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Doctoral Thesis

Three-dimensional simulation and experimental verification of areverberation chamber

Author(s): Bruns, Christian

Publication Date: 2005

Permanent Link: https://doi.org/10.3929/ethz-a-005027300

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DISS. ETH No. 16041

THREE-DIMENSIONAL SIMULATION

AND EXPERIMENTAL VERIFICATION

OF A REVERBERATION CHAMBER

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH

for the degree of

Doctor of Sciences

presented by

CHRISTIAN BRUNS

Dipl.-Ing., Universitat Fridericiana Karlsruhe (TH), Germany

born December 19, 1973

citizen of Germany

accepted on the recommendation of

Prof. Dr. R. Vahldieck, examinerProf. Dr. F. Canavero, co-examiner

2005

http://www.ethz.ch

mailto:[email protected]

http://www.uni-karlsruhe.de

If you can’t do it better — why bother doing it at all?

Michael E. Porter

Abstract

Electronic products must be designed so that they do not disturb the proper operationof other products and inversely withstand electromagnetic radiation emitted from sur-rounding devices. A crucial aspect of successful product development is therefore theeffective and efficient testing of the electromagnetic compatibility (EMC). Reverberationchambers (RCs) enjoy growing popularity as a complement or replacement to the wellestablished methods for radiated interference. RC testing exhibits several competitiveadvantages over existing methods, such as an enhanced test repeatability and a morerealistic as well as rigorous test environment. The importance of testing for EMC in RCsas an alternative measurement technique has recently been recognized in the IEC 61000-4-21 standard.The three-dimensional simulation of an RC is presented in this thesis. In the begin-ning, fundamental concepts and key parameters of an RC are introduced, among themthe mode distribution, mode density, modal gaps, and the quality factor. Furthermore,the RC is described as a statistical electromagnetic test environment and characterizedby distribution functions, correlation, uncertainty, and field uniformity. It is pointedout that it is crucial to select a suitable numerical method to perform meaningful RCsimulations. A chosen numerical technique must be able to deliver results over a widefrequency range without using excessive computational resources; the method must beable to handle large, irregular structures, and a varying geometry without introducingerrors. Furthermore, there must be a possibility to account for finite metal conductiv-ity as well as highly resonant structures. The computation of near fields at an arbitrarynumber of chamber locations should be possible without adding too much computationaloverhead. A frequency-domain electric field integral equation-based method-of-momentstechnique is chosen for the RC simulations.A prototype RC with a measurement system is built and used later on for simulationvalidations. Measurement errors originating from field probes, antennas, and stirrers areassessed for their impact on deviations between simulated and measured results. Thebiggest deviations are found to result from the antenna tripods and position inaccuraciesof the field probe head or the antennas. The prototype RC including the door, stirrers,several antennas, and an equipment under test is modeled for the electromagnetic sim-ulation. Suitable electrical conductivity values are derived for material as it is used ina shielded room construction. In addition to the prototype RC, cubic and corrugatedchambers, an offset-wall RC as well as several vertical and horizontal stirrers are mod-eled.Simulation results of a detailed asymmetric RC model are benchmarked against mea-surements and exhibit a good agreement in the lower-to-medium frequency range. It isshown that a proper validation of the simulation must be performed with direct compar-isons against measured near fields without further data processing or statistical analysis.Furthermore, a deeper analysis of various chamber geometries, TX/RX antennas, differ-

i

ii ABSTRACT

ent stirrer designs, and equipments under test is performed. The importance of smallgeometric details and the agreement between actual prototype and simulated RC dimen-sions is discussed. It is shown that the type, position, and alignment of the excitationsource in the simulation model change the field pattern significantly. In addition, theeffect of various stirrers on the fields, correlation, and uniformity inside the chamber arevisualized. The 6-paddle stirrer developed for this thesis and the commercially availableZ-fold stirrer have the best performance.A comparison between the standard rectangular RC with a cubic and a corrugated cham-ber revealed that the two latter chamber geometries do not offer significant advantagesconcerning correlation and field uniformity. On the other hand, the cubic RC does notperform as bad as always alleged. The presence of a stirring device shifts the modesin frequency depending on their respective field distribution away from the analyticallycalculated resonance frequencies. Therefore the usually observed problem of degeneratemodes is found not come into play within a cubic RC – contrary to the widely acceptedRC design guidelines, a cubic RC does not exhibit worse performance than other rec-tangular RCs. Three special multipath/direct path coupling scenarios are simulated(Gaussian, Rice, and Rayleigh statistical distributions). This investigation reveals thatthe usage of a Hertzian dipole in an RC simulation leads to undesirable strong directcoupling between an equipment under test and the excitation. Through the usage ofrealistic antennas with higher directivity, this unwanted direct coupling can be consid-erably reduced.In this thesis it is shown that for frequencies much smaller than the lowest usable fre-quency, the simulation of an RC is possible, the chamber however becomes electricallytoo small compared to the operational wavelength, which prevents sufficient statisticalfield uniformity (an optimization is therefore not possible). Conversely, at frequenciesmuch above the lowest usable frequency, where a high number of modes is above cutoff,almost any RC works well regardless of its particular design (hence, there is no optimiza-tion needed). With increasing frequency, the field within an RC becomes more and moresensitive to even small geometric details, which makes proper modeling numerically notfeasible at high frequencies. The possibilities for RC design optimizations significantlybelow or above the lowest usable frequency are therefore limited. At frequencies aroundthe lowest usable frequency, however, stirrer shapes or wall geometries can be optimizedusing the results presented in this thesis in order to improve field uniformity and toextend the operating frequency for a given RC to lower frequencies.

Zusammenfassung

Elektronische Produkte mussen heute so entwickelt werden, dass sie einerseits andereProdukte nicht in ihrer Funktion beeintrachtigen und gleichzeitig selbst weitgehend im-mun gegen elektromagnetische Einstrahlung anderer Gerate sind. Daher besteht einwichtiger Aspekt der erfolgreichen Produktentwicklung darin, neue Gerate in effizienterund effektiver Weise auf ihre elektromagnetische Vertraglichkeit (EMV) hin zu unter-suchen. Reverberation Chambers (RCs) erfreuen sich seit einiger Zeit steigender Be-liebtheit und stellen eine Erganzung bzw. einen Ersatz bestehender EMV-Testmethodendar. Tests in RCs weisen gegenuber bestehenden Verfahren verschiedene Vorteile auf, sieermoglichen z.B. eine bessere Wiederholgenauigkeit sowie ein realistischeres und stren-geres Prufverfahren. Der zunehmenden Bedeutung von RCs wurde durch den Entwurfund die kurzliche Veroffentlichung des IEC 61000-4-21 Standards Rechnung getragen.Diese Arbeit behandelt die dreidimensionale Simulation einer RC. Zunachst werden diegrundlegenden Aspekte sowie die wichtigsten Parameter einer RC behandelt. Dazu geho-ren die Verteilung der Moden, die Modendichte, die Modenlucken und der Gutefaktor.RCs werden im weiteren Verlauf als statistische Testumgebung beschrieben und charak-terisiert durch Verteilungsfunktionen, Korrelation, statistische Unsicherheit und Fel-duniformitat. Es wird aufgezeigt, dass die Wahl einer geeigneten numerischen Meth-ode entscheidend ist, um sinnvolle RC-Simulationen durchzufuhren. Die jeweilige nu-merische Methode muss einerseits Simulationen uber einen weiten Frequenzbereich er-lauben, ohne jedoch exorbitante Rechenleistung zu benotigen; andererseits muss dieMethode in der Lage sein, grosse unregelmassige Strukturen zu berechnen, bei denen sichTeile der Geometrie bewegen. Ausserdem muss es ohne grossen Mehraufwand moglichsein, die endliche Leitfahigkeit des Materials zu berucksichtigen. Die eingesetzte Sim-ulationstechnik sollte das Feld an sehr vielen raumlich verteilten Punkten bestimmenkonnen, ohne den numerischen Aufwand signifikant zu vergrossern. In dieser Arbeitwird fur die RC-Simulationen die auf der elektrischen Feldintegralgleichung basierendeMomentenmethode im Frequenzbereich verwendet.Ein RC-Prototyp wird konstruiert und zusammen mit einem Messsystem fur die Uber-prufung der Simulationsergebnisse eingesetzt. Durch Feldsonden, Antennen und Ruhrerentstehende Messfehler werden hinsichtlich ihres Einflusses auf die Ubereinstimmung vonMess- und Simulationsergebnissen beurteilt. Die grossten Abweichungen sind auf dieAntennenstative und die Positionierungenauigkeit der Feldsonden sowie der Antennenzuruckzufuhren. Der aus Wanden, Tur, Ruhrern, mehreren Antennen und einem Test-objekt bestehende RC-Prototyp wird fur die elektromagnetische Simulation modelliert.Brauchbare Leitfahigkeitswerte fur die verwendeten Materialien werden durch Messun-gen ermittelt, so dass die praktischen Verhaltnisse in einer Schirmkabine reproduziertwerden konnen. Neben dem RC-Prototyp werden kubische und gerippte Kammern, eineRC mit einer versetzten Wand sowie verschiedene vertikal und horizontal angeordneteRuhrer modelliert.

iii

iv ZUSAMMENFASSUNG

Die Simulationsergebnisse eines detailgetreuen, asymmetrischen RC-Modells werden ver-glichen mit Messresultaten. Dabei ergibt sich eine gute Ubereinstimmung im unterensowie mittleren Frequenzbereich. Es wird dargelegt, dass eine sinnvolle Validierungder Simulationsergebnisse nur uber direkte Vergleiche mit Messergebnissen durchgefuhrtwerden kann. Sowohl die Simulations- wie auch die Messergebnisse sollten dabei wederweiterverarbeitet werden zu abgeleiteten Grossen noch mittels statistischer Kennzahlenbeschrieben werden. Die modellierten RCs mit den darin befindlichen Sende- und Emp-fangsantennen, verschiedenen Ruhrern und Testobjekten werden mit Hilfe der Simulationuntersucht. Der Einfluss unscheinbarer Details sowie von geringen geometrischen Ab-weichungen zwischen Prototyp und Modell einer RC wird behandelt. Es wird gezeigt,dass die Art, Position und Ausrichtung der Anregung im Simulationsmodell die Feld-verteilung in der RC erheblich beeinflussen. Weiterhin wird der Effekt verschiedenerRuhrertypen auf das elektromagnetische Feld, die Korrelation und die Gleichmassigkeitder Feldverteilung in der Kammer untersucht. Die beste Leistung konnte mit demin dieser Arbeit entwickelten 6-Flugel-Ruhrer sowie dem kommerziell erhaltlichen Z-Faltung-Ruhrer erzielt werden.Der Vergleich einer gewohnlichen, rechteckigen RC mit einer kubischen bzw. geripptenRC zeigt, dass die beiden letztgenannten Varianten keinerlei Vor- oder Nachteile im Hin-blick auf Korrelation oder eine gleichmassige Feldverteilung aufweisen. Insbesondere istdie kubische Kammer nicht so schlecht wie oft behauptet: Durch den Ruhrer werden dieeinzelnen Moden (je nach zugehoriger Feldverteilung) weg von der analytisch berechnetenResonanzfrequenz verschoben. Dadurch tritt das Problem der Modendegeneration in-nerhalb einer kubischen RC praktisch nicht auf. Im Gegensatz zur weit verbreiteten Mei-nung (sowie Konstruktionsempfehlungen) weist eine kubische RC somit keine signifikantschlechtere Felduniformitat im Vergleich zu anderen rechteckigen Kammern auf. Weiter-hin werden drei Spezialfalle von Kopplungspfaden und Mehrwegeausbreitung untersucht(resultierend in Gauss, Rice und Rayleigh Verteilungen). Diese Untersuchung ergab,dass sich bei Verwendung eines Hertzschen Dipols in der Simulation eine unerwunschte,starke direkte Kopplung zwischen Testobjekt und Anregung ergibt. Durch praxisnaheAntennen mit hoherer Richtwirkung lasst sich diese unerwunschte direkte Kopplung er-heblich reduzieren.In dieser Arbeit wird gezeigt, dass die Simulation einer RC bei Frequenzen kleiner alsder niedrigsten Betriebsfrequenz zwar moglich ist, die Kammer allerdings fur eine aus-reichend gleichmassige Feldverteilung (und damit auch Optimierung) elektrisch zu kleinwird. Im Gegensatz dazu funktioniert unabhangig vom Design jede RC bei hohen Fre-quenzen (d.h. genugend grosse Anzahl Moden ausbreitungsfahig) ausreichend gut, womitsich die Optimierung erubrigt. Fur eine korrekte Simulation gestaltet sich die geeigneteModellierung bei hohen Frequenzen sehr schwierig, da das Feld in einer RC prinzip-bedingt stark auf kleinste Geometrieanderungen reagiert. Die Moglichkeiten fur dieRC-Optimierung sind damit sowohl im unteren wie auch im oberen Frequenzbereicheingeschrankt. Bei mittleren Frequenzen (um den Bereich der niedrigsten Betriebsfre-quenz herum) lasst sich eine RC aufbauend auf den in dieser Arbeit vorgestellten Ergeb-nissen optimieren. Durch Verwendung z.B. neuartiger Ruhrerformen oder Wandgeome-trien in der Simulation kann die Gleichmassigkeit der Feldverteilung verbessert werdenund damit der Einsatzbereich einer RC zu niedrigeren Frequenzen erweitert werden.

Contents

1 Introduction 11.1 Motivation and objective of this thesis . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Reverberation Chamber Theory 52.1 Electromagnetic fields in a reverberation chamber . . . . . . . . . . . . . . 5

2.1.1 Modes inside an ideal cavity . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Modes inside a lossy cavity . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Field distribution inside a reverberation chamber . . . . . . . . . . 9

2.2 Lowest usable frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Number of cavity modes . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Quality factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Stirring ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Field anisotropy and inhomogeneity . . . . . . . . . . . . . . . . . . . . . 182.3.1 Field anisotropy coefficients . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Field inhomogeneity coefficients . . . . . . . . . . . . . . . . . . . . 19

2.4 Field statistics and probability density functions . . . . . . . . . . . . . . 202.4.1 Quadrature and in-phase part statistics . . . . . . . . . . . . . . . 202.4.2 Magnitude statistics for single components and total field . . . . . 202.4.3 Power statistics for single components and total field . . . . . . . . 212.4.4 Statistical goodness-of-fit χ2-test . . . . . . . . . . . . . . . . . . . 23

2.5 Correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.1 Definition of correlation . . . . . . . . . . . . . . . . . . . . . . . . 252.5.2 Significance of correlation . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Statistical uncertainty and estimator accuracy . . . . . . . . . . . . . . . . 272.7 Field uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.8 Caveats for statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Numerical Procedure 373.1 Initial considerations of reverberation chamber simulations . . . . . . . . . 37

3.1.1 Wide operational frequency range . . . . . . . . . . . . . . . . . . 373.1.2 Large, varying, and irregular geometry . . . . . . . . . . . . . . . . 383.1.3 Finite conductivity and entirely closed structure . . . . . . . . . . 383.1.4 Highly resonant chamber . . . . . . . . . . . . . . . . . . . . . . . 393.1.5 Large number of spatial near field positions . . . . . . . . . . . . . 39

3.2 Computation of electromagnetic fields . . . . . . . . . . . . . . . . . . . . 393.2.1 Incident and scattered field . . . . . . . . . . . . . . . . . . . . . . 40

v

vi CONTENTS

3.2.2 Integral equation approach . . . . . . . . . . . . . . . . . . . . . . 403.2.3 Solution of integral equations . . . . . . . . . . . . . . . . . . . . . 413.2.4 Approximation of currents and current density . . . . . . . . . . . 413.2.5 Computation of line and surface current coefficients . . . . . . . . 43

3.3 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.1 Point matching and weighting functions . . . . . . . . . . . . . . . 443.3.2 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.3 Symmetry considerations . . . . . . . . . . . . . . . . . . . . . . . 463.3.4 Modeling of finite conductivity . . . . . . . . . . . . . . . . . . . . 47

3.4 Computational requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.1 Simulation memory . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.2 Simulation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Extensions to the method of moments . . . . . . . . . . . . . . . . . . . . 523.5.1 Field integral equation resonance problem . . . . . . . . . . . . . . 523.5.2 Iterative solution techniques . . . . . . . . . . . . . . . . . . . . . . 523.5.3 Method of moments and physical optics hybridization . . . . . . . 543.5.4 Method of moments and fast multipole method hybridization . . . 54

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Literature Overview 574.1 Historic reverberation chamber publications and patents . . . . . . . . . . 574.2 Reverberation chamber standards . . . . . . . . . . . . . . . . . . . . . . . 584.3 Previous reverberation chamber simulations . . . . . . . . . . . . . . . . . 58

4.3.1 Time-domain simulations . . . . . . . . . . . . . . . . . . . . . . . 594.3.2 Frequency-domain simulations . . . . . . . . . . . . . . . . . . . . 624.3.3 Statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Alternative stirring methods . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4.1 Moving walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.2 Electronic stirring . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Practical reverberation chamber applications . . . . . . . . . . . . . . . . 664.5.1 Automotive and aircraft avionics . . . . . . . . . . . . . . . . . . . 664.5.2 Antenna measurements and mobile communications . . . . . . . . 68

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Prototype and Measurement System Development 715.1 Reverberation chamber prototype . . . . . . . . . . . . . . . . . . . . . . . 71

5.1.1 Walls and door . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.2 Stirrer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.1.3 Auxiliary installations and electromagnetic leakage . . . . . . . . . 75

5.2 Measurement system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2.1 Transmit and receive measurement equipment . . . . . . . . . . . . 785.2.2 Field probe system . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.3 Data acquisition and interfacing . . . . . . . . . . . . . . . . . . . 83

5.3 Measurement errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.1 Field probe system . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

CONTENTS vii

5.3.2 Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3.3 Chamber and stirrer . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4 Measurement uncertainty budget . . . . . . . . . . . . . . . . . . . . . . . 915.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Modeling of the Reverberation Chamber 936.1 Chamber models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1.1 Modeling procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 936.1.2 Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.1.3 Prototype reverberation chamber . . . . . . . . . . . . . . . . . . . 956.1.4 Corrugated, cubic, and offset-wall reverberation chambers . . . . . 986.1.5 Other reverberation chambers . . . . . . . . . . . . . . . . . . . . . 100

6.2 Stirrer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2.1 Vertical stirrers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.2.2 Horizontal stirrers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.3 Stirrers used in other reverberation chambers . . . . . . . . . . . . 104

6.3 Wall and stirrer conductivities . . . . . . . . . . . . . . . . . . . . . . . . 1056.4 Transmit and receive antenna models . . . . . . . . . . . . . . . . . . . . . 106

6.4.1 Ideal Hertzian and realistic λ/2-dipole . . . . . . . . . . . . . . . . 1076.4.2 Biconical antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.4.3 Logarithmic-periodic antenna . . . . . . . . . . . . . . . . . . . . . 1096.4.4 Horn antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.5 Canonical equipment under test . . . . . . . . . . . . . . . . . . . . . . . . 1106.5.1 Practical canonical EUTs . . . . . . . . . . . . . . . . . . . . . . . 1106.5.2 Canonical EUT modeling . . . . . . . . . . . . . . . . . . . . . . . 112

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 Reverberation Chamber Simulation and Measurement 1157.1 Simulation and measurement workflow . . . . . . . . . . . . . . . . . . . . 1157.2 Cavity simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.2.1 Effect of the chamber door . . . . . . . . . . . . . . . . . . . . . . 1187.2.2 Insertion of a stirrer . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3 Prototype reverberation chamber analysis . . . . . . . . . . . . . . . . . . 1207.3.1 Different chamber geometries . . . . . . . . . . . . . . . . . . . . . 1217.3.2 Effect of a rotating stirrer . . . . . . . . . . . . . . . . . . . . . . . 1237.3.3 Different reverberation chamber excitations . . . . . . . . . . . . . 123

7.4 Measurement versus simulation . . . . . . . . . . . . . . . . . . . . . . . . 1277.4.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.4.2 Near field based simulation validation . . . . . . . . . . . . . . . . 1287.4.3 Statistical benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.5 Corrugated and cubic reverberation chamber . . . . . . . . . . . . . . . . 1327.5.1 Simulated near field distribution . . . . . . . . . . . . . . . . . . . 1347.5.2 Correlation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.5.3 Field uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.6 Equipment under test simulation . . . . . . . . . . . . . . . . . . . . . . . 138

viii CONTENTS

7.6.1 Simulated near field distribution . . . . . . . . . . . . . . . . . . . 1387.6.2 Field uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.6.3 TX/RX antenna coupling . . . . . . . . . . . . . . . . . . . . . . . 141

7.7 Comparison of different stirrers . . . . . . . . . . . . . . . . . . . . . . . . 1437.7.1 Simulated near field distribution . . . . . . . . . . . . . . . . . . . 1437.7.2 Correlation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.7.3 Field uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.7.4 Final performance evaluation . . . . . . . . . . . . . . . . . . . . . 1497.7.5 Plane-wave-based stirrer comparisons . . . . . . . . . . . . . . . . 149

7.8 Simulation and measurement time budget . . . . . . . . . . . . . . . . . . 1527.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8 Conclusion 157

9 Outlook 161

A Electromagnetic Simulation Software FEKO 163A.1 Special execution commands . . . . . . . . . . . . . . . . . . . . . . . . . . 163A.2 Memory considerations and bugs . . . . . . . . . . . . . . . . . . . . . . . 164

B Reverberation Chamber Measurement System 165B.1 Antenna placement: tripod vs. suspension . . . . . . . . . . . . . . . . . . 165B.2 Data acquisition and interfacing . . . . . . . . . . . . . . . . . . . . . . . . 168

C Reverberation Chamber Statistics 169C.1 Field uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169C.2 Probability distribution functions . . . . . . . . . . . . . . . . . . . . . . . 171

D Reverberation Chamber Simulation Data 173D.1 Spatial measurement positions . . . . . . . . . . . . . . . . . . . . . . . . 173D.2 Input power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174D.3 Different coupling paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174D.4 Field uniformity in prototype, cubic, and corrugated RC . . . . . . . . . . 175D.5 Field uniformity for different stirrers . . . . . . . . . . . . . . . . . . . . . 176D.6 Field uniformity for different canonical EUTs . . . . . . . . . . . . . . . . 179

Bibliography 183

Acknowledgments 199

List of Publications 201

Curriculum Vitae 203

List of Figures

1.1 Typical reverberation chamber . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Schematic reverberation chamber test setup . . . . . . . . . . . . . . . . . 102.2 Number of modes N above cutoff in standard and cubic cavity . . . . . . 122.3 Mode density ∂N/∂f in standard and cubic cavity . . . . . . . . . . . . . 132.4 Modal gap in standard and cubic cavity . . . . . . . . . . . . . . . . . . . 142.5 Number of modes above cutoff per 10 MHz in standard and cubic cavity . 152.6 Exemplary Gaussian normal, χ(2), χ(6), χ2

(6) statistical distribution . . . . 222.7 Standard deviation multiples of a Gaussian normal distribution . . . . . . 282.8 Independent stirrer positions and field uncertainty (1 and 3 components) . 292.9 Independent stirrer positions and field uncertainty (2 components) . . . . 302.10 Required EUT spacing from RC walls . . . . . . . . . . . . . . . . . . . . 32

3.1 Memory requirements for an RC simulation (50 . . . 300 MHz) . . . . . . . . 493.2 Memory requirements for an RC simulation (50 . . . 1000 MHz) . . . . . . . 50

5.1 RC geometry and dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Vertical 6-paddle stirrer mounting and motor drive . . . . . . . . . . . . . 735.3 RC apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4 Overview of the RC measurement system setup . . . . . . . . . . . . . . . 775.5 RC measurement setup and photo . . . . . . . . . . . . . . . . . . . . . . 795.6 Field probe with stand, TX/RX antenna, and measurement grid . . . . . 815.7 Schematic measurement and data acquisition procedure . . . . . . . . . . 835.8 Isotropy of the electric field probe . . . . . . . . . . . . . . . . . . . . . . 855.9 Schematic RC measurement grid . . . . . . . . . . . . . . . . . . . . . . . 875.10 Tripod-mounted and suspended TX antennas . . . . . . . . . . . . . . . . 885.11 Broadband effect of tripod on |E| (f = 50 . . .300 MHz) . . . . . . . . . . . 895.12 Spatial effect of tripod on |E| (x = 0.57 m, y = −1.2 . . .1.2 m, z = 0.47 m) 90

6.1 Schematic modeling and simulation preprocessing flowchart . . . . . . . . 946.2 Partly symmetric simulation model of the RC . . . . . . . . . . . . . . . . 956.3 Detailed fully asymmetric simulation model of the RC . . . . . . . . . . . 966.4 Photo of the RC door with gasket . . . . . . . . . . . . . . . . . . . . . . 976.5 Basic, wider, corrugated, and cubic simulation model of the RC . . . . . . 996.6 Vertical stirrer models with triangular discretization . . . . . . . . . . . . 1016.7 Horizontal stirrer models with triangular discretization . . . . . . . . . . . 1046.8 TX/RX antenna models with far field pattern . . . . . . . . . . . . . . . . 1086.9 Simulation model of the RC with canonical box-type EUT . . . . . . . . . 111

ix

x LIST OF FIGURES

6.10 Models and simulated far field patterns of different canonical EUTs . . . . 112

7.1 Schematic simulation, analysis, measurement, and benchmark procedure . 1177.2 Chamber door effect within a cavity . . . . . . . . . . . . . . . . . . . . . 1197.3 Near field comparison cavity against reverberation chamber . . . . . . . . 1207.4 Effect of different prototype reverberation chamber geometries . . . . . . 1227.5 Impact of a rotating stirrer within a reverberation chamber . . . . . . . . 1247.6 Different excitations in a reverberation chamber . . . . . . . . . . . . . . . 1267.7 Measurement vs. simulation at f = 300 MHz and f = 500 MHz . . . . . . 1287.8 Measurement vs. simulation at f = 700 MHz and f = 1000 MHz . . . . . . 1297.9 Influence of the RC door at f = 200 MHz and f = 250 MHz . . . . . . . . 1317.10 Statistical distribution of the electric field strength . . . . . . . . . . . . . 1337.11 Near field within prototype, cubic, and corrugated RC . . . . . . . . . . . 1357.12 Correlation for prototype, cubic, and corrugated RC . . . . . . . . . . . . 1377.13 Field uniformity envelopes in prototype, corrugated, and cubic RC . . . . 1387.14 Near field with canonical EUTs in a reverberation chamber . . . . . . . . 1397.15 Field uniformity envelopes for canonical EUTs . . . . . . . . . . . . . . . 1407.16 Coupling statistics for different excitations . . . . . . . . . . . . . . . . . . 1427.17 Near field patterns for different stirrers . . . . . . . . . . . . . . . . . . . . 1447.18 Correlation for stirrers with different shapes . . . . . . . . . . . . . . . . . 1467.19 Correlation for vertical/horizontal stirrers with and without gaps . . . . . 1477.20 Field uniformity envelopes for vertical and horizontal stirrers . . . . . . . 1487.21 Field uniformity envelopes for Z-fold and cross-plate stirrers . . . . . . . . 1497.22 Stirrer radar cross section calculations . . . . . . . . . . . . . . . . . . . . 1517.23 RCS-based correlation for the 6-paddle stirrer . . . . . . . . . . . . . . . . 152

B.1 Broadband effect of tripod on |E| at x = 0.77 m, y = 0.64 m, z = 0.47 m . 165B.2 Broadband effect of tripod on |E| at x = 0.57 m, y = −0.36 m, z = 0.47 m 166B.3 Spatial effect of tripod on |E| for 50 MHz and 100 MHz . . . . . . . . . . . 167B.4 Spatial effect of tripod on |E| for 150 MHz and 200 MHz . . . . . . . . . . 167B.5 Spatial effect of tripod on |E| for 250 MHz and 300 MHz . . . . . . . . . . 168

C.1 Independent stirrer positions and field uncertainty (1 component) . . . . . 169C.2 Independent stirrer positions and field uncertainty (2 components) . . . . 170C.3 Independent stirrer positions and field uncertainty (3 components) . . . . 170

D.1 Electric field pattern for 1 V and 10 V excitation source . . . . . . . . . . 174D.2 Antenna orientation within the RC for different coupling paths . . . . . . 174D.3 Field uniformity in prototype RC . . . . . . . . . . . . . . . . . . . . . . . 175D.4 Field uniformity in corrugated RC . . . . . . . . . . . . . . . . . . . . . . 175D.5 Field uniformity in cubic RC . . . . . . . . . . . . . . . . . . . . . . . . . 176D.6 Field uniformity for vertical 6-paddle stirrer . . . . . . . . . . . . . . . . . 176D.7 Field uniformity for vertical 6-paddle stirrer without gaps . . . . . . . . . 177D.8 Field uniformity for horizontal 6-paddle stirrer . . . . . . . . . . . . . . . 177D.9 Field uniformity for vertical cross-plate stirrer . . . . . . . . . . . . . . . . 178

LIST OF FIGURES xi

D.10 Field uniformity for vertical Z-fold stirrer . . . . . . . . . . . . . . . . . . 178D.11 Field uniformity for vertical Z-fold stirrer with gaps . . . . . . . . . . . . 179D.12 Field uniformity without canonical EUT . . . . . . . . . . . . . . . . . . . 179D.13 Field uniformity with canonical loop EUT . . . . . . . . . . . . . . . . . . 180D.14 Field uniformity with canonical box EUT . . . . . . . . . . . . . . . . . . 180D.15 Field uniformity with large canonical box EUT . . . . . . . . . . . . . . . 181

List of Tables

2.1 Typical values of the total field anisotropy coefficient 〈Atot〉 . . . . . . . . 192.2 Probability Pχ2 (χ2 ≥ χ2

0) for hypothesis rejection in a χ2-test . . . . . . . 242.3 Probability PN (|ρ| ≥ |ρ0)| for the correlation coefficient . . . . . . . . . . 27

3.1 Comparison of runtime and memory requirements of RC simulations . . . 51

4.1 Summary of previously published RC simulations . . . . . . . . . . . . . . 614.2 Basic differences between AC and RC test environment . . . . . . . . . . 67

6.1 Discretization data of chambers used in the RC simulations . . . . . . . . 1006.2 Discretization data of vertical stirrer simulation models . . . . . . . . . . 1036.3 Discretization data of horizontal stirrer simulation models . . . . . . . . . 1056.4 Typical electrical conductivity values . . . . . . . . . . . . . . . . . . . . . 1066.5 Electrical conductivity values used in the RC simulations . . . . . . . . . 1076.6 Discretization of transmit (TX) and receive (RX) antennas . . . . . . . . 1096.7 Discretization of the canonical emission EUTs (CEUTEs) . . . . . . . . . 113

7.1 Different reverberation chamber geometries . . . . . . . . . . . . . . . . . 1217.2 Different types of reverberation chamber excitations . . . . . . . . . . . . 1257.3 Normalized spatial 2-norm measurement vs. simulation . . . . . . . . . . . 1307.4 Performance comparison of different stirrers . . . . . . . . . . . . . . . . . 1507.5 Time expenditure comparison simulations versus measurements . . . . . . 153

D.1 Field points used in measurement and simulation for uniformity analysis . 173

xiii

List of Acronyms and Abbreviations

Numerical methods

BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . boundary-element method

CFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coupled field integral equation

CGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conjugate gradient squared method

EFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electric field integral equation

FDTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . finite-difference time-domain

FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . finite-element method

FIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . finite-integration technique

FMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fast multipole method

LU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lower upper

MFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . magnetic field integral equation

MLFMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . multilevel fast multipole method

MoM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . method-of-moments

NEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Electromagnetics Code

NGF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . numerical Green’s function

PEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . perfectly electrically conducting

PO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . physical optics

RWG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rao-Wilson-Glisson

TLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transmission-line-matrix method

xv

xvi LIST OF ACRONYMS

Reverberation chamber and electromagnetics terminology

AC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . anechoic chamber

AWGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . additive white Gaussian noise

CDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cumulative distribution function

CEUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . canonical equipment under test

CEUTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . canonical equipment under test for emission

CEUTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . canonical equipment under test for immunity

CNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . comparison noise emitter

dof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degrees of freedom

EM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electromagnetic

EMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electromagnetic compatibility

EMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . electromagnetic interference

EUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equipment under test

FAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fully anechoic room

GTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gigahertz transverse electromagnetic

i.i.d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . independent identically distributed

KS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kolmogorov-Smirnov

LOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lowest overmoded frequency

logper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . logarithmic-periodic

LUF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lowest usable frequency

MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . maximum likelihood estimator

OATS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . open area test site

PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . probability density function

RC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . reverberation chamber

RCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radar cross section

RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radio frequency

RX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . receive

SAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . semi anechoic chamber

LIST OF ACRONYMS xvii

SE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shielding effectiveness

SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . signal-to-noise ratio

SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stirring ratio

TE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transverse electric

TEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transverse electromagnetic

TM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transverse magnetic

TX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transmit

VIRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vibrating intrinsic reverberation chamber

Software

AdaptFEKO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . adaptive FEKO

CAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . computer aided design

Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . software for emission & immunity testing

DB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . database

EMSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Software and Systems Ltd.

FEKO . . . . . . . . . . . . . . . . . . . . . . . . . Feldberechnung bei Korpern mit beliebiger Oberflache

GPIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . general purpose interface bus

GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . graphical user interface

MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . message digest

MS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft

ODBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . open database connectivity

OS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . operating system

PreFEKO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . preprocessor for FEKO

SQL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . structured query language

WinFEKO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . user interface for FEKO

xviii LIST OF ACRONYMS

Organizations and official terms

CDV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Committee Draft for Vote

CISPR . . . . . . . . . . . . . . . Comite International Special des Perturbations Radioelectriques

DSTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Defence Science and Technology Organisation

FCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Federal Communications Commission

FDIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final Draft for International Standard

GUM . . . . . . . . . . . . . . . . . . . . . . . . . Guide to the expression of uncertainty in measurement

IEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Electrotechnical Commission

IEEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Institute of Electrical and Electronics Engineers

IFH . . . . . . . . . . . . . . . . Laboratory for Electromagnetic Fields and Microwave Electronics

ISO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Standardization Organization

NASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Aeronautics and Space Administration

NAWCWD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Naval Air Warfare Center Weapons Division

NBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Bureau of Standards

NIST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Institute of Standards and Technology

NPL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Physical Laboratory

NSWCDD . . . . . . . . . . . . . . . . . . . . . . . . . . . Naval Surface Warfare Center Dahlgren Division

SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . subcommittee

List of Symbols

α attenuation constant [1/m]

α statistical shape parameter

Axy, Ayz, Axz planar anisotropy coefficient

A area [m2]

Atot total anisotropy coefficient

A triangle area [m2]

B magnetic flux density [Vs/m2]

b number of dimensions

β propagation constant [rad/m]

β statistical scale parameter

c speed of an electromagnetic wave within a medium [m/s]

c0 vacuum speed of an electromagnetic wave [m/s]

χ random variable

D electric (displacement) flux density [As/m2]

d depth [m]

d distance [m]

d material thickness [m]

d statistical uncertainty

d triangular element size parameter [m]

d waveguide diameter [m]

δs skin depth [m]

dmax maximum triangle edge length [m]

dmin minimum triangle edge length [m]

xix

xx LIST OF SYMBOLS

E electric field strength [V/m]

e number of expected samples

εr relative dielectric permittivity

ε0 dielectric permittivity constant [As/(Vm)]

η surface charge density [As/m2]

F cumulative distribution function (CDF)

f probability density function (PDF)

ϕ angle []

fLOF lowest overmoded frequency (LOF) [Hz]

fLUF lowest usable frequency (LUF) [Hz]

f frequency [Hz]

fc cutoff frequency [Hz]

∆f modal frequency gap [Hz]

g basis function

G Green’s function

g number of observed samples

Γ Gamma function

γ triangle apex angle []

H magnetic field strength [A/m]

h reverberation chamber height [m]

I current coefficient vector

In line current [A]

i, i′, j index number

Iα planar inhomogeneity coefficient

Itot total inhomogeneity coefficient

J current density [A/m2]

Jm Bessel function of order m

LIST OF SYMBOLS xxi

JS surface current density [A/m]

k wave number vector [rad/m]

κ electric conductivity [S/m]

k statistical confidence level factor

l reverberation chamber length [m]

λ wavelength [m]

M memory [Byte]

m, n, p mode number

M number of spatial positions

µ0 magnetic permeability constant [Vs/(Am)]

µr relative magnetic permeability

µ statistical mean

N number of unknowns

NI number of line current basis functions

N J number of surface current density basis functions

n number of columns or rows in a matrix

N cumulated number of modes

n normal vector

NS number of segments

N number of samples

N number of stirrer steps

NT number of triangles

ν degrees of freedom (dof)

ω angular frequency [rad/s]

p confidence level percentage fraction

Pi input power [W]

P probability

xxii LIST OF SYMBOLS

PRX received power [W]

PTX transmitted power [W]

q number of calculated parameters

Q quality factor

r, r ′, r ′′ position vector [m]

ρ correlation coefficient

volume charge density [As/m3]

s variance

σ radar cross section (RCS) [m2]

σ standard deviation

SR stirring ratio

t time [s]

V volume [m3]

V excitation vector

W energy [J]

w reverberation chamber width [m]

w weighting function

X, Y, Z random variable

x, y, z axis variable of a general Cartesian coordinate system [m]

xm, ym, zm axis variable of the Cartesian measurement coordinate system [m]

xs, ys, zs axis variable of the Cartesian simulation coordinate system [m]

ξ axis variable of a general coordinate system [m]

Z basis / weighting function coupling matrix

Z impedance [Ω]

Mathematical Notation

x scalar

x vector

ex unit vector

n12 normal vector pointing from region 1 towards region 2

X matrix

x∗ conjugate complex

Re · real part

Im · imaginary part

‖ · ‖ norm

X· operator

< · > inner product

〈·〉 ensemble averaging

∇x gradient of x

∇ · x divergence of x

∇A · x surface divergence of x

∇× x curl of x

O(·) order of

∀ for all

∨ or

∼ proportional to

xxiii

1 Introduction

1.1 Motivation and objective of this thesis

Today’s electronic products must be designed so that they do not disturb the properoperation of other products and inversely withstand electromagnetic (EM) radiationemitted from all kinds of equipment. A crucial aspect of successful product developmentis therefore the fast, effective, and efficient testing of the electromagnetic compatibil-ity (EMC). EMC can be formally defined as “the ability of an equipment, subsystemor system to share the EM spectrum, and perform at the same time its desired functionwithout unacceptable degradation from or to the environment in which it exists” [1].EMC always involves two parties: the source of the interference (emissions) and the vic-tim (immunity/susceptibility). Many objects are simultaneously a source and a victim,i.e. they emit signals, which have an adverse effect on other items in the surroundingenvironment whilst at the same time being susceptible to noise generated by that envi-ronment. A system is therefore said to be electromagnetically compatible if it does notinterfere with other systems, it is not susceptible to emissions from other systems, and itdoes not cause interference with itself. Interferences can be either transmitted via cables(“conducted interference”) or through the surrounding media (“radiated interference”).The EMC testing community is continually searching for more reliable, repeatable, andeconomical test procedures. Reverberation chambers (RCs) enjoy growing popularityas a complement or replacement to well established methods for radiated interferencesuch as open area test sites (OATSs), (semi-) anechoic chambers (ACs) or transverseelectromagnetic (TEM)-cells. Those methods rely to a great extent on a “well-behaved”equipment under test (EUT) radiation pattern, assuming implicitly that the EUT radi-ates or receives similar to a monopole, dipole, or quadrupole. A typical EMC test insidea reverberation chamber (RC) is shown in Fig. 1.1. Particularly with regard to electronicdevices with complex radiation patterns, RC tests are expected to deliver more accu-rate and rigorous measurement results than the more traditional methods mentionedbefore [2, 3, 4, 5]. The importance of testing for EMC in RCs as an alternative mea-surement technique has recently been recognized in the IEC 61000-4-21 standard [6],published in August 2003. Standardization of RC measurements will lead to more wide-spread use of this technique. RC users will need to fully understand the RC workingprinciples in order to interpret the measurement results correctly and to optimize theperformance for various measurement tasks. This requires a good understanding of theEM field inside the chamber, for example, how the field depends on the finite metalconductivity of chamber walls, the type of transmit and receive antennas, the shape ofstirrers and the EUT. It will be shown that even small wall irregularities caused by theRC door, for example, or the position and the type of antenna used, have a profoundeffect on the field within the RC and therefore must be accurately modeled in a sim-ulation. To account for all these effects is a challenging task in EM field analysis and

1

2 1 INTRODUCTION

Figure 1.1: Reverberation chamber setup with logper and horn TX/RX antennas, field probes,stirrer, and motorcycle as EUT. Copyright c© Institut fur Grundlagen der Elektrotechnikund Elektromagnetische Vertraglichkeit, Universitat Magdeburg, Germany.

only few simulation techniques are able to cope with such problems. Among those arefrequency-domain method-of-moments (MoM) based techniques used in the investiga-tion presented in this thesis.Faced with the task of performing EMC tests in an RC, one will usually turn to the well-known IEC 61000-4-21 [6] standard as a first reference. Inside this standard one can finddetailed information regarding the modes of operation of an RC, chamber calibration,radiated immunity, emission, shielding effectiveness test procedures, and examples ofmeasurement data. However, when it comes to designing and constructing an RC, theIEC 61000-4-21 standard provides only basic guidelines, e.g. that “stirrers should havedimensions of a significant fraction of the chamber dimensions and of the wavelengthat the lowest useable frequency” or that “a reverberation chamber is an electricallylarge, highly conductive enclosed cavity”. Clearly, it is beyond the scope of fundamentalpublications such as [6] to provide precise instructions on how to build highly efficientstirrers or information on the exact material to be used for the RC walls. Many of theseconstruction suggestions were derived from years of practical experience in combinationwith applied basic physical principles (some examples are [7, 8, 9]). Rules-of-thumb

1.2 OUTLINE 3

guidelines were successfully established for general RC design, often obtained through atime-consuming trial-and-error approach until a chamber finally fulfilled desired specifi-cations.In contrast to this cumbersome procedure, three-dimensional (3-D) simulations facilitatethe thorough EM analysis of RCs and could speed up their development time. The ob-vious goal of simulations would be the complete design, evaluation and optimization ofan RC until all target specifications are met prior to physical construction. Among thetypical questions before the construction of an RC starts are those related to the impactof the chamber geometry as well as the conductivity of the sheet metal on the maximumachievable field level and the mode spacing. Is a rectangular cavity superior to a cubicone or are corrugations on the walls useful to decrease the lowest usable frequency (LUF,fLUF)? With regard to the stirrer one might ask: How should an optimum stirrer lookin a particular RC? What is the statistically uniform testing volume for a certain stirrerat a predefined uncertainty and how does it change with frequency? Does it matterwhere the stirrers are positioned? Will an effective stirrer in one chamber show simi-lar performance in a different RC? Of great interest are also questions such as: Couldthe installation of a second stirrer decrease fLUF even further and if yes, how is thisstirrer to be rotated with respect to the first one? Is the directivity of a logarithmic-periodic (logper) antenna pointing towards a stirrer sufficient at lower frequencies toavoid direct illumination of the EUT? These issues are addressed in detail in this thesis.In addition, the simulation of an RC serves as an educational tool which visualizes thecomplex EM field structure inside the chamber and therefore makes RC operation easierto understand. With a simulation it is possible to verify the above mentioned rules-of-thumb for chamber dimensions, stirrer size, shape, and position, the relation betweenchamber quality factor and mode spacing and so forth. Once a reliable simulation modelhas been established, the final goal is of course RC optimization, which – dependingon the simulation runtime – may not always be economically feasible. One must clearlystate, that in some cases the effects of structural chamber modifications are more quicklytested experimentally than with elaborate simulations; e.g. the effect of a very irregularlyshaped piece of aluminum foil attached to the stirrer can be analyzed rapidly with mea-surements, provided EM field data at only few locations is sufficient. An EM simulationtool can only be an aid in the design of RCs and does not replace a solid understandingof the subject matter (including EM fields) by the design engineer.

1.2 Outline

The outline of this thesis is as follows: At the beginning, the theoretical backgroundand basic foundation of RCs is introduced in Chapter 2. This chapter establishes themost important parameters that should be considered for the design, construction, andoperation of an RC: starting with an abstraction of RCs to ideal cavities, the conceptof modes, the field distribution, the number of modes, and the mode density are de-rived. As the focus is moved from an ideal cavity via a lossy cavity towards a realisticRC, the lowest usable frequency, the quality factor, and statistical field distributionsare addressed. The EM fields inside an RC are characterized by correlation coefficients,

4 1 INTRODUCTION

statistical uncertainty, goodness-of-fit tests, and field uniformity.After developing the rationale for the need of EM simulations of RCs in Chapter 3,the requirements that simulation tools must meet to be suitable for the analysis of RCsare deduced. A synopsis of the computational challenges particular for RCs is described.After contrasting these requirements against advantages and drawbacks of several numer-ical field solver techniques, it is outlined why for this thesis finally a frequency-domainelectric field integral equation (EFIE)-based method-of-moments (MoM) code was cho-sen for the RC simulations. The basic concept of the EFIE is outlined in Chapter 3and the MoM solution methodology used in the employed simulation tools introduced.Computational requirements regarding simulation time and memory are estimated andan outlook on MoM extensions and solver techniques is given.An overview of historic RC papers and patents as well as past publications dealing withRC simulations is given in Chapter 4. This chapter summarizes both the most significantaccomplishments and also potential shortcomings of various RC simulation approaches.In addition, alternative stirring methods (moving walls, electronic stirring, etc.), thepractical application of RCs to EMC testing, and a short qualitative comparison be-tween RCs and ACs are addressed.Chapter 5 decribes the construction and setup of the RC prototype including walls, door,stirrers, and auxiliary equipment. Special features of the measurement system utilizedfor data acquisition are explained. Measurement errors leading to deviations betweensimulated and measured results are outlined and strategies for error minimization pro-posed.Modeling of the RC is illustrated in detail in Chapter 6. Starting with a basic cavity,a comprehensive chamber model resembling the prototype RC is elaborated. Further-more, cubic and corrugated chambers, various vertical and horizontal stirrers, transmitand receive antennas, and several EUTs are designed and modeled.3-D simulation results are presented in Chapter 7. In the beginning, the procedure usedto perform RC data analysis is presented. The necessity of a rigorous simulation vali-dation is emphasized and different validation methods are compared. The electric fieldinside the chamber is computed and the influence of small geometric details and asymme-tries is investigated as well as the effect of different excitations and stirrers. It is demon-strated that a statistics-based validation of RC simulations is insufficient. To validatesimulation results, extensive near field measurements inside the prototype RC are per-formed. The effect of a rotating stirrer, the door, and several transmit (TX)/receive (RX)antenna types within the RC are analyzed. Comparisons of different chamber geome-tries (cubic, corrugated) versus the prototype RC are carried out based on near field,correlation, and field uniformity. Various stirrer designs are evaluated with respect totheir performance within the prototype RC. The presence of different EUTs is inves-tigated, and a loading, field uniformity, and coupling path analysis is performed. Thecomplete 3-D RC simulation, considering stirrers, door, and various practical excitations,accurately predicts the fields within the chamber in the important lower-to-medium fre-quency range and thus represents a reliable tool facilitating RC optimization.At the end of this thesis, those aspects which could not be addressed or finally resolved,are summarized and suggestions on how to proceed further with these issues in the futureare proposed.

2 Reverberation Chamber Theory

Abstract— This chapter starts with the abstraction of a reverberation chamber to a simple cavity

in order to explain basic, but important concepts such as EM modes, the number of modes, and

the modal density. As the focus is moved from an ideal cavity via a lossy cavity towards a realistic

reverberation chamber, the lowest usable frequency, the quality factor, and statistical field distribu-

tions are addressed. The EM fields inside a reverberation chamber are characterized by correlation

coefficients, statistical uncertainty, and field uniformity.

2.1 Electromagnetic fields in a reverberation chamber

A fully functional reverberation chamber (RC) consists of a metallic shielded room offinite conductivity with a stirring device, antennas, an EUT, and other devices inside.In order to understand its basic operating principles, the RC can be abstracted in thebeginning to an empty, rectangular cavity resonator with perfectly electrically conducting(PEC) walls.

2.1.1 Modes inside an ideal cavity

It is well known that cavity resonators can be formed by short-circuiting a rectangularwaveguide at two sufficiently separated ends [10, 11]. If the geometrical dimensions ofthis resonator reach certain lengths, at a given frequency an EM field within this res-onator forms a standing wave pattern.This standing wave pattern can be mathematically described by solving Maxwell’s equa-tions which are given in differential form as

∇× E = − B (2.1)

∇× H = D + J (2.2)∇ · D = (2.3)∇ · B = 0 (2.4)

wherein E is the electric and H the magnetic field strength, D is the electric and B themagnetic flux density. J denotes the electric current density and the volume chargedensity. For the derivation of the numerical formulation valid inside an ideal cavityresonator, it is assumed that there are no charges inside the computational volume V ,i.e. = 0 in (2.3). Furthermore the properties of the utilized materials are taken to be

5

6 2 REVERBERATION CHAMBER THEORY

linear, homogeneous, isotropic, and without memory so that

D = ε E (2.5)B = µ H (2.6)J = κE (2.7)

is obtained for the material equations. Herein ε denotes the dielectric permittivity, µ themagnetic permeability, and κ the electrical conductivity. If time-harmonic fields with anejωt-dependence are assumed and the material equations (2.5)-(2.7) utilized, Maxwell’sequations as given above by (2.1)-(2.4) can be simplified to

∇× E = −jωµ H (2.8)∇× H = jωε E + J (2.9)∇ · E = 0 (2.10)∇ · H = 0 (2.11)

Applying the vector identity [12]

∇×(

∇× X)

= ∇(

∇ · X)− ∆ X (2.12)

to (2.8) and (2.9) allows to derive the electrical and magnetic wave equations

∆ E =1c2

∂2 E

∂t2(2.13)

∆ H =1c2

∂2 H

∂t2(2.14)

which can be used to describe the fields within a cavity. c denotes the propagation speedof the EM waves in the cavity resonator and is given by

c =c0√εrµr

(2.15)

with c0 being the speed of an EM wave in vacuum. With e.g. a product separationapproach [12], (2.13) and (2.14) can be solved using boundary conditions, which can bederived for the tangential components of the electric and the magnetic field, respectively,as

∇× E = n12 ×(

E2 − E1

)= 0

⇔ Etan2 − Etan1 = 0 (2.16)

∇× H = n12 ×(

H2 − H1

)=

0JS

forκ < ∞κ → ∞

⇔ Htan2 − Htan1 =

0JS

forκ < ∞κ → ∞ (2.17)

2.1 ELECTROMAGNETIC FIELDS IN A REVERBERATION CHAMBER 7

wherein JS is the surface current density. The vector n12 represents a normal vectorwhich points from region 1 into region 2. The boundary conditions for the normalcomponents of the electric and magnetic field are enforced by

∇ · D = n12 ·(

D2 − D1

)= η

⇔ Dnor2 − Dnor1 = η (2.18)

∇ · B = n12 ·(

B2 − B1

)= 0

⇔ Bnor2 − Bnor1 = 0 (2.19)

wherein η is the surface charge. For an ideal cavity, (2.16) and (2.19) can be simplifiedto

Etan|∂V = 0 (2.20)Hnor|∂V = 0 (2.21)

valid on the PEC wall surface ∂V of the cavity for the tangential components of theelectrical field and the normal component of the magnetic field. Applying (2.20) and(2.21) to the rectangular geometry of an ideal cavity resonator yields

x = 0 ∨ x = w : Ey = 0, Ez = 0, Hx = 0 (2.22)y = 0 ∨ y = l : Ex = 0, Ez = 0, Hy = 0 (2.23)z = 0 ∨ z = h : Ex = 0, Ey = 0, Hz = 0 (2.24)

Using the boundary conditions (2.22)–(2.24), the wave equations (2.13) and (2.14) canbe fulfilled by certain EM field standing wave patterns within the cavity, the so-called“cavity modes”. These cavity modes can be classified into two main categories: modeswhich do not have an electrical field component into z-direction (Ez = 0) are saidto be of the transverse electric (TE)-type, modes with Hz = 0 are called transversemagnetic (TM). As a result, for the field components of TMmnp modes in an idealrectangular cavity resonator

Ex(x, y, z) = − 1k2

mn

(mπ

w

)(pπ

l

)E0 cos

(mπ

wx)

sin(nπ

hy)

sin(pπ

lz)

(2.25)

Ey(x, y, z) = − 1k2

mn

(nπ

h

)(pπ

l

)E0 sin

(mπ

wx)

cos(nπ

hy)

sin(pπ

lz)

(2.26)

Ez(x, y, z) = E0 sin(mπ

wx)

sin(nπ

hy)

cos(pπ

lz)

(2.27)

Hx(x, y, z) =jωε

k2mn

(nπ

h

)E0 sin

(mπ

wx)

cos(nπ

hy)

cos(pπ

lz)

(2.28)

Hy(x, y, z) = − jωε

k2mn

(mπ

w

)E0 cos

(mπ

wx)

sin(nπ

hy)

cos(pπ

lz)

(2.29)

Hz(x, y, z) = 0 (2.30)

is obtained with the integer numbers m, n = 1, 2, 3, . . . and p = 0, 1, 2, . . . . The indicesm, n, and p denote the number of half wavelengths in x-, y-, and z-direction, respectively.


The cavity dimensions are w (width), h (height), and l (length) in x-, y-, and z-direction– this order is due to the original waveguide notations. Similarly for TEmnp modes, thefollowing equations can be derived

Ex(x, y, z) =jωµ

k2mn

(nπ

h

)H0 cos

(mπ

wx)

sin(nπ

hy)

sin(pπ

lz)

(2.31)

Ey(x, y, z) = − jωµ

k2mn

(mπ

w

)H0 sin

(mπ

wx)

cos(nπ

hy)

sin(pπ

lz)

(2.32)

Ez(x, y, z) = 0 (2.33)

Hx(x, y, z) = − 1k2

mn

(mπ

w

)(pπ

l

)H0 sin

(mπ

wx)

cos(nπ

hy)

cos(pπ

lz)

(2.34)

Hy(x, y, z) = − 1k2

mn

(nπ

h

)(pπ

l

)H0 cos

(mπ

wx)

sin(nπ

hy)

cos(pπ

lz)

(2.35)

Hz(x, y, z) = H0 cos(mπ

wx)

cos(nπ

hy)

sin(pπ

lz)

(2.36)

with m, n = 0, 1, 2, . . . (but always only m ∨ n = 0) and p = 1, 2, . . . . The constant kmn

is utilized as an abbreviation in (2.25)–(2.30) and (2.31)–(2.36), which is given as

kmn =

√(mπ

w

)2

+(nπ

h

)2

(2.37)

The angular frequency ω as employed in (2.25)-(2.36) can be calculated from

ω

c=

2πf

c= kmnp =

√(mπ

w

)2

+(nπ

h

)2

+(pπ

l

)2

(2.38)

with c as given by (2.15). In an ideal cavity (PEC walls, no further losses throughdissipative objects inside) the cut-off frequencies for the individual modes are describedby

fmnp =c

2π

√(mπ

w

)2

+(nπ

h

)2

+(pπ

l

)2

(2.39)

Depending on the actual dimensions of a cavity resonator (i.e. the relation between w,h, and l), the modes with the lowest cutoff-frequencies are therefore either the TM110,the TE011, or the TE101. As shown in Section 2.2.1, the total number of modes abovecutoff at a certain frequency fmnp can be calculated by counting all (m, n, p)-tuples untilf = fmnp is reached, whereby always at least two of the three indices are not equal tozero. It is important to note that there can be several modes having the same cutoff-frequency – this is true for e.g. all TEmnp and TMmnp cavity modes with m ≥ 1, n ≥ 1,p ≥ 1. If several modes exhibit the same cutoff frequency they are called “degeneratemodes”.

2.1.2 Modes inside a lossy cavity

Whereas for an ideal, lossless cavity resonator the mode spectrum is discrete – i.e. aresonance only occurs at a certain frequency f0 – a finitely conducting, non-PEC-wall

2.2 LOWEST USABLE FREQUENCY 9

cavity (κ < ∞) allows the existence of a mode over a certain “modal bandwidth” ∆fQ.For the sake of simplicity it is assumed that modes can only be excited within the finitebandwidth f0 ± ∆fQ/2, hence the mode spectrum is not fully discrete anymore [11].Outside of its modal bandwidth f0 ± ∆fQ/2, the contribution of a mode to the overallfield distribution is taken to be negligible. From a certain frequency on, at a singlefrequency several modes can be excited, since their respective modal bandwidths startto overlap. Depending on the quality factor Q of the lossy cavity, more or less modesare excited simultaneously at a given frequency. At this point, the mono-mode regime ofthe cavity turns into multi-mode operation. The field distribution obtained within thecavity for multi-mode operation can be computed by carrying out a superposition of theindividual modes.

2.1.3 Field distribution inside a reverberation chamber

As long as the loading of the cavity is dominated by losses in the walls and κ ωε isvalid, the field distribution within the cavity does not change in its shape but only withrespect to its magnitude. In other words the field distribution with κ < ∞ is essentiallya scaled version of the field distribution obtained for κ → ∞. Unfortunately, as soon asany other scattering (or strongly absorbing) objects exist inside the lossy cavity, the fielddistribution cannot be represented accurately anymore by analytically calculated modesas given by (2.25)–(2.30) and (2.31)–(2.36) [13]. This is discussed in detail in Chapter 7.An RC as shown in Fig. 2.1 features several objects (such as one or more so-called stir-ring devices, antennas, and EUTs) disturbing considerably a “cavity-like” appearance.Moreover, the stirrers are explicitly designed so that they change the field distributionwithin an RC and modify the modal cutoff frequencies during a stirrer rotation. Theabstraction of an RC to a simple cavity initially put forth at the beginning of this chapteris therefore not fully valid. However, certain fundamental and important RC parameterssuch as e.g. the number of modes above cutoff or the modal density (cf. Section 2.2.1)can be derived using the RC-cavity-abstraction as a rough guideline. Nevertheless ithas to be stressed that the correct near field distribution inside an RC with a stirrer inoperation can only be computed employing a rigorous EM simulation.

2.2 Lowest usable frequency

The lowest usable frequency (LUF) fLUF is commonly understood to be the frequencyfrom which on an RC meets basic operational requirements [6, 15]. The LUF is often alsoreferred to as “lowest overmoded frequency (LOF)” [8]). There are several definitionsfor the LUF:

• the LUF equals three times the cutoff frequency fc of the fundamental mode of acavity with the same dimensions as the RC under investigation, i.e. fLUF = 3fc [6]

• fLUF is defined as the frequency at which 60 . . . 100 modes within an ideal cavity ofthe same size as the RC are above cutoff and at least 1.5 modes/MHz are present [6]


Tuner/Stirrerassembly

Drive motor

Incoming powermains filter

Non-conductivesupport Volume of

uniform field

Tuner/Stirrerassembly

Alternate positionfor tuner/stirrer

at lowest

useable frequency

Field generatingantenna pointed intocorner with tuner

Chamber penetration

Interconnection filterField generationequipment

EUTmeasurementinstrumentation

Figure 2.1: Schematic reverberation chamber test setup including multiple stirrers, equipmentunder test, and transmit antenna. Partly extracted from [14] and IEC 61000-4-21 [6] (Copy-right c© International Electrotechnical Commission (IEC), Geneva, Switzerland).

• the LUF is understood as being the lowest frequency at which a specified fielduniformity can be achieved over a volume defined by an eight location calibrationdata set [6]

It is important to note that the first two definitions (which rely again on the cavity ab-straction) are very qualitative requirements which give only a rough overview on whethera chamber of certain dimensions might be suitable for operation as an RC. The thirddefinition is much more stringent, since it involves measurements within the chamberand forces the user to think about the desired measurement uncertainties and confidenceintervals to be obtained for a given number of stirrer steps.

2.2.1 Number of cavity modes

In order to evaluate from which frequency fLUF on a chamber complies with fundamentalRC requirements [6], the cumulated number of modes, the mode density and the “modalgap” must be known. Computation of these parameters implicitly assumes an emptyRC without a stirrer, i.e. a rectangular cavity. The total number of cavity modes abovecutoff at a given frequency can be calculated by imaging kmnp (2.38) as a point in thethree-dimensional k-space. |kmnp| therefore resembles the distance in space between thepoint kmnp and the origin. In this geometrical model, the number of modes can becomputed by counting all discrete “nodes” in the k-space for which kmnp < k. When


counting the modes above cutoff, a summation needs to be made of the number

• N1(k) of TEmnp modes with m ≥ 1, n ≥ 1, p ≥ 1

• N2(k) of TMmnp modes with m ≥ 1, n ≥ 1, p ≥ 1

• N3(k) of TEmnp modes with m = 0, n ≥ 1, p ≥ 1

• N4(k) of TEmnp modes with m ≥ 1, n = 0, p ≥ 1

• N5(k) of TMmnp modes with m ≥ 1, n ≥ 1, p = 0

The exact total number of modes above cutoff at a given k is then given by

N =5∑

i=1

Ni(k) = 2N1(k) + N3(k) + N4(k) + N5(k) (2.40)

taking into consideration that the number of modes N1(k) = N2(k). A fairly complicatedcalculation outlined in [16] yields finally the approximate cumulated number of modesN above cutoff for a frequency f as

N(f) ≈ 8π

3· l w h ·

(f

c0

)3

− (l + w + h) · f

c0+

12

(2.41)

where c0 denotes the speed of light [16]. It can be see that the RC volume has the biggestimpact on the cumulated number of modes, as shown by the first part of (2.41). Thesecond part of (2.41) resembles the combined edge length of an RC. As noted above, for aproper operation of an RC usually at least 60 . . . 100 modes above cutoff are required [6].The mode density ∂N/∂f (number of modes per frequency interval) can be calculatedfrom (2.41) to be

∂N

∂f≈ 8π · l w h · f2

c30

− (l + w + h) · 1c0

(2.42)

To achieve sufficient statistical field uniformity and isotropy, a common RC specificationis to have at least ∂N/∂f = 1.5 modes/MHz above cutoff [16]. The cutoff frequencies ofthe individual modes inside an ideal rectangular cavity are given by (2.39), which wasslightly reformulated to facilitate mode-ordering into

f i(m, n, p) =

c0

2

√(m

l

)2

+( n

w

)2

+( p

h

)2

(2.43)

where m, n, p are integers with m ≥ 0, n ≥ 0, p ≥ 0 but assuming only (m ∨ n ∨ p) = 0.The superscript i is a positive integer used to consecutively index a set of mode numbers(m, n, p) and the corresponding cutoff frequencies. After sorting all f i

(m, n, p) so that thecutoff frequencies are arranged in an ascending order and renumbering the indices i againwith a new consecutive index i′

f i′+1(m, n, p) ≥ f i′

(m, n, p)′ ∀ i′ (2.44)


600

0

200

300

400

500

100

800

700

900

N

f [MHz]

100 150 300 400 500200 250 450350

Cubic RCPrototype RC

Nmin

Figure 2.2: Theoretical number of modes N above cutoff in a rectangular cavity with the samedimensions as the prototype RC or the cubic RC (see Sections 6.1.3 and 6.1.4 for chambergeometry details). An RC is required to have at least 60 . . . 100 modes above cutoff at theLUF [6].

is obtained, where the set (m, n, p)′ = (m, n, p) denotes a different mode (which mighthowever have the same cutoff frequency f(m, n, p)′ = f(m, n, p) in the case of degeneration).The “modal gap” ∆f i′+1; i′ between consecutive modes can be computed by

∆f i′+1; i′ = f i′+1(m, n, p) − f i′

(m, n, p)′ (2.45)

and serves as an important parameter in the evaluation of different RCs (for a good RCperformance the modal gap should be as small as possible).In the past, some authors have claimed that cubic RCs exhibit superior performance over“standard” rectangular, non-cubic RCs [17]. To investigate this claim, in this chaptera comparison between two RCs is carried out based on the cavity mode distribution.Examining first of all the cumulated number of modes N above cutoff obtained from(2.41), there is no significant difference between rectangular cavities resembling the pro-totype RC built during the course of this thesis (see Chapter 5) and the cubic RC: Inthe critical lower frequency range, Fig. 2.2 shows very similar values for the theoreticalcumulated number of modes above cutoff (N ≈ 50; 180; 420 for the prototype RC versusN ≈ 50; 190; 450 for the cubic RC at frequencies f ≈ 200; 300; 400 MHz). Furthermorealso the theoretical mode density ∂N/∂f computed with (2.42) and shown in Fig. 2.3exhibits a similar behavior for both chambers: As mentioned above, an RC is required tohave at least ∂N/∂f ≈ 1.5 modes/MHz above cutoff at the LUF, and both the prototype


0

N/ f [1/MHz]

f [MHz]

100 150 300 400 500200 250 450350

1

2

3

4

5

6Cubic RCPrototype RC

N / fmin

Figure 2.3: Theoretical mode density ∂N/∂f in a rectangular cavity with the same dimensionsas the prototype RC or the cubic RC (see Sections 6.1.3 and 6.1.4 for chamber geometrydetails). An RC is required to have at least 1.5 modes/MHz at the LUF [6].

and the cubic chamber pass this limit at around the same frequency. However not onlya sufficient number of cumulated modes N as well as an adequate mode density ∂N/∂fabove cutoff are needed. Due to the much more rapidly decreasing modal gap ∆f i′+1; i′

for the prototype RC compared with the cubic chamber (Fig. 2.4), from a modal analysispoint of view the prototype chamber is considerably better suited for RC operation thanthe cubic one [18]. As expected, above the fundamental mode cutoff frequency in thebeginning the modal gap ∆f i′+1; i′ also decreases quite fast for the cubic chamber – thecubic chamber though shows the existence of multiple degenerate modes (appearing as“spikes” in Fig. 2.4) as the frequency increases. A similar behavior can be seen if in-stead of the theoretical, “smooth” mode density ∂N/∂f derived from the approximation(2.42), the actual number of modes per 10 MHz interval is counted (Fig. 2.5). Althoughthe cubic RC exhibits at certain frequency intervals an even greater number of modesabove cutoff, in-between the mode density ∆N/10 MHz is – resulting from the modedegeneracies – significantly lower for the cubic chamber as compared to the rectangularprototype RC [19]. From (2.43) it is evident, that in order to avoid mode degeneracies, itis important that the ratio between squared dimensions of the chamber is a non-rationalnumber. Whether these mode degeneracies also significantly deteriorate field uniformityand isotropy inside the RC is investigated in Chapter 7.


0

10

5

f [MHz]50 100 150 300 400 500200 250 450350

15

20

25

30

f i +1;i [MHz]

* = Degenerate modes

*

* *

** *

**

*

*

** * * *


Figure 2.4: Modal gap ∆f i′+1; i′ in a rectangular cavity with the same dimensions as theprototype RC or the cubic RC (see Sections 6.1.3 and 6.1.4 for chamber geometry details).

2.2.2 Quality factor

The quality factor Q describes the ability of a system (such as an RC) to store energy.A high Q indicates that an RC has low losses and is therefore very efficient in storingenergy. The chamber Q is an important quantity because it allows prediction of the meanfield strength resulting for a given input power. In addition, it provides an estimate ofthe chamber shielding effectiveness (SE) and the RC time constant. Analytical qualityfactor derivations of mono-mode resonators for each individual TEmnp and TMmnp modeare outlined in [11].

Theoretical quality factorThe highly overmoded RC makes Q calculations based on resonant bandwidths (such asQ = f0/∆fQ, see Section 2.1.2) difficult, if not meaningless [20, 21]. A better approachis to use the basic definition of the quality factor based on the time-averaged storedenergy Ws and the energy dissipated during one period Wd within a resonator

Q = 2πWs

Wd=

ωWs

Pd(2.46)

with Pd being the dissipated power. Ws can be computed from

Ws =12

∫∫∫V

D · E dv =12ε

∫∫∫V

∣∣∣ E∣∣∣2 dv (2.47)


0

10

5

f [MHz]50 100 150 300 400 500200 250 450350

15

20

25

30

N/f [1/10MHz]

35

40

45


Figure 2.5: Number of modes ∆N per ∆f = 10MHz interval above cutoff in a rectangularcavity with the same dimensions as the prototype RC or the cubic RC (see Sections 6.1.3and 6.1.4 for chamber geometry details).

Inserting (2.47) into (2.46) and taking into account that the dissipated power equals thenet input power Pin (i.e. forward power minus reflected power) leads to

Q =ωε

2Pin

∫∫∫V

∣∣∣E∣∣∣2 dv (2.48)

To evaluate (2.48) in practice requires knowledge of all individual loss mechanisms withinan RC: wall losses, absorption due to e.g. an EUT, aperture leakage (doors, interconnec-tions between sheet metal panels, see Section 5.1), and losses introduced by the finitelyconducting RX and TX antennas. Whereas (2.48) includes per definitionem all theselosses, the simpler formula

Q =3V

2µrδsA

1[1 + 3λ

16

(1l + 1

w + 1h

)] (2.49)

accounting only for ohmic losses in the walls is often used for RCs, where V is thechamber volume and A the inner RC surface [16]. δs in (2.49) denotes the skin depth ofthe metal which is defined as

δs =1√

πfµκ(2.50)

The Q-factor estimate of (2.49) can be used as long as the walls are highly conducting(i.e. κ ωε) and as long as losses in the RC walls are the dominant absorption mecha-


nism. Expression (2.49) can be further reduced for typical values of l, w, h (such as thedimensions of the prototype RC of this thesis l = 2.48 m, w = 2.86 m, h = 3.06 m) andwavelengths λ ≤ 1 m to

Q ≈ 3V

2µrδsA(2.51)

which is also the Q-factor estimate proposed in the IEC 61000-4-12 standard [6]. Un-fortunately the derivation of the quality factor according to (2.51) is of little practicaluse (although surprisingly very often utilized), since theoretical Q values calculated with(2.49) or (2.51) prove to be consistently too high by a factor of 10. . . 500 when com-pared with measured ones [22, 20]. In addition (2.51) suggests a variation of Q with

√f

(because of its proportionality to 1/δs), which has not been observed in RC measure-ments [23, 24]. As [24] already notes, measurements indicate that “ [. . . ] loss mechanismsother than Joule heating [in the walls] are important”.

Measured quality factorOnce an RC is built, the chamber-Q can be determined by measurements. Q-factormeasurements in RCs are commonly carried out with a TX and RX antenna inside thechamber and recording the received and transmitted power PRX and PTX [20]. Usingthis approach, the chamber quality factor Q can be estimated as

Q =16π2V

λ3

PRX

PTX(2.52)

The relation (2.52) can be evaluated over one full stirrer rotation (ϕj = ϕ1 . . . ϕN ), whichallows the computation of the mean chamber quality factor

Q′ = 〈Q〉ϕj=ϕ1...ϕN(2.53)

It must be emphasized that (2.52) only represents a rough estimate of the actual Qof an RC. The application of (2.52) is particularly difficult for very high-Q chambers,where PTX should ideally be zero, since there are almost no losses (which implies alsofor the same reason PRX ≈ PTX). However due to the dominant absorption of theantennas (which are geometrically large) PTX is greater than zero resulting in a Qwhich is characteristic for the antennas, but not for the measured RC. A remedy to thisproblem is to measure the quality factor with very small probes, which do not lower thechamber-Q by loading the chamber.When designing an RC often the question arises, which type of material is to be used inorder to get a good chamber performance in terms of the quality factor Q. As mentionedshortly in the beginning of this section, Q tends to be influenced mostly by

• intrinsic chamber properties, i.e. conductivity κ of the wall material and the overallSE which is to a large extent defined by how the material was processed (soldered,welded, screwed, bolted, etc.) and the particular construction (apertures, feed-through panels, doors, ventilation ducts)

• loading introduced by TX and RX antennas along with tripods, probe stands,cables


• loading through the presence of the EUT together with supporting equipment

The trade-off that has to be made is essentially between field uniformity at lower frequen-cies (where only few modes are above cutoff) and maximum achievable field strength “perWatt” input power. These two extremes transform physically into a trade-off betweena chamber quality bandwidth ∆fQ large enough (i.e. Q small enough) for a sufficientnumber of modes to propagate at a fixed frequency versus a ∆fQ small enough (i.e. Qlarge enough) for a high average field strength within the RC (see Section 2.1.2 for anexplanation of ∆fQ). A comparison of the maximum achievable field strength between achamber built from aluminum against an RC made out of galvanized steel was presentedin [25] – with the not very surprising conclusion that the average field strength in thealuminum RC is higher – but unfortunately there was no analysis of the impact on fielduniformity performed.

2.2.3 Stirring ratio

The stirring ratio (SR) provides a global parameter to quantify changes of the field distri-bution induced by a rotating stirrer. This method essentially measures the effectivenessof the stirrer w.r.t. “moving the maxima and nulls within the chamber”. The SR iscommonly defined as

SR = maxϕj=ϕ1...ϕN

PRx(x0, y0, z0) − minϕj=ϕ1...ϕN

PRx(x0, y0, z0) (2.54)

which requires that the power received by an antenna within the RC is measured overa certain number of stirrer angles at a fixed spatial position [8, 9]. The input poweris kept constant PTx = const. for all rotational stirrer angles ϕj . Subsequently, fromthe recorded power data the maximum and minimum value is calculated. Then theSR is obtained by subtracting the minimum from the maximum value. The methodproposed in (2.54) relies on a mode-tuned operation of the RC with discrete stirrersteps, since mode-stirring with a continuously rotating stirrer would introduce somesort of “time-averaging”. In the common RC terminology, the SR is often expressedin terms of decibels. A high SR suggests that both maximum and minimum E- or H-field values occur at the same position for different stirrer steps; this indicates a moreeffective stirrer. If the SR was measured at all points throughout the chamber, theoptimal condition would be if it was uniform, as this would indicate that the stirrer waseffectively changing the boundary conditions evenly throughout the chamber. The lowerlimit normally accepted for the SR is 20 dB [26], and a sufficiently large SR is often takenas a prerequisite in achieving a good field uniformity within an RC.It should be noted that there are several other SR definitions used in the literature: somepublications (e.g. [23]) introduce the SR as

SR =max

ϕj=ϕ1...ϕN

|E(x0, y0, z0)|min

ϕj=ϕ1...ϕN

|E(x0, y0, z0)| (2.55)

i.e. the ratio of the maximum to minimum electric field strength measured at a fixed pointwithin the chamber over one revolution of the stirrer ϕj = ϕ1 . . . ϕN . Still other authors


(e.g. [22]) use the ratio between the input power of a TX antenna and the received powerof a RX antenna in the chamber. It must be emphasized that the various SR definitionsare not interchangeable and that especially the two latter expressions do not agree withthe SR statements found in fundamental RC literature such as [8, 9]. Generally, the SRparameter is not used in formal RC tests as specified by [6], but serves within certainlimitations as a means to compare different stirrers against each other.

2.3 Field anisotropy and inhomogeneity

The planar and total field anisotropy coefficients Aαβ and Atot as well as the inhomo-geneity coefficients Iα and Itot were introduced for RCs by L. Arnaut [27, 28]. Thesecoefficients yield specific performance measures for field homogeneity and randomness ofpolarization within an RC.

2.3.1 Field anisotropy coefficients

The field anisotropy coefficients Aαβ and Atot are defined according to [6] as

〈Aαβ〉 =

⟨ |Eα|2Pi

− |Eβ |2Pi

|Eα|2Pi

+ |Eβ |2Pi

⟩(2.56)

〈Atot〉 =

⟨√A2

xy + A2yz + A2

zx

3

⟩(2.57)

|Eα| and |Eβ | represent the respective single measured or simulated electric field strengthcomponent, with α or β = x, y, z for a given angular stirrer position. Pi is the net (i.e.forward minus reflected) input power injected into the RC for the same stirrer position.The 〈·〉 operator denotes ensemble averaging over all angular stirrer positions.The main reason why Pi is used in (2.56) is for cases when measurements along only asingle axis are being performed during one stirrer rotation. When re-orienting a single-axis probing sensor into two remaining different orientations in turn, the TX antenna“sees” slightly different configurations (plus different cable layouts, etc.), so Pi changesfrom measurement of one axis to another. If all field locations to be evaluated in (2.56)were simulated or measured using identical input power, Pi can be set to any arbitraryvalue [29]. Using e.g. Pi = 1 W for simplicity leads to

〈Aαβ〉 =⟨ |Eα|2 − |Eβ |2|Eα|2 + |Eβ |2

⟩(2.58)

The pointwise planar field anisotropy coefficients Aαβ are self-normalized quantities,taking values between −1 and +1 for each stirrer state, irrespective of the value ofthe input power Pi [6]. For perfect reverberation conditions (i.e. ideal statistical fieldisotropy), the random variable Aαβ can be shown to exhibit a uniform (rectangular)distribution, whose theoretical cumulative distribution function (CDF) is given by

FAαβ(aαβ) =

1 + aαβ

2(2.59)

2.3 FIELD ANISOTROPY AND INHOMOGENEITY 19

Number of stirrer steps N

“Stirring quality” N = 10 N = 30 N = 100 N = 300

“Medium” −2.5 dB −5.0 dB −7.5 dB −10.0 dB

“Good” −5.0 dB −10.0 dB −12.5 dB −15.0 dB

Table 2.1: Typical values of the total field anisotropy coefficient 〈Atot〉 for “medium” and “good”RC performance [6, 30, 31].

i.e. a straight line with unit slope. The maximum distance between the simulated ormeasured and the theoretical CDFs from (2.59) serves as an indirect measure for thefield anisotropy (i.e. for a bias in the statistical field polarization towards a certain direc-tion) [30]. Typical values obtained in RC simulations and measurements for “medium”and “good” stirring quality are shown in Table 2.1 [6].

2.3.2 Field inhomogeneity coefficients

In analogy to the anisotropy coefficients introduced in Section 2.3.1, the field inhomo-geneity coefficients are defined as

〈Iα(r1, r2)〉 =

⟨ |Eα(r1)|2Pi

− |Eα(r2)|2Pi

|Eα(r1)|2Pi

+ |Eα(r2)|2Pi

⟩(2.60)

〈Itot(r1, r2)〉 =

⟨√I2x + I2

y + I2z

3

⟩(2.61)

Similarly as for the field anisotropy coefficients, Pi can be set to any arbitrary value aslong as all field locations to be evaluated in (2.60) were simulated or measured usingidentical input power. For the case of constant input power, (2.60) simplifies to

〈Iα(r1, r2)〉 =⟨ |Eα(r1)|2 − |Eα(r2)|2|Eα(r1)|2 + |Eα(r2)|2

⟩(2.62)

When evaluating (2.60) care should be taken to avoid selecting locations r1 and r2 thatare separated by an integral number of half wavelengths or excessively small distancesmuch below this wavelength for which the fields are highly correlated. A minimumdistance corresponding to one wavelength is recommended [6].It is usually sufficient to investigate either Aαβ , Atot or Iα, Itot since field anisotropyand field inhomogeneity coefficients show highly correlated statistics. The distributionsof the Iα, Itot however, are usually more sensitive to mode-stirring imperfections [31].


2.4 Field statistics and probability density functions

The EM field at a given position in the RC can be decomposed into three components andeach of these components can be described by its real and imaginary part (or equivalentlyby its phase as well as its magnitude). Therefore in total six parameters are required tofully describe the field. These six parameters are called in-phase and quadrature partsin each of the three orthogonal directions x, y, and z

Ex = Re Ex + j Im Ex (2.63)Ey = Re Ey + j Im Ey (2.64)Ez = Re Ez + j Im Ez (2.65)

As the RC stirrer rotates from one angular position to the next, each of these in-phaseand quadrature parts is recorded and forms a data ensemble. The same statisticalconsiderations apply to both the electric and the magnetic field within an RC; for thesake of simplicity, the following sections only deal with the electric field.

2.4.1 Quadrature and in-phase part statistics

The six per-part ensembles can be statistically described as a compilation of a largenumber of random variables X which are – by the central limit theorem [32] – Gaussiannormally distributed

f(X |µ, σ) =1

σ√

2π· e−(X−µ)2

2σ2 (2.66)

where µ is the mean value and σ the standard deviation. If the RC is operated correctly(i.e. significantly above the LUF), a Gaussian distribution with the probability densityfunction (PDF) shown in Fig. 2.6 is obtained for each quadrature/in-phase part of thethree components (2.63)-(2.65). For sufficiently low correlation ρ between EM fieldssampled at different stirrer angles (see Section 2.5) and provided that a normal distrib-ution prevails, it can be concluded that the fields are also statistically independent [32].It is therefore reasonable to assume that all six quadrature/in-phase parts of the threecomponents are independent identically distributed (i.i.d.). Finally, the mean values µof these distributions can be assumed to be zero if there is not a significant direct pathfrom the TX antenna to the sampling point. This is a good assumption if the antennais near and pointed into a corner or directed towards the stirrer (see Section 7.6.3 for ananalysis of different coupling paths).

2.4.2 Magnitude statistics for single components and total field

Since the statistical ensemble of each quadrature/in-phase part is Gaussian normallydistributed, each of the three field component ensembles Ex, Ey, Ez exhibits a 2-DGaussian distribution over one full stirrer rotation. Therefore the magnitude of a single

2.4 FIELD STATISTICS AND PROBABILITY DENSITY FUNCTIONS 21

component

|Ex| =√

(Re Ex)2 + (Im Ex)2 (2.67)

|Ey| =√

(Re Ey)2 + (Im Ey)2 (2.68)

|Ez | =√

(Re Ez)2 + (Im Ez)2 (2.69)

follows each a χ-distribution with two degrees of freedom (i.e. χ(2)-distribution) [32].Generally, the PDF of the χ-distribution with ν degrees of freedom (χ(ν)) is given by

f(X | ν) =2

2ν2 · σν · Γ

(ν2

) · Xν−1e−X

2σ2 (2.70)

from which the individual distribution functions can be derived by setting ν to theappropriate value (e.g. ν = 2, see Fig. 2.6). Values of the Γ-function used in (2.70) canbe obtained numerically by

Γ(x) =

∞∫0

e−ttx−1dt (2.71)

or tabulated in e.g. [12]. A χ(2) distribution (obtained e.g. for a single component of theelectric field E consisting of in-phase and quadrature part) is also known in the literatureas a Rayleigh distribution [33]. An exemplary χ(2) distribution is shown in Fig. 2.6. Themagnitude of the resultant vector sum of the components for three dimensions (2.67)-(2.69) is the square root of the sum of the squares of six i.i.d., zero mean, normal randomvariables Xi

Y =√

X21 + X2

2 + X23 + X2

4 + X25 + X2

6 (2.72)

with each Xi resembling a quadrature/in-phase part of the three field components Ex,Ey, and Ez (2.63)-(2.65). The magnitude of the electric field

|E| =√|Ex|2 + |Ey|2 + |Ez|2 (2.73)

is therefore χ-distributed with six degrees of freedom (χ(6)-distributed) [28, 32]

f(Y | ν = 6) =1

4 · σ6 · Γ (3)· Y 5e−

Y2σ2 (2.74)

The χ(6) distribution for the magnitude of the EM field within an RC is depicted inFig. 2.6.

2.4.3 Power statistics for single components and total field

For RC measurements or simulations of power-related quantities the square of the ran-dom variable summation shown in (2.72)

Z = Y 2 = X21 + X2

2 + X23 + X2

4 + X25 + X2

6 (2.75)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Re ,E E|, |P|x [a.u.]|E , |x|

N(0.4, 0.1) M M/ max

Figure 2.6: Exemplary Gaussian normal N(0.4, 0.1), χ(2), χ(6), χ2(6) distribution. These statis-

tical distributions are obtained when counting the occurrence of e.g. the real part of a singlefield component (Gaussian normal N), the magnitude of a single field component (χ(2)), themagnitude of the total field (χ(6)), or the magnitude of the power (χ2

(6)) over a full stirrerrotation. M denotes a part of the total ensemble Mmax.

is of interest. For ensembles of the power random variable Z, the χ2-distribution with νdegrees of freedom (χ2

(ν)) is obtained. Its PDF is defined through

f(Z | ν) = 2ν2 · Γ

(ν

2

)· Z ν

2−1e−Z2 (2.76)

A χ2(2) distribution (which is the same as an exponential distribution) is obtained e.g. for a

single component of the received power as measured at the terminals of a standard logperantenna, which responds to only a single polarization (see Section 7.6.3). Similarly,the total power with all three components will follow a χ2

(6) distribution inside a well-operating RC (see Fig. 2.6). Additional details on the observed PDFs applicable to RCscan be found in Section 7.6.3 and Appendix C, Section C.2.

2.4 FIELD STATISTICS AND PROBABILITY DENSITY FUNCTIONS 23

2.4.4 Statistical goodness-of-fit χ2-test

Whenever measurement or simulation data of an RC is analyzed, it is good practice tomake a statement if and how well the data actually follows the theoretically predictedstatistical distributions. This is especially necessary, if the amount of gathered data isvery limited, so that a visual comparison against a theoretically expected distributionis impossible. To evaluate how good the agreement between prediction and reality is, aso-called goodness-of-fit test can be used.A commonly used test to check for fit with a theoretical distribution function is the“Chi-square test” (χ2-test) [32, 34]. The χ2-test is based on the random variable

χ2 =N∑

i=1

(gi − ei)2

ei(2.77)

where N is the total number of samples, gi is the number of observed samples in the i-thinterval (also called i-th class), and ei is the expected number of samples in this interval(or class) if the hypothesized distribution is correct. The expected number of samples ei

can be computed fromei = N · Pi (2.78)

with Pi being the probability that a particular sample is part of the i-th class. Theunderlying distribution must be divided into i intervals such that

in central classes ni · Pi ≥ 5 (2.79)in boundary classes ni · Pi ≥ 1 (2.80)

is satisfied for a given number of samples ni. The χ2 variable will be χ2-distributedwith ν = N − q degrees of freedom, q being the number of parameters in the assumeddistribution that are calculated from the data (calculated parameters are estimators forthe mean or the standard deviation obtained from the underlying data, as the actualvalues are not exactly known). For ν > 1 degree of freedom, (2.77) must be modifiedinto

χ2 =1ν

N∑i=1

(gi − ei)2

ei(2.81)

The theoretically expected mean of χ2 is 1. If χ2 1 the observed samples do notfit the a priori hypothesized distribution, for χ2 ≈ 1 the agreement is “satisfactory”.Similar to Section 2.5 it is desirable to quantify how e.g. a “satisfactory” agreementcan be translated to some sort of “numerical” confidence in a given hypothesis. Anexpression of the quantitative significance of a certain χ2 value can be evaluated bycalculating the probability Pχ2 to get a certain value of χ2 which is equal or greater as thesampled χ2

0 if the sampled distribution actually matches the hypothesized distribution.If e.g. Pχ2 (χ2 ≥ χ2

0) is large, the hypothesized and the sampled (measured or simulated)distributions seem to be identical; conversely for a small probability Pχ2(χ2 ≥ χ2

0),chances are high that hypothesized and sampled distribution do not match, i.e. there isa certain deviation between them, which leads to a rejection of the hypothesis.


χ20

ν 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

2 1 0.82 0.67 0.55 0.45 0.37 0.30 0.25 0.20 0.17

6 1 0.98 0.88 0.73 0.57 0.42 0.30 0.21 0.14 0.095

χ20

ν 2.0 2.2 2.4 2.6 2.8 3.0 3.5 4.0 4.5 5.0

2 0.14 0.11 0.09 0.074 0.061 0.05 0.03 0.018 0.011 0.007

6 0.06 0.04 0.03 0.016 0.01 0.006 0.002 — — —

Table 2.2: Probability Pχ2(χ2 ≥ χ20) that measurement or simulation samples taken out of

an ensemble with ν degrees of freedom would result in χ2 ≥ χ20 (— indicate probabilities

Pχ2 ≤ 0.0005).

In order to compute the probability Pχ2 (χ2 ≥ χ20), the χ2-PDF (2.70) can be integrated

(with σ = 1)

Pχ2(χ2 ≥ χ20) =

∞∫χ0

22

ν2 · Γ

(ν2

) · xν−1e−x22 dx (2.82)

General tabulated values of (2.82) can be found in e.g. [35, 12], the cases most relevantto RCs with ν = 2 and ν = 6 degrees of freedom are listed in Table 2.2. In particular, adeviation between a hypothesized and sampled distribution is said to be

significant if Pχ2(χ2 ≥ χ20) ≤ 0.05 (2.83)

andhighly significant if Pχ2(χ2 ≥ χ2

0) ≤ 0.01 (2.84)

For example, the probability to obtain χ20 = 3.5 from an experiment with ν = 2 degrees

of freedom is with (2.82) Pχ2(χ2 ≥ χ20) = 0.03 (see Table 2.2). This means that there

is a significant deviation between the hypothesized and the actual, sampled distributionwhich would lead to a rejection of the hypothesis.

2.5 Correlation coefficient

An ideal stirrer would be expected to generate EM field distributions with no similaritybetween one rotational stirrer position and the next [6]; a quantitive statement con-

2.5 CORRELATION COEFFICIENT 25

cerning the similarity is made by the correlation coefficient. Therefore the correlationcoefficient serves as a very important parameter in RC stirrer and uncertainty analysis.

2.5.1 Definition of correlation

In general, correlation can be visualized as a measure to evaluate how well N data points(x1, y1), . . . , (xN , yN ) fit a straight line, i.e. exhibit a linear dependence. The correlationcoefficient ρxy (or more precisely an estimate for the correlation coefficient) between twodiscrete ensembles X and Y with N samples each can be calculated as

ρxy =

1N

N∑i=1

(xi − 〈x〉)(yi − 〈y〉)√s2

x s2y

(2.85)

where 〈x〉 and 〈y〉 are the corresponding arithmetic mean values of these two ensembles.They are defined as

〈x〉 =1N

N∑i=1

xi and 〈y〉 =1N

N∑i=1

yi (2.86)

under the assumption that the two discrete ensembles are two random samples takenout of a very large underlying basic population (such as one created by a large numberof repeated measurements of identical parameters under consistent conditions) whereinthe ensembles themselves are uniformly distributed. sx and sy are the variances of theensembles and can be obtained from

sx =1

N − 1

N∑i=1

|xi − 〈x〉|2 and sy =1

N − 1

N∑i=1

|yi − 〈y〉|2 (2.87)

The correlation function ρxy can assume any value −1 ≤ ρxy ≤ 1, with values of ρxy = ±1indicating a good linear correlation and values of |ρ| ≈ 0 little or no correlation atall. Applied to RCs, ρxy relates e.g. the magnitude of a component of the electricfield (say |Ex|) at a fixed position (x0, y0, z0) for the angular position ϕ1 of the tuneragainst the magnitude of the same electric field component |Ex| at the same location(x0, y0, z0) for the angular position ϕ2. As mentioned above, for a well-operating RCa low correlation between the EM fields obtained at the two angular stirrer positionsϕ1, ϕ2 is desirable [36]. As noted before in Section 2.4, for normally distributed ensembles|ρ| ≈ 0 implies that consecutively taken samples are also mutually independent.

2.5.2 Significance of correlation

Since it is in practice very unlikely to obtain exactly ρ = 0 for a finite number ofsamples N < ∞ out of finitely long ensembles (ρ = 0 would be found between a largenumber of randomly picked, non-identical samples from a perfect additive white Gaussiannoise (AWGN) process of infinite bandwidth), the key question is which value of ρ wouldindicate that the correlation between two ensembles (i.e. the similarity) is “sufficiently


low” in order to classify the ensembles as “uncorrelated”. As an answer to this question,the quantitative significance of a certain correlation coefficient ρ0 can be evaluated bycalculating the probability PN that N samples taken out of two fully uncorrelated (i.e.ρ = 0) ensembles would seem to have a correlation ρ as large or larger than ρ0. Inaccordance with [35] the probability PN (|ρ| ≥ |ρ0|) is given by

PN (|ρ| ≥ |ρ0|) =2 Γ(

N−12

)√

π Γ(

N−22

)1∫

|ρ0|

(1 − ρ2

)N−44 dρ (2.88)

where the Γ-function can be calculated with (2.71). Table 2.3 shows PN (|ρ| ≥ |ρ0)| forvarious numbers of samples N and correlation coefficients ρ0. The number of samples Nis for RCs equal to the number of angular stirrer positions. If a correlation of ρ0 betweentwo ensembles is obtained, for which PN (|ρ| ≥ |ρ0|) is small, then there is a high chancethat the two ensembles are uncorrelated. In particular, a correlation is said to be

significant if PN (|ρ| ≥ |ρ0|) ≤ 0.05 (2.89)highly significant if PN (|ρ| ≥ |ρ0|) ≤ 0.01 (2.90)

For example, with (2.88) and Table 2.3 the probability that 50 samples (N = 50) oftwo uncorrelated ensembles will result in a correlation coefficient |ρ| ≥ 0.3 is around0.034. This means if 50 samples yield ρ = ±0.3, there is no evidence of a linear corre-lation between the two ensembles. In RC theory often once a correlation coefficient of|ρ0| < 1/e ≈ 0.37 is observed in a measurement or simulation series, two ensembles areconsidered uncorrelated [6]. As can be deduced from Table 2.3, the probability that theunderlying ensembles would actually have a higher correlation |ρ0| ≥ 1/e is for N ≥ 50stirrer steps less than 1%. In other words, there is highly significant evidence, that theensembles are actually not correlated. Obviously, the earlier this criterion is met (i.e.after as few stirrer steps as possible), the better is the effectiveness of the tuner.As the frequency f is increased, an RC becomes more and more sensitive to even smallgeometrical changes. Whereas, e.g. at f = 100 MHz, a stirrer rotation of ∆ϕ = 5 wouldnot affect the field distribution within the chamber at all, the same ∆ϕ will change thefield completely at a frequency of f = 400 MHz (see Chapter 7). Therefore with risingfrequency, EM fields in an RC become increasingly uncorrelated from one stirrer stepto the next. Since reasonably uncorrelated fields are always a prerequisite for sufficientstatistical field uniformity and isotropy, at higher frequencies the same stirrer can pro-vide more uncorrelated samples than at lower frequencies close to the LUF. In order toreduce the time spent for an RC test, the number of stirrer steps required can thereforebe reduced at higher frequencies resulting in a correlation which remains approximatelyconstant as the frequency is increased [36, 37].Care should be exercised when evaluating correlations, since a correlation coefficient canbe completely distorted by a single maverick (xi, yi)-pair, i.e. two ensembles appear tobe correlated although they are not or vice versa [38]. Therefore a scatter plot shouldalways be used for the analysis of the correlation coefficient ρxy where each individual(xi, yi)-pair is shown and possible outliers, anomalies, or systematic tendencies can bepinpointed.

2.6 STATISTICAL UNCERTAINTY AND ESTIMATOR ACCURACY 27

|ρ0|

N 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

5 1 0.87 0.75 0.62 0.50 0.39 0.28 0.19 0.10 0.037

10 1 0.78 0.58 0.40 0.25 0.14 0.067 0.024 0.005 —

20 1 0.67 0.40 0.20 0.081 0.003 0.005 0.001 — —

30 1 0.60 0.29 0.11 0.029 0.005 0.001 — — —

40 1 0.54 0.22 0.06 0.011 0.001 — — — —

|ρ0|

N 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

50 1 0.73 0.49 0.30 0.16 0.08 0.034 0.013 0.004 0.001

60 1 0.70 0.45 0.25 0.13 0.054 0.02 0.006 0.002 —

70 1 0.68 0.41 0.22 0.097 0.037 0.012 0.003 0.001 —

80 1 0.66 0.38 0.18 0.075 0.025 0.007 0.001 — —

90 1 0.64 0.35 0.16 0.059 0.017 0.004 0.001 — —

100 1 0.62 0.32 0.14 0.046 0.012 0.002 — — —

Table 2.3: Probability PN (|ρ| ≥ |ρ0)| that N samples taken out of two uncorrelated ensembleswould result in a correlation coefficient |ρ| ≥ |ρ0| (— indicate probabilities PN ≤ 0.0005).

2.6 Statistical uncertainty and estimator accuracy

In order to be able to quantify the uncertainty in a test performed in an RC, knowledgeis needed about the statistical behavior of the EM fields in the chamber. A typicalquestion when using an RC is: What is the number of statistically independent stirrersteps necessary to state with a certain confidence that the mean, minimum, or maximumfield level at the EUT is within ±3 dB (i.e. the uncertainty) of the corresponding valuesdetected by a field probe at a different location? Using the PDFs for the EM field givenin Section 2.4, estimators and their accuracy for e.g. the mean field within an RC can be


0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

p

Figure 2.7: ±k ·σ standard deviation multiples contain p fractions of all values from a Gaussiannormal distribution (e.g. ±2σ contain approx. 95% of all values). p is also known as theconfidence level.

calculated. In RC simulations and measurements there is often the interest to calculatethe interval that contains a certain percentage p of the values from a standard Gaussiannormal distribution. This can be done by solving the integral equation

p = F (x, µ, σ) =1

σ√

2π

x∫−∞

e−(t−µ)2

2σ2 dt (2.91)

obtained from (2.66). Fig. 2.7 shows a plot of (2.91): as k · σ goes towards infinity,more values of the sample distribution are taken into consideration, and therefore theconfidence level gets higher (in the limit for k · σ → ∞, p approaches 1). In accordancewith [24], the so-called maximum likelihood estimator (MLE) is employed as an estimatorof the EM field. This has several advantages over other estimators: the MLE is alwaysasymptotically unbiased – i.e. its mean is the true value for large amounts of data –, andits accuracy can be easily calculated [35]. As outlined in [35, 24], the amount of dataneeded to achieve a desired estimator accuracy can be determined by

d = 10 log10

1 + k√bN

1 − k√bN

(2.92)

2.6 STATISTICAL UNCERTAINTY AND ESTIMATOR ACCURACY 29

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

Number of stirrer positions N

Uncert

ain

ty[d

B]

d~

Confidence level p68%75%95%99%

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6


Uncert

ain

ty[d

B]

d~


a)

b)

Figure 2.8: Number of statistically independent stirrer positions N required to achieve theuncertainty interval ±d for a) a single and b) three EM field components at a confidence ofp (see Fig. 2.7 for corresponding standard deviation multiples).

in the “dB notation”. k determines the desired confidence level (e.g. k ≈ ±1.96σ forp = 0.95, i.e. 95%) as given by (2.91). b is the number of dimensions of the field data tobe estimated (usually 1 or 3) and N is the required number of statistically independentstirrer positions. If the field probe responds to only one dimension of the field in this


0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6


Uncert

ain

ty[d

B]

d~


Figure 2.9: Number of statistically independent stirrer positions N required to achieve theuncertainty interval ±d for two EM field components at a confidence of p (see Fig. 2.7 forcorresponding standard deviation multiples).

case b = 1. Solving for the required number of statistically independent stirrer positionsN results in

N =k2

b

(10d/10 + 110d/10 − 1

)2

(2.93)

Equation (2.93) is plotted for different confidence levels in Figs. 2.8 and 2.9. If, for ex-ample, the uncertainty interval should be d = ±1 dB and the desired level of confidenceis 90% (corresponding to k ≈ ±1.65σ), then one would obtain N ≈ 69 or N ≈ 207 forb = 3 and b = 1 dimensions, respectively (see Fig. 2.8). Conversely, if a determination ofthe average field is made using N independent stirrer positions, the resulting uncertaintyis given by the interval ±d (2.92). Further plots of the uncertainty interval for variousnumbers of field components can be found in Appendix C, Fig. C.1. . . Fig. C.3.Obviously, the best (i.e. lowest) achievable uncertainty ±d directly depends on the num-ber of statistically independent field distributions N generated by the rotating stirrer.Therefore it is essential to make sure that a given stirrer-RC-combination is actuallycapable of providing at least N independent distributions. To evaluate that a stirrerprovides independent field conditions, i.e. independent samples, the correlation coeffi-cient as presented in Section 2.5 is usually calculated for the chosen step angle of thestirrer. Thus, it is assumed that uncorrelated stirrer positions yield independent sam-ples. This assumption is valid since (as outlined in Section 2.4) the underlying sixEM field quadrature component ensembles are normally i.i.d.; for a normal distributionuncorrelated samples are also independent with respect to each other. Therefore the

2.7 FIELD UNIFORMITY 31

requirements statistically independent and statistically uncorrelated can be used inter-changeably. It is important to understand that if a stirrer is not capable of providinga sufficient number N of uncorrelated field distribution over a full rotation – this canbe challenging especially at low frequencies (see Chapter 7) – it may not be possible toestimate the EM chamber field with a desired uncertainty ±d. In an ideal RC, ensembleaveraging (i.e. sampling the EM field at a fixed location (x0, y0, z0) for several differentstirrer angles ϕi = ϕ1 . . . ϕN ) will result in the same mean, maximum, and minimum esti-mators as sampling the field over a certain space (xi, yi, zi) = (x0, y0, z0) . . . (xN , yN , zN)at a fixed stirrer angle ϕ1.

2.7 Field uniformity

The most important RC performance parameter is the statistical field uniformity, whichcan be achieved with a given stirrer in a chamber over its operating frequency range. Asmentioned in Section 2.2, the field uniformity is significantly better suited to determineat which point an RC reaches its LUF than the “number of modes above cutoff” and“modal density” criteria. These criteria serve as necessary prerequisites, but they donot guarantee a sufficient field uniformity. The field uniformity within an RC is ex-pressed in terms of the combined three-axis standard deviation σxyz and the single-axisstandard deviations σx, σy, and σz of the EM field as proposed in the IEC 61000-4-21standard [6]. Due to better availability of measurement equipment, normally the electricfield is used to compute the standard deviations. These quantities are calculated fromthe three components of the electric field Ex(xi, yi, zi), Ey(xi, yi, zi), and Ez(xi, yi, zi),which are related to the magnitude of the electric field |E(xi, yi, zi)|ϕj at the spatialposition (xi, yi, zi) with the stirrer fixed at the angle ϕj (j = 1 . . .N : j-th stirrer stepout of a maximum of N steps) by

|E|ϕj = |E(xi, yi, zi)|ϕj =√|Ex|2 + |Ey|2 + |Ez|2

∣∣∣∣(xi,yi,zi)

ϕj

(2.94)

For practical reasons, usually the electric field in the eight corner points (i = 1, . . . , 8)of the so-called “volume of uniform field” is employed as a means to predict RC perfor-mance. Two definitions exist describing the layout of this “volume of uniform field”, inwhich an EUT has to be placed during a test:

• at least λ/4 (with λ taken at the lowest frequency used for a particular test)

• at least 1 m (regardless of the operating frequency of the RC) [6]

away from RC walls, stirrers, antennas, and any other electromagnetically relevant ob-ject. For the IEC 61000-4-21 standard the second requirement has been adopted, it ishowever still a good idea to place the EUT sufficiently far away with respect to theoperating wavelength from e.g. the metallic walls in order to avoid parasitic couplingeffects. Fig. 2.10 shows how the λ/4 (as well as λ/2 and λ) criteria translate into acertain minimum distance at different frequencies.For a field uniformity analysis there is no obvious reason why not more than eight points


50 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

f [MHz]

d [m]

Figure 2.10: Required EUT spacing from RC walls versus operating frequency for differentdistance criteria (λ/4, λ/2, λ).

should be used for the analysis of an RC, since especially in an RC simulation, anyarbitrary number of field points is readily available [39]. The more points are used (e.g.instead of eight corner points of the field uniformity volume a much larger number ofdata points within this volume), the more accurate the estimators for the actual mean,minimum, and maximum field as well as its distribution function and standard devia-tion get. The restriction to eight points originates from field uniformity measurementmethods in order to minimize the time expenditure.In compliance with [6], initially the maximum of each single-axis electric field componentoccurring during one full rotation of the stirrer (ϕj = ϕ1 . . . ϕN ) is determined and thennormalized to the mean net input power Pi

Eξ,i =max

ϕj=ϕ1...ϕN

|Eξ(xi, yi, zi)|ϕj

√Pi

(2.95)

This will result in eight values for each individual axis ξ = x ∨ y ∨ z, i.e. 24 values intotal. If Pi is kept constant – either by employing a stress sensor and a leveling algorithmin a measurement setup or by definition in RC simulations – it can be arbitrarily set toPi = 1 W for simplicity, which reduces (2.95) to

Eξ,i = maxϕj=ϕ1...ϕN

|Eξ(xi, yi, zi)|ϕj (2.96)

2.7 FIELD UNIFORMITY 33

In order to compute the individual and the combined standard deviations, starting from(2.96), the three arithmetic per-axis means

⟨Eξ

⟩=

18

8∑i=1

Eξ,i (2.97)

and the combined arithmetic mean

⟨Eξ

⟩=

124

∑ξ=x,y,z

8∑i=1

Eξ,i (2.98)

using the three series’ of the x-, y-, and z-components is calculated. As a result, theindividual per-axis standard deviations

σξ =

√√√√√ 8∑i=1

Ei −⟨Eξ

⟩8 − 1

(2.99)

as well as the combined standard deviation

σxyz =

√√√√√ ∑ξ=x,y,z

8∑i=1

Eξ,i −⟨Exyz

⟩24 − 1

(2.100)

can be derived. The per-axis standard deviations can be expressed in the more familiar“decibel notation” through

σξ = 20 log10

σξ +⟨Eξ

⟩⟨Eξ

⟩ (2.101)

and for the combined standard deviation, respectively, through

σxyz = 20 log10

σxyz +⟨Exyz

⟩⟨Exyz

⟩ (2.102)

For both the per-axis standard deviations σξ as well as the combined standard deviationσxyz, [6] requires for a “well operating” RC with sufficient statistical field uniformityand a given uncertainty within all frequencies 80 MHz≤ f ≤ 100 MHz

σξ ≤ 4 dB and σxyz ≤ 4 dB (2.103)

For frequencies 100 MHz≤ f ≤ 400 MHz, the limits for σξ and σxyz decrease linearlyfrom 4 dB to 3 dB. Finally,

σξ ≤ 3 dB and σxyz ≤ 3 dB (2.104)


is required for all frequencies f ≥ 400 MHz. The above mentioned limits may be exceededat one single frequency (out of a predefined subset as outlined in [6]) within an octaveband of operation by 1 dB. Other standards such as the DO160 or the GMW have similar,slightly different limits.Prerequisites for a good statistical field uniformity are a sufficient number of modesabove cutoff as well as mode density (see Section 2.2.1) and weakly correlated (andhence independent) field distributions (see Section 2.5) within the RC. As shown inChapter 7, all these requirements are increasingly difficult to fulfill at low frequenciesaround the LUF.

2.8 Caveats for statistics

Using the procedures and quantities introduced in Section 2.4 (probability functions),Section 2.5 (correlation), Section 2.6 (uncertainty), and Section 2.7 (field uniformity) astatistical description of RCs can be performed. However, some precautions need to betaken when RC data is analyzed using statistical methods (in the context of this thesis,this applies especially to the results presented in Chapter 7):

• Class width in histograms can greatly influence the visual appearance of a graphand the outcome of a hypothesis test, which leads to a decision on agreement ordisagreement with a benchmark analytical distribution (see Section 7.6.3) [34].

• Residual classes with small expectation values should not be plotted, as they distortthe overall graph – instead a number referring to the number of samples withinthe residual classes is to be displayed. In Chi-square tests, residual classes mustbe combined.

• Few outliers should not be taken into consideration – a significant number of out-liers is however a strong indication that “something went wrong” during the mea-surement or simulation process [38].

• It must be always distinguished between “robust” (e.g. median) and “non-robust”(e.g. mean) statistical quantities: whereas a single outlier can completely offset themean of a data series, a quantity such as the median is influenced only very little.Therefore “robust” quantities should be used preferably.

• Correlation is a non-robust parameter; in addition, it is very easy to find somesort of correlation between two ensembles (especially if both exhibit some sort of alinear trend), although they are not correlated at all or even physically irrelevantfor each other [40].

• Chi-square tests can only be conducted for data which is a priori uncorrelated andstatistically independent and identically distributed (i.i.d.).

• In the end, almost everything has some sort of a normal distribution – chances tofind a good agreement between two quantities (e.g. a simulated electric field and ameasured one) based on the similarity to a normal distribution are therefore highand can be purely coincidental (see Section 7.4).

2.9 CONCLUSION 35

2.9 Conclusion

The fundamental concepts and key parameters of a reverberation chamber (RC) wereintroduced in this chapter. In the beginning, an RC was abstracted to a simple rectangu-lar cavity in order to explain the existence of TE and TM modes, the number of modes,and the modal density. It was shown that cavity modes are not sufficient to analyze theactual EM field distribution within an RC, but with their help guidelines for the lowestusable frequency (LUF) of an RC were derived. Using the proposed procedure to calcu-late the cumulated number of modes above cutoff and the mode density, an RC is likelyto perform well if at least 60 . . .100 modes within an ideal cavity of the same size as theRC are above cutoff and at least 1.5 modes/MHz are present. The mode distribution andLUF of the prototype RC built for this thesis and a cubic RC were computed. A morestringent, but time-consuming approach is to simulate or measure the lowest frequencyat which a specified field uniformity can be achieved over a volume defined by an eightlocation calibration data set. This method was carried out in Chapter 7.Methods for the derivation of the quality factor were established; the quality factorserves as an important parameter as it directly influences the field uniformity withinthe chamber, but it is difficult to measure and theoretical values are often several or-ders of magnitude too high due to the multimode-nature of an RC. Briefly the fieldanisotropy and inhomogeneity coefficients were mentioned, which are useful to assessRC performance and more sensitive than the standard field uniformity evaluation ap-proach. These coefficients were however little used in this thesis due to the unavailabilityof truly broadband field data (which is an inherent limitation of the frequency-domainsimulation technique).In a final step, the RC was described as a statistical test environment and the EMfields were characterized by distribution functions, correlation coefficients, statisticaluncertainty, and field uniformity. These parameters are used extensively throughout thefollowing chapters and establish the foundation on which the RC analysis by simulationand measurement is built.

3 Numerical Procedure

Abstract— In the beginning, this chapter summarizes fundamental requirements for a numerical

method to be suitable for reverberation chamber simulations. After contrasting these requirements

against advantages and drawbacks of several numerical field solver techniques, a frequency-domain

electrical field integral equation based method-of-moments was selected. The basic concept of

electromagnetic integral equations is outlined and the method-of-moments solution methodology is

introduced. Computational requirements regarding simulation time and memory are estimated and

an outlook on method-of-moment extensions and solver techniques is given.

3.1 Initial considerations of reverberation chamber simulations

Before deciding which EM simulation tool is most suited for the analysis of RCs, one mustidentify their critical design parameters. Based on these parameters, particular chal-lenges for different EM simulation tools will quickly become apparent. In the followingsections the frequency-domain method-of-moments (MoM) and the finite-difference time-domain (FDTD) method are used as two typical EM field solver techniques to demon-strate fundamental challenges. Problems typical for MoM occur also in the boundary-element method (BEM); difficulties existing in finite-difference time-domain (FDTD) arelikely to appear also in the finite-integration technique (FIT) and the transmission-line-matrix method (TLM) method. The following sections identify the key issues, whichneed to be taken into consideration as common requirements to achieve meaningful sim-ulation results.

3.1.1 Wide operational frequency range

RC test systems are designed to cover a broad frequency range, typically from around100 MHz to a few GHz (e.g. 80 MHz. . . 6 GHz). To minimize the computational effort andhence the simulation time, usually the chamber geometry is discretized by an adaptivefrequency-dependent mesh. Attention must be paid to the proper discretization: For ex-ample, insufficient mesh resolution may introduce numerical artifacts, viz. the computedfield values depend strongly on the discretization of the geometry and are affected bynumerical dispersion.For a broadband simulation response – in particular if high frequency resolution is re-quired – frequency-domain methods such as MoM are at a disadvantage compared totime-domain methods. While the latter require only one simulation run irrespective ofthe frequency resolution [41], the former need to be run at many, sometimes tightlyspaced discrete frequencies [42].

37

38 3 NUMERICAL PROCEDURE

3.1.2 Large, varying, and irregular geometry

By its physical principle of operation, an RC must be electrically large in terms of thewavelength at fLUF to achieve sufficient statistical field uniformity, i.e. the standard de-viation for both the three individual field components and the total data set are withina specified tolerance for a full stirrer rotation [6]. This means that an RC suitable fortesting of EUTs at frequencies as low as 80 MHz easily surpasses a volume of 10 000λ3

at a frequency of 1 GHz assuming a chamber size of 6 m·13 m·5 m. To discretize a com-putational domain of this size considering a mesh dimension of at least λ/10 in eachdimension, requires a huge number of mesh cells (i.e. triangles in MoM or volume ele-ments in FDTD). Methods using surface discretization such as MoM have an advantageover volume discretization based methods, because the large space comprising air doesnot have to be discretized.A particular characteristic which is rarely encountered in ordinary EM problems is thevarying geometry of the RC during operation: Since the stirrers inside an RC are ro-tating, the chosen simulation method must be able to accommodate a varying geometrywithout introducing additional errors. This problem is primarily found in methods us-ing volume discretization, where the mesh needs to be changed at least locally from onestirrer step to another, which might create additional numerical artifacts (for FDTDcf. chapters 10 and 11 in [41]). To achieve an optimal EM effectiveness, the stirrersare usually very irregular and designed asymmetrically, which is particularly difficult toconsider when structured meshes are used.

3.1.3 Finite conductivity and entirely closed structure

To obtain reliable and physically meaningful results, the chamber walls, stirrers, EUT,and antennas of an RC must be modeled taking finite conductivity, κ < ∞, into ac-count. Otherwise (for a perfectly electrically conducting structure with κ → ∞) the RCquality factor bandwidth ∆fQ [6] would be zero, i.e. at a given frequency f only a singlemode (and possibly respective degenerate modes) could be excited (see Section 2.1.2).In this case, coupling between different non-degenerate modes would not exist and thefield level within the cavity could reach unrealistically high values, only limited by thechosen numerical method. Realistic RCs however can only achieve sufficient statisticalfield uniformity if ∆fQ > 0 so that several modes are excited at a given frequency.The technique to implement lossy materials varies from one numerical method to another(see Section 3.3.4). Whether a specific numerical conductivity formulation is appropriatefor the simulation of an RC can only be determined through a validation of the simulatedresults by measurements.From the computational point of view, RCs represent an entirely bounded domain, whichdoes not radiate to the exterior. Several simulation techniques are numerically problem-atic if applied to computational domains that form a closed, non-radiating structure:Depending on the losses in the RC, a time-domain method such as FDTD might havedifficulties to achieve numerical convergence [41] so that e.g. “artificial” losses within theair volume have to be introduced or the wall conductivity has to be modeled unrealis-tically low (e.g. κ < 100 S/m [43]). A frequency-domain solver can exhibit the problem

3.2 COMPUTATION OF ELECTROMAGNETIC FIELDS 39

that in the resonance case there is either no unique solution for the system of equationsor the solution of this system can have large errors [44].

3.1.4 Highly resonant chamber

Depending on the actual losses introduced by walls of finite conductivity, stirrers, EUT,and antennas, EM resonances will be more or less pronounced and varying in bandwidth.In the RC operation mode f > fLUF a large number of resonances will exist, makingadaptive (i.e. resonance-dependent) frequency sampling and interpolation in-betweenawkward. This problem has been considerably reduced with the introduction of adaptivefrequency sampling algorithms that set simulation frequencies according to the presenceof resonances. For the simulation of an RC, however, adaptive frequency sampling onlymakes sense up to or slightly beyond fLUF where the number of modes above cutoff is stillsmall (60. . . 100 modes [6]). For frequencies f fLUF, too many discrete frequencieswould need to be sampled to compute a truly broadband response. Highly resonantstructures pose additional problems for a simulation method: in time-domain codes suchas FDTD the correct derivation of the quality factor Q of an RC can be difficult, sincethe underlying algorithm introduces phase errors due to numerical dispersion resultingin shifts of resonance frequencies (cf. chapter 4 in [41]). In addition, identifying narrowresonance peaks requires rather long simulation times.

3.1.5 Large number of spatial near field positions

In RC simulations it is desirable to have EM near field data available anywhere, or atleast on multiple planes within the chamber to analyze e.g. the effect of a stirrer or anEUT. In a method such as FDTD, the user must decide before launching a simulation, atwhich positions the near field is to be monitored and stored. If near field data throughoutthe entire RC is to be monitored, memory and computational requirements for FDTDincrease significantly, especially for a fine frequency resolution.For MoM the computationally expensive part is the inversion of a full matrix to obtainthe surface and line currents. Once however this system of equations is solved, thenear field at any arbitrary point can be computed a posteriori from the stored currentswithout re-running the whole solution process again.

3.2 Computation of electromagnetic fields

The simulation of a reverberation chamber (RC) requires the calculation of the electro-magnetic (EM) fields and currents which are excited by a source (usually a TX antenna)on and around a scattering structure. Part of this scattering structure are the walls, thestirrers, the TX antenna itself, the RX antenna, and an EUT. The resulting fields aredefined by Maxwell’s equations which are given in differential form by (2.1)–(2.4). Theseequations are further simplified by assuming that there are no free charges inside thecomputational volume V , i.e. = 0 in (2.3). In addition, the properties of the utilizedmaterials are assumed to be linear, homogeneous, isotropic, and without memory. Thisled to the simpler equations (2.8)–(2.11), which are used in the following sections.


3.2.1 Incident and scattered field

If EM waves of electric field strength Ei and magnetic field strength Hi are incidentonto a scattering structure, electric currents I on metallic wires and surface currents Jalong metallic surfaces will be excited. These currents are in return responsible for theresulting scattered fields Es and Hs. The total field consists of the superposition of theincident and the scattered field

Etot(r) = Ei(r) + Es(r) (3.1)Htot(r) = Hi(r) + Hs(r) (3.2)

The scattered field Es, Hs can be calculated from

Es(r) = LEJ

J(r, r ′)

+ L

EI

I(r, r ′)

(3.3)

Hs(r) = LHJ

J(r, r ′)

+ L

HI

I(r, r ′)

(3.4)

where LEJ

, LEI , L

HJ

, and LHI are linear integro-differential operators relating to the line

and surface currents I and J [45, 46]. For the sake of simplicity, (3.3) and (3.4) donot take into account electric volume currents (there is no current flowing within thecomputational volume V , that is J = 0, but currents JS flow on the surface ∂V = A)as well as magnetic surface and line currents, which can be used to model dielectricmaterials. An extension of (3.3) and (3.4) to dielectric materials including these currentscan be found in e.g. [47, 48].

3.2.2 Integral equation approach

The operators used in (3.3) and (3.4) are defined by

LEJ

J(r, r ′)

= − j

4πεω∇∫∫A′

(∇′

A · J(r ′))· G(r, r ′) dA′

−jωµ

4π

∫∫A′

J(r ′) · G(r, r ′) dA′ (3.5)

LHJ

J(r, r ′)

=

14π

∇×∫∫A′

J(r ′) · G(r, r ′) dA′ (3.6)

LEI

I(r, r ′)

= − j

4πεω∇∫L′

∂I(r ′)∂l′

· G(r, r ′) dl′

−jωµ

4π

∫L′

I(r ′) · el′ · G(r, r ′) dl′ (3.7)

LHI

I(r, r ′)

=

14π

∇×∫L′

I(r ′) · el′ · G(r, r ′) dl′ (3.8)


where the unit vector el′ denotes the direction of the current flow I along a wire segment.The vector r ′ denotes the location of the observation point. The surface divergence ofthe current density over the primed coordinate system is given by

∇′A · J(r ′) = ∇ · J(r ′) − en · ∂ J(r ′)

∂n(3.9)

with the normal unit vector en and the partial derivative ∂ J(r ′)∂n taken normal to the

surface. G(r, r ′) is the free space Green’s function

G(r, r ′) =e−jk|r−r ′|

|r − r ′| (3.10)

It is important to note that for the computation of the scattered fields using (3.3) and(3.4) the evaluation of the integrals in (3.5)-(3.8) require significant computational re-sources. Special formulations have been developed rendering these computations morefeasible [48, 42, 45].

3.2.3 Solution of integral equations

In order to solve (3.5)-(3.8), the surface current density J(r ′) on the surface dA′ and theline current I(r ′) on the wire of length dl′ must be known. A direct solution of theseequations is however not possible in most practical cases – this includes the simulationof RCs – as J(r ′) and I(r ′) are either not known at all or very complicated over a largearea dA′ and a long length dl′. The solution to this problem is to divide the computa-tional domain into small (compared to the operating wavelength λ) parts, wherein thecurrent density J(r ′) and the line current I(r ′) can be described by rather simple ap-proximations. An expansion of the current distribution within these small parts allowsthe computation of (3.5)-(3.8) and hence the calculation of the electric and magneticfield (3.1) and (3.2) from (3.3) and (3.4).

3.2.4 Approximation of currents and current density

The currents and the current density on a scatterer structure are approximated in theform of a series of a-priori known basis (sometimes also called expansion) functions. Ingeneral, one chooses as basis functions a set that has the ability to accurately representand resemble the anticipated unknown current density, while minimizing at the sametime the computational effort required to employ it. The sets of basis functions may bedivided into two general classes: the subdomain basis functions, which are nonzero onlyover a part of the structure (with a subdomain being either a small part of a larger wirestructure or a metallic surface); and the entire domain basis functions that exist over theentire structure to be simulated. For the latter basis functions, there is no discretizationof the structure under consideration involved, but there is a priori knowledge of theanticipated current distribution to be modeled required (which is e.g. feasible for thecomputation of the current distribution on a dipole, but not for more complex structuressuch as an RC). Of these two classes, subdomain functions are the most common [49].


Unlike entire domain functions, they may be used without prior knowledge of the natureof the current density function they must approximate. The subdomains are createdby discretizing the overall geometric structure: wires are subdivided into smaller parts(“segments” of length dl′), surfaces into smaller elements (“patches” of area dA′) suchas triangles or quadrangles. It is obvious that more sophisticated basis functions canapproximate arbitrary current distributions more accurately, resulting in a smootherrepresentation – this however comes at the cost of increased computational complexity.

Basis functions for wire structuresAn example for a set of subdomain basis functions g used to approximate the line currentIn on wire segments is given by

gIn(r ′) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1l+n

· |r ′ − a+2,n| for r ′ ∈ S+

n

1l−n

· |r ′ − a−2,n| for r ′ ∈ S−

n

0 for r ′ ∈ S+n , S−

n

(3.11)

wherein l+n , l−n are the lengths of the segments S+n , S−

n . a+2,n and a−

2,n refer to theend points of the two segments S+

n , S−n which are connected at the node a1,n. Since

gIn(r ′) is defined only on the center axis of the segments, (3.11) resembles the “thin wireapproximation”, assuming that the segments are infinitely thin; for “thicker” segmentsa more accurate formulation can be used [48].Using a superposition of the basis functions for wire segments (3.11), the actual currentin a segment will be expanded by

I(r ′) =NI∑n=1

In · gIn(r ′) (3.12)

where NI is the number of basis functions needed to cover the wire segments. With(3.11) and (3.12) the current in the wire segments is approximated in a piecewise-linearcontinuous manner. The currents In are the unknown coefficients which need to becomputed.

Basis functions for metallic surfacesIt is possible to model metallic surfaces through tightly spaced wire segments by creatinga so-called “wire grid”. This method is used e.g. in the freely available version of the“Numerical Electromagnetics Code (NEC)” [50]. Since the accuracy of this approach isnot very satisfying, vectorial basis functions for an approximation of the current densityJS on surface patches were introduced. As patch elements, often triangles are used sincethey have the ability to conform to any geometrical surface. Due to the derivatives andthe kernel in the integral equations of the EFIE, there are complications in using vec-torial basis functions for patches: In order to achieve physically consisting results, basisfunctions must be constructed such that the normal component of the current is con-tinuous across surface boundaries. The vectorial basis functions, which eliminate theseproblems (and are used by FEKO ), are the so-called “Rao-Wilson-Glisson (RWG)” ba-sis functions [51]. The RWG basis functions are not simply coincident with each triangle


face, but consist of pairs of triangular faces along an adjacent edge, similar to “roof top”functions. RWG functions are defined as

g Jn(r ′) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ln2A+

nr+

n = ln2A+

n· (r ′ − a+

3,n

)for r ′ ∈ T +

n

ln2A−

nr−n = − ln

2A+n· (r ′ − a−

3,n

)for r ′ ∈ T−

n

0 for r ′ ∈ T +n , T−

n

(3.13)

and are associated to the n-th edge that two adjacent triangles have in common. ln isthe length of the n-th “inner” edge, A+

n , A−n resemble the areas of triangles T +

n , T−n , and

r+n , r−n are the position vectors with respect to the vertices opposite from the n-th inner

edge [48]. The position vector r+n is directed from the free vertex of T +

n towards points inT +

n , whereas r−n is directed towards T−n . a+

3,n and a−3,n denote the position vectors of the

two free triangle vertices which are not part of the inner edge. These triangle-pair basisfunctions are free from fictitious line or point charges at their subdomain boundaries [45].They overlap, so that, except for boundary edges, each edge of the triangulated surfaceis a common edge between the two triangular faces of a subdomain. Hence, up to threebasis functions will be superimposed within each face of the triangulated surface, allowinga constant vector of arbitrary magnitude and direction to be synthesized on each face.Using a superposition of the basis functions for triangles (3.13), the actual surface currenton a triangle will be expanded by

J(r ′) =N J∑n=1

Jn · g Jn(r ′) (3.14)

where N J is the number of basis functions needed to cover the wire segments. With(3.13) and (3.14) the current density in the triangular patches is approximated. Thecurrent densities Jn are the unknown coefficients which need to be computed.Special basis functions are used to model the transitions between wire segments andtriangular surface elements [48]. Considerable advantages in computation time and re-duction of approximation errors can be gained by a careful choice of the basis functions.

3.2.5 Computation of line and surface current coefficients

In order to approximate the actual line currents I(r ′) and surface currents J(r ′), the lineand surface current coefficients In and Jn of (3.12) and (3.14) must be computed. Thesecoefficients can be determined by considering the boundary conditions for the EM fieldobtained from Maxwell’s equations (2.1)-(2.4) which can be derived for the tangentialand normal components of the electric and the magnetic field, respectively, as shown in(2.16)–(2.19).Two methods are common to calculate the line and surface current coefficients In andJn with the boundary conditions for the tangential field components (2.16) and (2.17):The so-called electric field integral equation (EFIE) enforces the boundary conditionson the tangential electric field (2.16) while the magnetic field integral equation (MFIE)enforces the boundary conditions on the tangential components of the magnetic field


(2.17). Also a combination of the EFIE and magnetic field integral equation (MFIE) isused and known as the coupled field integral equation (CFIE). The CFIE uses both theboundary conditions for the electric (2.16) and the magnetic (2.17) field.For the example of a PEC surface, (2.16) simplifies to

n12 × Etot = 0 (3.15)

For non-PEC structures, (3.15) needs to be reformulated so that the boundary conditionsimposed by (2.16) are satisfied. Substituting (3.3) into (3.1), (3.15) can be rewritten as

n12 ×(

Ei(r) + LEJ

J(r, r ′)

+ L

EI

I(r, r ′)

)= 0 (3.16)

Inserting the approximations for the surface current density (3.14) and the line currents(3.12) in (3.16) yields

n12 ×⎛⎝ Ei(r) + L

EJ

N J∑n=1

Jn · g Jn(r ′)

+ L

EI

NI∑

n=1

In · gIn(r ′)

⎞⎠ = 0 (3.17)

Rearranging (3.17) so that the known incident field Ei(r) is on the righthand side resultsin

n12 ×⎛⎝L

EJ

N J∑n=1

Jn · g Jn(r ′)

+ L

EI

NI∑n=1

In · gIn(r ′)

⎞⎠ = −n12 × Ei(r) (3.18)

3.3 Method of Moments

The EFIE (3.18) (and also the MFIE) are effectively solved using the method-of-moments(MoM), which is done for the EFIE in the computational kernel of the simulation softwareFEKO . It has to be outlined that the actual EM solution method as described inSection 3.2.5 is commonly incorrectly referred to as “the MoM”. The MoM is merelyused as a numerical technique for the solution of the EFIE (and respectively MFIE)describing the EM problem [42, 52]. With (3.18) it is in principle possible to findthe unknown constant line and surface current coefficients In and Jn. Since however(3.18) represents only one equation, it is alone not sufficient to determine the N J + NI

unknowns Jn and In. To solve for the N J +NI unknowns it is necessary to have N J +NI

linearly independent equations. The MoM transforms the EFIE (3.18) containing linearoperators into a system of linear equations. It owes its name to the process of takingmoments by multiplying (3.18) with appropriate weighting functions and integrating [53,46, 45].

3.3.1 Point matching and weighting functions

Having N J +NI linearly independent equations can be accomplished by evaluating (3.18)– i.e. applying the boundary conditions – at N J + NI different locations on surface

3.3 METHOD OF MOMENTS 45

patches and wire segments. This is usually referred to as the collocation method orpoint matching. Applying point matching eventually means that boundary conditions(2.16) and (2.17) are satisfied only at discrete points on a given structure. Between thesepoints the boundary conditions may not be satisfied, which implies that, for example,the tangential electric field will be nonzero on a PEC structure and a residual will be left.To minimize the residual in such a way that its overall average over the entire geometricstructure approaches zero, the method of weighted residuals is utilized in conjunctionwith a so-called inner scalar product

< . . . , w νm >=

∫∫∂V =A′

. . . · w νm dA′ (3.19)

for the surface current density and

< . . . , w νm >=

∫∂A=L′

. . . · w νm dl′ (3.20)

for the line currents [48]. w νm are suitable weighting functions, which are defined on the

N J surfaces and NI line segments. This technique does not lead to a vanishing residual atevery point on a PEC surface, but it forces the boundary conditions (2.16) and (2.17) tobe satisfied in an average sense over the entire surface. The choice of weighting functionsis important in that the elements of w ν

m must be linearly independent, so that the N J +NI

equations in (3.19) and (3.20) will be linearly independent, too. Furthermore, it will beadvantageous to choose weighting functions that minimize the computations requiredto evaluate the inner products. Since the linear independence between elements andthe computational efficiency are also important requirements of basis functions, similartypes of functions are often used for both weighting and basis functions.A particular choice of functions in the literature commonly referred to as “Galerkin’smethod” is to let the basis and the weighting function be the same. Other choices for w ν

m

include Dirac δ-functions, which reduce the “average” boundary condition matching backto point-wise matching. FEKO uses a computationally advantageous Quasi-Galerkinapproach with an adaptive integration technique, which distinguishes between spatiallynear and far boundary condition matching points [46].

3.3.2 Matrix formulation

Evaluating (3.18) at N J + NI different points and satisfying the boundary conditions(2.16) and (2.17) “on average” by the application of the weighted residuals (3.19) and(3.20) leads to N J + NI linear equations, which can be expressed in the commonly usedMoM matrix formulation

Z · I = V (3.21)

where I is an N = (N J + NI)-column vector which contains the expansion coefficientsIn and Jn of the MoM basis functions. Z is the N × N system matrix, containing thecoupling between basis and weighting functions; V is an N -column vector relating to


the impressed fields originating from sources such as an incident wave radiated by a TXantenna. Solving (3.21) for the vector I

I = Z−1 · V (3.22)

provides the unknown expansion coefficients In and Jn of the current approximation(3.12) and (3.14), thus giving the resulting surface current density and line currents.With (3.1) and (3.2) the total near fields can be straightforwardly calculated from thiscurrent approximation. In order to calculate the inverse of the system matrix Z−1

it is necessary to evaluate N2 terms in (3.22), with each term requiring two or moreintegrations. When these integrations are to be done numerically, depending on the sizeof Z, vast amounts of computation time and memory may be necessary (see Section 3.4).

3.3.3 Symmetry considerations

In order to reduce the calculation time and memory usage, symmetries can be utilized.Up to three coordinate planes may be defined as planes of symmetry. For each plane,there are three different types of symmetry that can be utilized.

Geometric SymmetryIf a complete structure or parts of a structure exhibit geometric symmetry in one ormore planes, the time for the computation of the elements of the system matrix Z can bereduced. The source, however, is not symmetric, thus a symmetric current distributiondoes not exist. This asymmetric current distribution leads to asymmetric electric andmagnetic fields. The order of the system matrix therefore remains the same, whichmeans that the computation time or the amount of memory needed for the solution ofthe equation system cannot be reduced [13].

Electric or Magnetic SymmetryIf – in addition to symmetry of the geometric structure – also the excitation is symmet-ric with respect to one or more planes, then these planes can be considered as beingplanes of electric (i.e. perfectly electrically conducting (PEC)) or magnetic (i.e. perfectlymagnetically conducting) symmetry. An electric (magnetic) symmetry plane is a planewhich can be replaced by an ideal electrically (magnetically) conducting wall withoutchanging the field distribution. For this situation, the corresponding coefficients of thecurrent basis functions (see Section 3.2.4) have either equal or equally negative values.This in turn means that the number of equations (i.e. the order of the system matrix)can be reduced, which results in a reduction of computation time and amount of mem-ory [46]. Each electric or magnetic symmetry reduces the number of unknowns in thesystem matrix by a factor of two, that is the required amount of memory by a factor offour.

Partial SymmetryNormally, a structure must be perfectly symmetric for FEKO in order to speed up thesolution by exploiting the symmetries mentioned above. Any asymmetric segments ortriangles, or ones that lie in a symmetry plane or on the axis of rotation, will destroy thesymmetry. If a geometry is not fully symmetric, one approach is to use the numerical

3.4 COMPUTATIONAL REQUIREMENTS 47

Green’s function (NGF) technique (see Section 3.5.2) to exploit at least partial symmetryin a structure. Partial symmetry can be exploited to reduce solution time by running asimulation of the symmetric part of the model first and saving this data. The asymmetricparts may then be added in a second run using the NGF technique.

3.3.4 Modeling of finite conductivity

The finite conductivity κ of materials can be taken into account in the EFIE by using(2.16) and (3.15) in a modified form so that

n12 × Etot = EZ (3.23)

at the boundary to a non-PEC structure. Finitely conducting surface materials areassumed to have a thickness d, permeability µ, and conductivity κ (ωε κ < ∞). Theyare modeled by a fictitious surface impedance [42]

ZS =1 − j

2κδs

1

tan(

(1 − j) d2δs

) (3.24)

where f denotes the simulation frequency and δs the skin depth of the metal as definedby (2.50). With (3.24) and (2.50), the tangential field on the non-PEC scatterer EZ

can be calculated. For finitely conducting wires, a similar surface impedance can becalculated as outlined in [48, 42, 45].

3.4 Computational requirements

The computational requirements needed for the simulation of RCs are significant athigher frequencies, both with respect to the main memory needed and the solution time.It is therefore useful to estimate up-front the memory and CPU-time it takes to completeone simulation run, which usually corresponds to one rotational stirrer position.

3.4.1 Simulation memory

The FEKO implementation of the EFIE (3.18) and direct solution using the MoM leadsto the following memory requirements for the storage of the system matrix Z:

• One basis function needs one variable of type “double complex” for storage, i.e.16 Byte [54].

• The surface current distribution on a single metallic triangle (3.14) is described withthree basis functions (3.13), provided that all edges of the triangle are connectedto other triangles [55].

• Dielectric structures need twice as many basis functions (to model electric andmagnetic currents) as metallic triangles.


• At the outer parts of a structure where there are triangles only connected on lessthan three edges to other triangles, one or two basis functions are used (dependingon the actual structure) [55].

• Each basis function (3.13) extends over two triangles.

• As discussed in Section 3.3.3, each electric or magnetic symmetry reduces the num-ber of unknowns in the system matrix by a factor of two; hence the required storagememory for the system matrix Z is reduced by a factor of four per electric/magneticsymmetry.

• One segment node which connects NS segments needs NS−1 basis functions (e.g. aloop wire without open ends and no other wires attached needs NS basis functions).

A useful estimate of the required memory M needed for storage of the MoM system ma-trix Z for NT triangles (no free triangle edges, no dielectrics involved) and NS segments(only nodes with exactly two segments attached) is therefore given by

M ≈ 16 Byte ·[(

32NT

)2

+ N2S

]=(36 N2

T + 16 N2S

)Byte (3.25)

Discretization with the CAD-software HyperMesh of the surface area A into trianglesof area A can be represented by

A = NT · A (3.26)

The “chordal deviation” discretization algorithm in HyperMesh is controlled via theparameters

• minimum and maximum element edge length dmin and dmax

• maximum chordal deviation

• maximum angle allowed between two adjacent elements

and allows a precise adjustment of the mesh generation. In the somewhat easier to handle“size and biasing” algorithm all of the parameters outlined above are combined into asingle user-adjustable parameter “element size” d [56]. The surface size of a triangle Acreated with HyperMesh can be approximated to

A ≈ 12d2 sin γ (3.27)

where γ accounts for the apex angle of the triangle. d is directly related to the dis-cretization requirements of the EFIE-MoM as implemented in FEKO . Values for dused throughout the RC simulations are

d =λ

5 . . . 10(3.28)


0

100

200

300

400

500

600

700

800

900

1000

50 100 150 200 250 300f [MHz]

Memory [GByte]M

Figure 3.1: Theoretical memory requirements of an RC simulation with A = 50 m2 of discretizedsurface and NS = 870 segments for four different triangular mesh resolutions A (3.25) inthe f = 50 . . . 300 MHz frequency range.

with λ denoting the free-space wavelength. This finally results in an estimate for thenumber of triangles

NT ≈ 2A

sin γ

(5 . . . 10 · f

c

)2

(3.29)

needed for a certain simulation frequency f (c being the speed of light). From (3.29)it is obvious, that the number of triangles NT exhibits a quadratic proportionality tothe frequency, i.e. NT ∼ f2. Using a similar derivation, it can be easily shown that thenumber of segments NS is directly proportional to the frequency, i.e. NS ∼ f [48].With (3.25) this means that the memory M needed for storage of the MoM systemmatrix Z depends on the frequency f as

M ∼ O(f4)

+ O(f2)

(3.30)

Memory requirements obtained in actual RC simulations with FEKO are listed inTable 3.1 and agree well with the approximate estimates of (3.25) and (3.30). It canbe clearly seen that with the computational approach presented in Section 3.2, a higher


100

10

1

0.1

0.01100 200 300 400 500 600 700 800 900 1000

f [MHz]

Memory [GByte]M

Figure 3.2: Theoretical memory requirements of an RC simulation with A = 50m2 of discretizedsurface and NS = 870 segments for four different triangular mesh resolutions A (3.25) inthe f = 50 . . . 1000 MHz frequency range.

operating frequency requires a finer discretization of the geometric structure, whichresults in a greater number of triangles NT and segments NS. Figs. 3.1 and 3.2 showthe memory requirements of an RC simulation with A = 50 m2 of discretized surface(corresponding to the prototype RC) and NS = 870 segments (two logper antennas)for four different triangular mesh resolutions (3.25) of A = λ2/30, λ2/50, λ2/70, andλ2/100. Whereas the data of Fig. 3.2 covers the f = 50 . . .1000 MHz frequency range(logarithmically scaled), Fig. 3.1 shows a zoomed section (linear scale) in the lowerf = 50 . . .300 MHz frequency range.

3.4.2 Simulation time

Before the inversion of the system matrix Z−1 is carried out, initially the matrix elementsmust be computed as described in Section 3.2. According to [46], the time tsetup for thedetermination of the system matrix elements can be estimated for segments as tsetup ∼ f2

and for metallic triangles as tsetup ∼ f4. The total determination of the system matrix


Frequency f[MHz]

Number oftriangles NT

MemoryM[MByte]

CPU runtime t[s (hh:mm)]

100 2 934 a 310 2 299 (00:38)

300 5 828 a 1 195 7 630 (02:07)

500 11 276 b 4 563 37 161 (10:19) c

700 13 486 b 6 481 93 679 (26:01) c

1000 19 112 b 12 980 655 311 (182:02) c

a in addition to the triangles, for the discretization of the biconical TX/RX antenna 162 seg-ments were usedb —— of the logper TX/RX antenna 435 segments were usedc for parallel computations the individual per-process run times were summed up

Table 3.1: Comparison of runtime and memory requirements of RC simulations for a singlefrequency and one stirrer position. A typical RC consists of walls, door, one 6-paddle verticalstirrer and one antenna. Simulations were run using the standard FEKO solver withoutsymmetries or approximative methods.

elements is therefore on the order of

tsetup ∼ O(f4)

+ O(f2)

(3.31)

With the implementation of a direct solver (e.g. LU-decomposition), the time neededfor the solution of the system of linear equations tsolve is for segments on the order oftsolve ∼ f2...3 and for metallic triangles tsolve ∼ f4...6. Under the assumption that thetime for initialization, checking of geometry, near and far field computation are negligible,the total simulation time can be estimated as

ttot = tsolve + tsetup ∼ O(f4...6

)+ O

(f4)

(3.32)

For small EM problems, at low frequencies, and depending on the numerical solver,tsetup can be a significant part of the total simulation time ttot (tsetup ≈ tsolve). In ahigh-frequency RC simulation with a fine discretization however, the matrix equationsolution time tsolve clearly dominates over the setup period tsetup. Computation timesobtained in actual RC simulations with FEKO are listed in Table 3.1: Due to the strongfrequency dependence of ttot (3.32), the rate at which the CPU runtime increases withgrowing frequencies is even more dramatic than the increase in required memory (3.30).The strong frequency dependence of both computational memory and solution time em-phasizes the importance to use as few triangles NT and segments NS as possible withinthe numerical restrictions of the MoM for the discretization. Therefore in order to mini-mize the number of triangles needed to mesh the large RC surface, a frequency-adaptive


and geometry-dependent discretization was used. Table 3.1 states also the typically re-quired numbers of triangles at different frequencies and a fixed stirrer position using astandard MoM approach without symmetries (Section 3.3.3) or approximative methods(Section 3.5.2). Simulations were performed on a distributed Sun Blade 1000/2000workstation cluster with up to three double-processor machines in parallel. Lookingat Table 3.1 it is clear that the excessive runtimes render RC simulations with today’savailable computer power and current numerical methods at high frequencies useless.

3.5 Extensions to the method of moments

3.5.1 Field integral equation resonance problem

A well-known problem with simple, standard EFIE-based MoM formulations is that inthe interior resonance case the system of equations can be “ill-conditioned” (which resultsin a very high condition number of Z) [57]. Practically however, this problem only showsup if the simulated structure is completely closed and in addition PEC. Furthermore, itis not of great relevance as long as direct solution methods involving a lower upper (LU)decomposition of the system matrix Z are used [58].A remedy to this problem might seem to employ the magnetic field integral equation(MFIE) or the coupled field integral equation (CFIE) which combines the EFIE and theMFIE. The MFIE, however, and hence also the CFIE, can only be used for regions thatare mathematically “simply connected” (“1-connected”) [48]. A simple cavity wouldsatisfy this requirement, whereas an RC with stirrers, antennas, and EUT certainly doesnot. In addition, MFIE solutions tend to diverge if applied to structures involving thinmetallic sheets [49]. Therefore integral equation methods based on MFIE or CFIE arenot applicable to the solution of most “real world” RC problems.Instead of using the MFIE or CFIE, EFIE-related problems can be eliminated (for mostpractical cases) through the usage of materials of finite conductivity κ < ∞ together withmodified forms of the EFIE which are still valid in the interior resonance case [59, 44]. Inaddition it is good practice to monitor the condition number of the system matrix andto test explicitly whether this matrix is singular. Both of these precautionary measuresare implemented in the FEKO -kernel, which warns the user if the matrix is singular orthe condition number reaches 1016 [54].

3.5.2 Iterative solution techniques

In the traditional subdomain-based MoM, it normally occurs that the resulting systemof equations (3.21) is described with a dense (i.e. full) matrix Z having complex-valuedelements . The matrix structure is usually optimized for specific numerical solution tech-niques by a preconditioner. Solving systems with full matrices is – as mentioned in Sec-tions 3.4.1 and 3.4.2 – computationally expensive, often to the point of being prohibitive.A remedy to this problem can be found in iterative methods or hybrid techniques com-bining the MoM with asymptotic methods such as the physical optics (PO) (current-/and ray-based) or the fast multipole method (FMM) mentioned in Sections 3.5.3 and3.5.4.

3.5 EXTENSIONS TO THE METHOD OF MOMENTS 53

Standard iterative schemesReducing the number of mathematical operations can be accomplished by using iterativeschemes such as

• stationary iterative methods: Jacobi, Gauss-Seidel, Successive-over-relaxation

• non-stationary iterative methods: Conjugate-gradient-on-normal-equations, Bi-conjugate gradient stabilized (BiCGstab) [60] or the transpose-free quasi minimumresidual (TFQMR) method [61]

which need fewer operations for the decomposition of the MoM system matrix Z. Sincethere were problems with the convergence of these numerical methods when applied tothe simulation of an RC, only direct solvers involving the LU decomposition were used.

Numerical Green’s function (NGF)

In RC simulations, modifications to small parts of the geometry (such as the rotationof a stirrer) result in changes to one or more small areas of the overall system matrixZ (3.21), leaving relatively large portions of Z unchanged [62]. Therefore it would behelpful to use techniques which avoid the repeated inversions of the original large, densesystem matrix. A solution to this problem is presented by [63] and [64] on the basisof the so-called “Sherman-Morrison-Woodbury expansion”. This approach is useful forconfigurations consisting of one or more large and one or more small objects

• where all of them do not change in size, shape, or electrical characteristics. How-ever, the small object(s) may move or change its orientation with respect to thelarge object(s).

• same as above, but where the small object(s) may change in size, shape, electricalcharacteristics, position, and/or orientation.

• where one of the objects is much larger and unchanging, and the other smallerobject is changing geometrically or electrically.

Obviously the RC fits well in these categories of problems, since the greater part of thegeometry (i.e. the cavity walls) do not change at all, whereas a small part of the geometry(i.e. the stirrers) change in orientation. This method is also known as “numerical Green’sfunction (NGF)”. The main purpose of the NGF is to avoid the unnecessary repetitionof calculations when a part of a model, such as e.g. the rotational position of a stirrer ina complex RC environment, will be modified one or more times while the remaining RCenvironment remains fixed. With the NGF, the so-called large “self-interaction matrix”for the fixed RC environment may be computed, factored for solution, and saved [62, 64].Solution for a new stirrer position then requires only the evaluation of the much smallerself-interaction matrix for the stirrer, the mutual stirrer-to-RC-environment interactions,and matrix manipulations for a partitioned-matrix solution. The NGF has not beenimplemented in FEKO due to considerable incompatibility issues with parallelizationand the out-of-core solver.


3.5.3 MoM and physical optics (PO) hybridization

In addition to the iterative techniques (Section 3.5.2), two other methods were consideredto speed up the RC simulations at higher frequencies: the MoM/physical optics (PO)hybridization [48] and the combination of MoM with the fast multipole method (FMM).The general requirements for the PO approach to compute the EM fields resulting froman excitation and a scatterer are [65]

• dimensions and radii of curvature of the illuminated object (i.e. the scatterer) mustbe large as compared to the wavelength.

• source point (i.e. excitation) is sufficiently far away (typically d > λ) so that anincident plane wave can be assumed locally.

• metallic region must be perfectly conducting, κ → ∞ (this restriction applies tothe PO implementation in FEKO [54]).

For complex geometric structures such as an RC, a large number of secondary reflectionsneeds to be taken into account. In order to accurately model the effects of secondaryreflections (e.g. for dihedrals at least two reflections and for trihedrals at least threereflections must be considered) it is essential to use the multiple reflection PO. Thereare however three problems associated with the PO applied to RCs:

• the ray tracing needed for the PO multiple reflection approach slows down thecomputation considerably.

• the PO is computationally very efficient only as long as PO and MoM regions aredecoupled – i.e. the MoM currents are the source for the currents in the PO region,but there is no effect of the PO region currents on the MoM currents. Due to thepronounced effects of the stirrer and the TX/RX antennas on the field structureinside the RC, decoupling was not feasible in RC simulations [66].

• the RC cannot be modeled with infinite conductivity κ → ∞; otherwise the RCquality factor bandwidth ∆fQ [6] would be zero, i.e. at a given frequency f only asingle mode (and possibly respective degenerate modes) could be excited [67].

3.5.4 MoM and Fast Multipole Method (FMM) hybridization

The FMM is a numerical method to compute efficiently convolution integrals and wasfirst used in fluid dynamics. Its derivative to Maxwell and Helmholtz equations wasinitiated by V. Rokhlin [68]. The FMM is highly beneficial from around 25 000 trian-gles on – where computation with a standard full-wave MoM approach is not feasiblew.r.t. computation time and memory requirements. Using the FMM, even structuresdiscretized with 500 000 triangles can be computed on a standard personal computer(2 GB RAM, 3 GHz processor) [54].

Single level FMMThe basic idea behind the FMM is to split the MoM system matrix Z into two partsZ = Znear + Zfar which describe separately near- and far-field interaction between seg-ments and triangles. Initially only the sparse matrix Znear is stored. The original MoM

3.6 CONCLUSION 55

scheme is accelerated by using the multipole transformations Zfar = LTG, wherein G(aggregation) transforms the basis functions into a group center with global multipole ex-pansion, T (translation) accumulates the global multipole expansions in a local one, andL (disaggregation) evaluates the local multipole expansions at the observation element.Then the whole system of equations can be solved using iterative solution techniquesmentioned in Section 3.5.2. This approach is known as the single level FMM.

Multilevel FMMThe multilevel FMM extends the idea of the single level FMM. Initially the currents inone region are grouped to act as a multipole, then several multipole regions are bundledtogether to act as another multipole and so forth – the whole EM problem is essentiallysubdivided into smaller “cells”; in each “cell” the above mentioned Zfar = LTG trans-formation is carried out. This hierarchical procedure leads to the name “multilevel”FMM [68, 69]. Using the multilevel fast multipole method (MLFMM) approach, thememory requirements for electromagnetically large problems with N unknowns can bedrastically reduced to O(N log N) and the computation time only grows as O(N [log N ]2),provided the size of the matrix Zfar is much smaller than that of the matrix Znear.The problem with an application of the FMM to RC simulations is that for lower-mediumfrequencies (large-medium wavelengths), the matrix Znear is much greater than the ma-trix Zfar. Consequently, the FMM does not exhibit a computational advantage overthe MoM at these frequencies. The frequencies at which the matrix Znear turns out tobe much smaller than Zfar are so high that an RC simulation is not reasonable due tovalidation problems (cf. Section 7.4).

3.6 Conclusion

It was outlined that a numerical method suitable for reverberation chamber (RC) sim-ulations must be able to compute the electromagnetic (EM) field over a broad range offrequencies, handle a large, varying, and irregular geometry, and model finite conduc-tivity. In addition, it must be able to cope with a highly resonant structure and shouldallow the field calculation at a large number of spatial field points without introducingtoo much computational overhead. In order to deliver broadband simulation data itwas shown that frequency-domain methods will need to compute the field at a lot ofdiscrete frequencies. Time-domain methods are at an advantage in this regard, since abroadband frequency response can be calculated from only a single run. Methods usingunstructured grids (triangles, tetrahedra) are advantageous over structured grids (quad-rangles, cubes) when it comes to modeling an irregular geometry.After contrasting these requirements against advantages and disadvantages of severalstate-of-the art methods, an electric field integral equation (EFIE)-based method-of-moments (MoM) technique as implemented in the commercial field solver FEKO waschosen. The MoM uses unstructured discretization elements and works in the frequency-domain. The basic concept of integral equations was outlined and the MoM solutionmethodology introduced. Basis functions for wires and metallic surfaces used for the ap-proximation of the currents in the RC simulations were described. It was explained howthe boundary conditions are satisfied in an average sense along a discretized structure


through the usage of point matching combined with weighting functions. Computationalrequirements regarding simulation time and memory were estimated; it was shown thatthe MoM is computationally very expensive for high frequencies, often to the point ofbeing prohibitive – the simulation of the prototype RC at a frequency of e.g. 1 GHzrequired more than 12 GByte of main memory and significant time expenditure. Theapplication of several types of symmetries was discussed and an outlook on MoM exten-sions and solver techniques (such as the NGF, PO and MLFMM) to reduce computationtime and memory was given. It was concluded that due to the geometric structure ofthe RC prototype built for this thesis, symmetries and the PO could not be used. Theapplication of the MLFMM was found not to be appropriate as the interaction betweentriangles (or segments) on one part of the RC and triangles (or segments) on another,distant part is predominantly “near field” as opposed to “far field”. The NGF has beensuccessfully used for a computationally efficient simulation of RCs by other authors, butwas not implemented in the MoM solver package utilized for this thesis.

4 Literature Overview

Abstract— This chapter provides an overview on parts of the reverberation chamber literature ref-

erenced in this thesis. Special consideration is given to publications dealing with simulations of

reverberation chambers. This overview is by no means a fully exhaustive compilation, but cites the

publications which were found to be most relevant in the context of this thesis. The state of the art

regarding reverberation chamber simulations at the beginning of this work, during its process, and

at the end of this thesis is summarized. It serves as a benchmark to value the achievements, but also

illuminates some of the questions which remain open and are subject to further research. In addition,

a short comparison between reverberation chamber and anechoic chamber EMC testing is presented.

4.1 Historic reverberation chamber publications and patents

One of the earliest documents dealing with the fundamental question on how to “dis-tribute energy evenly” inside a metallic cavity is a patent which dates back to 1947 [71].This patent was assigned to W. M. Hall of Raytheon Co. and is titled “Heating Appa-ratus”. It already described some of the problems that are still among the main issuesaddressed in today’s RC research and development:

• “. . . apparatus for producing a substantially uniform integrated radio-frequencyheat pattern. . . ” [71] (RC problem: field uniformity)

• “. . . cooking large volumes of food with the expenditure of a minimum amountof input power . . . ” [71] (RC problem: large volume of uniform field, high fieldstrength with modest input power)

• “. . . means for producing periodic changes in the field distribution in a radio-frequency cavity, whereby the integrated heating effect of the field is made sub-stantially uniform [71]” (RC problem: need for a device which is able to changethe field distribution efficiently and to provide a volume of uniform field)

• “. . . in which food masses, whose linear dimensions are large compared to the wave-length of the microwave energy used, may be cooked in a substantially uniformmanner [71]” (RC problem: EUT dimensions larger than its operating wavelength)

• “. . . to accomplish the above objects in a simple yet effective manner [71]”

In the following 20 years, only papers and patents dealing exclusively with “food heat-ing” (i.e. microwave cooking) were published – using such a device to test for EMCsimply did not seem to be an issue, because EMC itself was not as important as today.It was not until the 1970s that interest in the application of “large, oversized cavities”for EMC measurements started to show up.

57

58 4 LITERATURE OVERVIEW

Crawford is the author of a 1972 technical report from the National Bureau of Stan-dards (NBS), Boulder (CO), U.S. (now called National Institute of Standards and Tech-nology (NIST)) which covers “Electromagnetic Field Measurements in Low Q Enclo-sures” [72]. In 1976, Corona, Latmiral, Paolini, and Piccioli of the University of Naples,Italy, published one of the earliest papers on the “Use of a reverberating enclosure formeasurement of radiated power in the microwave range” [73].John and Hall are among the first to describe a practical RC testing setup in their paperdealing with “Electromagnetic susceptibility measurements using a mode-stirred cham-ber” from 1978 [74]. This paper is followed in 1980 by another publication of Corona etal. dealing with the “Performance and analysis of a reverberating enclosure with variablegeometry” [75].A historical overview on some parts of the above mentioned RC research is given in apaper published in 2002 by Corona and Ladbury on the occasion of M. Kanda’s obitu-ary [76].

4.2 Reverberation chamber standards

At the start of this thesis in December 2000, the general RC standard IEC 61000-4-21 was still in the stage of “Committee Draft for Vote (CDV)” [6]. The CDV stageimplies that both technical and editorial revisions can still be made to the preliminarystandard document as requested by organizations and countries participating in thestandardization process. In January 2001, the IEC 61000-4-21 passed the last objectionsfrom the International Electrotechnical Commission (IEC) member states and enteredthe “Final Draft for International Standard (FDIS)” phase. In the FDIS stage, finaleditorial changes to the standard are done – changes regarding the technical content canonly be performed in a major revision at a later point of time once the standard waspublished. After few editorial changes and voting the final standard was published inAugust 2003 and is available since then from the IEC [6].There is another IEC standard dealing with RCs available already since 1999, whichdescribes shielding efficiency measurements of components such as cable assemblies orconnectors: the IEC 61726 standard [77]. Contrary to the IEC 61000-4-21 mentionedabove, the IEC 61726 standard features a very detailed proposal of a suitable stirrerdesign, almost in a “cookbook” style. Surprisingly, the IEC 61726 standard is rarelycited in RC publications or mentioned in RC-related conferences.Large parts of the IEC 61000-4-21 and IEC 61726 standard are based on the technicalreports published by Crawford, Koepke [9, 78], and Hill [8, 79, 80, 81, 20] of NISTas well as original work by Hatfield [82, 83, 5], Freyer [84, 85], Ladbury [37], Lunden,Backstrom [36, 4], and Arnaut [28, 27, 30, 31].

4.3 Previous reverberation chamber simulations

In recent years, there has been growing interest in the simulation of RCs to addresssome of the questions outlined in Chapter 1. Because of the difficulties in the RCsimulations, published results are often limited to two dimensions only, use “analytical”

4.3 PREVIOUS REVERBERATION CHAMBER SIMULATIONS 59

excitations such as a Hertzian dipole, assume PEC walls, or impose restrictions on thecomplexity of the stirrer design. The following Sections 4.3.1-4.3.3 summarize publishedRC simulations performed in the time-domain, frequency-domain, and with statisticalmodels. Table 4.1 on page 61 provides a short overview and abstracts important keyfacts on these simulation approaches: listed are the main author, whether the simulationwas carried out in 2-D or 3-D, which numerical method was used, and if it was a time-or a frequency-domain simulation. Furthermore, Table 4.1 shows at a glance whether acommercial or a “home-made” solver was employed, which type of excitation was used,and if (and how) the simulated results were validated.

4.3.1 Time-domain simulations

This section summarizes significant RC simulations using the most popular time-domaintechniques, viz. the finite-difference time-domain (FDTD) method and the transmission-line-matrix method (TLM) method.

Finite-Difference Time-Domain (FDTD) method

A Ph. D. thesis on an RC simulation using the FDTD method was presented by Petit [86]:it discusses extensively the problems that occur when standard, self-made FDTD is usedfor the simulation, i.e. staircasing, errors introduced by the Fourier transform, difficultieswith modeling of losses, etc. In [86] also measurements are presented, but unfortunatelynone of them measures quantities that were simulated. The simulation itself is exclu-sively benchmarked against a statistical analysis of computed results. Excitation of theEM field is accomplished with an infinitely small ideal dipole. It should be noted thatmost of the numerical problems described in [86] are already solved in commerciallyavailable field solvers, such as e.g. CST’s FIT-based Microwave Studio (using its “per-fect boundary approximation” approach) or the FDTD code ASERIS-FD (see below).The approach used in [86] appears to be putting the cart before the horse: Instead ofemploying a numerical method which is well-suited to the irregular stirrer design, thestirrer shapes used in [86] are adapted to the cubic cell structure of standard FDTD.This – by today’s state of the art EM solvers unnecessary – modeling constraint is alsopursued by other authors employing FDTD-based simulation codes: Using two planarstirrers that were well adapted to the rectangular grid of non-conformal FDTD, Harimaand Yamanaka [87, 88] investigated the impact of the numerical reflection coefficientof the wall surface on the field distribution. Bai et al. [89, 90] studied the influenceof different stirrer mounting positions inside an RC on the field uniformity. A similaranalysis was pursued by Zhang and Song [91] with a commercially available field solver.Also these authors mention problems that occur when FDTD is used in conjunctionwith structural staircasing or modeling of losses. A 2-D FDTD simulation of an RCinvestigating the loading effect of different periodic “rat-like” EUT-configurations waspresented by Lammers et al. [92, 93].Kouveliotis et al. [94] simulated a vibrating intrinsic reverberation chamber (VIRC) [95,96] (see also Section 4.4.1) with a Hertzian dipole as excitation source. The VIRC wassimulated as a rectangular “moving-wall” cavity, with two one-dimensionally, in a stag-gered manner oscillating walls. In [94] no realistic VIRC could be simulated as the FDTD


code of the authors was not capable of modeling slanted, irregularly shaped walls. Thissomewhat questionable procedure is only seconded by making the oscillations of the wallsa priori in a random fashion and then concluding a posteriori that the simulations workwell, because the obtained field structure is also random. In other words: Kouveliotis etal. use randomly changing boundary conditions and validate their simulation by findingrandomly changing EM fields – the simulations were not verified by measurements atall, only a statistical analysis is presented. Also other authors used the “moving-wall”approach to simulate standard RCs (non-VIRC) [97].Ritter and Rothenhausler presented in their paper the FDTD simulation of a medium-sized RC (9.4 m · 6.5 m · 5.3 m) [98]. Their intention was to extend this simulation to avery large RC measuring 40 m · 20 m · 30 m suitable for full-size aircraft testing. Exci-tation of the RC was carried out using the impressed field of a logper antenna obtainedfrom earlier simulations of this antenna through a far-to-near-field transformation. ThisRC simulation was not benchmarked against measurements, but only against statisticalresults. In [98], the performance of several tuners was compared, the results howeverexhibited unrealistically small differences from one tuner to another w.r.t. the correlation(cf. Section 7.7).Moglie showed in a recent paper that achieving stability in an FDTD code can be verydifficult for high-Q devices such as RCs [99, 43]. In his simulation, he needed to lowerthe conductivity of the chamber walls to physically unrealistic values (e.g. κ < 100 S/m)in order to achieve convergence. Furthermore he introduced artificial losses in the airvolume within the RC to improve the convergence speed. Also Petit’s FDTD simula-tions make use of somewhat inappropriate conductivities on the order of κ = 100 S/mfor metallic surfaces [86].A group at EADS Airbus used the commercial codes ASERIS-FD (FDTD) and ASERIS-BE (BEM) to model and simulate a 3.7 m · 5 m · 2.5 m RC [100, 101]. This RC is currentlybeing used for immunity and emission testing of avionic equipment. They employed theBEM code to run an RC simulation for the computation of the near field, while theirFDTD code is used for the estimation of the chamber quality factor Q. While thisapproach seems to be promising, the simulation results were not validated by measure-ments, but instead only against statistical data. The RC-related problem of achievingsufficient statistical field uniformity within a predefined volume has been addressed alsoin FDTD-based simulations of microwave heating devices [102, 103].

Transmission-Line-Matrix (TLM) method

In [104, 105] Clegg et al. tried to optimize stirrer designs in an RC (4.7 m · 3.0 m · 2.4 m)using time-domain TLM in combination with a genetic algorithm. After an initial at-tempt to optimize the stirrer inside the RC they concluded that this was computationallynot feasible (simulation runtime typically 4 days for one stirrer revolution) and thereforeproposed to do the optimization in free space with plane wave illumination of the stirrer(runtime typically a few minutes). As a measure of stirrer efficiency they defined thechange in the Poynting vector for a certain number of spherically distributed samplepoints as the stirrer rotates. Although the genetic algorithm approach was not veryconvincing, the general idea of optimizing a stirrer without the RC seems promising (cf.Section 7.7.5).

4.3

PR

EV

IOU

SR

EV

ER

BE

RAT

ION

CH

AM

BE

RSIM

ULAT

ION

S61

Authors 2-D/3-D a TD/FD b Method c H/C d Excitation Validation e Year f

Wu et al. [113] 2-D TD TLM H HD g None ‘89

Bunting et al. [114]–[117] 2-D FD FEM H HD g Statistics h ‘98/‘99/‘02

Harima et al. [87, 88] 3-D TD FDTD H HD g None ‘98/‘99

Bai et al. [89, 90] 3-D TD FDTD H HD g None ‘99

Petirsch et al. [23] 3-D TD TLM H HD g None ‘99

Zhang et al. [91] 3-D TD FDTD C HD g None ‘00

Hoijer et al. [118] 3-D TD FDTD C HD g Resonances i ‘00

Hoeppe et al. [101] 3-D TD FDTD C HD g/Horn Statistics h ‘01

Hoeppe et al. [100] 3-D FD BEM C HD g/Horn Statistics h ‘01

Clegg et al. [104, 105] 3-D TD TLM H HD g None ‘02/‘04

Coates et al. [106] 3-D TD TLM C HD g Statistics i ‘02

Petit et al. [86] 3-D TD FDTD H HD g Statistics h ‘02

Laermans et al. [119] 2-D FD MoM H Dipole Statistics h ‘02

Asander et al. [120] 3-D FD BEM H HD g None ‘02

Kouveliotis et al. [94] 3-D TD FDTD H HD g Statistics h ‘03

Moglie et al. [121, 99, 43] 3-D TD FDTD H HD g None ‘03/‘04

Ritter et al. [98] 3-D TD FDTD C logper Statistics h ‘03

Lammers et al. [92, 93] 2-D TD FDTD H HD g Statistics h ‘04

Weinzierl et al. [111] 3-D TD TLM H HD g None ‘04a 2-dimensional / 3-dimensional simulation b time-domain / frequency-domain c numerical method d “home-made” / commercial solvere validation of the simulated results f year published g Hertzian dipole h simulation vs. theoretical data i simulation vs. measurements

Table 4.1: Summary of previously published RC simulations.


Other TLM-based simulations by Coates et al. [106] as well as by Duffy et al. [107]used the commercially available code Microstripes (now part of the Flomerics soft-ware package) to simulate the effects of different vane heights of a simple stirrer on theSR (cf. section 2.2.3). Their modeled RC is 5.0 m · 2.9 m · 2.4 m large. Both publica-tions [106, 107] presented simulations and measurements and compared them againsteach other, but only on the basis of the SR. As outlined in [108], a validation of RCsimulations using “highly processed” data such as the SR is not possible, as completelydifferent EM fields can have identical SR values [108]. Therefore it remains unclearwhether the aforementioned TLM simulations actually model the EM fields inside theRC correctly.Petirsch et al. [109] investigated the effect of diffusors placed inside an RC on the fieldhomogeneity using their own TLM code. Most of the results shown in [109] remainquestionable, since the utilized performance measures were based exclusively on non-validated simulation data. As noted correctly in a comment to [109], Arnaut criticizesthe results as somehow “random”, since diffusors were used completely beyond theiroriginally intended operation conditions [110]. The improvement in field homogeneityreported in [109] is most probably merely due to increased loading of the chamber byabsorbing material of the diffusors.Weinzierl et al. [111] carried out a TLM-based simulation to find out whether an ideaproposed earlier by Perini [112] (and discussed very controversially among RC experts)of a two-wire line RC excitation actually works. As shown in Section 2.2, RCs are limitedin their operation to frequencies f > fLUF, i.e. a large number of modes must be abovecutoff in order to achieve sufficient field uniformity within the chamber. The propaga-tion of quasi-TEM waves on a two-wire line within an RC was intended to remove thislow-frequency-limitation by exciting EM fields below the fundamental RC mode [112].Unfortunately, [111] only shows that there are quasi-TEM waves excited between thetwo wires – whether they serve as an appropriate RC excitation in providing sufficientlyuniform fields remains unclear. It is evident however, that below the cutoff frequency ofthe fundamental RC mode, the field distribution and magnitude within an RC does nothave anything in common with the distribution at or above the LUF.

4.3.2 Frequency-domain simulations

Method-of-Moments (MoM)A 2-D MoM-based RC simulation and statistical analysis was described by Laermansand De Zutter in [119]. In their work, an ideal line current source was used as excitationand boundaries were modeled as PEC. Whether these results are of any practical useis questionable, since the three-dimensional nature of EM fields in an RC was not takeninto account and the authors relied only on the evaluation of statistically processed data(cf. Section 7.4.3 and [108]).

Finite-Element Method (FEM)Bunting et al. [114] reported a statistical characterization and simulation of an RC usingthe finite-element method (FEM). His results could not be validated by measurementsof e.g. the near field or other readily accessible parameters, as the simulation was only

4.4 ALTERNATIVE STIRRING METHODS 63

carried out in two dimensions. Fields inside the RC were excited with an ideal Hertziandipole. The paper mentioned is based on earlier publications [116, 115, 117] and relies asmany other papers heavily on the validation by comparison against statistical or “highlyprocessed” data (cf. Section 7.4.3).

Boundary-Element Method (BEM)Asander et al. [120] presented a BEM-based simulation of an RC (4.9 m · 2.5 m · 3.0 m)in the frequency-domain. Simulations were carried out at a single frequency of 300 MHz.In [120], it was explicitly pointed out that with a simulation “far more data can be gener-ated and analyzed than is possible if measurements are used instead” and “that surfacediscretization has some advantages over methods relying on volume discretization whenit comes to model irregularly shaped, rotating stirrers”. A validation of the simulationusing measurements was not performed.Using the commercial frequency-domain BEM simulation tool ASERIS-BE, Hoeppe etal. [101, 100] investigated the RC working volume of statistical field uniformity as wellas the effect of chamber loading. ASERIS-BE features a technique similar to the NGFtechnique, so that from one stirrer step to another only small parts of a large system ofequations need to be re-computed and solved (see Section 3.5.2).

4.3.3 Statistical models

In the past, several statistical models describing EM effects inside an RC were developedto avoid the long computation runtimes and large memory requirements associated withtime- and frequency-domain simulation methods mentioned above. Most of them arebased on the assumption of local plane waves in the RC and try to facilitate the under-standing of effects such as coupling between TX and RX antennas or simple EUTs (e.g.transmission lines) and antennas [122, 123, 84, 85, 24, 124].A detailed description of the statistical properties of EM fields inside overmoded cav-ities can be found in [33], out of which some parts were condensed into a journal pa-per [125]. Furthermore, most books dealing with mobile communications serve as anexcellent source for a better understanding of the statistics observed in RCs, since thestatistical properties of the EM field within an RC have a lot in common with a mixeddirect-path/multi-path environment [126, 127, 32]. Vice versa there is also a numberof publications dealing with RCs which make use of the statistical analogies to mobilecommunications and employ them for propagation environment modeling [17, 128].

4.4 Alternative stirring methods

Changing the EM field distribution within an RC is usually accomplished with a me-chanical stirrer. The presence of this stirrer is undesirable for two reasons: first of all,with the stirrer in the chamber the space available for testing of an EUT is reduced. Themore important aspect, however, is that the stirrer (or tuner) represents a somehow dis-turbing mechanical element in this otherwise purely electrical testing environment. Thefeed-through bearing for the stirrer axle is difficult to shield properly, care must be takennot to move the stirrer manually since otherwise the precision gear may be damaged,


and rotating the stirrer from one position to another is time-consuming (acceleration,deceleration, decay of stirrer oscillations). The following sections summarize the mostimportant concepts presented in the open literature on how to eliminate the mechanicalstirrer.

4.4.1 Moving walls

One possibility to remove the mechanical stirrer inside the RC is to change the fielddistribution by actually moving the walls of the RC. A method that uses this approachwas proposed by Leferink et al. [95]. The “VIRC” is essentially a “tent” which is madeout of electromagnetically conducting cloth. Small motors with eccentric drive fixtureswobble the cloth sufficiently so that the field is changed and a certain field uniformity isachieved within the “tent” [96]. This approach does not get rid of mechanical problemscompletely, but it removes the stirrer from the RC’s interior. It was successfully appliedto build up “mobile” RCs in order to test equipment which is e.g. too large to be moved.There are however problems with achieving sufficient radio frequency (RF) SE throughelectromagnetically conducting cloth.

4.4.2 Electronic stirring

Methods of field stirring without any mechanical stirring devices within (or around) anRC are commonly referred to as “electronic stirring”. One limiting factor of electronicstirring is that until now it is only applicable – if at all – to immunity testing of EUTsin RCs. The major problem arising for these methods in emission measurements isthat one has no control over the emitted spectrum of a test object. This renders itvery difficult to ensure proper interaction between the electronic stirring equipmentand the EUT. Furthermore, since changes of the EM field distribution occur virtuallyinstantly (compared to the very slow changes using mechanical stirrers), the responsetime of an EUT in immunity testing must be known [31]. In addition, the advantage ofeliminating (rather cheap) mechanical stirrers by electronic means (as discussed below)comes at a considerable price: expensive RF equipment, such as additional antennas,power combiners, mixers, frequency modulators, up/down converters, etc. needs to beused.

Frequency and Gaussian noise stirringA method of changing the field distribution was proposed by Hill in 1994, the so-called“electronic mode stirring for RCs” [129]. Although theoretically only analyzed for thetwo-dimensional case in [129], this technique had been applied one year earlier alreadyin “real-world”, three-dimensional RCs [130]. The underlying principle of “frequencystirring” is to acknowledge that the change of the resonance frequencies of the cavitymodes by a rotating mechanical stirrer has some similarity to the frequency modulationof the source [113].Instead of changing the frequency “monochromatically” by standard frequency modu-lation, another proposed method uses additive white Gaussian noise (AWGN) which ismixed onto a periodically changing center frequency [131]. This approach claims thatthe field uniformity is increased compared to pure frequency modulation, while the test

4.4 ALTERNATIVE STIRRING METHODS 65

time is considerably shortened [132]. A combination of mechanical and electronic stirringis investigated in [133]. This procedure leads to a larger number of independent samplesand hence lower measurement uncertainties for tests in the RC.

Multiple sources and phase stirringInstead of frequency stirring, the usage of multiple source antennas is advantageous forsingle-frequency high-power excitation of RCs, because they eliminate the need for com-bining high-power signals through external RF components. In [129] it was investigatedwhether multiple sources alone (without any stirring) would lead to sufficient field uni-formity: [129] concludes that the improvement in field uniformity is rather marginal,and nevertheless a mechanical stirring device is needed. This was found to be true evenif the sources were incoherent or varied in phase (so called “phase stirring”) [134].

Three-dimensional TEM cellOne of the advantages of an RC is that there is no need to rotate the EUT during testing;a problematic aspect is however that RCs are limited in low-frequency-operation due totheir physical dimensions. An alternative to RCs at lower frequencies could be possiblythree-dimensional TEM cells. A 3-D TEM cell is a combination of three individual TEMcells in one, in such a way that the TEM coupling planes created by each plate are notparallel to each other in the center of the test volume. A particular case is when thecoupling planes are orthogonally arranged two-by-two: Each plate creates an electricand magnetic coupling which defines its TEM coupling plane [135]. Problems with 3-DTEM cells are mainly due to the field distribution and gradients in the central region ofthe structure, strongly limiting the test volume to a much smaller volume as comparedto RCs. The operating range of 3-D TEM cells is limited to frequencies below the firstfundamental resonance of the metallic cavity-like structure they are built into [136]. Ifa 3-D TEM cell and an RC are combined, one is left with an EMC test device whichcan be operated in the low frequency and the high frequency region – using completelydifferent operating principles –, but which fails to function in the intermediate regionbetween the first fundamental RC resonance and the LUF of the RC.

Two-wire TEM excitationIn a paper by Perini and Cohen published in 2000, it is proposed to use several wiresinside an RC to excite quasi-TEM modes [137]. As shown in Section 2.2, RCs are limitedin their operation to frequencies f > fLUF, i.e. a large number of modes must be abovecutoff in order to achieve sufficient field uniformity within the chamber. The propagationof quasi-TEM waves on a two-wire line within an RC was intended to remove thislow-frequency-limitation by exciting EM fields below the fundamental RC mode [137].According to the authors [137], TEM modes do not have a lower cutoff frequency (whichis undoubtedly true) and therefore a bundle of correctly placed wires within an RC willextend the operating range down to f = 0 Hz. Unfortunately, [137] (and later [111]) onlyshow that there are quasi-TEM waves excited between the two wires – whether theyserve as an appropriate RC excitation in providing sufficiently uniform and isotropicfields remains unclear. Since until now nobody was able to exploit the proposed effectsin practice, the two-wire TEM excitation is discussed very controversially among RCexperts [112].


4.5 Practical reverberation chamber applications

Tests in RCs are usually appropriate for EMC measurements where

• the radiation pattern of the EUT is a priori unknown (i.e. the EUT does notradiate like a dipole or quadrupole with known broadside direction)

• the EUT is large compared to the wavelength (such as e.g. a personal computeroperating at a clock speed of 3 GHz)

• a realistic, non-plane-wave test environment is desirable

• measurements must be carried out in a repeatable manner

• reliable and meaningful results are “mission-critical”, such as in medical, automo-tive, or avionic equipment

These requirements are difficult to meet with established, widely accepted EMC test en-vironments such as ACs, OATS, or Gigahertz transverse electromagnetic (GTEM) cellsusing the current testing methods as laid out in the respective standards. The follow-ing sections summarize important applications of RC testing for EUTs meeting theserequirements. A qualitative (and by no means fully exhaustive) comparison highlightingthe advantages and disadvantages of EMC tests in RCs versus ACs is given in Table 4.2.

4.5.1 Automotive and aircraft avionics

The characterization of an RC for automotive susceptibility is carried out in [140]. Thepaper describes the implementation and measurement of an RC suitable for susceptibilitytesting of automotive electronics in the 200 . . .1000 MHz range. The original problemwas that in this frequency range, required power levels have entailed prohibitive testingcosts if the test is carried out within an AC or a similar EMC testing environment. It isshown that the RC test method can be substituted for the AC method thereby effectivelysolving this problem [141].For the past several years there has been an increasing interest in the possibility oftesting large items such as aircrafts in an RC. As the use of electronic systems toperform critical flight functions steadily increases, the application of RCs to testing ofaircraft and avionics systems is discussed by Hatfield et. al. in [142]: data is presented onthe statistical characteristics of the EM environments in aircraft cavities and comparedwith those in RCs. In order to investigate the suitability of RCs for these large objects,scaled versions of aircrafts were made and tested in smaller chambers. One of the relatedconcerns is the scalability of the operational characteristics of RCs. For this reason, theNaval Surface Warfare Center Dahlgren Division (NSWCDD) maintains a database (DB)on the performance of operating RCs worldwide [83]. This data consists mainly of SRdata at 1 GHz. The volumes of chambers in the DB range from less than 1 m3 toabout 200 m3. It was found that the characteristics are scalable over this two orders ofmagnitude variation in volume. To test a reasonably large aircraft, a chamber would

4.5 PRACTICAL REVERBERATION CHAMBER APPLICATIONS 67

Anechoic chamber(AC)

Reverberation chamber(RC)

Mechanical setup- Shielded room- Absorbers- TX/RX antennas

- Shielded room- Stirrers- TX/RX antennas

Type of field Plane wave/Single path Multi mode/Multi path

Polarization Linear, fixed Arbitrary, not known

E, H phase relation Fixed Not fixed

Direction of incidence Known, fixed All directions, “isotropic”

Field impedance 377 Ω Unknown

EUT radiation patternAssumptions:- “well-behaved”- “dipole-like”

No assumptions made

Emission testing- extensive scanningneeded to get peak

- one direction at a time

- “integral approach”- omnidirectional testing

Immunity testing- uncertainty aboutEUT directivity

- one direction at a time

- “isotropic approach”- omnidirectional testing

Calibration Simple Elaborate

Test software Simple, not mandatory Complex, “mission critical”

Production line testing Slow, impossible Fast, automated

Test repeatability Bad (e.g. ±20 dB) [138] Good (e.g. ±3 dB) [22]

High field strengths. . . . . . need large amplifiers . . . need small amplifiers

Table 4.2: Basic differences between the AC and RC EMC test environment (partly extractedfrom [139]).

have to have a volume of 5 · 104 m3 and greater [143]. Measurements of the RF energycoupled to instrumented avionics boxes from a large transport aircraft and a simulatedavionics box were presented in [144]. These measurements were made at NSWCDDwhen the avionics bay, cockpit, and passenger cabin were internally excited with swept


RF energy from 100 MHz to 6 GHz and mechanical mode-stirring techniques were used.The tests were intended to demonstrate that the coupling characteristics of an aircraftand simulated avionics boxes measured in an RC constitute valid descriptions of thesame boxes when installed and operating in an aircraft.Extensive measurements of radiated emissions in RCs are adressed in a Ph. D. thesis byKurner [22]. In addition, the SE of various EUTs is analyzed, general recommendationsfor proper EMC testing (loading, EUT position and orientation, etc.) in RCs are given,and a comparison between tests in ACs and RCs is performed.

4.5.2 Antenna measurements and mobile communications

In order to compare the results of tests in ACs against tests in RCs, identical antennaswere measured once in an RC and once in an AC [22, 23]. Measurements in an RC tocompute the radiation efficiency and return loss of an electrically small antenna werecarried out by Carlsson et. al. in [145]. Madsen et. al. investigated whether RCs aresuitable for mobile phone antenna tests [146]. They conclude that an RC could beused for production line testing to check periodically mobile phones for compliance withspecific absorption rate requirements.Since antenna diversity in mobile terminals is likely to be commonly used in future mobilecommunication systems, methods for characterization of antenna diversity performanceare therefore of interest. A new characterization method using an RC, with severaladvantages over existing techniques, was proposed in [147, 148]. It was found thatwhen performing diversity measurements in an RC, the correlation between the signalsreceived by the diversity antenna elements was similar to the correlation in a real multi-path environment. Hence an RC could be used to effectively model the propagationcharacteristics commonly encountered in mobile communication systems.

4.6 Conclusion

An extensive overview on the literature treating the early development of reverberationchambers (RCs), RC standards, and the simulation of RCs was given. On the orderof 35 papers were published in the past dealing with RC simulations, prepared by ap-proximately 20 different research groups. The earliest published RC simulation datesback to 1989. Most groups used a self-made numerical code, with FDTD being clearlythe preferred method (probably because this technique is rather easy to understandand comparatively simple to implement – the latter statement is certainly only true forvery basic, non-conformal, non-subgridding FDTD). With the notable exception of onegroup, all published papers employed a Hertzian dipole as excitation. Whereas earliersimulations were still carried out in two dimensions, more recent RC analyses made useof three-dimensional simulation tools. It is important to note that until today all butone publication did not use appropriate means of validation. Half of the groups chosenot to validate their simulated results at all, the remainder performed only a statisticalbenchmark (this problematic issue is discussed extensively in Section 7.4).Several ideas were proposed in the past to eliminate the mechanical stirrer in the RC and

4.6 CONCLUSION 69

to extend the operational range of RCs to lower frequencies. Among them are the vibrat-ing intrinsic reverberation chamber (VIRC), which consists of conductive fabric forminga tent, sophisticated electronic stirring techniques, and the three-dimensional TEM cell– as of today, none of them have found widespread use, either due to prohibitive costs,electromagnetic issues, or other implementation problems. Finally, a short compari-son between the RC and anechoic chamber (AC) EMC test environment was presentedhighlighting advantages and disadvantages of each methodology.

5 Prototype and Measurement System Development

Abstract— This chapter describes the construction and setup of the reverberation chamber pro-

totype including walls, door, stirrers, and auxiliary equipment. Details of the measurement system

developed for data acquisition are outlined. Measurement errors originating from field probes, an-

tennas, and stirrers are discussed and assessed for their impact on deviations between simulated and

measured results.

5.1 Reverberation chamber prototype

For this thesis, an RC prototype was built having inner dimensions of 2.86 · 2.48 · 3.06 m3

(width w · length l · height h). This chamber features several geometrical details, such asa door, a coaxial feed-through panel, a circular waveguide, three stirrer motor mounts,two honeycomb ventilation ducts, and two lights. The total RC wall and door surfaceamounts to approximately 50 m2.

5.1.1 Walls and door

The chamber walls consist of a sandwich-type construction made of chip board woodbetween two galvanized sheet steel layers. The wood provides for sufficient mechanicalstability of the RC construction. The steel layers are made of several separate panels ofoverlapping flat stock, which are connected by I-profiles. The main material componentsof commercially available galvanized steel sheets are typically iron (97. . . 99 wt.-%), zinccoating (0.5. . . 2 wt.-%), copper (0.4 wt.-%), and manganese (0.4 wt.-%) [149]. Thesematerial properties are important and will be needed in Chapter 6 to accurately modelthe RC for a simulation. Within the RC up to three stirrers can be mounted and operatedsimultaneously on different axes. A schematic overview of the basic RC structure isshown in Fig. 5.1: Depicted are two horizontal (marked as I and II) and one vertical(marked as III) stirrer axes. Stirrer axes mounting points are ∆x = ∆y = ∆z =0.6 m and respectively ∆xI = ∆zI = 0.8 m spaced from the walls. In Fig. 5.1 alsotwo different coordinate systems are indicated, where the right-handed (x, y, z) is usedin the simulations and the left-handed (xm, ym, zm) for measurements. A coordinatetransformation from one system to the other is given by

xm = x − w

2(5.1)

ym =l

2− y (5.2)

zm = z − h

2(5.3)

and must be used in order to compare measurement with simulation results.The RC chamber door has Beryllium copper contact finger strips as gasket to prevent

71

72 5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT

xm

ym

zm

x

z

y

2.86m

3.0

6m

2.48m

III

x

y

I

II

Door

xI

z I

z

Figure 5.1: RC geometry and dimensions with three different stirrer axes (I, II, III) and chamberdoor. Stirrer axes mounting points are ∆x = ∆y = ∆z = 0.6 m and ∆xI = ∆zI = 0.8 m.Note the different right- and left-handed coordinate system (x, y, z) and (xm, ym, zm).

EM leakage. The dimensions of the door are 2 m in height and 0.92 m in width with a0.07 m recess of the door frame from the chamber wall. The doorstep is elevated 0.07 mfrom the RC bottom and spaced 0.17 m away from the outer chamber edge. Although,at first glance, these geometric details seem to be small, it is shown in Chapter 7 thatthey affect the chamber fields significantly and thus cannot be neglected. Details onmodeling of the chamber door can be found in Section 6.1.3.

5.1.2 Stirrer

This section outlines basic design guidelines and shows how they were applied to thestirrers built for the RC prototype. Furthermore, details on the mechanical and electronicpart of the stirrer drive and controller are given.

Stirrer designFor the analysis of the RC prototype shown in Fig. 5.2a), initially one vertically mountedstirrer was built. This vertical stirrer consists of six rectangular paddles of size 0.60 m· 0.60 m, rotationally offset around the stirrer axis by 60. The slanting angle betweeneach paddle and stirrer axis is 45. The horizontal spacing between the paddle centersis approx. 0.45 m, the distance from the lowest and the topmost paddle edge is 0.15 mto the RC floor and the ceiling, respectively. The stirrer was built according to thegenerally accepted design principles available at the time of the thesis’ start, i.e.: thestirrer should be electrically large at the LUF (which is around f = 300 MHz as shown inSection 2.2) and the overall stirrer structure must not be rotationally symmetric [9, 78].The first design principle ensures that the stirrer will be effective in modifying the field

5.1 REVERBERATION CHAMBER PROTOTYPE 73

a) b)

c) d)

Figure 5.2: a) 6-paddle stirrer, logper TX antenna, and field probe; b) zoomed view: stirrer’saxle with anti-flexing fixture mechanism; c) stirrer motor drive unit with copper-shieldingaround mounting aperture and d) pyramidal absorber placed on top.

distribution within an RC sufficiently. The asymmetry-criterion provides for dissimilarfield distributions, i.e. field distributions which are statistically weakly correlated. Amongthe stirrers developed in the past exhibiting a good performance were the so-called “Z-fold” stirrer [150], the triangular-base stirrer with perpendicularly mounted paddles [139],the “Rot-Z” stirrer [151, 109], and the fan-style stirrer used in the RC of the Defence


Science and Technology Organisation (DSTO), Australia [7]. These stirrer designs followthe basic guidelines mentioned above, and so does the stirrer designed for this thesis.There is however one aspect controversially discussed among RC experts with respect tothe requirement for stirrers being electrically large: What does electrically large mean fora rotating object? In an ordinary sense, one would refer to an object being electricallylarge if it extends in one or more dimensions to a length greater than a significantfraction of the wavelength (such as d > λ/2). The stirrer used in this thesis has e.g. atotal geometrical height (distance from upper edge of the topmost paddle to lower edge oflowest paddle) of 2.76 m – its electrical height is however lower, as the individual stirrerpaddles are not connected with each other. Whether the performance of this stirrer canbe enhanced by manufacturing it with all paddles being connected (in order to increaseits electrical size in the traditional sense), is discussed in detail in Section 7.7. Takinginto considerations published results, it can be stated already that both stirrer typesmade out of one single piece (e.g. “Z-fold” stirrer [150]) as well as stirrers consisting ofseveral separate parts (e.g. “Rot-Z” stirrer [151, 109]) exhibit good performance.

Stirrer drive and controllerThe stirrer paddles are mounted on a plastic rod with a diameter of 50 mm using plasticspacers cut at an angle of 45; these spacers allow the paddles to be rotated around therod’s axis so that all paddles can be aligned at any arbitrary angle with respect to eachother. In order to check proper alignment of the paddles on the rod, a rotary angularscale is mounted on each paddle’s spacer. The rod itself is attached on the upper end tothe stirrer motor gearbox and on the lower end supported by a ball bearing on the cham-ber floor. The rod was fitted with the anti-flexing fixture mechanism shown in Fig. 5.2b)in order to achieve a maximum stiffness of the whole stirrer structure; this reduces thesettling time between different angular steps for the stirrer paddles considerably, whichallows faster measurement cycles [152].The stirrer drive depicted in Fig. 5.2c) is an electrical 24 V DC, 20 W brush servo mo-tor equipped with a planetary gearhead and an electronic encoder. The gearhead hasa gear reduction ratio of 128 : 1, the encoder is used to provide feedback on the ac-tual angular stirrer position to the motor controller [153]. A pulse-width-modulationtechnique is used to control the motor drive. The motor controller is a fully digitalposition, speed, and current control unit, which can receive high-level motion commandsand provide continuously updated status feedback to the measurement system [154] (seeSection 5.2). Parameters for acceleration and deceleration ramps are stored in the con-troller’s firmware and need to be adjusted to the load inertia represented by the stirrer(the whole stirrer assembly weighs approx. 30 kg). The maximum permissible stirrerspeed is 30 min−1. An important parameter is the angular resolution of the stirrer drive,since the field distribution depends strongly on the rotational stirrer position, whichdirectly affects comparisons of measurements with simulations (see Section 7.7). Thetheoretically achievable angular resolution in the RC prototype can be calculated asfollows:

• since there is a direct connection between load and load-side of the gearhead,one load revolution (i.e. one revolution of the stirrer) equals one gearhead outputrevolution (1:1)

5.1 REVERBERATION CHAMBER PROTOTYPE 75

• with the above-mentioned reduction ratio, one revolution on the gearhead load-siderequires 128 revolutions on the gearhead motor-side (128:1)

• the encoder outputs 500 impulses to the controller per motor revolution (500:1)

• with the help of a bi-phase detection technique, one single encoder impulse isconverted to four so-called “quadcounts” (4:1) in the controller [153]

Since one quadcount is the smallest detectable movement, the theoretically achievableangular resolution amounts to 256 000:1 or ∆ϕ ≈ 0.0014. The practically achievableangular resolution is mainly limited by the quality of the planetary gearhead, i.e. essen-tially its inherent back-lash. With specified back-lash values on the order of ∆ϕ ≤ 1

the practically achievable angular resolution is significantly worse than 0.0014. Consid-ering the strong dependence between rotational stirrer position ϕ and EM field withinthe RC, this needs to be taken into account when evaluating simulation vs. measurementbenchmarks.

5.1.3 Auxiliary installations and electromagnetic leakage

In addition to the walls, stirrer, and door, the RC features a coaxial feed-through panel,a circular waveguide, three stirrer motor mounts, two honeycomb ventilation ducts, twoceiling-mounted lights, a power line filter, and cable ducts. The feed-through panel shownin Fig. 5.3a) is used to connect the measurement equipment outside of the RC to theTX/RX antennas placed within the chamber. This panel is equipped with two Type Nfeed-through connectors suitable for frequencies up to 18 GHz and a circular waveguide,which is utilized for the optical fibers of the field probe system (cf. Section 5.2.2).The three stirrer motor mounts – out of which one is shown in Fig. 5.3b) – correspondto the three axes I, II, and III depicted in Fig. 5.1 and allow for an installation of thestirrer at any of these positions. The length of the stirrers was designed so that they canbe mounted on all axes using appropriate rods without changing the mechanical paddleconfigurations.Fig. 5.3c) shows the power line filter, which is used to transfer AC power to any electricalequipment operated within the RC. This filter prevents transmission of RF energy cou-pled to power lines running within the RC to the outside environment. The honeycombventilation duct depicted in Fig. 5.3d) acts as an array of small waveguides operated be-low cutoff, providing high RF attenuation and at the same time means for air exchangebetween chamber interior and exterior.In order to achieve high field strengths within an RC and to prevent electromagneticinterference (EMI) with devices outside of the RC, it is necessary that the chamberstructure guarantees sufficient SE for the frequency range of operation. The overall SEwas measured with the two-antenna-method, i.e. one antenna is placed inside the emptyRC and a second antenna is used outside the chamber. With the second antenna, thewalls of the RC are scanned, such that the worst-case SE is found. Using this measure-ment method, all the apertures depicted in Fig. 5.3 exhibited an SE well above 100 dBwithin 50 MHz. . . 4.2 GHz. Initially, the stirrer drive mounting aperture had some EMleakage; as shown in Fig. 5.2c) and d), this problem was solved with a copper tape bypass


a) b)

c) d)

Figure 5.3: Apertures in the RC: a) feed-through panel with coaxial Type N connectors andcircular waveguide for optical fiber (black bundle); b) stirrer mounting plate; c) externalpower line filter; d) honeycomb duct for ventilation and interior metal fixtures.

between chamber wall and mounting plate in addition to the placement of a pyramidalabsorber on top of the stirrer drive. After retrofitting the RC with special corner ele-ments (the original ones exhibited poor RF shielding), the overall chamber is capable ofproviding an SE of 80 . . .90 dB over the 50 MHz. . . 4.2 GHz frequency range.

5.2 MEASUREMENT SYSTEM 77

Figure 5.4: Overview of the RC measurement system setup. Control and data acquisitioncomputer (left); RF amplifiers, signal generator, power meter, power supplies, and stirrermotor controller (right).

5.2 Measurement system

The RC measurement system is used to acquire information on the electric field withinthe chamber as well as the power being transmitted and received by the TX/RX anten-nas. One part of the system consists of the equipment used to generate and measure theEM field in the RC (Section 5.2.1), the other part is needed to control this equipment aswell as to record and process the gathered data (Section 5.2.3). As in any measurementsystem, it is important to keep the influence of the measurement sensors on the quan-tities to be measured as small as possible (Section 5.3). Fig. 5.4 gives an overview ofthe RC measurement system setup: the control and data acquisition computer is shownon the left side, the measurement equipment consisting of RF amplifiers, signal gener-ator, power meter, power supplies for the amplifiers and the stirrer drive, and stirrermotor controller on the right. Two effects (which are not limited to RCs only) must beparticularly taken into consideration when measurements are performed:

• by its physically finite size, the measurement equipment performs spatial averagingof the EM field – i.e. the field is not measured at a single, infinitely small point inspace, but rather over the geometric extension of the field sensor. The nonzero fieldsensor size may mask an actually poor reverberation performance [28, 27]. Thiseffect becomes very pronounced if conventional, spatially large RX antennas areused for field measurements, instead of spatially small field probe systems [155].

• if measurements are taken in mode-stirred operation of an RC (i.e. the stirrercontinuously rotates), the EM field changes very rapidly over time. As today’sfield probe systems have a rather slow response time and are therefore not capableof measuring fast fluctuations of the EM field (settling time typically 0.5 s [156]),the measured field will be a time-averaged version of the physical field [157].


Both spatial- and time-averaging result in deviations of the measured vs. the actual EMfield and make the gathered measurement data appear smoother. In the mode-tunedoperation of an RC (the stirrer is stepped from one angle to the next with comparativelylong pauses in-between), the time-averaging effect does not occur since the field is onlymeasured once steady-state is reached. For this reason, the RC analysis presented inthis thesis was limited to mode-tuning.

5.2.1 Transmit (TX) and receive (RX) measurement equipment

This section provides an overview on the general measurement system and details on theactually used equipment. The field probe system is explained in a separate section below(Section 5.2.2). Fig. 5.5 shows the equipment setup used for measurements of the elec-tric near field as well as the forward and reverse power at the TX and RX antennas. Asoutlined in Section 5.2, these antennas were not used for direct near field measurementsas they provide insufficient spatial field resolution. Furthermore it is almost impossi-ble to measure three-components near field data reliably with conventional antennas(they would need to be rotated into different axes around a virtual center position in ahighly repeatable procedure and without changing the field distribution). The followingequipment was used for EM field generation and antenna-based measurements:

Signal generatorMarconi (now IFR Test Systems) generator, type 2024, output level adjustable between−137dBm. . . +13 dBm, frequency range f = 100 kHz. . . 2 GHz

AmplifiersThree broadband power amplifiers with a maximum input level of 10 dBm:

• Mini Circuits LZY-1, frequency range f = 10 MHz. . . 512 MHz, minimum gain39 dB, maximum output power Pmax = 47 dBm

• Mini Circuits LZY-2, frequency range f = 440 MHz. . . 1 GHz, minimum gain 40 dB,maximum output power Pmax = 44 dBm

• Schaffner CBA 9428, frequency range f = 1 GHz. . . 3 GHz, minimum gain 46 dB,maximum output power Pmax = 43 dBm

All amplifiers feature built-in forced air cooling; the LZY-1 and LZY-2 require a sepa-rate power supply (28 V, 10 A) and an external input/output stage protection againstexcessively large input signals or highly reflective loads.

Directional couplersThe (bi-)directional couplers are used to measure the forward and reflected power atthe TX/RX antenna terminals. Two different couplers were employed to cover the wideoperating frequency range:

• Werlatone C3946, 40 dB attenuation, f = 100 kHz. . . 1 GHz

• Hewlett-Packard 788B, 20 dB attenuation, f = 100 MHz. . . 3 GHz


Reverberation Chamber

2 3 4

5

6

7

8

9

10

11

12

13

14

15

1

16

Data signal path

RF signal path

1

3

11

3

5

86

4

2

7 9

17

17

Figure 5.5: RC measurement setup and photo. 1: signal generator; 2, 4: attenuators; 3: RFamplifier; 5: bi-directional coupler; 6-9: power meter with probe heads; 10: electric fieldprobe set; 11: stirrer drive controller; 12: network analyzer; 13, 14: TX/RX antennas; 15:EUT; 16: control and data acquisition computer, 17: power supplies.

Power meterIn order to quantify the forward and reflected power provided by the bidirectional cou-plers, a Rohde & Schwarz NRVD dual-channel power meter is used [158]. The twochannels are completely independent from each other and can measure power simulta-neously. In combination with a bidirectional coupler, the net power delivered to the RC


can be calculated from these measurements. The specified frequency range of the NRVDunit is f = 0 Hz. . . 26.5 GHz, the actual operating frequency range is however defined bythe external probe heads. To measure the power, two Rohde & Schwarz insertion unitsURV5-Z4 are used as probe heads, which can operate within f = 100 kHz. . . 3 GHz [159].These insertion units are diode-based sensors (thermal sensors would be too slow withsettling times on the order of 10 s) shunted on the output side with a 50 Ω resistor. Dueto their minimum insertion loss, these units leave the 50 Ω-line connected to the bidi-rectional coupler virtually unaffected. The NRVD power meter is fed with the currentoperating frequency by the measurement system, so that the probe heads can providecalibration data feedback to the NRVD using a correction factor lookup table.

TX/RX antennasTo cover the broad operational frequency range of the RC, several different antennatypes were used:

• One precision conical dipole antenna, manufactured by Austrian Research CenterSeibersdorf, type PCD8250, f = 80 MHz. . . 2.5 GHz

• One biconical antenna by Ailtech, type AT-200, f = 20 MHz. . . 200 MHz

• Two biconical antennas by A.H. Systems, type SAS-541, f = 20 MHz. . . 330 MHz

• Two Schwarzbeck logper antennas, type USLP 9143, f = 300 MHz. . . 5.2 GHz

• Two Hewlett-Packard standard gain horns, f = 2.2 GHz. . . 3.3 GHz

The conical dipole antenna was only used for TX measurements in the RC withoutstirrers to validate the simulations in the lower frequency range, the other antennaswere utilized for all further TX/RX measurements in the RC. As shown in Section 5.3and Fig. 5.10, the antennas were specially mounted to reduce the influence on the fielddistribution within the RC.

Spectrum analyzerFor scattering parameter measurements and EUT emission tests, a Rohde & SchwarzESI 40 spectrum analyzer (f = 20 Hz. . . 40 GHz) was used.

Attenuators, cables, and adaptorsA 10 dB attenuator was inserted between the output of the signal generator and theamplifier’s input to protect the input stage of the amplifier. Against excessively reflectiveloads connected to the amplifier’s output, a 3 dB attenuator was used between the outputand the directional coupler’s input to protect the amplifier’s output stage. All coaxialcables were Sucoflex 104 by Huber+Suhner with N and SMA connectors and adaptors.

5.2.2 Field probe system

The field probe system is utilized to measure the three components of the electric field.Its main purpose in this project was to allow a validation of the simulation results;furthermore it is used during calibration of the RC. For actual EMC measurements, afield probe is not necessarily needed, but still recommended [6]. Since the calibration of


Figure 5.6: Field probe (sensor head and processing unit) with non-conductive stand, suspendedTX/RX antenna, and measurement grid on the chamber floor.

an RC requires recording of the electric field in eight or respectively nine points (thesepoints form the so-called “volume of uniform field”), it is highly advantageous to employa system which is capable to measure field data in all these points simultaneously –otherwise the field probe needs to be moved from one point to the next in a time-consuming, sequential procedure.The field probe system as shown in Fig. 5.6 can be usually separated into three parts:the actual probe sensor head which measures the field (mostly electrically, in some newersystems also optically [160]), the probe’s processing unit (the “electronics box”) whichreads out and converts this data, and the probe’s mechanical support (this can be a simplestand, a rack, or in sophisticated systems a remote-controllable 3-D manipulator).

Field probe sensor head

The probe’s sensor head (a Type-8 device manufactured by Narda Safety Test SolutionsSystems, formerly Wandel & Goltermann) is a combination of three electrically short(l λ) dipoles mounted in a special orientation on a prism, so that they are capa-ble of providing simultaneously three-component electric field data [161]. Each of themeasured electric field components is rectified inside the sensor head using diodes and


transmitted to the processing unit through a high-impedance (resistance several MΩ/m)transmission line. This is done in order not to disturb the field distribution around theprobe system [155]. The distance between the sensor head and the processing unit iskept constant at 30 cm (see Fig. 5.6), which allows a repeatable calibration of the wholesystem. The Type-8 probe head can be operated between f = 100 kHz and 3 GHz.

Field probe processing unitThe rectified voltages of the three short dipoles are transferred to the field probe’s mainprocessing unit EMR-20 [156] (the box with the six buttons in Fig. 5.6). The EMR-20performs all data processing which includes auto-zeroing (needed to correct for offsetvoltages and temperature drift of the impedance transformers and the analog-to-digitalconverters), sending/receiving of data to the equipment control PC, and power manage-ment. The dynamic range of the sensor head in combination with the processing unit isgreater than 60 dB, allowing to detect field strengths ranging from 0.6 . . .800 V/m with-out range-switching. This is accomplished using precision analog-to-digital converterswith a dynamic range of 120 dB in the processing unit.In order to eliminate the influence of the probe’s data link on the EM field, the datais transmitted to the measurement PC with a bidirectional serial RS232 optical fiberconnection.

Optical fiber feed-through waveguideIn order to provide sufficient SE of the RC, a cylindrical waveguide is used to feed theabove-mentioned optical fiber of the field probe system through one of the RC’s walls.The waveguide is operated below its fundamental cutoff frequency fcmn

, which can becalculated for the TEmn mode from

fcmn=

J ′mn

πd√

µ0ε0εr(5.4)

wherein J ′mn is the n-th zero of the first derivative of the Bessel function Jm, d is the

diameter and εr the dielectric filling of the waveguide section [162]. The propagationconstant for f < fcmn

is given by

βz =√

β20 − β2

mn =

√(2πf

co

)2

− β2mn = ±j

√√√√( 2π√εr

πdJ′

mn

)2

−(

2πf

co

)2

(5.5)

The fundamental mode (i.e. the mode with the lowest cutoff frequency) of a circularwaveguide is TE11 which results in

βz = ±j

√(2J ′

11√εrd

)2

−(

2πf

co

)2

(5.6)

Using J ′11 ≈ 1.84, the attenuation of a circular waveguide of length l for the fundamental

TE11 mode is therefore equal to

α · l =

√(2J ′

11√εrd

)2

−(

2πf

co

)2

· l ≈√(

2 · 1.84√εrd

)2

−(

2πf

co

)2

· l (5.7)


Field Solver&

Graphical UserInterface

SimulationPostprocessing

&Measurement

Data Acquisition

GeometryModeling

&Simulation

Preprocessing

MS Access®

DatabaseSystem

Compliance®

MeasurementSystem

Interface

DataExtraction

InterfaceMATLAB®

Statistics &Benchmarks

Figure 5.7: Schematic measurement and data acquisition procedure.

For the prototype RC of this thesis, an air-filled (i.e. εr = 1), l = 0.1 m long circularwaveguide with a diameter of d = 0.013 m was used, resulting in a cutoff frequencyfor the fundamental TE11 mode of fc11 ≈ 13.5 GHz and a theoretical attenuation ofα · l ≥ 200 dB for frequencies f ≤ 6 GHz – this is considerably more attenuation thanthe chamber itself can provide (cf. Section 5.1.3).

5.2.3 Data acquisition and interfacing

As shown in Fig. 5.5, all active devices of the RC equipment setup (signal generator,power meter, field probe system, spectrum analyzer, and stirrer drive controller) areremote controlled from a computer via the GPIB and RS232 bus. The RC control anddata acquisition programs were written using the Schaffner Compliance C3i softwaresuite [154] . Compliance provides the necessary equipment drivers enabling a high-levelcontrol of measurement functions. The developed RC programs are specially adapted tothe requirements of measurements used for simulation validation: they allow to define arectangular test grid for the field probe system by setting spatial (x, y, z) start and stoppositions within the RC, measurement grid spacing, the frequency range of interest, andthe desired rotational stirrer angles.The acquired measurement data is initially stored in the Compliance database (DB)


system. In order to perform extended measurement-vs.-simulation benchmarks and touse advanced analysis tools, the procedure shown in Fig. 5.7 is needed to transfer thedata from Compliance : To facilitate data handling, measurement (and also simula-tion) results are gathered through a MATLAB -based data extraction interface fromCompliance (and respectively FEKO ) and fed directly into a common MS AccessDB. With this procedure, benchmarks, statistical analyses, and 2-D/3-D visualizationsof simulated and measured data are performed by retrieving the results from the DBwithout the need to manipulate (e.g. scale, offset, etc.) underlying data – this in turnsignificantly reduces the probability of accidentally introducing errors. Data transfers areaccomplished using SQL expressions from MATLAB via the ODBC application pro-gramming interface included with MS Windows . Further details on the Compliance -to-Access DB and FEKO -to-Access DB interfacing system can be found in Appendix B,Section B.2.

5.3 Measurement errors

Deviations between measurements and simulations can have two reasons: either thesimulation does not reproduce the physical phenomena inside the RC accurately or themeasured data is incorrect and additionally varying in repeated measurement trials. Thefirst issue is extensively discussed in Chapter 7, reasons for measurement errors are ad-dressed in this section. It is essential to distinguish the term “error” (in a measurementresult) from the term “uncertainty”: error is the measurement result minus the truevalue of the measurand [163, 164]. Whenever possible a correction equal and of oppositesign to an error is applied to the result. Because true values are never known exactly,corrections are always approximate and therefore a residual error remains [165, 166].The uncertainty in this residual error will contribute to the uncertainty of the reportedresult. Uncertainty can be characterized in terms of the spread of the probability dis-tribution for the residual error; further references concerning measurement uncertaintiescan be found in Section 5.4.The sources of measurement errors can be attributed to the field probe system, theTX/RX antennas, and the chamber prototype itself. A general problem for the determi-nation of errors related to near-field measurements is the strongly frequency-dependentimpact on the results – if e.g. the field probe system is capable of a spatial resolution of0.05 m, the resulting error will be very different at 50 MHz (free-space wavelength 6 m)compared against the one at 1 GHz (free-space wavelength 0.3 m). Whereas in the firstcase measured field values are not going to change much as the probe is displaced by0.05 m, at 1 GHz a significant difference in the displayed field values can be expected.

5.3.1 Field probe system

For the field probe system, it is useful to make a distinction between inherent, quasi-inherent, and non-inherent errors. Inherent errors are understood to be fundamentalerrors that – with today’s available techniques – cannot be eliminated as they are couplede.g. to the physical measurement principle. Although one “has to live with” inherenterrors, it is important to be aware of these elementary limitations. Errors are referred

5.3 MEASUREMENT ERRORS 85

| |E [V/m]

60

50

40

30

20

10

050 100 150 200 250 300

Frequency [MHz]f

135 180 225 27090450 315

Figure 5.8: Isotropy of the electric field probe between f = 50 . . . 300 MHz for eight differentprobe head orientations [156]. Shown is the magnitude of the measured electric field |E|.

to as being quasi-inherent, if a deviation only occurs because of a certain equipmentbeing used. Differently designed equipment may not exhibit a particular effect to thesame extent, so in principle it might be possible to get rid of this error contribution.As the measurements in this thesis were to be performed with test equipment at hand,the quasi-inherent errors have to be accepted “as is”. Finally, non-inherent errors areinaccuracies which can be significantly reduced by using rather straightforward methods(e.g. a protective radome on the probe’s sensor head that slightly distorts the fielddistribution, but which can be removed without problems).

Inherent errors• isotropy of the field probe: an ideal field probe would measure EM fields in a

perfectly isotropic manner, i.e. there is no preferred spatial orientation.

• sensitivity of the field probe: the EM fields within the RC need to be sufficientlylarge in order to have an accurate response of the probe’s dipole sensors.

• finite probe size: theoretically the field should be measured at one, infinitely smallpoint in space. Practically this is not possible, instead the sensor “spatially aver-ages” the EM field over its finite geometric extent. The nonzero field probe sensorsize may mask an actually poor reverberation performance [28]. This effect of“spatial averaging” becomes extremely pronounced if antennas are used as sensorsinstead of field probes to quantify the EM fields. Usage of antennas for near fieldmeasurements is therefore strongly discouraged.


• offset between x-, y-, and z-dipole loops: theoretically all three field componentsshould be measured at the same location. As mentioned in Section 5.2.2, thedipoles are mounted on a prism and therefore measure the field at slightly differentpositions.

• mutual coupling between x-, y-, and z-dipole loops: ideally, all loops should mea-sure the individual field components separately. In all practical field probes, thereis however mutual coupling by physical design constraints.

• response time/sampling rate of the field probe: an ideal field probe would measurethe field instantaneously and respond to changes without any delay. In practice,a certain settling time is needed; sampling rates on the order of 20 . . .40 Hz arecommon [156, 167].

All the errors mentioned above cannot be eliminated and will be more or less pronounceddepending on the particular field probe system used. Compared with the quasi-inherentand the non-inherent errors outlined below, however, they can be classified to a firstapproximation as negligible in the EMR-20 field probe system. The probe’s sensorType-8 head exhibited a good isotropy, one exemplary result where the probe head isrotated in 45 steps over 360 is shown in Fig. 5.8. The lower detectable electric fieldmagnitude of the field probe system is specified as 0.6 V/m, actually measured valuesused for further analysis were always greater than 10 V/m. As in this thesis only themode-tuned RC operation is investigated with non-transient, steady-state fields, theresponse time/sampling rate issue does not need to be taken into account.

Quasi-inherent errors• distortion of the field to be measured: by the physical presence of the field probe

system (sensor head, processing unit, transmission lines between sensor head andprocessing unit, probe stand) the EM field is disturbed. Several field probe manu-facturers e.g. recommend to measure the field without placing the processing unitnearby. Instead it is suggested to separate the probe’s sensor part from the elec-tronic processing unit and to transfer the sensor data via a fiber optic link. Theprocessing unit is to be placed outside of the RC.

• general application of field probes in near field conditions with possibly strong fieldgradients: several field probes are not suitable for use in near field conditions, theymay only be employed sufficiently far away from an excitation antenna, e.g. onlyonce plane wave propagation is predominant [168, 169].

The quasi-inherent errors can be a significant contribution to the overall measurementerror budget, especially if field probe systems are used beyond their intended range ofoperation (cf. far field limitation). On the contrary, the EMR-20 system with the Type-8sensor head is explicitly recommended for use in near field conditions such as the onesencountered in RCs. Field distortion will always be an issue with the EMR-20 system,as the sensor head cannot be operated without the bulky processing unit and the probestand attached to it. At least the mechanical probe stand is made out of non-conductive,“electromagnetically transparent” material.


1.1

7

-1.2

3

0.9

7

0.7

7

0.5

7

0.3

7

0.1

7

-0.0

3

-0.2

3

-0.4

3

-0.6

3

-0.8

3

-1.0

3

-1.16

-0.96

-0.76

-0.56

-0.36

-0.16

0.04

0.24

0.44

0.64

0.84

1.04

1.3

7

1.24

-1.4

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0

0.2

0.4

0.6

0.81

1.2

1.4

1.6

1.82

2.2

2.4

2.6

2.8

x

y

z

zm

ym

xm

Figure 5.9: Schematic RC measurement grid with measurement coordinate system (xm, ym, zm)and simulation coordinate system (x, y, z) overlay, TX antennas and stirrer (cf. also Fig. 5.1).

Non-inherent errors

• due to non-ideal probe response, the measured field strength (e.g. displayed is1.1 V/m) is not equal to the actual physical field strength (e.g. 1 V/m). Througha proper broadband calibration, this effect can be reduced.

• accuracy of the x, y, z measurement position: a difference from the measured to thesimulated results can occur simply because the spatial measurement and simulationposition are not identical. This problem already starts with the difficulty of clearlydefining spatial measurement positions within a prototype RC, since the walls arenot completely flat or the field probe stand might be slightly tilted.

In the case of the field probe system, the non-inherent errors have by far the biggestimpact on an agreement between simulation and measurement, especially at higher fre-quencies. In order to position the field probe system correctly within the RC, twomethods were used: first of all, a fine measurement grid (spatial resolution 0.05 m) wasprinted on the chamber bottom (see Fig. 5.9 for a schematic overview) for a coarse pre-positioning and a plummet attached to probe head for proper alignment with the grid.Secondly, a laser range distance metering device can be attached to the probe head,which ensures a more precise and repeatable positioning of the field probe system.


a) b)

Figure 5.10: Biconical TX antenna a) tripod-mounted and b) suspended from the ceiling withplastic wires and Velcro. Attached is a plummet for precision positioning.

5.3.2 Antennas

The influence of the TX/RX antennas on the agreement between simulated and measuredresults can be summarized by essentially three issues:

• accuracy of the antenna position (x, y, z) and alignment angle θ: experiments inthe prototype RC revealed that the EM field distribution is very sensitive to theexcitation antenna position. The argument here is very similar to the one discussedabove for the positioning accuracy of the field probe system (cf. Section 5.3.1).

• presence of the tripod in the RC prototype: whereas the antennas in the RCmeasurements need to be somehow mechanically supported, in the simulationsthey are “floating in the air”.

• coaxial cable feed: the simulated antennas are excited by an impressed voltageacross a gap or an impressed current flowing within the source segment. In practice,a coaxial cable is needed to connect the antennas to the amplifiers, it was howeveralways routed as close as possible to the RC’s walls.

Antennas in the prototype RC were carefully positioned using the measurement gridprinted on the chamber bottom, a plummet was attached to the antenna, and the laser


50 100 150 200 250 300

Frequency [MHz]f

| |E [V/m]

60

50

40

30

20

10

0

70

80Tripod Plastic wires and Velcro

Figure 5.11: Magnitude of the electric field |E| measured at a fixed position (x = −0.63 m,y = 0.64 m, z = 0.47 m) for a biconical antenna mounted on tripod vs. suspended withplastic ropes and Velcro from the chamber ceiling (f = 50 . . . 300 MHz).

range distance metering device employed which had been introduced for the field probesystem. This ensures the greatest possible agreement between TX/RX antenna positionand angular alignment in the measurement setup and the simulation.Initial comparisons between simulations and measurements were carried out with theexcitation antenna supported by a tripod. With this setup, at higher frequencies con-siderable differences between measured and simulated results occurred. A closer exam-ination revealed that the tripod had metallic feet, “legs” made out of wood, a plasticmounting rod, head, and antenna clamp. It was found that especially the highly ab-sorbing wooden “legs” caused a significant distortion of the field distribution withinthe prototype RC. In order to avoid further unintentional loading and distortion ofthe EM field, the tripod was removed and the TX/RX antennas either suspended fromthe chamber ceiling using nylon ropes and Velcro or placed onto styrofoam blocks (seeFig. 5.10). A comparison between the magnitude of the electric field |E| measured at afixed position (x = −0.63 m, y = 0.64 m, z = 0.47 m) for a biconical antenna mountedon the tripod against the same antenna suspended by plastic ropes and Velcro from thechamber ceiling (f = 50 . . . 300 MHz, ∆f = 1 MHz frequency resolution) is depicted inFig. 5.11. It can be seen that the measured field values are quite dissimilar concerningtheir magnitudes and that – contrary to common believe – without the presence of theabsorbing tripod in the RC the magnitude is not necessarily greater at a fixed location.This effect can be explained by looking at the field distribution along a line within theRC, rather than at a fixed position only: Fig. 5.12 depicts the magnitude of the electric


0

10

20

30

40

50

-1.2 -0.8 -0.4 0 0.4 0.8 1.2y [m]

| |E [V/m] Tripod Plastic wires and Velcro

150 MHz

200 MHz

0

10

20

30

40

50

-1.2 -0.8 -0.4 0 0.4 0.8 1.2y [m]


250 MHz

300 MHz

a)

b)

Figure 5.12: Magnitude of the electric field |E| measured along a line (x = 0.57 m, y =−1.2 . . . 1.2 m, z = 0.47 m) for a biconical antenna supported by a tripod vs. suspendedwith plastic ropes and Velcro from the chamber ceiling (f = 150 MHz, 200 MHz, 250 MHz,and 300 MHz).

5.4 MEASUREMENT UNCERTAINTY BUDGET 91

field |E| measured along a line (x = 0.57 m, y = −1.2 . . .1.2 m, z = 0.47 m) for a biconi-cal antenna supported by a tripod vs. suspended with plastic ropes and Velcro from thechamber ceiling at four frequencies (f = 150 MHz, 200 MHz, 250 MHz, and 300 MHz).Obviously the presence of the tripod does not simply attenuate (i.e. linearly scale) theelectric field, it also distorts the field distribution. This implies that a peak measured ata certain point with the tripod in the chamber will be shifted spatially once the tripodis removed, which explains the behavior shown in Fig. 5.12. The impact of the tripodcan be seen at other spatial locations in Fig. B.1, B.2 and for additional frequencies inAppendix B, Figs. B.3-B.5.

5.3.3 Chamber and stirrer

Once the tripod was removed and the antennas suspended from the chamber ceiling,the RC was further analyzed concerning differences between the actual measurementprototype and the simulation model. The majority of these differences are of the type“geometry/position problem” and similar measures as for the non-inherent errors of thefield probe system can be applied. Worthwhile mentioning are the following issues withrespect to the stirrer and chamber for modeling of the RC:

• the RC door (frame, door handle, gasket) exhibited a strong impact at certainfrequencies and needed to be taken into consideration in the simulation model (seeSection 6.1.3)

• flexing of the stirrer rod (into a curved shape, which offsets the paddle positions)was minimized by adding a special fixture, see Fig. 5.2b); the stirrer rod itself wasfound to be electromagnetically irrelevant and was hence not simulated

• unnecessary cable channels for power lines were removed in the prototype RC

• chamber imperfections (screws, stirrer fixation panels, special corner mounts), ven-tilation ducts, lighting, and the RF feed-through panel proved to have an onlyminor effect on the measured EM field in the lower-to-medium frequency rangeand were therefore neglected in the RC simulation model

It should be emphasized that the sensitivity of both simulations and measurements on theabove-mentioned issues (e.g. the door or chamber imperfections) aggravates substantiallywith increasing frequency (see Section 7.4).

5.4 Measurement uncertainty budget

It is beyond the scope of this thesis to investigate the reliability and significance of EMCtests of EUTs inside an RC. An introduction to the field strength uncertainties to beexpected for a given number of stirrer steps within an RC was given in Section 2.5. Inorder to establish an uncertainty budget, the “Guide to the expression of uncertaintyin measurement (GUM)” can be advantageously used [170]. This standard evaluationapproach is in the form of a cookbook and comes with the widely used “GUM Work-bench” software. Further general application notes for uncertainty budgets are provided


in e.g. [163, 171, 164]. Very detailed information on the assessment of special RC-relatedmeasurement uncertainties for chamber calibration and EUT emission and immunitytesting can be found in [22] and in the IEC 61000-4-21 standard [6].

5.5 Conclusion

The construction and setup of the reverberation chamber (RC) prototype having innerdimensions of 2.86 · 2.48 · 3.06 m3 (width · length · height) including the door, stirrers,and auxiliary equipment was described. The issue “What does electrically large meanfor a rotating object such as a stirrer in an RC?” versus “What does electrically largemean in the traditional electromagnetic sense?” was brought up and will be addressed inSection 7.7. Specifications of the measurement system (consisting of the transmit/receiveequipment and the field probe unit) developed for data acquisition were outlined. Thefield probe system with its probe head and processing unit was analyzed in detail, sinceit forms the most important part of the measurement system for validation of simulationresults. A system capable of providing simultaneously three-component electric fielddata without distorting the field distribution significantly was chosen.Measurement errors originating from field probes, antennas, and stirrers were discussedand assessed for their impact on deviations between simulated and measured results.The biggest deviations were found to result from the antenna tripods and position inac-curacies of the field probe head or the antennas. For validations of the measurements,tripods were entirely removed from the RC and antennas suspended with plastic ropesand Velcro from the ceiling. A positioning and alignment system comprised of an opticalgrid and a laser range distance metering device was proposed. Other measurement errorsresulting from anisotropy, sensitivity, mutual coupling and finite extension of the fieldprobe were found to be negligible. The fundamental limitations concerning the measure-ment system addressed in this chapter were taken into consideration when benchmarkingsimulated against measured results in Chapter 7.

6 Modeling of the Reverberation Chamber

Abstract—This chapter describes the modeling procedure that was used for reverberation chamber

simulations. Starting with modeling of a basic cavity, a comprehensive chamber model resembling

the prototype reverberation chamber is elaborated. An analysis of electrical conductivities appro-

priate for the materials in use is presented. Furthermore, cubic and corrugated chambers, various

vertical and horizontal stirrers, transmit and receive antennas, and EUTs are designed and modeled.

6.1 Chamber models

6.1.1 Modeling procedure

Different types of RCs were modeled in order to investigate their performance and tofind out which geometric features would be the most important for an optimally de-signed chamber. One particular challenge in RC simulations is the generation of a largenumber (on the order of several hundreds) of chamber models where the only change inthe geometric structure is the step-by-step rotation of the stirrer. Following the require-ments outlined in Section 2.5, at least 50 rotational stirrer angles need to be simulatedper frequency to achieve sufficiently low statistical uncertainty. The EM simulation soft-ware FEKO requires for each simulation involving a change in the RC geometry a newPreFEKO input file. To speed up the modeling phase and to reduce the number of mod-eling errors, these input files were generated automatically with minimum user inputemploying a MATLAB -based tool specially developed for the project of this thesis.Initially, an empty RC is designed with a 3-D CAD system [56]. Stirrers, doors, EUTs,TX/RX antennas, and other objects within the RC are created separately in differentCAD models. To set up an RC simulation, parameters such as linear/logarithmic fre-quency stepping and triangle discretization area are generated with the MATLAB tooland passed on to the mesh generator and the preprocessor (Fig. 6.1). The mesh gen-erator uses these input parameters together with the rules laid out in Section 3.4.1 tocompute an estimate of the appropriate discretization for all structures. All surfacesare discretized with triangles and all wires with segments which facilitate an accuraterepresentation of arbitrarily shaped structures and are well suited for the EFIE-basedMoM technique (see Chapter 3). Furthermore, the stirrer positions are set up as wellas the position of the EUT, the TX and RX antennas and the location at which thenear field is to be computed. The procedure mentioned above allows the reliable, au-tomated setup of a large number of similar simulations, where only certain parts of theRC geometry are varying. The final RC geometry along with discretization-related datais transferred back to the CAD system, which finally creates a mesh by generating amixed triangle/segment discretization [56]. The discretization process’ output data iscombined with the simulation settings in the preprocessor, which checks the validity

93

94 6 MODELING OF THE REVERBERATION CHAMBER

3D CADMesher

HyperMesh®

GeometryConversion

InterfacePreFEKO

SimulationData Process

GeometryModeling

&Simulation

Preprocessing

Field Solver&



&Measurement

Data Acquisition

Interface

DataExtraction

MATLAB®

ParameterGenerator

Figure 6.1: Schematic geometry modeling and simulation preprocessing flowchart.

of the data (e.g. violation of certain discretization constraints) and passes it on to thesimulation kernel (Fig. 6.1). Before a simulation is finally started, the required memory(which depends mostly on the desired frequency range) is estimated: the MATLAB -based tool automatically selects whether to run a sequential, single processor simulationor to distribute the computational load equally across several machines using the parallelFEKO solver. A scheduler is utilized to run all computations belonging to the samebase model over one stirrer rotation in a row.

6.1.2 Cavity

Reasonable conductivity values for the RC walls and stirrers as listed in Table 6.4 wereobtained from initial simulations considering a simple RC model without stirrers, doorsor other geometrical details – i.e. a cavity. Triple geometric symmetries were used forcavity modeling and during the computations (see Section 3.3.3). Also stirrers in theprototype RC (see Section 5.1) were removed during measurements to match the sim-ulation model. Furthermore, these initial simulations and measurements were used toidentify chamber details having the biggest impact on the simulated and measured nearfield (e.g. ventilation honeycomb ducts, antenna cable and power line routing, or leak-age through the stirrer motor bearings). The reason not to include the stirrer in this

6.1 CHAMBER MODELS 95

x

z

y

xz-symmetry

Chamber walls

6-paddle V-stirrer

xy-symmetry

yz-symmetry

Hertzian dipoleexcitation antenna

V-stirrer axis

yz-s

y

Figure 6.2: Partly symmetric simulation model of the RC. Three geometrical symmetries areutilized to construct the RC model by mirror imaging one eighth of the chamber walls (de-picted in dark grey). The stirrer is moved to its designated position by a mathematicaltranslate/rotate operation. Note that this chamber (contrary to the RC in Fig. 6.3) does notfeature a door.

first comparison was to eliminate the possibility of deviations between simulations andmeasurements resulting merely from the rotational positioning accuracy of the stirrerpaddles. Results of the cavity simulations are shown in Section 7.2. The cavity simula-tions and measurements were also used to investigate the loading effect of the TX/RXantenna tripods mentioned in Section B.1.

6.1.3 Prototype reverberation chamber

Due to less powerful computational resources available in the past and to facilitatemodeling, simulations of RCs were commonly restricted to two dimensions – full-wave3-D simulations of RCs have only become feasible in the most recent years. Although thisis a significant step forward compared to the 2-D models used earlier, 3-D simulationsare computationally still very slow (cf. Table 3.1). Therefore significant simplificationsare usually made in RC simulation models: Double or triple symmetry (such as in thecavity simulations above) is often utilized and small geometrical details are neglected inorder to speed up simulations.

Chamber model without door

As a starting point, a partly symmetric RC was modeled using similar simplifications: Asdepicted in Fig. 6.2, the walls are constructed by designing only one eighth of the chamber


Chamber door

Excitationantenna

xyz

6-paddle V-stirrer

V-stirrer axis

H-stirrer axis

2

H-s

tirre

r axis

1

Receiveantenna

Chamber walls

Figure 6.3: Detailed fully asymmetric simulation model of the RC. Excitation source is alogper antenna located in front of the door. Shown is its 3-D free space radiation patternsuperimposed for illustrative purposes. The logper antenna depicted in the right corner isused for emission testing setups in the simulation model. E and H fields can be calculatedat any arbitrary point, as an example one of the near field computation planes is shown. Allmetallic parts are of finite conductivity.

and mirror imaging it at the xy-, xz-, and yz-plane. The geometry mirroring process isdone by the simulation preprocessor (Fig. 6.1). To exploit triple geometric symmetry, thevertical 6-paddle stirrer is discretized separately and subsequently moved to its intendedposition using a mathematical shift/rotate operation. Using this approach, there is areduction in computation time possible when the elements of the MoM system matrixare determined – however, the time and memory needed for solution of this system ofequations is not reduced as the RC is only partly symmetric (cf. Section 3.3.3). Thisbasic chamber has dimensions of 3.06 m · 2.86 m · 2.48 m (heighth ·width w · length l).To minimize the number of triangles needed to mesh the large RC surface, a frequency-adaptive and geometry-dependent discretization was used (see Table 6.1 for details). Forease of modeling, all walls and stirrers are assumed to be of single-layered metal. Asopposed to the RC model below, this chamber does not feature any apertures.


Figure 6.4: Photo of the RC door with gasket on frame and contact finger strips.

Chamber model including doorTwo RC models were designed and simulated, the “basic” one with flat walls describedabove and a more “detailed” one including the chamber door as shown in Fig. 6.3. Thedimensions of the door are 2 m in height and 0.92 m in width with a 0.07 m recess of thedoor frame from the chamber wall (see Fig. 6.5a)). Although, at first glance, these detailsseem to be small, it is shown in Sections 7.2.1 and 7.3 that they affect the chamber fieldssignificantly and thus cannot be neglected. Since the RC model including the door doesnot exhibit any symmetries, no reduction at all in memory or computation time can beachieved (cf. Section 3.3.3).As mentioned in Section 5.1.1, the prototype RC features beryllium-copper contact fingerstrips which are used as gaskets to prevent EM leakage from the door (Fig. 6.3). Leakagethrough the chamber door in the simulation would require the modeling of extremelytiny gaps between the door and the chamber, which would introduce numerical artifactsinto the RC simulation and increase considerably the number of triangles (which would


need to have edge lengths on the order of the gap’s width). Modeling of the imperfectdoor seal in FEKO can be carried out with two methods:

• Using a distributed series impedance between the edge of the chamber door cutoutand the chamber door itself (“LE-loading”): with this technique, the current distri-bution J around and on the door surface is modified. In the simulation, however,there is no EM radiation leakage out of the RC.

• If radiation leakage through the door gasket is to be simulated, a tiny gap must bemodeled in the RC model in FEKO between the door cutout and the door. Theproblem is that this tiny gap introduces numerical artifacts in the simulation dueto the scale of dimensions (size of the door vs. size of the gap). This prohibits ameaningful inclusion of the door gap in the RC simulation model.

Therefore, modeling of the door gasket was accomplished by introducing a distributed se-ries impedance between the edge of the chamber door cutout and the door itself (Fig. 6.3).The detailed fully asymmetric RC model including chamber door and gasket was simu-lated using different stirrers (Section 6.2) and antenna models (Section 6.4) and assumingfinite conductivity (Section 6.3) for all structures. This chamber is used as a benchmarkmodel for comparisons against all other RCs (see Fig. 6.5 and Table 6.1).

6.1.4 Corrugated, cubic, and offset-wall reverberation chambers

To investigate the influence of a particular RC design on the EM near field and on typicalRC parameters (stirring efficiency, field uniformity, correlation, etc.) corrugated, cubic,and offset-wall chambers were modeled and simulated for performance comparisons.

Corrugated RCCorrugations were introduced for use in RCs in analogy with acoustic ray theory andwere expected to exhibit similarly beneficial effects as in e.g. corrugated horns. Thedriving force behind corrugations applied to RCs was the anticipated improvement offield uniformity at lower frequencies close to the LUF. In [173] the corrugations have anamplitude of 1.5 inch (i.e. 0.0254 m) and a valley-to-valley distance of 2 inch. Their RC israther compact and has dimensions of 1.2 m · 0.8 m ·1.8 m (h·w·l). To investigate whethercorrugations are indeed beneficial, the dimensions of the corrugations from [173] werescaled to fit the size of the prototype RC used in this thesis. The scaled corrugationshave an amplitude of 0.1 m and a valley-to-valley distance of 0.15 m, see Fig. 6.5c).Normally the space between the topmost part of the stirrer and the chamber ceilingis 0.165 m, and between the walls and the outermost stirrer part 0.24 m. Employingthe corrugations also on the ceiling and the side walls of the RC would reduce the topspace to 0.065 m and the side space to 0.14 m. Since these spacings are rather small,the neighboring walls of the RC remained flat, see Fig. 6.5c). This chamber features thesame door as the one in Section 6.1.3. Simulations of the corrugated RC are discussedin Section 7.5.

Cubic and offset-wall RCAs shown in Section 2.2.1 and Fig. 2.4, cubic chambers suffer from mode degeneration sothat the usually required “∂N/∂f = 1.5 modes/MHz above cutoff”-criterion is reached


a) b)

c) d)

Figure 6.5: Different simulation models of the RC: a) basic RC resembling the prototypechamber (cf. Fig. 6.3), b) RC with width w = 2.96 m, c) corrugated RC, d) cubic RC.

consistently only at much higher frequencies compared to an RC of the same volume,but non-cubic shape. Several authors however claim that cubic chambers may generatea more uniform field than standard rectangular RCs. For this reason, a cubic chamberwas modeled having dimensions of 2.86 m · 2.86 m · 2.86 m (h ·w · l, shown in Fig. 6.5d)).Another “offset-wall” chamber was designed similar to the cubic RC, but 0.1 m wider,


Number of triangles for discretization

Chamber type 30. . . 250MHz a

30. . . 400MHz b

30. . . 600MHz c

30. . . 1000MHz d

Standard (no door) 2 304 4 928 9 696 12 512

Standard (w. door) 2 490 5 360 10 832 13 118

Cubic (w. door) 2 557 5 385 10 653 12 815

Corrugated (w. door) 4 829 10 091 11 553 13 699

Offset wall (w. door) 2 526 5 166 10 278 12 506

a for this frequency range the HyperMesh element edge size was set to 0.200b —— to 0.140 c — — to 0.100 d — — to 0.089

Table 6.1: Discretization data of chambers used in the RC simulations.

see Fig. 6.5b). The offset-wall chamber has dimensions of 2.86 m · 2.96 m · 2.86 m (h ·w · l). Both the cubic and the offset-wall chamber feature the same door as the one inSection 6.1.3. Simulations of the cubic and the offset-wall RC are discussed in Section 7.5.

6.1.5 Other reverberation chambers

Two other existing RCs were modeled and simulated for comparisons and to investigateperformance scaling:

• IEH chamber at the Universitat Karlsruhe (Germany): this RC has dimensions of2.34 m · 5.24 m · 2.39 m (h · w · l) and was initially simulated at frequencies lowerthan 500 MHz to compare field uniformity levels of similar “Rot-Z” stirrers.

• Medium-sized (SMART200) and large (SMART80) ETS Lindgren chambers: thesechambers are 3.05 m · 4.83 m · 3.61 m and respectively 4.90 m · 13.40 m · 6.10 m (h ·w · l) in size. As the names imply, they are designed to operate from 200 MHzand, respectively, 80 MHz on. Simulation requirements (time and memory) es-pecially for the larger chamber grow quickly to astronomical levels, since for aproper discretization of the chamber walls at frequencies beyond 500 MHz morethan 20 000 triangles are needed (cf. Section 3.4).

6.2 Stirrer models

Contrary to the RC models, the stirrers were only modeled with two different discretiza-tion levels to cover both the lower and higher frequency range. Discretization data

6.2 STIRRER MODELS 101

a) b) c) d)

Figure 6.6: Vertical stirrer models with triangular discretization: a) 6-paddle stirrer, b) cross-plate stirrer, c) 6-paddle connected stirrer, d) upset Z-fold stirrer.

details can be found in Table 6.2 for vertical and in Table 6.3 for horizontal stirrers.Most of the vertical and all of the horizontal stirrers are shown in Fig. 6.6 and Fig. 6.7.The standard mounting position for all stirrers is 0.60 m away from the back and 0.60 maway from the right side wall (designated as “position III” in Figure 5.1). Alternativemounting positions shown in Figure 5.1 are “position I” (horizontal) and “position II”(horizontal), both spaced 0.80 m away from the neighboring walls. Details on the stirrerdesign procedure are outlined in Section 5.1.2.

6.2.1 Vertical stirrers

In order to facilitate relative performance comparisons, all simulated vertical stirrerscan be circumscribed by a cylinder with a diameter of 0.735m and 2.76 m height, i.e. allstirrers have the same “rotational diameter” and “rotational height” and therefore alsothe same “rotational volume” of roughly 2 m3. Vertical stirrers were always mounted inposition III.

Vertical paddle-type stirrers• 6-paddle stirrer: simulation replica of the stirrer physically existing in the proto-

type RC (see Section 5.1.2 and Fig. 5.2). Consists of six rectangular paddles ofsize 0.60 m · 0.60 m, rotationally offset by 60 with a slanting angle of 45 for eachpaddle. Horizontal spacing between the paddle centers is approx. 0.46 m. Distancefrom lower and upper paddle edge is 0.15 m to the bottom floor and the ceiling,respectively. The nearest edge-to-edge distance between two paddles is 0.04 m.


The stirrer rod (height 3.06 m), which supports the paddles, is not simulated asit was found to be electromagnetically irrelevant (see Section A.1 for modeling).This stirrer is shown in Fig. 6.6a).

• 6-paddle stirrer without gaps: similar to the stirrer above, but all paddles connectedwith spline surfaces, so that the paddles become one single structure. This stirrerwas used to analyze the impact of the electrical stirrer size on the field uniformityand is shown in Fig. 6.6c).

• 4-paddle stirrer: essentially the same as the conventional vertical 6-paddle stir-rer, but the top and the bottom paddle are missing. This stirrer has a smaller“rotational volume” as the stirrers above.

• 4-paddle stirrer with double gap: almost identical to the vertical 6-paddle stirrer,however two adjacent paddles in the middle of the stirrer are missing; this impliesthat the rotational “volume” is the same as for all other stirrers.

Vertical single- and cross-plate stirrersThese stirrers were designed to compare “fancy, irregularly shaped” stirrers with veryrudimentary ones.

• cross-plate stirrer: two rectangular plates of equal size that intersect at an angleof 90. The plates measure 2.76 m · 0.735 m (h · w).

• stacked cross-plate stirrer: two rectangular plates where one is mounted on top ofthe other so that the same stirrer height is achieved as for all other stirrer types.The upper plate is rotated with respect to the lower plate by an angle of 90. Eachplate measures 1.33 m · 0.735 m (h · w). The gap between the two stirrer plates is0.09 m. This stirrer is shown in Fig. 6.6b).

• single-plate stirrer: the most basic stirrer, consisting of a single rectangular platemeasuring 2.745 m · 0.735 m (h · w).

Vertical upset Z-fold stirrersThe so-called Z-fold stirrers used in the prototype RC simulations are upset versions ofthe original stirrers built by ETS Lindgren (described in Section 6.2.3).

• upset Z-fold stirrer: this stirrer is used for performance benchmarks against thestandard 6-paddle stirrers mentioned in Section 6.2.1. The upset version of theoriginal Z-fold stirrer was scaled so that it fits into an identical rotational volumeas the other stirrers. This stirrer is shown in Fig. 6.6d).

• upset Z-fold stirrer with gaps: for the most part identical to the upset Z-fold stirrermentioned above, however the metal “Z-fold”-part is not a single piece, but brokeninto three separate parts with two gaps of approx. 0.09 m between each other.

6.2 STIRRER MODELS 103

Number of trianglesfor discretization

Stirrer type 30. . . 600MHz a 30. . . 1000MHz c

6-paddle standard 443 599

6-paddle without gaps 695 926


4-paddle with double gap 295 350

Double cross-plate 879 1 039

Stacked cross-plate 376 592

Single-plate 439 525

Upset Z-fold 391 (+ 1 495 c) 475 (+ 2 049 c)

Upset Z-fold with gaps 363 (+ 1 495 c) 389 (+ 2 049 c)

a for this frequency range the HyperMesh element edge size was set to 0.100b — — to 0.085c supporting sidewall structure (electromagnetically irrelevant)

Table 6.2: Discretization data of vertical stirrers used in the RC simulations.

6.2.2 Horizontal stirrers

• 4-paddle stirrer: consists of four rectangular paddles (size 0.80 m · 0.80 m), rota-tionally offset by 90 with a slanting angle of 45. The nearest edge-to-edge dis-tance between two paddles is 0.02 m. This stirrer is similar to the one described inSection 6.2.3 and is shown in Fig. 6.7a).

• 5-paddle stirrer: features five rectangular paddles of 0.60 m · 0.60 m, rotationallyoffset by 72 with a slanting angle of 45. The nearest edge-to-edge distancebetween two paddles is 0.04 m. This stirrer is shown in Fig. 6.7b)

• 6-paddle stirrer: identical to the “vertical 6-paddle stirrer” described in Sec-tion 6.2.1, but mounted horizontally at position II (see Fig. 5.1). This stirreris shown in Fig. 6.7c).


a)

b)

c)

Figure 6.7: Horizontal stirrer models with triangular discretization: a) 4-paddle stirrer,b) 5-paddle stirrer, c) 6-paddle stirrer.

6.2.3 Stirrers used in other reverberation chambers

The following stirrers are employed in the ETS Lindgren and IEH RCs mentioned inSection 6.1.5 and were utilized to design stirrers needed for performance benchmarks:

• vertical “Rot-Z” stirrer: the stirrer in the IEH RC chamber is a slightly modifiedversion of the original one used in a small RC at the NSWCDD [173]. This stirrerconsists of four rectangular paddles of size 0.80 m · 0.80 m, rotationally offset by90 with a slanting angle of 45. The nearest edge-to-edge distance between twopaddles is 0.04 m.

• original vertical and horizontal “Z-fold” stirrer: stirrers employed in ETS Lind-gren’s SMART80 and SMART200 chambers are very similar in their geometry(one is simply an upset version of the other). Both of them have the patented“Z-fold” design [150], which was originally developed together with Hatfield andSlocum [174], and measure 4.40 m · 1.52 m · 1.21 m (h · w · d). They consist of afolded metal sheet (“Z-fold”), have a small plate with an aperture on the top(in the patent called “radiation-leakage device” [150]) and walls supporting the“Z-fold” structure. The latter is electromagnetically irrelevant and therefore notincluded in the simulation (see Section A.1). It is used for illustrative purposeshowever in the preprocessor PreFEKO.

6.3 WALL AND STIRRER CONDUCTIVITIES 105

Number of trianglesfor discretization

Stirrer type 30. . . 600MHz a 30. . . 1000MHz b




a for this frequency range the HyperMesh element edge size was set to 0.100b —— to 0.085

Table 6.3: Discretization data of horizontal stirrer models used in the RC simulations.

6.3 Wall and stirrer conductivities

The main materials used for construction of RCs are usually galvanized steel, aluminum,and copper (cf. Section 5.1). For the simulation of the materials in the RC prototype,a relative magnetic permeability of µr = 1 Vs/(Am) was used. Electrical conductivityvalues for all metallic structures were assumed to be equal to DC conductivities and toremain constant over the 50 MHz. . . 1 GHz range (according to NIST, DC conductivityvalues can be safely used for metals at frequencies up to 5 GHz [175]). For the aluminumstructure of the stirrers, the tabulated value of σ = 27 · 106 S/m in Table 6.4 was used.Obtaining reasonable values for galvanized steel walls proved to be rather cumbersome:the problem was to find conductivity values for this material as it is used in a shieldedroom construction, i.e. walls consisting of several interconnected sheets with intermedi-ate overlapping flat stock.The main material components of commercially available galvanized steel sheets are typ-ically iron (97. . . 99 wt.-%), zinc coating (0.5. . . 2 wt.-%), copper (0.4 wt.-%), manganese(0.4 wt.-%), phosphorus (0.1 wt.-%), carbon (0.05 wt.-%), and sulfur (0.05 wt.-%). Asthe skin depth in the zinc coating goes down with rising frequency (2.50) from approx.18 µm at f = 50 MHz to 4 µm at f = 1 GHz, it is logical to assume that the surfaceimpedance ZS (3.24) of the chamber walls tends to be increasingly defined by the con-ductivity of the zinc coating alone rather than the conductivity of the zinc combinedwith the underlying iron. Commercially available iron has conductivities on the order ofκ = 10 · 106 S/m, commercially available galvanized steel sheets have κ = 0.95 · 106 S/m.Pure zinc is listed in [149] with κ = 16 · 106 S/m, [24] reports the measured conductivityof zinc-coatings at frequencies of 1. . . 6 GHz used for galvanized steel as κ = 12 ·106 S/m.A detailed overview on conductivity data of commonly used materials can be found inTable 6.4. Measurements of the chamber quality factor Q however always exhibit onerepeating pattern: the theoretically predicted Q (2.51) is much higher than the mea-sured one (difference of a factor 10. . . 500 depending on the frequency) [24, 9]. This in


MaterialConductivityκ[

106 S/m] Conductivity

κ/κAg (relative c)

Silver (Ag) a 61 1

Copper (Cu) a 58 0.95

Aluminum (Al) a 37 0.61

Aluminum b 27 0.44

Zinc (Zn) a 16 0.26

Zinc b 12 0.20

Iron b 8 0.13

Galvanized steel (GS) b 3 0.049

Stainless steel b 1 0.016

a pure material at T = 300 K [149, 175, 176]b commercially available material at T = 300 K [149, 177]c normalized to the conductivity of pure silver

Table 6.4: Typical electrical conductivity values for materials used in EMC applications.

turn means that the RC’s wall and stirrer conductivity is consistently estimated too highor, vice versa, the chamber loading too low. In any case, assuming overall conductivityvalues of zinc in an RC simulation, results in unrealistically high field strengths withinthe chamber.Taking into account interconnections between the different metal sheet panels, screwsand cutouts as well as dirt and occasional oxidation spots, the best agreement betweensimulations and measurements (see Chapter 7) was achieved with conductivities in the“Simulation Medium” (κ = 0.09 · 106 S/m) and “Simulation High” (κ = 1.1 · 106 S/m)range (see Table 6.5). These values were used throughout the MoM simulations in thesurface impedance approach introduced in Section 3.3.4.

6.4 Transmit and receive antenna models

The transmit (TX) (i.e. excitation) and receive (RX) antennas used in the RC simula-tions were either ideal, infinitely small Hertzian dipoles, realistic λ/2-dipoles, biconical(50 . . .350 MHz), logper (300 MHz. . . 5.2 GHz), or horn antennas (2.2 . . .3.3 GHz). With

6.4 TRANSMIT AND RECEIVE ANTENNA MODELS 107

SimulationMaterial

Conductivityκ[

106 S/m] Conductivity

κ/κGS (relative b)

“High” 1.1 0.37

“Medium”a 0.09 0.03

“Low” 0.05 0.017

a “Medium” is the default conductivity used in the RC simulationsb normalized to the conductivity of galvanized steel (see Table 6.4)

Table 6.5: Electrical conductivity values used in the RC simulations.

the exception of the Hertzian dipole, the same antennas were also used during the mea-surements. Depending on the testing scenario in the RC (emission, immunity, benchmarksimulation-measurement) either one or two antennas were operated within the chamber(one in the TX, one in the RX mode). A setup with two modeled logper antennas in TXand RX operation is depicted in Fig. 6.3. An overview of all utilized antennas along withtheir respective 3-D far field radiation pattern is given in Fig. 6.8. Before employing theTX and RX antennas in the RC simulation, their far field patterns were simulated sep-arately and validated by measured or analytical results. All antennas can be positionedin the simulation at any arbitrary location and in any orientation within the RC.Antenna wire structures (such as the logper antenna or the feed section of the horn)were discretized using λ/15 . . . λ/10-long segments of finite conductivity σ = 1.1·106 S/m(“High”, cf. Table 6.5) and finite diameter. The waveguide section and the flares of thehorn antenna were modeled with triangles of finite conductivity σ = 1.1 · 106 S/m. Sinceinside a typical RC with well-conducting walls there is EM field generated with virtuallyzero input power and the coupling between antennas and the RC itself is very strong,the active power at the TX (excitation) antenna terminals appears as almost zero orin some cases even slightly negative [178]. The reactive power at the antenna ports ishowever fairly large, and therefore it is not possible to compute scattering parameterdata correctly with EFIE-based MoM simulations in an RC [179].

6.4.1 Ideal Hertzian and realistic λ/2-dipole

In the first RC simulations, an ideal Hertzian dipole was used for the whole frequencyrange of interest, i.e. 50 MHz. . . 1 GHz (see Table 6.6). This was done for two reasons:initially for the sake of simplicity, since the Hertzian dipole can be implemented analyt-ically in the numerical code. Secondly, at the start of this project in June 2001, all (butone) published RC simulations had used Hertzian dipoles as an excitation source (seeTable 4.1). Due to significant disagreement between measured and simulated EM fields(see Section 7.4.2), the usage of Hertzian dipoles in the simulations was not considered.The free-space far field pattern of the Hertzian dipole is shown in Fig. 6.8a).


a) b)

c) d)

Figure 6.8: Antenna models with corresponding 3-D free-space far field radiation patterns usedin the RC simulations as TX and RX antennas: a) ideal Hertzian dipole (infinitely small),b) biconical antenna (f = 50 . . . 350 MHz), c) logper antenna (f = 300 MHz. . . 5.2 GHz),d) horn antenna (f = 2.2 . . . 3.3 GHz).

However in order to investigate further the effect of a basic (but for EMC testing unre-alistic) RC excitation, a λ/2-dipole was modeled and simulated. This half-wavelengthdipole (discretized with 11 wire segments) was used as a model for the adjustable preci-sion conical dipole used in initial RC measurement setups (Section 5.2.1).

6.4.2 Biconical antenna

The biconical antenna which was modeled for the RC simulations is the A.H. Systems,type SAS-541 antenna used in the measurements (Section 5.2.1). This antenna consists

6.4 TRANSMIT AND RECEIVE ANTENNA MODELS 109

Frequency rangeTX/RXantenna

Number ofsegments / triangles

Arbitrary Hertzian dipole 0 / 0

50 MHz. . . 3 GHz λ2 -dipole 11 a / 0

50 MHz. . . 350 MHz biconical 162 b / 0

300 MHz. . . 3 GHz c logper 435 a / 0

2.2 GHz. . . 3.3 GHz horn 1 a / 2 486

a excitation source was modeled with one segment b —— with two segmentsc discretization limits the maximum frequency of the logper antenna to 3GHz

Table 6.6: Discretization of transmit (TX) and receive (RX) antennas.

of a pair of six trapezoidally-shaped thin rods that are rotated by 60 against each otherwith a 0.1 m long feed line in-between. The largest overall dimension of the biconicalantenna is 1.32 m. The biconical antenna was modeled by 162 wire segments (with twosource segments for symmetry reasons), its free-space far field pattern (simulated gain1.5. . . 2.1 dBi) is shown in Fig. 6.8b).

6.4.3 Logarithmic-periodic antenna

The logarithmic-periodic (logper) antennas used in the measurements (Schwarzbeck typeUSLP 9143) were also modeled for the simulations. Although these logper antennas are inpractice useable up to f = 5.2 GHz, the discretization limits their maximum operationalfrequency to 3 GHz. This is due to the very short, but comparatively “thick” segmentsutilized for the tip dipoles of the antenna. Given a certain wavelength λ and geometricwire radius r, the conditions imposed on the discretized segments of length ∆l using theEFIE with MoM (Section 3.2) are such that

r < ∆l <λ

10(6.1)

As outlined in Section 3.2.4, in order to be able to accurately approximate the line cur-rents on a wire structure, the segment length ∆l must be smaller than λ/10. Furthermoreas (6.1) suggests, ∆l has at the same time a lower limit of r (Section 3.4). Large radii rtherefore limit ∆l to a certain minimum length greater than r. Since higher frequenciesf correspond to smaller wavelengths λ (c = λ · f), and for small λ a finer discretizationis needed (i.e. segments with sufficiently small ∆l). This implies that the wire radiusr defines a maximum simulation frequency. With the “thick” segments of the logperantenna, this maximum frequency is at around 3 GHz, even though the physical antenna


is specified to work up to 5.2 GHz. In total 435 wire segments (including one sourcesegment) are needed to model the logper antenna (Table 6.6). Its free-space far fieldpattern (simulated gain 5.2. . . 7 dBi) is shown in Fig. 6.8c).

6.4.4 Horn antenna

The TX/RX horns mentioned in Chapter 5 were almost exclusively used for RC mea-surements (with the exception of a few special RC/horn simulations used for a couplinganalysis). Since the frequency range of most of the simulations in this thesis was limitedto 50 MHz. . . 1 GHz, horn antennas are not considered in the simulation vs. measure-ment comparisons discussed in Chapter 7. Reasons for the restriction to frequenciesf ≤ 1 GHz were the prohibitive computational runtimes and memory requirements(cf. Section 3.4), along with the electromagnetically extremely sensitive RC structureat higher frequencies (discussed in Section 7.4.2). The horn antenna has an apertureof 0.21 m · 0.28 m with a rectangular waveguide feeding section of 0.088 m · 0.044 m andan overall length (waveguide shorting back-plate to horn aperture) of 0.445 m. It wasmodeled with 2 486 triangles and one segment for the excitation in the waveguide part.The free-space far field pattern (simulated gain 15.3. . . 17.1 dBi) of the horn antenna isshown in Fig. 6.8d).

6.5 Canonical equipment under test (CEUT)

The term “canonical equipment under test (CEUT)” is normally encountered in so-called“round-robin-tests” where the aim is to compare emissions or immunity test results invarious testing environments [138]. The CEUT serves as a “standardized” EUT whichexhibits a consistent and reproducible radiation pattern. For simple CEUTs, the emis-sion/immunity test response can be calculated even analytically. Comparisons are usu-ally made between different laboratories (“inter-laboratory comparison”) where the testsare made in either similar (e.g. different ACs against each other) or dissimilar testingenvironments (e.g. OATS against AC). CEUTs are commonly also referred to as “imi-tated equipment” [180], “tightly specified test device” [138], “reference radiator” [181],“representative EUT” [182], or “artificial EUT emitter/receiver” [183]. A typical EMCtest setup showing a box-type CEUT inside an RC is shown in Fig. 6.9.

6.5.1 Practical CEUT

The following sections provide an introduction to CEUTs as they are used in practicalemission and immunity EMC tests. Details on CEUTs modeled in the course of the RCsimulations are given in Section 6.5.2.

Emission CEUTIn the emission case, the CEUT is typically a battery-powered device, so that feedingcables are not needed. With this approach, the EM field is not influenced by the layoutof cables attached to the CEUT and their strong effect on the radiation characteristics iseliminated. CEUTs can be operated either in an autonomous or in an interactive mode,

6.5 CANONICAL EQUIPMENT UNDER TEST 111

xyz

Chamber door 6-paddle V-stirrer

RX/TXantenna 1

V-stirrer axis

Near field

computatio

n plane

Chamber walls RX/TXantenna 2

EUT

Figure 6.9: Simulation model of the RC with EUT. Shown are two TX/RX logper antennas,which can be used as excitation or receive antenna depending on the test setup (immu-nity/emission). A canonical box-type EUT is measured in the RC.

which allows certain settings to be controlled remotely by the user. Remote controlcapability is in practice usually provided through a fiber optic system (Section 5.2.2) withwhich the output power, frequency spectrum, or radiation pattern can be adjusted [180].Fixed frequency as well as comb generators are used as the main active componentsinside a CEUT, where the first generate a very narrow and the latter a broad frequencyspectrum (e.g. 5 MHz spacing with 30 . . .2000 MHz detectable output). Other possiblebroadband excitation sources are comparison noise emitters (CNEs) [184]. As explainedin detail below in Section 6.5.2, EM energy is radiated through one or more simple wireantennas (dipoles, loops) or one or more slots and gaps [182].

Immunity CEUT

CEUTs for immunity testing are significantly less widely used and “standardized” thanthe ones for emissions mentioned above. Often devices for radiated immunity testsare purpose-built for a very specific setup. Examples of special CEUTs for referenceimmunity measurements are integrated circuit timers and comparators; combined withsome simple circuitry it is possible to monitor from which field level threshold on they are

112 6 MODELING OF THE REVERBERATION CHAMBERM

odels

f=

300

MH

zf=

600

MH

z

Figure 6.10: Different canonical EUTs models (top) with their corresponding simulated 3-Dfree space radiation patterns at f = 300 MHz (middle) and f = 600 MHz (bottom). Shownare (from left to right) a realistic dipole (finite length), a loop, and a box EUT operated inslot mode (left) as well as in gap mode (right).

“disturbed” in their normal operation [185, 186, 187]. There are also box-type immunityCEUTs where EM energy is coupled into the box through slots or gaps and picked upby a metal rod. Voltage across the ends of the rod is measured and modulates inside theCEUT an optical signal. This signal is output through a similar optic system as in theemission EUT [188]. The optical signal is detected outside the testing environment andserves as a measure of the field strength that the canonical immunity EUT is exposedto [84].

6.5.2 CEUT modeling

For both emission and immunity CEUT simulations, test objects were used which werealready successfully employed in several round-robin tests (such as the FAR project ofthe European Union [189, 138, 183] and the RC, GTEM, FAR, and OATS comparisonscarried out by the FCC in the U.S. [182, 190]). These box EUTs are made of thin brasssheets soldered together at the edges. The following CEUTs were modeled and simulatedin both free-space (for validation purposes) and in the RC:

6.5 CANONICAL EQUIPMENT UNDER TEST 113

• realistic dipole, measuring 0.4 m in length

• loop EUT, measuring 0.3 m · 0.3 m

• box EUT, measuring 0.48 m ·0.48 m ·0.16 m (“gap mode”) or 0.48 m ·0.48 m ·0.12 mwith a slot of 0.12 m · 0.04 m on the front panel (“slot mode”)

For the box-type CEUT, EM radiation from the inside of the box to the outside (or viceversa) is possible through either the slot or the gap between the side panels and the topor through a combination of both slot and gap. Simulation models of the CEUTEs andCEUTIs along with their corresponding 3-D free space radiation patterns at 300 MHzand 600 MHz are shown in Fig. 6.10.Common requirements for EMC tests in RCs are that the EUT should take up lessthan 8%. . . 10% of the chamber volume [6]. This recommendation aims at preventingexcessive RC loading and hence deterioration of field uniformity within the chamber. Toreduce direct coupling between EUT walls and RC, the EUT must be located betweenλ/4 . . . λ/2 away from every conducting object (chamber walls, stirrers, antennas, etc.)at the LUF (see Fig. 2.10). These values correspond to maximum EUT volumes of1.7. . . 2.2m3 and a minimum spacing towards any metallic object of 0.25. . . 0.5m in theprototype RC. The actual volume of the CEUT is significantly below these limits, itsposition was always adjusted to meet the requirements outlined above. All geometricalspecifications and discretization details of these CEUTs are summarized in Table 6.7.

Slot ModeIn the so-called “slot mode”, the CEUT features a slot of 0.12 m · 0.04 m in one of the sidewalls (defined as the “front” of the CEUT, see Figure 6.10), through which radiation fromthe inside of the box to the outside is possible. The EM field is excited by connecting the

Frequency range Canonical emissionEUT type

Number ofsegments / triangles

50 MHz. . . 3 GHz Realistic dipole (0.4 m) 38 a / 0

50 MHz. . . 3 GHz Loop (0.3 · 0.3 m2) 120 a / 0

50 MHz. . . 1.2 GHzBox with slot c

(0.48 · 0.48 · 0.12m3) 1 b / 922

50 MHz. . . 1.2 GHzBox with gap c

(0.48 · 0.48 · 0.16m3) 1 b / 900

50 MHz. . . 1.2 GHzBox with slot and gap c

(0.48 · 0.48 · 0.16m3) 1 b / 906

a excitation source was modeled with two segments b —— one segmentc height of the box EUT is either 0.12m (slot mode) or 0.16m (gap mode)

Table 6.7: Discretization of the canonical emission EUTs (CEUTEs).


outer conductor of a coaxial feeding cable to the center of the lower slot edge, runningthe inner conductor across the slot, and connecting it to the center of the upper slotedge. Measured and simulated values for the gain in this mode of operation are on theorder of 6. . . 8 dBi in the 600. . . 1200MHz frequency range [22, 186].

Gap ModeBy closing the aforementioned slot and mounting the top side of the CEUT 0.04 m spacedaway from the side walls, a gap is created which permits radiation of EM energy fromcircuitry placed inside the box CEUT to the outside. The coaxial cable that was usedfor the “slot mode” excitation is mounted in a similar fashion for the “gap mode”: theouter conductor of the coaxial cable is connected to the center of the lower gap edge;the inner conductor runs across the gap and is connected to the center of the upper gapedge, i.e. to the top side. Measured and simulated values for the directivity of the CEUTare on the order of 2. . . 3 dBi in the 600. . . 1200 MHz frequency range [22, 186].

Slot and Gap ModeThe “slot and gap mode” CEUT combines the two apertures mentioned above. Excita-tion of the EM field is carried out by using the same configuration as described in the“gap mode” section.

6.6 Conclusion

The general modeling procedure for reverberation chamber (RC) geometries was intro-duced along with a tool automating the generation of RC simulation input data. Thistool allows the repeated, accurate, and consistent creation of a large number (on theorder of several hundreds) of RC geometries where the only change in the structure isthe step-by-step rotation of the stirrer. Once the stage of simple startup simulations hadpassed, the usage of this automated tool turned out to be an absolute necessity.Starting with a basic cavity, a comprehensive chamber model resembling the prototypeRC was elaborated. Discretization of the structure was performed using triangular sur-face patches and wire segments which were chosen according to the frequency rangeof interest to minimize the computational effort. Obtaining and assigning reasonablevalues for materials used in RCs proved to be rather cumbersome: the problem was tofind conductivity values for material as it is used in a shielded room construction, i.e.walls consisting of several interconnected sheets with intermediate overlapping flat stock.Therefore, an initial cavity model was used to obtain reasonable conductivity values forthe RC walls and stirrers in the simulations.A prototype RC with and without door, cubic and corrugated chambers, and an offset-wall RC were modeled. Furthermore, eleven vertical and three horizontal stirrers aswell as various transmit and receive antennas (Hertzian dipole, λ/2-dipole, biconical an-tenna, logper antenna, horn) were designed. The concept of a canonical equipment undertest (CEUT) was introduced and three different CEUTs (dipole, loop, and box) weremodeled in the style of the EUTs employed in several international round-robin tests.All structures designed and modeled in this chapter were used extensively throughoutthe simulations performed in the course of this thesis.

7 Reverberation Chamber Simulation and Measurement

Abstract—This chapter summarizes the most significant results of reverberation chamber measure-

ments and simulations. In the beginning, the procedure used to perform reverberation chamber data

analysis is presented. The necessity of a rigorous simulation validation is emphasized and different

methods along with their particular advantages and drawbacks are described. Measurements were

chosen in this thesis to validate the simulation results. Firstly, cavity simulations are performed to

investigate the influence of the door and to derive suitable conductivity values. These initial results

are extended to reverberation chamber simulations, which are benchmarked against measurements.

The effect of a rotating stirrer, the door, and several TX/RX antenna types within the reverberation

chamber are analyzed. Comparisons of different chamber geometries (cubic, corrugated) versus the

prototype reverberation chamber are carried out based on near field, correlation, and field uniformity.

Various stirrer designs are evaluated with respect to their performance within the prototype rever-

beration chamber. The presence of different EUTs is investigated, and a loading, field uniformity,

and coupling path analysis is performed.

7.1 Simulation and measurement workflow

Separate sub-procedures structuring the modeling/simulation process and the measure-ment procedure were introduced in detail in Sections 5.2 and 6.1 (Fig. 5.7 and Fig. 6.1).This section integrates all procedures and summarizes the complete process to simulateRCs and to perform measurements in Fig 7.1: Following the guidelines in Section 6.1,initially the RC is designed with a 3-D CAD system [56]. A parameter generator isthen employed to set the simulation frequencies and compute the data for the geometrydiscretization. Furthermore, the stirrer positions are selected as well as the position ofthe EUT, the TX and RX antennas and the location at which the near field is to becomputed. This generator allows the automated setup of a large number of similar sim-ulations, where e.g. only the rotational stirrer angle ϕ is varying. The final RC geometryalong with discretization-related data is transferred back to the CAD system, which cre-ates a mesh by generating a mixed triangle/segment discretization complying with therequirements laid out in Section 3.4. Discretization data is combined with the simulationsettings in the preprocessor and passed on to the simulation kernel. After evaluating thepros and cons of different numerical methods as discussed in Section 3.1, an EFIE-basedfrequency-domain MoM field solver was chosen as the simulation kernel [54]. Once thesurface and line currents are computed and stored (see Section 3.2), near field data andscattering parameters are calculated from the currents.Simulations without any indication for validity are inherently problematic: at best theyare by chance correct, more often they are flawed in some way, and in the worst casethe results represent expensive and possibly utter nonsense. A thorough validation ofsimulations by benchmarks is therefore absolutely necessary [108, 191]. Surprisingly in

115

116 7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT

the majority of published RC simulations (cf. Section 4.3) the choice was made eithernot to use any means of validation at all or to perform a validation utilizing unsuitablequantities. Especially the latter approach will be addressed in detail in Section 7.4.3in this chapter. In general, there are three possible validation options for simulationresults:

• benchmark against analytically calculated results; this includes also an evaluationwhether a simulated result “makes any sense” from a purely theoretical point ofview

• comparison with the results obtained with other EM solvers (preferably utilizinga completely different numerical method)

• validation measurements (EM near field, far field, scattering parameters, etc.)

These three approaches have each its particular advantages and disadvantages; the big(and essentially only) advantage of the analytical method is its simplicity and hencespeed. For the validation of RC simulations, however the problematic aspects of ana-lytical results dominate: first of all, the EM fields can only be computed in an idealcavity without stirrers, antennas or an EUT inside. Secondly, analytic approaches usu-ally assume infinite conductivity for the cavity (PEC); this implies that the quality factorbandwidth ∆fQ (Section 2.1.2) would be zero, i.e. at a given frequency f only a singlemode (and possibly respective degenerate modes) could be excited and coupling betweendifferent non-degenerate modes does not exist. Realistic RCs however can only achievesufficient statistical field uniformity if ∆fQ > 0 so that several modes are excited ata given frequency. Also lumped-elements circuit theory formulations cannot reproducerealistic RCs [13].Using the results obtained with other EM solvers represents a suitable validation option.For comparisons, preferably solvers should be used based on a completely different nu-merical technique and with one solver operating in the frequency- and the other in thetime-domain (e.g. MoM vs. FDTD). There are however also several problems associatedwith solver-to-solver benchmarks: a particular numerical method might be well-adaptedto simulate an RC whereas an other may not. Some methods might not be able tosimulate an RC simply because of the numerical size of the problem. In addition, RCsimulations are time-consuming to run, expensive software needs to be bought, and thereis a lot of experience required by the end-user (high-end EM field simulations still tendto be more “art” than just plugging in some numbers [178]).In this thesis, benchmarks by doing measurements were chosen as a validation technique.Although measurements are – similar to simulations – time-consuming too, and, in addi-tion, expensive equipment is needed (cf. Section 5.2), they provide an useful insight intothe “reality of the chamber physics”. This allows to identify critical parameters whichare important when performing RC tests in practice, but which would be neglected oth-erwise in RC simulations. Certainly, as outlined in Section 5.3, special care needs to betaken of additional errors associated with measurements.During the course of this thesis, measurements performed in the prototype RC usuallyincluded the electric near field as well as forward and reflected power on all antennas.

7.2 CAVITY SIMULATION 117

3D CADMesher

HyperMesh®

GeometryConversion

InterfacePreFEKO

SimulationData Process

WinFEKOGraphFEKO

J E H S, , ,Visualization

GeometryModeling

&Simulation

Preprocessing

Field Solver&



&Measurement

Data Acquisition

MS Access®

DatabaseSystem

Compliance®

MeasurementSystem

Interface

DataExtraction

FEKO®

MoM KernelMLFMM / PO

Interface

DataExtraction

MATLAB®

Statistics &Benchmarks

Interface

DataExtraction

MATLAB®

ParameterGenerator

Figure 7.1: Schematic simulation, analysis, measurement, and benchmark procedure.

Extensive details on the measurement system with a comprehensive error analysis canbe found in Chapter 5. To facilitate data handling, both simulation and measurementresults were gathered and fed directly into a database (cf. Fig. 7.1). With this procedure,benchmarks, statistical analyses, and 2-D/3-D visualizations of simulated and measureddata are performed by retrieving the results from the database without the need to ma-nipulate (e.g. scale, offset, etc.) underlying data – this in turn significantly reduces theprobability of accidentally introducing errors.

7.2 Cavity simulation

Initially, simulations considering a simple RC model without stirrers, doors or othergeometrical details (in other words: a cavity) were performed to obtain reasonable con-ductivity values for the RC walls and stirrers as listed in Table 6.4. The cavity modelused in the simulations is described in Section 6.1.2. For this purpose also stirrers in the


prototype RC were removed during measurements to match the simulation model. Fur-thermore, these preliminary simulations and measurements were used to identify cham-ber details having the biggest impact on the simulated and measured near field (e.g.ventilation honeycomb ducts, antenna cable and power line routing, or leakage throughthe stirrer motor bearings). The reason not to include the stirrer in this first comparisonwas to eliminate the possibility of deviations between simulations and measurementsresulting merely from the rotational positioning accuracy of the stirrer paddles [192, 13].The stability of the simulations was checked by simulating the same cavity at the samefrequency using greatly different mesh discretizations of the walls: decreasing the trian-gular mesh size from 2 498 triangles to 13 236 triangles did not change the field distri-bution within the cavity at all [191]. None of the changes in discretization introducedany “numerical leakage” of EM energy from the inside to the outside of the chamber.Whether the field distribution made sense from a straightforward physical point of viewwas verified by examining if and how well boundary conditions for the EM fields weremet at the cavity walls. The computational region was extended beyond the cavity wallsso that the EM field could be easily checked for violations of the boundary conditions.In addition, the current distribution was checked against continuity errors [178]. Thisbasic analysis did not bring up any major surprises, therefore the cavity model was usedas a basis for the RC simulations shown from Section 7.4 on.

7.2.1 Effect of the chamber door

First measurements taken in the cavity showed in the lower frequency range an ex-cellent agreement with simulations, whereas for several higher frequencies considerabledifferences started to appear. Therefore the cavity was closely examined to identify thegeometrical details causing the differences between the simulated and the real cavity.The prototype RC without stirrer was modified with different coaxial cable and powerline routing, the simulated cavity was modeled with several modifications (walls with dif-ferent conductivities, inclusion of the stirrer motor mount, etc.). This analysis revealedthat by far the biggest perturbance seemed to be caused by the door. To facilitatemodeling of the cavity, the door (among with other apertures) was neglected in the firstsimulation models. In subsequent simulations, the cavity door was included in a newmodel and proved to have a significant effect on the simulated results. Figure 7.2 clearlyshows the strong, very frequency selective impact of the door on the simulated EM nearfield results: whereas at f = 200 MHz the field in the cavity with and without door isvirtually identical, at f = 250 MHz a significant difference can be seen. As shown belowin Section 7.3.1, a similar effect was noticed in later comparisons of RC measurementsand simulations; for this reason, the door was also included in the final simulation modelresembling the RC prototype.

7.2.2 Insertion of a stirrer

After initial “door-modified” cavity simulations proved to make sense, a stirrer was mod-eled for the simulations and installed into the prototype RC (see Section 6.2). Figure 7.3depicts a comparison between an RC without (left) and including (right) a six-paddle

7.2 CAVITY SIMULATION 119

a) b)

c) d)

xyz

xyz

xyz

xyz

Withoutchamber door

Withchamber door

Withoutchamber door

Withchamber door

Figure 7.2: Effect of the chamber door on the magnitude of the electric field |E| within a cavityat a) and b) f = 200 MHz and c) and d) at f = 250 MHz. Whereas at f = 200 MHz thefield in the cavity with and without door is virtually identical, at f = 250 MHz a significantdifference can be seen. Excitation is a biconical antenna.


a) b)

xyz

xyz

Figure 7.3: Comparison of a) an empty cavity-like chamber against b) a reverberation chamberwith a single vertical six-paddle stirrer. Depicted is the magnitude of the electric field |E| inthe xy-plane at y = 1.73 m above the chamber bottom. The simulation frequency is 200 MHz,which is close to the TE312/TM312 resonance in an ideal cavity. Excitation is a λ/2-dipole.

stirrer at f = 200 MHz. Since at the start of this thesis measurements were not readilyavailable, the empty cavity simulations were used to verify the simulation by bench-marking against the analytical results of a cavity. For the setup shown in Figure 7.3,benchmarking was performed against the ideal TE312/TM312 resonance. Since the the-oretical cavity has PEC walls contrary to the simulated one, this benchmark is only anestimate for the correctness of the numerical full-wave simulation. Rigorous measure-ments as presented from Section 7.4 onwards needed to be performed to clarify if thesimulations can pass a benchmark test satisfactorily. Figure 7.3 clearly illustrates thatthe insertion of the large (in terms of wavelength λ) RC stirrer significantly changes thefield distribution in the chamber, even at this comparatively low frequency [192].

7.3 Prototype reverberation chamber analysis

Considering the large variety of possible antenna and EUT locations, combinations ofstirrer structures and different chambers, any thorough RC simulation will generate largeamounts of data. In the following sections only the most important results are illustratedand discussed.

7.3 PROTOTYPE REVERBERATION CHAMBER ANALYSIS 121

Chamber Geometry

Figure Wall Conductivity Door Dimensions

Fig. 7.4a), 7.4e) “Medium” a No w × l × h

Fig. 7.4b), 7.4f) “Medium” a Yes w × l × h

Fig. 7.4c), 7.4g) “Medium” a Yes (w + 0.1 m) × l × h

Fig. 7.4d), 7.4h) “High” a Yes w × l × h

a see Table 6.4 for corresponding conductivity values

Table 7.1: Simulation parameter overview for different RC geometries shown in Fig. 7.4.

7.3.1 Different reverberation chamber geometries

After establishing a suitable model for the RC as noted above, the effect of the cham-ber door already observed in the cavity simulations could be reproduced also in theRC simulations: the simulated and measured results were in good agreement in the lowfrequency range f ≤ 250 MHz, which is still below fLUF. Similar as in the cavity sim-ulations at certain frequencies f ≥ 300 MHz however considerable differences betweenRC measurements and simulations appeared. The difference in the magnitude of thesimulated electric field strength in the RC model with and without the door for a fixedrotational stirrer angle of ϕ = 225 can be clearly seen: Figs. 7.4a) and 7.4e) show|E| computed in the xy-plane at a height of z = 2 m above the chamber bottom with-out and Figs. 7.4b) and 7.4f) including the door. Whereas the effect at a frequencyof f = 250 MHz (Figs. 7.4a) vs. 7.4b)) is just a slight distortion of |E| in the nearestvicinity of the door, at f = 300 MHz in Figs. 7.4e) vs. 7.4f) a considerable change ofthe field distribution throughout the RC can be seen. As a result of this unexpected,yet significant effect, the RC door including the gasket is accounted for in the detailedmodel (cf. Section 6.1). The influence of the door along with other small geometricdetails increases with rising frequency (see also Section 7.4).Along with the necessity to consider small structural details it is essential to utilizecorrect inner dimensions in the RC simulations: The simulated chambers depicted inFigs. 7.4c) (f = 250 MHz) and 7.4g) (f = 300 MHz) are ∆w = 0.1 m wider than theother simulation models and the RC prototype. The length l and height h were keptidentical to the dimensions in the chamber prototype. Compared with the RC simulationmodels shown in Figs. 7.4b) and 7.4f) which match all prototype dimensions w, l, and h,the electric field pattern changes completely for both frequencies, even if the correspond-ing free-space wavelength λ0 is much larger than the small geometrical modification ∆wof the RC.

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a) b) c) d)

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Figure 7.4: Magnitude of the electric field strength |E| simulated in the xy-plane at z = 2m above the chamber bottom at f = 250 MHzin a). . . d) and f = 300 MHz in e). . . h). Analyzed is the effect of different RC geometries on the electric field: a) and e) simplepartly symmetric RC without door, b) and f) detailed asymmetric RC with door, c) and g) RC which is ∆W = 0.1 m wider thanthe other RCs, d) and h) RC walls are of conductivity “High” (see Table 6.4). Excitation is a biconical antenna. Table 7.1 showsthe differences between the various geometries.


To investigate the numerical stability of the simulations, the RCs were modeled withvarious conductivity values. A comparison for two different conductivities is shown inFig. 7.4: The chambers in Figs. 7.4d) and 7.4h) were simulated with the conductivity“High” (Table 6.4) at f = 250 MHz and f = 300 MHz. “Medium” conductivity resultsare shown in Figs. 7.4b) and 7.4f). As expected, for a higher conductivity value themagnitude of the electric field increases whereas the overall field pattern shape staysthe same. Similar results are obtained for other conductivities, i.e. higher conductivityvalues consistently lead to higher, and lower conductivities to lower field magnitudes.Stability problems reported by other authors (e.g. [43, 86]) using different numericaltechniques were not encountered.

7.3.2 Effect of a rotating stirrer

In Section 7.2.2 it was already shown that the insertion of a non-rotating stirrer into arectangular cavity changes significantly the field distribution. In an RC analysis it is ofparticular interest to analyze how the field distribution changes as the stirrer rotates.This effect inside the detailed RC with door is displayed in Fig. 7.5. The vertical six-paddle stirrer is rotated incrementally in steps of ∆ϕ = 5 between ϕ = 0 and ϕ = 355.A subset of this data with ∆ϕ = 30 (between ϕ = 0 and ϕ = 210) is shown in Fig. 7.5:The near field is computed at a height of z = 2 m above the chamber bottom in the xy-plane at a frequency of f = 400 MHz. It can be clearly seen how the magnitude ofthe electric field varies. The excitation is a logper antenna positioned in front of thechamber door and pointing towards the stirrer. Large E-field variations over one stirrerrevolution verify the effectiveness of this stirrer even for relatively low frequencies [193].This stirrers’ performance is compared below in Section 7.7 against five other stirrersoperated within the simulated prototype RC.

7.3.3 Different reverberation chamber excitations

An important issue for reliable RC simulations is the implementation of the excitationsource in the numerical code, as mentioned before in Section 4.3. To facilitate thediscretization of the RC structure and the simulation setup, often an ideal Hertziandipole is used, implicitly assuming that the effect on the actual field distribution insidethe chamber will be rather small. Whereas the assumption that different antennas willlead to similar results in a statistical sense (i.e. a large number of samples taken froma large number of stirrer positions, see Section 2.4) is certainly true, the results for agiven, fixed stirrer position are strongly dependent on a particular antenna: Fig. 7.6a)depicts the magnitude of the electric field obtained inside the detailed asymmetric RC ata frequency of f = 300 MHz using an ideal Hertzian dipole compared with a broadbandlogper antenna in Fig. 7.6b). Both the Hertzian dipole and the center of the active regionof the logper antenna at f = 300 MHz were positioned at the same location inside theRC. The difference in the field pattern is quite remarkable and can be attributed to thestrong coupling between the excitation antenna and the RC itself.Moreover, not only the type, but also the correct (in the sense that it agrees with theantenna setup in the measurement prototype) positioning and alignment of an excitation

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= 30° = 60° = 0° = 90°

= 120° = 150° = 180° = 210°

xyz

Figure 7.5: Magnitude of the electric field strength |E| at f = 400 MHz simulated in the xy-plane at a height of z = 2m above thechamber bottom. The vertical six-paddle stirrer is rotated from one position to the next by ∆ϕ = 30. Excitation is a logperantenna in front of the chamber door.


antenna are important for the simulation: As a reference configuration, the setup inFig. 7.6b) is used, where the logper antenna is positioned at x0 = −0.80 m, y0 = 0.70 m,z0 = −0.08 m – with (x, y, z) = (0, 0, 0) being the geometric center of the RC – and atan alignment angle of 0. The rotational stirrer angle in Fig. 7.6 remained fixed for allsimulations at ϕ = 45. Compared to the reference position and alignment of the logperantenna in Fig. 7.6b), Fig. 7.6c) reveals that the field pattern changes significantly if theexcitation antenna is moved by ∆x = 0.2 m into +x-direction. Instead of e.g. four peaksin the xz-cut plane at y = −L/2 in Fig. 7.6b), only three maxima occur in Fig. 7.6c).A somewhat different effect can be seen in Fig. 7.6d) if the excitation antenna remainsat (x0, y0, z0), but is aligned at an angle of 90 perpendicularly instead of parallel tothe RC side walls: similar as in Fig. 7.6b) four peaks of the electric field magnitude inthe xz-cut plane exist, they are however slightly shifted in Fig. 7.6d). In addition, theoverall field pattern throughout the RC in both Fig. 7.6c) (shifted excitation antenna)and Fig. 7.6d) (perpendicular excitation antenna) is quite different compared with thereference configuration depicted in Fig. 7.6b).The results shown in Figs. 7.4, 7.5, and 7.6 clearly indicate that a correct validation ofthe simulated results through measured field data is only possible with an RC simulationmodel accounting for seemingly insignificant small geometric details (indentations suchas a door or protrusions from stirrer mounts) and utilizing appropriate conductivityvalues for the chamber materials. Furthermore, realistic as well as correctly positionedand aligned TX/RX antennas, which resemble the actual antennas employed in theprototype chamber, must be considered in any meaningful RC simulation model.

Excitation Antenna

Figure Type Position Angle

Fig. 7.2 biconical (x0, y0, z0) 0

Fig. 7.3a), 7.3b) λ/2-dipole (x0, y0, z0) 0

Fig. 7.6a), 7.10a) Hertzian dipole (x0, y0, z0) 0

Fig. 7.6b), 7.10b) logper (x0, y0, z0) 0

Fig. 7.6c) logper (x0 + 0.2 m, y0, z0) 0

Fig. 7.6d), 7.10c) logper (x0, y0, z0) 90

Table 7.2: Simulation parameter overview for the different types of reverberation chamberexcitations shown in Figs. 7.3, 7.6 and 7.10.


a) b)

c) d)

xyz

Figure 7.6: Magnitude of the electric field strength |E| simulated in the reverberation chamberat f = 300 MHz for different excitations: a) Hertzian dipole, b) logper antenna, c) logperantenna shifted by ∆x = 0.2 m into +x-direction, d) logper antenna pivoted by 90, i.e.aligned perpendicularly instead of parallel to the chamber side walls. Table 7.2 shows at aglance the details of the different excitation settings.

7.4 MEASUREMENT VERSUS SIMULATION 127

7.4 Measurement versus simulation

The RC simulations were validated by extensive measurements; only a subset of thetotal amount of comparison data is presented in this section. Due to unavailability ofmeasurement equipment no validation has been carried out based on the magnetic fieldH .

7.4.1 Measurement setup

Fig. 5.7 shows the equipment setup used for measurements of the electric near field as wellas the forward and reverse power of the TX and RX antennas. As noted in Section 5.2,these antennas were not used for near field measurements as they provide insufficientspatial field resolution. Measurements of the near field were taken using diode-equippedfield probes (Section 5.2.2) measuring simultaneously |Ex|, |Ey |, and |Ez | componentsof the electric field. Comparisons between simulated and measured results in this thesisare based on the absolute value of the electric field strength |E| as defined in (2.73).The following issues were found to have the biggest impact on the agreement betweenmeasured and simulated results:

• position and alignment of the field probe in the measurement is not the same asin the simulation

• small geometric details of the prototype RC are not accurately modeled in thesimulation (Section 7.3.1)

• the excitation antenna used in the simulation is different from the actual antennain the prototype (Section 7.3.3)

• unintentional loading of the prototype RC (Section 5.3.2)

• rotational stirrer angle in the simulation deviates from the measurement setup

• RF cable routing from the coaxial feed-through panel to the TX and RX antennas(Section 5.3.2)

As mentioned in Section 5.3, the first issue was resolved by positioning the field probesinside the RC using a coarse optical measurement grid of 0.1 m×0.1 m on the chamberfloor in combination with laser range distance metering for final precision alignment. Byincluding the RC door into the simulation model, a better agreement between simulatedand measured results was achieved. To avoid unintentional loading and distortion ofthe EM field, tripods were removed and antennas either suspended from the chamberceiling with plastic ropes and Velcro or placed onto styrofoam blocks (see Fig. 5.10). Anencoder-controlled servo-motor and a special anti-backlash gearbox facilitate an angularpositioning accuracy of the vertical 6-paddle stirrer of ∆ϕ < 1. Coaxial RF cables wererouted as close as possible to the walls of the prototype RC to reduce field distortion,unnecessary power lines and cable ducts were removed.


a) b)

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Figure 7.7: Comparison between measurement and simulation of the magnitude of the electricfield strength |E| along a line in the reverberation chamber (x = 0.77 m, y = −1.2 . . . 1.2 m,z = 2 m). |E| is shown for two rotational stirrer positions of ϕ = 90 and ϕ = 135 atfrequencies of a), b) f = 300 MHz and c), d) f = 500 MHz. Excitation source is a biconicalantenna at f = 300 MHz and a logper antenna at f = 500 MHz. Note the good agreementbetween measurements and simulations.

7.4.2 Near field based simulation validation

In an RC immunity or emission test there is usually very little, if any, interest in actualmeasured near field data. The laborious procedure of near field measurements is onlycarried out for the purpose of chamber calibration and the gathered data is immedi-ately “processed” to compute specific performance measures of the RC such as statisti-cal field homogeneity (Section 2.7), randomness of polarization, correlation coefficients


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70

-1.2 -0.8 -0.4 0.4 0.8 1.2

90

100

0

E [V/m]

f = 700MHz

Simulation

Measurement

90

y [m]

60

0

20

30

40

50

10

80

70

-1.2 -0.8 -0.4 0.4 0.8 1.2

90

100

0

E [V/m] 90

f = 1000MHz

Simulation

Measurement

y [m]

60

0

20

30

40

50

10

80

70

-1.2 -0.8 -0.4 0.4 0.8 1.2

90

100

0

E [V/m]

f = 700MHz

Measurement

Simulation

135

y [m]

60

0

20

30

40

50

10

80

70

-1.2 -0.8 -0.4 0.4 0.8 1.2

90

100

0

E [V/m] 135

f = 1000MHz

Measurement

Simulation

d)c)

Figure 7.8: Comparison between measurement and simulation of the magnitude of the electricfield strength |E| along a line in the reverberation chamber (x = 0.77 m, y = −1.2 . . . 1.2 m,z = 2m). |E| is shown for two rotational stirrer positions of ϕ = 90 and ϕ = 135

at frequencies of a), b) f = 700 MHz and c), d) f = 1000 MHz. Excitation source is alogper antenna. Compared with Fig. 7.7, note how the agreement between measurementsand simulations progressively deteriorates as the frequency increases.

(Section 2.5), standard deviations (Section 2.7), or field anisotropy and inhomogeneitycoefficients (Section 2.3) [6]. These parameters are perfectly suited to analyze the per-formances of different RCs – they are however not suitable at all for the comparisonof RC simulation results against measurements: Since completely different EM fieldscan still generate an identical field uniformity, the same correlation coefficient or equalanisotropy coefficients, a thorough validation of RC simulation results cannot be accom-plished using this type of “processed data”. “Processed data” can be classified as any


result obtained from large sets of values (e.g. electric field values measured or simulatedin several points over one stirrer revolution) to create one single metric (such as a corre-lation coefficient). Generally, any kind of data which cannot be uniquely “mapped” toa corresponding near field or current distribution, should not be used for the validationof an RC simulation. Therefore it was decided to use “raw”, “unprocessed” near fielddata to benchmark simulated results.Fig. 7.7 and Fig. 7.8 depict the mangitude of the electric field |E| measured along a linein the y-direction (x = 0.77 m, y = −1.2 . . . 1.2 m, z = 2 m) within the RC comparedagainst the simulation. |E| is shown for two rotational stirrer positions of ϕ = 90

and ϕ = 135 at frequencies of f = 300 MHz, 500 MHz, 700 MHz, and 1000 MHz. Inaddition, Fig. 7.9 exhibits a comparison between measurement and simulation, wherethe RC simulation was carried out once with and once without door. Excitation sourceis a biconical antenna for f = 200 MHz, f = 250 MHz, f = 300 MHz and a logper an-tenna in the f = 500 . . .1000 MHz range. For lower frequencies (Fig. 7.7, f = 300 MHz,500 MHz) simulation and measurement agree well (with the exception of some field val-ues taken next to the RC walls, which might be due to limitations in the simulationmethod or proximity coupling effects between the field probes and the chamber walls).As the frequency is increased to f = 700 MHz and 1000 MHz (Fig. 7.8), the agreementbetween measurements and simulations progressively deteriorates: Whereas some mea-sured peaks in Fig. 7.8 can still be reproduced by the simulation, others are shifted intheir location or appear significantly distorted.In order to quantify the (dis-)agreement between simulated (Es) and measured (Em)electric field, a general norm can be used rather than employing a visual “quality agree-ment” [12]. The spatial p-norm in its most general form is denoted by

‖x‖p = p√|x1| + |x2| + · · · + |xn|p p ≥ 1 (7.1)

Stirrer position ϕ

Figure Frequency 90 135

Fig. 7.7a), b) 300 MHz 6.6 V/m 4.6 V/m

Fig. 7.7c), d) 500 MHz 9.7 V/m 9.9 V/m

Fig. 7.8a), b) 700 MHz 21.5 V/m 23.1 V/m

Fig. 7.8c), d) 1000 MHz 23.6 V/m 16.7 V/m

Table 7.3: Agreement between simulation and measurement (Fig. 7.7 and Fig. 7.8) expressedby a normalized spatial 2-norm ‖∆E‖ as defined by (7.3).


Adapted to EM fields, the 2-norm (also called Euclidean vector norm) is defined as

‖E‖2 =√‖Es| − |Em‖2 (7.2)

wherein |Es| and |Em| are the magnitude of the simulated and measured electric field,respectively. More specifically applied to RCs, the normalized spatial 2-norm

‖∆E‖ =‖∆E‖√

M=

√M∑i=1

(|Es(xi, yi, zi)| − |Em(xi, yi, zi)|)2√

M(7.3)

can be used, where M denotes the total number of spatial positions (xi, yi, zi) used forcomparison. A perfect agreement between measurement and simulation would result in‖∆E‖ = 0. The normalized spatial 2-norm ‖∆E‖ is listed in Table 7.3. ‖∆E‖ wascomputed for both stirrer angles ϕ = 90 and ϕ = 135 using the data shown in Fig. 7.7and Fig. 7.8 and confirms the good agreement between simulation and measurement forlower frequencies as well as the progressive deterioration in the higher frequency range.This “breakdown” of the simulation starting from f > 600 . . .700 MHz is related to thefact that the EM field inside an RC becomes extremely sensitive to even tiny geometricdetails. Ironically this sensitivity is highly desirable for the proper operation of an RC(even a very small stirrer step angle will change the field distribution), but renders asimulation practically not feasible at higher frequencies f fLUF – unless one is willingto undertake the challenge to model and discretize virtually every nut and bolt of thechamber [67].

a) b)

Biconicalfeed antenna

y [m]

60

0

20

30

40

50

10

80

70

-1.2 -0.8 -0.4 0.4 0.8 1.2

90

100

0

E [V/m]

f = 200MHz

SimulationMeasurement

RC without door RC with door

Biconicalfeed antenna

60

80

70

90

100

E [V/m]

f = 250MHz

SimulationMeasurement

RC without door RC with door

y [m]

0

20

30

40

50

10

-1.2 -0.8 -0.4 0.4 0.8 1.20

Figure 7.9: Influence of the chamber door: comparison between measurement and simulation ofthe magnitude of the electric field strength |E|. |E| is shown at a rotational stirrer positionof ϕ = 225 and frequencies of a) f = 200 MHz and b) f = 250 MHz.


7.4.3 Statistical benchmarks

As the EM field within an RC is completely deterministically defined for a fixed rotationalstirrer angle ϕ = ϕ0 and at a given position (x0, y0, z0), near field based validations of asimulation can be carried out as presented in Section 7.4 as well as Fig. 7.7, Fig. 7.8, andFig. 7.9. Since RCs are usually regarded as a “statistical” EMC test environment, mostpapers report a simulation validation based exclusively on statistics (see Section 4.3).The problem with this approach is that chances are high to find an excellent agreementbetween statistics of simulated and theoretical or measured data even if there is totaldisagreement between the field simulation and the measurement. Fig. 7.10 shows thestatistical distribution of the simulated electric field strength |E|/|Emax| sampled in arectangular test volume of 0.5 m× 1.0 m× 0.5 m (width ∆w× length ∆l× height ∆h)within the RC. These results were obtained with three different excitation antennas:a) ideal Hertzian dipole, b) logper antenna, and c) logper antenna pivoted by 90. Al-though the near field excited in the RC by the three antenna configurations has onlyvery little in common (cf. Fig. 7.6), all electric field histograms exhibit a fairly goodagreement with the theoretically expected χ(6) distribution for the field magnitude. Thisis especially true for the logper versus the pivoted logper antenna. The Hertzian dipoleexhibits from a visual point of view a slight deviation from the analytical χ(6) distribu-tion (this effect is addressed also in Section 7.6.3 below). A statistical goodness-of-fit test(see Section 2.4.4) indicates that the hypothesis of a χ(6) EM field distribution obtainedwith a Hertzian dipole excitation as shown in Fig. 7.10 will be rejected. For the twologper antenna orientations however, a goodness-of-fit test accepts the hypothesis of aχ(6) distribution. Looking only at the simulated EM fields processed to a statistical dis-tribution, one would accept the two RC simulation results using the logper TX antennaas “correct”, possibly also the one with the Hertzian dipole.Contrary to this, the only simulated near field that matches measurements in the RCprototype, results from the logper antenna in its standard position as indicated in Fig. 7.7and Fig. 7.8. Therefore one cannot conclude that a simulation showing the χ(6) (respec-tively χ2

(6)) behavior is correct in the sense that it approximates the actual field insidean RC. Validations exclusively based on statistics allow only the conclusion that the EMsimulator works well as a rather sophisticated random number generator producing 2-DGaussian-distributed Ex, Ey, and Ez field components – they do not reveal how wellreality is reproduced in a simulation.

7.5 Corrugated and cubic reverberation chamber

To investigate the influence of a particular RC design on the EM near field and ontypical RC parameters (field uniformity, correlation, etc.), two other RCs were modeledin addition to the prototype RC:

• Cubic RC, 2.90 m×2.90 m×2.90 m (see Section 6.1.4)

• Corrugated RC, 2.70 m×2.30 m×2.90 m “mean” inner dimensions taking into ac-count the height of the corrugations (see Section 6.1.4)

7.5 CORRUGATED AND CUBIC REVERBERATION CHAMBER 133

0

100

200

300

400

500Number of samples Analytical -distribution(6) Distribution of simulated fields

Hertziandipole

0 0.2 0.6 0.8 10.4 0 0.2 0.6 0.8 10.4

Logperantenna

a) b)

E

Emax

0

100

200

300

400

500Number of samples

0 0.2 0.6 0.8 10.4

Logper90 pivoted°

c)

E

Emax

Figure 7.10: Statistical distribution of the normalized magnitude of the simulated electricfield strength |E|/|Emax| within the RC. Results shown were calculated at a frequencyf = 500 MHz and cumulated from angular stirrer positions of ϕ = 0 . . . 355 with 5 stepangle using different excitation antenna types and orientations: a) ideal Hertzian dipole,b) logper antenna, c) logper antenna pivoted by 90. Regardless of the excitation antenna,the simulation results match well an analytical χ(6) distribution – although the simulated nearfield of a) and c) differs greatly from b) which is the only one agreeing with measurements.In total, 6 000 samples were used to plot each histogram.

The prototype RC is used in this thesis as a reference chamber for benchmarks againstthe other RCs. In all RCs the same vertical 6-paddle stirrer is utilized, consisting of sixsquare plates of size 0.60 m× 0.60 m, rotationally offset around the stirrer axis by 60.The slanting angle of each plate is 45 vs. the stirrer axis (see Section 6.2). The choice touse identical stirrers was made to facilitate the comparison between different RCs, withthe shape and dimensions of the chamber walls being the only variables. The wedges inthe corrugated RC have an amplitude of 0.1 m and a valley-to-valley distance of 0.15 m.


7.5.1 Simulated near field distribution

For the prototype versus cubic versus corrugated RC comparisons, the three chamberswere simulated over a frequency range of 50 . . . 500 MHz (frequency resolution 10, 25 and50 MHz) with a rotational stirrer increment angle of 5 resulting in 72 stirrer steps. Thethree-component electric and magnetic near field was computed in ten equally spacedplanes parallel to the xy-plane with a spatial resolution of 0.05 m, i.e. near field datais available at 27 440 points throughout the chamber. With Fig. 7.11 it is possible tocompare qualitatively the different chambers: depicted is the magnitude of the simulatedelectric field strength |E| computed at a height of z = 2 m above the chamber bottom.Results shown were calculated at a frequency of f = 300 MHz and three angular positionsof ϕ = 0, 10, 20. The chamber performance can be qualitatively analyzed by lookingat the change of the overall field distribution and by examining how the field varies inthe cut planes at x = −1.45 m and y = −1.25 m from one stirrer step to the next. Thesimulated EM near field within the RCs is utilized to investigate further the followingtwo controversially discussed statements:

• The idea to decrease the LUF by using corrugated chamber walls to enhance thefield uniformity was suggested by several authors (e.g. [173]) – however the ap-proach presented in [173] is somewhat questionable since the “supporting data”was achieved by changing twice antenna locations, shifting the frequency twiceand neglecting the worst 25% of the totally obtained data set. In addition thereis literature stating that great surface irregularities, such as corrugations, tendto increase the difficulty of providing uniform EM fields within devices similar toRCs. Already in e.g. [71] (see Section 4.1) it is outlined that for the application ofmicrowave food heating “it has been found that great surface irregularities [. . . ],for example deep corrugations, tend to increase the difficulty of providing uniformheating”. This in turn supports the theory that corrugations do not enhance EMfield uniformity within an RC.

• As shown in the modal analysis in Section 2.2, when choosing a rectangular roomas a basis for building an RC, ideally the dimensions should not be simple mul-tiples or rational fractions of each other. The idea governing this statementis that this choice will result in the largest number of (non-degenerate) modeswith different resonance frequencies, which is usually thought to improve cham-ber performance particularly at lower frequencies [6]. As shown in Section 2.2.1(Fig. 2.2. . . Fig. 2.5), cubic cavities suffer from mode degeneration so that the usu-ally required “∂N/∂f = 1.5 modes/MHz above cutoff”-criterion is reached consis-tently only at much higher frequencies compared to a cavity of similar rectangular,but non-cubic shape. Nevertheless several authors claim that cubic chambers maygenerate a more uniform field than standard rectangular RCs (e.g. [75]).

Visually inspecting the near field however does not yield a final conclusive (and espe-cially quantitative) answer – it is necessary to take a closer look at the two key RCparameters: field correlation and spatial field uniformity over a broad frequency range.Results presented in this thesis focus on the frequency range close to the LUF where thecorrugations or the cubic shape should show the biggest impact.

7.5 CORRUGATED AND CUBIC REVERBERATION CHAMBER 135P

roto

typ

eR

C

a) b) c)

0° 10° 20°

Cu

bic

RC

d) e) f)

0° 10° 20°

Co

rru

ga

ted

RC

g) h) i)

0° 10° 20°

xyz

xyz

xyz

xyz

xyz

xyz

Figure 7.11: Magnitude of the simulated electric field strength |E| computed in the xy-plane at aheight z = 2m above the chamber bottom. Results shown were calculated at a frequency f =300 MHz and at angular stirrer positions of ϕ = 0 . . . 20 for different chamber geometries:a). . . c) prototype RC; d). . . f) cubic RC; g). . . i) corrugated RC. Excitation antenna was alogper antenna in all chambers, see Fig. 6.8c).


7.5.2 Correlation analysis

As opposed to the qualitative field-pattern-based analysis of different stirrers shown be-fore, the correlation allows a quantitative comparison. The correlation coefficient ρ(ϕ)is calculated from the magnitude of the electric field |E| sampled at 8 corner points onthe top and bottom side of a 0.4 · 1.0 · 1.0 m3 test volume located z = 1 m above thechamber floor. Each data point in Fig. 7.12 corresponds to the absolute value of thecorrelation |ρ(ϕj)| between the simulated |E| for the reference stirrer angle ϕ0 = 0 and|E| for ϕj = 0 . . . 355 calculated as described in Section 2.5.The moderate slopes of |ρ(ϕj)| in Fig. 7.12 at f = 100 MHz and f = 150 MHz forall RCs indicates that the stirrer is still rather ineffective in providing a large numberof sufficiently uncorrelated samples at low frequencies. The cubic RC shows the bestcorrelation-based performance at these frequencies. At f = 250 MHz and f = 300 MHzhowever the corrugated RC exhibits a good performance whereas in the cubic RC onlyrelatively high correlation values are obtained (Fig. 7.12). |ρ(ϕj)| of the prototype cham-ber reaches intermediate levels at both frequencies. From a correlation point of view,neither the cubic nor the corrugated RC exhibit convincing results across all frequencieswhich would clearly outclass the standard rectangular prototype RC [19].

7.5.3 Field uniformity

As introduced in Section 2.7, the field uniformity within an RC is expressed in termsof the combined three-axis standard deviation σxyz and the single-axis standard devia-tions σx, σy, and σz as proposed in [6]. These quantities are calculated from the threecomponents of the electric field Ex(xi, yi, zi), Ey(xi, yi, zi), and Ez(xi, yi, zi). For boththe per-axis standard deviations σξ as well as the combined standard deviation σxyz, theIEC 61000-4-21 standard [6] requires for a “well operating” RC with sufficient statisticalfield uniformity and a given uncertainty within all frequencies 80 MHz≤ f ≤ 100 MHz

σξ ≤ 4 dB and σxyz ≤ 4 dB (7.4)

For frequencies 100 MHz≤ f ≤ 400 MHz, the limits for σξ and σxyz decrease linearlyfrom 4 dB to 3 dB. Finally,

σξ ≤ 3 dB and σxyz ≤ 3 dB (7.5)

is required for all frequencies f ≥ 400 MHz. Fig. 7.13 shows the statistical field uni-formity envelopes of σξ and σxyz for all three RCs. As expected, at frequencies in the50 . . .250 MHz range, the field uniformity is clearly insufficient in all chambers, whereasthe cubic RC performs the worst. Starting from f > 300 MHz however, sufficient statis-tical field uniformity is achieved, indicated by electrical field standard deviations on theorder of σ ≤ 3 dB – with the exception of the cubic RC exhibiting significantly highervalues at some frequencies, e.g. at f ≈ 400 MHz. Detailed per-component field unifor-mities can be found in Appendix D, Fig. D.3. . . Fig. D.5.As a summary, neither a cubic chamber nor an RC with corrugations on the walls ex-hibits consistently superior or inferior field uniformity performance. Especially the cubic

7.5 CORRUGATED AND CUBIC REVERBERATION CHAMBER 137

Stirrer angle [ ]

0.6

0

0.2

0.3

0.4

0.5

0.1

0.8

0.7

0 60 120 240 300 360

0.9

1

180

Correlation

Prototype RC

Cubic RC

Corrugated RC

f = 150 MHz

Stirrer angle [ ]

0.6

0

0.2

0.3

0.4

0.5

0.1

0.8

0.7

0 60 120 240 300 360

0.9

1

180

Correlation

Prototype RC

Cubic RC

Corrugated RC

f = 200 MHz

Stirrer angle [ ]

0.6

0

0.2

0.3

0.4

0.5

0.1

0.8

0.7

0 60 120 240 300 360

0.9

1

180

Correlation

Prototype RC

Cubic RC

Corrugated RC

f = 300 MHz

Stirrer angle [ ]

0.6

0

0.2

0.3

0.4

0.5

0.1

0.8

0.7

0 60 120 240 300 360

0.9

1

180

Correlation

Prototype RC

Cubic RC

Corrugated RC

f = 100 MHz

Figure 7.12: Absolute value of the correlation coefficient |ρ(ϕ)| as a function of stirrer angleϕ = 0 . . . 355 in the prototype RC, the corrugated RC, and the cubic RC at a frequency off = 100 MHz and f = 150 MHz (top) as well as f = 200 MHz and f = 300 MHz (bottom).

RC does not perform as bad as always alleged, mainly due to the fact that the fielddistribution within a cubic RC (including a stirrer) does not have anything in commonwith the fields observed in a cubic cavity. The presence of a stirring device shifts themodes in frequency depending on their respective field distribution away from the ana-lytically calculated resonance frequencies [19]. Therefore the usually observed problemof degenerate modes does not come into play within a cubic RC and contrary to thewidely accepted RC design guidelines [6] a cubic RC will not exhibit worse (or better)performance than other rectangular RCs.


1

2

3

4

5

6IEC limit lineStandard deviation [dB]

Prototype RC

Cubic RC

Corrugated RC

0

Frequency [MHz]f

100 300 400 5002000

Figure 7.13: Envelopes of the statistical field uniformities σxyz and σξ in the prototype RC, thecorrugated RC, and the cubic RC obtained according to the procedure outlined in Section 2.7(- - - IEC limit line). Corresponding detailed per-component field uniformities are shown inFig. D.3. . . Fig. D.5.

7.6 Equipment under test simulation

For the analysis of EUTs within an RC, the prototype RC as shown in Fig. 6.9 was usedalong with the canonical EUTs (CEUTs) modeled in Section 6.5.2.


Near field simulations were performed over a frequency range of 50. . . 1000 MHz (fre-quency resolution 10, 25 and 50 MHz) with a rotational stirrer increment angle of 5

resulting in 72 stirrer steps. The three component electric and magnetic near field wascomputed in ten equally spaced planes parallel to the xy-plane with a spatial resolutionof 0.05 m, i.e. near field data is available at 27 440 points throughout the chamber. WithFig. 7.14 it is possible to investigate the effects caused by placing different EUTs withinthe RC: depicted is the magnitude of the simulated electric field strength |E| computedat a height of z = 2 m above the chamber bottom. The chamber is operated in theimmunity testing mode. Results shown were calculated at a frequency of f = 400 MHzand a fixed stirrer position of ϕ = 210. The chamber loading can be qualitatively an-alyzed by looking at the change of the overall field distribution and by examining howthe field varies in the cut planes at x = −1.45 m and y = −1.25 m: shown is the emptyRC without EUT in Fig. 7.14a), the RC with the loop EUT in Fig. 7.14b), and thebox EUT operated in gap mode configuration in Fig. 7.14c). These CEUTs are testedfor immunity, i.e. they are “passive” and the excitation is the logper antenna in front

7.6 EQUIPMENT UNDER TEST SIMULATION 139

a) b)

c)

xyz

d)

xyz

Figure 7.14: Magnitude of the electric field strength |E| simulated in the reverberation chamberat f = 400 MHz with different canonical EUTs: a) RC without CEUT, b) RC with loopCEUT (immunity), c) RC with box CEUT operated in gap mode (immunity), d) RC withbox CEUT operated in gap mode (emission).

the RC door. When comparing Fig. 7.14a) and Fig. 7.14b) (i.e. empty and loop EUTloaded RC) it is immediately apparent that the loading introduced by the loop EUT isvery small at f = 400 MHz: the change of the electric field pattern and magnitude isalmost negligible, which is due to the small size of the loop EUT together with its highconductivity of 1.1 ·106 S/m. Increased loading of the chamber can be seen in Fig. 7.14c):As the box EUT is placed inside the RC, the magnitude of the electric field is reduced,the overall field distribution however remains similar to the one in the unloaded RCdepicted in Fig. 7.14a). The latter is an indication that the box EUT (conductivity also1.1 · 106 S/m) loads the chamber only moderately - simulations performed during the


No EUT

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit lineStandard deviation [dB]

600

No EUT

Box EUT

Loop EUT

Figure 7.15: Envelopes of the statistical field uniformities σxyz and σξ without an EUT, withthe canonical loop EUTs, and the box EUT, obtained according to the procedure outlined inSection 2.7 (- - - IEC limit line).

course of this thesis involving larger EUTs and hence higher losses showed that strongloading of the chamber is exhibited by a reduction in field magnitude and at the sametime a significant change in the field distribution, starting from locally around the posi-tion of the EUT to globally throughout the entire chamber with increasing EUT-inducedloading [194]. Fig. 7.14d) exhibits the electric field distribution obtained when the boxCEUT is tested for emissions, so that the logper antenna serves as a receiving antennapicking up EM fields radiated by the “active” CEUT. It can be seen that the fields closeto the box CEUT show some similarity for the emission and the immunity configuration;further away from the CEUT however, this resemblance gradually diminishes. As thefrequency is increased, this effect becomes more and more pronounced [194].


The field uniformity with the CEUTs placed inside the RC was computed as outlined inSection 2.7 and [6]. Field uniformity computations are based on the simulated electricnear field with the respective stirrer in operation in the prototype RC and are shownin Fig. 7.15. The “volume of uniform field” has dimensions of ∆w = 0.4 m · ∆l =1 m ·∆h = 1 m. As expected, at frequencies in the 50. . . 250 MHz range, the field unifor-mity is insufficient. This is due to the chamber size as well as geometry (the “at least1.5 modes/MHz” and “more than 100 modes above cutoff” criteria are passed aroundf = 270 MHz) and the stirrer effectiveness.Starting from 350 MHz, sufficient statistical field uniformity is achieved for all CEUTswithin the testing volume, indicated by electrical field standard deviations of σ ≤ 3 dB.

7.6 EQUIPMENT UNDER TEST SIMULATION 141

Without any EUT in the RC the field uniformity exhibits the best performance; as theloading is increased from the simple loop EUT to the box EUT, field uniformity val-ues slightly worse than in the “empty RC” setup are obtained in the simulation. Forthe whole frequency range of f = 350 . . .600 MHz (and also higher), however, the per-component and the combined field uniformity σ remains below 3 dB, confirming thatthe loading of the RC even with the relatively large box EUT is still within an accept-able level. With the near field distribution investigated in Section 7.6.1 and shown inFig. 7.14, this result is a logical implication. Detailed per-component and the com-bined field uniformity σ plots for all canonical EUTs can be found in Appendix D,Fig. D.12. . . Fig. D.14. Fig. D.15 also exhibits the field uniformity obtained with a muchlarger EUT within the RC, which clearly loads the chamber beyond its maximum limit.In order to perform EMC tests complying with [6], either a larger RC needs to be utilizedor the requirements on the field uniformity and hence uncertainty of the results must berelaxed.

7.6.3 TX/RX antenna coupling

As mentioned in Section 7.3.3, often analytical point or line sources and ideal Hertziandipoles are employed for RC simulations, because they are easy to implement in a numer-ical code. This is problematic due to undesirable coupling effects between an EUT andthe TX/RX antenna setup. Generally, the following classification can be made for typi-cal RC operation modes and the resulting magnitude of the electric field |E| (it is alwaysassumed that the underlying EM field ensembles are statistically independent) [126]

• Strong dominant direct (i.e. deterministic) coupling path and comparatively smallmulti-path propagation (this implies in terms of the traditional signal-to-noise ratio(SNR)→ ∞): |E| is Gaussian distributed with nonzero mean (see Section C.2).

• Little direct (i.e. deterministic) coupling and mostly multi-path propagation (e.g.SNR= 10): |E| is Rice distributed (also known as noncentral χ(6), Section C.2).

• No direct coupling, only multi-path propagation (SNR→ 0): |E| is Rayleigh dis-tributed (also known as central χ(6), see Section C.2).

Fig. 7.16a) proves that the usage of a Hertzian dipole in an RC simulation leads tothe highly undesirable result of strong coupling between an EUT within the chamberand the excitation: Due to its very low directivity (1.76 dB), the |E|/|Emax| distrib-ution resembles almost a Gaussian distribution with nonzero mean, which is a clearindication of a dominant coupling path in an EM environment with only little multi-path propagation [195] – the exact opposite of a “well-behaved” RC, where for a properoperation implicitly “pure” multi-path propagation is assumed. Through the usage ofantennas with higher directivity, this unwanted direct coupling can be considerably re-duced: Fig. 7.16b) and Fig. 7.16c) show the respective |E|/|Emax| distributions obtainedwith a biconical and a logper antenna. With their higher directivity (biconical 3.5 dB,logper antenna 6 dB) the statistical distributions are shifted towards the origin and re-semble more a Rice distribution where direct, non-dominant coupling paths still exist,but multi-path propagation is clearly dominant [195, 126].


Biconical antennaHertzian dipole

0 0.2 0.4 0.6 0.8 1

E E/ max

M M/ max

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Logper antenna(toward corner)

Logper antenna(toward EUT)M M/ max

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

E E/ max

a) b)

c) d)

0

0.2

0.4

0.6

0.8

1

Figure 7.16: Statistical distribution (normalized number of samples M/Mmax) of the magni-tude of the simulated electric field strength |E|/|Emax| within the RC. Results shown werecalculated at a frequency f = 300 MHz and cumulated from angular stirrer positions ofϕ = 0 . . . 355 with 5 step angle using different excitation antenna types and orientations:a) ideal Hertzian dipole; b) biconical antenna; c) logper antenna pointing toward the testvolume V ; d) logper antenna pointing toward an RC corner opposite the vertical stirrer. Intotal, 186 000 samples were used to plot each histogram.

In contrast to the positioning of the logper antenna within the RC in Section 7.3.3, theorientation of this antenna was changed such that it points into a corner opposite to thestirrer (as shown in Appendix D, Fig. D.2). According to theoretical considerations, thisrepresents the situation of a multi-path environment without any dominant direct cou-pling path, which results in a central χ(6) distribution for the magnitude of the electricfield strength (also called Rayleigh distribution with a signal-to-noise ratio of 0, which

7.7 COMPARISON OF DIFFERENT STIRRERS 143

represents the ideal RC mode of operation). This effect can be seen when comparingFig. 7.16c) with Fig. 7.16d): In Fig. 7.16c) the |E|/|Emax| distribution appears to beslightly offset from the origin for the logper antenna oriented towards the rectangulartest volume. Once however the antenna points away from this test volume towards achamber corner, the offset vanishes and Fig. 7.16d) shows a very good agreement withthe theoretical χ(6) distribution. The latter antenna orientation is also recommendedin the IEC 61000-4-21 standard [6] in order to prevent direct “illumination” of the vol-ume where an EUT will be placed during RC testing. It is important to note that inthe graphical representation of Fig. 7.16, the number of samples M per class has beennormalized to the maximum number of samples of all classes Mmax – Fig. 7.16a). . . d)all extend along the ordinate to 1. In addition to the usage of different class widths (asmentioned in Section 2.8), this normalization together with a much larger of samplesare the main reason why the visual appearance of Fig. 7.16a) differs considerably fromFig. 7.10a).

7.7 Comparison of different stirrers

In this section the simulation and performance analysis of various types of EM stirrersinside the RC prototype is presented. Several different stirrer designs and sizes arecompared against each other, and the influence of the stirrer axis orientation (verticalvs. horizontal) within the chamber is shown. The following stirrers were compared:

• Vertical and horizontal 6-paddle stirrers, see Fig. 6.6a) and Fig. 6.7c)

• Stacked cross-plate stirrer, as shown in Fig. 6.6b)

• 6-paddle connected stirrer, see Fig. 6.6c)

• Upset Z-fold stirrer with and without gaps, as shown in Fig. 6.6d)

The stirrer simulation models utilized for the performance comparisons were described indetail in Section 6.2. It should be noted that all simulated stirrers can be circumscribedby a cylinder with a diameter of 0.735 m and 2.76 m height, i.e. all stirrers have the same“rotational diameter” and “rotational height” and therefore also the same “rotationalvolume”. This was done in order to make the comparison “even-handed”, since froma basic understanding of electromagnetics it is obvious that a much larger stirrer willintroduce much greater changes of the field distribution as it is rotated and thereforeexhibits a better performance than a smaller one.


Performance comparisons between all stirrer types were carried out based on the simu-lated electric near field with the respective stirrer in operation in the prototype RC. Thesame RC was simulated with the six stirrers mentioned above over a frequency range of50 . . .500 MHz (frequency resolution 10, 25 and 50 MHz) with a rotational stirrer incre-ment angle of 5 resulting in 72 stirrer steps. The three-component electric and magnetic

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Figure 7.17: Magnitude of the simulated electric field strength |E| computed in the xy-plane at a height z = 2m above the chamberbottom. Results shown were calculated at a frequency f = 300 MHz and an angular position of ϕ = 200 in a). . . d) and ϕ = 210

in e). . . h) for different vertical stirrer geometries: a), e) 6-paddle V-stirrer; b), f) stacked cross-plate V-stirrer; c), g) 6-paddleconnected V-stirrer; d), h) upset Z-fold V-stirrer. Excitation source is a logper antenna.


near field was computed in ten equally spaced planes parallel to the xy-plane with a spa-tial resolution of 0.05 m in x- and y-direction, i.e. near field data is available at 27 440points throughout the chamber. In order to compare more intuitively the impact of thedifferent stirrers on the field distribution, results data shown here was chosen from thelower to medium frequency range (close to the LOF/LUF), where it is easier to visualizehow effective a particular stirrer is.Fig. 7.17 depicts the magnitude of the simulated electric field strength |E| computed ata height of z = 2 m above the chamber bottom for four different stirrers (the horizontaland the upset Z-fold without gaps are not shown in this figure). These results were cal-culated at a frequency of f = 300 MHz and for a fixed angular position of ϕ = 200 (toprow) and ϕ = 210 (bottom row). The stirrer performance can be qualitatively analyzedby looking at the change of the overall field distribution and by examining how the fieldvaries in the cut planes at x = −1.45 m and y = −1.25 m from one stirrer step to thenext but one. Immediately apparent is that the resulting chamber field is quite differentamong all stirrers. The only moderate similarity can be seen between the field patternexcited with the “standard 6-paddle stirrer” in place as shown in Fig. 7.17a), Fig. 7.17e),and the one obtained for the “6-paddle connected stirrer” depicted in Fig. 7.17c) andFig. 7.17g). This similarity however does not prevail at other frequencies, which sug-gests that very similar stirrer designs (here in terms of the larger electrical length ofthe “6-paddle connected stirrer”) do not necessarily result in a similar EM behavior.As intuitively expected, the minimum change of the field among all stirrers for a ro-tation of 10 occurs for the “stacked cross-plate V-stirrer” in Fig. 7.17b) and f): Boththe spatial field distribution as well as the field in the two cut planes at x = −1.45 mand y = −1.25 m change only marginally, which is due to the rather simple geometricstructure and symmetry of this particular stirrer.A near field analysis at very low frequencies (around f = 100 MHz – these field distri-butions are not shown in this thesis) revealed that all stirrers are equally (in-)effectivefar below the LUF: shape, orientation, and electrical size do not matter [193]. As shownin Fig. 7.17, at f = 300 MHz all stirrers are “somewhat similarly effective” and by look-ing at the near field distribution it is difficult to judge whether one stirrer outperformsanother. Furthermore the impact of the gaps (two models of the 6-paddle and upsetZ-fold stirrer) remains unclear. Summing up, evaluating directly the near field withoutprocessing it to a more useful metric does not help in classifying stirrers with respect totheir performance.

7.7.2 Correlation analysis

As opposed to the qualitative field-pattern-based analysis of different stirrers shownbefore, the computation of the correlation allows a quantitative comparison of the stirrerperformance. The correlation coefficient ρ(ϕ) is calculated from the magnitude of theelectric field |E| sampled at 5 788 field points on the top and bottom side of a 0.4 ·1.0 · 1.0 m3 test volume located z = 1 m above the chamber floor. Each data point inFigs. 7.18 and 7.19 corresponds to the correlation between the simulated |E| for thereference stirrer angle ϕ = 0 and |E| for ϕ = 0 . . . 355. Fig. 7.18 compares |ρ(ϕ)| forfour stirrers (out of which three have completely different designs), but mounted in the


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Figure 7.18: Absolute value of the correlation coefficient |ρ(ϕ)| as a function of stirrer angleϕ at a frequency f = 300 MHz: vertical 6-paddle stirrer vs. vertical cross-plate stirrer vs.vertical Z-fold stirrer with and without gaps.

same vertical position. Initially (for ϕ = 0 . . . 60) the correlation |ρ(ϕ)| drops rapidlyfor all stirrers (except for the Z-fold with gaps), which indicates that they are electricallysufficiently large in order to achieve substantial changes of the field pattern within theRC. For both the 6-paddle and the Z-fold stirrer without gaps |ρ(ϕ)| remains relativelysmall (with some oscillations), whereas the cross-plate stirrer reaches |ρ(ϕ) = 1| again atϕ = 180, which is due to its rotational symmetry. Clearly, this stirrer is not suitable forthe application within an RC and is outperformed by the much better 6-paddle stirrerand the very well operating Z-fold stirrer without gaps. It is interesting to note that thesmall “gap modification” of the Z-fold stirrer changes completely the correlation. The(in the traditional sense) electrically much larger Z-fold stirrer without gaps performssignificantly better than the stirrer with the artificially introduced gaps.When designing an RC, typical questions often are: In which orientation is a stirrer tobe mounted and does it matter whether the stirrer is made from one piece or consistsout of several separate parts? Fig. 7.19 depicts the effect of the standard 6-paddle stirrermounted in two orientations in the RC: For this particular RC test volume it can beseen that the vertically mounted stirrer works slightly better than the horizontal oneby exhibiting a lower |ρ(ϕ)| throughout the ϕ = 0 . . . 355 range. Fig. 7.19 also showsthat surprisingly the 6-paddle connected stirrer (although electrically larger than thestandard 6-paddle stirrer) performs significantly worse than the 6-paddle stirrer made


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Figure 7.19: Absolute value of the correlation coefficient |ρ(ϕ)| as a function of stirrer angle ϕat a frequency f = 300 MHz: vertical 6-paddle stirrer vs. vertical 6-paddle connected stirrervs. horizontal 6-paddle stirrer.

of separate, not-connected parts. This effect needs further analysis in order to clarifywhether the considerable performance difference appears consistently for a wide rangeof frequencies or if it is only a narrow band phenomenon. Contrary to the frequency-domain EFIE-based MoM method utilized in this thesis, a numerical technique providingbroadband simulation data with one computation run such as e.g. FDTD should be used.Evaluating the correlation for all stirrers shows that – as expected – at low frequencies(around f = 100 MHz) the overall correlation is high. This is due to the fact that anyintentional rotational asymmetry is too small compared to the wavelength.The quantitative significance of a simulated or measured correlation coefficient |ρ(ϕ)|can be evaluated by calculating the probability PN that N samples of two uncorrelated,i.e. |ρ(ϕ)| = 0, variables would give a correlation coefficient as large as or larger thana certain predefined value. The procedure to compute the probability PN (|ρ| ≥ |ρ0|)was introduced in Section 2.5.2, tabulated values are listed in Table 2.3. Among allstirrers the Z-fold stirrer however exhibits the best performance in the lower frequencyregion, followed by the cross-plate and the vertical/horizontal 6-paddle stirrers. As thefrequency is increased, the overall correlation tends to be low (with the exception of thecross-plate stirrer; this effect is due to its structural symmetry), the vertical 6-paddleand the Z-fold stirrer perform the best, followed by the horizontally mounted 6-paddlestirrer. With respect to the impact of gaps between individual stirrer elements (tochange the electrical size of a stirrer), their effect remains still unclear: whereas gaps are


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Figure 7.20: Envelopes of the statistical field uniformities σxyz and σξ for the vertical 6-paddlestirrer, the vertical 6-paddle connected stirrer, and the horizontal 6-paddle stirrer, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).

beneficial in the 6-paddle stirrer (lower correlation than the connected version), they arenot beneficial for the upset Z-fold stirrer (the Z-fold stirrer made out of one single pieceperforms better in terms of correlation than the model with gaps). It seems impossibleto clearly state that a large stirrer made out of one piece of metal performs significantlybetter or worse than a replica version with small gaps in-between individual plates. Forthis reason, unfortunately, a case-by-case analysis has to be carried out.


Evaluating the field uniformity within the RC leads to a similar conclusion with respectto the stirrer gaps as the correlation analysis: as shown in Fig. 7.20 and Fig. 7.21 for somestirrers, gaps have a positive effect on the field uniformity (e.g. 6-paddle stirrer), whereasfor others their effect is negative (e.g. Z-fold). It can be concluded that “complicated”stirrers exhibit consistently superior performance than stirrers with a simple geometry(e.g. cross-plate stirrer) and that all “complicated” stirrers analyzed in this thesis performin a similar way over a broad range of frequencies. It is interesting to note that the stirrerorientation plays an important role for the field uniformity, as the horizontal 6-paddlestirrer performs significantly worse than the vertically mounted 6-paddle stirrer. Thiseffect might be due to the TX antenna orientation and strong, orientation-dependentcoupling between the antenna and the 6-paddle stirrers. Detailed per-component fielduniformities can be found in Appendix D, Fig. D.6. . . Fig. D.11.


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Figure 7.21: Envelopes of the statistical field uniformities σxyz and σξ for the vertical 6-paddlestirrer, vertical cross-plate stirrer, and vertical Z-fold stirrer, obtained according to the pro-cedure outlined in Section 2.7 (- - - IEC limit line).

7.7.4 Final performance evaluation

It was found that stirrer performance comparisons are valid to a great extent only forthe RC from which the performance data originated, especially at frequencies aroundthe LOF/LUF. Generally, a stirrer exhibiting superior performance over other stirrers(in terms of statistical field uniformity or isotropy, correlation, and especially the unfor-tunate SR), may perform also better in another chamber with a different geometry. Asshown here, there is however no obvious reason, why it must mandatorily outperformthese “other stirrers” in any other chamber, as the stirrer performance depends stronglyon parameters such as the position or the orientation of the stirrer within the chamberand on the field distribution (which is strongly influenced by the chamber geometryitself). As a conclusion resulting from the near field, correlation, and field uniformityanalysis of the stirrers spanning 50 . . .500 MHz, the best performance in the prototypeRC is exhibited by the two Z-fold stirrers (with and without gaps), closely followed bythe vertically mounted 6-paddle stirrers (with and without gaps). These four stirrersperform significantly better than the symmetric cross-plate stirrer, which is still bet-ter than the horizontal six-paddle stirrer. Table 7.4 summarizes the gathered stirrerperformance data for frequencies of 100 MHz and 300 MHz.

7.7.5 Plane-wave-based stirrer comparisons

Comparisons of several stirrer types within an RC generally result in very long simula-tion runtimes (on the order of several weeks to months). Therefore an idea originallypublished in [104, 105] was taken up, proposing to do a stirrer analysis and optimization

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Stirrerperformance NF a |ρ(ϕ)| σ NF a |ρ(ϕ)| σ

Best Z b (gaps) Z b Z b (gaps) Z b (gaps) 6V d 6V d

↑ – X-plate c X-plate c Z b Z b Z b (gaps)

– Z b (gaps) Z b 6V d 6H e Z b

– 6H e 6V d (no gaps) 6V d (no gaps) 6V d (no gaps) 6V d (no gaps)

↓ – 6V d 6V d 6H e Z b (gaps) X-plate c

Worst all others 6V d (no gaps) 6H e X-plate c X-plate c 6H e

a near field b Z-fold c Cross-plate d 6-paddle vertical e 6-paddle horizontal

Table 7.4: Performance comparison of different stirrers with respect to the near field (NF), the correlation coefficient |ρ(ϕ)|, and thefield uniformity standard deviation σ at 100 MHz and 300 MHz.


i =60 E i

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Figure 7.22: Stirrer RCS calculations using circularly polarized plane wave excitation. Thedirection of the plane wave incidence was varied from 0 . . . 180 with increments of 20 (i.e.10 directions) and the stirrer was rotated in 5 angular steps.

in free space with plane wave illumination of the stirrer. A radar cross section (RCS)-based simulation of the stirrer models was carried out to compute the scattered far fieldduring a stirrer rotation (see Fig. 7.22 for the computational setup).The RCS σ is the measure of a target’s ability to reflect radar signals in the directionof the radar receiver, i.e. it is a measure of the ratio of backscatter power per unit solidangle in the direction of the incident wave (from the target) to the power density thatis intercepted by the target [196]. Mathematically the RCS can be expressed as

σ = A · R · D (7.6)

where A is the projected cross section, R denotes the reflectivity, and D the directivityof the target. The RCS was calculated by using the plane wave excitation (circularlypolarized) mentioned above and by varying the direction of the plane wave incidencefrom 0 . . . 180 with increments of 20 (i.e. 10 directions, see Fig. 7.22). The stirrer wasrotated in angular steps of 5. Far field calculations utilized for the RCS are based on37×73 = 2701 data points and take only the scattered field into consideration (resolutionin both azimuthal and elevational direction 5).Based on the RCS results, it was tried to compare the individual stirrer performancesagainst each other and to map their free space behavior to their effect inside the RC.This approach proved to be rather problematic for several reasons:

• stirrers with completely different geometric designs exhibited a very similar RCS

• throughout the whole lower frequency range (100 MHz. . . 1 GHz) for a given stirrer,


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the RCS changes only minimally from one rotational stirrer position to the next –this is a very significant difference compared to the stirrer effect in the RC on theEM field (compare e.g. with Fig. 7.5)

• consequently, also processing the gathered RCS data further and computing thecorrelation did not help much, as the correlation coefficient is almost unity re-gardless of the stirrer shape and the angle of incidence for the RCS over a broadrange of frequencies (see Fig. 7.23 for a comparison of the correlation coefficientcomputed for the 6-paddle stirrer within the prototype RC and the same stirrer infree-space using the RCS).

Although very attractive in terms of required computational power and time expenditure,the performance analysis of stirrers in free-space using RCS calculations was not furtherpursued.

7.8 Simulation and measurement time budget

Measurements of the electric field only provide limited information on the complex physi-cal phenomena within an RC, whereas simulations can supply a wealth of EM field data.Still, it is very important to always keep in mind the substantial time expenditure ittakes to compute one single metric such as correlation or field uniformity over a broader

7.8 SIMULATION AND MEASUREMENT TIME BUDGET 153

E-field at. . . Simulation time Measurement time

1 spatial position1 stirrer angle10 frequencies

1 hour. . . 7 days a 60 s

100 spatial positions1 stirrer angle1 frequency

10 s b 4 hours

1 spatial position10 stirrer angles1 frequency

1 hour. . . 7 days a 180 s

a depending on the frequency range b E-, H-field and currents I, J readily available

Table 7.5: Time expenditure comparison to derive a certain number of samples of the electricfield in RC simulations based on the EFIE MoM versus measurements.

frequency range [178]. Table 7.5 gives an overview on the average time expenditurerequired to simulate or measure the electric field at a given number of spatial positionswithin the prototype (or any similarly sized) RC over several frequencies with a certainnumber of rotational stirrer angles. It is interesting to note how much a particular EMfield analysis differs in time between simulation versus measurement: computing e.g. theelectric field within the prototype RC at ten frequencies and one stirrer position takesbetween one hour up to several days depending on the actual frequency (see Section 3.4and Table 3.1). Performing the same task in a measurement setup takes only about oneminute at one spatial position, irrespective of the frequency. Measuring however theelectric field throughout the entire chamber at 10 000 spatial positions for one frequencycan easily take two or three weeks (depending on the actual field probe system and thenumber of field probes used in the measurement setup). Contrary to this, in the simu-lation – compared to the total simulation time – once the currents are computed it doesnot really matter whether the field is calculated at one single or 10 000 spatial positions(cf. Section 3.2).Comparing the EFIE and MoM simulation technique applied to RCs with near field mea-surements yields a reciprocal behavior for the time expenditure: analysis tasks whichcan be quickly performed in a simulation require a lot of time in measurements and viceversa (see Table 7.5). Especially frequency sweeps and a rotation of the stirrer are time-consuming tasks in a frequency-domain MoM simulation compared with measurements.This is a problematic issue for several typical RC performance parameters: if, for exam-ple, the correlation coefficient or field uniformity is of interest at 20 frequencies and witha stirrer angle resolution of 5 (i.e. 72 rotational stirrer positions), this translates into1440 simulations which need to be performed. Generating plots such as e.g. Fig. 7.21or Fig. 7.19 therefore easily adds up to combined CPU simulation run times of several


months. As long as the required number of spatial field points remains small and thechamber geometry does not change, and if there is only an interest in the electric field,measurements can provide this data significantly faster. Once however fields need to bevisualized at a large number of spatial positions in order to get a deeper understandingof the physical processes within an RC, or the RC geometry has to be changed (e.g. froma standard rectangular chamber to a corrugated RC), EM simulations are advantageousand clearly better for the task at hand.

7.9 Conclusion

In the beginning, the procedure used to carry out an RC data analysis was presented.The necessity of a rigorous simulation validation was emphasized and the particularadvantages and drawbacks of a comparison with analytically calculated results, resultsobtained by a different numerical technique, and measurements were described. Mea-surements were utilized in this thesis to validate the simulation results, as they providean additional insight into the “reality of the chamber physics”.Cavity simulations were performed to investigate the influence of the chamber door andto derive suitable conductivity values. This analysis revealed that by far the biggest per-turbance seemed to be caused by the door. To facilitate modeling of the cavity and theRC, the door was neglected in the first simulation models. In subsequent simulations,the door was included and proved to have a significant effect on the simulated results.In a next step, a stirrer was inserted into the cavity to build a fully functional RC. Thesimulation allowed a thorough analysis of the change in the field distribution with andwithout the stirrer in the chamber and showed the effect of the stirrer as it is rotated.The influence of different wall conductivities, the door, and small modifications of thechamber geometry was investigated. As expected, the choice of a reasonable conductiv-ity is important if the absolute field values in the chamber are of interest; if only therelative field distribution needs to be known, absolute conductivity values do not playa very important role (as long as metallic parts of the RC still appear as “metallic”rather than “semiconducting”). The simulations showed that even small changes in thegeometry (for example the recess of the door) can have a significant impact on the fieldwithin the RC.The effect of Hertzian dipoles, λ/2-dipoles, biconical, logper, and horn antennas wasexamined in the RC simulations. It was found that the actual antenna type employed inan RC strongly influences the field distribution. Whereas the assumption that differentantennas will lead to similar results in a statistical sense (i.e. a large number of samplestaken from a large number of stirrer positions) was confirmed, the results for a given,fixed stirrer position were strongly dependent on a particular antenna. Therefore, if agood agreement between measurements and simulations is desired, the correct modelingof the TX/RX antennas as used in the measurements is of prime importance. This doesnot only apply to the type, but also to the position and alignment of an antenna.In order to make sure that the modeled RC represented a good approximation of theactual prototype RC, simulations were validated by extensive measurements. It waspointed out that completely different electromagnetic (EM) fields can still generate an

7.9 CONCLUSION 155

identical field uniformity, the same correlation coefficient or equal anisotropy coefficients– therefore a thorough validation of RC simulation results cannot be accomplished usingthis type of “processed data”. For lower frequencies (f = 50 MHz. . . 500 MHz) simula-tion and measurement were found to agree well, however as the frequency was increasedto f = 700 MHz. . . 1000 MHz, the agreement between measurements and simulationsprogressively deteriorated. It was concluded that this “breakdown” of the simulationstarting from f > 600 . . .700 MHz is related to the fact that the EM field inside an RCbecomes extremely sensitive to even tiny geometric details. This (from an RC applica-tion’s point of view desirable) phenomenon renders a simulation practically not feasibleat frequencies much greater than the lowest usable frequency – unless one is willingto undertake the challenge to model and discretize virtually every nut and bolt of thechamber.The simulated near field distribution within the RC was used as a basis to compute thecorrelation coefficients, statistical field uniformity, and coupling paths for the analysis ofdifferent RCs, stirrers, and canonical equipment under tests (EUTs). The unprocessednear field was useful to get a rough overview on how the field distribution (e.g. localmaxima and minima) changes as the stirrer rotates. Otherwise, raw near field data didnot provide very conclusive insights into whether an RC is particularly good or bad andwas therefore of little practical use.To investigate the influence of special chamber designs on the RC performance, a corru-gated and a cubic RC were modeled in addition to the prototype RC. From a correlationpoint of view, neither the cubic nor the corrugated RC exhibited convincing results acrossall frequencies which would clearly outclass the standard rectangular prototype RC. Thefield uniformity at frequencies in the lower range was insufficient in all chambers, withthe cubic RC performing the worst. Starting from f > 300 MHz, however, sufficientstatistical field uniformity was achieved – with the exception of the cubic RC exhibitingsignificantly higher values at e.g. f ≈ 400 MHz. As a summary, neither a cubic chambernor an RC with corrugations on the walls exhibited consistently superior (or inferior)field uniformity performance. It is interesting to note that the cubic RC does not performas bad as always alleged, mainly due to the fact that the field distribution within a cubicRC (including a stirrer) does not have anything in common with the fields observed ina cubic cavity. The presence of a stirring device shifts the modes in frequency depend-ing on their respective field distribution away from the analytically calculated resonancefrequencies. Therefore the usually observed problem of degenerate modes does not comeinto play within a cubic RC and contrary to the widely accepted RC design guidelines acubic RC will not exhibit worse (or better) performance than other rectangular RCs.Three different canonical EUTs were simulated within the prototype RC to do a load-ing, field uniformity, and coupling path analysis. As the loading of the chamber wasgradually increased from a dipole via a simple loop EUT to the box EUT, field uni-formity values worse than in the “empty RC” setup were obtained in the simulation.The per-component and the combined field uniformity however confirmed that the load-ing of the RC even with the relatively large box EUT was still within an acceptablelevel. The investigation of different coupling paths revealed that the usage of a Hertziandipole in an RC simulation leads to the highly undesirable result of strong direct cou-pling between an EUT and the excitation. Through the usage of realistic antennas with


higher directivity, this unwanted direct coupling can be considerably reduced. Threespecial multipath/direct path scenarios were simulated resulting in Gaussian, Rice, andRayleigh statistical distributions.Several different stirrer designs and sizes were compared against each other, and theinfluence of the stirrer axis orientation within the chamber was shown. In a correlationanalysis of all stirrers, the Z-fold stirrer exhibited the best performance in the lower fre-quency region, followed by the cross-plate and the vertical/horizontal 6-paddle stirrers.As the frequency was increased, the vertical 6-paddle and the Z-fold stirrer performedthe best, followed by the horizontally mounted 6-paddle stirrer. By using only correla-tion data it was however impossible to clearly state that a large stirrer made out of onepiece of metal performed significantly better or worse than a replica version with smallgaps in-between individual parts. This was also true for the field uniformity: in somestirrers, gaps had a positive effect, whereas in others their effect was negative. It canbe concluded that “complicated” stirrers exhibit consistently superior performance thanstirrers with a simple geometry – which is not surprising. In addition, all “complicated”stirrers analyzed in this thesis performed in a similar way over a broad range of frequen-cies. It is interesting to note that the stirrer orientation played an important role for thefield uniformity, as the 6-paddle stirrer performed significantly worse in the horizontalthan in the vertical position. Summing up, it can be stated that stirrer performancecomparisons are valid to a great extent only for the RC from which the performancedata originated. This makes the design of a universal “high performance” stirrer verydifficult, if not impossible. The much faster stirrer performance analysis in free spaceproved to be unsuccessful and was therefore not further pursued.

8 Conclusion

The three-dimensional simulation of a reverberation chamber (RC) was presented inthis thesis. In the beginning, fundamental concepts and key parameters of an RC wereintroduced in Chapter 2. These included the mode distribution, mode density, modalgaps, and the quality factor. Furthermore, the RC was described as a statistical elec-tromagnetic test environment and characterized by distribution functions, correlation,uncertainty, and field uniformity. In Chapter 3 it was shown that it is crucial to select asuitable numerical method to perform meaningful RC simulations. A chosen numericaltechnique must be able to deliver results over a wide frequency range without usingexcessive computational resources; the method must be able to handle large, irregu-lar structures, and a varying geometry without introducing errors. Furthermore, theremust be a possibility to account for finite metal conductivity as well as highly reso-nant structures. The computation of near fields at an arbitrary number of chamberlocations should be possible without adding too much computational overhead. Forthis thesis, a frequency-domain electric field integral equation (EFIE)-based method-of-moments (MoM) technique was chosen. Chapter 4 put the work accomplished duringthis thesis into perspective with previous publications on RC simulations. One strikingresult of this literature survey was that in the majority of the published material a thor-ough simulation validation tends to be neglected. Chapter 5 described the constructionof a prototype RC used later on for simulation validations and explained the setup of themeasurement system. Measurement errors originating from field probes, antennas, andstirrers were discussed and assessed for their impact on deviations between simulatedand measured results. The biggest deviations were found to result from the antennatripods and position inaccuracies of the field probe head or the antennas. Chapter 6outlined how the practical prototype RC including the door, a stirrer, several differentantennas, and EUTs was modeled for the electromagnetic simulation. Electrical con-ductivity values were defined for material as it is used in a shielded room construction,i.e. walls consisting of several interconnected sheets with intermediate overlapping flatstock. In addition to the prototype RC, cubic and corrugated chambers, an offset-wallRC as well as several vertical and horizontal stirrers were modeled.Whereas Chapters 2. . . 6 laid the ground for the comprehensive simulation of the RC,Chapter 7 makes use of the previous work and presents a thorough analysis of RCs.Simulation results of a detailed asymmetric RC model were benchmarked against mea-surements and exhibited a good agreement in the lower-to-medium frequency range (atfrequencies less or equal to twice the lowest usable frequency). It was shown that aproper validation of the simulation must be performed with direct comparisons againstmeasured near fields without further data processing or statistical analysis. Furthermore,a deeper analysis of different chamber geometries, TX/RX antennas, various stirrer de-signs, and EUTs was performed. The importance of small geometric details and theagreement between actual prototype and simulated RC dimensions was discussed. It

157

158 8 CONCLUSION

was shown that the type, position, and alignment of the excitation source in the sim-ulation model change the field pattern significantly. In addition, the effect of variousstirrers on the fields, correlation, and uniformity inside the chamber was visualized. The6-paddle stirrer developed for this thesis and the commercially available Z-fold stirrerexhibited the best performance. The insertion of gaps between parts of a stirrer didnot show a consistently “good” or “bad” effect – some stirrers performed better with,some better without geometrical gaps. The stirrer analysis using a simulation approachonly confirmed the well-known conclusion that “complicated, asymmetric” stirrers willoutperform “simple, symmetric” stirrers over a broad frequency range. In addition, itwas found that the performance of a stirrer is linked to a particular RC, which rendersthe design of a universal “high performance” stirrer very difficult, if not impossible.A comparison between the standard rectangular RC with a cubic and a corrugated cham-ber revealed that the two latter chamber geometries do not offer significant advantagesconcerning correlation and field uniformity. On the other hand, the cubic RC does notperform as bad as always alleged, mainly due to the fact that the field distribution withina cubic RC (including a stirrer) does not have anything in common with the fields ob-served in a cubic cavity. The presence of a stirring device shifts the modes in frequencydepending on their respective field distribution away from the analytically calculatedresonance frequencies. Therefore the usually observed problem of degenerate modesdoes not come into play within a cubic RC and contrary to the widely accepted RCdesign guidelines a cubic RC will not exhibit worse (or better) performance than otherrectangular RCs. Three special multipath/direct path coupling scenarios were simulated(Gaussian, Rice, and Rayleigh statistical distributions). This investigation revealed thatthe usage of a Hertzian dipole in an RC simulation leads to undesirable strong directcoupling between an EUT and the excitation. Through the usage of realistic antennaswith higher directivity, this unwanted direct coupling can be considerably reduced.Which lessons can be learned from this thesis? One should always keep in mind that theluxury of having the knowledge about the electromagnetic field at any arbitrary positionwithin the RC and being able to visualize the field distribution is very instructive, butjust one part of the whole story. For most, if not all, important RC quantities such ascorrelation, anisotropy coefficients, or uniformity, knowledge of the field throughout thechamber at only one stirrer step is essentially useless. In order to calculate these quan-tities in a meaningful manner and to come up with technically sound recommendations,often the field distribution within an RC needs to be known at 50. . . 100 angular stirrerpositions. To complicate matters more, knowledge of the correlation coefficient or fielduniformity at only a single frequency is of not much use either. Essentially needed isknowledge of the electromagnetic field at a lot of spatial positions within the chamber,computed for many different angular stirrer positions over a broad range of frequencieswith a fine frequency resolution. Starting with these requirements, as shown at theend of Chapter 7, the time it takes to compute one single metric such as correlation orfield uniformity is at minimum substantial, possibly even prohibitive. If, for example, afrequency-domain solver is used and the correlation is of interest over a bandwidth of500 MHz with 5 MHz steps at an angular stirrer resolution of 5 (i.e. 72 rotational stirrerpositions), 7200 simulations need to be performed, assuming that no interpolations canbe done. Depending on the actual frequency range, this task might be computationally

159

not feasible. Even at lower frequencies, where fewer discretization elements can be usedin a simulation, generating plots as the ones presented in Chapter 7 easily adds up tocombined run times of several months – especially if two or more chambers or stirrersare to be compared with each other.For frequencies much smaller than the lowest usable frequency, the simulation of an RCis possible, the chamber however becomes electrically too small compared to the op-erational wavelength, which prevents sufficient statistical field uniformity – the laws ofphysics do not permit an optimization of RCs. Conversely, at frequencies much above thelowest usable frequency, where a high number of modes is above cutoff, almost any RCworks well regardless of its particular design (hence, there is no optimization needed).In addition, as shown in this thesis, with increasing frequency the field within an RCbecomes more and more sensitive to even small geometric details, which makes propermodeling numerically not feasible at high frequencies. The possibilities for RC designoptimizations significantly below or above the lowest usable frequency are therefore lim-ited. At frequencies around the lowest usable frequency, however, stirrer shapes or wallgeometries can be optimized with an electromagnetic simulation and the effect of mul-tiple stirrers or other means for an improvement of field uniformity can be investigatedin order to extend the operating frequency for a given RC to lower frequencies. For asuccessful optimization it is therefore crucial to have a numerical tool at hand, which canaccurately simulate an RC around the lowest usable frequency, accommodate a rotatingstirrer, and provide broadband simulation data with as few computations as possible.

9 Outlook

In this thesis a simulation model of a reverberation chamber (RC) was developed, whichallows to accurately reproduce the electromagnetic fields within a prototype chamber inthe lower to medium frequency range. This model of the RC can be used for furtherresearch. The biggest problem with frequency-domain RC simulations are the notori-ously long computation runtimes; although time-domain simulation codes are known tobe problematic with highly resonant structures, it would be advisable to compare the re-sults obtained in the frequency-domain simulations of this thesis with broadband resultsfrom an electromagnetic time-domain solver. In order not to “reinvent the wheel”, forthis purpose preferably a commercial, state-of-the-art solver (such as for example CST’sMicrowave Studio ) should be utilized.Two parameters were identified as crucial for the performance of an RC in terms ofcorrelation and field uniformity: the stirrer effectiveness and the quality factor of thechamber. With respect to the first issue, it was shown in this thesis that a comprehen-sive stirrer analysis with the stirrer operating inside the RC is computationally extremelyexpensive and hence time-consuming. A stirrer analysis and optimization in free-spaceusing multiple-angle plane wave illumination was attempted, but proved to be ratherunsuccessful as the metric “far field / radar cross section correlation” is too insensitive.Nevertheless, the idea to optimize a stirrer without having to simulate the whole RC istempting and should be pursued further with other more suitable metrics.Secondly, an extension to this thesis could deal with a rigorous analysis of the qualityfactor of an RC. This is a challenging topic, as the usually applied analytical formulas formultimode resonators cannot be easily applied to RCs due to their irregular geometry,the presence of one or more stirrers, and the strong coupling between the excitation andthe chamber. Furthermore, the electromagnetic modes will be shifted and widened intheir bandwidth by the stirrer rotation. Once the quality factor in a multimode RC isderived with the help of a simulation tool, it needs to be validated by measurements ina practical chamber. However, quality factor measurements tend to be inherently diffi-cult in highly resonant structures. Conductivity and leakage in the RC model must beadjusted accordingly in order to achieve a good agreement between quality factor valuesof the measurement setup and computed from the simulation. Using the simulated andmeasured results, suitable techniques can be developed allowing to influence the qualityfactor which facilitates a control of the field uniformity within an RC.An evaluation of the quality factor should also take into consideration the additionalloading of the chamber introduced through the presence of different EUTs. The CEUTresults shown in this thesis were mostly benchmarked against theoretical or previouslypublished measurement results. To gain more insight into the effects caused by an EUTwithin an RC, a CEUT should be built (or borrowed from a round robin test) andtested for immunity and emissions in a prototype RC. Results from these tests shouldbe compared with the immunity and emission simulations presented in this thesis. To

161

162 9 OUTLOOK

measure the electric near field in the prototype RC, efficient methods for field measure-ments should be investigated allowing to sample the field distribution without distortingit significantly (e.g. by electro-optical techniques). The measurement time needed for asimulation validation could be considerably reduced by developing an automated nearfield scanning system with multiple sensors and high positioning accuracy, thus facilitat-ing fast sampling of the electric field at a large number of measurement positions. Asproposed in this thesis, it is advisable to feed both measured and simulated data into acommon database for fast, error-free, and convenient access.Finally, an extension to this thesis could deal with novel TX/RX systems in an RC. Thiscould include research on electronic stirring methods aiming at complete removal of thestirring device (this issue was briefly touched in Section 4.4) as well as spatially efficientantennas resulting in larger maximum EUT volumes. Reproducing the electromagneticeffect of a rotating stirrer by electronic means could lead to a significant reduction inoverall RC test time for EMC. Electronic stirring techniques together with the abilityto effectively control the quality factor would also help in reducing the lowest usablefrequency of today’s RCs, thus extending their range of applications.

A Electromagnetic Simulation Software FEKO

A.1 Special execution commands

Parallel solver execution mode

Using the command runfeko filename -np x --machines file machname calls thepreprocessor PreFEKO to generate the simulation model from the file filename (if itdoes not already exist) and launches ‘x’ parallel FEKO processes on the machines listedin the file machname. Load distribution and communication between these machines ismanaged by FEKO .

Model geometry validation

To quickly check the geometry for errors such as inhomogeneous discretization, dupli-cate, badly connected, or degenerated elements, and label cross-referencing, FEKO canbe invoked with the option runfeko...--feko-options --check-only. FEKO willcheck simply the geometry and try to allocate and de-allocate the required memory. Itwill not start to set up and solve the system matrix however.

Setting priorities for sequential and parallel solver processes

Prioritizing biased by the user can be achieved by starting sequential FEKO processeswith the standard UNIX command nice +10. However, parallel processes cannot be“niced” using this procedure, as nice does not propagate to the individual solver process.To resolve this problem, FEKO can be called with runfeko...--priority 1 whichsets the execution priority on all parallel processes to 10. Note that adjusting processpriorities influences load sharing only to a small degree.

Model geometry input is different for preprocessor and field solver

Some stirrer and EUT setups in the practical RC feature certain parts that are addedfor mechanical stability reasons only. An example is a styrofoam block used to serve asa “table” onto which an EUT is placed. Simulating a styrofoam table would require asignificant number of dielectric triangles or cubes, although it is a priori known that theEM effect of this table is negligible (since in practice exactly for this reason styrofoamhas been chosen).Nevertheless, the presence of the styrofoam table is needed for illustrative purposes inthe simulation geometry model. In other words, for PreFEKO the table is “visible”whereas for FEKO the table is “invisible”. Since the FEKO package does not includethis capability by itself, a small trick has to be applied: The desired behavior can beachieved by framing the code which describes the styrofoam parts with !!if... and!!endif. For PreFEKO the !!if... statement will be set to “true” and for FEKO itwill appear to be “false”. When displaying the simulated geometry, currents, and fieldsin WinFEKO the “Non-matching MD5 checksums” warning can be ignored.

163

164 APPENDIX A ELECTROMAGNETIC SIMULATION SOFTWARE FEKO

A.2 Memory considerations and bugs

32 bit versus 64 bit FEKO versionAt the start of this thesis, FEKO was only distributed in a 32 bit version for the Sun

operating system (OS). Currently, a full 64 bit version of FEKO is available whichunfortunately shows an error that could not be resolved by EMSS until the end of thisthesis. If run in 64 bit mode, FEKO crashes in a fully repeatable manner after the LUdecomposition of the system matrix with an “EOF on socket: 1” error message [197].For this reason – although running on a 64 bit Sun system – FEKO must be invokedin 32 bit mode using runfeko-4.1.32 due to an internal bug.Usually a 32 bit OS would allow FEKO to allocate up to 2 GByte of memory per processfor the solution of the system equations. Sun however implemented a special memoryallocation mechanism in Solaris 8 which allows a single process to allocate and manageup to 4 GByte of memory. This is referred to as “XMEM” (extended memory) support.

MLFMM versus MoMThe solutions for the near field obtained by using the MLFMM (introduced in Sec-tion 3.5.4) and MoM differ by a factor of approx. 4.5. The current distribution andconsequently the near field patterns are however completely identical for both solutionmethods, i.e. the MLFMM solution is a scaled version of the MoM solution and viceversa. This discrepancy could not be resolved with EMSS during the course of this the-sis [198]. Therefore all results presented in this thesis were calculated with the reliablefull wave MoM solver.

B Reverberation Chamber Measurement System

B.1 Antenna placement: tripod vs. suspension

This section shows the loading effect of a tripod made out of wood, metal, and plasticson the electric field within an RC without a stirrer (i.e. a cavity). Each measurementof the magnitude of the electric field |E| was performed once with the TX/excitationantenna mounted on the tripod and once without the tripod and the antenna suspendedfrom the RC’s ceiling with plastic ropes and Velcro. The setup is depicted in Fig. 5.10.Two types of measurements are shown

• |E| measured at a single, fixed position over a broad frequency range with a highfrequency resolution of 1 MHz (Fig. B.1 and Fig. B.2)

• |E| measured along a line with a spatial resolution of 0.1 m at certain, discretefrequencies (Fig. B.3. . . Fig. B.5)

Broadband effect at lower frequencies

50 100 150 200 250 300

Frequency [MHz]f

| |E [V/m]

60

50

40

30

20

10

0

70


Figure B.1: Magnitude of the electric field |E| measured at a fixed position (x = 0.77 m,y = 0.64 m, z = 0.47 m) for a biconical antenna mounted on tripod vs. suspended withplastic ropes and Velcro from the chamber ceiling (f = 50 . . . 300 MHz).

165

166 APPENDIX B REVERBERATION CHAMBER MEASUREMENT SYSTEM

From Fig. B.1. . . Fig. B.5 it is obvious that the electric field with the tripod installedin the chamber is not just a linearly scaled version of the field measured without thetripod. In addition to being scaled, the EM field is also strongly distorted, as apparentfrom Fig. B.3. . . Fig. B.5.

50 100 150 200 250 300

Frequency [MHz]f

| |E [V/m]

60

50

40

30

20

10

0

70


Figure B.2: Magnitude of the electric field |E| measured at a fixed position (x = 0.57 m,y = −0.36 m, z = 0.47 m) for a biconical antenna mounted on tripod vs. suspended withplastic ropes and Velcro from the chamber ceiling (f = 50 . . . 300 MHz).

APPENDIX B REVERBERATION CHAMBER MEASUREMENT SYSTEM 167

Spatial effect at lower frequencies

0

10

20

30

40

50

-1.2 -0.8 -0.4 0 0.4 0.8 1.2y [m]


50 MHz

100 MHz

Figure B.3: Magnitude of the electric field |E| measured along a line (x = 0.57 m, y =−1.2 . . . 1.2 m, z = 0.47 m) for a biconical antenna mounted on tripod vs. suspended withplastic ropes and Velcro from the chamber ceiling (50 MHz and 100 MHz).

0

10

20

30

40

50

-1.2 -0.8 -0.4 0 0.4 0.8 1.2y [m]


150 MHz

200 MHz


168 APPENDIX B REVERBERATION CHAMBER MEASUREMENT SYSTEM

0

10

20

30

40

50

-1.2 -0.8 -0.4 0 0.4 0.8 1.2y [m]


250 MHz

300 MHz


B.2 Data acquisition and interfacing

Simulation data can be transferred automatically from FEKO into a DB system via aMATLAB -based tool. Due to the lack of an import/export interface of the measure-ment software Compliance C3i (this problem has been fixed for a short time), mea-surement data had to be transferred manually into the DB system via MS Excel [154].To be able be to use the MS Access “Get external data...” procedure, the originalCompliance measurement data must be copied and pasted manually into Excel andthen preprocessed. The columns in the Excel worksheet must have exactly the samenames as the field names in Access; it is not enough that the Excel columns bear thesame name as the Access captions (which are essentially just to pretty-print the datatables). To further complicate matters, problems arise if list boxes together with under-lying lookup tables are used in the Access DB: it is not sufficient to have the values inExcel as displayed by Access – one has to back-reference the displayed list box valuesto the primary keys of the underlying lookup table. If e.g. the list box in the DB haspossible options “Measurement” and “Simulation” and the corresponding lookup tableprimary keys are “1” and “2”, in the Excel table the value “2” must be entered in orderto get “Measurement” in the DB. Entering “Measurement” will not import the valuefrom the Excel worksheet into the Access DB. If alternatively the “Paste append” com-mand from Access is used, back-referencing is not needed and the entry “Measurement”in the Excel sheet will be matched to the primary keys in Access automatically.Data transfers to and from the DB are accomplished using SQL expressions of MATLAB

via the ODBC application programming interface included with MS Windows .

C Reverberation Chamber Statistics

C.1 Field uncertainties

As outlined in Section 2.6, the amount of data needed to achieve a desired estimatoraccuracy can be determined by (2.92). k determines the desired confidence level (e.g.k ≈ ±1.96σ for p = 0.95, i.e. 95%) as given by (2.91). b is the number of dimensions of thefield data to be estimated (usually 1 or 3) and N is the required number of statisticallyindependent stirrer positions. If the field probe responds to only one dimension of thefield in this case b = 1. Solving for the required number of statistically independentstirrer positions N results in

N =k2

b

(10d/10 + 110d/10 − 1

)2

(C.1)

Equation (2.93) is plotted for different confidence levels in Fig. C.1. . . Fig. C.3. If, forexample, the uncertainty interval should be d = ±1 dB and the desired level of confidenceis 90% (corresponding to k ≈ ±1.65σ), then one would obtain N ≈ 69 or N ≈ 207 forb = 3 and b = 1 dimensions, respectively (see Fig. C.1 and Fig. C.3).

0 50 100 150 200 250 300 350 400-6

-4

-2

0

2

4

6


Uncert

ain

ty[d

B]

d~


Figure C.1: Number of statistically independent stirrer positions N required to achieve theuncertainty interval ±d for one EM field component at a confidence of p (see Fig. 2.7 forcorresponding standard deviation multiples).

169

170 APPENDIX C REVERBERATION CHAMBER STATISTICS

0 50 100 150 200 250 300 350 400-6

-4

-2

0

2

4

6


Uncert

ain

ty[d

B]

d~


Figure C.2: Number of statistically independent stirrer positions N required to achieve theuncertainty interval ±d for two EM field components at a confidence of p (see Fig. 2.7 forcorresponding standard deviation multiples).

0 50 100 150 200 250 300 350 400-6

-4

-2

0

2

4

6


Uncert

ain

ty[d

B]

d~


Figure C.3: Number of statistically independent stirrer positions N required to achieve theuncertainty interval ±d for three EM field components at a confidence of p (see Fig. 2.7 forcorresponding standard deviation multiples).

APPENDIX C REVERBERATION CHAMBER STATISTICS 171

C.2 Probability distribution functions

This section summarizes the PDFs and CDFs of probability distribution functions com-monly encountered in RC analysis.

Gaussian distributionThe PDF of a Gaussian distributed random variable X is given by

f(X |µ, σ) =1

σ√

2π· e− (X−µ)2

2σ2 (C.2)

with the mean µ and standard deviation σ. Its corresponding CDF is given by

F (X |µ, σ) =

X∫−∞

f(u) du =12

[1 + erf

(X − µ√

2σ

)](C.3)

with erf(·) denoting the error function as given by [12, 195].

Chi-square distributionA Chi-square (χ2) distributed random variable is related to a Gaussian-distributed ran-dom variable in the sense that the former can be viewed as a transformation of the latter.If X is a Gaussian random variable and Y = X2, then Y has a Chi-square distribution.Two types of Chi-square distributions are distinguished: the central and the non-centralChi-square distribution.The central Chi-square distribution is obtained if the underlying Gaussian distributionhas zero mean µ = 0. The PDF of the central Chi-square distribution is given by

f(Y |σ) =1√

2πY σ· e− Y

2σ2 Y ≥ 0 (C.4)

If, however, X is Gaussian distributed with non-zero mean µ = 0 and variance σ, thenthe PDF of the non-central Chi-square distribution resulting from Y = X2 is given by

f(Y |µ, σ) =1√

2πY σ· e−Y +µ2

2σ2 cosh

(√Y µ

σ2

)Y ≥ 0 (C.5)

The corresponding CDF for the central Chi-square distribution is given by

F (Y |σ) =

Y∫0

f(u) du =1√

2πY σ

Y∫0

1√u· e− u

2σ2 du (C.6)

The integral in (C.6) cannot be expressed in closed form.For the CDF of the non-central Chi-square distribution (C.5) see [195].

Rayleigh distributionThe Rayleigh distribution is closely related to the central Chi-square distribution. If X1

and X2 are two zero-mean i.i.d. Gaussian random variables, each having the common

172 APPENDIX C REVERBERATION CHAMBER STATISTICS

variance σ2, then Y = X21 + X2

2 is central Chi-square distributed with two degrees offreedom as given by (C.4). By introducing the new random variable

R =√

X21 + X2

2 =√

Y (C.7)

the Rayleigh PDF

f(R |σ) =R

σ2· e− R2

2σ2 R ≥ 0 (C.8)

is obtained. The corresponding CDF for the central Rayleigh distribution is given by

F (R |σ) =

R∫0

f(u) du = 1 − e−R2

2σ2 (C.9)

Rice distributionJust as the Rayleigh distribution is related to the central Chi-square distribution, theRice distribution is closely related to the non-central Chi-square distribution. If X1 andX2 are two non-zero-mean (with means µ1 and µ2) i.i.d. Gaussian random variables, eachhaving the common variance σ2, then Y = X2

1 +X22 is non-central Chi-square distributed

with two degrees of freedom as given by (C.5). By introducing the new random variable

R =√

X21 + X2

2 =√

Y (C.10)

the Rice PDF

f(R |µ1, µ2, σ) =R

σ2· e−

R2+µ21+µ2

22σ2 J0

(R√

µ1µ2

σ2

)R ≥ 0 (C.11)

as given by [195] is obtained with J0 being the 0-th order Bessel function of the firstkind. The corresponding CDF for (C.11) can be found in [195].

Gamma distributionThe Gamma (Γ) distribution is a generalization of the Chi-square distribution (C.4) toν degrees of freedom. The PDF of the Gamma distribution is given by

f(X |σ, ν) =1

Γ(

ν2

)σν2

ν2· X ν

2−1e−X

2σ2 X ≥ 0 (C.12)

The case ν = 2 yields the exponential distribution. For ν = 1 and by utilizing Γ(12 ) =

√π

the central χ2-distribution (C.4) is obtained. The CDF of the Gamma distribution canbe found in [195].

Nakagami-m and multivariate Gaussian distributionFor the PDF and, respectively, CDF of the multivariate Gaussian distribution and theNakagami-m distribution see [195].

D Reverberation Chamber Simulation Data

D.1 Spatial measurement positions

In Fig. 5.1, the two different coordinate systems relevant for simulation validations areindicated, where the right-handed (x, y, z) is used in the simulations and the left-handed(xm, ym, zm) for measurements. A coordinate transformation from one system to theother is given by (5.1)-(5.3) by letting xs = x, ys = y, and zs = z. Spatial coordinatesof the field points used for the measurement and simulation 8-point field uniformityanalysis complying with IEC 61000-4-21 [6] are listed in Table D.1.

Measurementpoints

Simulationpoints

Field pointnumber

xm

[m]ym[m]

zm

[m]xs

[m]ys[m]

zs

[m]

1 0.57 0.44 0.47 2.0 0.8 2.0

2 0.57 -0.56 0.47 2.0 1.8 2.0

3 0.97 0.44 0.47 2.4 0.8 2.0

4 0.97 -0.56 0.47 2.4 1.8 2.0

5 0.57 0.44 -0.53 2.0 0.8 1.0

6 0.57 -0.56 -0.53 2.0 1.8 1.0

7 0.97 0.44 -0.53 2.4 0.8 1.0

8 0.97 -0.56 -0.53 2.4 1.8 1.0

Table D.1: Field points used in measurement and simulation for uniformity analysis (defaultvalues). Simulation and measurement points are based on different coordinate systems.

173

174 APPENDIX D REVERBERATION CHAMBER SIMULATION DATA

D.2 Input power

a) b)

Figure D.1: Comparison between the electric field pattern inside the RC at f = 250 MHzexcited by a) a 1V and b) a 10 V source at the feeding element of the biconical antenna.Both representations are normalized to their respective maximum value.

D.3 Different coupling paths

a) b)

Figure D.2: Different antenna orientations within the RC to investigate several coupling sce-narios: a) towards the stirrer, b) towards the corner of the RC.

APPENDIX D REVERBERATION CHAMBER SIMULATION DATA 175

D.4 Field uniformity in prototype, cubic, and corrugated RC

This section shows the detailed per-component and combined field uniformity for theprototype RC (Fig. D.3), the corrugated RC (Fig. D.4), and the cubic RC (Fig. D.5).

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs

Standard deviation [dB]

Figure D.3: Statistical field uniformities σxyz and σξ in the prototype RC obtained accordingto the procedure outlined in Section 2.7 (- - - IEC limit line).

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs


Figure D.4: Statistical field uniformities σxyz and σξ in the corrugated RC obtained accordingto the procedure outlined in Section 2.7 (- - - IEC limit line).


0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs


Figure D.5: Statistical field uniformities σxyz and σξ in the cubic RC obtained according to theprocedure outlined in Section 2.7 (- - - IEC limit line).

D.5 Field uniformity for different stirrers

This section shows the detailed per-component and combined field uniformity for thedifferent RC stirrers introduced in Section 7.7 (see Fig. D.6. . . Fig. D.11).

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs


Figure D.6: Statistical field uniformities σxyz and σξ for the vertical 6-paddle stirrer, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).


0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs


Figure D.7: Statistical field uniformities σxyz and σξ for the vertical 6-paddle connected stirrer,obtained according to the procedure outlined in Section 2.7 (- - - IEC limit line).

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0


xyz

x

y

z

abs

Figure D.8: Statistical field uniformities σxyz and σξ for the horizontal 6-paddle stirrer, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).


0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs


Figure D.9: Statistical field uniformities σxyz and σξ for the vertical cross-plate stirrer, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs


Figure D.10: Statistical field uniformities σxyz and σξ for the vertical Z-fold stirrer, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).


0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0

IEC limit line

xyz

x

y

z

abs


Figure D.11: Statistical field uniformities σxyz and σξ for the vertical Z-fold stirrer with gaps,obtained according to the procedure outlined in Section 2.7 (- - - IEC limit line).

D.6 Field uniformity for different canonical EUTs

This section shows the detailed per-component and combined field uniformity with dif-ferent canonical EUTs placed inside the RC (see Fig. D.12. . . Fig. D.15).

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0


600

xyz

x

y

z

Figure D.12: Statistical field uniformities σxyz and σξ without an EUT, obtained according tothe procedure outlined in Section 2.7 (- - - IEC limit line).


0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0


600

xyz

x

y

z

Figure D.13: Statistical field uniformities σxyz and σξ with the canonical loop EUT, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).

0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0


600

xyz

x

y

z

Figure D.14: Statistical field uniformities σxyz and σξ with the canonical box EUT, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).


0

Frequency [MHz]f

100 300 400 500200

1

2

3

4

5

6

0


600

xyz

x

y

z

Figure D.15: Statistical field uniformities σxyz and σξ with a large canonical box EUT, obtainedaccording to the procedure outlined in Section 2.7 (- - - IEC limit line).

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Acknowledgments

Work for this thesis was done at the Laboratory for Electromagnetic Fields and Mi-crowave Electronics (IFH), ETH Zurich, Switzerland and Schaffner EMV AG, Luter-bach, Switzerland.First of all I would like to thank Prof. Rudiger Vahldieck of IFH, ETH Zurich who gaveme the opportunity to carry out my Ph.D. thesis in his research group. He offered mea highly interesting topic, bridging the gap between academia and industry, and gaveme the freedom to define independently by myself the path leading through this thesis.Despite his tight office schedule, he was accessible for me to discuss both topics relatedto my thesis as well as issues as diverse as teaching, publishing and reviewing of papers,or the “dos and donts” of talks. In addition, I always had the resources, lab environment,and support available to do my research projects effectively and efficiently, be it people,computers, software, tools, prototypes, or access to literature. I also had the uniquepossibility to attend virtually any conference related to my thesis topic and of interestto me, present my research results, get to know other key people in the field of EMC,and publish both conference as well as journal papers.Furthermore, I owe sincere thanks to my co-examiner Prof. Flavio Canavero of thePolitecnico di Torino, Torino, Italy, for the thorough review of this thesis, his very con-structive comments, and his personal commitment.I want to thank my supervisor Dr. Pascal Leuchtmann for his guidance during my Ph.D.thesis. His experience and knowledge in the field of electromagnetics was very helpfulduring this project and contributed to the results presented in this thesis.Several big “Merci viiielmool” are due to the IFH measurement guru Hansruedi Benedick-ter, Aldo Rossi of the IFH electronics workshop, Ray Ballisti for help in computer mattersas well as Stephen Wheeler and Claudio Maccio of the IFH mechanics workshop.I would like to express my gratitude to Heinrich Kunz, Dr. Jan Sroka, John Dear-ing, Dan Hamblin, Richard Davy, David Riley, Uwe Karsten, and Michael Rehfeldt ofSchaffner EMV AG, who co-initiated this project, helped making it progress throughseveral stages, spent considerable time on the prototype measurements and software de-velopment, discussed new ideas, and allowed me to participate in the day-to-day businessof the company.Dr. Ulrich Jakobus of EMSS GmbH, Boblingen, Germany, assisted me a lot with theelectromagnetic simulation software package FEKO and its peculiarities.I enjoyed the fruitful discussions regarding reverberation chamber theory, simulations,and measurements with Michael Hatfield of NSWCDD, Dahlgren (VA), USA, John Lad-bury of NIST, Boulder (CO), USA, Dr. Luk Arnaut and Martin Alexander, both of NPL,Teddington, UK, Dr. Wolfgang Kurner of EADS AG, Hamburg, Germany, Dr. HansGeorg Krauthauser of the Universitat Magdeburg, Magdeburg, Germany, Dr. Nils Euligof the Universitat Braunschweig, Braunschweig, Germany, Albin Maridet and Frederic

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200 ACKNOWLEDGMENTS

Hoeppe, both of EADS CCR S.A., Suresnes, France, and Gerard Orjubin of the Univer-site Marne-La-Vallee, Marne-La-Vallee, France.During the course of this thesis and at several conferences I had the pleasure to share theresults of my research with Dr. Mats Backstrom, Magnus Hoijer, and Olof Lunden, allof the Swedish Defence Research Agency (FOI), Linkoping, Sweden, Magnus Otterskogfrom the Orebro University, Orebro, Sweden, Tim Harrington of the FCC, Laurel (MD),USA, Gus Freyer, Monument (CO), USA, Michael Windler of Underwriters LaboratoriesInc., Northbrook (IL), USA, Peter Landgren of Saab Bofors Dynamics AB, Kent Madsenof Flextronics International AB, Linkoping, Sweden as well as Nico van Dijk of PhilipsResearch B.V., Eindhoven, The Netherlands.Special thanks to all members of the IFH laboratory who made work, lunch and coffeebreaks more pleasant, with whom it was fun to accomplish tasks as diverse as teaching,conference and Christmas party organization, building Ph.D. candidates’ hats, going tothe gym to work out, skiing, hiking, biking or discussing political, economic, cultural orscientific topics – and who became true friends instead of just colleagues and thus mademy time at ETH Zurich a great experience.Apart from the scientific results presented in this thesis, what did I learn personally frompursuing a Ph.D.? “If you can’t do it better – why bother doing it at all?” is an excellentguideline to select what you do for your thesis and how you do it. If it is clear from thebeginning that you cannot solve the key problem of your thesis in a better way anyhow,then there is no point in going further, as your contribution to the scientific world willbe simply a repetition of someone else’s work. On top of this, you will waste yours andother people’s time. Establish the starting point of your thesis by performing a thoroughsearch on what has been done up to now by others around the world – not just in theoffices next door. This approach will help you a lot when it comes to publishing yourresults later on. Usage of readily available proven tools, perfectly suited to solve a certainproblem of your thesis, will keep you from reinventing the wheel, thus greatly accelerateyour progress. Identify your strongest scientific competitors – again worldwide, not justlocally – and learn from their achievements instead of ignoring them. Benchmark yourwork against theirs and publish in a language which is universally understood. Choosenot to do simply another “Me-too” Ph.D. thesis. Cheers!

Copyright c© 2003 Scott Adams Inc., distributed by United Feature Syndicate, Inc.

List of Publications

Publications related to this thesis

Journal papers

P1 C. Bruns and R. Vahldieck, “A closer look at reverberation chambers – 3-Dsimulation and experimental verification,” accepted for publication in IEEETrans. Electromagn. Compat., Aug. 2005.

Conference papers

P2 C. Bruns, P. Leuchtmann, and R. Vahldieck, “Introduction to reverbera-tion chamber simulation,” in Proc. 2nd NPL FREEMET meeting at MIRA,Nuneaton. Teddington, UK: National Physical Laboratory (NPL), 2002, [Elec-tronic].

P3 ——, “Three-dimensional method of moments simulation of a reverberationchamber in the frequency domain,” in Proc. 15th Int. Zurich Symp. and Tech-nical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: SwissFederal Inst. Technol. Zurich, 2003, pp. 229–232.

P4 ——, “Challenges and results of realistic reverberation chamber simulations andmeasurements,” in Proc. 2003 Reverberation Chamber, Anechoic Chamber andOATS Users Meeting, Austin, TX, 2003.

P5 ——, “Broadband method of moment simulation and measurement of a mediumsized reverberation chamber,” in Proc. IEEE Int. Symp. on ElectromagneticCompatibility. Piscataway, NJ: IEEE, 2003, pp. 844–849.

P6 P. Leuchtmann, C. Bruns, and R. Vahldieck, “On the validation of simulatedfields in a reverberation chamber,” in Proc. European Microwave Conference2003. London, UK: Horizon House Publ. Ltd., 2003, [Electronic].

P7 C. Bruns, P. Leuchtmann, and R. Vahldieck, “Comparison of various reverber-ation chamber geometries and excitations using a frequency domain methodof moments simulation,” in Proc. 17th Int. Wroclaw Symp. and Exhibition onElectromagnetic Compatibility. Wroclaw, Poland: Politechniki Wroclawskiej,2004, pp. 97–102.

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202 LIST OF PUBLICATIONS

P8 C. Bruns, P. Leuchtmann, and R. Vahldieck, “Simulation and comparison ofdifferent stirrer types inside a reverberation chamber,” in Proc. IEEE Int. Symp.on Electromagnetic Compatibility. Piscataway, NJ: IEEE, 2004, pp. 241–244.

P9 ——, “Modeling and simulation of a canonical equipment under test inside amedium-sized reverberation chamber,” in Proc. Int. Symp. on ElectromagneticCompatibility. Eindhoven, The Netherlands: Technische Universiteit Eind-hoven, 2004, pp. 744–749.

P10 ——, “Cubic and corrugated reverberation chambers: mode distribution, cor-relation, and field uniformity,” in Proc. 16th Int. Zurich Symp. and TechnicalExhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Fed-eral Inst. Technol. Zurich, 2005, pp. 539–542.

P11 R. Vahldieck and C. Bruns, “Statistical characterization of reverberation cham-bers,” accepted for publication in Proc. 9th Int. Conference on Electromagneticsin Advanced Applications. Torino, Italy: Politecnico di Torino, 2005.

Publications related to previous work

Journal papers

P12 C. Bruns, P. Leuchtmann, and R. Vahldieck, “Comprehensive analysis and simu-lation of a 1-18 GHz broadband parabolic reflector horn antenna system,” IEEETrans. Antennas Propagat., vol. 51, no. 6, pp. 1418–1422, June 2003.

P13 ——, “Analysis and simulation of a 1–18-GHz broadband double-ridged hornantenna,” IEEE Trans. Electromagn. Compat., vol. 45, no. 1, pp. 55–60, Feb.2003.

Conference papers

P14 C. Bruns, P. Leuchtmann, and R. Vahldieck, “Full wave analysis and experi-mental verification of a broadband ridged horn antenna system with parabolicreflector,” in Proc. IEEE Antennas and Propagat. Society Int. Symp., vol. 4.Piscataway, NJ: IEEE, 2001, pp. 230–233.

P15 ——, “Full field calculation of a 1–18 GHz broadband ridged horn antenna,” inProc. URSI Int. Symp. on Electromagn. Theory. Ghent, Belgium: Int. Unionof Radio Science (URSI), 2001, pp. 621–623.

Curriculum Vitae

Personal data

Name: Christian Bruns

Nationality: German

Date of birth: December 19, 1973

E-mail: [email protected]

Professional experience

11/04 – 03/05: Huber+Suhner AG, Herisau, SwitzerlandStrategic evaluation of IEEE 802.16/WiMAX standard (diploma thesis)

11/00 – 05/05: ETH Zurich, Zurich, SwitzerlandLaboratory for Electromagnetic Fields and Microwave ElectronicsDoctorate in Electrical EngineeringProject with Schaffner EMV AG, Luterbach, Switzerland

05/97 – 10/97: Robert Bosch GmbH, Stuttgart, GermanyGasoline engine management systems group (trainee)

01/96 – 08/96: Energy and High Voltage Systems Institute, Karlsruhe, GermanyElectromagnetic compatibility group (consultant, research assistant)

01/93 – 12/95: DATEC Elektroanlagen GmbH, Karlsruhe, GermanyPlanning and setup of IT networks (trainee, freelance)

01/92 – 06/96: Bruns + Gerst GbR, Karlsruhe, GermanyProfessional light and audio systems (founder and co-owner)

04/90 – 12/93: Radio Badenia GmbH, Karlsruhe, GermanyInterviews, data mining, reports, pre-/post-production (freelance)

Education10/01 – 02/05: ETH Zurich, Zurich, Switzerland

Post-graduate studies in business, economics, and finance

08/99 – 04/00: ETH Zurich, Zurich, SwitzerlandAnalysis and simulation of a broadband horn antenna (diploma thesis)

08/96 – 05/97: Massachusetts Institute of Technology, Cambridge (MA), USAUniversity of Massachusetts, Dartmouth (MA), USAGraduate study of Electrical Engineering (scholarship)

10/93 – 04/00: Universitat Karlsruhe (TH), Karlsruhe, GermanyDipl.-Ing. Electrical Engineering

09/84 – 05/93: High School Max-Planck-Gymnasium, Karlsruhe, Germany

203

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