Field Analysis of Ridged Waveguides Using Transfer Matrix

The mode matching method is applied to analyze generalized ridged waveguides. The tangential fields in each region are expressed in terms of the product of several matrices, i.e., a functional matrix about x-F(x), a functional matrix about y-G(y) and a column vector of amplitudes. The boundary conditions are transformed into a set of linear equations by taking the inner products of each element of G(y) with weight functions. Two types of ridged waveguide are calculated to validate the theory. Several new modes not reported in previous analysis are presented.


Introduction
Ridged waveguides have many applications in microwave and millimeter wave devices, owing to the well known fact that the cutoff frequency of their fundamental mode is far lower than that of the rectangular waveguide with the same outer dimensions [1][2][3][4]. Theoretical research on them has been continued steadily [1,2,[4][5][6][7][8][9]. The analysis methods involve mode-matching technique [1,2,5,6], quasi-static method [4], variational method [7] and integral equations technique [8,9]. All these approaches are only for the analysis of ridged waveguide with a certain structure, can not be used to analyze a group of ridged waveguides with different shapes. In this paper, a general mode matching method is presented, applying to different types of generalized ridged waveguides (shown in Figure  1). The waveguide is constructed by inserting arbitrary number of rectangular ridges in the rectangular waveguide.
To analyze the waveguide, the structure is divided into I regions. For each region i (xi≤x≤x i+1 , y i ≤y≤y i +b i ), we express the tangential fields in each region in terms of the multiplication of several matrices, i.e., a functional matrix about x-F(x), a functional matrix about y-G(y) and a column vector of amplitudes of the individual region. The boundary conditions of each pair of adjacent regions are accomplished by taking the inner products of each element of G(y) with weight functions, and a set of linear equations are obtained. While neither the eigenvalues nor the amplitudes can be solved directly from the equations. The boundary conditions of the waveguide's wall are involved to solve the problem. Two types of ridged waveguide are calculated to validate the theory. Several new modes not reported in previous analysis are presented.

Matrix Formulation Of Field Components
In general, the modes in ridged waveguide with inhomogenous dielectric filled are neither TE nor TM to the guide axis. In each region the fields can be expressed as a superposition of TE and TM modes of parallel planes. We denote ( ) where the upper symbol of (i) represents region i. ( ) The amplitude coefficients are all unknown in previous analysis. But in this paper, for regions 1 and I, some of the coefficients are of zero, determined by the boundary conditions at x=x 1 , x I+1 (which is perfect electric or magnetic walls). This will be discussed in section 4. Then the tangential field components in each region are written as follows Because of the limitation of computing time and storage requirements, the infinite series terms in (4) are restricted to finite number N i . The expression in (4) can be written in matrix notation as The components of ( ) ( )  .

Field Matching
We define then the boundary conditions at ( 2,3, , ) i are given by (9.d) To transform the boundary conditions into an algebraic system, we take the inner products of both sides of (9) with weight functions. Usually the eigenfunctions of the guide with larger height are chosen as the weight functions for enforcing the electric continuity in (9.a) and (9.b), and the eigenfunctions of the smaller guide are chosen for magnetic continuity in (9.c) and (9.d).
Taking the inner products of both sides of (9.a~d) with ( ) where i represents i x x = , j represents the region j.

Eigenvalue Equation
The eigenvalues and unknown amplitudes cannot be solved directly from the linear equations of (10), because the number of equation is less than that of the amplitude. The boundary conditions at x=x 1 , x I+1 are used to determine the eigenvalues and amplitudes. Let ) (i T is the transfer matrix considering higher order modes for the region i. When cascading the transfer matrix of every pair of adjacent regions, we get Equation (18) is the eigenvalue equation.

Numerical Results
Two examples are calculated to validate the theory. Ridged Waveguide with Inhomogeneous Dielectric-Slab loading. The waveguide is shown in Figure 2. It has an aspect ratio of b/a=0.5, and a dielectric slab of cross-sectional dimensions a ' and b ' is placed inside the gap between the ridges, whereas the outer parts of the waveguide's cross section remain empty. Figure 3 shows gap-height dependence of the cutoff wavelength λ c of the dominant mode normalized to the waveguide broad dimension a with a ' =0.5a. The permittivity ε r severs as parameter. The solid line is obtained with N=5 and the dots are from [6]. We can see in Fig.3 that the data agree well with those in [6].
Materials Science and Information Technology Single-Ridged Waveguide. The waveguide is filled with air, with dimensions of a=b=9.5 mm, s=0.15 mm, and d=1.7 mm (see Figure 4). Using the symmetry of the waveguide, the modes can be classified into two types: 1) odd modes with a perfect magnetic wall at x=0; 2) even modes with a perfect electric wall at x=0. We call the kth odd mode as mode 2k-1, and the kth even mode as mode 2k (k=1,2,3…). Tables 1 and 2 show the cutoff wavenumbers of the first eleven modes of even and odd modes respectively, along with the data presented in [7,8]. The waveguide is studied in [7,8] by use of variational method and integral equations technique respectively, whereas only odd modes are considered in [7]. Cross section of single-ridged waveguide As shown in Table 1, the agreement is good, except for mode 19. The calculated result of mode 19 is similar to that of TM mode 5 in Table III of [8], whereas the datum is absent in [7]. In Table 2, the computed data of even mode have much difference from [8]. There are corresponding data in [8] for mode 8, 16, 20 and 22, while no data for other modes are presented in [8]. It is found that the first higher order mode is an even mode and its cutoff number is 0.3297, different from the value of 0.3332 in [7,8].

Conclusions
The advantage of the approach is that the formulations can be used to analyze ridged waveguides with any shape. And the matching equations can be easily got from boundary conditions by simple matrix operation.

Acknowledgments
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