CC BY
3
621.372
Rotations of Cylindrical Dielectric Resonators in a Rectangular Waveguide
Trubin A. A.
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine
E-mail: alrubin'J&gmail. com
The coupling coefficients of cylindrical dielectric resonators with a regular rectangular waveguide under the condition their axes rotation are calculated. The dependences of the coupling coefficients on the angles of rotation and transverse coordinates of the resonator in the case of excitation of the main magnetic types of natural oscillations in them are considered. The dependence of the coupling value on the angles of rotation at the points of circular polarization of the fundamental wave of a rectangular waveguide is shown. The condition for the angle of rotation of the axis of the resonator, determined by the dimensions of the cross section of the waveguide and the frequency of the main type of natural oscillations, is also established, when fulfilled, the coupling coefficient becomes constant in the transverse plane of symmetry of the waveguide. New analytical expressions are derived for the mutual coupling coefficients of identical cylindrical dielectric resonators when their axes rotate relative to a rectangular waveguide. The dependences of the mutual coupling coefficients on the angles of rotation and coordinates of the resonators are investigated. Conditions are found under which the mutual coupling coefficients of two cylindrical resonators are independent of their transverse coordinate in the plane of symmetry of the waveguide. The reasons for the change in the sign of the coupling coefficients of the resonators during their rotation are discussed. The effect of the emergence of coupling ext.rema for different relative orientations of dielectric resonators is noted. In particular cases of parallelism of the resonator axes of one of the coordinate axes of the waveguide, the analytical expressions found in the work coincide with those obtained earlier. The obtained analytical results make it possible to construct analytical models of bandpass and notch filters, significantly reduce the computation time and optimize complex multi-cavity structures of microwave and optical communication systems.
Key words: coupling coefficient: mutual coupling coefficient: rotation: cylindrical dielectric resonator: waveguide
DOI: 10.20535/RADAP. 2021.86.22-28
Introduction
The rotation of dielectric resonators (DRs) can be used to reduce the size of filters fl 3], improve the radiation parameters of antennas [4 9,11], and control the scattering characteristics of various metasurfaces [10, 12 14]. To calculate and optimize the scattering characteristics of these devices, it is necessary to study the coupling coefficients of dielectric resonators in various waveguides and open space. For rotating dielectric resonators, such a calculation of the coupling coefficients usually leads to extremely cumbersome analytical expressions. However, in some cases, obtaining analytical expressions is possible. The presence of the specified analytical formulas allows to establish previously unknown properties of the interaction of resonators with each other and the transmission line, as well as to build an electromagnetic scattering models on a large number of coupled elements.
1 Statement of the problem
The purpose of this article is to calculation and analysis of coupling coefficients of the cylindrical dielectric resonators in a rectangular propagating as well as a cut-off waveguide in the case of rotation of the resonator axes. Derivation of analytical formulas for the coupling coefficients of resonators with their possible rotation relative to one of the coordinate axes of a rectangular waveguide. Search for new patterns of change in the coupling of cylindrical dielectric resonators with the main magnetic types of resonances when they rotate relative to each other and a rectangular waveguide.
2 Coupling coefficients of rotation Cylindrical DRs with rectangular waveguide
To calculate the coupling coefficient of a resonator with a regular waveguide, we need to find the projection c± of the resonator fie Id (e, h) onto a propagating waves In the case of the main magnetic
type of natural oscillations H101, the field of a cylindrical dielectric resonator in the local coordinate system (p', ip', z') associated with the axis of the dielectric cylinder (Fig. 1) is represented as:
Fig. 1. Cylindrical dielectric resonator in the local cylindrical coordinate system (p',<p',z')
In the general case of rotations.
c± = i/2w(ei - £0) (e, (E±) )dv.
(2)
-hl ^ JlW Jsin &,
1 ß 1(HH )\cosßzz'j ,
(1)
The integral (2) takes on a very cumbersome form, therefore, in this work, we will consider only rotations relatively one selected coordinate axis of a rectangular waveguide.
In the case of propagating waves, the coupling coefficient of the resonator with the waveguide, taking into account the normalization, takes the form:
I c± I 2 \ct \
uW ,
here hi is the amplitude; w is the circular frequency; k =
p0 is the magnetic permeability; ) are the wave numbers ft2 + $2 = k2; k\ = w^p0elk0 = w^p0s0, where W- is the energy, stored in the DR material.
(3)
0
space and resonator, respectively.
The results of calculating the coupling coefficients (3) are most conveniently presented in the form:
a) in the case of rotation of the resonator axis about the x axis of the waveguide (Fig. , a)
k = 4À;ox2œcos2^2Cos2(xixxo)\u(xix, r sinß2, rcosß2)| ;
(4)
k =ko I (xix cos a1 ± r sin a1)v((x1x cos a1 ± r sin a1), 0, (x1x sin a1 ^ r cos a1))e +
' I 2
+ (X1x cos a1 ^ r sin a1)w((x1x cos a1 ^ r sin a1), 0, (x1x sin a1 ± r cos a1))e*XlxX° | ;
(5)
k = 4ko|r|2cos2a1sin2(x1x^o)|^(r,X1x sina1,X1x cosa1)|2,
(6)
where
pLJ1^sl + S2r^j Jo(p±) - ysX + syroJo(^J4 + s2yro) J1(PL) w*(sz)
( «X + « y)- p\
s X + s 2
complex conjugate symbol; Jn(z) - Bessel functions.
p
z
*
Fig. 2. Rotation of the dielectric resonator relatively the x-^is (a); y-axis (b); z-axis (c). Dependence of the DR coupling coefficient on the rotation angle (d): (y0 = 0, 56; 1 - x0 = 0, 2a; 2- x0 = 0, 3a; 3 - x0 = 0, 4a). Dependence of the DR coupling coefficient on the rotation angle a\ relatively the y-axis (e): (y0 = 0, 56; 1 -x0 = 7, 341mm; 2 - x0 = 0, 3a; 3 - x0 = 0, 4a). The dependence of the coupling coefficient of the DR on the angle of rotation « for the initial position z' || x (f): (y0 = 0, 56; 1 - x0 = 0, 2a; 2 - x0 = 0, 3a; 3 - x0 =0, 4a). Dependence of the coupling coefficient on the DR coordinate x0(y0 = 0,56) (g): (1 - =0, 1; 2 - #2 =0,6; 3 -= 1, 2); (h): (1 - «1 =0,1; 2 - «1 =0, 659; 3 - « = 1, 2); (i): (1 - «1 =0, 1; 2 - « =0, 7; 3 - «1 = 1, 2).
As can be seen from (7)
^X, Sy7
«y7 7 ^z
(8)
For basic magnetic oscillations ff101 ( ), (ev' = — J1(yS//)cos,0zz') the function ( ) also is even
in all arguments:
^y , ) Sx, ^y, ^z )
= w(Sx, Sy, — sz) = w(Sx, Sy, sz). (9) Here and below Xsx = s^/a; x«y = w^/6; r =
\J(Xsx)2 + (x«y)2 — ^o is the wave numbers of a rectangular waveguide with a cross section a x 6 (Fig. ,a); £1r = £1/e0; = ^r0; pz = ^zL/2 and g± = fc0r0; </z =fc0L/2 is the characteristic parameters; r0 - radius; L - height of the resonator (Fig. );
( Sz ) = -ttX
z ( z ) fZ — ( - zL/2)2 ( —i [pz cospz sin(s ZL/2) — szL/2 sinpz cos(szL/2)j 1 _ | [pz sin pz cos( szL/2) — szL/2cos pz sin( szL/2)j J'
2
fco = 4w;
(e ir - 1) L
£ir |r|a6 vo
vo = [Ji(p±) - Jo(p±)Ji(p±)] (2pz +sin2pz).
The results of calculating the dependences of the coupling coefficients on the coordinates and the angle of rotation of the resonator are shown in Fig. 2 for #ioi oscillations and £1r = 36; L/2r0 = 0,4; Zo = 7 GHz; a x 6 = 35 x 15 mm2. In Fig. , d,f shows the possibility of changing the coupling coefficient by rotating the resonator about the x mid y axes, as well as by varying the coordinate of the resonator center along the x-axis.
But the most nontrivial results are obtained by the rotation of the resonator about the y-axis (Fig. , e, h). In the first case (Fig. 2,h; e curve 1), we see that the coupling coefficient does not depend on the orientation of the DR axis at the points of "circular polarization" of the magnetic field of the waveguide:
x01 and
1 |r|
-ar cctg-,
Xix Xix
(10)
X02
a — xoi-
In the second case (Fig. 2,e), the coupling coefficient does not depend on the transverse coordinates
(Fig. 2. h curvo 2) when the axis of the DR is rotated by an angle:
|r|
ac = arcctg-. (11)
X1x
In limiting cases = 0, ±ir/2, expressions ( -
C) coincide with the coupling coefficients for standard positions of the resonator in the waveguide.
3 Calculating mutual coupling coefficients of rotation Cylindrical DRs in the rectangular waveguide
Calculation of the mutual coupling coefficients of identical cylindrical DRs with oscillations (1) in a cutoff rectangular waveguide leads to the expressions in the waveguide coordinate system (x,y, z) (Fig. ,a-c):
&12 = k
°2 ft t> tM
f!(Ti F)(/t2(± r))vr|z2-zi1,
(12)
here
ko2 = -2tT ,
12 o o
qjPz (e 1r - 1) L _
£ 1r
o
x
ft1 (Tr) =cos(x1x^n){[E*xo(xuy cos ßn T «r sinßn) - E*yoXsx cosßn + iE*oXsx sin^„]x x e-1 Xu»ynv(xsx, (Xuy cos ßn T ir sin ßn), (xuy sin ßn ± iV cos ßn)) + + [E*yo (xuy cos ßn ± tTsin^n) - E*oXsx cos ßn + iEyoXsx sin^njx x etx^ynv(xsx, (xuy cos ßn ± ir sin ßn), -(xuy sin ßn T ¿rcos^n)^ ;
(13)
ft (T ir) = cos(xuyVn){ [^XoXuy cos an - S*o(Xsx cos an T «r sin an) - iE^Xuy sin an] x x Xsxxnv((xsx cos an T «r sin an), Xuy, (Xsx sin an ± iV cos a„))+ + [E*yoXuy cos an - E*o(xsx cos an ± iT sin an) + iE^Xuy sin an] x xetXsœ'xnv((x.sx cos an ± iV sin an), xuy, -(xsx sin an T ^ cos a„)^;
(14)
ft (T i r) =[± i r(Eyo sin an + E*o cosan) - «(xsx sin an + xuy cosaJK^x
x sin(xsx^n + xuyUn )v( ¿r, (xsx sin an + xuy cosan), (xsx cosan - xuy« ina.n))-- [±«r^o sin an - E*yo cos an) - «(xsx sin an - xuy cos an)Eyo] x x sin(xsx^n - xuyVn)v(-ir, (xsx sin an - xuy cos an), (xsx cos an + xuySinari)),
(15)
where
Exo = «xuy hou; Eyo = -xsx |wfu| Ku
for H-waves:
Exo = Txsxeüsu; Eyo = Txuy¿ou; Ezo = x2/ |r| e^ for E-waves of the waveguide:
ho = -
h su = s u x
|r|
(¿H^ah
1/2
(1 + ¿so + ¿u0 )1/2:
e s = 2 6su x
|r| •
we^ab
1/2
(1 - ^o - ¿uo)1/2;
= wp/r; x = [x2sX + xUy]1/2;
) is the coordinates of the center of the n-th resonator in the waveguide coordinate system; ¡3n, an is the angles between the z'-axes of the local coordi-
x n
1
x
waveguide. Dependence of mutual coupling coefficients on the DR longitudinal coordinate A z = | z1 — z2| for y1 = y2 = 0,56; (d) = 0; 1 - y32 = 0; 2 - ,02 = n: (e, f) «1 = 0; 1 - «2 = 0; 2 - «2 = n. Dependence of the
x1 = x2 = 0, 5 a 1 = 2 = 0, 5 A = 10 - «1 = 0; 2 - «1 = n/4; 3 - «1 = n/2. Dependence of the coupling coefficients on the rotation angle (0) = (k-1 ) «1 = «2) of the DRs for x1 = 0, 25a; y1 = y2 = 0, 56; Az = 10 mm, and (j-1) 1 - x2 =0, 25a;
x2 = 0, 5 a x2 = 0, 75 a
4 Analysis of mutual coupling coefficients
Relations (12-15) make it possible to calculate the mutual coupling coefficients of the DRs in the cases of the considered rotations of their axes. The found expressions in particular cases when the axes of the resonators become parallel to one of the coordinate axes of the waveguide coincide with those obtained earlier.
In Fig. 3 shows the dependences of the coupling coefficients on the coordinates and angles of rotation of the resonators. As follows from the results obtained, the sign of the coupling coefficient can change with a smooth change in the structure parameters. This
reason has a twofold nature: purely mathematical due to a change in the direction of the field in one of the resonators and physical, due to different values of the coupling coefficients for different relative positions of the resonators in the waveguide. An example of a
purely mathematical reason for a change in the sign
'
of the resonators changes its direction by an angle n, the direction of the field changes sign with respect to the direction of the local coordinate system. In this case, the structure remains the same, but the sign of the coupling coefficient changes to the opposite (see for example curves 2 in Fig. 3,d-f).
In the second case, as is known, the sign of the coupling is determined by the mutual positi-
on of the resonators. For coaxial arrangement of resonators if Az > L and for example Ap =
\J(xi - X2)2 + (yi - i/2)2 < 2ro, then fen < 0 (see curves 1 in Fig. 3,d-e).
If the axes of the resonators are parallel, for example, the x axis and Az > 2ro, then fci2 > 0 (see curve 1 in Fig. 3,f). This is a purely physical property of coupled oscillations, determined by the nature of the spatial distribution of the resonator fields.
In Fig. 3,g-l shows the calculated dependences of the coupling on the mutual orientations of the axes of the DR and waveguide. The effect of the independence of the value of the mutual coupling from the transverse coordinate of the second resonator at a certain angle of inclination of their axes, shown in Fig. 3,.j, in the case x
As follows from the data shown in Fig. 3,g-k, for each resonator 1 position, there is an optimal resonator 2 position at which the coupling coefficient between them reaches a maximum.
Discussion and Conclusion
In this paper, we obtained analytical expressions for the coupling coefficients, as well as the mutual coupling coefficients of cylindrical DRs in the case of their rotation about one of the axes of a rectangular waveguide.
1) The conditions are established under which the coupling coefficient with the main propagating wave
io
"circular polarization" of the magnetic field does not depend on the angle of rotation of the resonator (10).
2) It is shown that the coupling coefficient with
io
transverse coordinates of the DR if the resonator axis is rotated relative to the axis of the waveguide in a plane parallel to the wide wall by an angle determined by the dimensions of the waveguide and the frequency of
ioi
formula is given for determining the angle of rotation of the resonator (11).
3) The existence of angles of rotation is established at which the coefficients of their mutual coupling do not depend on the transverse coordinates of cylindrical DGs in a rectangular waveguide (Fig. 3,.j).
4) For each resonator 1 position, there is an optimal resonator 2 position at which the mutual coupling coefficient between them reaches a maximum.
The proposed theory can be used to calculate and analyze complex band-pass or band-stop filters, multiplexers, and other communication devices built on dielectric resonators in the microwave, infrared and optical wavelength ranges.
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Обертання ци л i нд p и ч них д!електри-чних резонатор!в в прямокутному хви-левод!
Трубгн О. О.
Розраховаш коефщ1епти зв'язку д1електричпих ре-зопатор!в цилтдричло! форми з рогулярпим прямоку-тпим хвилеводом за умови обертаппя i'x осей. Розгляпу-то залежпоста коефщ1еттв зв'язку в!д кутав обертаппя i поперечпих координат резонатора в раз! збуджеппя в них осповпих магштпих тишв власпих коливапь. Показана пезалежшсть величипи зв'язку в!д кутав обертаппя в точках кругово! иоляризацп ociiobiioi хвшн прямоку-тпого хвилеводу. Також встаповлепо умови для кута обертаппя oci резонатора, визпачаемого розм!рами поперечного перер!зу хвилеводу i частотою основного типу власпих коливапь, при викопапш якого, коефкцепт зв'язку стае постайпим в поперечшй площиш симотрп хвилеводу. Виведепо нов! апалггичш вирази для кое-фщ1ептав взаемпого зв'язку одпакових д!електричпих резопатор!в цилтдричло! форми при обертапш i'x осей щодо прямокутпого хвилеводу. Досл1джепо залежпоста коефкцептав взаемпого зв'язку в!д кутав оберпешш та координат резопаторш. Встаповлепо умови при вико-naiuii яких коефщ1епти взаемпого зв'язку двох цилт-дричпих резопатор!в по залежать в!д i'x поперечно! коор-дипати у площппн симотрп хвилеводу. Обговорюються причшш змпш зпашв коефщ1ептав зв'язку резопато-р!в при i'x обертапш. В1дзпачаеться ефект вшшкпеппя екстремум!в зв'язку для р1зпих в1дпоспих ор!ептацш д1електричпих резопатор!в. В окремих випадках пара-лелыюста осей резопатор!в одше! з коордипатпих осей хвилеводу, зпайдеш в робот! апал!тичш вирази зб!гаю-ться з отримапими рашше. Зпайдеш апал1тичш резуль-тати дозволяють будувати апалиичш модел! смугових i режекторпих фгльтр1в, що зпачпо скорочуе час роз-рахупк!в та оптим!зацп складпих багаторезопаторпих структур мшрохвильових i оптичпих систем зв'язку.
Ключовг слова: коефцрепт зв'язку: коефщ!епт взаемпого зв'язку: обертаппя: цилшдричпий д1електричпий резонатор: хвилев!д
Вращения цилиндрических диэлектрических резонаторов в прямоугольном волноводе
Трубин А. А.
Рассчитаны коэффициенты связи диэлектрических резонаторов цилиндрической формы с регулярным прямоугольным волноводом при условии вращения их осей. Рассмотрены зависимости коэффициентов связи от углов вращения и поперечных коордгшат резонатора в случае возбуждения в mix основных магнитных тгшов собственных колебаний. Показана независимость величины связи от углов вращения в точках круговой поляризации основной волны прямоугольного волновода. Также установлено условие для угла вращения оси резонатора, определяемое размерами поперечного сечения волновода и частотой основного типа собственных колебаний, при выполнении которого, коэффициент связи становится постоянным в поперечной плоскости симметрии волновода. Выведены новые аналитические выражения для коэффициентов взаимной связи одинаковых диэлектрических резонаторов цилиндрической формы при вращении их осей относительно прямоугольного волновода. Исследованы зависимости коэффициентов взаимной связи от углов вращения и координат резонаторов. Установлены условия при выполнении которых коэффициенты взаимной связи двух цилиндрических резонаторов не зависят от их поперечной координаты в плоскости симметрии волновода. Обсуждаются причины изменения знака коэффициентов связи резонаторов при их вращении. Отмечается эффект возникновения экстремумов связи для различных относительных ориептаций диэлектрических резонаторов. В частных случаях параллельности осей резонаторов одной из координатных осей волновода найденные в работе аналитические выражения совпадают с полученными ранее. Полученные аналитические результаты позволяют строить аналитические модели полосовых и режекторпых фильтров, значительно сокращать время вычислений и оптимизировать сложные мпогорезопа-торпые структуры микроволновых и оптических систем связи.
Ключевые слова: коэффициент связи: коэффициент взаимной связи: вращение: цилиндрический диэлектрический резонатор: волновод
