Second-order statistics of complex observables in fully stochastic electromagnetic interactions: Applications to EMC

Abstract

[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).

Higher-Order Statistics for Stochastic Electromagnetic Interactions: Applications to a Thin-Wire Frame

“…The slopes of higher-order bounds become steeper as the order of the normalized moments increases. Higher-order Chebychev bounds become meaningful, i.e., below 1, later than lower-order bounds, as expected from Lyapunov's inequality (11). At f 1 , for instance, if r = 2, the only bounds that are lower than 1 and hence exploitable are b 2 , b 4 and b 6 .…”

“…This model is mainly relevant for problems where the actual shape and length of the wire are not entirely known. The indetermination of the wire length has a major influence on the frequencies at which resonances occur [11]. In addition, the load R influences strongly the presence of resonance as well as the value of the observable at those frequencies: while an adapted line will exhibit little or no resonance, a mismatched impedance will produce resonances with large quality factors, particularly if the line is terminated by an open-or short-circuit [21].…”

“…Further, when multiple integrals need to be computed simultaneously, the same quadrature rule is employed. Moreover, the quadrature rule is chosen to efficiently handle integrals over the multidimensional domain U, as is the case with a sparse-grid [26], a MonteCarlo [25, p. 205] or a space-filling-curve rule [11]. The convergence of the quadrature rule is monitored through the relative modification of Q N [h(γ)] as the number of samples N is increased.…”

“…However, the limitations of such an assumption are clearly visible once resonances appear, i.e., at f 1 and f 3 . The occurrence and intensity of resonances requires specific conditions both in terms of the electrical length of the thin wire and in the value of the impedance R [19, p. 254-257], [11]. Hence, at f 1 and f 3 , due to the uncertainty of the geometrical and physical properties of the wire and the polarization variations of the incident field, only a few configurations will produce resonances, i.e., large values of |V e |.…”

“…We have successfully applied this modus operandi to thin-wire problems by evaluating the first four statistical moments of the observable with the aid of quadrature algorithms. The thus obtained statistics could be post-processed using Chebychev's inequality to obtain general bounds of the distribution of the observable via its second-order moments [11] or to quantify the likelihood of observing extreme samples using the fourth-order moment, or kurtosis [12].…”