Dominant Mode Wave Impedance of Regular Polygonal Waveguides

Polygonal metal waveguides are analyzed analytically and numerically. Classical equation for the wave impedance of arbitrary shaped waveguides is completed with approximate expression for the cutoff wavelength of the dominant mode. Proposed approach is tested with the help of 3D finite difference time domain models of microwave waveguides junctions. Obtained data are used for computer-aided design of microwave transition from coaxial line to cylindrical waveguide.


Introduction
Regular polygonal waveguides (RPW) with different number of side walls () find application in microwave engineering as basic units of antennas [1], orthomode transducers [2], Tjunctions [3], mode transformers [4], heating devices [5], and power combiners [6].Distribution of the TE-and TM-modes in such waveguides can be obtained from the Helmholtz equation solution with Neumann and Dirichlet boundary conditions on metal walls.But the exact analytic solution of this problem is possible only for the triangular ( = 3) [7] and square ( = 4) waveguides.Approximate analytical approaches based, for example, on perturbation method [8], method of analytical regularization [9], method of analogy [10], or some others [11] are also used for calculation of the eigenvalues and eigenfields of RPW.Some of these methods are restricted either by the waveguide shape [7] or by the mode type [11].Numerical techniques implemented in commercial packages, HFSS, CST MWS, and MATLAB, are the alternative tools of modeling electromagnetic fields in RPW [9,10,12].
Most of publications mentioned above are devoted to the computation of the cutoff wavenumbers of hollow RPW with 3 ≤  ≤ 8. Closed-form expressions for approximate calculation of the characteristic impedance, attenuation, and phase constant have been derived in [10], where a very simplified equation for wave impedance without any examples of its application is represented.
The objective of the present study was to check the applicability of metal waveguides theory to the modeling of wave impedance of RPW.

Wave Impedance of RPW
Electromagnetic characteristics of any RPW depend on the metal wall size (  ), which can be determined using another geometrical parameter-outer radius (  ): As it is known,   >   , for 3 ≤  ≤ 5;  6 =  6 , while for the rest waveguides   <   when  ≥ 7 (Figure 1).
The wave impedance of the lowest TE-mode of any shaped metal waveguide: where  is the wavenumber;  is the free space wave impedance;  is the phase constant;  is the wavelength in free space;   is the cutoff wavelength of the dominant 2 International Journal of Microwave Science and Technology mode.The eigenvalues of RPW can be calculated using the results of numerical modeling obtained with the help of the finite element method (FEM) implemented in PDE Toolbox of MATLAB software [13].[12,13].

Numerical Verification
Numerical simulation tool-commercial software Quick-Wave 3D [14] based on the finite difference time domain (FDTD) method-has been utilized in this study for checking (4).Comparative analysis of two numerical techniques, FEM and FDTD, and analytical approach described in [10] are given in Table 1.Junctions of RPW can be successfully employed for design of microwave transitions from widely available coaxial line to RPW.One of such approaches is analyzed in present study.
Let us consider standard cylindrical waveguide (CW) with radius  = 4.181 cm and the TE 11 -mode propagating at ISM-frequency 2.45 GHz.Matching of this waveguide with RPW is achieved when where  11 is the reflection coefficient; Z is the wave impedance of the TE 11 -mode of CW at 2.45 GHz' and according to (2), Z = 733.37Ω. Condition (6) will be satisfied if  TE  → Z.Then, in order to find sizes of RPW which correspond to (6) we can rewrite (2) as And now employing (4) we obtain International Journal of Microwave Science and Technology  Six 3D FDTD models of CW-RPW junctions for 3 ≤  ≤ 8 have been built applying QuickWave 3D and (8).One of such models is shown in Figure 2.
Input signal in the form of a pulse of spectrum from 2.3 GHz to 2.6 GHz and TE 11 -mode was selected for CW.Analogous signal but for arbitrary shaped transmission line was used in the output port.Simulation results: absolute values of reflection coefficient | 11 | at 2.45 GHz are listed in Table 2.
Obtained data demonstrate that proposed approach to the RPW wave impedance definition works well when  ≥ 4. For the  = 3, | 11 | value is higher than expected.Additional calculations have shown that matching condition ( 6) is satisfied for  3 = 4.9 cm and | 11 | = 0.115.

Example of Microwave Transition Design
Analytical approach described in present study has been used for building the model of coaxial waveguide transition shown in Figure 3 [15].Both techniques agree well near the ISM-frequency 2.45 GHz.And now taking into account results in Table 2, RPW with any number of side walls can be used instead of CW to design transition from coaxial line to the selected RPW.

Conclusion
It has been proved that the classical approach to the wave impedance definition previously applied for rectangular and cylindrical waveguides is also applicable for RPW.Exception observed for the triangular waveguide should be a subject of a separate study.Obtained data can be employed in design of various microwave components, for example, transitions, on RPW.

Table 1 :
Normalized cutoff wavelengths of the lowest TE-mode of RPW.

Table 2 :
Reflection coefficient of waveguide junctions.