# Dominant Mode Wave Impedance of Regular Polygonal Waveguides

Authors: Vyacheslav V. Komarov
Journal: International Journal of Microwave Science and Technology
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

Regular polygonal waveguides (RPW) with different number of side walls(n) find application in microwave engineering as basic units of antennas [1], orthomode transducers [2], T-junctions [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 (n=3) [7] and square (n=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 with3≤n≤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.


## 2. Wave Impedance of RPW

Electromagnetic characteristics of any RPW depend on the metal wall size (an), which can be determined using another geometrical parameter—outer radius (Rn):
(1)an=2Rnsin(πn).As it is known,an>Rn, for 3≤n≤5; a6 = R6, while for the rest waveguides an<Rn when n≥7 (Figure 1).Triangular (a) and octagonal (b) waveguides.
(a)
(b)The wave impedance of the lowest TE-mode of any shaped metal waveguide:(2)ZnTE=kηβ=120π1-(λ/λcn)2,
where k is the wavenumber; η is the free space wave impedance; β is the phase constant; λ is the wavelength in free space; λcn is the cutoff wavelength of the dominant 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]. Longitudinal formulation of 2D Helmholtz equation with Neumann boundary conditions was utilized in finite element modeling of the lowest TE-mode of each RPW. Automated mesh generator and adaptive mesh refiner based on Rosenberg-Stenger scheme are realized in PDE Toolbox for flexible triangulation of arbitrary shaped 2D domains. Electromagnetic field components were approximated by the first-order polynomial functions and generalized eigenvalue problem was solved by Arnoldi algorithm. Number of triangular elements in the mesh for all RPW did not exceed 40,000. Preliminary testing of numerical solution on examples of square (n=4) and cylindrical (n≥32) waveguides has shown a good agreement with analytical solutions for these classical transmission lines available from the literature. Additional details of RPW simulations by means of the FEM are described in [12, 13].Curve-fitting procedures of MATLAB and MS Excel have been applied to derive a closed-form expressions using numerically obtained dependenciesλcn(n), where 3≤n≤100:
(3)λcn=(0.5529n-0.1863)an,3≤n≤8,(4)λcn=(0.0019n3-0.0555n2+0.5561n+1.3777)Rn=qRn,3≤n≤8,(5)λcn=(0.0018n+3.2362)Rn,8≤n≤100.Determination coefficient for (3)–(5) was R2≥0.9998.


## 3. Numerical Verification

Numerical simulation tool—commercial softwareQuickWave 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.Table 1
Normalized cutoff wavelengths of the lowest TE-mode of RPW.
n
λ
c
n
/
R
n
FEM
FDTD
Anal. [10]
3
2.599
2.587
2.275
4
2.828
2.83
2.828
5
3.0155
3.014
3.053
6
3.127
3.128
3.168
7
3.2
3.2
3.235
8
3.25
3.251
3.273Junctions 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 radiusR=4.181 cm and the TE11-mode propagating at ISM-frequency 2.45 GHz. Matching of this waveguide with RPW is achieved when
(6)S11=Z~-ZnTEZ~+ZnTE⟶min,
where S11 is the reflection coefficient; Z~ is the wave impedance of the TE11-mode of CW at 2.45 GHz’ and according to (2), Z~=733.37 Ω.Condition (6) will be satisfied if ZnTE→Z~. Then, in order to find sizes of RPW which correspond to (6) we can rewrite (2) as
(7)λcn=λ1-(120π/Z~)2.And now employing (4) we obtain
(8)Rn=λq1-(120π/Z~)2.Six 3D FDTD models of CW-RPW junctions for3≤n≤8 have been built applying QuickWave 3D and (8). One of such models is shown in Figure 2.Figure 2
Junction of cylindrical and hexagonal waveguides.Input signal in the form of a pulse of spectrum from 2.3 GHz to 2.6 GHz and TE11-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 |S11| at 2.45 GHz are listed in Table 2.Table 2
Reflection coefficient of waveguide junctions.
n
3
4
5
6
7
8
R
n, cm
5,488
5,045
4,732
4,563
4,459
4,390
|
S
11
|
0.251
0.062
0.024
0.012
0.0076
0.005Obtained data demonstrate that proposed approach to the RPW wave impedance definition works well whenn≥4. For the n=3, |S11| value is higher than expected. Additional calculations have shown that matching condition (6) is satisfied for R3 = 4.9 cm and |S11| = 0.115.


## 4. Example of Microwave Transition Design

Analytical approach described in present study has been used for building the model of coaxial waveguide transition shown in Figure3. Transition is consisted of two sections: coaxial square waveguide (SQW) adapter and SQW-CW junction. Coaxial line with sizes D = 7 mm and d = 3.04 mm excites TE10 mode in SQW 100 mm length with side wall size a4 = 71.347 mm. The dominant mode of SQW is transformed in TE11-mode of CW 15 mm length and 83.62 mm diameter in waveguide junction.Figure 3
Configuration of microwave transition under study.Cylindrical metal probet=2 mm connected to the coaxial feeder is placed at the distance w=30 mm from short circuited wall of SQW. Optimization problem solution has allowed determining the probe height value s=25 mm.Results of numerical modeling: absolute values of reflection coefficient in frequency range2.2≤f, GHz≤2.7 are represented in Figure 4. FDTD model has been verified by 3D FEM model developed with the help of COMSOL software [15]. Both techniques agree well near the ISM-frequency 2.45 GHz.Figure 4
Absolute values of reflection coefficient for microwave coaxial SQW adapter(1) and complete transition coaxial SQW-CW (2).And now taking into account results in Table2, RPW with any number of side walls can be used instead of CW to design transition from coaxial line to the selected RPW.


## 5. 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.



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