Statistical characterization and the simulation of a reverberation chamber using finite-element...

214 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 1, FEBRUARY 2002

Statistical Characterization and the Simulationof a Reverberation Chamber Using

Finite-Element TechniquesCharles F. Bunting

Abstract—The statistical characterization of a simulation rever-beration chamber is performed by considering a two-dimensionalfinite element model. This model includes a source to study theparticular modal fields that couple into either a transverse elec-tric or transverse magnetic configuration. The analysis includes acharacterization of the basic field statistics, max-to-average ratio,normalized standard deviation, stirring ratio, and field uniformity.The shielding effectiveness of an aperture will be studied that willprovide some insight into the nature of the fields coupled from acomplex to a noncomplex environment.

Index Terms—Electromagnetic compatibility, finite-elementmethods, mode-stirred chambers, reverberation chambers, statis-tical electromagnetics.

I. INTRODUCTION

A REVERBERATION chamber is an enclosure consistingof metal walls with a metallic paddle wheel (denoted a

“stirrer” or “tuner”) forming a high quality factor ( ) cavitywith continuously variable boundary conditions. Reverberationchambers have attained increased importance in the determi-nation of electromagnetic susceptibility of avionics equipment.This importance will become even more critical as advancedhigh-speed transport aircraft are developed that increasingly de-pend on electronic sensors and computer control of flight sur-faces to manage the flight parameters.

The fields in a reverberation chamber are typically charac-terized by statistical means. Specifically the probability densityfunctions (pdfs) for the real and imaginary components of a par-ticular polarization of the electric and magnetic fields are nor-mally distributed. The field magnitudes are Rayleigh distributedand the power is exponentially distributed. Reasonable statis-tical agreement for a source-free two-dimensional (2-D) finiteelement model has been obtained and was the primary emphasisof the work presented by the author [1].

There are two focus areas that will be addressed in this work.The first problem to be briefly addressed in this paper is anexamination of a 2-D finite element model for both the trans-verse electric (TE) and transverse magnetic (TM) solutions in-side a reverberation chamber. A 2-D approach to the analysisof reverberation chambers was initially suggested by Wu [2] for

Manuscript received October 8, 2000; revised August 9, 2001. This work wassupported by NASA under Grant NAG-1-1982 at NASA Langley in Hampton,VA.

The author is with the Department of Electrical and Computer Engineering,Oklahoma State University, Stillwater, OK 74078 USA (e-mail: [email protected]).

Publisher Item Identifier S 0018-9375(02)01434-5.

Fig. 1. The 2-D reverberation geometry.

mechanical stirring using the transmission line matrix (TLM)method. Hill [3] examined frequency stirring for an empty 2-Dstructure supporting TM modes. For the current work, the fields(both TE and TM) in the cavity will be simulated with an em-phasis on the tuner effects on the modal structure and the re-sulting statistics of the field distribution. The second focus isshielding effectiveness in a reverberating environment. Typicalshielding effectiveness measurements are performed in a planewave environment under various angles of incidence. The re-verberating environment may provide additional insight into theshielding properties in a statistical sense.

The following section presents an overview of the compu-tational tools used to simulate the reverberation environment.The TE results for the simulation “chamber” are then presentedwith an examination of coupling (fields inside versus outside)shielding effectiveness through an aperture. The TM results arethen presented for the simulation chamber. The application ofa 2-D analysis tool may lead to useful investigations of stirrerefficiency, field homogeneity, and shielding effectiveness.

II. FINITE ELEMENT FUNDAMENTALS FOR TE AND TM FIELDS

The finite element method is a deterministic approach to thesolution of Maxwell’s equations using a weighted residual for-mulation over a set of compact-support basis functions to solvefor the fields. Consider the geometry of Fig. 1 with a source lo-cated at ( ).

The electromagnetic field behavior is governed by Maxwell’sequations as given by

(1)

(2)

0018–9375/02$17.00 © 2002 IEEE

BUNTING: STATISTICAL CHARACTERIZATION AND THE SIMULATION OF A REVERBERATION CHAMBER 215

The inclusion of the source will be accomplished through thecharacteristics of the electric current density,, in (2). By takingthe curl of (1) and substituting (2) and assuming nonmagneticmedia, the following inhomogeneous vector wave equation isobtained:

(3)

where is the electric field intensity in volts/meter, is thewavenumber with , at a radian frequency, withpermittivity and permeability , and intrinsic impedance .The behavior of the electric field is of the form

Setting for cutoff and expressing the del operator as, it is possible to write two separate equa-

tions—one for the transverse part and another for thecompo-nent. The transverse field behavior is modeled with the use ofedge elements and the-directed fields are modeled using tradi-tional node-based elements. The expression of (3) will dependon whether a TE or TM field will be considered to exist withinthe structure. This dependence completely rests in the expres-sion of the electric current density,.

A. TE Fundamentals—A Dipole Source

The TE case can be expressed by considering a TE currentthat can be thought of as a short dipole antenna with a uniformcurrent density. In the TE case,can be expressed as

from which (3) is written as

(4)

(5)

Equation (4) represents the transverse variation and (5) rep-resents the -directed variation. The TE current exists only onthe edge of a given element and is zero elsewhere. Obtainingthe weighted residual form involves multiplying by an appro-priate testing function, for edge-based expansion functionsand for node-based expansion functions, and integrating overthe element support and assigning the continuity conditions. In-tegrating on an elemental basis is accomplished by

(6)

and

(7)

Assigning continuity conditions and setting the fields to zero onthe perfectly conducting walls and tuner yields a matrix equa-tion of the form for the transverse fields. Note that, forthis system, and .

Fig. 2. Discretized geometry of Fig. 1. Chamber size 3.96 m� 7.01 m.Aperture is 0.25 m. The left- and right-hand sides are same size.

B. TM Fundamentals—An Infinite Line Source

Following directly the work of Hill [3], the TM case uses aninfinite line source located at of the form

(8)

so that (4) can be written as

(9)

(10)

leading to the following weighted residual form:

(11)

and

(12)

Note, that in obtaining (12), the integration required to obtainthe right-hand side is trivial. The resulting linear system of equa-tions is of the form . Note that, for this system,

and and .

III. TE DIPOLE SOURCERESULTS

The application of the finite element technique to fields thatare TE will be presented in this section. The resulting fields,

and , will be used in the 2-D TE finite element modelof the reverberation chamber. Some measures of reverberationchamber performance include the statistical characterization ofthe fields, stirring ratio, field uniformity, the maximum to av-erage ratio, and the normalized standard deviation [4]. Severalof these measures will be applied to provide basic validation ofthe reverberating characteristics of the simulation “chamber.”The shielding effectiveness in the simulation chamber will alsobe presented in the next subsection. The 2-D dipole source re-sults are obtained by the rotation of the tuner for 225 steps in1.6 increments, thus providing a full mechanical rotation ofthe tuner. The results represent the solution of the matrix equa-tion corresponding to (6) for each of the tuner positions at eachfrequency of interest. Consider the geometry of Fig. 2 depictingthe discretized structure to be analyzed with a typical result in


Fig. 3. Fields for Fig. 2 for a fixed tuner position.

Fig. 3. Note that the results of Fig. 3 are logarithmic and empha-size the field structure rather than the absolute field levels. Thesource is in the right-hand region at location (2,1).

A. Reverberation in the TE Environment

There are several key factors governing reverberation charac-teristics, and these factors can be used to determine if a chamberis operating properly. These factors include the stirring ratio,field uniformity, the ratio of the maximum field to average field,and the underlying statistical characterization of the fields. Ofthese measures of reverberation characteristics, the stirring ratioand field uniformity will not be addressed in this paper. As thevalues of the field at a particular point are plotted, it was notedthat strong numerical resonances occur throughout the com-putational domain in the lossless structure. These resonancescorrespond to the eigenvalues of the source-free problem [1].In order to improve the numerical stability, the quality factor

can be altered by filling the chamber with lossy material( ). The field statistics at a given point for 225tuner steps is depicted in Fig. 4. The total squared electric fieldstatistics at a point inside the shielded left-hand side are de-picted with a of 1000 at a frequency of 400 MHz. The Weibullprobability plot [1] shown at the bottom of Fig. 4 is a strongindicator regarding the nature of a sample statistic. The prob-ability plot helps eliminate ambiguities that may arise in inter-preting agreement by comparing cumulative distribution func-tions (cdfs). The probability plot maps the parent distributionto a straight line and emphasizes differences in the tails of thesample distribution. Note that the symbols on the plot rep-resent the sample and all but approximately 2% of the dataare nearly collinear with the parent Weibull distribution. TheWeibull density function is a more general density function thatcan fit a variety of bounded continuous distributions includingthe exponential distribution. The square of a particular compo-nent of the electric field, say , can be written

(13)

If the underlying distributions for the real and the imaginaryparts are , then the squared sum is exponentially dis-tributed (chi-squared with two degrees of freedom). The pdf foran exponential distribution is

(14)

Fig. 4. The statistical characteristics of the total TE field for a fixed point for225 tuner positions.

with parameter . The exponential distribution is a special caseof a Weibull distribution given by the pdf

.(15)

The exponential distribution corresponds to in (15). Con-sidering the component of the electric field as depicted inFig. 4, the maximum likelihood estimate of isa good indicator of agreement with the expected distribution.It strongly suggests that the underlying distributions are indeednormally distributed. The CDF is plotted in Fig. 5 showing astrong agreement with the expected Weibull distribution and thechi-squared distribution with two degrees of freedom. Anothermeasure of reverberation quality is the normalized standard de-viation is defined by

where is the sample standard deviation andis the samplemean. The point shown (70, 40) in the right-hand side (RHS),which is the complex portion of the structure, has a normalizedstandard deviation of 0.991. The ratio of the maximum to av-erage field is 7.93 dB is shown in Fig. 6 which agrees well theexpected value of 8 dB. The stirrer ratio is 33.89 dB which is in-dicative of excellent stirring within the complex RHS. A sum-mary of example statistics for a few points in the structure isshown in Table I. Table I portrays the Weibull parameters, thenormalized standard deviation, the max-to-average ratio, andthe stirring ratio for two points in each of the two major regionswithin the simulation chamber. The plots of Figs. 4–6 representthe data for the point (70, 40). Note the general excellent agree-ment for each of the points shown. Of particular interest is thatthe LHS is unstirred and it has generally identical statistical andreverberation characteristics as the complex RHS. This inter-esting result is suggestive of a form of “source” stirring whereinthe fields appearing at the aperture joining the two regions acts


TABLE ISUMMARY OF STATISTICAL CHARACTERISTICS FOR THETE FIELDS

Fig. 5. The cumulative distribution of the total TE field for a fixed point for225 tuner positions.

Fig. 6. The total TE field for a fixed point for 225 tuner positions highlightingthe max-to-average ratio and stirrer ratio.

with sufficiently random amplitude and structure as to excite acomplex field in the noncomplex region.

Considering the field uniformity within each region can fur-ther highlight the idea of “source” stirring. In this study the fielduniformity will be explored by considering a test region in whichthe ensemble average of the fields is sampled and compared toother points within the test region. The fields will be consid-ered to be uniform if the sample is within3 dB of the average

Fig. 7. The�3-dB field uniformity of the total TE field for 100 points in theRHS for 225 tuner positions.

Fig. 8. The�3-dB field uniformity of the total TE field for 100 points in theLHS for 225 tuner positions.

of all sample points. The test region is considered uniform if75% of the sample field points are within the3-dB region.The total electric field in the RHS is studied in Fig. 7. The top ofFig. 7 depicts the sample region wherein 100 points are chosenat random. Note that the wall with the aperture is not shown, butexists for reference points (50,). The uniformity is depictedin the bottom of Fig. 7 with the circled data representing pointswithin 3 dB of the average value of approximately7 dB. Thetriangles indicate points that are outside the specified limits. As


Fig. 9. Shielding effectiveness of the maximums for the TE field.

Fig. 10. The statistical characteristics of the total TM field for a fixed pointfor 225 tuner positions.

can be seen, 81 out of 100 points are within the limits and there-fore the test region in the RHS can be considered to have a uni-form field. The uniformity in the LHS is shown in Fig. 8 with91% of the points falling within 3 dB of the average. The goodresult for the field uniformity in the noncomplex LHS furthersuggests that “source” stirring can provide excellent reverbera-tion characteristics.

B. Shielding Effectiveness for the TE Structure

The application to the problem of coupling shielding effec-tiveness via an aperture will be examined in this section. Thegeometry of Fig. 1 is the geometry under consideration forthis examination. Consider a set of 100 random pointsinthe source region (RHS). For those points, there will be somemaximum field level that will exist over all tuner positions,

RHS . This level represents the maximumlevel in the source region. Consider taking another set of 100random points in the test region (LHS). This set consistsof the individual maximums for each test point for all tuner

Fig. 11. The cumulative distribution of the total TM field for a fixed point for225 tuner positions.

Fig. 12. The total field total TM field for a fixed point for 225 tuner positionshighlighting the max-to-average ratio and stirrer ratio.

positions given by LHS . The shielding effectivenessat each test point can be expressed as the ratio

RHS

LHS(16)

A typical result for the shielding effectiveness of the maximumscan be seen in Fig. 9. Interestingly, there are 22 occurrences ofnegative shielding effectiveness out of the total of 100 randompoints chosen. This is indicative of phenomena wherein the fieldis larger in the shielded test region than in the source region. Theaverage shielding effectiveness of the maximums was 2.19 dBfor 400 MHz.

IV. TM I NFINITE LINE-SOURCERESULTS

The work presented in this section will examine the statisticsof the TM fields in the two-dimensional reverberation structure.


TABLE IISUMMARY OF STATISTICAL CHARACTERISTICS FOR THETM FIELDS

Fig. 13. The�3-dB field uniformity of the total TM field for 100 points in theRHS for 225 tuner positions.

This work closely follows the work of Hill in that the only sur-viving field component is . The results presented in this sec-tion will exactly parallel the work on the TE fields previouslyshown, so developmental discussions will be omitted.

A. Reverberation in the TM Environment

The field solution for the TM problem consists of the-component of the electric field with real and imaginary parts.

The field statistics for the TM case are shown in Fig. 10 forreference point (70, 30). The estimate of the shape parameter

for the Weibull distribution indicates thatone or more of the underlying distributions is not normallydistributed. This result is consistent with Hill’s results whenthe frequency chosen is such that an insufficient number ofmodes exist within the structure. It is significant to note that thefield levels for the TM problem are several orders of magnitudelarger than the TE problem. The cdf of the sample is depictedin Fig. 11 in comparison with the Weibull distribution and thechi-squared distribution with the normalized standard deviationof 2.3413. The maximum to average ratio is shown in Fig. 12and is 12.34 dB. The stirrer ratio is also portrayed in Fig. 12and has a value of 49.56 dB. Table II depicts several pointsin various points within the geometry similar to those shownearlier with similar results for the fields in the complex RHSand the noncomplex LHS. The field uniformity in the RHS andLHS are studied in Figs. 13 and 14. Note that the RHS has 63%uniformity in Fig. 13 and the LHS has 59% uniformity. Theseresults combine to raise some basic questions regarding thestatistical characteristics of the TM model at the frequencies

Fig. 14. The�3-dB field uniformity of the total TM field for 100 points in theLHS for 225 tuner positions.

used, and are the subject of a more detailed exploration in thefollowing paragraph.

Further examination using statistical simulation software(Expert Fit version 2.2) revealed that the real and imaginaryparts of are distributed according to a Johnson SU distri-bution [5], [6]. The Johnson distribution is an extreme valuedistribution denoted and is a four-parameter distributionwith a pdf

(17)

for all real numbers . The parameters are a location parameter, scale parameter , and shape parameters, and . The cdf is given by

(18)

with as the standard normal distribution function. Theskewness of the density function is represented in the shape pa-rameter with the density function skewed left when ,symmetrical when , and skewed right when .


TABLE IIISUMMARY OF STATISTICAL CHARACTERISTICS FOR THEREAL PART OF THE TM FIELDS

TABLE IVSUMMARY OF STATISTICAL CHARACTERISTICS FOR THEIMAGINARY PART OF THE TM FIELDS

Fig. 15. Shielding effectiveness of the maximums for the TM field.

Tables III and IV depict the summary statistics for the real andimaginary parts, respectively. These results depict a wide varia-tion of the parameters for various locations within the structure.Equations (17) and (18) may be useful in determining the max-imum-to-average ratio of an undermoded chamber and may helpin extending the current reverberation chamber theory to lowerfrequencies. Work is ongoing to ascertain the connection thatcan be made between these results and an undermoded rever-beration chamber.

B. Shielding Effectiveness for the TM Structure

The shielding effectiveness of the maximums for the TM fieldis depicted in Fig. 15. The resulting coupling appears to bemuch stronger for an identical test configuration. The averageshielding effectiveness is seen to be 0.991 dB with 40 points outof 100 demonstrating negative shielding effectiveness.

V. CONCLUSION

A 2-D finite element model for both the TE and TM solutionsinside a simulation reverberation chamber has been presented.Tuner effects on the modal structure and the resulting statis-

tics of the field distribution have been explored. It was demon-strated that the TE simulation reverberation chamber providedexcellent statistical and reverberation characteristics. The mea-sures of reverberation characteristics included normalized stan-dard deviation, the max-to-average ratio, the stirrer ratio, andthe field uniformity. In addition, the notion of “source” stirringwas introduced wherein the fields that were coupled into thenoncomplex geometry were shown to be statistically similar tothose in the complex environment. The idea of source stirringmay shed light on the question of whether it is absolutely nec-essary to stir in all regions that a reverberation type test is beingperformed. The practical issues are enormous when consideringthe demands that internal stirring may place on the testing reg-imen. The TM simulation chamber did not satisfy any of therequired reverberation characteristics, and the field componentswere shown to be distributed according to a Johnsondistri-bution. Further study may link the results of the TM simulationchamber to an undermoded reverberation chamber and may beuseful in extending the useful low frequency of the reverberationchamber. The shielding effectiveness in a reverberating environ-ment was examined. Coupling shielding effectiveness for TEand TM results was presented. Future work may include explo-ration aperture size and its influence on the resulting statisticalrepresentation of the fields and the corresponding shielding ef-fectiveness. The reverberating environment provides additionalinsight into the shielding properties in a statistical sense.

REFERENCES

[1] C. F. Bunting, K. J. Moeller, C. J. Reddy, and S. A. Scearce, “A two-dimensional finite element analysis of reverberation chambers,”IEEETrans. Electromagn. Compat., vol. 41, pp. 280–289, May 1999.

[2] D. L. Wu and D. C. Chang, “The effect of an electrically large stirrer ina mode-stirred chamber,”IEEE Trans. Electromagn. Compat., vol. 31,pp. 164–170, May 1989.

[3] D. A. Hill, “Electronic mode stirring for reverberation chambers,”IEEETrans. Electromagn. Compat., vol. 36, pp. 294–299, May 1994.

[4] M. O. Hatfield, G. J. Freyer, and M. B. Slocum, “Reverberation charac-teristics of large welded steel shielded enclosures,” inProc. IEEE 1997Int. Symp. on Electromag. Compat., Austin, TX, 1997, pp. 38–43.

[5] A. M. Law and W. D. Kelton,Simulation Modeling and Analysis, 3rded. New York: McGraw-Hill, 2000.

[6] N. L. Johnson, S. Kotz, and N. Balakrishnan,Continuous UnivariateDistributions, 2nd ed. Boston, MA: Houghton Mifflin, 1995, vol. 2.


Charles F. Bunting (S’89–M’94) was born inVirginia Beach, VA in 1962. He received the A.A.S.degree in electronics technology from TidewaterCommunicy College, Portsmouth, VA, in 1985, theB.S. degree in engineering technology from OldDominion University, Norfolk VA, in 1989, and theM.S. and Ph.D. degrees in electrical engineering,both from the Virginia Polytechnic Institute andState University, Blacksburg, VA, in 1992, and 1994,respectively.

He was an apprentice, Electronics Mechanic,and Electronics Measurement Equipment Mechanic with the Naval AviationDepot, Norfolk, VA, between 1981 and 1989. From 1994 to 2001, he was anAssistant/Associate Professor with the Old Dominion University’s Departmentof Engineering Technology, and worked closely with the NASA LangleyResearch Center in the area of electromagnetic field penetration in aircraftstructures and reverberation chamber simulation using finite-element tech-niques. Since Fall 2001, Dr. Bunting is an Associate Editor with the OklahomaState University. Stillwater, OK. His chief interests are fundamental variationalprinciples and computational electromagnetics, statistical electromagnetics,characterization and application of reverberation chambers, and the analysisof optical and microwave structures using numerical methods includingfinite-element techniques.

Dr. Bunting is member of Tau Alpha Pi, Phi Kappa Phi, and Alpha Chi.

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