Analysis of the electrostratic field generated by a charge distribution on a dielectric layer loading a rectangular waveguide
A. Berenguer, A. Coves, F. Mesa, E. Bronchalo, B. Gimeno, V. Boria
AAnalysis of the electrostratic field generated by acharge distribution on a dielectric layer loading arectangular waveguide
A. Berenguer (1) , A. Coves (1) , F. Mesa (2) , E. Bronchalo (1) , B. Gimeno (3) and V. Boria (4) [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] (1)
Dpto. de Ingenier´ıa de Comunicaciones. Univ. Miguel Hern´andez de Elche. 03203, Elche (Alicante), Spain. (2)
Dpto. de F´ısica Aplicada I. Univ. de Sevilla. 41012, Sevilla, Spain. (3)
Dpto. de F´ısica Aplicada y Electromagnetismo-Inst. de Ciencia de Materiales. Univ. de Valencia. 46100, Valencia, Spain. (4)
Dpto. de Comunicaciones. Univ. Polit´ecnica de Valencia. 46022, Valencia, Spain.
Abstract —The goal of this paper is to study the electrostaticfield due to an arbitrary charge distribution on a dielectriclayer in a dielectric-loaded rectangular waveguide. In orderto obtain this electrostatic field, the potential due to a pointcharge on the dielectric layer is solved in advance. The highcomputational complexity of this problem requires the use ofdifferent numerical integration techniques (e.g. Filon, Gauss-Kronrod, Lobatto, ...) and interpolation methods. Using theprinciple of superposition, the potential due to an arbitrarycharge distribution on a dielectric layer is obtained by addingthe individual contribution of each point charge. Finally, anumerical differentiation of the potential is carried out to obtainthe electrostatic field in the waveguide. The results of thiselectrostatic problem are going to be extended to model themultipactor effect, which is a problem of great interest in thespace industry.
I. I
NTRODUCTION
The calculation of the electrostatic field, E dc , in adielectric-loaded waveguide due to an arbitrary charge dis-tribution on the dielectric layer is a problem of great interestin the space industry, because of the lack of rigorous studiesabout the multipactor effect appearing in dielectric loadedwaveguide-based microwave devices in satellite on-boardequipment. When dealing with a partially dielectric-loadedrectangular waveguide, the electrons emitted by the dielectricsurface charge the dielectric material positively, whereas theelectrons absorbed by the dielectric layer charge it negatively.This charge gives rise to an electrostatic field which has to betaken into account in order to obtain an accurate trajectory ofthe electrons in the structure.Lots of works have studied the electrostatic field appearingon RF dielectric windows [1]–[8], but not so many works havestudied the electrostatic field appearing during a multipactordischarge in dielectric loaded waveguides [9]–[11].Although the problem of obtaining the electrostatic fieldoriginated by an arbitrary electron charge distribution has beensolved in many electromagnetism books [12]–[14], this is thefirst time that the problem under consideration in this work isrigorously solved, to the best of the authors’ knowledge.The paper is organized as follows: Section II describesthe theory and fundamental principles underlying the problemunder investigation. In Section III, the results obtained for thesolution of the electrostatic problem, including three differentcharge distributions on the dielectric layer are shown. A fewconcluding remarks are made in Section IV. II. T HEORY
In Fig.1 it is shown the transverse section of the waveg-uide under study, consisting on a partially dielectric-loadedrectangular waveguide, whose dielectric layer has relativepermittivity (cid:15) r and thickness h . The aim is to compute the Fig. 1. Geometry and dimensions of the problem under investigation. electric field at the observation point (cid:126)r = ( x, y, z ) , which isassumed to be located in the air region of a waveguide withtranslational symmetry along the longitudinal direction z , dueto a point charge on the dielectric layer at (cid:126)r (cid:48) = ( x (cid:48) , , .In order to determine the electric field (cid:126)E ( x, y, z ) = −∇ φ ( x, y, z ) , first the potential due to the point charge iscalculated according to Laplace’s equation for the electrostaticGreen’s function [15], ∇ · (cid:15) r ( (cid:126)r ) ∇ G ( (cid:126)r ) = − (cid:15) δ ( x − x (cid:48) ) δ ( y ) δ ( z ) (1)The geometric characteristics and the linear nature of theproblem makes that the Dirac delta functions can be expressedas, δ ( x − x (cid:48) ) = 2 a ∞ (cid:88) n =1 sin( k xn x ) sin( k xn x (cid:48) ) (2) δ ( z ) = 12 π (cid:90) ∞−∞ e − jk z z dk z (3)where k xn = nπa . The above expressions come from thefact that the eigenfunctions of the differential operator aresinusoidal functions along x -axis and complex exponential a r X i v : . [ phy s i c s . c l a ss - ph ] M a y unctions along z -axis, respectively. This is equivalent toapply the discrete sine transform (DST) along x -axis and theintegral transform along z -axis, G = 1 πa (cid:90) ∞−∞ dk z e − jk z z ∞ (cid:88) n =1 sin( k xn x ) sin( k xn x (cid:48) ) ˜ G (4) ˜ G = (cid:90) ∞−∞ dze jk z z ∞ (cid:88) n =1 sin( k xn x ) sin( k xn x (cid:48) ) G (5)where G = G ( x, x (cid:48) , y, z ) and ˜ G = ˜ G ( k xn , k z ; y ) .According to the above considerations, Eq.(1) can be ex-pressed as, (cid:26) ∂∂y (cid:15) r ( y ) ∂∂y − k t (cid:27) ˜ G = − δ ( y ) (cid:15) (6a) ˜ G ( y = − h ) = 0 (6b) ˜ G ( y = H ) = 0 (6c)where k t = k xn + k z . Solving Eq.(6), the following spectralGreen’s function, ˜ G , is obtained in the air region y ≥ , ˜ G = sinh[ k t ( H − y )] (cid:15) k t [ (cid:15) r coth( k t h ) + coth( k t H )] sinh( k t H ) (7)and the Green’s function, G , is achieved by replacing Eq.(7)into Eq.(4). G = 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 H )] sinh( k t H ) dk z (8)The high computational complexity of Eq.(8) requires theuse of different numerical integration techniques (e.g., Filon,Gauss-Kronrod, Lobatto, ...). Because of the rapid oscillationof the integrand for large values of z Filon’s integrationmethod is chosen since it is desirable for integrals [16], (cid:90) ab f ( x ) cos( kx ) dx (9)Using superposition, the potential due to an arbitrary chargedistribution on a dielectric layer is obtained by adding theindividual contribution of each point charge. φ ( x, y, z ) = (cid:90) G ( x − x (cid:48) , y, z ) ρ ( x (cid:48) ) dx (cid:48) (10)Finally, a numerical differentiation of the potential is carriedout to obtain the electrostatic field in the waveguide.III. N UMERICAL RESULTS AND DISCUSSION
This section shows the results obtained for a dielectric-loaded rectangular waveguide in Fig.1. An algorithm basedon the expressions given in Section II has been programmedusing Matlab to provide the results outlined next.First, in order to validate Eq.(8), the potential in the air re-gion due to a point charge between two infinite homogeneousmediums ( (cid:15) r = 1 and (cid:15) r ) is used as a benchmark, φ = 14 π(cid:15) (cid:15) r (cid:112) ( x − a ) + y + z (11) The results of Eq.(8) should approach Eq.(11) if the di-mensions a , h and H are chosen so that the point chargeand the observation point are far enough from the walls ofthe waveguide. In this case, the following parameters areconsidered: a = 600 mm, H = 250 mm, h = 250 mm, x = 305 mm, y values from mm to mm with mmwidthstep, z = 5 mm, x (cid:48) = 300 mm and (cid:15) r = 2 . . Asshown in Fig.2, the results of Eq.(8) and Eq.(11) agree aslong as the observation point is far enough away from the topwall, i.e. y = 25 mm approximately. However, beyond this y value, as the observation point approaches the top wall, theapproximation is no longer valid and discrepancies appear. Fig. 2. Comparison of the potential in the air region due to a point chargebetween two infinite homogeneous mediums (dashed line) vs Green’s functionof the problem under study (black line).
As discussed in the previous section, the high computa-tional complexity of this problem requires a detailed analysisof the parts forming the solution. In particular, it is useful tounderstand the spectral Green’s function, Eq.(7), with respectto the integration variable k z . In this case, the followingparameters are considered: a = 20 mm, H = 5 mm, h = 5 mm, z = 3 mm, n = 1 and (cid:15) r = 2 . . In terms of the rateof convergence, the worst scenarios are for the cases of low y values and high n values. In Fig.3, (cid:12)(cid:12)(cid:12) (cid:15) ˜ G (cid:12)(cid:12)(cid:12) for the case of y = 0 . mm and n = 1 is plotted. As it is shown, k z ≥ × has to be considered to reach convergence. (cid:12)(cid:12)(cid:12) (cid:15) ˜ G (cid:12)(cid:12)(cid:12) for the caseof y = 0 . mm, and n = 500 is plotted in Fig.4. In thiscase, k z ≥ × is needed. The asymptotic behavior of theintegrand, determined by the term e − k t y , allows us to establisha condition to stop the computation when the convergence isreached. It involves calculating the relative value of the i-th summand of the integral with respect to the accumulatedvalue of the integral until this iteration. If this relative value isless than a particular convergence tolerance, the computationof the integral is stopped. On the other hand, regarding theconvergence of the series in Eq.(8), it depends on the productof two sinusoidal functions and no asymptotic behavior canbe observed in this case. For this reason, in order to ensurethat the convergence is reached, the series is decomposed intoa sum of partial series of ten terms each of them. The relativevalue of the i-th partial series with respect to the accumulatedvalue provides the stop condition.Once the solution of the electrostatic problem, Eq.(8), has ig. 3. (cid:12)(cid:12)(cid:12) (cid:15) ˜ G (cid:12)(cid:12)(cid:12) for the case a = 20 mm, H = 5 mm, h = 5 mm, y = 0 . mm, z = 3 mm, n = 1 and (cid:15) r = 2 . .Fig. 4. (cid:12)(cid:12)(cid:12) (cid:15) ˜ G (cid:12)(cid:12)(cid:12) for the case a = 20 mm, H = 5 mm, h = 5 mm, y = 0 . mm, z = 3 mm, n = 500 and (cid:15) r = 2 . . been validated and a convergence study has been carriedout, the next step is to consider a problem with real di-mensions. For this particular case, the parameters chosenare: a = 19 . mm, H = 10 . mm, h = 0 . mmand (cid:15) r = 2 . . The origin has been located in the cen-ter of the waveguide. A total of eleven equidistant pointcharges has been considered in the calculation. The elec-trostatic field in a z-plane constant containing the pointcharges, z = 0 , due to an uniform charge distribution [ q i =(1 , , , , , , , , , , · e ], a triangular charge distribution[ q i = (1 , , , , , , , , , , · e ] and a gaussian chargedistribution [ q i = (0 , , , . , . , , . , . , , , · e ] onthe dielectric layer are shown in Fig.5, Fig.6 and Fig.7respectively, where e is the electron charge. As the observationpoints approach the dielectric layer where the point chargesare located, the electrostatic field intensity becomes higher.Furthermore, as expected, a symmetrical behavior with respectto the central x-axis, xa = 0 , is observed in all cases.IV. C ONCLUSION
In this work, a method for calculating the electrostatic fieldin a dielectric-loaded waveguide due to an arbitrary chargedistribution on the dielectric layer has been shown. The highcomputational complexity of this problem requires the use ofdifferent mathematical techniques to minimize computationtime. For this purpose, it is recommendable to carry out aconvergence study of the problem under study. There may becritical points, e.g. close to the walls or the dielectric layer, inthe structure in which the solution does not converge properly.In these cases, it is proposed to calculate the solution at pointsclose to them and apply extrapolation techniques.The results of this electrostatic problem are going to beextended to model the multipactor effect, which is a problemof great interest in the space industry.
Fig. 5. Electrostatic field due to an uniform charge distribution in a dielectric-loaded WG90.Fig. 6. Electrostatic field due to a triangular charge distribution in a dielectric-loaded WG90.Fig. 7. Electrostatic field due to a gaussian charge distribution in a dielectric-loaded WG90. A CKNOWLEDGMENT
This work was supported by the Ministerio de Econom´ıay Competitividad, Spanish Government, under the coordi-nated project TEC2013-47037-C5-4-R, TEC2013-47037-C5-1-R and TEC2013-41913-P.
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