Vector complex eigenmode analysis of microwave cavities

N.A.Golias T.V. Yi o u Its i s T.D.Tsiboukis

Indexing terms: Edge analysis, Eigenvalue analysis, Finite-element method, Microwave cavities, Resonance

Abstract: A vector complex eigenmode analysis procedure of microwave cavities is presented. The resonance problem of arbitrary shaped microwave cavities loaded with lossy dielectric material is formulated with the finite-element method in either E or H fields through a Galerkin-weighted residual procedure. Discretisation of the problem results in a very large, highly sparse, complex symmetric generalised eigenvalue problem. Inverse iteration with shifting in combination with a preconditioned conjugate gradient procedure is employed for the efficient solution of the complex eigenvalue problem. The method, being very efficient in terms of time and memory, is applied in the determination of the resonance of a cylindrical cavity loaded with lossy dielectric material. The obtained results are in very good agreement with experimental ones. A duality in the resonant frequencies is observed between the E and H formulations. Furthermore, the method can be used in the solution of the inverse problem of determining the dielectric properties of the loaded material.

1 Introduction

The resonance of loaded cavities is an important prob- lem in the analysis of microwave devices. The calcula- tion of the resonant frequencies of loaded cavities is an efficient way of determining the dielectric properties of the loaded material. The finite-element method is suita- ble for the simulation of electromagnetic field propaga- tion through microwave devices and the determination of their performance.

Following the standard steps of the finite-element analysis the problem domain is subdivided into finite elements and an approximation model is defined in each element. Then the problem is formulated through a variational or Galerkin procedure. The contribution of each element is added in a global assembly and a highly sparse linear algebraic problem is formed. Due to the nature of the finite element method it can be

0 IEE, 1995 IEE Proceedings online no 19952261 Paper first received 6th April 1995 and in revised form 10th August 1995 The authors are with the Department of Electrical Engineering, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece

IEE Proc.-Microw. Antennas Propag., Vol. 142, No. 6, December 1995

applied to model arbitrary geometries, provided a suffi- cient number of elements is used for the approximation of the problem. During recent years it has been applied to the solution of high-frequency electromagnetic prob- lems [1,2]. Solution can be obtained in terms of poten- tials or the fields E and H directly. Two formulations employing the electric and the magnetic intensity directly are well known [ 3 ] .

A problem in the application of the finite-element method to the solution of high-frequency problems is the appearance of nonphysical solutions, the spurious modes. These solutions contaminate the true solutions and they are totally undesired. A possible reason for the appearance of spurious modes is the overcontinuity that nodal elements impose on the field variables. Nodal elements impose full continuity of the approxi- mated vector fields, while the electric or the magnetic intensity present only tangential continuity across inter- faces with different material properties. Edge elements, on the other hand, impose only tangential continuity on the approximated fields, which is just what is needed. Unknowns on these elements are the circula- tions of the field on the edges of the elements [4]. It has been shown that with the use of edge elements, spuri- ous modes do not appear [5-71. Edge elements are employed in this paper for the discretisation of two weak formulations in terms of the electric and the mag- netic intensity.

In a cavity with no lossy material, the angular fre- quency of oscillation w is real and the modes in the cavity are excited with an infinite amplitude at reso- nance. When a lossy material is inserted in the cavity, then the resonant mode is excited with a large, albeit finite amplitude. The electromagnetic wave attenuates when it is travelling through a lossy dielectric and a continuous flow of energy should be supplied so that resonance is maintained. The quantities E and H are complex and the same holds for the angular frequency o. The complex frequency w can be interpreted as a composite frequency of the wave variation, incorporat- ing the attenuation due to propagation through the lossy material [8]. Therefore the discretisation of the problem with the finite-element method results in a complex generalised eigenvalue problem.

The solution of large complex eigenvalue problems consists one of the most difficult areas of numerical analysis [9,10]. Special techniques must be employed to fully exploit the sparse structure of the matrices [ll]. An efficient technique, suitable for the solution of the highly sparse complex generalised eigenproblem obtained from the application of the finite-element

457

method, is the inverse iteration with shifting. The implementation of the method in conjunction with a preconditioned conjugate gradient technique [ 12-14] that exploits fully the sparsity and symmetry of finite- element matrices is presented in this paper.

The numerical formulation of the method is pre- sented, followed by a description of the inverse itera- tion technique employed in the solution of the complex generalised eigenvalue problem. Then the method is applied in the determination of the complex resonant frequencies of a cylindrical cavity either unloaded or loaded with a lossy dielectric. Values obtained of the resonant frequencies with the present method are in very good agreement with experimental results. Fur- ther, the method is used in the determination of the dielectric properties of the loaded material.

n x E = O

n x H , t O n x E , t O

2 Numerical formulation

Consider the solution of the resonance problem of a cavity, loaded with a lossy dielectric, as is shown in Fig. 1. Determination of the resonant frequencies is obtained with the finite-element method in terms of the fields E or H. The vector wave equations that the elec- tric intensity E and the magnetic intensity H satisfy are

P x ( p v x E) - -%E=o

P x ( E - ~ O x H) - w2pH = 0

(1)

(2) where o is the complex angular frequency, E the com- plex electric permittivity and p the magnetic permeabil- ity. The corresponding weak forms to these equations are

J ( p ‘ C x E) . (V x E’)dv - w’ E E . E’dw R J

62 J

R - {[E’ x (pulV x E)] nds = 0 ( 3 ) dR

/(s-’Y x H) . (V x H‘)dv - J’ pH H’dv ( 2

- ![HI x (&-IC x H)] . nds = 0 (4)

Boundary conditions are the vanishing of the tangential components of the electric intensity on electric walls (n x E = 0), and the vanishing of the tangential compo- nents of the magnetic intensity on magnetic walls (n x H = 0). In this way boundary conditions for the E for- mulation are homogeneous Dirichlet on electric walls and homogeneous Neumann on magnetic walls, while

dR

the reverse is true for the H formulation. Symmetry planes are modelled as electric or magnetic walls, according to the distribution of the electromagnetic field in the cavity [15].

Vector finite elements are used for the numerical approximation of the electric and the magnetic inten- sity. Unknowns on these elements are the circulations of the field on the element’s edges, hence called edge elements [4,5]. Edge elements impose continuity of the tangential components only across element interfaces and are used for the discretisation of the weak formula- tions. The basis functions for the first-order I-form Whitney tetrahedral elements for a typical edge e = { i , j } connecting vertices i andj, are given by

we = StPSJ - S,V& (5) where 5, (i = I , 2, 3 , 4) are the simplex or local co- ordinates in the tetrahedron. The electric and the mag- netic intensity are approximated as

6 6

E = C E , W , H = C H , W , (6) a = 1 2=1

where the coefficients E, and H, are the circulations of the field intensities along the edges of the tetrahedron. When no input field is prescribed the problem corre- sponds to the resonance of the cavity, which is a typical eigenvalue problem. If the edge shape functions are used as weighting functions in the weak formulations, the following generalised eigenvalue problem is obtained

[AIIXI = XPl[Xl ( 7 ) where [A] and [B] are matrices with elements

R

b,, = EW, . W,dv R J

for the E formulation and

(9)

b,, = pW, . W,dv R s

for the H formulation.

3 Inverse iteration with shifting

The solution of the complex sparse symmetric general- ised eigenvalue problem (eqn. 7) is obtained with inverse iteration with shifting implemented in conjunc- tion with a preconditioned conjugate gradient tech- nique. By employing a shift constant p the following modified generalised eigenvalue problem is obtained

{[AI - PIBIWI = (A - P)[BI[Xl (12)

By applying inverse iteration on the generalised eigen- value problem the eigenvalue that is closer to p as well as the corresponding eigenvector are obtained. An ini- tial vector X ’ O ) is used as input to the iterative proce- dure. Successive approximations to the eigenvector X

458 IEE Prw-Mii~run. . Antennus Propug., Vol. 142. Nu. 6 , Deceitrhcr I995

are calculated as follows

{[A] - p[B]}[X(” l ) ] = [ B ] [ X ( ” ]

First, a sparse matrix-vector multiplication routine is employed for the multiplication of the eigenvector [Si)] with the matrix [B] . Then the resulting linear system of equations is solved with a preconditioned conjugate gradient method, exploiting fully the sparsity and sym- metry of the finite element matrices. The associated eigenvalue h is calculated as

1 max ( [ X ( i+ 1 ) ] ) X = p +

and the eigenvector [#‘+‘)I is normalized to its maxi- mum component. The process is repeated until suffi- cient convergence to the eigenvalue has been obtained, as shown in Fig. 2 where the flow diagram of the appli- cation of the inverse iteration with shifting is presented. The closer the constant p is to the eigenvalue, the faster is the convergence of the iterative procedure. Thus, a good guess of the shift constant p accelerates the con- vergence of the method. Successive values of the eigen- value can be obtained by solving the problem in a coarse mesh and then use the obtained eigenvalue as input to the solution on the finer mesh. Furthermore, the convergence of the method is accelerated with the use of the preconditioned conjugate gradient technique, since the already obtained eigenvectors are used as ini- tial vectors in the next iteration. In this way, every aspect of economy is exploited and the solution of very large complex eigenproblems (in this work with dimen- sion 33820) is possible on a workstation.

4 Application

The resonance of a cylindrical cavity with the TMO10 mode is examined with the present method. The cavity is excited by a rectangular waveguide connected to the cavity through an iris. The travelling wave in the waveguide is inserted into the cavity for a very narrow band of frequencies and the cavity is said to be criti- cally coupled to the waveguide.The diameter of the

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cavity is 90mm, its height is 85mm, the waveguide is a standard 86.36 x 43.18mm and the iris has a width I = 15mm and height 43.18mm as is shown in Fig. 3, where the lower half of the structure is presented. The cavity may be empty or loaded with a vertical rod with diameter d = 7 or d = 9mm of either Plexiglass E, = 2.7 - jO.01 or PVC E, = 4.0 - j0.05. Experimental results for the eigenfrequencies of this problem have been obtained for the cases mentioned of dielectric material [ 161 and are repeated for convenience on Table 1.

Fig. 3 Cylindrical cavity excited by rectungulur waveguide

There are two symmetry planes one horizontal and one vertical, the horizontal plane is an electric wall and the vertical plane is a magnetic wall. Therefore only one fourth of the problem domain needs to be discrc- tised for computational economy. Three finite-element meshes with 655, 4580 and 33820 unknowns (edges) are used.

Table 1: Experimental results for for d 15mm (MHz)

Load Resonant frequency (MHz)

Empty cavity 2544.5

7 m m Plexi rod 2496.0

7 m m PVC rod 2486.6

9 m m Plexi rod 2463.5

9 m m PVC rod 2449.5

Table 2: Resonant frequencies for I 15mm and empty cavity (MHz)

Edges fo (E formulation) f, (H formulation)

655 2504.73 2617.18

4580 2536.78 2562.53

33820 2547.83 2548.80

Table 3: Resonant frequencies for I = 15mm and d = 7mm Plexi rod (MHz)

Edges fo (E formulation) fo (H formulation)

655 2461.38+j0.25 2563.83+j0.33

4580 2486.1 3+j0.27 251 3.13+j0.29

33820 2496.25+j0.28 2500.96+j0.29

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Table 4: Resonant frequencies for I 15mm and d = 7 mm PVC rod (MHz)

Edges fa (E formulation) fo (H formulation)

655 2425.09+j1.32 2519.00+j1.76 4580 2449.01+j1.44 2473.20+j1.55 33820 2458.17+j1.47 2462.07+j1.51

Table 5: Resonant frequencies for I 15mm and d = 9mm Plexi rod (MHz)

~~~

Edges fo (E formulation) fa (H formulation)

655 2415.72+j0.43 2528.10+j0.54 4580 2456.17+j0.46 2480.91+j0.49 33820 2465.29+j0.46 2469.14+j0.48

Table 6: Resonant frequencies for I = 15mm and d = 9mm PVC rod (MHzl

Edges fa (E formulation) fo (H formulation)

655 2375.82+j2.18 2455.65+j2.81 4580 2395.28+j2.35 2415.71+j2.51 33820 2403.58+j2.39 2405.92+j2.43

The problem is solved for all loaded cases and results for the resonant frequencies are presented on Tables 2- 6. The results presented in the tables are in very good agreement with the experimental results [ 161 presented in Table 1. However, there is a discrepancy in the results with the second dielectric, namely the PVC with given relative dielectric permittivity E , = 4.0 - j0.05. The obtained frequencies are lower than the resonant frequencies obtained experimentally. By solving the problem with various values of rod permittivity it is found that the results agree with the experimental for a value of the relative dielectric permittivity of about E , = 3.2 -j0.05. In this way, the numerical solution in com- bination with the experimental measurements can be used for the determination of the dielectric properties of materials.

2 . 7

2.65 1

2.45 1 2 . 4 1

0 10 20 30 40 edges x 1000

Fig.4 Resonant frequencies for I = 15 mm and empty cavity

Dual bounds of the values of the resonant frequen- cies are observed in the graphical presentation of results (Fig. 4). An upper bound is observed for the H solution and a lower bound for the E solution, and in this way the character of the dual bounds in the present problem can give a better estimate of the resonant fre- quency. Although, dual bounds appear in the present application, there is no theoretical proof for the exist-

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ence of dual bounds in time-varying problems. Numer- ical evidence, in fact, gives proof to the contrary [15]. The distribution of the electric intensity E and the mag- netic intensity H in the cavity are presented in Fig. 5.

a

b

Fig.5 a Electric intensity E b Magnetic intensity H

Electromagnetic field distribution in cavity

5 Conclusions

Complex eigenmode analysis was presented for the case of microwave cavities loaded with lossy dielectrics. The finite-element method has been employed successfully in the solution of the resonance problem of loaded microwave cavities. Vector finite elements have been used for the discretisation of the problem to avoid the appearance of nonphysical spurious modes. The com- plex sparse generalised eigenvalue problem has been handled efficiently with inverse iteration with shifting, exploiting fully the symmetry and sparsity of the finite- element matrices. Application to a microwave cavity for which experimental data exist, has showed very good agreement. Furthermore, the method has been used for the solution of the inverse problem of deter- mining the properties of the lossy material loaded into the cavity.

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