Multiple scattering of microwaves from septum discontinuities in a circular bend

doi:10.1016/j.wavemoti.2004.12.002

Wave Motion

Volume 42, Issue 2, August 2005, Pages 109-125

https://doi.org/10.1016/j.wavemoti.2004.12.002 Get rights and content

Abstract

A wave splitting method is introduced to solve the problem of multiple scattering of microwaves from septum discontinuities in a circular waveguide bend of rectangular cross section. The splitting method allows that the scattering matrices for the septum discontinuities are determined by mode-matching and cascade techniques in a stable manner. We also demonstrate that the scattering coefficients converge as the number of expansion terms increases, and give criteria for the existence of trapped modes at the septum junction in the bend. The uniqueness in the solution is lost for a discrete set of frequencies where these trapped modes are present. For this case regularization methods are applied to make the solution numerically stable. Finally, numerical results are given for a model problem of two septum discontinuities in a bend. The algorithm presented here is suitable in the design of E-plane filters, where bends are present within the band pass filter.

Introduction

The classical problem of wave propagation in curved waveguide bends has been analyzed by many authors during the last century. The interest in waveguide bends is of course due to their importance in many technical applications, e.g. fan and car silencers and microwave filters. To be able to design a waveguide that fulfills various demands on its geometry as well as its performance concerning wave transmission, we need to have a fast and accurate algorithm for the wave characteristics of bends. One of the difficulties in the analysis of circularly bent waveguides is the problem of matching the solution of the wave equation in the bend to the wave modes in the straight waveguide. Various methods have been developed and used over the past decades to overcome this difficulty.

To mention some of the more recent contributions concerning scattering of microwaves in bends, Weisshaar et al. [1], use a method of moment solution and a mode-matching technique, where a Fourier expansion of the eigenfunctions of the Helmholtz equation is employed. An extension of this work has been done by San Blas et al. [2], where scattering by bends under full-wave incidence is considered. Cornet et al. [3], use a tensorial covariant formalism of the Maxwell’s equations to determine the scattering characteristics of the bend. Their method is more efficient compared to the method of [1] for simulations of bends with zero inner radius. MacPhie and Wu [4], use Bessel–Fourier modal functions to derive solutions for bends having zero inner radius. Their method is an extension of the boundary contour mode-matching technique developed by Reiter and Arndt [5]. This method is fast, accurate and can be used to analyze a wide class of waveguide discontinuities with both planar and circular cylindrical boundaries. However, the methods presented by [1], [2], [3], [4], [5] are not easily generalized to include multiple scattering of septum discontinuities, parallel to the propagation direction, in the bend; though the case of including one thin matching diaphragm perpendicular to the propagation direction in the bend is considered by Mongiardo et al. [6].

The finite element method (FEM) is well known for its flexibility and versatility in solving waveguide scattering problems. Still, this method is much more time consuming than the more analytical solution methods presented by [1], [2], [3], [4], [5], [6], especially if the inverse problem is to be solved, i.e. to determine the interior geometry of the waveguide in order to satisfy some specified demands concerning the wave transmission properties.

Nilsson [7], presents a theory to calculate the transmission of acoustic waves in basically two-dimensional waveguides having varying cross sections, as well as corners. First, the waveguide is transformed from curved to straight using a conformal or local orthogonal transformation. Then, wave splitting and invariant embedding methods are used to calculate the scattering operators in a stable manner for the straight waveguide now having a non-uniform index of refraction.

Like Nilsson [7], a local orthogonal transformation and a splitting technique are used in our formulation. However, the splitting is formally exact here and is calculated by numerical linear algebra and the numerically more complex invariant embedding method is not required.

In contrast to the investigations above, the present work deals with multiple scattering in waveguides. More precisely, we solve Maxwell’s equations in a waveguide, having a rectangular cross section, which is straight except for one portion that is a circular bend; see Fig. 1, Fig. 2. For waveguides, like the present, having piecewise constant values of curvature and cross section the formulation is particularly transparent for analyzing multiple scattering. In addition to being technically interesting, the solution of this problem is therefore an important step towards solving multiple scattering from internal objects in waveguides with more complicated boundaries. The basis for the transparency of the method is that the wave splitting is formally exact as the exponential of the square root of a differential operator.

Two or more circular bends can be connected such that coupled equations appear, showing the polarization effects of the electromagnetic fields. Combining the introduced method and cascade techniques this problem can be converted to the solution of uncoupled equations only. However, such a trick is not effective for bends of circular cross section. We believe that the Maxwell’s equations in the toroidal geometry can be analyzed by further developing the introduced method.

The splitting method introduced here gives the solution in terms of waves propagating to the right and to the left in the bend. Thus, the reflection and transmission properties of bends with septum discontinuities can be determined for full-wave incidence. The basis functions in the transverse variables are chosen as the eigenfunctions of the straight waveguide, i.e. trigonometric functions. These basis functions satisfy the boundary conditions in the radial and vertical cylindrical variables. This implies that matching procedures with straight guides are easy to handle. Also, matching of septum discontinuities in the bend can be performed in a similar way as in the straight waveguide.

There are two main techniques to design and produce waveguide band pass filters. They are the so-called H- and E-plane filter techniques, see e.g. [8], [9]. The concept of an H-plane filter is used for the case when the $H$ field is in the plane defined by the possible wave propagation directions. For the E-plane filter we have the condition that the $E$ field is in the plane defined by the possible wave propagation directions. The manufacturing of H-plane filters requires that a miller is used to produce the resonance cavities involved, while the manufacturing of E-plane filters involves metal insert of septum discontinuities in an uniform waveguide. This difference in production technique implies that the milling cost is substantially reduced for the E-plane filters and consequently they are, in many cases, much cheaper to produce than H-plane filters. To have the flexibility that the design of modern diplexers requires it is crucial to be able to incorporate bends within the filter structure. These bends are easy to design for an H-plane filter but not for the E-plane case. The purpose of this work is therefore to solve the multiple scattering problem of septum discontinuities in a bend. Our solution could then be incorporated into a filter design software and be used by the industry in the design of E-plane filters.

The outline of this paper is that the formulation is given in Section 2. In Section 3 the exact wave splitting method in terms of the exponential of the square root of a differential operator is introduced. In Sections 4 Matching at the bend interface, 5 Matching at the septum discontinuity, 6 Cascade techniques matching and cascade techniques are used to solve the scattering problem. In Section 7 we discuss convergence of the scattering coefficients and give criteria for the existence of trapped modes at the septum junction in the bend. In Section 8 we present some numerical results for our model problem. We end this work in Section 9 with concluding remarks.

Section snippets

Formulation

We solve the problem of wave propagation in rectangular E-plane bends with septum discontinuities, connected to straight waveguides of rectangular cross section, see Fig. 2. The concept of an E-plane bend is used for the case when the $E$ field is in the plane defined by the bend. The transverse rectangular measures of the waveguide are denoted by a and b, where a is the waveguide width in the plane defined by the bend. If $a < b$ then the bend is an E-plane bend. In our geometry, the fundamental

The wave splitting method

We start the analysis by looking at wave propagation in straight waveguides of rectangular cross sections. From (2) we have for the $E$ -plane modes $(\nabla^{2} + k^{2}) H_{y} = 0 .$ Let us denote the operator $\partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2} + k^{2} = K^{2}$ . Then (4) for $H_{y}$ can be written as $(\frac{\partial^{2}}{\partial z^{2}} + K^{2}) H_{y} = 0 .$ It is well known from quantum mechanics that the solution of the time dependent Schrödinger equation $(i ℏ \partial / \partial t - H) Ψ (x, t) = 0$ is represented as $Ψ (x, t) = ɛ^{- i H t / ℏ} Ψ (x, 0),$ where H is the Hamiltonian operator. Inspired by the functional calculus of quantum

Matching at the bend interface

To get complete sets of equations for the scattered wave amplitudes we use the continuity condition for the transverse electric field components at the bend interfaces with the straight waveguides as well as at the septum junctions in the bend. The electric field components $E_{ρ} (ρ, ϕ, y)$ in the bend and $E_{x} (x, y, z)$ in the straight waveguide, are determined from (11), (19) by using (3), and we get $E_{ρ} (ρ, ϕ, y) = - i ω μ {(\frac{\partial^{2}}{\partial y^{2}} + k^{2})}^{- 1} \frac{1}{ρ} \frac{\partial}{\partial ϕ} H_{y} (ρ, ϕ, y), E_{x} (x, y, z) = - i ω μ {(\frac{\partial^{2}}{\partial y^{2}} + k^{2})}^{- 1} \frac{\partial}{\partial z} H_{y} (x, y, z) .$ It is sufficient to use the

Matching at the septum discontinuity

The magnetic and electric field components $H_{y}$ and $E_{ρ}$ at the septum junction, see Fig. 3, are determined by using (11), (3). By extending the mode-matching method given by Shih in [11], from a two-dimensional case to a three-dimensional case, we get the following equations:

For Region I, $R - a / 2 < ρ < R + a / 2, ϕ < 0, 0 < y < b$ , $H_{y} = \sum_{l = 1}^{L} X^{T} \cdot (ɛ^{i B_{l}^{b} ϕ} \cdot H_{l}^{+} + ɛ^{- i B_{l}^{b} ϕ} \cdot H_{l}^{-}) y_{l}^{b}, E_{ρ} = {(\frac{\partial^{2}}{\partial y^{2}} + k^{2})}^{- 1} \frac{ω μ}{ρ} \sum_{l = 1}^{L} X^{T} \cdot B_{l}^{b} \cdot (ɛ^{i B_{l}^{b} ϕ} \cdot H_{l}^{+} - ɛ^{- i B_{l}^{b} ϕ} \cdot H_{l}^{-}) y_{l}^{b} .$ For Region II, $R - a / 2 < ρ < R + a / 2, 0 < ϕ, 0 < y < c_{1}$ $H_{y} = \sum_{l = 1}^{L^{c_{1}}} X^{T} \cdot (ɛ^{i B_{l}^{c_{1}} ϕ} \cdot C_{l}^{1 +} + ɛ^{- i B_{l}^{c_{1}} ϕ} \cdot C_{l}^{1 -}) y_{l}^{c_{1}}, E_{ρ} = (\frac{\partial^{2}}{\partial y^{2}} +)$

Cascade techniques

The four continuity equations for the magnetic and electric field components, $H_{y}$ and $E_{ρ}$ , at the septum interfaces $ϕ = 0$ and $ϕ = ϕ_{s}$ , where $ϕ_{s}$ is the angular measure of one septum discontinuity in Fig. 2, are determined from (36) by using cascade techniques and we get $T_{s} \cdot (H_{1}^{+} + H_{1}^{-}) = D^{+} + ɛ^{i B^{c} ϕ_{s}} \cdot D^{-}, T_{s} \cdot (H_{2}^{+} + H_{2}^{-}) = ɛ^{i B^{c} ϕ_{s}} \cdot D^{+} + D^{-}, S_{s} \cdot (H_{1}^{+} - H_{1}^{-}) = D^{+} - ɛ^{i B^{c} ϕ_{s}} \cdot D^{-}, S_{s} \cdot (H_{2}^{+} - H_{2}^{-}) = ɛ^{i B^{c} ϕ_{s}} \cdot D^{+} - D^{-},$ where the indices 1 and 2 on $H$ refer to the first and second septum interface encountered by a wave propagating in the positive $ϕ$

Convergence and trapped modes

To perform numerical calculations of the scattering coefficients for a bend with septum discontinuities, we must, as mentioned before, truncate the infinite matrices that result from the projections of the operators involved. The question of convergence of the scattering coefficients must thereby be handled. Since the scattering matrices are representations of their corresponding operators in the basis functions, we proceed by analyzing the properties of the operators involved.

By giving upper

Results

Although the knowledge of convergence for the scattering coefficients is important it is of little practical use unless it is fast enough. Otherwise the size of the matrices in this three-dimensional problem will be too large, and the algorithm will be too slow to be of any use in the design problem. The easiest way to investigate how many expansion terms we need to reach convergence is to perform simulations. We see that for our model problem it is enough to have two expansion terms in the

Conclusion

The multiple scattering problem of microwaves from septum discontinuities in a circular bend is solved by a wave splitting method. The wave splitting is given in terms of the exponential of the square root of a differential operator. In this way a transparent presentation is achieved that opens up the possibility to solve the scattering problem by mode-matching and cascade techniques.

We demonstrate that the scattering coefficients converge as the number of expansion terms increases. It is shown

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Multiple scattering of microwaves from septum discontinuities in a circular bend

Abstract

Introduction

Section snippets

Formulation

The wave splitting method

Matching at the bend interface

Matching at the septum discontinuity

Cascade techniques

Convergence and trapped modes

Results

Conclusion

A rigorous and efficient method of moments solution for curved waveguide bends

IEEE Trans. Microwave Theory Technol.

A rigorous and efficient full-wave analysis of uniform bends in rectangular waveguide under arbitrary incidence

IEEE Trans. Microwave Theory Technol.

Wave propagation in curved waveguides of rectangular cross section

IEEE Trans. Microwave Theory Technol.

A full-wave modal analysis of inhomogeneous waveguide discontinuities with both planar and circular cylindrical boundaries

IEEE Trans. Microwave Theory Technol.

Rigorous analysis of arbitrarily shaped H- and E-plane discontinuities in rectangular waveguides by full-wave boundary contour mode-matching method

IEEE Trans. Microwave Theory Technol.

Analysis and design of full-band matchedwaveguide bends

IEEE Trans. Microwave Theory Technol.

Acoustic transmission in curved ducts with varying cross-section

Proc. Roy. Soc. Lond.

Millimeter-wave Ka-band H-plane diplexers and multiplexers

IEEE Trans. Microwave Theory Technol.