SEARCH

SEARCH BY CITATION

Keywords:

  • stochastic;
  • electromagnetic;
  • principal components

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

[1] The usefulness of stochastic methods to efficiently quantify uncertainties in computational models of electromagnetic interactions is illustrated. A refined study of the second-order moments of a complex-valued Thévenin model, which represents the coupling between a wire structure and a time-harmonic electromagnetic field, is presented. The configuration of a stochastically undulating thin wire illuminated by a stochastic incident plane wave is investigated in detail. Three computational methods are used to evaluate the mean values and covariance coefficients of the observable: a straightforward Cartesian-product quadrature method, a Monte-Carlo method, and a space-filling-curve method. The underlying patterns of the randomness are revealed by analyzing the covariance matrix' principal components. The study of this interaction configuration shows some general characteristics, which are expected to show up in any stochastic electromagnetic interaction problem. In particular, the results indicate that fluctuations in self-interaction coefficients (impedances) have distinct features and are quite different from the coefficients describing the interaction with externally generated fields (voltages).

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

[2] Modeling the interaction between electromagnetic waves and material objects is a topic of interest in fields as diverse as radio astronomy, biomedical engineering, and electromagnetic compatibility (EMC). In EMC, the study of such interactions is a crucial part of the design of electronic devices, to investigate their immunity to intentional or parasitic electromagnetic fields emanating from internal components of highly integrated circuits, or from external electromagnetic sources [Sun et al., 2007; Perez, 2008].

[3] Generally, the models of these interactions are parameterized by a set of inputs describing the interaction configuration. The outputs, also known as the observables, can be as diverse as scattered-field amplitudes, impedances, or the induced voltage at the port of an electronic interconnect system.

[4] In practice the input parameters can exhibit a variability due to, e.g., noise in the measurement of the inputs, to general model assumptions, or to manufacturing defects. Modeling the corresponding variation of the observables by a repeated execution of the model for each possible configuration, can be tedious, firstly due to the numerical effort required by these simulations, secondly because of the need to postprocess the collected data.

[5] A stochastic analysis, where the variations of the unknown inputs are assumed to be random, is an appealing alternative. It implies that the observables become random variables and probability theory can be used, in principle, to compute the distribution of their values. In practice, the mathematics of the explicit dependence of the observables on the configuration parameters are often too complicated to carry out, particularly in the presence of numerical models. Restricting the stochastic configuration to small variations of the inputs around a nominal situation gives more opportunities to construct the probability distributions of the observables [Ajayi et al., 2008; de Menezes et al., 2008]. However the “small-variation” hypothesis limits the scope of the conclusions drawn from such an analysis.

[6] Instead of aiming for the total probability distribution of the observable, its statistical moments can be computed by quadrature rules over the probability space parameterizing the interaction configuration. Hence, rather than precomputing and subsequently postprocessing a large amount of data, the objective of such a stochastic approach is to efficiently compute the statistical moments via the study of a limited number of sample configurations. Through such a rationale, stochastic fluctuations can be handled without restrictions on their amplitude. Once these moments are available, they provide valuable information on the observable, which is generally valid, i.e., the same information would be obtained if a large amount of configurations would have been studied in detail.

[7] Stochastic approaches have previously been proposed in rough-surface scattering problems [Brown, 1985], and in mode-stirred-chamber theory [Hill, 1998]. Interactions between electromagnetic waves and configurations of wires, which form a topic of prime interest in EMC, have also been studied from a stochastic point of view [Bellan and Pignari, 2001; Michielsen, 2005; Pignari, 2006]. These configurations are for instance present in wiring systems of integrated circuits or in harnesses of vehicles.

[8] However, all these works aim either at characterizing real-valued observables such as the amplitude of the voltage or the current induced at a given port, or at studying complex-valued observables only via their mean and their variance. In our case, where the interactions are formulated in the frequency domain, the observables are exclusively complex-valued. Although the average and the variance provide valuable information on the distribution of the observable values, the variance does not inform on finer details of the spread of the values in the complex plane. In the most general case, where real and imaginary parts are not related, such an approach leads to some loss of statistical information on the original complex variables. Such loss of information can prove penalizing for instance in impedance adaptation studies, which require a distinction between the resistive and reactive parts of the impedance to optimize power transfers.

[9] In the present paper, the aim is to refine the statistical analysis based on the computation of first- and second-order statistical moments of a complex random observable. To this end, the observable is handled as a random equation image2 vector, with its real and imaginary parts as the vector components. The average vector and the covariance matrix of this vector are computed efficiently by a space-filling-curve quadrature rule [Cukier et al., 1973], which is tailor-made for higher-dimensional parameter spaces. This rule offers advantageous performances both in terms of complexity and in terms of convergence properties, when compared to a deterministic Cartesian-product rule and a Monte-Carlo approach. With the covariance matrix at hand, its underlying patterns can be revealed by determining and analysis its principal components.

[10] The outline of this paper is as follows. The configuration of a deterministic electromagnetic interaction between a thin-wire structure and an incident field is first described in section 2. The observables are chosen to be the coefficients of the equivalent Thévenin network model, i.e., a voltage source Ve and an impedance Ze. In section 3, the parameters of the interaction are randomized by regarding the wire geometry, the incident field, as well as the observables Ve and Ze as stochastic objects.

[11] This stochastic parametrization allows for the definition of the average and the covariance of the observables. Next, the covariance matrices are spectrally analyzed to derive an optimal decomposition of Ve and Ze, which eases the expression and the interpretation of their statistical properties. Section 4 presents the quadrature methods employed to efficiently compute the statistical moments of the observable. The quadrature rules range from a deterministic Cartesian-product rule and a Monte-Carlo rule, to a space-filling-curve rule. The results provided in section 5 refer to a fully stochastic interaction that involves a randomly undulating thin wire under a random incident field. Conclusions are then drawn in section 6.

2. Deterministic Setup

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

2.1. Interaction Configuration

[12] As displayed in Figure 1, the interaction studied in this paper involves a perfectly conducting thin wire over a metallic ground plane and an electromagnetic field representative of the excitation originating from the environment of the scatterer. This setup lies in free-space and is studied in the time-harmonic regime.

image

Figure 1. Interaction configuration: scattering wire Sα and observation region P under illumination Eβi.

Download figure to PowerPoint

[13] The scatterer consists of two 5 cm long vertical thin wires, one of which contains a 2 cm port region denoted by P. These thin wires, which have a circular cross section with a diameter of 1 mm, are connected below to an infinite metallic ground plane lying in the xy plane, and above by a thin wire. Over a distance of 1 m, the axis of this thin wire undulates and is described with respect to the ground plane D by a smooth mapping ρα defined by

  • equation image

in Cartesian coordinates. The mapping ρα depends on the vector α ∈ �� ⊂ equation imagem, which parameterizes the deformation of the domain D. The wire is called Sα to mark its dependence on α.

[14] In a similar way, a family of incident fields is defined, which is parameterized by the vector β ∈ ℬ ⊂ equation imagen, determining the polarization and the direction of propagation of the incident field

  • equation image

We shall write Eβi(r) for the incident electric field corresponding to the parameter values βequation image, as a function of the spatial coordinates r.

[15] A given interaction configuration is therefore entirely determined by the specification of the input vector denoted γ = (α, β) ∈ �� × ℬ, which contains the parameters of Sα and of Eβi.

2.2. Response Parameters: Equivalent Thévenin Network

[16] At the port P of the wire, the interaction between Sα and Eβi is observed through the equivalent Thévenin network, which is composed of an ideal voltage source Ve in series with an impedance Ze, as depicted in Figure 2.

image

Figure 2. Thévenin equivalent circuit.

Download figure to PowerPoint

[17] This generic network replaces the entire configuration made of Sα, the ground plane and Eβi. It is very helpful when trying to connect an electronic circuit to the port P. In an EMC context, the equivalent network allows the study of the immunity of Sα to voltages induced at P by external sources, and the impedance matching of a device connected at P. For antenna design purposes, the equivalent voltage can be used to determine the radiation pattern of the scatterer considered as an emitting antenna, whereas the impedance acts like an antenna parameter informing on the stored and radiated power.

[18] Given a configuration γ = (α, β) ∈ ��, the Thévenin voltage Ve(γ) corresponds to the voltage induced by Eβi at the port P, which is in an open-circuit state. Under the assumption that the dimensions of P are small compared to the wavelength λ, and by using Lorentz' reciprocity theorem [Rumsey, 1954], Ve(γ) can be written as the follows

  • equation image

The transmitting-state current jα is induced on Sα, in absence of Eβi, when a current IP is applied at P, with uz the unit vector of the z axis.

[19] The equivalent impedance seen from P is also defined by considering the thin wire in a transmitting state. In this state, the voltage VZ existing at P is linked to the current IP at P and to the equivalent impedance Ze by Ohm's law

  • equation image

The voltage VZ is obtained similarly to Ve by replacing Eβi by the field E[IP], radiated by the current IP, in equation (3), which results in

  • equation image

Since Ze is defined in a configuration where Eβi is absent, it only describes the inherent electromagnetic properties of the scatterer Sα. Thus, a model in which Eβi and Ze are statistically independent is consistent with electromagnetic interaction theory. On the other hand, the Thévenin voltage source Ve and the impedance Ze are statistically dependent stochastic variables, in general.

2.3. Transmitting-State Current jα

[20] Both Ve and Ze depend on the transmitting-state current distribution jα, which is unknown. The value of jα on Sα is obtained by solving a Pocklington electric-field integral equation (EFIE), which is commonly used for thin-wire problems [see Mei, 1965; van Beurden and Tijhuis, 2007; Tijhuis and Peng, 1991]. This equation is established by writing the nullity of the total tangential electric field on the axis of the perfectly conducting wire. The resulting reduced-kernel EFIE is solved in its discrete form by a Galerkin method, which uses quadratic-segment basis functions defined on Sα [Champagne et al., 1992]. The choice for the reduced kernel and for these basis functions is motivated by the need for a time-efficient model for curved wires. The resulting approximation of jα reads

  • equation image

where [Z(α)] is the impedance matrix, which depends on the parameters α of the geometry, and which results from the discretization of the electromagnetic radiation operator on the set of expansion and testing basis functions. The vectors equation image and equation image contain the coefficients of jα and E[IP] on the basis.

[21] For a given α ∈ ��, the filling of [Z(α)] and the solution time for equation (6) has a given cost which should not be overlooked as it represents the essence of the numerical effort involved in the deterministic model.

2.4. Presence of Resonances

[22] The geometrical undulations of Sα modify both the total length Lα and the characteristic impedance of the thin wire. These modifications give rise to resonances that can be estimated via the theory of transmission lines. The wire setup depicted in Figure 1 can be regarded as a transmission line, which sees an open circuit at one end and a short circuit at the other end. At the port region, resonances characterized by large magnitudes of Ve and Ze will occur whenever [Grant and Phillips, 1990, p. 338]

  • equation image

Conversely, resonances characterized by low magnitudes of Ve and Ze will arise whenever [Grant and Phillips, 1990, p. 338]

  • equation image

3. Stochastic Parameterization of the Problem

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

[23] For the sake of simplicity, the stochastic method will be established for the observable Ve. The same rationale applies for the impedance Ze.

3.1. Randomization

[24] The deterministic model established in the previous section can be used to compute Ve(γ) for any given configuration specified by the input vector γ = (α, β) ∈ �� = �� × ℬ. When uncertainties affect the knowledge of the value of γ in ��, systematically computing all the possible values of Ve(γ), which form the set ��e = Ve(��), can be numerically costly, if not impossible and inefficient.

[25] Instead, γ's uncertainty in �� is treated as random according to a known probability distribution equation imageγ, or a known probability distribution function (pdf) fγ, when it exists. The components of γ can be made to be mutually statistically independent by applying a Karhunen-Loève transformation [Papoulis, 1991, p. 413].

[26] All the functions depending on γ then become random variables. The randomness of α induces the randomness of Sα, of jα and of the observables Ve and Ze. On the other hand, due to the randomness of β, the incident field Eβi, and, once again, Ve become random. The random fluctuations of Ve(γ) in ��e are fully characterized by the probability distribution equation imageequation image, which is unknown a priori and defined in terms of equation imageγ as [Bharucha-Reid, 1972, p. 36]

  • equation image

Given the definition of Ve in terms of γ via the solution of a boundary value problem, it is generally impossible to explicitly express the mapping Ve−1. In contrast, transmission-line theory provides analytical expressions linking Ve to γ, thereby easing the expression of equation imageequation image as a function of equation imageγ [Rannou et al., 2001; Bellan and Pignari, 2001].

[27] Nonetheless, the average of any measurable function h of Ve can be expressed as [Gubner, 2006, p. 83]

  • equation image

The right-hand side of equation (10) is an integral over a computable integrand h(Ve(γ′)) fγ(γ′) and a known domain of integration ��. It can be determined numerically, by applying a suitably chosen quadrature rule to the domain ��, as will be further discussed in section 4.

3.2. Scalar Second-Order Statistical Moments

[28] Since Ve is complex-valued, its average equation image[Ve] and its variance var[Ve] are defined as [Papoulis, 1991, p. 188]

  • equation image
  • equation image

The standard deviation σ[Ve] = equation image, which has the same dimension as Ve, physically quantifies the dispersion of Ve around equation image[Ve].

[29] From these moments, the normalized dimensionless variable Vn is defined as

  • equation image

with equation image[Vn] = 0 and σ[Vn] = 1. This normalization preserves the randomness of the original random variable, i.e., Ve and Vn have the same type of probability distribution. The normalization also enables comparisons on a common ground between the spreads of random voltages that have different statistical moments. For instance, the randomness of Ve at two different frequencies can be mutually compared as well as the spreads of Ve and Ze, even though they have different physical dimensions.

[30] Although equation image[Ve] and σ[Ve] fully characterize Gaussian random variables, the Gaussian nature of Ve cannot be taken for granted. Chebychev's inequality [Papoulis, 1991, p. 114] provides a more general bound for the distribution of Ve as it states that

  • equation image

for any m > 0. Chebychev's bounds are generally very loose for most probability distributions. However, the strength of this inequality resides in its general validity for random variables that have finite variances.

[31] Based on Chebychev's inequality, confidence domains equation imagem = {Ve∣ ∣Vequation image[Ve]∣ ≤ [Ve]} are defined as concentric discs centered around equation image[Ve] and with radii proportional to σ[Ve]. The normalization of equation imagem leads to a disk Cm centered around the origin and with a radius m. The circular shape of these domains highlights that σ[Ve] is an “isotropic” measure of the spread of Ve around equation image[Ve]. Thus, the randomness of Re(Ve) and Im(Ve) cannot be characterized individually by σ[Ve]. Such a characterization would however be very helpful, e.g., in impedance adaptation studies where both Re(Ze) and Im(Ze) are needed to match the equivalent impedance.

3.3. Vectorial Second-Order Statistical Moments

[32] Rather than handling Ve as a complex scalar, it is regarded as the real-valued random vector Ve = (Re(Ve), Im(Ve)). The first two statistical moments then correspond to the mean vector equation image[Ve] and the covariance matrix Cequation image, with [Papoulis, 1991, p. 190]

  • equation image
  • equation image

where Cov[X, Y] = equation image[XY] − equation image[X]equation image[Y] = Cov[Y, X]. The correlation coefficient ρ[Ve], which assesses the linear correlation between Re(Ve) and Im(Ve), is defined as [Papoulis, 1991, p. 152]

  • equation image

for σ[Re(Ve)] > 0 and σ[Im(Ve)] > 0.

3.4. Principal Components

[33] An optimal representation of Ve, on a basis of uncorrelated random variables, can be established by determining the principal components of Cequation image, the covariance matrix of Vn. Owing to its positive definite nature, Cequation image can be decomposed by applying an eigenvalue decomposition (EVD), i.e.,

  • equation image

with ordered eigenvalues λn,1λn,2 ≥ 0. The unit vectors u(1) = (u1(1), u2(1)) and u(2) = (u1(2), u2(2)) define orthogonal principal directions. The normalized voltage Vn is then projected on the basis (u(1), u(2))

  • equation image

with Vn(1) = u1(1) Re(Vn) + u2(1) Im(Vn) and Vn(2) = u1(2) Re(Vn) + u2(2) Im(Vn). The random variables Vn(1) and Vn(2) are uncorrelated and their variances are equal to λn,12 and λn,22, respectively. Hence, u(1) represents the direction with the highest dispersion, and, λn,12 (resp. λn,22) is the contribution of Vn(1) (resp. Vn(2)) to (σ[Vn])2.

[34] The orthogonal frame (equation image[Ve], u(1), u(2)) can be deduced from the canonical frame (equation image[Ve], Re(Ve), Im(Ve)) by a rotation over an angle

  • equation image

Chebychev's theorem applied to Vn(1) and Vn(2) leads to the definition of the interior of ellipses equation imagem as

  • equation image

for any m > 0. These ellipses are centered around the origin and have their axes parallel to u(1) and u(2), with lengths equation imageλn,1 and equation imageλn,2. Chebychev's inequality can then be recast into the form

  • equation image

If λ1equation imageλ2, the values of Ve are mainly distributed along a line parallel to u(1), and passing through equation image[Ve]. Then, the sole study of Vn(1) already yields a suitable picture of the spread of Ve around equation image[Ve]. Hence, when compared to σ[Ve], the analysis of Cequation image refines the quantification of the uncertainty of Ve.

4. Computation of the Moments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

[35] The statistical moments equation image[Ve], σ[Ve] and Cequation image are computed numerically by a quadrature rule ��L of level L that approximates the integral in equation (10) by a discrete sum

  • equation image

where Lequation image and the number of samples NL is an increasing function of L. The quadrature rule is fully defined by the abscissae ��L = {γnn = 1,…, NL} ⊂ ��, and the positive weights ��L = {wnn = 1,…, NL}. For stable quadrature rules, increasing L ensures a higher accuracy in the approximation of equation image[h(Ve)] by ��L. At the same time, NL represents the complexity of the quadrature formula as it corresponds to the number of evaluations Ve which, as previously mentioned, bears a certain numerical cost.

[36] A first step toward limiting NL consists in taking advantage of the definition of all statistical moments as integrals over the same support ��. The same samples Ve(��L) = {Ve(γn) ∣ n = 1,…, NL} can thus be reused to compute the various integrals, provided that the same quadrature rules are used. This procedure requires a simultaneous convergence of all the integrals being computed.

[37] Secondly, the abscissae are chosen in a nested manner by setting NL=0 = 1 and NL = 2L + 1, for Lequation image*. Such a nesting reduces the effort necessary to increase the level of the quadrature rule: only NL new function evaluations are needed to obtain equation imageL+1 from ��L, instead of NL+1 = 2NL + 1, in the nonnested case.

[38] Ideally, the accuracy of the approximation in equation (22) would be evaluated via the absolute error

  • equation image

However, due to the unavailability of equation image[h(Ve)], alternative error indicators are employed. The relative error E(L) of ��L is used instead, to track the variations of ��L as a function of the level L, with

  • equation image

when ��L[h(Ve) fγ] ≠ 0. As equation imageL[h(Ve) fγ] converges to equation image[h(Ve)], E(L) will gradually decrease to zero.

4.1. Monte-Carlo Quadrature Rule

[39] One of the most popular multidimensional quadrature approaches is Monte-Carlo's (MC) rule, which is defined as

  • equation image

The abscissae γn are random, statistically independent, and drawn from �� by a random-number generator, which uses the pdf fγ. The convergence rate of this rule can be obtained through the Central Limit Theorem [Krommer and Ueberhuber, 1998, p. 254], and it evolves as 1/equation image, which is very slow. This convergence rate depends only on the size of the set Ve(��L) and not on the dimension of γ, which makes the rule very robust. However, when aiming for a certain accuracy in NL steps, 100 times more samples are required to increase the accuracy by a single digit. Moreover, the smoothness of the integrand h(Ve(γ)) as a function of γ has little influence on the convergence rate of the MC rule, which is an advantage for the integration of roughly behaved integrands. Owing to these properties, the MC rule is taken as a robust reference, but alternative quadrature rules are employed as well.

4.2. Deterministic Cartesian-Product Quadrature Rule

[40] As a first alternative, a deterministic Cartesian-product (DCP) rule is considered. If the input vector γ = (γ1,…, γd) is d-dimensional, and �� is the Cartesian product of one-dimensional domains ��1,…, ��d, equation (10) becomes

  • equation image

A 1-D rule ��equation image of level L1 is then applied to approximate the integral over ��1

  • equation image

where the Nequation image abscissae {γequation image ∈ ��1, n1 = 1,…, Nequation image} are all deterministic.

[41] For a repeated trapezoidal quadrature rule, the convergence rate of ��equation image is given by Euler-MacLaurin's formula [Krommer and Ueberhuber, 1998, p. 146]. This rate takes advantage of the smoothness of the integrand and it evolves at least as 1/Nequation image2, i.e., much faster than the order 1/equation image of a MC rule.

[42] In the d-dimensional case, ��equation image is repeatedly applied to each of the other integrals, which produces to the following approximation

  • equation image

As such, the d-dimensional rule ��Ld benefits from the advantageous property of favoring smooth integrands over rough ones. This rule however results in a grid that requires NL = (Nequation image)d evaluations of Ve(γ). Such an exponentially increasing complexity in terms of the dimension d of ��, also known as the “curse of dimensionality,” is extremely penalizing for high dimensions, i.e., d ≥ 3.

4.3. Space-Filling-Curve Quadrature Rule

[43] For higher dimensions (d ≥ 3), a space-filling-curve (SFC) quadrature rule has been implemented. It is based on the transformation of the multidimensional integral over ��, into a one-dimensional curvilinear integral. This is achieved by constructing a Peano curve denoted χγ. If all components of γ = (γ1,…, γd) are statistically independent, they can be expressed in terms of a single scalar s ∈ [−π, +π] as follows [see Saltelli et al., 1999]

  • equation image

The functions Gi are the solutions to a differential equation involving fγ and ensure that, for any region ℛ0 ⊂ ��, the length of χγ contained in ℛ0 equals the probability of having samples γ belonging to ℛ0. As s varies in [−π, +π], the points γ(s) describe a curve χγ, which can be made to go arbitrarily closely to any point of �� by properly selecting the frequencies wi. For practical reasons, the frequencies wi are chosen as integers that form an incommensurate set of order M [Schaibly and Shuler, 1973], i.e.,

  • equation image

The frequencies wi constitute a linearly independent set of order M. By applying Weyl's ergodicity theorem [Weyl, 1938], equation (10) is cast into a 1-D integral

  • equation image

The one-dimensional integral in equation (31) is evaluated numerically by a 1-D quadrature rule

  • equation image

The abscissae sn are equally spaced in [−π, +π] to ensure an exponential convergence rate for analytic functions. Regarding the complexity of this rule, Nyquist's criterion yields a lower bound for NL as

  • equation image

Cukier et al. [1975] have established an empirical formula that links NL to the dimension d of ��, i.e.,

  • equation image

where �� is Landau's symbol and ɛ > 0. In particular, for M = 4, NL ≈ 2.6 d2.5, which is significantly lower than the exponential complexity of the DCP rule. The accuracy of the SFC rule depends on the ability of the search curve to fill the space ��, which follows from the incommensurability of the frequencies wi. The convergence rate of the SFC rule exploits the smoothness of the integrand through the coefficient M: this coefficient Fourier determines the level beyond which the interferences in the spectrum of the integrand are rejected.

[44] Similarities can be found between the SFC rule and so-called lattice rules [Sloan and Joe, 1994], e.g., regarding their constructions based on the Fourier analysis of the integrand. The main difference lies however in the presence of the probability distribution equation imageγ in the definition of the search curve χγ.

5. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

5.1. Setup

[45] The stochastic method presented above is now applied to study a thin-wire configuration Sα. With reference to Figure 1, the axis of the undulating portion of Sα is described by

  • equation image
  • equation image

for y ∈ [ym, yM], where α1, α2, α3, α4 are statistically independent and uniformly distributed in ��1 = ��2 = ��3 = ��4 = [−0.02, +0.02] m. The random variations of the coefficients αi lead to transverse modifications of Sα, with the straight wire Sα=0 as the average geometry. This is illustrated in Figure 3, where 10 sample geometries are shown.

image

Figure 3. Sample geometries of the thin wire.

Download figure to PowerPoint

[46] The total length Lα of Sα varies between 1.120 m, for Sα=0, and 1.241 m. For Lα = 1.120 m, consecutive resonance frequencies are spaced by 134 MHz, whereas when Lα = 1.241 m, the spacing reduces to 121 MHz. Hence, the geometrical variations of Sα shift the resonance frequencies.

[47] The wire is meshed into 224 quadratic segments [Champagne et al., 1992] among which 200 segments are assigned to the undulating part of the wire. To give an idea of the numerical load, the evaluation of the Thévenin network for a single configuration amounts to 0.162 s, the detail of which is provided in Table 1. These computation times correspond to a DELL PWS690 personal computer with a 3 GHz processor.

Table 1. Detail of the Computation Time for One Evaluation of Ve and Ze
 Time (ms)Percent of Total Time
Building Sα86
Building [Z(α)]10676
Computing jα1914
Computing Ve and Ze64
Total time139100

[48] The incident field Eβi is a parallel-polarized plane wave with amplitude 1 V.m−1, propagating along the direction (θi = 45°, ϕi = β1), where the azimuth angle ϕi = β1 is uniformly distributed in the interval ℬ1 = [0°, 90°]. Therefore, the average propagation direction of Eβi is (θi = 45°, ϕi = 45°).

[49] This stochastic problem involves 5 random inputs that are mutually statistically independent and gathered in the vector γ = (γ1,…, γ5) = (α1, α2, α3, α4, β1). The voltage Ve depends on the five components of γ, while Ze, which is independent of β1, depends on the four components of α.

5.2. Computational Cost

[50] The statistical moments of Ve and Ze are computed by using the deterministic-Cartesian-product (DCP), the Monte-Carlo (MC) and the space-filling-curve (SFC) rules presented in section 4. The search curve χγ of the SFC rule is constructed with the following functions Gi

  • equation image

where mi and li are the center and the length of the interval ��i, respectively. The frequencies ωi ∈ {11, 21, 27, 35, 39} are chosen from the optimal set specified by Cukier et al. [1978]. The convergence of the quadrature rules is controlled via their relative errors, as defined in equation (24). Figure 4 displays the relative errors of the approximations of equation image[∣Ve∣] and var[Ve], at f = 300 MHz.

image

Figure 4. Relative error of the approximations of equation image[∣Ve∣] and var[Ve] at f = 300 MHz.

Download figure to PowerPoint

[51] The convergence of all the rules is faster for the evaluation of equation image[∣Ve∣] than for var[Ve]. This is due to the fact that the definition of var[Ve] involves a higher-order polynomial of Ve than the definition of equation image[∣Ve∣]. The convergence of the DCP rule is notably slower than convergence of the MC and the SFC rules. This behavior is caused by the prohibitive complexity of the DCP rule for such a 5D stochastic problem.

[52] For limited levels of accuracy, e.g., E(L) ≥ 0.01, the relative errors of the SFC and the MC rules are comparable. However, for a higher accuracy, i.e., E(L) ≤ 0.001, a faster convergence is attained via the SFC rule. This feature is confirmed by Table 2, which provides the complexity N(Emax) needed to reach a relative error lower than Emax in the approximation of var[Ve]. Similar convergence properties are observed in Figure 5 and Table 3, which concern the computation of equation image[∣Ze∣] and var[Ze].

image

Figure 5. Relative error of the approximations of equation image[∣Ze∣] and var[Ze] at f = 300 MHz.

Download figure to PowerPoint

Table 2. Evaluation of var[Ve] With a Relative Error Lower Than Emax: Complexities NDCP, NMC, and NSFC of the DCP, MC, and SFC Rules, Respectively
 Emax = 10−2Emax = 10−3Emax = 10−4
NDCP>105>105>105
NMC1,02532,769>105
NSFC2,0494,09716,385
Table 3. Evaluation of var[Ze] With a Relative Error Lower Than Emax: Complexities NDCP, NMC, and NSFC of the DCP, MC, and SFC Rules, Respectively
 Emax = 10−2Emax = 10−3Emax = 10−4
NDCP59,049>105>105
NMC3332,049
NSFC171291,025

[53] Compared to Figure 4, the error levels of the statistics of Ze are lower. This is due to the independence of Ze from ϕi, which implies that equation image[∣Ze∣] and var[Ze] are four-dimensional integrals, rather than five-dimensional ones. Further, the convergence rates of the quadrature rules also benefit from the absence of the oscillations caused by the phase variations of Eβi, due to changing values of ϕi.

[54] As stated in section 4, the same quadrature rule is employed to compute the statistical moments of Ve and Ze simultaneously. Hence, the total complexity of the quadrature rule is dictated by the slowest converging integral, i.e., the statistical moments of Ve in the present case.

[55] This strategy is adopted in the sequel, to evaluate the different statistical moments of Ve and Ze for various equidistantly spaced frequencies (spacing of 6 MHz) in the range f ∈ [100 MHz, 500 MHz]. The upper bound for the complexity NL is fixed at 105 and the maximum relative error is set at 0.01. The average complexity per frequency is equal to 5695 samples for the SFC rule, whereas 14120 samples are necessary for the MC rule. The median value of NL, i.e., the complexity to compute the integrals for half of the frequencies in [100 MHz, 500 MHz], is equal to 513 samples for the SFC rule as opposed to 1025 samples for the MC rule.

5.3. Average and Variance of Ve and Ze Versus f

[56] The magnitudes of equation image[Ve] and equation image[Ze] are plotted in Figures 6 and 7, as well as the magnitudes ∣Ve(0)∣ and ∣Ze(0)∣ of the Thévenin response of the average geometry S0 to the average incident field, i.e., (θi = 45°, ϕi = 45°).

image

Figure 6. Average ∣equation image[Ve]∣ (solid line), voltage ∣Ve(0)∣ on the average unperturbed wire (dashed line), and standard deviation σ[Ve] (circled line).

Download figure to PowerPoint

image

Figure 7. Average ∣equation image[Ze]∣ (solid line), impedance ∣Ze(0)∣ on the average unperturbed wire (dashed line), and standard deviation σ[Ze] (circled line).

Download figure to PowerPoint

[57] At first sight, (∣equation image[Ve]∣, ∣equation image[Ze]∣) and (∣Ve(0)∣, ∣Ze(0)∣) show similar behavior in terms of the frequency, such as the presence of three resonance peaks in the domain of frequencies studied. However, differences appear between the two models. Figure 6 clearly reveals the differences between ∣equation image[Ve]∣ and ∣Ve(0)∣, whereas the differences between ∣equation image[Ze]∣ and ∣Ze(0)∣ are more moderate and appear mainly above 195 MHz. Thus, the randomness of Eβi has a drastic effect on the randomness of Ve. Further, the resonance peaks between (∣equation image[Ve]∣, ∣equation image[Ze]∣) and (∣Ve(0)∣, ∣Ze(0)∣) are shifted. These differences are caused by the geometrical fluctuations of the wire, as explained in section 5.1, and they modify the total length of the wire, and therewith the values of the resonance frequencies. Assuming that the Thévenin response of the average geometry S0 is equal to the average (equation image[Ve], equation image[Ze]), as is done in first-order perturbation methods [Sy et al., 2007], would therefore lead to erroneous results.

[58] The standard deviations σ[Ve] and σ[Ze] are also plotted in Figures 6 and 7. Away from resonances, σ[Ve] and σ[Ze] are typically comprised in the ranges [70 mV; 250 mV] and [17Ω;100Ω], respectively. As resonances occur, σ[Ve] and σ[Ze] increase significantly and reach values larger than 4 V and 10 kΩ, respectively. This increase in standard deviation indicates a wider spread of the samples of Ve and Ze in the complex plane.

[59] Since resonances lead to an increase in the magnitudes of (equation image[Ve], equation image[Ze]), at first sight, one could expect that all the values of (Ve, Ze) increase accordingly and assume values close to (equation image[Ve], equation image[Ze]), thereby yielding a moderate standard deviation. However, such is not the case due to the high sensitivity of the resonance conditions to the geometry of the configuration. The dispersion of the observable around resonances could be analyzed in detail by investigating the sensitivity of the resonance frequency to the geometry of the setup. Such a study of the derivative of the resonance frequency can be challenging and is not considered in the scope of this article. Instead an approximate interpretation is given, which hinges on the ratio between the wavelength λ and the total length Lα of the wire Sα. Given a configuration α1 ∈ �� such that Sequation image leads to a high-quality factor resonance, i.e., large values of Ve and Ze, other wire configurations α2equation image, with Lequation imageLequation image ± /4 for any kequation image, will generally produce resonances with different quality factors, or no resonance at all, i.e., moderate values of Ve and Ze. The resulting wide dynamic range of (Ve, Ze) explains the larger dispersion of the observable around resonance frequencies, which in turn causes the increase of σ[Ve] and σ[Ze].

5.4. Covariance Matrices

[60] The elements of the normalized covariance matrices Cequation image and Cequation image are presented in Figures 8 and 9. Since σ[Re(Vn)]2 + σ[Im(Vn)]2 = 1, it suffices to depict the variance of Re(Vn), together with the correlation coefficient ρ[Ve].

image

Figure 8. Elements of Cequation image: var[Re(Vn)] (solid line) and ρ[Ve] (dashed line).

Download figure to PowerPoint

image

Figure 9. Elements of Cequation image: var[Re(Zn)] (solid line) and ρ[Ze] (dashed line).

Download figure to PowerPoint

[61] The variances of Re(Vn) and Im(Vn) are comparable in magnitude since 0.27 ≤ var[Re(Vn)] ≤ 0.8. In other words, the variabilities of the real and imaginary parts of the samples Ve are equally important. It is however not easy to detect the resonances from the graph of Re(Vn) alone. These resonances are however clearly marked by the plot of ρ[Ve], as ρ[Ve] drop and varies rapidly in the resonance regions [175 MHz, 225 MHz], [310 MHz, 360 MHz], and [440 MHz, 480 MHz]. The relatively low values of ρ[Ve] ≤ 0.7 demonstrate the limited correlation between Re(Ve) and Im(Ve).

[62] On the contrary, the behavior of var[Re(Zn)], in Figure 9, distinctly signals the resonance regions. At intermittent frequencies, var[Re(Zn)] takes very small values below 0.05. In these cases, the uncertainty of Im(Ze) plays a significant role in the variations of Ze. In the presence of resonances, var[Re(Zn)] increases to approximately 0.5, which demonstrates the equal spread of Re(Ze) and Im(Ze).

[63] Away from resonances, the correlation between Re(Ze) and Im(Ze) is nonnegligible, as indicated by the values of ρ[Ve] comprised between 0.5 and 0.9.

[64] The nonvanishing values of ρ[Ve] and ρ[Ze] highlight the presence of a statistical correlation between real and imaginary parts of Ve and Ze. It is however not possible, at this stage of the discussion, to identify the direction that contains the majority of the samples of the observable, particularly for Ve. This direction will be determined through the analysis of the principal components of the covariance matrices.

5.5. Principal Components and Angle

[65] To obtain more insight into the randomness of Ve and Ze, their principal components, as plotted in Figures 10 and 11, are now studied. These graphs represent the first principal component λn,12 which is given in percentage of the total variance, as well as the principal angle that belongs to the interval [−90°, +90°].

image

Figure 10. Principal components of Ve, λn,12 in % of σ[Ve] (solid line), and θequation image in (°) (dashed line).

Download figure to PowerPoint

image

Figure 11. Principal components of Ze, λn,12 in % of σ[Ze] (solid line), and θequation image in (°) (dashed line).

Download figure to PowerPoint

[66] The influence of λn,22 on Ve is nonnegligible at regular frequencies where it accounts for 15% to 48% of the total variance. The resonances are characterized by λn,12λn,22 ≈ 50%, and by rapid oscillations of θequation image. These variations of θequation image are rooted in the fact that the spread of the observable in the complex plane no longer follows a preferential direction, meaning that θequation image can no longer be defined.

[67] Between 360 MHz and 425 MHz, a rotation of the direction of highest spread, caused by the frequency variation, is apparent. Moreover, around 250 MHz, a sudden change of the principal direction is revealed as θequation image drops from +78° to −62°. The information of this rotation exemplifies the refinement of the principal component analysis, as it is not revealed by the graph of ρ[Ve] shown in Figure 8.

[68] The general profile of the principal components of Ze is such that λn,12 represents more than 98% of the total variance at regular frequencies and drops to approximately 55% when resonances occur. At regular frequencies, θequation image is stable around +85°, which implies that the direction of highest variation is close to the imaginary axis. This result is in agreement with Figure 9, which underscores the dominant role of the variations of Im(Ze). Similarly to θequation image, θequation image is no longer defined when resonances occur, which signals the absence of a direction of highest spread.

[69] These plots show the robustness in the patterns of the randomness of Ze, despite the changes in frequency f. Since Ze is independent of Eβi, the aforementioned robustness actually originates from the robustness of the current distribution jα, which represents the effect of the random geometry of Sα in the definitions of Ve and Ze via equations (3) and (5).

[70] This result can be explained by interpreting jα through transmission-line theory. Since the transmission line Sα sees an open circuit at one end (the port P) and a short circuit at the other end, the waves traveling through this line undergo simple reflections at both ends of the thin wire. Thus, jα follows a standing-wave pattern that depends mainly on the propagation delay. This pattern is robust owing to the fact that only the first three resonances are studied (with λ ≥ 60 cm), whereas the wire length is restricted to the interval [1.120 m, 1.241 m]. To compute Ze via equation (5), jα is integrated over Sα with weights that correspond to the trace of the deterministic field E[IP] along Sα. Hence the randomness appears only in the propagation delay, thus the robustness of Ze. On the other hand, to obtain Ve via equation (3), the values of jα along Sα are summed with random weights that correspond to the values of the trace Eβi along Sα. The important changes in the angle of incidence of Eβi significantly modify the way in which the field polarization faces the thin wire, making this problem much more sensitive to the actual field configuration.

5.6. Chebychev Circles Versus Chebychev Ellipses

[71] The conclusions based on the analysis of the statistical moments of Ve and Ze are now confronted with the actual sample distribution of the Thévenin parameters. To this end, 105 randomly drawn configurations are analyzed at the frequencies specified in Table 4. The observables (Ve, Ze) are then normalized according to equation (13), and compared to the confidence domains ��m and ℰm deduced from Chebychev's inequalities (14) and (21), where mequation image.

Table 4. Statistical Results at f = 268 MHz and f = 331 MHz
f (MHz)equation image[Ve]σ[Ve]ρ[Ve]λn,12(%)equation imageequation image
268−0.046 − j 0.081 V0.141 V0.5479−57°
331−0.235 − j 0.337 V2.077 V0.025365°
f (MHz)equation image[Ze]σ[Ze]ρ[Ze]λn,12(%)θequation image
268−1 + j 28 Ω40 Ω0.639985°
331133.5 + j 569 Ω3901 Ω0.115855°

[72] The normalized samples of Ve and Ze at the regular frequency f = 268 MHz are shown in Figures 12 and 13, as well as the confidence domains ��m and ℰm, for 1 ≤ m ≤ 4.

image

Figure 12. The 105 normalized samples of Ve at the regular frequency f = 268 MHz.

Download figure to PowerPoint

image

Figure 13. The 105 normalized samples of Ze at the regular frequency f = 268 MHz.

Download figure to PowerPoint

[73] Almost all the samples of (Ve, Ze) are within 4σ's of the average (equation image[Ve], equation image[Ze]), which corresponds to the normalized domains ��4 and ℰ4. Although the concentric circles yield a good quantification of the spread of the samples, they tend to exaggerate the actual distribution of the samples, which occupy a narrow subdomain of these circles. In this sense, the analysis based on the isotropic variance leads to a worst-case type of conclusion. On the other hand, the ellipses provide a more conforming type of quantification by accurately specifying the directions of extreme dispersion both for Ve and Ze.

[74] The samples of Ve are more scattered in the complex plane, as expected from the importance of the second principal component λn,22, which represents 21% of the variance of Ve. Regarding Ze, the ellipses are narrower and the main axis of the ellipses is close to the imaginary axis, albeit slightly tilted. This backs the observation, based on the analysis of CZe, that Im(Ze) exhibits the highest dispersion.

[75] The dominance of λn,1 (λn,12 = 99% of (σ[Ve])2) suggests that the analysis of the randomness of Ze can be satisfactorily performed by studying the principal component Ze(1) only. From this stance, a compression of information is achieved as the study of the single random variable Ze(1) bears the information for the random vector Ze = (Re(Ze), Im(Ze)).

[76] The number of samples contained in the Chebychev circles and ellipses is detailed in Tables 5 and 6, for Ve and Ze, respectively. These numbers are compared to the worst-case bounds provided by Chebychev's inequality. The latter bounds are such that, for any m > 1, ��m and ℰm should contain at least NCheby(m) = (1 − 1/m2) % of all the samples.

Table 5. Number of Samples of Ve in the Chebychev Circles, equation image(m), and Ellipses, equation image(m), Versus Chebychev's Worst-Case Bounds, NCheby(m), at f = 268 MHz
mequation image(m)equation image(m)NCheby(m)
152,17460,336-
297,66597,43975,000
399,79699,73988,000
4100,00099,99893,000
5100,000100,00096,000
Table 6. Number of Samples of Ze in the Chebychev Circles, equation image(m), and Ellipses, equation image(m), Versus Chebychev's Worst-Case Bounds, NCheby(m), at f = 268 MHz
mequation image(m)equation image(m)NCheby(m)
163,58063,660-
290,02090,62075,000
396,50096,72088,000
498,30198,28393,000
599,36098,84096,000

[77] The circle ��4 contains all the values of Ve, whereas there are still a few samples beyond ℰ4. Similarly, almost all the samples of Ze are contained in ��5 and ℰ5. The loose nature of Chebychev's bounds is also highlighted by Tables 5 and 6, particularly for m ≤ 3.

[78] At the frequency f = 331 MHz in the resonance region, the samples of Ve and Ze, displayed in Figures 14 and 15, are completely scattered in the complex plane.

image

Figure 14. The 105 normalized samples of Ve at the frequency f = 331 MHz in the resonance region.

Download figure to PowerPoint

image

Figure 15. The 105 normalized samples of Ze at the frequency f = 331 MHz in the resonance region.

Download figure to PowerPoint

[79] The rounded shape of the ellipses ℰm is consistent with the high values of λn,22 ≈ 45% ≈ λn,12 both for Ve and Ze. Although ��5 contains most of the samples of Ve and Ze, there are still extreme values that lie more than 8 σ's away from the mean, as confirmed by Tables 7 and 8.

Table 7. Number of Samples of Ve in the Chebychev Circles, equation image(m), and Ellipses, equation image(m), Versus Chebychev's Worst-Case Bounds, NCheby(m), at f = 331 MHz in the Resonance Region
mequation image(m)equation image(m)NCheby(m)
185,36085,415-
296,14096,14275,000
398,40898,40488,000
499,05399,05193,000
899,72899,73096,000
Table 8. Number of Samples of Ze in the Chebychev Circles, equation image(m), and Ellipses, equation image(m), Versus Chebychev's Worst-Case Bounds, NCheby(m), at f = 331 MHz in the Resonance Region
mequation image(m)equation image(m)NCheby(m)
184,24084,460-
295,68095,60075,000
398,10098,00088,000
498,60098,56093,000
899,66099,66096,000

[80] In this resonant case, since the bounds obtained via the circles and the ellipses are very close, the analysis of the variance is sufficient to obtain a precise picture of the distribution of the observables.

[81] In comparison with the sample distributions at f = 268 MHz, at f = 300 MHz the majority of the samples of Ve and Ze is located in the vicinity of the mean, despite the presence of extreme samples. This can be observed by comparing the values of equation image(1) and equation image(1) for f = 268 MHz and f = 300 MHz. This statistical result is in agreement with the physical explanation provided in section 5.3, viz. at resonance frequencies, under the effect of the random geometry, only a few configurations are in actual resonance conditions and give rise to important values of the observable, while the other configurations lead to nonresonance moderate values that cluster around the mean.

[82] The patterns of the samples of Ze in the complex plane can also be interpreted in terms of the input current Iin. To this end, the thin wire is considered in the absence of the incident field Eβi, and in an open-circuit state. Using the equivalent Thévenin model and given a voltage value Vin at the port, the current Iin flowing across Ze follows by Ohm's law as Vin = ZeIin. Around the resonance frequencies such as f = 331 MHz, due to the large values of Ze, Iin will have a very small magnitude and a phase that is highly sensitive to modifications caused by the geometrical randomness of the wire. Conversely, in the vicinity of so-called antiresonance frequencies, i.e., frequencies where Ze assumes very small values, Iin will have a large magnitude and a well defined phase that is relatively insensitive to the geometrical modifications of Sα.

6. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

[83] A probabilistic approach has been presented to statistically quantify uncertainties in electromagnetic interactions. This method relies on the efficient use of quadrature rules to compute statistical moments of the observables. For higher dimensions, the space-filling-curve rule proves to be very efficient in terms of complexity and convergence rate, when compared to the Monte-Carlo rule, which in turn is far more efficient than a deterministic Cartesian-product rule.

[84] The approach has been applied to a fully stochastic interaction between a random plane wave and a random wire geometry. The complex-valued observables, chosen as the coefficients of a Thévenin model, have been handled as real-valued vectors through their real and imaginary components. Chebychev's inequality has highlighted the isotropic and strict nature of the variance when the spread of complex observables is measured. A finer statistical characterization of both components of the observable was possible with the aid of the average and the covariance of the observables. The correlation coefficient has highlighted the statistical coupling that generally exists between the components of complex observables. The principal component representation provided a refined, decoupled and conformal quantification of the stochastic parameters studied.

[85] Particular differences have been noted between the distribution of the impedance coefficients, which measure the self-interaction of the stochastic geometry, and the distribution of the induced voltage sources, which measure the interaction of the stochastic geometry with external sources. This is probably not specific for the thin-wire example but reveals that when the geometry plays a double role of emitter and receiver, as is the case when the impedance coefficients are computed, it induces additional correlations between real and imaginary parts of the observable.

[86] The determination of additional qualitative information on the probability distribution of the Thévenin parameters requires the analysis of higher-order statistical moments such as the skewness and the kurtosis. This is the subject of our future work.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information

[87] This work was funded by the Dutch Ministry of Economic Affairs, in the Innovation Research Program (IOP) (EMVT 04302). We thank the reviewers for their helpful comments regarding the physical interpretation of our results.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information
  • Ajayi, A., P. Ingrey, P. Sewell, and C. Christopoulos (2008), Direct computation of statistical variations in electromagnetic problems, IEEE Trans. EMC, 50(2), 325332.
  • Bellan, D., and S. Pignari (2001), A probabilistic model for the response of an electrically short two-conductor transmission line driven by a random plane wave field, IEEE Trans. EMC, 43(2), 130139.
  • Bharucha-Reid, A. T. (1972), Analysis and regularization of the thin-wire integral equation with reduced kernel, in Random Integral Equations, vol. 96, Mathematics in Science and Engineering, Academic, New York.
  • Brown, G. S. (1985), Simplifications in the stochastic Fourier transform approach to random surface scattering, IEEE Trans. EMC, 33(1), 4855.
  • Champagne, N. J.II, J. T. Williams, and D. R. Wilton (1992), The use of curved segments for numerically modeling thin wire antennas and scatterers, IEEE Trans. Antennas Propag., 40(6), 682689.
  • Cukier, R. I., C. M. Fortuin, K. E. Shuler, A. G. Petschek, and J. H. Schaibly (1973), Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I. Theory, J. Chem. Phys., 59(8), 11401149.
  • Cukier, R. I., J. H. Schaibly, and K. E. Shuler (1975), Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. III. Analysis of the approximations, J. Chem. Phys., 63(3), 11401149.
  • Cukier, R. I., H. B. Levine, and K. E. Shuler (1978), Nonlinear sensitivity analysis of multiparameter model systems, J. Comput. Phys., 26, 142.
  • de Menezes, L. R. A. X., A. Ajayi, C. Christopoulos, P. Sewell, and G. A. Borges (2008), Efficient computation of stochastic electromagnetic problems using unscented transforms, IET Sci. Meas. Technol., 2, 8895.
  • Grant, I. S., and W. R. Phillips (1990), Electromagnetism, John Wiley, Hoboken, N. J.
  • Gubner, J. A. (2006), Probability and Random Processes for Electrical and Computer Engineers, Cambridge Univ. Press, New York.
  • Hill, D. A. (1998), Plane wave integral representation for fields in reverberation chambers, IEEE Trans. EMC, 40(3), 209217.
  • Krommer, A. R., and C. W. Ueberhuber (1998), Computational integration, SIAM, 40(3), 209217.
  • Mei, K. K. (1965), On the integral equation of thin wire antennas, IEEE Trans. Antennas Propag., 13(3), 374378.
  • Michielsen, B. L. (2005), Analysis of the coupling of a deterministic plane wave to a stochastic twisted pair of wires, paper presented at 16th International Zurich Symposium on EMC, Swiss Fed. Inst. of Technol., Zurich, Switzerland.
  • Papoulis, A. (1991), Probability, Random Variables and Stochastic Processes, pp. 439442, McGraw-Hill, New York.
  • Perez, R. (2008), Methods for spacecraft avionics protection against space radiation in the form of single-event transients, IEEE Trans. EMC, 50(3), 455465.
  • Pignari, S. A. (2006), Statistics and EMC, Radio Sci. Bull., 316, 1326.
  • Rannou, V., F. Brouaye, P. De Doncker, M. Helier, and W. Tabbara (2001), Statistical analysis of the end current of a transmission line illuminated by an elementary current source at random orientation and position, Proc. IEEE Int. Symp. EMC, 2, 10781083.
  • Rumsey, V. H. (1954), Reaction concept in electromagnetic theory, Phys. Rev., 94(6), 14831491.
  • Saltelli, A., S. Tarantola, and K. P.-S. Chan (1999), Global sensitivity analysis—A computational implementation of the Fourier Amplitude Sensitivity Test (FAST) reaction concept in electromagnetic theory, Technometrics, 41(1), 3956.
  • Schaibly, J. H., and K. E. Shuler (1973), Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. II. Applications, J. Chem. Phys., 59(8), 38793888.
  • Sloan, I. H., and S. Joe (1994), Lattice Methods for Multiple Integration, pp. 38793888, Clarendon, Oxford, U. K.
  • Sun, S., G. Liu, J. L. Drewniak, and D. J. Pommerenke (2007), Hand-assembled cable bundle modeling for crosstalk and common-mode radiation prediction, IEEE Trans. EMC, 49(3), 708718.
  • Sy, O. O., J. A. H. M. Vaessen, M. C. van Beurden, A. G. Tijhiuis, and B. L. Michielsen (2007), Probabilistic study of the coupling between deterministic electromagnetic fields and a stochastic thin-wire over a PEC plane, Proc. Int. Conf. Electromagn. Adv. Appl., 49(3), 637640.
  • Tijhuis, A. G., and Z. Q. Peng (1991), Marching-on-in-frequency method for solving integral equations in transient electromagnetic scattering, IEE Proc. H Microwaves Antennas Propag., 138(4), 347355.
  • van Beurden, M. C., and A. G. Tijhuis (2007), Analysis and regularization of the thin-wire integral equation with reduced kernel, IEEE Trans. Antennas Propag., 55(1), 120129.
  • Weyl, H. (1938), Mean motion, Am. J. Math., 60(4), 889896.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Deterministic Setup
  5. 3. Stochastic Parameterization of the Problem
  6. 4. Computation of the Moments
  7. 5. Results
  8. 6. Conclusion
  9. Acknowledgments
  10. References
  11. Supporting Information
FilenameFormatSizeDescription
rds5712-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5712-sup-0002-t02.txtplain text document0KTab-delimited Table 2.
rds5712-sup-0003-t03.txtplain text document0KTab-delimited Table 3.
rds5712-sup-0004-t04.txtplain text document0KTab-delimited Table 4.
rds5712-sup-0005-t05.txtplain text document0KTab-delimited Table 5.
rds5712-sup-0006-t06.txtplain text document0KTab-delimited Table 6.
rds5712-sup-0007-t07.txtplain text document0KTab-delimited Table 7.
rds5712-sup-0008-t08.txtplain text document0KTab-delimited Table 8.

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.