Analysis of mathematical techniques for the calculation of the electrostatic field in a dielectric-loaded waveguide
11 Analysis of mathematical techniques for the calculation of theelectrostatic field in a dielectric-loaded waveguide
A. Berenguer , A. Coves , E. Bronchalo and F. Mesa Dpto. de Ingenier´ıa de Comunicaciones. Univ. Miguel Hern´andez de Elche, Spain Dpto. de F´ısica Aplicada I. Univ. de Sevilla, Spain
Abstract — The resolution of the Green’s function for obtaining the electrostatic potentialgenerated by the charges located in the dielectric layer of a rectangular waveguide requires efficientintegration techniques. Due to the characteristics and the oscillatory nature of the function tobe integrated, the use of the Filon’s method together with a convergence analysis is adequate.However, the application of this numerical integration technique can lead to numerical instabilitiesin the result. For this reason, in this research work we have presented two different methods todeal with these numerical errors of integration: SSA/MSSA and GPR. In this way it is possible toclean the electrostatic potential avoiding subsequent errors in the calculation of the electrostaticfield.
1. INTRODUCTION
The calculation of the electrostatic field, Edc, in a dielectric-loaded waveguide due to an arbitrarycharge distribution on the dielectric layer is a problem of great interest in the space industry, becauseof the lack of rigorous studies about the multipactor effect appearing in dielectric loaded waveguide-based microwave devices in satellite on-board equipment. When dealing with a partially dielectric-loaded rectangular waveguide, the electrons emitted by the dielectric surface charge the dielectricmaterial positively, whereas the electrons absorbed by the dielectric layer charge it negatively.This charge gives rise to an electrostatic field which has to be taken into account in order toobtain an accurate trajectory of the electrons in the structure. While many works have studied theelectrostatic field appearing on RF dielectric windows [1, 2, 3, 4, 5, 6, 7, 8], less attention has receivedthe analgous problem in dielectric loaded waveguides [9, 10, 11]. Although the problem of obtainingthe electrostatic field originated by an arbitrary electron charge distribution has been solved in manyelectromagnetism books [12, 13, 14], the high computational complexity of this problem requiresthe use of different mathematical techniques to minimize computation time. In addition, theremay be critical points, e.g. close to the walls or the dielectric layer, in the structure in which thesolution does not converge properly. The convergence of the solution to the problem of calculatingthe electrostatic potential necessary for the calculation of the electrostatic field sometimes presentsnumerical instabilities generating false peaks. In this work, several numerical algorithms have beenanalysed and compared for the detection of these outliers in the electrostatic potential. Theseinclude, among others, Spectrum Analysis Models (e.g. SSA/MSSA) or Non-parametric BayessianModels (e.g. GPRs).The paper is organized as follows: Section 2 describes the theory and fundamental principlesunderlying the problem under investigation. In Section 3, the results obtained for a particularrectangular waveguide is shown, including some examples of the application of SSA/MSSA andGPR methods. A few concluding remarks are made in Section 4.
2. THEORY2.1. Electrostatic field in a dielectric-loaded waveguide due to an arbitrary charge distributionon the dielectric layer
In Fig. 1 it is shown the section of the waveguide under study, consisting on a partially dielectric-loaded rectangular waveguide, whose dielectric layer has relative permittivity (cid:15) r and thickness h .The aim is to compute the electrostatic field at the observation point r = ( x, y, z ), which is assumedto be located in the air region of a waveguide with translational symmetry along the longitudinaldirection z , due to a point charge on the dielectric layer at r (cid:48) = ( x (cid:48) , , E DC generated by the charges on the dielectric can be obtained as, E DC ( x, y, z ) = −∇ φ ( x, y, z ) (1) a r X i v : . [ phy s i c s . c o m p - ph ] M a y Figure 1: Geometry and dimensions of the problem under investigation. where φ ( x, y, z ) is the potential inside the waveguide. Using superposition, this potential due to theset of charges Q i on the dielectric surface can be obtained by adding the individual contribution ofeach charge, φ ( x, y, z ) = (cid:88) i G ( x − x (cid:48) i , y, (cid:12)(cid:12) z − z (cid:48) i (cid:12)(cid:12) ) Q i ( x (cid:48) i , , z (cid:48) i ) (2)where G ( x, y, z ) is the electrostatic potential due to a unit point charge, that is, the Green’s functionfor this problem. Due to the geometric characteristics and the linear nature of the problem underconsideration it is straightforward to demonstrate, as explained in [15, 16], that the Green’s functionin the spatial domain G can be obtained in the air region y ≥ G ( x, x (cid:48) , y, z ) = 2 (cid:15) πa ∞ (cid:88) n =1 sin( k xn x ) sin( k xn x (cid:48) ) × (cid:90) ∞ sinh[ k t ( H − y )] cos( k z z ) k t [ (cid:15) r coth( k t h ) + coth( k t d )] sinh( k t d ) d k z . (3)In (3), if the point charge is placed at z (cid:48) (cid:54) = 0, z must be replaced by ( z − z (cid:48) ). Here it is worthnoting that very efficient numerical summation and integration techniques have to be employed tocompute the Green’s function with sufficient accuracy and tolerable CPU times. Because of therapid oscillation of the integrand for large values of z , Filon’s integration method is chosen since itis desirable for integrals that has the form [17], (cid:90) ba f ( x ) cos( kx )d x (4)Using superposition, the potential due to an arbitrary charge distribution on a dielectric layeris obtained by adding the individual contribution of each point charge, φ ( x, y, z ) = (cid:90) G ( x − x (cid:48) , y, z ) ρ ( x (cid:48) )d x (cid:48) (5)Once the Green’s function has been calculated, the E DC field is obtained by using (5) andcalculating numerical differentiation of (1) by means of the central difference technique. The Singular Spectrum Analysis (SSA), is a relatively novel but powerful technique in data analysis,which has been developed and applied to many practical problems across different fields. The SSAtechnique is a decomposition-based approach and its usefulness lies in extracting information fromthe (auto)covariance structure of the data. In the original formulation of SSA it was assumedthat the data under analysis has a deterministic component with noise superimposed and thatthe deterministic component can be successfully extracted from the noise. This formulation isnot, of course, confined to SSA; the decomposition-based approach to data analysis is very old.What SSA brings into the picture, that identify it as a novel method, is that it accounts for the(auto)covariance structure of the data without imposing a parametric model for it. It is thus adata adaptive, non-parametric method based on embedding a data in a vector space, and froma practical perspective, a model-free approach for data analysis. The SSA method proceeds bydiagonalizing the lag-covariance matrix to obtain spectral information on the data. The data can then be analyzed as a sum of simpler, elementary series which correspond to different sub groupsof eigentriples (each eigentriple is composed of an eigenvalue and its associated left and righteigenvectors) of the lag-covariance matrix.
The first step, called embedding, is to transform a one-dimensional data { x , ..., x n } into a multidi-mensional trajectory matrix of lagged vectors X = [ x , ..., x k ] where k = n − m + 1 and each laggedvector is defined as x i = ( x i , ..., x i + m − ) T for i = 1 , ..., k . Each of these vectors corresponds to apartial view of the original data, seen through a window of length m . Choosing the lag windowsize m is a matter of balancing the retrieval of information on the structure of the underlying dataand the degree of statistical confidence in the results. The trajectory matrix X is a rectangularHankel matrix of the form, X = x x · · · x k x x · · · x k +1 ... ... ... ... x m x m +1 · · · x n (6) The second step consists on the Singular Value Decomposition (SVD) of the trajectory matrix X as, X = U Σ V T (7)where U is an orthogonal matrix of size m × m , Σ is a rectangular diagonal matrix of size m × k and V is an orthogonal matrix of size k × k . The elements of Σ, called singular values, are thesquare roots of the eigenvalues of the covariance matrix X X T . The rows of U are the eigenvectorsof X T X and are referred to as the left singular vectors. The columns of V T are the eigenvectorsof X X T . For SSA, the singular values are organized in decreasing order. Then any subset ofthe d eigentriples 1 ≤ d ≤ m , for which the singular values are strictly positive provides the bestrepresentation of the X matrix as a sum of X i matrices for i = 1 , ..., d . The third step involves the partitioning of these d eigentriples into p disjoint sub groups andsumming them within each group, such that represents a component series described by distinctsubsets of eigentriples. The last step of the SSA algorithm, known as diagonal averaging , aims at transforming the compo-nent matrices X i into Hankel matrices, which then become the trajectory matrices of the underlyingdata, in such a way that the original data can be reconstructed as a sum of these components. Theentire procedure aims at defining in some optimal way what those components are.For multivariate data, the Multichannel Singular Spectrum Analysis (MSSA) gap filling algo-rithm takes advantage of both spatial (cross multiple data) and (auto)correlation. Gaussian Processes (GPs) provide an alternative approach to regression problems. The GP ap-proach is a non-parametric approach, in a way that it finds a distribution over the possible functions f ( x ) that are consistent with the observed data, y = f ( x ) + (cid:15) (8)As with all Bayesian methods it, begins with a prior distribution and updates this as data points areobserved, producing the posterior distribution over functions. A GP defines a prior over functions,which can be converted into a posterior over functions once we have seen some data. Although itmight seem difficult to represent a distrubtion over a function, it turns out that we only need tobe able to define a distribution over the function’s values at a finite, but arbitrary, set of points { x , ..., x n } . A GP assumes that p ( f ( x )) , ..., p ( f ( x n )) is jointly Gaussian, with some mean µ ( x ) andcovariance matrix Σ( x, x (cid:48) ) given by Σ ij = k ( x i , x j ), where k is a positive definite kernel function, f ( x )... f ( x n ) ∼ N µ ( x )... µ ( x n ) , k ( x , x ) · · · k ( x , x n )... ... ... k ( x n , x ) · · · k ( x n , x n ) (9) The key idea is that if x i and x j are deemed by the kernel to be similar, then we expect the outputof the function at those points to be similar, too. The mathematical crux of GPs is the multivariateGaussian distribution. The covariance matrix, along with a mean function to output the expectedvalue of f ( x ) defines the Gaussian Process.Since the key assumption in GP modelling is that our data can be represented as a sample froma multivariate Gaussian distribution, we have that, (cid:20) ff ∗ (cid:21) ∼ N (cid:18)(cid:20) µµ ∗ (cid:21) , (cid:20) K K T ∗ K ∗ K ∗∗ (cid:21)(cid:19) (10)where K is the matrix we get by applying the kernel function to our observed x values (i.e. trainingdata), K ∗ gets us the similarity of the observed x values to the values whose output we’re tryingto estimate (i.e. test data) and K ∗∗ gives the similarity of the estimated values to each other. T indicates matrix transposition.We are interested in the conditional probability p ( f ∗ | f ): ”given the data, how likely is a certainprediction for f ∗ ?”. The probability follows a Gaussian distribution, f ∗ | f ∼ N ( K ∗ K − f, K ∗∗ − K ∗ K − K T ∗ ) (11)Our best estimate for f ∗ is the mean of this distribution,¯ f ∗ = K ∗ K − f (12)and the uncertainty in our estimate is captured in its variance,var( f ∗ ) = K ∗∗ − K ∗ K − K T ∗ (13)
3. NUMERICAL RESULTS AND DISCUSSION
This section shows the results obtained for a particular rectangular waveguide. In this case, thefollowing parameters for the geometry and materials are considered: a = 19 .
05 mm, H = 0 . h = 0 .
25 mm, z = 1 mm and (cid:15) r = 2 . k z has been analyzed into detail. Fig. 2 shows the shape of the function to be integrated for the firstterm of the summation, n = 1, and a distance y = 0 . Figure 2: The integrated function for the first term of the summation, n = 1, and a distance y = 0 . In order to achive accurate results, it is relevant to carry out a convergence study on theintegrand. In terms of the rate of convergence, the worst scenarios are for the cases of low y valuesand high n values. f ( k z ) for the case of y = 0 . n = 1 is plotted in Fig. 3. As it is shown, k z ≥ × has to be considered to reach convergence. The asymptotic behavior of the integrand,determined by the term e − k t y , allows us to establish a condition to stop the computation when Figure 3: f ( k z ) for the case a = 19 .
05 mm, H = 0 .
375 mm, h = 0 .
25 mm, y = 0 . z = 1 mm, n = 1and (cid:15) r = 2 . the convergence is reached. It involves calculating the relative value of the i-th summand of theintegral with respect to the accumulated value of the integral until this iteration. If this relativevalue is less than a particular convergence tolerance, the computation of the integral is stopped.As shown in Fig. 4, false peaks are observed in the the Green’s function calculated by usingFilon’s numerical integration. To fix these errors and remove them, the SSA/MSSA technique isused. To do this, a lag window of size m = 4 is considered for calculating the trajectory matrixof the electrostatic potential. The reason for choosing this size is that the objective is to minimizeor eliminate false peaks when possible, impacting the correct values of the potential as little aspossible. Choosing a higher value of the window lag m would lead to reduce these undesirableeffects in the electrostatic potential, however it would impact the real value of the centered peak.In spite of obtaining accurate results with SSA/MSSA technique, a second method of datacleaning is used to compare the results. In this case the GPR method is applied to the potentialdata. In this case, the results for the electrostatic potential improve slightly the ones obtained fromthe SSA/MSSA method. Figure 4: Comparasion of SSA/MSSA and GPR methods for data cleaning process of the electrostaticpotential.
4. CONCLUSION
In this work, two different numerical methods, SSA/MSSA and GPR, has been shown for dealingwith numerical errors when calculating the Green’s function. Although Filon’s method is suitablefor integrating oscillating functions, as is the case of the problem under investigation, for certainpoints, false peaks in the potential are obtained which need to be corrected to avoid their propaga-tion in the calculation of the electrical field generated by a charge distribution on the surface of thedielectric layer. The GPR behaves slightly better than the MSSA/SSA method in the examplesanalyzed in this work. However, the results obtained with both of them are accurate enough tosolve this issue.
ACKNOWLEDGMENT
This work was supported by the Agencia Estatal de Investigaci´on (AEI) and by the Uni´on Europeathrough the Fondo Europeo de Desarrollo Regional - FEDER - Una manera de hacer Europa(AEI/FEDER, UE), under the Research Project TEC2016-75934-C4-2-R.
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