Reverberating chamber electromagnetic field in ... - IEEE Xplore

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 42, NO. 2, MAY 2000

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Reverberating Chamber Electromagnetic Field in Presence of an Unstirred Component Paolo Corona, Fellow, IEEE, Giuseppe Ferrara, and Maurizio Migliaccio, Member, IEEE

Abstract—In this paper, a theoretical model for the electromagnetic field within a reverberating chamber in presence of an unstirred component is presented. Due to the stochastic nature of the electromagnetic field within the chamber, a statistical electromagnetic approach is developed and tested versus measurements. Index Terms—Electromagnetic compatibility, mode-stirred chambers, reverberating chambers, statistical electromagnetic fields.

I. INTRODUCTION

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LECTROMAGNETIC compatibility (EMC) is a discipline that deals with the needs of an electronic apparatus (system) to be safely operated in a complex electromagnetic environment and the demand that the apparatus must not create an environment that is unsafe, either for other equipment or for human beings that might be present. In order to acquire the best comprehension of complex electromagnetic environments, it is highly desirable to depict some effective theoretical models as well as some testing procedures. To deal with these needs statistical electromagnetic models can be effectively exploited and some appropriate tests and procedures must be outlined. An effective general model for real complex electromagnetic environments must embody both stochastic and deterministic components. This is, for instance, the case of urban environments. We note, however, that in order to enforce electromagnetic compatibility requirements, it is mandatory to conceive a realistic and manageable testing procedure. In fact, it is important to have available an accurate, reliable, and economically convenient test procedure which synthesizes the real environment (or some real-like reference situation) in which the equipment under test (EUT) is likely to be operated. Since it is well known that the electromagnetic field within a well-operated reverberating chamber is stochastic in nature it is straightforward to consider it as a reference. Reverberating chambers are large overmoded cavities wherein the electromagnetic field is stochastically uniform and isotropic [1]–[7]. They were first introduced by acoustical engineers as an economical alternative to anechoic chambers. In order to ensure a sufficient degree of uniformity and isotropy, the electromagnetic field must be properly perturbed. This is Manuscript received July 12, 1999; revised November 14, 1999. P. Corona and G. Ferrara are with the Istituto Universitario Navale, Istituto Teoria e Tecnica delle Onde Elettromagnetiche, 80133 Napoli, Italy. M. Migliaccio is with the Università di Cagliari, Dipartimento di Ingegneria Elettrica ed Elettronica, 09123 Cagliari, Italy (e-mail: [email protected]). Publisher Item Identifier S 0018-9375(00)04682-2.

usually accomplished electronically [1] or physically [2]–[7]. In the latter class fall, the step-mode reverberating chambers [2]–[4] and the mode-stirred ones [5]–[7]. In the first case, a sufficient set of electromagnetic field data relevant to different cavity geometries are collected, in the second case, the cavity geometry is continuously perturbed during measurements and an electromagnetic field data series is obtained. As a result, in both cases, the electromagnetic field within a well-operating reverberating chamber is a circular complex Gaussian field as predicted by independent theoretical models and measurements [1]–[7]. In this paper, a statistical electromagnetic theory, which generally models the electromagnetic field within a reverberating chamber, is presented. In particular, we focus our attention on the case that some unstirred component of the electromagnetic field is present. This is for instance the case whenever a direct antenna-to-antenna coupling occurs [8], [9]. We show that such a case can be seen as a modification of the popular reference thermodynamic case [3], [6]. The statistical model we present is able to relate the behavior of a well-operating reverberating chamber with the case that an unstirred electromagnetic field component is present. Some pioneering conference papers on the case that a direct antenna-to-antenna coupling is present have been published in the past [8], [9]. The new theory which is presented in this paper embodies the classical thermodynamic case and is validated by a set of appropriately tailored measurements performed at the Istituto Universitario Navale (IUN) mode-stirred reverberating chamber. This paper is structured as follows. In Section II, the theory which explains the characteristics of the transmission coefficient of a reverberating chamber whenever an unstirred component of the electromagnetic field is present is compared with the one in a well-operating chamber is given. In Section III, a set of relevant experiments are presented and discussed with reference to the former theory. Finally, in Section IV, some concluding remarks are reported. II. THEORY In this section, an effective theory that models the electromagnetic field in a reverberating chamber is illustrated. Due to the stochastic nature of the electromagnetic field within the chamber, a statistical theoretical model is developed here. At the basis of this theory, we suppose that the electromagnetic field within a well-operating reverberating chamber is a circular complex Gaussian field. This has been predicted by different theoretical models and supported by many independent experimental studies [2]–[7].

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Hence, the electromagnetic field within a well-operating reverberating chamber can be simply expressed by two real invariance. The dependent Gaussian fields of zero-mean and amplitude of the complex electromagnetic field is Rayleigh distributed and the phase is uniformly distributed. Namely the electromagnetic field amplitude is distributed as follows [2]–[7]: (1) is the step function and is the variance of wherein the two Gaussian fields composing . It is useful to recall the [10] Rayleigh normalized moments

in order to determine the pdf of the field amplitude which now results as [10] (6) is the zeroth-order modified Bessel function of the wherein first order and is the amplitude of the unstirred component of the field. Equation (6) is a Rice pdf which has been first derived by Rice when modeling the pdf of a single harmonic component corrupted by an independent Gaussian noise. Evaluation of its ’s is not at all straightforward [11]

(2)

represents the statistical mean operator. wherein It is important to note that (1) corresponds to a probability density function (pdf) with one parameter. In fact, the statistical and its variance VAR are by no means indepenmean dent and are given by

(7) is the Eulerian Gamma function and wherein is the degenerate hypergeometric function, which is also called the Kummer function [12]. Again, it is also useful to provide the mean and variance of the field in this case which are now given by [11]

(3) (8)

and VAR

(4)

respectively. It can be of practical utility to evaluate the ratio of the variance to the squared mean VAR

and VAR

(5)

which, in the case in question, is a constant equal to . This statistical model is sufficient to describe (at the first within order) the amplitude of the complex insertion loss a well-operating reverberating chamber [2]–[7]. In particular, if we denote as the amplitude of the transmission coefficient, , we can i.e., the amplitude of the complex insertion loss employ the former statistical model to characterize it1 [7]. Let us now consider the electromagnetic field in the case there is an unstirred component of the field. The model is similar to the one formerly employed but for an important factor. Due as generated by to the unstirred field we cannot consider two independent equally distributed Gaussian fields; we have to consider the deterministic component that is superimposed on the stochastic electromagnetic field due to the presence of the unstirred field. Consequently, the corresponding statistical model calls for two real independent Gaussian fields with the same variance but nonzero mean values. We can combine them 1In the following, we keep using E instead of T . Physically, we always make reference to the output field normalized to the input one.

(9) respectively. It is important to analyze the case that the unstirred component of the field is large. In particular, if we suppose that we can asymptotically expand as follows [12]: (10) Therefore, if we only consider the first-order term in and insert it in (6) we get to (11) which resembles a Gaussian pdf. On a theoretical basis, it is important to note that (11), apart from the first term, cannot be a Gaussian pdf for the important fact which constrains with probability 1 to be nonnegative. On a practical basis, with some

CORONA et al.: REVERBERATING CHAMBER EM FIELD IN PRESENCE OF UNSTIRRED COMPONENT

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Fig. 1. The complex insertion loss S and the pdf of the amplitude insertion loss T are shown for three cases. All experiments have been accomplished at 10 GHz at the IUN mode-stirred reverberating chamber with all stirrers operating. The transmitting and receiving antennas are both X -band WR90 flanges. Case I corresponds to the transmitting antenna located on the left side of the entrance door and the receiving antenna located at the center of the chamber facing the second stirrer [7]. The two antennas are cross polarized. Case II corresponds to the transmitting and receiving (copolarized) antennas faced at a distance of 50 cm. Case III corresponds to the case that the transmitting and receiving (copolarized) antennas are faced at a distance of 75 cm. In (a)–(d), S is shown in a polar format, where horizontal is the real axis and vertical the imaginary one. In (b)–(d) and (f) the measured and theoretical pdf model regarding T are shown. Dots corresponds to measurements, solid lines to theoretical models. (a) and (b) correspond to Case I; (c) and (d) correspond to the Case II; (e) and (f) correspond to the Case III. In Case I, the theory calls for a Rayleigh pdf whereas in Cases II and III, i.e., whenever a direct antenna-to-antenna coupling is present, the theory calls for a Rice pdf. Theoretical pdf parameters have been estimated as in [7].

cautions, it can be useful to approximate (11) with a Gaussian one. than the cusEquation (6) is a more general model for tomary Rayleigh one. In fact, as soon as we set to 0 we get (1). As a further verification we can recover the normalized mo[12] and making ments of (2) by noting that some manipulation of (7). The suitability of this general model is tested in Section III. III. EXPERIMENTS In this section, we present and discuss some experiments which aim at providing some evidence in support of the new

general theory for in a reverberating chamber presented in Section II. The theory relates the electromagnetic field in a well-operating reverberating chamber and the electromagnetic field in presence of an unstirred component of the field. Such a case is experimentally reproduced examining the case that an antenna-to-antenna coupling is present. This latter case is related to the corresponding reference case of a well-operating chamber. The experiments have been conducted at the mode-stirred reverberating chamber of the IUN which is a 2 m 2 m 2 m room internally coated by aluminum wherein three electrically large rotating plates are present [5]–[7]. The three stirrers are located at the left of the entrance door, in front of it, and on the top

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of the chamber [5]–[7]. In the referred experiments the transmitting and receiving antennas are both -band WR90 flanges, the operating frequency was 10 GHz and all three stirrers were in motion. An HP-8510C Network Analyzer was used to perform the measurements. All data sets consisted of 3000 points and have been achieved by sampling the time series at a rate of 0.23 ms. A full calibration of the system was a priori undertaken. was measured and an off-line The complex insertion loss data analysis was performed. Three cases have been considered. In the first case (Case I) the transmitting antenna was located on the left side of the entrance door whereas the receiving antenna was located at the center of the chamber facing the second stirrer [7]. The two antennas were cross polarized. Results are shown in Fig. 1(a) wherein a polar plot format is employed. The corresponding complex data set is centered on the origin. Such a result is consistent with the one published in [7]. Focusing on we have that it is Rayleigh distributed with mean equal to 3.465e-2 and variance equal to 3.290e-4 [see Fig. 1(b)]. As a consequence the two zero-mean real Gaussian processes generating the complex insertion loss have a equal about to 0.0277 [see (3) and (4)]. The corresponding figure is plotted in Fig. 1(b): dots correspond to measurements, line to the model. The -test of hypothesis with a 95% confidence level was applied and passed. More simply, if we evaluate the ratio of the variance with the squared mean, we obtain 0.274, which is in total agreement with the theory (see Section II). Let us examine two other cases (Cases II and III), which show the presence of some direct antenna-to-antenna coupling. In these latter cases, the transmitting and receiving antennas are located at the center of the chamber and are faced at a distance of 50 (Case II) and 75 cm (Case III) and are copolarized. The corresponding complex data sets are off centered and are shown in Fig. 1(c) and (e), respectively. In particular, as expected, if we compare the two data sets, the largest offset corresponds to the case of 50 cm [see Fig. 1(c)]. Note also that you can read these latter cases as generated by a proper shift of the data of Case I (thermodynamic case). This is in agreement with the theory of Section II. Let us now focus on . The theory presented in Section II predicts a Rice distribution. As a matter of fact, in the 50-cm we can estimate the mean and variance of case [7]. We have that the mean and the variance are equal to 1.837e-1 and 8.483e-4, respectively. The antenna coupling can be simply determined by analyzing the means of the real and imaginary part of the complex electromagnetic field (Section II). As a consequence, the corresponding Rice pdf can now be properly defined and tested versus measurements -test [6]. In order to test the model, we again used the of hypothesis with a 95% confidence level [7], which was passed. This is also visually witnessed in Fig. 1(d). Dots correspond to measurements and the solid line to the Rice pdf. Let us finally move to the details of the 75-cm case (Case III). Again, we have the presence of some direct antenna-to-antenna coupling emphasized by the off centering of the complex data, see Fig. 1(e). Data analysis determined a mean equal to 1.316e-1 and a variance of 8.045e-4. Proceeding as before, we can test

with the behavior predicted by the theory of Section II, see Fig. 1(f). The figure format is the same of Fig. 1(d). Again quantitative and qualitative analysis were positive showing the validity of the proposed theory. IV. CONCLUDING REMARKS A general statistical electromagnetic theory, which models the electromagnetic field within a reverberating chamber, has been presented. It applies both in the thermodynamic limit (or Rayleigh case) [3], [4], [6], [7] and whenever an unstirred component of the field is present. A set of measurements performed at the IUN mode-stirred reverberating chamber validated the theory. On the theoretical side, such a physical model is representative of many complex electromagnetic environmental situations, e.g., urban environments, and, therefore, it is of great relevance in the EMC field. On the applicative side, the model can be furthered in order to characterize a reverberating chamber whether it is well operated or not and to set up an environment with a specific amount of direct coupling in conjunction with a stochastic field. Therefore, the new theory is amenable to study the influence of a device under test on the electromagnetic field in a reverberating chamber as well as the effect of absorbers. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments which improved the quality of the paper. REFERENCES [1] D. A. Hill, “Electronic mode stirring for reverberating chambers,” IEEE Trans. Electromagn. Compat., vol. 36, pp. 294–299, Nov. 1994. [2] M. T. Ma, M. Kanda, M. L. Crawford, and E. B. Larsen, “A review of electromagnetic compatibility/interference measurement methodologies,” Proc. IEEE, vol. 73, pp. 388–411, Mar. 1985. [3] J. G. Kostas and B. Boverie, “Statistical model for a mode-stirred chamber,” IEEE Trans. Electromagn. Compat., vol. 33, pp. 366–370, Nov. 1991. [4] J. Ladbury, G. Koepke, and D. Camell, “Evaluating of the NASA Langley Research Center mode-stirred chamber facility,”, NIST Tech. Note 1508, 1999. [5] P. Corona, G. Latmiral, E. Paolini, and L. Piccioli, “Use of a reverberating enclosure for measurements of radiated power in the microwave range,” IEEE Trans. Electromagn. Compat., vol. EMC-18, pp. 54–59, May 1976. [6] P. Corona and G. Latmiral, “Approccio termodinamico allo studio di una camera riverberante elettromagnetica a geometria variabile,” (in Italian), , Istituto Univ. Navale Rep., 1978. [7] P. Corona, G. Ferrara, and M. Migliaccio, “Reverberating chambers as sources of stochastic electromagnetic fields,” IEEE Trans. Electromagn. Compat., vol. 38, pp. 348–356, Aug. 1996. [8] P. Corona and G. Latmiral, “Valutazione ed impiego normativo della camera riverberante dell’Istituto Universitario Navale,” in Atti I Riunione Nazionale di Elettromagnetismo Applicato, L’Aquila, Rome, Italy, 1976, pp. 103–108. [9] R. R. Lentz and H. C. Anderson, “Reverberating chamber for EMC measurements,” in Proc. EMC Symp., San Diego, CA, Oct. 1979, pp. 446–451. [10] A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill, 1984. [11] J. Omura and T. Kailath, “Some useful probability distributions,” Syst. Theory Lab., Stanford Electron. Lab., Stanford Univ., Stanford, CA, Tech. Rep. 7050-6, 1965. [12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972.

CORONA et al.: REVERBERATING CHAMBER EM FIELD IN PRESENCE OF UNSTIRRED COMPONENT

Paolo Corona (M’78–SM’90–F’92) received the Laurea degree in electronic engineering from the University of Napoli, Italy. He is a Full Professor of electromagnetic waves at the Istituto Universitario Navale (IUN), Napoli, Italy. He has also taught radar and radio aids, antennas and propagation, remote sensing, measurement techniques, and electromagnetic compatibility. He has been a Visiting Scientist at Colorado University, Boulder. He has served as a Consultant for many Italian and European companies. His main research activities include antennas, radar cross section, material characterization, and electromagnetic compatibility. Prof. Corona is chairman of the European Union COST Action 243 on Electromagnetic Compatibility and a member of the Steering Committee of the CNR Electromagnetic Group and AEI Electromagnetic Compatibility Group, the IEC Working Group on Reverberating Chambers, the CT 110, and the CEI Working Group on Anechoic Chambers. He is an Associate Editor of IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY.

Giuseppe Ferrara received the Laurea degree in electronic engineering from the University of Napoli, Italy. He is an Associate Professor of radar theory at the Istituto Universitario Navale, Napoli, Italy. He was previously an Associate Professor (microwaves) at the Università di Catania, Sicily, Italy. He was formerly involved in electromagnetic biological effects. He is now involved in many projects in the fields of electromagnetic high-frequency methods, microwave-power techniques, radar cross sections modeling and experimentation, microwave measurements, and electromagnetic compatibility.

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Maurizio Migliaccio (M’91) was born in Napoli, Italy. He received the Laurea degree in electronic engineering (summa cum laude) from the Universita di Napoli Federico II, Italy, in 1987. In 1987, he joined the Department of Electronic Engineering (Electromagnetic Division) at the Universita di Napoli Federico II. In 1990 he became University Researcher at the Instituo Universitario Navale (IUN), Napoli, Italy, where he joined the Electromagnetic Waves Institute. Since 1993 he taught microwave remote sensing at the IUN. In 1998 he became Associate Professor at the Universita di Cagliari, Sardinia, Italy, as a member of the Engineering Faculty, where he teaches microwave remote sensing and electromagnetic compatibility. In 1993, 1998, 1999, and 2000, he was Adjoint Scientific Researcher of the Italian National Council of Research (CNR)—IRECE in the remote sensing field. He has been participating in an International NASA-ASI-DARA project on interferometry. He has been a Visiting Scientist at the DLR, Germany. His main research activities have been regarding active microwave remote sensing, electromagnetic modeling of natural senses, electromagnetic fractal modeling, stochastic electromagnetic fields sampling, and electromagnetic reverberating chambers. He has been a member of the IUN AdCom and European Union secretary of the past COST 243 project on electromagnetic compatibility. He is a member of the board of Ph.D.’s in Computer Science, Applied Electromagnetics, and Telecommunications Engineering held at the University of Salerno, Italy. Mr. Migliaccio is a member of the AGU.