Luk R. Arnaut

Operation of electromagnetic reverberation chambers with wave diffractors at relatively low frequencies

The dependence of the modal density on chamber geometry is analyzed, for general unstirred reverberation chambers at relatively low frequencies of operation. An expression for the lowest usable frequency is derived based on the chamber mode density. The use of curved wave diffractors is investigated as a method for increasing the average mode density. The specific effect of corner, edge, surface and volume diffractors on the mode density and quality factor of a nominally rectangular shielded room is analyzed as a function of diffractor size, density, and operating frequency. This paper complements earlier studies on field statistics of undermoded reverberation chambers and the performance of acoustic-type diffusers in electromagnetic reverberation chambers.

Statistics of the quality factor of a rectangular reverberation chamber

The mean and standard deviation of the theoretical quality (Q) factor of a rectangular reverberation chamber are considered. Different averaging methods are investigated for deriving alternative expressions for its bandwidth-averaged value. Alternative basic assumptions are made regarding the distribution of the total energy density inside the cavity across the participating eigenmodes, thus providing alternatives for the assumption of equipartition of excitation energy. The physical reasons for such possible departures are explained on the basis of the stored and dissipated modal energy. For a given volume-to-surface ratio of a rectangular cavity, the theoretical arithmetic average Q (unlike the harmonic average) exhibits an explicit asymptotic dependence on the aspect ratios of the cavity. In the asymptotic high-frequency limit, the first-order dependence of the arithmetic Q on inverse frequency is governed by the imbalance between the TM and TE quality factors and by the aspect ratios of the cavity. Simulation results indicate better agreement between actual and smoothed theoretical arithmetic averages, particularly at lower frequencies, in comparison with those for the harmonic mean values. An expression for the distribution function of the arithmetic Q is formulated based on its statistical moments. We furthermore analyze the Q of a chamber with dynamically varying walls but constant average mode density. Such a chamber may serve as a model for mode stirring using flexible walls. The existence of a mode bunching effect which varies with tuner state but stabilizes with increasing frequency is shown. Effects of continuous dynamics of the cavity deformation on Q are discussed.

Limit distributions for imperfect electromagnetic reverberation

Investigates departures from ideal circular Gauss normality for the statistical distribution of complex fields in undermoded mode-tuned or mode-stirred reverberation chambers. A physical nonlinear model is developed, in which the energy density of the interior field is expressed as a functional of energy densities for equivalent boundary sources associated with a partitioning of the cavity surface. This results in Meijer G and Bessel K limit distributions which extend the well-known Gauss normal, chi and chi-squared asymptotic distributions for ideal reverberation fields, their magnitude and energy density, respectively. The additional distribution parameter permits graduation of the distribution, allowing one to incorporate chamber imperfections and operating at lower frequencies. The role of entropy, mode migrations, mode mutations, and the link with doubly stochastic random walks and cavity ensembles are outlined. In an alternative model based on cavity feedback, a powered gamma distribution is obtained for the field magnitude and energy density, exhibiting one further distribution parameter. Theoretical results are illustrated with measured and numerically simulated data, and include results on eigenfrequency dynamics.

Effect of local stir and spatial averaging on measurement and testing in mode-tuned and mode-stirred reverberation chambers

The effect of local averaging in mode-tuned and mode-stirred reverberation chambers is investigated, under the assumption of a wide-sense stationary statistical cavity field. The analysis is based on the characterization of the tuning or stirring process in the spectral spatial domain and in the spectral stir domain. The variance function, scale of fluctuation, optimum sampling rate, and normalized spectral bandwidths are computed for each case, based on the modeled power spectral-density function. This second-order analysis enables the effect of local averaging on the point-interval correlation, mean and standard deviation of the maximum test level, upward threshold crossing frequency, time to first passage and mean excursion length to be quantified. The theoretical results for quantifying EUT reliability are illustrated and compared with measured data. The results provide guidelines for the maximum tolerable sensor aperture and stirrer step sizes for mode tuning, and sampling rate and sampling width for mode stirring in chamber calibration and EMC emissions and susceptibility testing.

Electromagnetic reverberation near a perfectly conducting boundary

We analyze the effect of an infinite planar perfectly conducting surface on the physical and statistical properties of an otherwise ideal reverberant field. The surface induces statistical uniaxial field anisotropy at a distance, and its effect can be characterized based on calculable local field polarization and anisotropy coefficients that exhibit a damped oscillatory behavior as a function of the distance to the surface. It is shown, both theoretically and experimentally, that compound exponential (CE) distribution functions with a degenerate polarization coefficient as parameter govern the statistics of the local energy density and amplitude of the vector field near the surface. The influence of adjacent walls is taken into account by a transverse field anisotropy coefficient and gives rise to bifurcation of the polarization coefficient. The effect of the transverse field anisotropy on the statistics of the energy density, field amplitude, and their sample maxima as a function of distance to the surface is quantified. It is found that the increase in the expected value of the maximum electric field strength caused by the presence of the surface is of the order of 1 dB. Theoretical results are validated against measured data. A theoretical derivation based on a spectral plane-wave expansion for this configuration is given. The results are relevant to applications in which a sensor, test artefact or critical component in immunity testing is relatively close to a conducting surface, in aperture coupling between cavities, and in emissions measurement of total radiated power.

Generalized Extreme-Value Distributions of Power Near a Boundary Inside Electromagnetic Reverberation Chambers

This paper presents results of an experimental investigation regarding the statistical distribution of the maximum field and analysis of statistical field inhomogeneities near a cavity wall inside a reverberation chamber. Measurements were performed for both undermoded and overmoded regimes. Departures from ideal isotropic random field behavior at relatively low frequencies were measured by placing a receiving antenna close to one cavity wall. The coexistence of field heterogeneity and anisotropy has been confirmed, together with departures from the ideal statistical distributions of field and energy. Empirical distributions of the maximum value were derived for hybrid (i.e., combined mechanical, frequency, and spatial) mode stirring. The maximum-value distribution is found to be of Fréchet type in undermoded regime and converges to a reverse Weibull distribution in highly overmoded operation, with a transit across the Gumbel distribution when the operation is weakly overmoded. Results exhibit good agreement with previous theoretical and numerical findings.

Effect of size, orientation, and eccentricity of mode stirrers on their performance in reverberation chambers

We analyze the effect of the dimensions, aspect ratio, orientation, and eccentricity of rotation of a rectangular surface element on its performance as a mechanical mode stirrer in a reverberation chamber. Stirring performance is quantified by the number of coherence cells traversed by this surface during the stirring process. Concentrically as well as eccentrically rotating elements are investigated. The dependence of stirring sensitivity coefficients on the width-to-wavelength ratio and the aspect ratio is different for both cases. The radius of the orbit and the orientation of the surface element relative to the orbit are found to have a major influence

Numerical convergence in periodic method of moments analysis of frequency-selective surfaces based on wire elements

We present a subdomain formulation of the periodic method of moments (PMM) with thin-wire kernel for analyzing frequency-selective surfaces (FSSs) with rectilinear wire-type elements. Analysis of the convergence of the impedance matrix for a FSS with aligned unidirectional elements indicates the effect of individual oscillatory and decaying components. For the individual impedance elements of this FSS, we prove and demonstrate the universality of their envelopes as a function of shell size in the spectral domain. For N wire segments, the PMM converges according to O(N/sup 4/). The dependence on the order of polynomial basis functions shows a geometric progression. The theory is also applied to a single-layer FSS having asymmetrically split segmented rings.

On the maximum rate of fluctuation in mode-stirred reverberation

Theoretical limits are derived for the maximum permissible rate of mechanical/electronic stirring or spatial/spectral scanning inside a reverberant cavity. These limits are based on upper bounds for the maximum rate of field fluctuations and are obtained by imposing the requirement of quasi-stationarity on the interior cavity field. For a sinusoidal excitation field, the interior field is represented as a narrowband random hybrid amplitude-plus-frequency modulation, as induced by the stirring or scanning process, and is based on an analytic field formulation. Distortion (nonlinearity) of the modal and effective field is quantified for first-order variations of amplitude and frequency. Its dependence on the random amplitude modulation index is demonstrated. The latter is estimated from macroscopic system parameters. The effect of compensation of net input power on the distortion and maximum stirring rate is analyzed. The maximum stirring rate exhibits an inverse power law dependence on the operating frequency. For a specified level of maximum distortion, maximum rates of electronic, mechanical, or electromechanical mode stirring are derived. The effect of the order of the EUT transfer function on the distortion, as well as the detuning, bandwidth, and amplitude distortion of the perceived test field caused by stirring or scanning are quantified and analyzed.

Time-Domain Measurement and Analysis of Mechanical Step Transitions in Mode-Tuned Reverberation: Characterization of Instantaneous Field

We report on time-domain measurement and statistical analysis of field transients that are caused by mechanical transitions between the stationary states of an overmoded resonant cavity. These transitions and field transients are produced by the stepwise rotation of a mode tuner (reflective paddle wheel) inside a reverberation chamber excited by an uninterrupted time-harmonic source. It is shown that the cross-correlation function between the so-called stir sequences enables the reconstruction of the averaged mechanical motion of the mode tuner. Evolutions of the probability distribution and statistics of the transmission S-parameter show correlation with tuner motion and demonstrate nonstationarity of the cavity field. The influence of the operating frequency and various motor parameters is investigated. In estimating the evolving maximum-to-average ratio, the variations in the underlying (parent) distribution are shown to be considerably more influential than the variations of the number of independent samples. Semi- and fully empirical estimates of this ratio are compared with the directly measured values, and fully empirical estimation allows for more accurate prediction. It is shown that the measured local minima and local maxima of this ratio can be identified with short- and long-tailed instantaneous parent distributions, respectively. The effect is physically interpreted as modal bunching and dilation. The results are relevant to EMC immunity testing in reverberation chambers.