COUPLING COEFFICIENTS OF DIFFERENT CYLINDRICAL DIELECTRIC RESONATORS IN THE OPEN SPACE 1

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.


Introduction
Cylindrical DRs are applying today in various microwave devices [1 -6]. For calculation and optimization of such devices is more convenient to use electrodynamic 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]:    .
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 1,0,1 H  , the mutual coupling coefficients can be obtained in the form: in the area: 12 z r r    : For the AB position ( fig. 2, a) in the area: 12 z r r    : For the AC position (see fig. 3, a) in the area: For the CC position (see fig. 4, a) in the area:     For the AA position: For the AB position: (2) 1,2 0 For the AC position: For the CC position: 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]. 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.

Coupling coefficient analysis
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 coefficient of two different cylindrical DR with the mode 1,0,1 H  will be represented as: where 1 1 1 h ( r, , )    -is the magnetic field of the first DR in the center of the second one; 2 2 2 2 n n ( , )    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.

Conclusions
Analytical relationships for mutual coupling coefficients for the 1,0,1 H  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.