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УДК 621.372
Scattering of Electromagnetic Waves on Different Dielectric Resonators of the Microwave Filters
Trubin A. A.
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
E-mail: [email protected]
The theory of scattering of electromagnetic waves by systems of various coupled the different shape and variant permittivity dielectric resonators is expanded. A new definition of the coupling coefficients of different dielectric resonators is given. The analytical expressions of coupling coefficient of different cylindrical and spherical dielectric resonators made from different dielectrics are obtained. The main regularities of the change in the coupling with the variation of the structure’s parameters are considered. The results of calculation of the transmission and the reflection coefficients for bandpass and bandstop filters on various dielectric resonators in the rectangular and circular waveguides are presented. Most optimal configurations, allowing to achieve the best scattering characteristics are determined.
Key words: different dielectric resonator; coupling coefficient; S-matrix; bandstop filter; bandpass filter
Introduction
It is well-known that in addition to a very high Q factors the dielectric resonators (DR) have a number of disadvantages, such as increased density of the spectrum, in some cases non-optimal coupling. The scattering parameters of a variety of devices can be significantly improved by using different forms of DR [1-8]. For the purposes of theory development all resonators are generally supposed to have the same shape and manufactured of the same dielectric [9]. In order to improve the parameters in some cases there is need to build filters on DRs with different shapes made of variant dielectrics. However, in this case the theory describing scattering processes becomes more complicated. In this article we developed electrodynamic theory for describing different DRs. The equation system for the unknown amplitudes of the DR coupled oscillations have been obtained. Total analytical solutions have been found. The research of electromagnetic wave scattering on different cylindrical and spherical DR structures have been conducted in the propagating waveguide and evanescent waveguide segment. In special case of identical resonators obtained solutions are simplified to the known ones [9].
1 Statement of the problem
The goal of the current article is the development of the theory of microwave filters, consisting of different DRs, that can be used in the modern communication systems.
2 Scattering theory
Consider the system of different DRs, consisting of different materials. We assume that the eigenosci-llation field of each isolated DR is known: (es,hs), (s = 1,2,..., N). Here es — is the electric field and hs — is the magnetic field intensity of the s-th isolated DR. The eigenoscillation field of the Ж-DR system (e, h) can be found as a superposition of fields of the isolated resonators:
N ^ N
e = J2 bses; h = ^ bshs (1)
s=1 s=1
As shown, the DR amplitudes bs should satisfy the equation system [10]:
N
J2 (ksn + iksn)bs - A6„ =0, (n =1, 2,..., N) (2)
S=1
where
A = 2 • (5ш/ш0 + iui''/шо), (3)
ш0 = Re[ws] is the real part of frequencies of isolated partial resonators (s = 1, 2, ...,N); 5ш = Re(ck — w0); w” = Im(ck); со — is the complex resonance frequency of the DR system; ksn, ksn are the coupling coefficients of the s-th and n-th DRs on damped and expanding waves of the transmission line [10] respectively:
к
sn
к
sn
-1
UQWn
E (Е)о(сГ)0е
t>tM
FAz.
1
u0wn
E (Е)о(Е)0е
t<t M
■iFAz
(4)
Here tM — is the ultimate multi index, determining the numbers of the expanding waves in the line, and a
(c®±)o — is the expansion coefficient of the s-th DR field on the t-th wave of the transmission line [10], calculated in the coordinate system, associated with s-th resonator center; Az = \zs — zn\; zs — is the longitudinal coordinate of the s-th DR; Г — is the longitudinal wave number of the transmission line; and
wn = 1 f (e„\ei„\2 + Mo|^in| )dv — is the energy, vn
stored in the dielectric of the s-th DR; e„ = Ree„; en = £n — *e" — is the complex dielectric permittivity and the ^0 is the permeability of the n-th DR.
In the case of different DRs the coupling coefficients (4) take different views: ksn = kns mid ksn = kns.
Generally, by providing the solution to the equation systems (2) for each obtained Xv, it is possible both to calculate approximately the complex frequency w” of the system coupling oscillation and to determine all amplitudes of partial resonators bv = (b\,b^2,...,bvN), (a = 1, 2 ,...,N).
The problem solution of the waveguide wave (^E+, j scattering on the DR system will be
searched in the form of expansion on coupling modes of the DR lattices (1):
^ ^ _ N E « E+ + E aa&; н « H+ +52 «shs, (5)
S=1 S=1
where (e®, hs) is the field of the A-DR systems (1), corresponding to the eigenvalue As (3).
By using perturbation theory, after the volume integration of each partial resonator, the equation system with the unknown coefficients as has been obtained in the form:
N
52 asbstQst(ix>) = —
s= 1
= _ К
(t =1, 2,..., N), (6)
where for different resonators, the functions Qst(ш) are dependent on the partial DR and the coupled oscillation numbers:
Qst(iv) = 2i^-^~ Q? + —, (7)
шо шо
Q® = iv0wt/PtD; PfD = ш0f \dt\2dv — is the loss
vt
power in the dielectric of t-th DR.
The transmission T and the reflection coefficient R of different DR system in the transmission line can be obtained froms (5), s (1) in the form:
N ( N
т = To + E E ^4 )au
u=1 \s = 1 N f N
R = Ro + E E busc- lau
U=1 \s = 1
R0 B(lo) E (ш)
S = 1 N
R0— вщ E в-(ш)
S = 1
Here T0, R0 are the transmission and reflection coefficients of the transmission line without DRs;
B±(P =
= det
Ъ1Яц(ш)
b^Qn(^)
bpjQ1N (Р
N
Q? E bsuk2+
u=1
N
Q? E ^и^и2+
U=1
N
Qn E bsu
u=1
... b?Qrn(u) ... b25Qn2(+)
... b^Qnn(Р
B(w) = det
Ъ1Яц(ш)
HQ 12 (и)
b5 Qn 1(^) b5 Qn 2(u)
b]^ Q1N (n>) ... b% Qnn (n>)
Kn = (c+ cri*)/(u’0+n) = (ksn)0e гГ(5 Zn); ks+ = (c~ct*)/(u0Wn) = (ksn)0^iT{Zs+Znh (ksn)0 — is the coupling coefficients (4), for the propagating wave, expressed without phase difference accounting in the transmission line (s, n = 1, 2,..., N).
3 Coupling coefficients of the different DRs calculation
In order to determining the S-matrix parameters: S21 = S21(v) = 20lg \TH\; G21 = G21H = arg[T(и)]; E1 = E 1p) = 20lg \ДН\; 1 =
С11(ш) = arg[R(w)], we have to calculate coupling coefficients of different DRs in the transmission line. Suppose we have two DRs of cylindrical shape with radius r1 and r2, height of L^id L2, respectively. Assume that each resonator is excited in the fundamental magnetic oscillation H+01 [10]. In this case, the coupling coefficient is of the form:
h 2
64n^2z П ft2 (к2 — к%)(к2 — к%)
V2
Г2 ftl
ЩаЪ
t (mx~
4x2
uy \ -TAz
1 + du
\Г\
)e
nsin XsxXt cos Xuyyt . (ft — xly + Г2)
t=1
X2 (Xly — Г2)
J0 (ptL )J1(rt^xly — —
\J xly — Г2 J1(Pt±)J0(rt^xly — Г2)
J 5lz — x2x
a ■ ■ Lt
Ptz sin Ptz COS Xsx 2 Xsx COS Ptz sin Xsx 2
(10)
1
1
In the case of coupling on propagating rectangular waveguide wave H1o:
к12
32n^2z n @2 (fcj - k°)(k° - feo) V2 T'2 Pi ЩаЬ
^—iTAz
Г
2
•П
t=1
sin Xsx^t
№ - Г2)
PtJo(pt±)Ji (Г rt)
- Г -h(pt±)Jo(Trt)
1
Hi - xl
■ Ptz sinptz CQSXsx
Lt
2
Lt
Xsx cosPtz sin Xsx Y
!■
(H)
where v = [J°(p2±)-Jo(p2±)J2 (P2±)](2p2z+sin2p2z);
ptx = Pt^t', ptz = PtzLt/2’1 kt = uj^lZtpo] is
the dielectric permittivity; p0 is the permeability and (Pt, Ptz) are the wave numbers of the t-th DR [10];
Xsx sn/a] xuy и'к/Ъ] x уX'zx + x'iy-. Г is
the longitudinal wave number; a x b — is the crosssectional dimensions of the waveguide; (xt, yt, zt) — are rectangular coordinates of the t-th DR (t = 1, 2); here Az = Izi - z21.
Fig. 1. Two different cylindrical (a) spherical (c) DRs in the propagating metal rectangular waveguide (e1r = 36;£2r = 82). Dependence of mutual coupling coefficients versus the distance between the cylindrical (b) or the spherical (d) DR centers (x1 = x2 = a/2; У1 = У2 = b/2; a = 58mm; b = 25 mm; f0 = 4GHz): Ai = L1/2r1 = 0,2 Ao = L2/2r2 =0,8 (b).
Example of mutual coupling coefficients calculations for two different cylindrical DRs in the rectangular waveguide, obtained on a basis of (10), (11), is showed in fig. 1 a-b. As follows from (10), the difference between the values of the coupling coefficients is mainly due to the different value of stored energy in the resonator material.
The use of different spherical DRs in some cases allow to improve the filter parameters. We have represented the coupling coefficients of different spherical DRs with magnetic oscillations H111 in the form:
ku = af (jp1,q!, p2,q2 )Р, (12)
where
(p1,q!,p2,^2)
31(p1) 31(p2)
У1(Ч1) Ч2У1(Ч2) 1
{[p2 - 2)]j°(p2) + [71^2) - p2io(p2)]2}’
T1 = 24a
1 ЩаЬ
{TO
£
s,u=(0)
|Г| x:
koX>
ly ”
^o(1 + buo)
|Г|
-TAz
1 2 \
— ^ sin XsxXt cos XuyPt 7, (13)
X t=1 J
e
(xt,yt,zt) — are rectangular coordinates of the t-th spherical DR centers in the waveguide: Az = |m - z2\i jn(z); yn(z) are the spherical Bessel and the Neumann functions, respectively [11]. The characteristic parameters pt = ktrt, qt = kort can be obtained from the equations for natural oscillations of the t-th spherical DR [12]. Here rt — is the radius of the t-th spherical DR.
For the coupling on propagating wave H1o:
кю
_12 X1xT
kob k°
af (p1,q1; p2,®)
_)ГД 7 . .
■ e sin X1zX1 sin X1x'X2.
(14)
The mutual coupling coefficients, calculated for two different spherical DRs in the rectangular waveguide, obtained from (12) - (14), are showed in fig. 1 c-d. As can be seen, the difference between the values of k12 and k21 in this case is small.
If two different spherical DRs located on the axis of the cylindrical metal waveguide [12], the mutual coupling coefficients becomes:
kn = af (p1 ,qi, p2,q2)F1, (15)
where
Fl = 3'
£j v bft)
-Гн Az
[1 + (Гв/ko)2] e
-Г e Az
Wt - 1)J1(jvs)2 |Г^/fco| \j1,sj'1(j1,s)i
f
2
e
Гh,E — is the longitudinal wave number for the magnetic, electrical cylindrical waveguide waves, respectively; jm^s, {j'ma) is the s-th root of the Bessel jm(z) (derivative of the Bessel jp(z)) function [8].
4 Bandstop Filters on different DRs
DRs in regular transmission lines represent the most interesting case from the point of view of the theory, because in this case all the resonators at the same time exchange fluctuations both propagating and by not extending waves.
Fig. 2. The structure of different cylindrical DRs on the symmetry axis of propagating rectangular metal waveguide (a). Scattering parameters (b - e) of 5 cylindrical DRs with e1r = 36; Qf = 2000; A1 = L\/2r\ = 0, 4; and 6 cylindrical DRs with e2r = 82;
Qf = 1500; A2 = L2/2r2 = 0, 8.
Fig. 2 shows scattering parameters of the bandstop filter, consisting of 5 cylindrical DRs characterized by the dielectric permittivity r = 36; Qf = 2000 and by the relative sizes Ai = L\/2r\ = 0,4 as well as 6 DRs with e2r = 82; Qf = 1500; A2 = L2/2r2 = 0, 8, calculated by the formula (2),(7), (8), (9) with help of the (10), (11). All resonators placed on the waveguide axis. The distance between adjacent DR centers was equal to Xw/4, where Xw — is the guided wavelength.
The result of the scattering of the rectangular waveguide waves Що on the structure of 9 different spherical DRs is shown in fig. 3 b - e. The coupling coefficients of the DRs were calculated by the formulas
(12)-(15).
As can be seen, the use of different alternating DRs in this case gives acceptable results for the frequency distribution of scattering parameters.
Fig. 3. The structure of different spherical DRs on the symmetry axis of propagating rectangular metal waveguide (a). Scattering parameters (b-e) of 9 DRs bandstop filter with £ir = 36; Qf = 2000; e2r = 82; Qf = 1000.
Fig. 4. Cylindrical DR on the symmetry axis of evanescent rectangular metal waveguide segment (a). Scattering parameters of the bandpass filter on 11 cylindrical DRs with £ir = 36; Qf = 2000; e2r = 82; Qf = 1500.
5 Bandpass filters on different DRs
The best results were obtained for the DR structures, located in the evanescent waveguide segment and forming bandpass filters. The filters containing the DRs should have bands free of spurious oscillations. A known solution to this problem is to use different forms of DR.
Fig. 4. shows scattering parameters of the filter, consisting of two lattices with different cylindrical DRs. First lattice contains 4 DR with e1r = 36, Д1 = 0, 8, the second lattice consists of 7 DR with e2r = 81, Д2 = 0,4. The distance between the centers of adjacent first type DRs was equal to 17 mm; for the second type DRs 21mm. All resonators placed on the waveguide axis.
Fig. 5. Different spherical DRs in the evanescent rectangular metal waveguide segment (a). Scattering parameters of the bandpass filter on 11 cylindrical DRs with e1r = 36; = 3000; e2r = 82; = 2000 (b,
c). Group delay (d) of the filter and comparative group velocity (e) as a function of the frequency.
Application of spherical DR with different dielectric permittivity allows us to increase the coupling coefficients of the outside resonators in different structures of bandpass filters and thereby reduce the loss in the pass band.
Fig. 5 shows 8 spherical DR bandpass filter scattering parameters. Recent resonators of the filter made of dielectric with r = 36; Qf = 3000, the rest are made of dielectric e2r = 81; Q-f = 2000. All resonators form symmetrical structure. The distance between the centers of first and second DR is 23,5 mm, between
other DRs is 20 mm. The cross section of the input and output waveguides a x b = 58 x 25 mm2; the cross section of the evanescent waveguide az x bz = 20 x 25 mm2.
Fig. 6. Different spherical DRs in the evanescent cylindrical metal waveguide segment (a). Scattering parameters of the bandpass filter on 11 spherical DRs with £1 r = 36; Q® = 3000; e2r = 82; = 2000 (b,
c). Group delay (d) and comparative group velocity (e) as a function of the frequency.
Fig. 6 shows 11 spherical DR bandpass filter arrangement in circular cylindrical metal waveguide. The cross section of the input and output waveguides axb = 7 x 3 m m2, the radius of the evanescent waveguide Rz = 1, 5 mm. We have calculated the dependence of group delay tg = —d/dw[G21(w)] and comparative group velocity: vg/с = |z^ — z1\fctg (fig. 1, fig. 6 d, e), where c — is the velocity of light; \z^ — z1\ — is the longitudinal length of the filter. As can be seen from Fig. 6 d, e, the results demonstrate a remarkable slowing of signal propagation, characteristic of the filters on DRs.
Proposed enhancement of the electrodynamic theory for the scattering electromagnetic waves on different dielectric resonators greatly enhances design of the filters and other devices. As shown from calculations, the developed model correctly describes the scattering processes in the system of different DRs for a variety of transmission lines. The obtained solutions makes it possible to calculate all scattering parameters of the filters, made in various DR. Such design has several advantages compared with identical resonators for filters, in particular the filters have a more clean stop band, and in the case of spherical cavities, produce better scattering characteristics due
to the wider coupling bands variations. The frequency dependence of the scattering S-matrix can be further improved even more after a fine optimization of the filter parameters.
Conclusion
A scattering theory on different dielectric resonator systems, based on perturbation theory, has been expanded.
Given new definitions of coupling coefficients for the different dielectric resonators in the transmission line.
New analytical relationships for the coupling coefficients of different spherical and cylindrical dielectric resonators has been obtained.
A new design of the bandstop and bandpass filters on different DRs are proposed.
References
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Розсйовання електромагштних хвиль на р!зних д!електричних резонаторах Апкрохвильових фшьтр!в
Трубт О. О.
Розширена теор!я розс!ювання електромагштних хвиль лши передачи на системах зв’язаних д!електри-чних резонатор!в р!зно!' форми та д!електрично!' про-никност!. Дано нове визначення коеф!ц!ент!в зв’язку р!зних д!електричних резонатор!в. Отриман! анал!тичн! вирази для коеф!ц!ент!в взаемного зв’язку д!електри-чних резонатор!в цшиндрично!' та сферично!' форми, виконаних 1з р!зних матер!ал!в. Розглянут! основн! зако-ном!рност1 зм!ни зв’язку при вар!ацй параметр!в стру-ктури. Приведен! результата розрахуншв коеф!ц!ент!в передачи та в!дбиття смугових та режекторних ф!ль-тр!в, побудованих на р!зних д!електричних резонаторах в прямокутному та круглому хвилеводах. Встановле-но найбгпын оптимальн! конф!гураци, як! дозволяють досягати найкращих характеристик розс!ювання.
Ключовг слова: д!електричний резонатор; коеф!ц!-ент зв’язку; S-матрица; режекторний ф!льтр; смуговий ф!льтр
Рассеяние электромагнитных волн на разных диэлектрических резонаторах микроволновых фильтров
Трубин А. А.
Расширена теория рассеяния электромагнитных волн линии передачи на системах связанных диэлектрических резонаторов разной формы и диэлектрической проницаемости. Дано новое определение коэффициентов связи различных диэлектрических резонаторов. Получены аналитические выражения для коэффициентов связи диэлектрических резонаторов цилиндрической и сферической формы, выполненных из различных материалов. Рассмотрены основные закономерности изменения связи при вариации параметров структуры. Приведены результаты расчета коэффициентов передачи и отражения полосовых и режекторных фильтров, построенных на различных диэлектрических резонаторах в прямоугольном и круглом волноводах. Установлены наиболее оптимальные конфигурации, позволяющие достигать наилучших характеристик рассеяния.
Ключевые слова: диэлектрический резонатор; коэффициент связи; S-матрица; режекторный фильтр; полосовой фильтр
