2]
qs!
(7)
[pz sin psz cos qs^1 -q2 -yj1 -q2 cos psz sin -q2]
Qs
(psz)2 - (Vw)2
qsz
and psl=Psrs; pSz =PSzLs/2; qsi= k0rs; Qsz = k0Ls/2; Ps, Psz - are the wave numbers of the eigenoscillation field of the s -th Cylindrical DR [10]; r - is the radius, L - is the height of the s -th DR (s = 1,2); H(2)(P0p) are the Hankel functions of the second kind; Jn (x) - are the Bessel functions of the first kind of the
n -th order; kj = ; k0 = «0 / c; ©0 - is the circular resonance frequency; c -
is the light velocity; ssr - is the relative dielectric permittivity of the s -th DR' material.
Fig. 3. Position AC of the Cylindrical DRs in Open space. Mutual coupling coefficients K12 (solid lines); K2i (dotted lines) (b - e) as a function coordinates of the DR centers with
elr= 36; e2r= 81 ; A1 = L1/2r1 = 0,4 ; A2 = L2/2r2 = 0,8. B, d: k0Ay = 0; k Az = 1,3k0(r + L / 2) ; c, e: k0 Ax = 2 ; k0 Ay = 0.
The integral convergence provides by choice of the radical signs for E > 1 :
VT-^2 = -i>/E2 -1 in the (2 - 4), as well as for k> 1: -k2 = -iVk2 -1 in the (5).
Note also that the function 9 (q) has no singularities in the region 0 < q < œ. It makes a major contribution to the integrals (2-6) only for small values q. Expand the function fs(q) = 9S (q) / q in a series of q at 1 and taking into account that f (1) = 9 (1) obtain:
9s(q) = 9s(1) -q+y-
dq
9s(q)
q
q(q-1)+•
q=1
Next, using the approximation:
9s(q) ~9s(1) -q.
we obtain
(8)
Fig. 4. CC position of two Cylindrical DRs in Open space. Mutual coupling coefficients (b -g) as a function coordinates of the DR centers with slr - 36 ; s2r - 81 ; \ - L !2rx - 0,4 ;
A2= L/2r -0,8. B, c, e, g: k0Ax -k0Ay = 0; d, f: k0Az = 1,7k(L + L)/2.
'Ap^ 2 v Ar y
For the AA position:
Ki2 -<Pi(l)<P2 (!)*(
( Ap-^Ay2 +Az2 )
For the AB position:
ku=-k0 - cp1(l)cp2(l) For the AC position:
h02)(koAr) - [1 - 3 — ] VAr )
^AxY2(2)(k0Ar). .
k0Ar
} ; (9)
AX^h22)(k0Ar),
(10)
k1;2 -k0 .(^(1)^(1)*——h<2)(k0Ar); (11)
AxAz Ar
For the CC position:
Ar
K.^^-cp.dMdni^l hif>(k0Ar)-[i-3Î^l ]h\(koAr)}; (12)
Ar
k Ar
V ¿A1 J \ ¿A1 J JS.QZ
( Ap = ^Ax2 + Ay2 )
In the (9 - 12) Ar = ^Ax2 + Ay2 +Az2 ; h(2)(z) - are the spherical Hankel functions
of the second kind [12]. Here we have also used the generalized Sommerfeld's integral [11].
Found relations allow us to calculate the coupling coefficients of different Cylindrical DRs in the open space. It is interesting that at the same time restrictions on the range DR coordinates (2 - 5) are removed. In the particular case of identical DRs the relations (2, 5) coincide with [13].
Coupling coefficient analysis
Fig. 1, 4, b - c shows the dependence of the coupling coefficients of the DR center coordinates, calculated according to the formulas (9, 12) (solid curves) as well as the numerical formulas (2, 5) (dashed curves). As can be seen from these curves, the use of approximation (8) gives a very good accuracy.
It is easy to verify that the coupling coefficients found (9 - 12) are proportional to the respective magnetic field components of the first resonator in the axis of symmetry direction of the second resonator.
Given this observation, we can assume that, in general, mutual coupling coef-
H+
ficient of two different cylindrical DR with the mode 101 will be represented as:
Ki,2 -(PiQ^C1)* •(h1(Ar,ei,(p1),ii2) (13)
where h, (Ar, 0,, cp,) - is the magnetic field of the first DR in the center of the second one; n2 =n2(02,(p2)is the unit vector directed from the DR center along the axis of second DR (fig. 5, a).
The relation (13) is exactly the same (9-12) in the case of AA; AB; AC and CC DR position. Fig. 5, b - e shows mutual coupling coefficients as a function of arbitrary relative DR orientation.
Fig. 5. General position of the Cylindrical DRs in the Open space. Mutual coupling coefficients (b - e) as a function of the relative DR orientation: k0 Ar = 3; b, c: ^ = = 0;
curve 1: ^ /2; curve 2: 0j /4; curve 3: 01 /8; d, e: ^ = 0; 02 /4; curve 1:
0!=^:/2; curve 2: 0j=tc/4; curve 3: /8.
Conclusions
Analytical relationships for mutual coupling coefficients for the H1+,0,1 modes
of different Cylindrical DR in the Open space has been obtained and investigated.
It stated, that mutual coupling coefficients are determined by the dependencies on the DR magnetic field and relative orientation of the DR axes.
The resulting ratio can be used for calculations of the DR natural oscillations, as well as the scattering parameters of the various element gratings in the communication devices with dielectric resonators.
References
1. Kajfez D. and Guillon P. (1998) Dielectric Resonators. Noble Publishing Corporation, Atlanta, 561 p.
2. Ueda T., Lai A. and Itoh T. (2007) Demonstration of Negative Refraction in a Cutoff Parallel-Plate Waveguide Loaded With 2-D Square Lattice of Dielectric Resonators. IEEE Trans. onMTT, Vol. 55, No. 6, pp. 1280-1287. DOI : 10.1109/tmtt.2007.897753
3. Kishk A. A. (2005) Directive Yagi-Uda dielectric resonator antennas. Microwave and optical technology letters, Vol. 44, No. 5, pp. 451-453. DOI: 10.1002/mop.20664
4. Benson T. M., Boriskina S. V., Sewell P., Vukovic A., Greedy S. C. and Nosich A. I. (2006) Micro-Optical Resonators_ for Microlasers and Integrated Optoelectronics. Frontiers in Planar Lightwave Circuit Technology, No 216, pp. 39. DOI : 10.1007/1-4020-4167-5_02
5. Preu S., Schwefel H. G. L., Malzer S., D' ohler G. H., Wang L. J., Hanson M., Zimmer-
man J. D. and Gossard A. C. (2008) Coupled whispering gallery mode resonators in the Terahertz frequency range. Optics Express, Vol. 16, No. 10, pp. 7336-7343. DOI: 10.1364/oe.16.007336
6. Semouchkina E. A., Semouchkin G. B., Lanagan M. and Randall C. A. (2005) FDTD Study of Resonance Processes in Metamaterials. IEEE Trans. onMTT, Vol. 53, No 4, pp. 14771487. DOI: 10.1109/tmtt.2005.845203
7. Trubin A. A. (1987) Perturbation theory analysis of the System Coupling Dielectric Resonators. Vestnik Kievskogo _politekhnicheskogo instituta. Radiotekhnika, No 24, pp. 42-45. (in Russian)
8. Trubin A.A. (1997) Scattering of electromagnetic waves on the Systems of Coupling Dielectric Resonators. Radioelectronics. No 2, pp. 35-42.
9. Trubin A. A. (2016) Lattices of Dielectric Resonators, Series in Advanced Microelectronics Vol 53, Springer International Publishing, 159 p. DOI : 10.1007/978-3-319-25148-6
10. Ilchenko M. E. and Trubin A. A. (2004) Electrodynamics' of Dielectric Resonators, Kiev, Naukova Dumka, 265 p. (in Russian)
11. Trubin A. A. Several specified integrals of special functions, Available at : http://eqworld.ipmnet.ru/ru/auxiliary/Trubin-EqWorld-2010_02_06.pdf
12. Abramowitz M. ed. and Stegun I. (1964) Handbook of mathematical functions. National bureau of standards.
13. Trubin, A. A., Shmyglyuk, G. S. (2006) The modeling of the antenna lattce parameters on cylindrical dielectric resonators. Visn. NTUUKPI, Ser. Radioteh. radioaparatobuduv., no. 33, pp. 101-108. (in Ukrainian)
Трубт О. О. Коефценти зв'язкур1зних цилтдричних д1електричнихрезонатор1в у в1дкритому простори Приведено результати розрахунк1в коефщент1в взаемного зв'язку цилтдричних д1електричнихрезонатор1вр1зних в1дноснихрозм1р1в з магнтними типами коливань. Знайдено загальна формула для коефщент1в взаемного зв'язку для довшьног ор1ентацИ цилтдричних ДР у в1дкритому простор1. Розглянут1 основт зако-ном1рност1 змти зв'язку при вар1аци в1дносного положеннярезонатор1в.
Ключов1 слова: цилтдричт д1електричт резонатори р1зних розм1р1в; коефщенти зв'язку; магштш типи коливань.
Трубин А. А. Коэффициенты связи различных цилиндрических диэлектрических резонаторов в открытом пространсте. Приведены результаты расчетов коэффициентов взаимной связи цилиндрических диэлектрических резонаторов различных относительных размеров с магнитными типами колебаний. Получена общая формула для коэффициентов взаимной связи для произвольной ориентации цилиндрических ДР в открытом пространстве. Рассмотрены основные закономерности изменения связи при вариации относительного положения резонаторов.
Ключевые слова: цилиндрические диэлектрические резонаторы различных размеров, коэффициенты связи, магнитные типы колебаний
Trubin A. A. Coupling coefficients of different cylindrical dielectric resonators in the open space. The calculation results of the mutual coupling coefficients of different relative sizes cylindrical dielectric resonators with magnetic modes are presented. A general formula for the mutual coupling coefficients for an arbitrary orientation of the cylindrical DRs in the open space are obtained. The basic patterns of coupling coefficient changing with the variation of the relative position of the resonators are examined.
Keywords: different cylindrical dielectric resonator, coupling coefficient, magnetic mode.
