MUTUAL COUPLING COEFFICIENTS OF ROTATING RECTANGULAR DIELECTRIC RESONATORS IN CUT-OFF RECTANGULAR WAVEGUIDE

Background. A further increase in the speed of information transfer is determined by more stringent requirements for the elements of communication devices. One of the most important components of such devices is various filters, which are often made on the basis of dielectric resonators. Calculation of the parameters of multi-section filters is impossible without further development of the theory of their design. The development of filter theory is based on electrodynamic modelling, which involves calculating the coupling coefficients of dielectric resonators in various transmission lines. Objective. The aim of the research is to calculate and study the coupling coefficients of rectangular dielectric resonators with a rectangular metal waveguide when their axes rotate. Investigation of new effects to improve the performance of filters and other devices based on them. Methods. Methods of technical electrodynamics are used to calculate and analyse the coupling coefficients. The end result is to obtain new analytical formulas for new structures with rectangular dielectric resonators, which make it possible to analyse and calculate their coupling coefficients. Results. New analytical expressions are found for the coupling coefficients of dielectric resonators with the rotation of their axes in a rectangular waveguide. Conclusions. The theory of designing filters based on new structures of dielectric resonators with rotation of their axes in metal waveguides has been expanded. New analytical relationships and new patterns of change in the coupling coefficients are found.


Introduction
Multi-section band-pass and band-stop filters based on dielectric resonators (DR) of different shapes are used in various devices of telecommunication systems [1][2][3][4][5][6][7][8]. Further improvement of filter characteristics can be achieved by applying less traditional structures, such as dielectric resonators with rotation of their axes relative to each other and the transmission line [1,2]. Theoretical analysis of the characteristics of such filters, it is required to calculate the mutual coupling coefficients of the DR located at an arbitrary angle with respect to the waveguides.

Statement of the problem
The purpose of this article is to calculation and study mutual coupling coefficients of the rectangular DRs in a rectangular metal cut-off waveguide.

Calculation of fields during rotation of dielectric resonators in a waveguide
The internal field (e, h)   of a rectangular dielectric resonator with magnetic type oscillations H nml in the local coordinate system (x , y , z )    ( Fig.1) with good accuracy can be represented as: -is the resonant frequency; 0  -is the magnetic permeability; 0  ; 1  -is the dielectric permittivity of the external space and resonator, respectively. An analytical expressions for the coupling coefficients obtained from [10] for cut-off waveguide 2 1 Here W -energy stored in the dielectric of the resonator;  -is the guided wavelength; t -is the multiindex, defining non-propagating line wave type; n zlongitudinal coordinate of the n -th DR centre ( n 1, 2  );  -is the complex conjugate symbol.
We need to calculate the expansion coefficients n t 0 (c )  of the field of natural oscillations of the DR over the waveguide field in the resonator centre coordinate system.
For a rectangular resonator, it is more convenient to use representation for expansion coefficients: Here t E   -electric field of the t-th eigen wave of a waveguide; n e  -is the field (1); V -is the resonator volume; n 1, 2  . For simplicity, we will rewrite (2), (3) in the waveguide coordinate system (x, y, z) ( Fig. 2 -a) in a more convenient form: where for rectangular DR x y z x y z 0 2 12 1r 0 p p p q q q 16 -is the characteristic parameters of the resonator.

Rotation of the dielectric resonator relatively x-
axis of the waveguide In the case of rotation of the resonator relatively to the x-axis in the waveguide coordinate system (Fig. 2, a), the functions n t f ( i )   of (4) take the form for n 1, 2  : uy n i y y u y n n z u y n n e ( cos i sin ) ( sin i cos ) uy n i y y uy n n z uy n n e ( ( cos i sin )) ( ( sin i cos )) numbers of a rectangular cut-off waveguide with a cross section  a b (Fig. 2, a);

Rotation of the dielectric resonator relatively yaxis of the waveguide
The coupling coefficient of the dielectric resonators rotated relatively the y-axis determines by the functions:  u n t y0 y x y y n u y n Here n  -is the angle between z -axes of the local coordinate system of the resonator (Fig. 1) and z -axes of the waveguide (Fig. 3, a) ( n 1, 2  ).

Rotations of the dielectric resonator relatively zaxis of the waveguide
In the case of rotation of the dielectric resonator relatively z-axis at the initial position of the resonator axis z parallel to the waveguide axis (Fig. 4, a), the coupling coefficient defined by functions: Here n  can be defined as the angle between the yaxis of the waveguide (Fig. 4, a) and the y -axis (Fig. 1) of the n -th resonator. In the case of rotation of the dielectric resonator relative z-axis at the initial position of the resonator y -axis parallel to the waveguide y -axis (Fig. 5, a):  Calculation and analysis of mutual coupling coefficients Relations (4-8) were used for calculations mutual coupling coefficient dependences.
In Fig. 2 (1)) and f 7  GHz. As follows from the indicated dependencies, the coupling coefficients can take both positive and negative values. This can be due to physical and mathematical reasons. If co-directional z -axes of the resonators with 111 H oscillations lie on the z -axis of the waveguide and z L   , the mutual coupling coefficient is negative. If the co-directed z -axes are parallel and the resonators are coupled along the side wall, the coupling coefficients are usually positive for 111 H oscillations. If there is continuous rotation between these two positions of the DR, it leads to a change in the sign of the coupling (see fig. 2, 3, b -e). However, the rotation of one of the resonators at an angle  can also lead to a change in the sign of the coupling. This purely mathematically compensates for the change in the direction of the natural oscillation field of one of the resonators (see, for example, fig. 2, b or fig. 5, b). As can be seen from the results of calculations, in spite of the fact that the coupling functions are complex (see (4) -(8)), all coupling coefficients of the resonators in the rectangular cut-off waveguide are purely real (see fig. 3, 4, c). As expected, when the transverse coordinates are varied, the maximum coupling is achieved at the walls of the waveguide with the longitudinal arrangement of the resonator axes (Fig. 2, 4, d), and with the transverse arrangement of the axes, this maximum lies near the waveguide axis (Fig. 5, c).

Conclusions
In the paper, new analytical expressions are obtained for the mutual coupling coefficients for rectangular dielectric resonators in the rectangular waveguide, when the resonators are rotated relative to one of the waveguide axis.
The dependences of the coupling coefficients on the angles of rotation of the axes of dielectric resonators are studied.
It is shown that the change in the sign of the coupling can be due to both the physical nature and the spatial change in the direction of the field of natural oscillations of the resonators.
In the case of rotation about the x axis, the existence of an angle is shown at which the coupling coefficients weakly depend on the second resonator coordinate in the waveguide symmetry plane y b / 2  . With the longitudinal arrangement of the axes of resonators with magnetic types of oscillations 111 H relative to the axis of the waveguide, a weak dependence of the coupling on the angles of rotation has been established.
The obtained analytical relations can be used to develop mathematical models for a wide class of elements of telecommunication systems, such as filters, channel dividers, multiplexers, to name a few.