A statistical characterization of resonant electromagnetic interactions with thin wires : variance and kurtosis analysis

A statistical characterization of resonant electromagnetic

interactions with thin wires : variance and kurtosis analysis

Citation for published version (APA):

Sy, O. O., Beurden, van, M. C., Michielsen, B. L., Vaessen, J. A. H. M., & Tijhuis, A. G. (2008). A statistical characterization of resonant electromagnetic interactions with thin wires : variance and kurtosis analysis. In J. Roos, & L. R. J. Costa (Eds.), Scientific computing in electrical engineering SCEE 2008 (pp. 117-124). (Mathematics in Industry; Vol. 14). Springer. https://doi.org/10.1007/978-3-642-12294-1_16

DOI:

10.1007/978-3-642-12294-1_16

Document status and date: Published: 01/01/2008

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the 3rd order TSE scheme is depicted in Fig. 5. We observe that the trajectory de viates clearly from a closed orbit and is damped over the time to the center of the separatrix. Increasing the Taylor expansion up to order~‘ 5 the scheme catch the

problem, resulting in a stable closed orbit solution as it is seen in Fig. 6. The shown particle orbit is identical with this one computed with the area preserving classical leap-frog scheme presented in [4].

4 Conclusion and Outlook

The phase space coordinates of charged particles driven by the Lorentz force are nu merically computed up to sixth order by a new high order particle (HIOP) method based on truncated Taylor series expansion (TSE) in time. Numerical results ob tained from three simulation experiments clearly demonstrate the great potential of the proposed TSE approach. For both non-relativistic and relativistic test cases the numerical TSE results for ,)t~ > 5, are in very good agreement with the available analytic solutions. The capability of the TSE schemes is also proved in the compli cated test case of non-linear electromagnetic field. Furthermore, we observe from experimental order of convergence studies that the design order of all schemes are very close to the formal order of the proposed approach. The test stage of the stand alone HIOP solver draw to a close and the module should be applied as an attractive alternative to the Boris leap-frog solver in the existing Maxwell-Vlasov module in near future. Clearly, this accounts for a multitude of numerical standard tests to enhance the status to a verified method for scientific application of the new TSE approach and to establish an attractive high order alternative to the second order classical leap-frog method.

Acknowledgements We gratefully acknowledge the Landesstiftung Baden-Wurttemberg who funded the development within the program “Modeling and Simulation on HighPerformance Com

puters”from 2003—2005 and theDeutscheForschungsgemeinschaft (DFG) for funding within the project “Numerische Modellierung und Simulation hochverdtinnter Plasmastromungen”.

References

1. T. Schwartzkopff, F Lorcher, C.-D. Munz, and R.Schneider, Arbitrary High Order Finite Volwne Methods for Electromagnetic Wave Propagation, Computer Physics Communications, 174:689—703, 2006.

2. C.-D. Munz, P. Omnes, R. Schneider, E. Sonnendrucker and U. Vo13, Divergence correction techniques for Maxwell solvers based on a hyperbolic model, J. Comput. Phys., 161:484—511,

2000.

3. J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1999.

4. V. Fuchs and JR Gunn, On the Integration of Equations of Motion for Particle-in-Cell Codes, J. Comput. Phys., 214:299—3 15,2006.

A Statistical Characterization of Resonant

Electromagnetic Interactions with Thin Wires:

Variance and Kurtosis Analysis

0.0. Sy, M.C. van Beurden, B.L. Michielsen, J.A.H.M. Vaessen, and A.G. Tijhuis

Abstract A statistical characterization of random electromagnetic interactions af fected by resonances is presented. It hinges on the analysis of the variance and the kurtosis to assess the intensity of the resonances. The method is illustrated by the study of a randomly varying thin wire modeled by a Pocklington integral equation.

1 Introduction

Interactions between electronic devices and electromagnetic sources in their envi ronment are of prime importance in EMC models for design or maintenance studies. A convenient way to model such interactions is based on the multi-port models of both the electronic components and the interconnect networks making up the com plete system. In principle, both types of multi-port models need extensions, in the form of Thdvenin or Norton sources, accounting for the presence of exterior sources of electromagnetic fields. In practice, the sources added to the interconnect sub system are the dominant ones because of the greater geometrical size of the printed wirings compared to the size of the electronic devices. This is even more so when exterior cables come into play.

The range of validity of these models depends on their ability to accurately rep resent an ensemble of configurations. For non-resonant systems, the study of a few configurations provides a good picture of the overall interaction. However, for res onant phenomena, the number of configurations needed can increase drastically. Instead, a stochastic approach yields a more suitable quantitative and qualitative

0.0. Sy, MC. vanBeurden,J.A.H.M. Vaessen, AG. Tijhuis

Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, The Netherlands, e-mail: o.o.sy@tue.nl, m.c.v.beurdenfgtue.nl, j.a.h.m.vaessen@tue.nl, a.g.tijhuis@tue.nl

B.L. Michielsen

ONERA, 2, avE. Belin, 31055 Toulouse Cedex, France, e-mail: bastiaan.michielsen@ onera. if

J. Roos and Luis R.J. Costa (eds.), Scientific Computing in Electrical Engineering S~EE

2008,Mathematics in Industry 14, DOl 10.10071978-3-642-12294-l_16,

© Springer-VerlagBerlin Heidelberg 2010

p

0.0. Syet al. model. Stochastic methods are frequently used in fields as diverse as rough-surface scattering problems [1] and Mode-stirred-Chamber theory [2]. In EMC, random models have been applied to undulating thin-wire setups modeled by transmission-line theory [3], [4], or by integral equations [5]. In all these cases the aim is to quantify the uncertainty of the response parameters, or “observables”, by their aver age and variance. Although these statistics provide bounds for the observable, they do not inform on the presence of extreme values beyond these bounds.

Estimating the probability that an observable will have values beyond a certain distance from the average is important in “risk assessment”. Reliable estimates need a good approximation of the entire probability distribution, which is generally im possible to obtain. Gaussian distributions can be fitted by looking only at the first two moments and therefore provide easy estimates. The next few moments are qual itative indicators of the suitability of such fits [6]. This paper shows that the kurtosis should be investigated to identify significant deviations from the Gaussian distribu tion near “risky” resonance conditions.

The outline of this paper is as follows. Section 2 describes the general setup which involves the integral-equation model of a thin wire over a ground-plane. A random parametrization of the problem in Section 3 allows for the definition of the statistical moments of interest, viz, the average, the variance and the kurtosis. All these moments are computed by a sparse-grid quadrature rule, which efficiently han dles integrals over multi-dimensional domains. The importance of these moments in characterizing electromagnetic interactions is illustrated in Section 4 through the example of a roughly undulating transmission line illuminated by a plane wave.

2 Deterministic Configuration

The purpose of this paper is to show that in electromagnetic interaction configu rations with stochastic geometries, the value distribution of observables shows a peculiar behavior near resonance conditions which necessitates the computation of higher order moments, like the kurtosis, before a reliable interpretation of the results can be established. For that purpose, we choose a simple one-port system, consisting of a perfectly conducting wire Sn over a ground plane, in an incident plane wave as shown in Figure 1. The vector a gathers all the variables controlling the geometry of the wire. The electromagnetic coupling itself is observed through the equivalent Thévenin voltage sourceVe(U) induced at the port of Sn and defined as

Ve(a)=_±/ jcx~E’,

‘~.

where Ja is the current distribution flowing on the device in absence of E’, when a current source ‘~ is applied at the port of the wire [7]. This current Ja follows by solving a frequency-domain electric-field integral equation (EFIE) representing the wire in a transmitting state [5]. The resonances appear at frequencies where a wave,

~Ve(~y4;~jy

1/

X/ ~im

propagating along the waveguide formed by the wire and the ground plane, becomes resonant due to the boundary conditions at the wire extremals.

In spite of its simplicity, this configuration, derived from an EMC benchmark [8], is representative for a large class of interaction problems, for example the common-mode interference appearing at the connection of a power cable to a printed circuit board or certain types of wire antenna problems.

3 Random Parameterization

When an ensemble A of configurations is considered, computing Ve(a) for each element a of A can be very costly numerically. Instead, the variations of a in A are viewed as random according to a known distribution p~. The voltage Ve(a) then becomes a random variable, with statistical moments, such as its mean E[Ve] and its standard deviationU[Vi], defined as

E[Ve] /Ve(a’)Pa(a’)da’~ VE[IVeI2j_IE[Ve]12≥0.

The standard deviation a[Ve] is a positive parameter measuring, in volts, the spread

of Ve around IE[Ve],as can be seen from Chebychev’s inequality [9].

Extreme values of Ve, at least 4a [Ve] away from E [Ve], are accounted for by the kurtosis

ic[IVeIj,

which is a dimensionless positive moment defined as

VeI

[IV~I]\~l

~[IVeI]=E[(

11>0.

U[IVeI]

I

j

—

Gaussian random variables, which have approximately 95% of their values within a distance of 2a to their average, have a kurtosisof 3. Hence, the higher the value

118 _{Stochastic Thin-Wire Electromagnetic Interactions: Variance and Kurtosis Analysis}

E’

119

Fig. 1: Undulating thin-wire over a PEC plane

(1)

(2)

(3)

ofIC[lVel] above 3, the more occurrences ofVe with very large magnitude are to be expected.

Equation (2) shows that all the statistical moments are defined by integrals in volving a known integrand which depends onVe,and over the same support A. These integrals can therefore be computed numerically by quadrature rules. Moreover, a significant gain in computation time is achieved by re-using the same samples ofVe

to compute the different integrals in Equations (2)-(4).

The most straightforward generalization to integration over higher dimensional spaces, consists in using the Cartesian tensor product of a univariate quadrature rule. However, this leads to a “curse of dimensionality” [10], i.e. exponentially grow ing numbers of grid points and hence prohibitive numbers of evaluations. More over, such Cartesian product rules are not isotropic, i.e. in certain directions of a d-dimensional space, the accuracy is of much higher degree than in other directions. Algorithms, such as Sparse grid (SG) methods, have been found which allow for the elimination of grid points while preserving exact integrals of polynomials up to a given degree in any direction. As such SG methods can be regarded as multidimen sional generalizations of Gaussian-type integration rules defined in one dimension. For integrals over moderately dimensioned spaces(d< 10), the convergence rate of

the SG rule is faster than a Monte-carlo approach. In addition, SG rules take advan tage of the smoothness of the integrand, unlike Monte-Carlo rules [11]. In this paper, a SG rule is employed which starts from a 1D Clenshaw-Curtis quadrature rule and applies Smolyak’s algorithm to build the multidimensional quadrature rule [12].

4 Results

With reference to Figure 1, a roughly undulating thin wire is studied with a geometry defined as

The vector of amplitudes a = (a1 ,a2) has independent and uniformly distributed

components in the domainsAj =A2 =[—3;3] cm. The average geometry therefore

corresponds to the straight wire So. The incident field is a 0-polarized plane wave with an amplitude of I V.m~, and propagating in the direction 0~= 450, ~ =00.

A single computation of the induced voltage amounts to 0.1 second. All the sta tistical moments are computed for 50 frequencies between 100 MHz and 500 MHz, with a relative error below 1%. The number of function evaluations ranges from

N,1111~ =321 (m 32 seconds) at regular frequencies, toN,,iat=7169 (m 12 minutes)

at resonance frequencies, with an average ofNa,,=3782 values per frequency (m 6

minutes). This appreciable performance is primarily dictated by the integral defining

~C[lVel],as it converges slower than a[Ve], which itself converges slower thanE[Ve].

4.1 Average E [Ve] and Standard Deviation a

[Ve]

First, the voltage Ve(0) corresponding to the average configuration is compared to the average of the voltage E[Ve]. In a perturbation-like approach, Ve(0) would be considered as the average ofVe, and local expansions would be performed around

Ve(0) to represent the global variations ofVe [5]. Figure 2 points out the clear differences between IVe(0) and

IE [V~]l,

mainly concerning the position of their extrema. These discrepancies back the need to take the true variations of Sa into account when computing the statistics ofVe. The effect of the variations of Sa on

Fig. 2:I~’e(0)~ (circled line), IS[V~J (dashed line) anda[V~1 (solid line)

vs

frequency

Ve is also indicated by the standard deviation which is depicted in Figure 2. At regular frequencies, a[Ve] is of the order of 30 mV, but increases by several orders of magnitude around the resonance frequencies. This plot reveals three resonance regions with increasing widths viz.~ [175; 2151 MHz,~2 [295; 3501 MHz and

[415; 480] MHz. The intensity of the resonances decreases with the frequency: The peaks of a[V~] go from 16.120 V in~, and2.227 V in~2to 0.666 V in~

High values of a EVe] indicate a high physical variability ofVe around its average E[Vei. However, the increased uncertainty ofVecan be caused either by a smooth distribution ofVe around E[Ve], or, by the presence of a few very large samples of

Vgcoexisting with a cluster of samples aroundE[Ve].The distinction between these two cases is possible thanks to the analysis of FC[l

Veil.

The kurtosis

~~[I

Veil is displayed in Figure 3 together with the standard deviation

a

[Vel.Since

FC[lVel]

is seldom equal to 3, the assumption of a Gaussian distribution ofVeis generally inaccurate. [V] x0(y)

a1

sin(Siry), 10100 150 200 250 300 350 400 450 500 f [MHz] za(y) 5 + a2sin(9ny) in cm. (5)

4.2 Kurtosis

1C[~Ve~]

0.0. Sy et al. _{Stochastic Thin Wire Electromagnetic Interactions: Variance and Kurtosis Analysis}

Table1: Statistical moments at given frequencies in~2

IEVe aV~[V~I1

f~ 300MHz 0.025-jO.lO7V 1.131 V 402

f2 342 MHz -0.239+j0.160 V 0.822 V 7

10100 150 200 250 300 350 400 450

50b0

f [MHz]

The behavior of 1C[iVeil roughly follows that of

a

[Ve]. Nevertheless, ‘C Ve pro vides a finer characterization ofVe than a Ve as it reveals the different types of sample distributions within a single resonance region. In.~?2for instance, between

295MHz and 320 MHz,

a

[Vej rises from 40 mV to 2.277 V indicating an increase in the physical uncertainty ofVe.However, the variations of

1C[I

Veil reveal that the ef fect of the extreme samples is mainly dominant at 306 MHz where IC[iVeil 5415. Between 330 MHz and 350 MHz, in spite of a high value of a[Ve] 1V, ~C[IVe

drops below 15, thereby highlighting a smoother distribution ofVearound E[Vel. A similar analysis can be conducted in~ and~.

4.3 Comparison with Deterministic Samples

To confirm the observations based on the analysis of Figure 3, ~ deterministic samples have been computed at the frequencies specified in Tables 1 and 2. These samples are normalized as follows

= Ve 1E[Vel with E[V,I} —0 and a[V,1] 1.

The statistical properties of the normalized samples can thus be compared on a com mon ground. In Figures 4a and 4b, concentric circles are shown, which correspond to the normalized samples with distances of 4a[Vel and 8a Ve} from IE V

First, two frequenciesfl=300 MHz andf2=342MHz are considered in the reso

nance domain.~2.As can be seen in Table 1,

a[Vel

has comparable values at the two frequencies, with

a[Velfi

>

a[Ve]f2.

Nonetheless 1C[i

Veil

is two orders of magnitude

larger at fi than atf2.

2C =409 MHz 15 3 =475 MHz 10 5 0 —5 0 —15 • ~2O—15 —10 —~

6

5 10 15 20 Re(V) (a) (b)

Fig. 4: Normalized samples V,, forfi andf2(a) and forf andf b

Next, the resonance domain.~ is analyzed at the two frequencies

f~

=409 MHz

and

f~

475 MHz as detailed in Table 2. The standard deviation a

[Velf3

is more

Table2: Statistical moments at given frequencies in ~ 1EV OV ~[VeI]

than 7 times smaller than a Velf4,thus the physical dispersion ofVeis more intense at f4. Conversely 1C V~ ~ is approximately 30 times larger than

1C[IVeIlf4

which implies a much wider statistical spread at f~. These predictions are confirmed in Figure 4b: all the samples corresponding to

f~

are clustered within 4a[Ve] of IE V

unlike the samples atf~,which can lie more than 20 a V away from the average. The non-negligible statistical uncertainty ofVeat

f~,

indicated by ic[iVeil,could not have been foreseen by the sole study of a V

102 io4 10’ .~ l0~ —~ 10°

1

“~ ~ l0~ > . —l . I 10 ~,

/

~. . . ‘ 10

Fig. 3: c~[Ve](dashed line)and K V~ (solid line) vsfrequency

The normalized samples depicted in Figure 4a confirm that the samples Ve are statistically more dispersed at fi than at f2: at fi Ve takes extreme values up to 20a V away from IE[Vel,whereas atf2,all the samples are within 5 a[Velof IE V

> C 20 15 l0 5 0 —5 —10 —‘5 ~-3:MH,

•~:.::~. f4

> a ~0—I5 0—5 0 5 Re(V) 10 15 20 (6) f~ 409 MHz -0.004-j0.065 V 0.093 V 89 fi 475 MHz -0.232+j0.096 V 0.632 V 3

5 Conclusion

Part II

Circuit Simulation

The results obtained for the varying thin-wire setup have revealed situations where, for high as well as low values of the standard deviation, a highly unsymmetrical dis tribution of the values around the average appears. Such casesarecorrectly signalled by high values of the kurtosis. Estimation of the probability of system failure con ditions in such situations must therefore account for significant deviations from the Gaussian distribution. These statistical indicators can be determined numerically by quadrature rules such as a sparse-grid rule which outperforms a Monte-Carlo rule, for integrations over domains having moderate dimensions (below 10). A hierar chy has been observed in the computation of the statistical moments, as the average converges faster than the variance which, in turn, converges faster than the kurtosis. The analyses of the standard deviation and of the kurtosis are complementary: the variance is useful in a dimensioning process as it measures the physical variations of the voltage, whereas the kurtosis is valuable in a protection stage to foretell extreme values of the response parameter, which could damage the receiving device.

Acknowledgements This workis funded by the DutchMinistry of EconomicAffairs, in the In novation Research Program (TOP) number EMVT 04302. The authors are also thankful to Dr. John Burkardt for his assistance in implementing the sparse-grid quadrature rule.

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