Computation of Electromagnetic fields inside arbitrary-shaped components by means of the BI-RME method Stephan Marini(1), Jordi Gil (2), Carlos Vicente(2) , Benito Gimeno(3), Vicente Boria(4) (1)
Department of Physics, Systems Engineering and Signal Theory, University of Alicante, Spain, Email:
[email protected] (2) Aurora Software and Testing S. L., Valencia, Spain (3) Department of Applied Physics and Electromagnetics, University of Valencia, Spain (4) Department of Communications, Technical University of Valencia, Spain
ABSTRACT In this paper, we extend the BI-RME (Boundary Integral-Resonant Mode Expansion) formulation in order to obtain the electromagnetic fields of arbitrary-shaped waveguides. The novel technique for considering the presence of circular and/or elliptical segments within the frame of the well-known boundary integral-resonant mode expansion (BI-RME) method is utilised. The accuracy and flexibility of the extended method has been verified with numerical results. 1.
INTRODUCTION
Microwave passive components for space applications are usually implemented in waveguide technology, because of its good electrical response, low ohmic losses, and high power handling capabilities [1]. However, the reduction of mass, size, and manufacturing costs related to such technology, as well as improvements in specs and performances, are increasingly tight requirements [2]. In this context, the use of flexible computer-aided design (CAD) tools based on fast and accurate electromagnetic simulators reveals necessary [3]. Arbitrarily shaped waveguides, whose cross-section is defined by linear, circular and/or elliptical arcs, are usually found in many passive waveguide devices used in real applications. For analysis purposes, it is required to know the complete modal chart of such arbitrary waveguides in a very accurate way. To reach this aim, the well-known BIRME method is proposed to be followed [4]. This method was developed to analyze waveguides with arbitrary cross sections, waveguides with inner perturbations (with one or more inner conductors) and planar microwave circuits with arbitrary geometry in the H- and E-plane. The classical implementation of such a method always defines the profile of the arbitrary waveguide by means of only straight arcs. Recently, this method has been revisited in order to cope with arbitrary profiles defined by the combination of linear, circular, and/or elliptical arcs [5]. In this work, we have extended the BI-RME method to the computation of the electromagnetic fields inside 2D arbitrary components in order to perform high power analysis. The software tool developed will be integrated into FEST3D CAD tool which will allow the full high power analysis (Corona and Multipactor) of complicated structures such as waffle-iron filters, bar lines, or waveguides with rounded corners very efficiently. This finally results in a software tool capable to cover the full design process of a specific component with low and high power specifications. 2.
THEORY
In this section we will outline the basics of this extended BI-RME technique. The arbitrarily shaped waveguide to be considered has a cross section defined by a combination of linear, circular, and/or elliptical arcs (see in Fig. 1). The arbitrary contour of such a waveguide (defined by the tangent vector and a suitable abscissa, also shown in Fig. 1) must be completely enclosed within a standard rectangular waveguide of cross section. The electric field at a generic observation point r inside can be obtained by
E ( r ) = − jηk ∫ G e ( r, s′, k ) ⋅ Jσ ( l ′ ) dl ′ σ
where s′ indicates a source point on σ,
(1)
η = μ ε is the characteristic impedance and k = ω με is the
wavenumber, G e is the two-dimensional dyadic Green’s function of the electric type for the two dimensional resonator of cross section Ω and Jσ is the current density.
Fig. 1. Waveguide with an arbitrary cross section S enclosed within a standard rectangular waveguide Ω
Splitting G e and Jσ into its transversal and longitudinal components, and imposing the corresponding boundary conditions on σ for TE modes, and for TM modes,
Et ( r ) ⋅ tˆ ( l ) = 0
(2)
Ez (r ) = 0
(3)
we obtain the integral equation for the transversal electric field
1 ∂ k 2 ∂l
∫σ
g ( r, s′ )
Et ( r ) (TE modes)
∂J t ( l ′ ) t ( l ) ⋅ em ( r ) dl ′ + ∫ tˆ ( l ) ⋅ G st ( r, s′ ) ⋅ tˆ ( l ′ ) J t ( l ′ ) dl ′ + ∑ am = 0 σ ∂l ′ km2 m
am =
k2 km2 − k 2
∫σ e ( s′) ⋅ tˆ ( l ′) J ( l ′) dl ′ m
(5)
t
where g is the rapidly convergent scalar two dimensional Green’s function and
(4)
G st is the rapidly convergent
solenoidal dyadic normal to the boundary described in [4], tˆ ( l ) is the unitary tangent vector on the arbitrary contour, km and
em ( r ) are respectively, the cut-off wavenumber and the normalised modal vector of the m-th TE mode of the
rectangular waveguide Ω. For TM modes we obtain the integral equation for the longitudinal electric field
∫σ g ( r, s′) J ( l ′) dl ′ + ∑ z
m
am′ = where km′ and
k ′2 km′2 − k ′2
ψm (r ) am′ = 0 km′2
∫σ ψ ( s′) J ( l ′) dl ′ m
z
E z (r ) (6)
(7)
ψ m ( r ) are, respectively, the cut-off wavenumber and the normalised scalar potential of the m-th TM
mode of the rectangular waveguide Ω. The next step in the BI-RME method is the segmentation of the problem: N
J z ( l ′ ) = ∑ b′j u j ( l ′ ) j =1
N
Jt ( l ′) = ∑ b j w j ( l ′)
(8)
j =1
where the basis and testing functions are chosen to be overlapping piece-wise parabolic splines. Applying the Galerkin approach of the MoM [6], the following general eigenvalue problem is obtained for the TE case
⎧ ⎡ U 0t ⎤ 2 ⎡ D R t ⎤ ⎫ ⎡ a ⎤ ⎨⎢ ⎥ − k ⎢ R L ⎥ ⎬ ⎢b ⎥ = 0 , ⎣ ⎦⎭ ⎣ ⎦ ⎩⎣ 0 C ⎦
(9)
and the following standard eigenvalue problem is reached for the TM case
( D′ − R ′ ⋅ L ′
−1
t
⋅ R ′ ) a ′ = k −2 a ′ , b ′ = −L ′−1 ⋅ R ′ ⋅ a ′
(10)
where the explicit expressions for the involved matrices are found in [4] and [5]. The solution of eqs. (9) and (10) provides as eigenvalues the TE ( k ) and TM ( k ′ ) cutoff wavenumbers, and as eigenvectors the modal expansion coefficients ( a , a′ ) and the amplitudes of the transversal ( b ) and longitudinal ( b′ ) components of the unknown current density. Once the TE and TM problem are solved, the electromagnetic fields of the q-th TE mode inside the arbitrary shaped waveguides are
e
◊ q TE
1 (r ) = kq
N
∑ b ∇ ∫σ n =1
q n
T
g ( r , s′ )
N M e ∂wn ( l ′ ) m TE ( r ) q q ˆ ′ ′ ′ ′ ′ am dl + kq ∑ bn ∫ G st ( r,s ) ⋅ t ( l ) wn ( l ) dl + kq ∑ σ km2 ∂l ′ n =1 m =1
h ◊q TE ( r ) = zˆ × e◊q TE ( r ) hz q◊ TE ( r ) =
◊ ◊ 1 1 ⎛ ∂e x q TE ( r ) ∂e y q TE ( r ) ⎞ ∇ T ⋅ ( zˆ × e ◊q TE ( r ) ) = ⎜ − ⎟⎟ kq kq ⎜⎝ ∂y ∂x ⎠
(11)
and the electromagnetic fields of the q-th TM mode inside the arbitrary shaped waveguides are
e
◊ q TM
N′
M′
( r ) = − ∑ bn′ ∫σ ∇ T g ( r , s′ ) u n ( l ′ ) dl ′ + ∑ n =1 m =1 q
h ◊q TM ( r ) = zˆ × e◊q TM ( r ) ⎡N ezq◊ TM ( r ) = kq′ ⎢ ∑ bn′q ⎣ n =1
a m′q e m TM ( r ) k m′ 2
M′ ⎤ ′ ′ ′ g r , s u l dl km′−2 ψ m ( r ) am′q ⎥ + ( ) ( ) ∑ n ∫σ m =1 ⎦
(12)
The TEM case, included for completeness, is totally independent of the above considerations. These fields are determined as gradients of the electrostatic potential, which may be expressed using the same scalar Green’s function g previously considered. N
e ◊ qTEM ( r ) = −∇ T Φ ( r ) = −∇ T ∫ g ( r, s′ )ρσ ( l ′ ) dl ′ = −∑ bn′′q ∫ ∇ T g ( r, s′ )un ( l ′ ) dl ′ σ
where ρσ
( l ′) are the charge density
n =1
(13)
σ
of inner conductor and the basis functions are chosen to be the same overlapping
piece-wise parabolic splines of the TM problem. Finally to obtain the complete field expression is necessary to add the time dependence, the wave propagation in the z direction, and the modal amplitudes (the equivalent voltage and current for the q-th modes).
3.
VALIDATION RESULTS
The new algorithm has been tested on different arbitrary geometries. All tests have been performed on a PC Intel Core 2 @ 2.4GHz and agreed with the results obtained using the commercial software Ansoft HFSS v9.2. The validity of this method is proved by representing the axial and transversal normalised electromagnetic field for some waveguides such as a ridge, circular, elliptical and coaxial waveguide. The computational efficiency of the novel modal analysis tool has been revealed as being very good. The total CPU time required to solve these examples has been of few minutes (depending of the number of computational points) which is quite good compared with the time required from the commercial software.
Fig. 2. (Left) Representation of the transversal electric and magnetic field (arrows) of the first TE mode of a ridge waveguide (4 x 2.976 mm) in a WR -75 (a=19.05, b=9.525 mm). The cut-off frequency of the mode is 6.71 GHz. (Right) Axial electric field of the first TM mode (fc=21.4 GHz). Cartesian mesh of 90 x 40 computational points.
Fig. 3. (Left) Representation of the transversal electric and magnetic field (arrows) of the first TE mode of a circular waveguide (radius r=4 mm, fc=21.96 GHz ). (Right) Representation of the transversal electric and magnetic field (arrows) of the second TM mode (fc=45.7 GHz ). Mesh of 60 x 60 computational points.
Fig. 4. Representation of the axial electric field of the second and fourth TM mode of a circular waveguide (r=4 mm, fc2=45.7 and fc4= 61.2 GHz ).
Fig. 5. (Left) Representation of the transversal electric and magnetic field (arrows) of the tenth TE mode of a elliptical waveguide (major semiaxis of 5 mm, and eccentricity e=0.5). (Right) Axial magnetic field of the same mode (fc=58.7 GHz).
Fig. 6. (Left) Representation of the transversal electric and magnetic field (arrows) of the first TM mode of a coaxial waveguide (internal radius of 0.25 mm, and external 2.4 mm ). (Right) Axial magnetic field of the same mode (fc=66.2 GHz). Mesh of 60 x 60 computational points.
The algorithm presented in this paper will be integrated into ESTEC FEST3D Software tool which will allow the analysis of high power effects in complicated waveguide filters such as dual mode filters based on coupling screws or waffle-iron filters. In particular, multipactor analysis in waffle-iron or evanescent filters are very interesting since parallel-plate approaches are too conservative since fringing field effects become critical. Additionally, corona issues close to metallic corners are also of interest and will be analyzed in the frame of dual mode filters making use of the software tool. 4.
CONCLUSIONS
In this paper, we have extended the BI-RME method in order to obtain the electromagnetic fields of arbitraryshaped waveguides. The novel technique for considering the presence of circular and/or elliptical segments within the frame of the well-known boundary integral-resonant mode expansion (BI-RME) method is utilised. The accuracy and flexibility of the extended method has been verified with numerical results. This is the first step to develop a software tool capable to analyse high power effects in complicated structures commonly used in satellite applications such as dual-mode or waffle-iron filters. REFERENCES
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