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., Universität Fridericiana Karlsruhe (TH), Germany
born December 19, 1973
citizen of Germany
accepted on the recommendation of
Prof. Dr. R. Vahldieck, examiner
Prof. Dr. F. Canavero, co-examiner
2005
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., Universität Fridericiana Karlsruhe (TH), Germany
born December 19, 1973
citizen of Germany
accepted on the recommendation of
Prof. Dr. R. Vahldieck, examiner
Prof. Dr. F. Canavero, co-examiner
2005
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 operation
of other products and inversely withstand electromagnetic radiation emitted from surrounding devices. A crucial aspect of successful product development is therefore the
effective and efficient testing of the electromagnetic compatibility (EMC). Reverberation
chambers (RCs) enjoy growing popularity as a complement or replacement to the well
established methods for radiated interference. RC testing exhibits several competitive
advantages over existing methods, such as an enhanced test repeatability and a more
realistic as well as rigorous test environment. The importance of testing for EMC in RCs
as an alternative measurement technique has recently been recognized in the IEC 610004-21 standard.
The three-dimensional simulation of an RC is presented in this thesis. In the beginning, fundamental concepts and key parameters of an RC are introduced, among them
the mode distribution, mode density, modal gaps, and the quality factor. Furthermore,
the RC is described as a statistical electromagnetic test environment and characterized
by distribution functions, correlation, uncertainty, and field uniformity. It is pointed
out that it is crucial to select a suitable numerical method to perform meaningful RC
simulations. A chosen numerical technique must be able to deliver results over a wide
frequency range without using excessive computational resources; the method must be
able to handle large, irregular structures, and a varying geometry without introducing
errors. Furthermore, there must be a possibility to account for finite metal conductivity as well as highly resonant structures. The computation of near fields at an arbitrary
number of chamber locations should be possible without adding too much computational
overhead. A frequency-domain electric field integral equation-based method-of-moments
technique is chosen for the RC simulations.
A prototype RC with a measurement system is built and used later on for simulation
validations. Measurement errors originating from field probes, antennas, and stirrers are
assessed for their impact on deviations between simulated and measured results. The
biggest deviations are found to result from the antenna tripods and position inaccuracies
of 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 simulation. Suitable electrical conductivity values are derived for material as it is used in
a shielded room construction. In addition to the prototype RC, cubic and corrugated
chambers, an offset-wall RC as well as several vertical and horizontal stirrers are modeled.
Simulation results of a detailed asymmetric RC model are benchmarked against measurements and exhibit a good agreement in the lower-to-medium frequency range. It is
shown that a proper validation of the simulation must be performed with direct comparisons 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 small
geometric details and the agreement between actual prototype and simulated RC dimensions is discussed. It is shown that the type, position, and alignment of the excitation
source in the simulation model change the field pattern significantly. In addition, the
effect of various stirrers on the fields, correlation, and uniformity inside the chamber are
visualized. The 6-paddle stirrer developed for this thesis and the commercially available
Z-fold stirrer have the best performance.
A comparison between the standard rectangular RC with a cubic and a corrugated chamber revealed that the two latter chamber geometries do not offer significant advantages
concerning correlation and field uniformity. On the other hand, the cubic RC does not
perform as bad as always alleged. The presence of a stirring device shifts the modes
in frequency depending on their respective field distribution away from the analytically
calculated resonance frequencies. Therefore the usually observed problem of degenerate
modes is found not come into play within a cubic RC – contrary to the widely accepted
RC design guidelines, a cubic RC does not exhibit worse performance than other rectangular RCs. Three special multipath/direct path coupling scenarios are simulated
(Gaussian, Rice, and Rayleigh statistical distributions). This investigation reveals that
the usage of a Hertzian dipole in an RC simulation leads to undesirable strong direct
coupling between an equipment under test and the excitation. Through the usage of
realistic antennas with higher directivity, this unwanted direct coupling can be considerably reduced.
In this thesis it is shown that for frequencies much smaller than the lowest usable frequency, the simulation of an RC is possible, the chamber however becomes electrically
too small compared to the operational wavelength, which prevents sufficient statistical
field uniformity (an optimization is therefore not possible). Conversely, at frequencies
much 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 optimization needed). With increasing frequency, the field within an RC becomes more and more
sensitive to even small geometric details, which makes proper modeling numerically not
feasible at high frequencies. The possibilities for RC design optimizations significantly
below or above the lowest usable frequency are therefore limited. At frequencies around
the lowest usable frequency, however, stirrer shapes or wall geometries can be optimized
using the results presented in this thesis in order to improve field uniformity and to
extend the operating frequency for a given RC to lower frequencies.
Zusammenfassung
Elektronische Produkte müssen heute so entwickelt werden, dass sie einerseits andere
Produkte nicht in ihrer Funktion beeinträchtigen und gleichzeitig selbst weitgehend immun gegen elektromagnetische Einstrahlung anderer Geräte sind. Daher besteht ein
wichtiger Aspekt der erfolgreichen Produktentwicklung darin, neue Geräte in effizienter
und effektiver Weise auf ihre elektromagnetische Verträglichkeit (EMV) hin zu untersuchen. Reverberation Chambers (RCs) erfreuen sich seit einiger Zeit steigender Beliebtheit und stellen eine Ergänzung bzw. einen Ersatz bestehender EMV-Testmethoden
dar. Tests in RCs weisen gegenüber bestehenden Verfahren verschiedene Vorteile auf, sie
ermöglichen z.B. eine bessere Wiederholgenauigkeit sowie ein realistischeres und strengeres Prüfverfahren. Der zunehmenden Bedeutung von RCs wurde durch den Entwurf
und die kürzliche Veröffentlichung des IEC 61000-4-21 Standards Rechnung getragen.
Diese Arbeit behandelt die dreidimensionale Simulation einer RC. Zunächst werden die
grundlegenden Aspekte sowie die wichtigsten Parameter einer RC behandelt. Dazu gehören die Verteilung der Moden, die Modendichte, die Modenlücken und der Gütefaktor.
RCs werden im weiteren Verlauf als statistische Testumgebung beschrieben und charakterisiert durch Verteilungsfunktionen, Korrelation, statistische Unsicherheit und Felduniformität. Es wird aufgezeigt, dass die Wahl einer geeigneten numerischen Methode entscheidend ist, um sinnvolle RC-Simulationen durchzuführen. Die jeweilige numerische Methode muss einerseits Simulationen über einen weiten Frequenzbereich erlauben, ohne jedoch exorbitante Rechenleistung zu benötigen; andererseits muss die
Methode in der Lage sein, grosse unregelmässige Strukturen zu berechnen, bei denen sich
Teile der Geometrie bewegen. Ausserdem muss es ohne grossen Mehraufwand möglich
sein, die endliche Leitfähigkeit des Materials zu berücksichtigen. Die eingesetzte Simulationstechnik sollte das Feld an sehr vielen räumlich verteilten Punkten bestimmen
können, ohne den numerischen Aufwand signifikant zu vergrössern. In dieser Arbeit
wird für die RC-Simulationen die auf der elektrischen Feldintegralgleichung basierende
Momentenmethode im Frequenzbereich verwendet.
Ein RC-Prototyp wird konstruiert und zusammen mit einem Messsystem für die Überprüfung der Simulationsergebnisse eingesetzt. Durch Feldsonden, Antennen und Rührer
entstehende Messfehler werden hinsichtlich ihres Einflusses auf die Übereinstimmung von
Mess- und Simulationsergebnissen beurteilt. Die grössten Abweichungen sind auf die
Antennenstative und die Positionierungenauigkeit der Feldsonden sowie der Antennen
zurückzuführen. Der aus Wänden, Tür, Rührern, mehreren Antennen und einem Testobjekt bestehende RC-Prototyp wird für die elektromagnetische Simulation modelliert.
Brauchbare Leitfähigkeitswerte für die verwendeten Materialien werden durch Messungen ermittelt, so dass die praktischen Verhältnisse in einer Schirmkabine reproduziert
werden können. Neben dem RC-Prototyp werden kubische und gerippte Kammern, eine
RC mit einer versetzten Wand sowie verschiedene vertikal und horizontal angeordnete
Rührer modelliert.
iii
iv
ZUSAMMENFASSUNG
Die Simulationsergebnisse eines detailgetreuen, asymmetrischen RC-Modells werden verglichen mit Messresultaten. Dabei ergibt sich eine gute Übereinstimmung im unteren
sowie mittleren Frequenzbereich. Es wird dargelegt, dass eine sinnvolle Validierung
der Simulationsergebnisse nur über direkte Vergleiche mit Messergebnissen durchgeführt
werden kann. Sowohl die Simulations- wie auch die Messergebnisse sollten dabei weder
weiterverarbeitet werden zu abgeleiteten Grössen noch mittels statistischer Kennzahlen
beschrieben werden. Die modellierten RCs mit den darin befindlichen Sende- und Empfangsantennen, verschiedenen Rührern und Testobjekten werden mit Hilfe der Simulation
untersucht. Der Einfluss unscheinbarer Details sowie von geringen geometrischen Abweichungen zwischen Prototyp und Modell einer RC wird behandelt. Es wird gezeigt,
dass die Art, Position und Ausrichtung der Anregung im Simulationsmodell die Feldverteilung in der RC erheblich beeinflussen. Weiterhin wird der Effekt verschiedener
Rührertypen auf das elektromagnetische Feld, die Korrelation und die Gleichmässigkeit
der Feldverteilung in der Kammer untersucht. Die beste Leistung konnte mit dem
in dieser Arbeit entwickelten 6-Flügel-Rührer sowie dem kommerziell erhältlichen ZFaltung-Rührer erzielt werden.
Der Vergleich einer gewöhnlichen, rechteckigen RC mit einer kubischen bzw. gerippten
RC zeigt, dass die beiden letztgenannten Varianten keinerlei Vor- oder Nachteile im Hinblick auf Korrelation oder eine gleichmässige Feldverteilung aufweisen. Insbesondere ist
die kubische Kammer nicht so schlecht wie oft behauptet: Durch den Rührer werden die
einzelnen Moden (je nach zugehöriger Feldverteilung) weg von der analytisch berechneten
Resonanzfrequenz verschoben. Dadurch tritt das Problem der Modendegeneration innerhalb einer kubischen RC praktisch nicht auf. Im Gegensatz zur weit verbreiteten Meinung (sowie Konstruktionsempfehlungen) weist eine kubische RC somit keine signifikant
schlechtere Felduniformität im Vergleich zu anderen rechteckigen Kammern auf. Weiterhin werden drei Spezialfälle 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 unerwünschte,
starke direkte Kopplung zwischen Testobjekt und Anregung ergibt. Durch praxisnahe
Antennen mit höherer Richtwirkung lässt sich diese unerwünschte direkte Kopplung erheblich reduzieren.
In dieser Arbeit wird gezeigt, dass die Simulation einer RC bei Frequenzen kleiner als
der niedrigsten Betriebsfrequenz zwar möglich ist, die Kammer allerdings für eine ausreichend gleichmässige Feldverteilung (und damit auch Optimierung) elektrisch zu klein
wird. Im Gegensatz dazu funktioniert unabhängig vom Design jede RC bei hohen Frequenzen (d.h. genügend grosse Anzahl Moden ausbreitungsfähig) ausreichend gut, womit
sich die Optimierung erübrigt. Für eine korrekte Simulation gestaltet sich die geeignete
Modellierung bei hohen Frequenzen sehr schwierig, da das Feld in einer RC prinzipbedingt stark auf kleinste Geometrieänderungen reagiert. Die Möglichkeiten für die
RC-Optimierung sind damit sowohl im unteren wie auch im oberen Frequenzbereich
eingeschränkt. Bei mittleren Frequenzen (um den Bereich der niedrigsten Betriebsfrequenz herum) lässt sich eine RC aufbauend auf den in dieser Arbeit vorgestellten Ergebnissen optimieren. Durch Verwendung z.B. neuartiger Rührerformen oder Wandgeometrien in der Simulation kann die Gleichmässigkeit der Feldverteilung verbessert werden
und damit der Einsatzbereich einer RC zu niedrigeren Frequenzen erweitert werden.
Contents
1 Introduction
1.1 Motivation and objective of this thesis . . . . . . . . . . . . . . . . . . . .
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
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3
2 Reverberation Chamber Theory
2.1 Electromagnetic fields in a reverberation chamber . . . . . . . . .
2.1.1 Modes inside an ideal cavity . . . . . . . . . . . . . . . . .
2.1.2 Modes inside a lossy cavity . . . . . . . . . . . . . . . . .
2.1.3 Field distribution inside a reverberation chamber . . . . .
2.2 Lowest usable frequency . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Number of cavity modes . . . . . . . . . . . . . . . . . . .
2.2.2 Quality factor . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Stirring ratio . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Field anisotropy and inhomogeneity . . . . . . . . . . . . . . . .
2.3.1 Field anisotropy coefficients . . . . . . . . . . . . . . . . .
2.3.2 Field inhomogeneity coefficients . . . . . . . . . . . . . . .
2.4 Field statistics and probability density functions . . . . . . . . .
2.4.1 Quadrature and in-phase part statistics . . . . . . . . . .
2.4.2 Magnitude statistics for single components and total field
2.4.3 Power statistics for single components and total field . . .
2.4.4 Statistical goodness-of-fit χ2 -test . . . . . . . . . . . . . .
2.5 Correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Definition of correlation . . . . . . . . . . . . . . . . . . .
2.5.2 Significance of correlation . . . . . . . . . . . . . . . . . .
2.6 Statistical uncertainty and estimator accuracy . . . . . . . . . . .
2.7 Field uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Caveats for statistics . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Numerical Procedure
3.1 Initial considerations of reverberation chamber simulations .
3.1.1 Wide operational frequency range . . . . . . . . . .
3.1.2 Large, varying, and irregular geometry . . . . . . . .
3.1.3 Finite conductivity and entirely closed structure . .
3.1.4 Highly resonant chamber . . . . . . . . . . . . . . .
3.1.5 Large number of spatial near field positions . . . . .
3.2 Computation of electromagnetic fields . . . . . . . . . . . .
3.2.1 Incident and scattered field . . . . . . . . . . . . . .
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v
CONTENTS
vi
3.3
3.4
3.5
3.6
3.2.2 Integral equation approach . . . . . . . . . . . . . . . . . . .
3.2.3 Solution of integral equations . . . . . . . . . . . . . . . . . .
3.2.4 Approximation of currents and current density . . . . . . . .
3.2.5 Computation of line and surface current coefficients . . . . .
Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Point matching and weighting functions . . . . . . . . . . . .
3.3.2 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Symmetry considerations . . . . . . . . . . . . . . . . . . . .
3.3.4 Modeling of finite conductivity . . . . . . . . . . . . . . . . .
Computational requirements . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Simulation memory . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Simulation time . . . . . . . . . . . . . . . . . . . . . . . . . .
Extensions to the method of moments . . . . . . . . . . . . . . . . .
3.5.1 Field integral equation resonance problem . . . . . . . . . . .
3.5.2 Iterative solution techniques . . . . . . . . . . . . . . . . . . .
3.5.3 Method of moments and physical optics hybridization . . . .
3.5.4 Method of moments and fast multipole method hybridization
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Literature Overview
4.1 Historic reverberation chamber publications and patents . .
4.2 Reverberation chamber standards . . . . . . . . . . . . . . .
4.3 Previous reverberation chamber simulations . . . . . . . . .
4.3.1 Time-domain simulations . . . . . . . . . . . . . . .
4.3.2 Frequency-domain simulations . . . . . . . . . . . .
4.3.3 Statistical models . . . . . . . . . . . . . . . . . . . .
4.4 Alternative stirring methods . . . . . . . . . . . . . . . . . .
4.4.1 Moving walls . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Electronic stirring . . . . . . . . . . . . . . . . . . .
4.5 Practical reverberation chamber applications . . . . . . . .
4.5.1 Automotive and aircraft avionics . . . . . . . . . . .
4.5.2 Antenna measurements and mobile communications
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Prototype and Measurement System Development
5.1 Reverberation chamber prototype . . . . . . . . . . . . . .
5.1.1 Walls and door . . . . . . . . . . . . . . . . . . . .
5.1.2 Stirrer . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Auxiliary installations and electromagnetic leakage
5.2 Measurement system . . . . . . . . . . . . . . . . . . . . .
5.2.1 Transmit and receive measurement equipment . . .
5.2.2 Field probe system . . . . . . . . . . . . . . . . . .
5.2.3 Data acquisition and interfacing . . . . . . . . . .
5.3 Measurement errors . . . . . . . . . . . . . . . . . . . . .
5.3.1 Field probe system . . . . . . . . . . . . . . . . . .
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CONTENTS
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6 Modeling of the Reverberation Chamber
6.1 Chamber models . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Modeling procedure . . . . . . . . . . . . . . . . . . . . .
6.1.2 Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.3 Prototype reverberation chamber . . . . . . . . . . . . . .
6.1.4 Corrugated, cubic, and offset-wall reverberation chambers
6.1.5 Other reverberation chambers . . . . . . . . . . . . . . . .
6.2 Stirrer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Vertical stirrers . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Horizontal stirrers . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Stirrers used in other reverberation chambers . . . . . . .
6.3 Wall and stirrer conductivities . . . . . . . . . . . . . . . . . . .
6.4 Transmit and receive antenna models . . . . . . . . . . . . . . . .
6.4.1 Ideal Hertzian and realistic λ/2-dipole . . . . . . . . . . .
6.4.2 Biconical antenna . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Logarithmic-periodic antenna . . . . . . . . . . . . . . . .
6.4.4 Horn antenna . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Canonical equipment under test . . . . . . . . . . . . . . . . . . .
6.5.1 Practical canonical EUTs . . . . . . . . . . . . . . . . . .
6.5.2 Canonical EUT modeling . . . . . . . . . . . . . . . . . .
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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93
93
93
94
95
98
100
100
101
103
104
105
106
107
108
109
110
110
110
112
114
7 Reverberation Chamber Simulation and Measurement
7.1 Simulation and measurement workflow . . . . . . .
7.2 Cavity simulation . . . . . . . . . . . . . . . . . . .
7.2.1 Effect of the chamber door . . . . . . . . .
7.2.2 Insertion of a stirrer . . . . . . . . . . . . .
7.3 Prototype reverberation chamber analysis . . . . .
7.3.1 Different chamber geometries . . . . . . . .
7.3.2 Effect of a rotating stirrer . . . . . . . . . .
7.3.3 Different reverberation chamber excitations
7.4 Measurement versus simulation . . . . . . . . . . .
7.4.1 Measurement setup . . . . . . . . . . . . . .
7.4.2 Near field based simulation validation . . .
7.4.3 Statistical benchmarks . . . . . . . . . . . .
7.5 Corrugated and cubic reverberation chamber . . .
7.5.1 Simulated near field distribution . . . . . .
7.5.2 Correlation analysis . . . . . . . . . . . . .
7.5.3 Field uniformity . . . . . . . . . . . . . . .
7.6 Equipment under test simulation . . . . . . . . . .
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115
115
117
118
118
120
121
123
123
127
127
128
132
132
134
136
136
138
5.4
5.5
5.3.2 Antennas . . . . . . . . .
5.3.3 Chamber and stirrer . . .
Measurement uncertainty budget
Conclusion . . . . . . . . . . . .
vii
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CONTENTS
viii
7.7
7.8
7.9
7.6.1 Simulated near field distribution . . .
7.6.2 Field uniformity . . . . . . . . . . . .
7.6.3 TX/RX antenna coupling . . . . . . .
Comparison of different stirrers . . . . . . . .
7.7.1 Simulated near field distribution . . .
7.7.2 Correlation analysis . . . . . . . . . .
7.7.3 Field uniformity . . . . . . . . . . . .
7.7.4 Final performance evaluation . . . . .
7.7.5 Plane-wave-based stirrer comparisons
Simulation and measurement time budget . .
Conclusion . . . . . . . . . . . . . . . . . . .
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138
140
141
143
143
145
148
149
149
152
154
8 Conclusion
157
9 Outlook
161
A Electromagnetic Simulation Software FEKO
163
A.1 Special execution commands . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.2 Memory considerations and bugs . . . . . . . . . . . . . . . . . . . . . . . 164
B Reverberation Chamber Measurement System
165
B.1 Antenna placement: tripod vs. suspension . . . . . . . . . . . . . . . . . . 165
B.2 Data acquisition and interfacing . . . . . . . . . . . . . . . . . . . . . . . . 168
C Reverberation Chamber Statistics
169
C.1 Field uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
C.2 Probability distribution functions . . . . . . . . . . . . . . . . . . . . . . . 171
D Reverberation Chamber Simulation Data
D.1 Spatial measurement positions . . . . . . . . . . . . . .
D.2 Input power . . . . . . . . . . . . . . . . . . . . . . . . .
D.3 Different coupling paths . . . . . . . . . . . . . . . . . .
D.4 Field uniformity in prototype, cubic, and corrugated RC
D.5 Field uniformity for different stirrers . . . . . . . . . . .
D.6 Field uniformity for different canonical EUTs . . . . . .
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173
173
174
174
175
176
179
Bibliography
183
Acknowledgments
199
List of Publications
201
Curriculum Vitae
203
List of Figures
1.1
Typical reverberation chamber . . . . . . . . . . . . . . . . . . . . . . . .
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Schematic reverberation chamber test setup . . . . . . . . . . . . . . . . .
Number of modes N above cutoff in standard and cubic cavity . . . . . .
Mode density ∂N/∂f in standard and cubic cavity . . . . . . . . . . . . .
Modal gap in standard and cubic cavity . . . . . . . . . . . . . . . . . . .
Number of modes above cutoff per 10 MHz in standard and cubic cavity .
Exemplary Gaussian normal, χ(2) , χ(6) , χ2(6) statistical distribution . . . .
Standard deviation multiples of a Gaussian normal distribution . . . . . .
Independent stirrer positions and field uncertainty (1 and 3 components) .
Independent stirrer positions and field uncertainty (2 components) . . . .
Required EUT spacing from RC walls . . . . . . . . . . . . . . . . . . . .
10
12
13
14
15
22
28
29
30
32
3.1
3.2
Memory requirements for an RC simulation (50 . . . 300 MHz) . . . . . . . . 49
Memory requirements for an RC simulation (50 . . . 1000 MHz) . . . . . . . 50
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
RC geometry and dimensions . . . . . . . . . . . . . . . . . . . . . . . . .
Vertical 6-paddle stirrer mounting and motor drive . . . . . . . . . . . . .
RC apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overview of the RC measurement system setup . . . . . . . . . . . . . . .
RC measurement setup and photo . . . . . . . . . . . . . . . . . . . . . .
Field probe with stand, TX/RX antenna, and measurement grid . . . . .
Schematic measurement and data acquisition procedure . . . . . . . . . .
Isotropy of the electric field probe . . . . . . . . . . . . . . . . . . . . . .
Schematic RC measurement grid . . . . . . . . . . . . . . . . . . . . . . .
Tripod-mounted and suspended TX antennas . . . . . . . . . . . . . . . .
Broadband effect of tripod on |E| (f = 50 . . . 300 MHz) . . . . . . . . . . .
Spatial effect of tripod on |E| (x = 0.57 m, y = −1.2 . . . 1.2 m, z = 0.47 m)
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
Schematic modeling and simulation preprocessing flowchart . .
Partly symmetric simulation model of the RC . . . . . . . . . .
Detailed fully asymmetric simulation model of the RC . . . . .
Photo of the RC door with gasket . . . . . . . . . . . . . . . .
Basic, wider, corrugated, and cubic simulation model of the RC
Vertical stirrer models with triangular discretization . . . . . .
Horizontal stirrer models with triangular discretization . . . . .
TX/RX antenna models with far field pattern . . . . . . . . . .
Simulation model of the RC with canonical box-type EUT . . .
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72
73
76
77
79
81
83
85
87
88
89
90
94
95
96
97
99
101
104
108
111
ix
LIST OF FIGURES
x
6.10 Models and simulated far field patterns of different canonical EUTs . . . . 112
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
7.19
7.20
7.21
7.22
7.23
Schematic simulation, analysis, measurement, and benchmark procedure
Chamber door effect within a cavity . . . . . . . . . . . . . . . . . . . .
Near field comparison cavity against reverberation chamber . . . . . . .
Effect of different prototype reverberation chamber geometries . . . . .
Impact of a rotating stirrer within a reverberation chamber . . . . . . .
Different excitations in a reverberation chamber . . . . . . . . . . . . . .
Measurement vs. simulation at f = 300 MHz and f = 500 MHz . . . . .
Measurement vs. simulation at f = 700 MHz and f = 1000 MHz . . . . .
Influence of the RC door at f = 200 MHz and f = 250 MHz . . . . . . .
Statistical distribution of the electric field strength . . . . . . . . . . . .
Near field within prototype, cubic, and corrugated RC . . . . . . . . . .
Correlation for prototype, cubic, and corrugated RC . . . . . . . . . . .
Field uniformity envelopes in prototype, corrugated, and cubic RC . . .
Near field with canonical EUTs in a reverberation chamber . . . . . . .
Field uniformity envelopes for canonical EUTs . . . . . . . . . . . . . .
Coupling statistics for different excitations . . . . . . . . . . . . . . . . .
Near field patterns for different stirrers . . . . . . . . . . . . . . . . . . .
Correlation for stirrers with different shapes . . . . . . . . . . . . . . . .
Correlation for vertical/horizontal stirrers with and without gaps . . . .
Field uniformity envelopes for vertical and horizontal stirrers . . . . . .
Field uniformity envelopes for Z-fold and cross-plate stirrers . . . . . . .
Stirrer radar cross section calculations . . . . . . . . . . . . . . . . . . .
RCS-based correlation for the 6-paddle stirrer . . . . . . . . . . . . . . .
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117
119
120
122
124
126
128
129
131
133
135
137
138
139
140
142
144
146
147
148
149
151
152
B.1
B.2
B.3
B.4
B.5
Broadband effect of tripod on |E| at x = 0.77 m, y = 0.64 m, z = 0.47 m .
Broadband effect of tripod on |E| at x = 0.57 m, y = −0.36 m, z = 0.47 m
Spatial effect of tripod on |E| for 50 MHz and 100 MHz . . . . . . . . . . .
Spatial effect of tripod on |E| for 150 MHz and 200 MHz . . . . . . . . . .
Spatial effect of tripod on |E| for 250 MHz and 300 MHz . . . . . . . . . .
165
166
167
167
168
C.1 Independent stirrer positions and field uncertainty (1 component) . . . . . 169
C.2 Independent stirrer positions and field uncertainty (2 components) . . . . 170
C.3 Independent stirrer positions and field uncertainty (3 components) . . . . 170
D.1
D.2
D.3
D.4
D.5
D.6
D.7
D.8
D.9
Electric field pattern for 1 V and 10 V excitation source . . . .
Antenna orientation within the RC for different coupling paths
Field uniformity in prototype RC . . . . . . . . . . . . . . . . .
Field uniformity in corrugated RC . . . . . . . . . . . . . . . .
Field uniformity in cubic RC . . . . . . . . . . . . . . . . . . .
Field uniformity for vertical 6-paddle stirrer . . . . . . . . . . .
Field uniformity for vertical 6-paddle stirrer without gaps . . .
Field uniformity for horizontal 6-paddle stirrer . . . . . . . . .
Field uniformity for vertical cross-plate stirrer . . . . . . . . . .
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174
174
175
175
176
176
177
177
178
LIST OF FIGURES
D.10 Field
D.11 Field
D.12 Field
D.13 Field
D.14 Field
D.15 Field
uniformity
uniformity
uniformity
uniformity
uniformity
uniformity
xi
for vertical Z-fold stirrer . . . . . .
for vertical Z-fold stirrer with gaps
without canonical EUT . . . . . . .
with canonical loop EUT . . . . . .
with canonical box EUT . . . . . .
with large canonical box EUT . . .
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178
179
179
180
180
181
xii
List of Tables
2.1
2.2
2.3
Typical values of the total field anisotropy coefficient Atot . . . . . . . . 19
Probability Pχ2 (χ2 ≥ χ20 ) for hypothesis rejection in a χ2 -test . . . . . . . 24
Probability PN (|ρ| ≥ |ρ0 )| for the correlation coefficient . . . . . . . . . . 27
3.1
Comparison of runtime and memory requirements of RC simulations . . . 51
4.1
4.2
Summary of previously published RC simulations . . . . . . . . . . . . . . 61
Basic differences between AC and RC test environment . . . . . . . . . . 67
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Discretization data of chambers used in the RC simulations
Discretization data of vertical stirrer simulation models . .
Discretization data of horizontal stirrer simulation models .
Typical electrical conductivity values . . . . . . . . . . . . .
Electrical conductivity values used in the RC simulations .
Discretization of transmit (TX) and receive (RX) antennas
Discretization of the canonical emission EUTs (CEUTEs) .
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100
103
105
106
107
109
113
7.1
7.2
7.3
7.4
7.5
Different reverberation chamber geometries . . . . . . . . . . .
Different types of reverberation chamber excitations . . . . . .
Normalized spatial 2-norm measurement vs. simulation . . . . .
Performance comparison of different stirrers . . . . . . . . . . .
Time expenditure comparison simulations versus measurements
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121
125
130
150
153
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D.1 Field points used in measurement and simulation for uniformity analysis . 173
xiii
xiv
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 Körpern mit beliebiger Oberfläche
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 . . . . . . . . . . . . . . . Comité International Spécial 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
Atot
total anisotropy coefficient
A
triangle area
B
magnetic flux density
b
number of dimensions
β
propagation constant
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
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]
[m2 ]
[m2 ]
[Vs/m2 ]
[rad/m]
[As/m2 ]
xix
LIST OF SYMBOLS
xx
E
electric field strength
e
number of expected samples
εr
relative dielectric permittivity
ε0
dielectric permittivity constant
η
surface charge density
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
h
reverberation chamber height
I
current coefficient vector
In
line current
i, i , j
index number
Iα
planar inhomogeneity coefficient
Itot
total inhomogeneity coefficient
J
current density
Jm
Bessel function of order m
[V/m]
[As/(Vm)]
[As/m2 ]
[◦ ]
[◦ ]
[A/m]
[m]
[A]
[A/m2 ]
LIST OF SYMBOLS
xxi
JS
surface current density
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
m, n, p
mode number
M
number of spatial positions
µ0
magnetic permeability constant
µr
relative magnetic permeability
µ
statistical mean
N
number of unknowns
NI
number of line current basis functions
NJ
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
p
confidence level percentage fraction
Pi
input power
P
probability
[A/m]
[Byte]
[Vs/(Am)]
[rad/s]
[W]
LIST OF SYMBOLS
xxii
PRX
received power
[W]
PTX
transmitted power
[W]
q
number of calculated parameters
Q
quality factor
r, r , r position vector
ρ
correlation coefficient
volume charge density
s
variance
σ
radar cross section (RCS)
σ
standard deviation
SR
stirring ratio
t
time
V
volume
V
excitation vector
W
energy
w
reverberation chamber width
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
[m]
[As/m3 ]
[m2 ]
[s]
[m3 ]
[J]
[m]
[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
xxiv
1 Introduction
1.1 Motivation and objective of this thesis
Today’s electronic products must be designed so that they do not disturb the proper
operation of other products and inversely withstand electromagnetic (EM) radiation
emitted from all kinds of equipment. A crucial aspect of successful product development
is therefore the fast, effective, and efficient testing of the electromagnetic compatibility (EMC). EMC can be formally defined as “the ability of an equipment, subsystem
or system to share the EM spectrum, and perform at the same time its desired function
without 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 victim (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 surrounding
environment whilst at the same time being susceptible to noise generated by that environment. A system is therefore said to be electromagnetically compatible if it does not
interfere with other systems, it is not susceptible to emissions from other systems, and it
does 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, and
economical test procedures. Reverberation chambers (RCs) enjoy growing popularity
as a complement or replacement to well established methods for radiated interference
such as open area test sites (OATSs), (semi-) anechoic chambers (ACs) or transverse
electromagnetic (TEM)-cells. Those methods rely to a great extent on a “well-behaved”
equipment under test (EUT) radiation pattern, assuming implicitly that the EUT radiates or receives similar to a monopole, dipole, or quadrupole. A typical EMC test inside
a reverberation chamber (RC) is shown in Fig. 1.1. Particularly with regard to electronic
devices with complex radiation patterns, RC tests are expected to deliver more accurate and rigorous measurement results than the more traditional methods mentioned
before [2, 3, 4, 5]. The importance of testing for EMC in RCs as an alternative measurement 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 widespread use of this technique. RC users will need to fully understand the RC working
principles in order to interpret the measurement results correctly and to optimize the
performance for various measurement tasks. This requires a good understanding of the
EM field inside the chamber, for example, how the field depends on the finite metal
conductivity of chamber walls, the type of transmit and receive antennas, the shape of
stirrers and the EUT. It will be shown that even small wall irregularities caused by the
RC door, for example, or the position and the type of antenna used, have a profound
effect on the field within the RC and therefore must be accurately modeled in a simulation. 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,
c Institut für Grundlagen der Elektrotechnik
stirrer, and motorcycle as EUT. Copyright und Elektromagnetische Verträglichkeit, Universität Magdeburg, Germany.
only few simulation techniques are able to cope with such problems. Among those are
frequency-domain method-of-moments (MoM) based techniques used in the investigation presented in this thesis.
Faced with the task of performing EMC tests in an RC, one will usually turn to the wellknown IEC 61000-4-21 [6] standard as a first reference. Inside this standard one can find
detailed information regarding the modes of operation of an RC, chamber calibration,
radiated immunity, emission, shielding effectiveness test procedures, and examples of
measurement data. However, when it comes to designing and constructing an RC, the
IEC 61000-4-21 standard provides only basic guidelines, e.g. that “stirrers should have
dimensions of a significant fraction of the chamber dimensions and of the wavelength
at the lowest useable frequency” or that “a reverberation chamber is an electrically
large, highly conductive enclosed cavity”. Clearly, it is beyond the scope of fundamental
publications such as [6] to provide precise instructions on how to build highly efficient
stirrers or information on the exact material to be used for the RC walls. Many of these
construction suggestions were derived from years of practical experience in combination
with 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 a
time-consuming trial-and-error approach until a chamber finally fulfilled desired specifications.
In contrast to this cumbersome procedure, three-dimensional (3-D) simulations facilitate
the thorough EM analysis of RCs and could speed up their development time. The obvious goal of simulations would be the complete design, evaluation and optimization of
an RC until all target specifications are met prior to physical construction. Among the
typical questions before the construction of an RC starts are those related to the impact
of the chamber geometry as well as the conductivity of the sheet metal on the maximum
achievable field level and the mode spacing. Is a rectangular cavity superior to a cubic
one 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 look
in a particular RC? What is the statistically uniform testing volume for a certain stirrer
at a predefined uncertainty and how does it change with frequency? Does it matter
where the stirrers are positioned? Will an effective stirrer in one chamber show similar performance in a different RC? Of great interest are also questions such as: Could
the installation of a second stirrer decrease fLUF even further and if yes, how is this
stirrer to be rotated with respect to the first one? Is the directivity of a logarithmicperiodic (logper) antenna pointing towards a stirrer sufficient at lower frequencies to
avoid 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 the
complex EM field structure inside the chamber and therefore makes RC operation easier
to understand. With a simulation it is possible to verify the above mentioned rules-ofthumb for chamber dimensions, stirrer size, shape, and position, the relation between
chamber quality factor and mode spacing and so forth. Once a reliable simulation model
has been established, the final goal is of course RC optimization, which – depending
on the simulation runtime – may not always be economically feasible. One must clearly
state, that in some cases the effects of structural chamber modifications are more quickly
tested experimentally than with elaborate simulations; e.g. the effect of a very irregularly
shaped piece of aluminum foil attached to the stirrer can be analyzed rapidly with measurements, provided EM field data at only few locations is sufficient. An EM simulation
tool can only be an aid in the design of RCs and does not replace a solid understanding
of 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 background
and basic foundation of RCs is introduced in Chapter 2. This chapter establishes the
most important parameters that should be considered for the design, construction, and
operation of an RC: starting with an abstraction of RCs to ideal cavities, the concept
of modes, the field distribution, the number of modes, and the mode density are derived. As the focus is moved from an ideal cavity via a lossy cavity towards a realistic
RC, the lowest usable frequency, the quality factor, and statistical field distributions
are 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 RCs
are deduced. A synopsis of the computational challenges particular for RCs is described.
After contrasting these requirements against advantages and drawbacks of several numerical field solver techniques, it is outlined why for this thesis finally a frequency-domain
electric field integral equation (EFIE)-based method-of-moments (MoM) code was chosen for the RC simulations. The basic concept of the EFIE is outlined in Chapter 3
and the MoM solution methodology used in the employed simulation tools introduced.
Computational requirements regarding simulation time and memory are estimated and
an outlook on MoM extensions and solver techniques is given.
An overview of historic RC papers and patents as well as past publications dealing with
RC simulations is given in Chapter 4. This chapter summarizes both the most significant
accomplishments and also potential shortcomings of various RC simulation approaches.
In addition, alternative stirring methods (moving walls, electronic stirring, etc.), the
practical application of RCs to EMC testing, and a short qualitative comparison between 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 utilized
for data acquisition are explained. Measurement errors leading to deviations between
simulated and measured results are outlined and strategies for error minimization proposed.
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. Furthermore, cubic and corrugated chambers, various vertical and horizontal stirrers, transmit
and receive antennas, and several EUTs are designed and modeled.
3-D simulation results are presented in Chapter 7. In the beginning, the procedure used
to perform RC data analysis is presented. The necessity of a rigorous simulation validation is emphasized and different validation methods are compared. The electric field
inside the chamber is computed and the influence of small geometric details and asymmetries is investigated as well as the effect of different excitations and stirrers. It is demonstrated that a statistics-based validation of RC simulations is insufficient. To validate
simulation results, extensive near field measurements inside the prototype RC are performed. 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 geometries (cubic, corrugated) versus the prototype RC are carried out based on near field,
correlation, and field uniformity. Various stirrer designs are evaluated with respect to
their performance within the prototype RC. The presence of different EUTs is investigated, and a loading, field uniformity, and coupling path analysis is performed. The
complete 3-D RC simulation, considering stirrers, door, and various practical excitations,
accurately predicts the fields within the chamber in the important lower-to-medium frequency 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 future
are 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 distributions 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 of
finite 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 the
beginning 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 rectangular
waveguide at two sufficiently separated ends [10, 11]. If the geometrical dimensions of
this resonator reach certain lengths, at a given frequency an EM field within this resonator forms a standing wave pattern.
This standing wave pattern can be mathematically described by solving Maxwell’s equations which are given in differential form as
×E
∇
=
×H
∇
·D
∇
=
˙
−B
˙ + J
D
=
(2.3)
·B
∇
=
0
(2.4)
(2.1)
(2.2)
is the electric and H
the magnetic field strength, D
is the electric and B
the
wherein E
magnetic flux density. J denotes the electric current density and the volume charge
density. For the derivation of the numerical formulation valid inside an ideal cavity
resonator, 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
2 REVERBERATION CHAMBER THEORY
6
linear, homogeneous, isotropic, and without memory so that
D
B
(2.5)
=
εE
µH
J =
κE
(2.7)
=
(2.6)
is obtained for the material equations. Herein ε denotes the dielectric permittivity, µ the
magnetic permeability, and κ the electrical conductivity. If time-harmonic fields with an
ejωt -dependence are assumed and the material equations (2.5)-(2.7) utilized, Maxwell’s
equations as given above by (2.1)-(2.4) can be simplified to
×E
∇
×H
∇
·E
∇
·H
∇
= −jωµH
+ J
= jωεE
(2.8)
(2.9)
= 0
(2.10)
= 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
1 ∂2E
2
2
c ∂t
2
= 1 ∂ H
∆H
2
c ∂t2
=
∆E
(2.13)
(2.14)
which can be used to describe the fields within a cavity. c denotes the propagation speed
of the EM waves in the cavity resonator and is given by
c0
c= √
εr µr
(2.15)
with c0 being the speed of an EM wave in vacuum. With e.g. a product separation
approach [12], (2.13) and (2.14) can be solved using boundary conditions, which can be
derived for the tangential components of the electric and the magnetic field, respectively,
as
2 − E
1
×E
= n12 × E
= 0
∇
tan2 − E
tan1
⇔E
×H
= n12 × H
2 − H
1
∇
= 0
=
tan2 − H
tan1
⇔H
=
(2.16)
0
JS
for
κ<∞
κ→∞
0
JS
for
κ<∞
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 vector
which points from region 1 into region 2. The boundary conditions for the normal
components of the electric and magnetic field are enforced by
·D
= n12 · D
2 −D
1 = η
∇
·B
∇
⇔ Dnor2 − Dnor1 = η
2 − B
1 = 0
= n12 · B
(2.18)
⇔ Bnor2 − Bnor1 = 0
(2.19)
wherein η is the surface charge. For an ideal cavity, (2.16) and (2.19) can be simplified
to
tan |∂V
E
Hnor |∂V
= 0
= 0
(2.20)
(2.21)
valid on the PEC wall surface ∂V of the cavity for the tangential components of the
electrical 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
y=0 ∨ y=l
:
:
Ey = 0, Ez = 0, Hx = 0
Ex = 0, Ez = 0, Hy = 0
(2.22)
(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) can
be 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: modes
which do not have an electrical field component into z-direction (Ez = 0) are said
to be of the transverse electric (TE)-type, modes with Hz = 0 are called transverse
magnetic (TM). As a result, for the field components of TMmnp modes in an ideal
rectangular cavity resonator
nπ pπ mπ 1 mπ pπ E0 cos
x sin
y sin
z (2.25)
Ex (x, y, z) = − 2
kmn w
l
w
h
l
nπ pπ mπ 1 nπ pπ Ey (x, y, z) = − 2
E0 sin
x cos
y sin
z
(2.26)
kmn h
l
w
h
l
nπ pπ mπ x sin
y cos
z
(2.27)
Ez (x, y, z) = E0 sin
w
h
l
nπ
pπ
jωε nπ
mπ
Hx (x, y, z) =
E0 sin
x cos
y cos
z
(2.28)
2
kmn h
w
h
l
mπ nπ pπ jωε mπ Hy (x, y, z) = − 2
E0 cos
x sin
y cos
z
(2.29)
kmn w
w
h
l
Hz (x, y, z) = 0
(2.30)
is obtained with the integer numbers m, n = 1, 2, 3, . . . and p = 0, 1, 2, . . . . The indices
m, n, and p denote the number of half wavelengths in x-, y-, and z-direction, respectively.
8
2 REVERBERATION CHAMBER THEORY
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, the
following equations can be derived
nπ pπ mπ jωµ nπ H
x
sin
y
sin
z
(2.31)
Ex (x, y, z) =
cos
0
2
kmn
h
w
h
l
nπ pπ mπ jωµ mπ Ey (x, y, z) = − 2
H0 sin
x cos
y sin
z
(2.32)
kmn w
w
h
l
Ez (x, y, z) = 0
(2.33)
nπ pπ mπ 1 mπ pπ H0 sin
x cos
y cos
z (2.34)
Hx (x, y, z) = − 2
kmn w
l
w
h
l
nπ pπ mπ 1 nπ pπ Hy (x, y, z) = − 2
H0 cos
x sin
y cos
z
(2.35)
kmn h
l
w
h
l
mπ nπ pπ Hz (x, y, z) = H0 cos
x cos
y sin
z
(2.36)
w
h
l
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
mπ 2 nπ 2
+
(2.37)
kmn =
w
h
The angular frequency ω as employed in (2.25)-(2.36) can be calculated from
2πf
ω
mπ 2 nπ 2 pπ 2
=
= kmnp =
+
+
c
c
w
h
l
(2.38)
with c as given by (2.15). In an ideal cavity (PEC walls, no further losses through
dissipative objects inside) the cut-off frequencies for the individual modes are described
by
c mπ 2 nπ 2 pπ 2
+
+
(2.39)
fmnp =
2π
w
h
l
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 above
cutoff at a certain frequency fmnp can be calculated by counting all (m, n, p)-tuples until
f = fmnp is reached, whereby always at least two of the three indices are not equal to
zero. It is important to note that there can be several modes having the same cutofffrequency – 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 “degenerate
modes”.
2.1.2 Modes inside a lossy cavity
Whereas for an ideal, lossless cavity resonator the mode spectrum is discrete – i.e. a
resonance 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 finite
bandwidth 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 overall
field distribution is taken to be negligible. From a certain frequency on, at a single
frequency several modes can be excited, since their respective modal bandwidths start
to overlap. Depending on the quality factor Q of the lossy cavity, more or less modes
are excited simultaneously at a given frequency. At this point, the mono-mode regime of
the cavity turns into multi-mode operation. The field distribution obtained within the
cavity for multi-mode operation can be computed by carrying out a superposition of the
individual 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 is
valid, the field distribution within the cavity does not change in its shape but only with
respect to its magnitude. In other words the field distribution with κ < ∞ is essentially
a scaled version of the field distribution obtained for κ → ∞. Unfortunately, as soon as
any other scattering (or strongly absorbing) objects exist inside the lossy cavity, the field
distribution cannot be represented accurately anymore by analytically calculated modes
as 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 stirring devices, antennas, and EUTs) disturbing considerably a “cavity-like” appearance.
Moreover, the stirrers are explicitly designed so that they change the field distribution
within an RC and modify the modal cutoff frequencies during a stirrer rotation. The
abstraction of an RC to a simple cavity initially put forth at the beginning of this chapter
is therefore not fully valid. However, certain fundamental and important RC parameters
such 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 it
has to be stressed that the correct near field distribution inside an RC with a stirrer in
operation 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 frequency
from which on an RC meets basic operational requirements [6, 15]. The LUF is often also
referred to as “lowest overmoded frequency (LOF)” [8]). There are several definitions
for the LUF:
• the LUF equals three times the cutoff frequency fc of the fundamental mode of a
cavity 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 of
the same size as the RC are above cutoff and at least 1.5 modes/MHz are present [6]
2 REVERBERATION CHAMBER THEORY
10
Tuner/Stirrer
Non-conductive
assembly
support
Volume of
Incoming power
uniform field
Alternate position
mains filter
for tuner/stirrer
Drive motor
Tuner/Stirrer
assembly
l/4 at lowest
useable frequency
EUT
measurement
instrumentation
Field generation
equipment
Field generating
antenna pointed into
corner with tuner
Chamber penetration
Interconnection filter
Figure 2.1: Schematic reverberation chamber test setup including multiple stirrers, equipment
under test, and transmit antenna. Partly extracted from [14] and IEC 61000-4-21 [6] (Copyc International Electrotechnical Commission (IEC), Geneva, Switzerland).
right • the LUF is understood as being the lowest frequency at which a specified field
uniformity can be achieved over a volume defined by an eight location calibration
data set [6]
It is important to note that the first two definitions (which rely again on the cavity abstraction) are very qualitative requirements which give only a rough overview on whether
a chamber of certain dimensions might be suitable for operation as an RC. The third
definition is much more stringent, since it involves measurements within the chamber
and forces the user to think about the desired measurement uncertainties and confidence
intervals 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 fundamental
RC requirements [6], the cumulated number of modes, the mode density and the “modal
gap” must be known. Computation of these parameters implicitly assumes an empty
RC without a stirrer, i.e. a rectangular cavity. The total number of cavity modes above
cutoff at a given frequency can be calculated by imaging kmnp (2.38) as a point in the
three-dimensional k-space. |kmnp | therefore resembles the distance in space between the
point kmnp and the origin. In this geometrical model, the number of modes can be
computed by counting all discrete “nodes” in the k-space for which kmnp < k. When
2.2 LOWEST USABLE FREQUENCY
11
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
Ni (k) = 2N1 (k) + N3 (k) + N4 (k) + N5 (k)
(2.40)
i=1
taking into consideration that the number of modes N1 (k) = N2 (k). A fairly complicated
calculation outlined in [16] yields finally the approximate cumulated number of modes
N above cutoff for a frequency f as
8π
·lwh·
N (f ) ≈
3
f
c0
3
− (l + w + h) ·
f
1
+
c0
2
(2.41)
where c0 denotes the speed of light [16]. It can be see that the RC volume has the biggest
impact on the cumulated number of modes, as shown by the first part of (2.41). The
second part of (2.41) resembles the combined edge length of an RC. As noted above, for a
proper 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 calculated
from (2.41) to be
f2
1
∂N
≈ 8π · l w h · 3 − (l + w + h) ·
(2.42)
∂f
c0
c0
To achieve sufficient statistical field uniformity and isotropy, a common RC specification
is to have at least ∂N/∂f = 1.5 modes/MHz above cutoff [16]. The cutoff frequencies of
the individual modes inside an ideal rectangular cavity are given by (2.39), which was
slightly reformulated to facilitate mode-ordering into
c0 m 2 n 2 p 2
i
+
+
(2.43)
f(m, n, p) =
2
l
w
h
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
i
(m, n, p) and the corresponding cutoff frequencies. After sorting all f(m,
n, p) so that the
cutoff frequencies are arranged in an ascending order and renumbering the indices i again
with a new consecutive index i
i +1
i
f(m,
n, p) ≥ f(m, n, p)
∀ i
(2.44)
2 REVERBERATION CHAMBER THEORY
12
Prototype RC
N
900
Cubic RC
800
700
600
500
400
300
200
100
Nmin
0
100
150
200
250
300
350
f [MHz]
400
450
500
Figure 2.2: Theoretical number of modes N above cutoff in a rectangular cavity with the same
dimensions as the prototype RC or the cubic RC (see Sections 6.1.3 and 6.1.4 for chamber
geometry details). An RC is required to have at least 60 . . . 100 modes above cutoff at the
LUF [6].
is obtained, where the set (m, n, p) = (m, n, p) denotes a different mode (which might
however 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
i +1
i
∆f i +1; i = f(m,
n, p) − f(m, n, p)
(2.45)
and serves as an important parameter in the evaluation of different RCs (for a good RC
performance 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 chapter
a 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 prototype RC built during the course of this thesis (see Chapter 5) and the cubic RC: In
the critical lower frequency range, Fig. 2.2 shows very similar values for the theoretical
cumulated number of modes above cutoff (N ≈ 50; 180; 420 for the prototype RC versus
N ≈ 50; 190; 450 for the cubic RC at frequencies f ≈ 200; 300; 400 MHz). Furthermore
also the theoretical mode density ∂N/∂f computed with (2.42) and shown in Fig. 2.3
exhibits a similar behavior for both chambers: As mentioned above, an RC is required to
have at least ∂N/∂f ≈ 1.5 modes/MHz above cutoff at the LUF, and both the prototype
2.2 LOWEST USABLE FREQUENCY
¶N/ ¶f [1/MHz]
6
13
Prototype RC
Cubic RC
5
4
3
2
¶Nmin / ¶f
1
0
100
150
200
250
300
350
f [MHz]
400
450
500
Figure 2.3: Theoretical mode density ∂N/∂f in a rectangular cavity with the same dimensions
as the prototype RC or the cubic RC (see Sections 6.1.3 and 6.1.4 for chamber geometry
details). 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 only
a sufficient number of cumulated modes N as well as an adequate mode density ∂N/∂f
above 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 analysis
point of view the prototype chamber is considerably better suited for RC operation than
the cubic one [18]. As expected, above the fundamental mode cutoff frequency in the
beginning the modal gap ∆f i +1; i also decreases quite fast for the cubic chamber – the
cubic 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 instead 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). Although
the cubic RC exhibits at certain frequency intervals an even greater number of modes
above cutoff, in-between the mode density ∆N/10 MHz is – resulting from the mode
degeneracies – significantly lower for the cubic chamber as compared to the rectangular
prototype RC [19]. From (2.43) it is evident, that in order to avoid mode degeneracies, it
is important that the ratio between squared dimensions of the chamber is a non-rational
number. Whether these mode degeneracies also significantly deteriorate field uniformity
and isotropy inside the RC is investigated in Chapter 7.
2 REVERBERATION CHAMBER THEORY
14
Df i¢+1;i¢ [MHz]
30
Prototype RC
Cubic RC
* = Degenerate modes
25
*
20
*
*
15
*
10
**
*
*
*
* *
5
0
50
100
200
150
300
250
f [MHz]
350
400
* * *
*
450
500
Figure 2.4: Modal gap ∆f i +1; i in a rectangular cavity with the same dimensions as the
prototype 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 storing
energy. The chamber Q is an important quantity because it allows prediction of the mean
field strength resulting for a given input power. In addition, it provides an estimate of
the chamber shielding effectiveness (SE) and the RC time constant. Analytical quality
factor derivations of mono-mode resonators for each individual TEmnp and TMmnp mode
are outlined in [11].
Theoretical quality factor
The highly overmoded RC makes Q calculations based on resonant bandwidths (such as
Q = f0 /∆fQ , see Section 2.1.2) difficult, if not meaningless [20, 21]. A better approach
is to use the basic definition of the quality factor based on the time-averaged stored
energy Ws and the energy dissipated during one period Wd within a resonator
Q = 2π
ωWs
Ws
=
Wd
Pd
with Pd being the dissipated power. Ws can be computed from
2
1
1
D · E dv = ε
Ws =
E dv
2
2
V
V
(2.46)
(2.47)
2.2 LOWEST USABLE FREQUENCY
DN/Df [1/10 MHz]
45
15
Prototype RC
Cubic RC
40
35
30
25
20
15
10
5
0
50
100
150
200
300
250
f [MHz]
350
400
450
500
Figure 2.5: Number of modes ∆N per ∆f = 10 MHz interval above cutoff in a rectangular
cavity with the same dimensions as the prototype RC or the cubic RC (see Sections 6.1.3
and 6.1.4 for chamber geometry details).
Inserting (2.47) into (2.46) and taking into account that the dissipated power equals the
net input power Pin (i.e. forward power minus reflected power) leads to
2
ωε
(2.48)
Q=
E dv
2Pin
V
To evaluate (2.48) in practice requires knowledge of all individual loss mechanisms within
an RC: wall losses, absorption due to e.g. an EUT, aperture leakage (doors, interconnections between sheet metal panels, see Section 5.1), and losses introduced by the finitely
conducting RX and TX antennas. Whereas (2.48) includes per definitionem all these
losses, the simpler formula
Q=
3V
2µr δs A 1 +
3λ
16
1
l +
1
1
w
+
1
h
(2.49)
accounting only for ohmic losses in the walls is often used for RCs, where V is the
chamber volume and A the inner RC surface [16]. δs in (2.49) denotes the skin depth of
the metal which is defined as
1
δs = √
(2.50)
πf µκ
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-
2 REVERBERATION CHAMBER THEORY
16
nism. Expression (2.49) can be further reduced for typical values of l, w, h (such as the
dimensions of the prototype RC of this thesis l = 2.48 m, w = 2.86 m, h = 3.06 m) and
wavelengths λ ≤ 1 m to
3V
(2.51)
Q≈
2µr δs A
which is also the Q-factor estimate proposed in the IEC 61000-4-12 standard [6]. Unfortunately the derivation of the quality factor according to (2.51) is of little practical
use (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 measurements [23, 24]. As [24] already notes, measurements indicate that “ [. . . ] loss mechanisms
other than Joule heating [in the walls] are important”.
Measured quality factor
Once an RC is built, the chamber-Q can be determined by measurements. Q-factor
measurements in RCs are commonly carried out with a TX and RX antenna inside the
chamber and recording the received and transmitted power P RX and P TX [20]. Using
this approach, the chamber quality factor Q can be estimated as
Q=
16π 2 V P RX
λ3 P TX
(2.52)
The relation (2.52) can be evaluated over one full stirrer rotation (ϕj = ϕ1 . . . ϕN ), which
allows the computation of the mean chamber quality factor
Q = Qϕj =ϕ1 ...ϕN
(2.53)
It must be emphasized that (2.52) only represents a rough estimate of the actual Q
of an RC. The application of (2.52) is particularly difficult for very high-Q chambers,
where P TX should ideally be zero, since there are almost no losses (which implies also
for the same reason P RX ≈ P TX ). However due to the dominant absorption of the
antennas (which are geometrically large) P TX is greater than zero resulting in a Q
which is characteristic for the antennas, but not for the measured RC. A remedy to this
problem is to measure the quality factor with very small probes, which do not lower the
chamber-Q by loading the chamber.
When designing an RC often the question arises, which type of material is to be used in
order to get a good chamber performance in terms of the quality factor Q. As mentioned
shortly 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 overall
SE which is to a large extent defined by how the material was processed (soldered,
welded, screwed, bolted, etc.) and the particular construction (apertures, feedthrough panels, doors, ventilation ducts)
• loading introduced by TX and RX antennas along with tripods, probe stands,
cables
2.2 LOWEST USABLE FREQUENCY
17
• 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 frequencies (where only few modes are above cutoff) and maximum achievable field strength “per
Watt” input power. These two extremes transform physically into a trade-off between
a chamber quality bandwidth ∆fQ large enough (i.e. Q small enough) for a sufficient
number of modes to propagate at a fixed frequency versus a ∆fQ small enough (i.e. Q
large enough) for a high average field strength within the RC (see Section 2.1.2 for an
explanation of ∆fQ ). A comparison of the maximum achievable field strength between a
chamber built from aluminum against an RC made out of galvanized steel was presented
in [25] – with the not very surprising conclusion that the average field strength in the
aluminum RC is higher – but unfortunately there was no analysis of the impact on field
uniformity performed.
2.2.3 Stirring ratio
The stirring ratio (SR) provides a global parameter to quantify changes of the field distribution induced by a rotating stirrer. This method essentially measures the effectiveness
of the stirrer w.r.t. “moving the maxima and nulls within the chamber”. The SR is
commonly defined as
SR =
max
ϕj =ϕ1 ...ϕN
PRx (x0 , y0 , z0 ) −
min
ϕj =ϕ1 ...ϕN
PRx (x0 , y0 , z0 )
(2.54)
which requires that the power received by an antenna within the RC is measured over
a certain number of stirrer angles at a fixed spatial position [8, 9]. The input power
is kept constant PTx = const. for all rotational stirrer angles ϕj . Subsequently, from
the recorded power data the maximum and minimum value is calculated. Then the
SR is obtained by subtracting the minimum from the maximum value. The method
proposed in (2.54) relies on a mode-tuned operation of the RC with discrete stirrer
steps, since mode-stirring with a continuously rotating stirrer would introduce some
sort of “time-averaging”. In the common RC terminology, the SR is often expressed
in terms of decibels. A high SR suggests that both maximum and minimum E- or Hfield values occur at the same position for different stirrer steps; this indicates a more
effective stirrer. If the SR was measured at all points throughout the chamber, the
optimal condition would be if it was uniform, as this would indicate that the stirrer was
effectively changing the boundary conditions evenly throughout the chamber. The lower
limit normally accepted for the SR is 20 dB [26], and a sufficiently large SR is often taken
as 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: some
publications (e.g. [23]) introduce the SR as
SR =
max
|E(x0 , y0 , z0 )|
min
|E(x0 , y0 , z0 )|
ϕj =ϕ1 ...ϕN
ϕj =ϕ1 ...ϕN
(2.55)
i.e. the ratio of the maximum to minimum electric field strength measured at a fixed point
within the chamber over one revolution of the stirrer ϕj = ϕ1 . . . ϕN . Still other authors
2 REVERBERATION CHAMBER THEORY
18
(e.g. [22]) use the ratio between the input power of a TX antenna and the received power
of a RX antenna in the chamber. It must be emphasized that the various SR definitions
are not interchangeable and that especially the two latter expressions do not agree with
the SR statements found in fundamental RC literature such as [8, 9]. Generally, the SR
parameter is not used in formal RC tests as specified by [6], but serves within certain
limitations 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 inhomogeneity coefficients Iα and Itot were introduced for RCs by L. Arnaut [27, 28]. These
coefficients yield specific performance measures for field homogeneity and randomness of
polarization within an RC.
2.3.1 Field anisotropy coefficients
The field anisotropy coefficients Aαβ and Atot are defined according to [6] as
|Eα |2
|Eβ |2
Pi − Pi
Aαβ =
|Eβ |2
|Eα |2
Pi + Pi
A2xy + A2yz + A2zx
Atot =
3
(2.56)
(2.57)
|Eα | and |Eβ | represent the respective single measured or simulated electric field strength
component, 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 a
single axis are being performed during one stirrer rotation. When re-orienting a singleaxis probing sensor into two remaining different orientations in turn, the TX antenna
“sees” slightly different configurations (plus different cable layouts, etc.), so Pi changes
from 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 arbitrary
value [29]. Using e.g. Pi = 1 W for simplicity leads to
|Eα |2 − |Eβ |2
Aαβ =
(2.58)
|Eα |2 + |Eβ |2
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 of
the input power Pi [6]. For perfect reverberation conditions (i.e. ideal statistical field
isotropy), 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 or
measured and the theoretical CDFs from (2.59) serves as an indirect measure for the
field anisotropy (i.e. for a bias in the statistical field polarization towards a certain direction) [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 inhomogeneity coefficients are defined as
|Eα (r1 )|2
Pi
|Eα (r1 )|2
Pi
Iα (r1 , r2 ) =
Itot (r1 , r2 ) =
−
+
|Eα (r2 )|2
Pi
|Eα (r2 )|2
Pi
Ix2 + Iy2 + Iz2
3
(2.60)
(2.61)
Similarly as for the field anisotropy coefficients, Pi can be set to any arbitrary value as
long as all field locations to be evaluated in (2.60) were simulated or measured using
identical 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 that
are separated by an integral number of half wavelengths or excessively small distances
much below this wavelength for which the fields are highly correlated. A minimum
distance corresponding to one wavelength is recommended [6].
It is usually sufficient to investigate either Aαβ , Atot or Iα , Itot since field anisotropy
and field inhomogeneity coefficients show highly correlated statistics. The distributions
of the Iα , Itot however, are usually more sensitive to mode-stirring imperfections [31].
2 REVERBERATION CHAMBER THEORY
20
2.4 Field statistics and probability density functions
The EM field at a given position in the RC can be decomposed into three components and
each of these components can be described by its real and imaginary part (or equivalently
by its phase as well as its magnitude). Therefore in total six parameters are required to
fully describe the field. These six parameters are called in-phase and quadrature parts
in each of the three orthogonal directions x, y, and z
Ex
= Re {Ex } + j Im {Ex }
(2.63)
Ey
Ez
= Re {Ey } + j Im {Ey }
= Re {Ez } + j Im {Ez }
(2.64)
(2.65)
As the RC stirrer rotates from one angular position to the next, each of these in-phase
and quadrature parts is recorded and forms a data ensemble. The same statistical
considerations apply to both the electric and the magnetic field within an RC; for the
sake 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 large
number of random variables X which are – by the central limit theorem [32] – Gaussian
normally distributed
−(X−µ)2
1
f (X | µ, σ) = √ · e 2σ2
(2.66)
σ 2π
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 density
function (PDF) shown in Fig. 2.6 is obtained for each quadrature/in-phase part of the
three components (2.63)-(2.65). For sufficiently low correlation ρ between EM fields
sampled at different stirrer angles (see Section 2.5) and provided that a normal distribution 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 three
components are independent identically distributed (i.i.d.). Finally, the mean values µ
of these distributions can be assumed to be zero if there is not a significant direct path
from the TX antenna to the sampling point. This is a good assumption if the antenna
is near and pointed into a corner or directed towards the stirrer (see Section 7.6.3 for an
analysis 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 normally
distributed, each of the three field component ensembles Ex , Ey , Ez exhibits a 2-D
Gaussian distribution over one full stirrer rotation. Therefore the magnitude of a single
2.4 FIELD STATISTICS AND PROBABILITY DENSITY FUNCTIONS
21
component
|Ex | =
|Ey | =
|Ez | =
2
2
(Re {Ex }) + (Im {Ex })
2
2
(Re {Ey }) + (Im {Ey })
2
2
(Re {Ez }) + (Im {Ez })
(2.67)
(2.68)
(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) =
X
2
· X ν−1 e− 2σ2
ν
2 2 · σ ν · Γ ν2
(2.70)
from which the individual distribution functions can be derived by setting ν to the
appropriate value (e.g. ν = 2, see Fig. 2.6). Values of the Γ-function used in (2.70) can
be obtained numerically by
∞
Γ(x) = e−t tx−1 dt
(2.71)
0
or tabulated in e.g. [12]. A χ(2) distribution (obtained e.g. for a single component of the
electric field E consisting of in-phase and quadrature part) is also known in the literature
as a Rayleigh distribution [33]. An exemplary χ(2) distribution is shown in Fig. 2.6. The
magnitude 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 random
variables Xi
Y =
X12 + X22 + X32 + X42 + X52 + X62
(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
2
2
2
|E| = |Ex | + |Ey | + |Ez |
(2.73)
is therefore χ-distributed with six degrees of freedom (χ(6) -distributed) [28, 32]
f (Y | ν = 6) =
4·
σ6
Y
1
· Y 5 e− 2σ2
· Γ (3)
(2.74)
The χ(6) distribution for the magnitude of the EM field within an RC is depicted in
Fig. 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 random variable summation shown in (2.72)
Z = Y 2 = X12 + X22 + X32 + X42 + X52 + X62
(2.75)
2 REVERBERATION CHAMBER THEORY
22
c(2)
N(0.4, 0.1)
M /Mmax
0.7
c(6)
c2(6)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Re{Ex}, |Ex|, |E|, |P| [a.u.]
0.8
0.9
1
Figure 2.6: Exemplary Gaussian normal N (0.4, 0.1), χ(2) , χ(6) , χ2(6) distribution. These statistical distributions are obtained when counting the occurrence of e.g. the real part of a single
field component (Gaussian normal N ), the magnitude of a single field component (χ(2) ), the
magnitude of the total field (χ(6) ), or the magnitude of the power (χ2(6) ) over a full stirrer
rotation. M denotes a part of the total ensemble Mmax .
is of interest. For ensembles of the power random variable Z, the χ2 -distribution with ν
degrees of freedom (χ2(ν) ) is obtained. Its PDF is defined through
ν
f (Z | ν) = 2 2 · Γ
ν 2
ν
Z
· Z 2 −1 e− 2
(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 logper
antenna, 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 welloperating RC (see Fig. 2.6). Additional details on the observed PDFs applicable to RCs
can 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 to
make a statement if and how well the data actually follows the theoretically predicted
statistical distributions. This is especially necessary, if the amount of gathered data is
very limited, so that a visual comparison against a theoretically expected distribution
is impossible. To evaluate how good the agreement between prediction and reality is, a
so-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
(gi − ei )2
i=1
(2.77)
ei
where N is the total number of samples, gi is the number of observed samples in the i-th
interval (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 from
ei = N · Pi
(2.78)
with Pi being the probability that a particular sample is part of the i-th class. The
underlying distribution must be divided into i intervals such that
in central classes ni · Pi
in boundary classes ni · Pi
≥ 5
≥ 1
(2.79)
(2.80)
is satisfied for a given number of samples ni . The χ2 variable will be χ2 -distributed
with ν = N − q degrees of freedom, q being the number of parameters in the assumed
distribution that are calculated from the data (calculated parameters are estimators for
the mean or the standard deviation obtained from the underlying data, as the actual
values are not exactly known). For ν > 1 degree of freedom, (2.77) must be modified
into
N
1 (gi − ei )2
(2.81)
χ2 =
ν i=1
ei
The theoretically expected mean of χ2 is 1. If χ2 1 the observed samples do not
fit 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” agreement
can be translated to some sort of “numerical” confidence in a given hypothesis. An
expression of the quantitative significance of a certain χ2 value can be evaluated by
calculating the probability Pχ2 to get a certain value of χ2 which is equal or greater as the
sampled χ20 if the sampled distribution actually matches the hypothesized distribution.
If e.g. Pχ2 (χ2 ≥ χ20 ) is large, the hypothesized and the sampled (measured or simulated)
distributions seem to be identical; conversely for a small probability Pχ2 (χ2 ≥ χ20 ),
chances are high that hypothesized and sampled distribution do not match, i.e. there is
a certain deviation between them, which leads to a rejection of the hypothesis.
2 REVERBERATION CHAMBER THEORY
24
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)
∞
x2
2
2
2
ν · xν−1 e− 2 dx
Pχ2 (χ ≥ χ0 ) =
(2.82)
ν
22 · Γ 2
χ0
General tabulated values of (2.82) can be found in e.g. [35, 12], the cases most relevant
to RCs with ν = 2 and ν = 6 degrees of freedom are listed in Table 2.2. In particular, a
deviation between a hypothesized and sampled distribution is said to be
significant
if Pχ2 (χ2 ≥ χ20 ) ≤ 0.05
(2.83)
and
highly significant
if
Pχ2 (χ2 ≥ χ20 ) ≤ 0.01
(2.84)
χ20
For example, the probability to obtain
= 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 distribution
which 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 similarity
between 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 correlation
coefficient 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 correlation
coefficient ρxy (or more precisely an estimate for the correlation coefficient) between two
discrete ensembles X and Y with N samples each can be calculated as
ρxy
N
1 N i=1(xi − x)(yi − y)
=
s2x s2y
(2.85)
where x and y are the corresponding arithmetic mean values of these two ensembles.
They are defined as
N
N
1 1 x =
xi and y =
yi
(2.86)
N i=1
N i=1
under the assumption that the two discrete ensembles are two random samples taken
out of a very large underlying basic population (such as one created by a large number
of repeated measurements of identical parameters under consistent conditions) wherein
the ensembles themselves are uniformly distributed. sx and sy are the variances of the
ensembles and can be obtained from
1 |xi − x|2
N − 1 i=1
N
sx =
1 |yi − y|2
N − 1 i=1
N
and sy =
(2.87)
The correlation function ρxy can assume any value −1 ≤ ρxy ≤ 1, with values of ρxy = ±1
indicating a good linear correlation and values of |ρ| ≈ 0 little or no correlation at
all. Applied to RCs, ρxy relates e.g. the magnitude of a component of the electric
field (say |Ex |) at a fixed position (x0 , y0 , z0 ) for the angular position ϕ1 of the tuner
against 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 RC
a low correlation between the EM fields obtained at the two angular stirrer positions
ϕ1 , 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 of
samples N < ∞ out of finitely long ensembles (ρ = 0 would be found between a large
number of randomly picked, non-identical samples from a perfect additive white Gaussian
noise (AWGN) process of infinite bandwidth), the key question is which value of ρ would
indicate that the correlation between two ensembles (i.e. the similarity) is “sufficiently
2 REVERBERATION CHAMBER THEORY
26
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 by
calculating 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 . In
accordance with [35] the probability PN (|ρ| ≥ |ρ0 |) is given by
1
N −4
2 Γ N2−1
N −2 1 − ρ2 4 dρ
PN (|ρ| ≥ |ρ0 |) = √
πΓ 2
(2.88)
|ρ0 |
where the Γ-function can be calculated with (2.71). Table 2.3 shows PN (|ρ| ≥ |ρ0 )| for
various numbers of samples N and correlation coefficients ρ0 . The number of samples N
is for RCs equal to the number of angular stirrer positions. If a correlation of ρ0 between
two ensembles is obtained, for which PN (|ρ| ≥ |ρ0 |) is small, then there is a high chance
that 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) of
two uncorrelated ensembles will result in a correlation coefficient |ρ| ≥ 0.3 is around
0.034. This means if 50 samples yield ρ = ±0.3, there is no evidence of a linear correlation 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 are
considered uncorrelated [6]. As can be deduced from Table 2.3, the probability that the
underlying ensembles would actually have a higher correlation |ρ0 | ≥ 1/e is for N ≥ 50
stirrer steps less than 1%. In other words, there is highly significant evidence, that the
ensembles 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 small
geometrical changes. Whereas, e.g. at f = 100 MHz, a stirrer rotation of ∆ϕ = 5◦ would
not affect the field distribution within the chamber at all, the same ∆ϕ will change the
field completely at a frequency of f = 400 MHz (see Chapter 7). Therefore with rising
frequency, EM fields in an RC become increasingly uncorrelated from one stirrer step
to the next. Since reasonably uncorrelated fields are always a prerequisite for sufficient
statistical field uniformity and isotropy, at higher frequencies the same stirrer can provide more uncorrelated samples than at lower frequencies close to the LUF. In order to
reduce the time spent for an RC test, the number of stirrer steps required can therefore
be reduced at higher frequencies resulting in a correlation which remains approximately
constant as the frequency is increased [36, 37].
Care should be exercised when evaluating correlations, since a correlation coefficient can
be completely distorted by a single maverick (xi , yi )-pair, i.e. two ensembles appear to
be correlated although they are not or vice versa [38]. Therefore a scatter plot should
always 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 be
pinpointed.
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 ensembles
would 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, knowledge
is needed about the statistical behavior of the EM fields in the chamber. A typical
question when using an RC is: What is the number of statistically independent stirrer
steps necessary to state with a certain confidence that the mean, minimum, or maximum
field level at the EUT is within ±3 dB (i.e. the uncertainty) of the corresponding values
detected by a field probe at a different location? Using the PDFs for the EM field given
in Section 2.4, estimators and their accuracy for e.g. the mean field within an RC can be
2 REVERBERATION CHAMBER THEORY
28
1
p
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2.5
2
k ×s
3
3.5
Figure 2.7: ±k · σ standard deviation multiples contain p fractions of all values from a Gaussian
normal distribution (e.g. ±2σ contain approx. 95% of all values). p is also known as the
confidence level.
calculated. In RC simulations and measurements there is often the interest to calculate
the interval that contains a certain percentage p of the values from a standard Gaussian
normal distribution. This can be done by solving the integral equation
1
p = F (x, µ, σ) = √
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 the
confidence level gets higher (in the limit for k · σ → ∞, p approaches 1). In accordance
with [24], the so-called maximum likelihood estimator (MLE) is employed as an estimator
of the EM field. This has several advantages over other estimators: the MLE is always
asymptotically unbiased – i.e. its mean is the true value for large amounts of data –, and
its accuracy can be easily calculated [35]. As outlined in [35, 24], the amount of data
needed to achieve a desired estimator accuracy can be determined by
d˜ = 10 log10
1+
1−
√k
bN
√k
bN
(2.92)
2.6 STATISTICAL UNCERTAINTY AND ESTIMATOR ACCURACY
29
6
Confidence level p
68%
75%
95%
99%
a)
~
Uncertainty d [dB]
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Number of stirrer positions N
6
Confidence level p
68%
75%
95%
99%
b)
~
Uncertainty d [dB]
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Number of stirrer positions N
Figure 2.8: Number of statistically independent stirrer positions N required to achieve the
uncertainty interval ±d˜ for a) a single and b) three EM field components at a confidence of
p (see Fig. 2.7 for corresponding standard deviation multiples).
in the “dB notation”. 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 the field data to
be estimated (usually 1 or 3) and N is the required number of statistically independent
stirrer positions. If the field probe responds to only one dimension of the field in this
2 REVERBERATION CHAMBER THEORY
30
6
Confidence level p
68%
75%
95%
99%
~
Uncertainty d [dB]
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Number of stirrer positions N
Figure 2.9: Number of statistically independent stirrer positions N required to achieve the
uncertainty interval ±d˜ for two EM field components at a confidence of p (see Fig. 2.7 for
corresponding standard deviation multiples).
case b = 1. Solving for the required number of statistically independent stirrer positions
N results in
2
k 2 10d̃/10 + 1
N=
(2.93)
b 10d̃/10 − 1
Equation (2.93) is plotted for different confidence levels in Figs. 2.8 and 2.9. If, for example, the uncertainty interval should be d˜ = ±1 dB and the desired level of confidence
is 90% (corresponding to k ≈ ±1.65σ), then one would obtain N ≈ 69 or N ≈ 207 for
b = 3 and b = 1 dimensions, respectively (see Fig. 2.8). Conversely, if a determination of
the average field is made using N independent stirrer positions, the resulting uncertainty
is given by the interval ±d˜ (2.92). Further plots of the uncertainty interval for various
numbers 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 number 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 actually
capable of providing at least N independent distributions. To evaluate that a stirrer
provides independent field conditions, i.e. independent samples, the correlation coefficient as presented in Section 2.5 is usually calculated for the chosen step angle of the
stirrer. Thus, it is assumed that uncorrelated stirrer positions yield independent samples. This assumption is valid since (as outlined in Section 2.4) the underlying six
EM field quadrature component ensembles are normally i.i.d.; for a normal distribution
uncorrelated 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 interchangeably. It is important to understand that if a stirrer is not capable of providing
a sufficient number N of uncorrelated field distribution over a full rotation – this can
be challenging especially at low frequencies (see Chapter 7) – it may not be possible to
˜ In an ideal RC, ensemble
estimate the EM chamber field with a desired uncertainty ±d.
averaging (i.e. sampling the EM field at a fixed location (x0 , y0 , z0 ) for several different
stirrer angles ϕi = ϕ1 . . . ϕN ) will result in the same mean, maximum, and minimum estimators 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, which
can be achieved with a given stirrer in a chamber over its operating frequency range. As
mentioned in Section 2.2, the field uniformity is significantly better suited to determine
at 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 do
not guarantee a sufficient field uniformity. The field uniformity within an RC is expressed in terms of the combined three-axis standard deviation σxyz and the single-axis
standard deviations σx , σy , and σz of the EM field as proposed in the IEC 61000-4-21
standard [6]. Due to better availability of measurement equipment, normally the electric
field is used to compute the standard deviations. These quantities are calculated from
the 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 spatial
position (xi , yi , zi ) with the stirrer fixed at the angle ϕj (j = 1 . . . N : j-th stirrer step
out of a maximum of N steps) by
|E|ϕj = |E(xi , yi , zi )|ϕj
(xi ,yi ,zi )
2
2
2
= |Ex | + |Ey | + |Ez | (2.94)
ϕj
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 performance. Two definitions exist describing the layout of this “volume of uniform field”, in
which 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 object. For the IEC 61000-4-21 standard the second requirement has been adopted, it is
however still a good idea to place the EUT sufficiently far away with respect to the
operating wavelength from e.g. the metallic walls in order to avoid parasitic coupling
effects. Fig. 2.10 shows how the λ/4 (as well as λ/2 and λ) criteria translate into a
certain minimum distance at different frequencies.
For a field uniformity analysis there is no obvious reason why not more than eight points
2 REVERBERATION CHAMBER THEORY
32
l/4
d [m]
6
l/2
l
5
4
3
2
1
0
50 100
200
300
400
500
600
f [MHz]
700
800
900
1000
Figure 2.10: Required EUT spacing from RC walls versus operating frequency for different
distance criteria (λ/4, λ/2, λ).
should be used for the analysis of an RC, since especially in an RC simulation, any
arbitrary 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 of
data 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 deviation get. The restriction to eight points originates from field uniformity measurement
methods in order to minimize the time expenditure.
In compliance with [6], initially the maximum of each single-axis electric field component
occurring during one full rotation of the stirrer (ϕj = ϕ1 . . . ϕN ) is determined and then
normalized to the mean net input power Pi
max
Ẽξ,i =
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 in
total. If Pi is kept constant – either by employing a stress sensor and a leveling algorithm
in a measurement setup or by definition in RC simulations – it can be arbitrarily set to
Pi = 1 W for simplicity, which reduces (2.95) to
Ẽξ,i =
max
ϕj =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
8
1
Ẽξ,i
Ẽξ =
8 i=1
(2.97)
and the combined arithmetic mean
1
Ẽξ =
24
8
Ẽξ,i
(2.98)
ξ={x,y,z} i=1
using the three series’ of the x-, y-, and z-components is calculated. As a result, the
individual per-axis standard deviations
8
Ẽi − Ẽξ
σξ = i=1
(2.99)
8−1
as well as the combined standard deviation
8
Ẽξ,i − Ẽxyz
ξ={x,y,z} i=1
σxyz =
24 − 1
(2.100)
can be derived. The per-axis standard deviations can be expressed in the more familiar
“decibel notation” through
σξ + Ẽξ
σ̃ξ = 20 log10 (2.101)
Ẽξ
and for the combined standard deviation, respectively, through
σxyz + Ẽxyz
σ̃xyz = 20 log10
Ẽxyz
(2.102)
For both the per-axis standard deviations σ̃ξ as well as the combined standard deviation
σ̃xyz , [6] requires for a “well operating” RC with sufficient statistical field uniformity
and a given uncertainty within all frequencies 80 MHz ≤ f ≤ 100 MHz
σ̃ξ ≤ 4 dB and σ̃xyz ≤ 4 dB
(2.103)
For frequencies 100 MHz ≤ f ≤ 400 MHz, the limits for σ̃ξ and σ̃xyz decrease linearly
from 4 dB to 3 dB. Finally,
σ̃ξ ≤ 3 dB and σ̃xyz ≤ 3 dB
(2.104)
2 REVERBERATION CHAMBER THEORY
34
is required for all frequencies f ≥ 400 MHz. The above mentioned limits may be exceeded
at one single frequency (out of a predefined subset as outlined in [6]) within an octave
band 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 modes
above cutoff as well as mode density (see Section 2.2.1) and weakly correlated (and
hence independent) field distributions (see Section 2.5) within the RC. As shown in
Chapter 7, all these requirements are increasingly difficult to fulfill at low frequencies
around 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) a
statistical description of RCs can be performed. However, some precautions need to be
taken 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 graph
and the outcome of a hypothesis test, which leads to a decision on agreement or
disagreement with a benchmark analytical distribution (see Section 7.6.3) [34].
• Residual classes with small expectation values should not be plotted, as they distort
the overall graph – instead a number referring to the number of samples within
the residual classes is to be displayed. In Chi-square tests, residual classes must
be combined.
• Few outliers should not be taken into consideration – a significant number of outliers is however a strong indication that “something went wrong” during the measurement 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 the
mean 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 some
sort of correlation between two ensembles (especially if both exhibit some sort of a
linear trend), although they are not correlated at all or even physically irrelevant
for each other [40].
• Chi-square tests can only be conducted for data which is a priori uncorrelated and
statistically independent and identically distributed (i.i.d.).
• In the end, almost everything has some sort of a normal distribution – chances to
find a good agreement between two quantities (e.g. a simulated electric field and a
measured one) based on the similarity to a normal distribution are therefore high
and 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) were
introduced in this chapter. In the beginning, an RC was abstracted to a simple rectangular 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 the
actual EM field distribution within an RC, but with their help guidelines for the lowest
usable frequency (LUF) of an RC were derived. Using the proposed procedure to calculate the cumulated number of modes above cutoff and the mode density, an RC is likely
to perform well if at least 60 . . . 100 modes within an ideal cavity of the same size as the
RC are above cutoff and at least 1.5 modes/MHz are present. The mode distribution and
LUF of the prototype RC built for this thesis and a cubic RC were computed. A more
stringent, but time-consuming approach is to simulate or measure the lowest frequency
at which a specified field uniformity can be achieved over a volume defined by an eight
location calibration data set. This method was carried out in Chapter 7.
Methods for the derivation of the quality factor were established; the quality factor
serves as an important parameter as it directly influences the field uniformity within
the chamber, but it is difficult to measure and theoretical values are often several orders of magnitude too high due to the multimode-nature of an RC. Briefly the field
anisotropy and inhomogeneity coefficients were mentioned, which are useful to assess
RC performance and more sensitive than the standard field uniformity evaluation approach. These coefficients were however little used in this thesis due to the unavailability
of truly broadband field data (which is an inherent limitation of the frequency-domain
simulation technique).
In a final step, the RC was described as a statistical test environment and the EM
fields were characterized by distribution functions, correlation coefficients, statistical
uncertainty, and field uniformity. These parameters are used extensively throughout the
following chapters and establish the foundation on which the RC analysis by simulation
and measurement is built.
36
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 must
identify their critical design parameters. Based on these parameters, particular challenges for different EM simulation tools will quickly become apparent. In the following
sections the frequency-domain method-of-moments (MoM) and the finite-difference timedomain (FDTD) method are used as two typical EM field solver techniques to demonstrate fundamental challenges. Problems typical for MoM occur also in the boundaryelement method (BEM); difficulties existing in finite-difference time-domain (FDTD) are
likely to appear also in the finite-integration technique (FIT) and the transmission-linematrix method (TLM) method. The following sections identify the key issues, which
need to be taken into consideration as common requirements to achieve meaningful simulation results.
3.1.1 Wide operational frequency range
RC test systems are designed to cover a broad frequency range, typically from around
100 MHz to a few GHz (e.g. 80 MHz. . . 6 GHz). To minimize the computational effort and
hence the simulation time, usually the chamber geometry is discretized by an adaptive
frequency-dependent mesh. Attention must be paid to the proper discretization: For example, insufficient mesh resolution may introduce numerical artifacts, viz. the computed
field values depend strongly on the discretization of the geometry and are affected by
numerical dispersion.
For a broadband simulation response – in particular if high frequency resolution is required – frequency-domain methods such as MoM are at a disadvantage compared to
time-domain methods. While the latter require only one simulation run irrespective of
the frequency resolution [41], the former need to be run at many, sometimes tightly
spaced 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 the
wavelength at fLUF to achieve sufficient statistical field uniformity, i.e. the standard deviation for both the three individual field components and the total data set are within
a specified tolerance for a full stirrer rotation [6]. This means that an RC suitable for
testing 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 computational domain of this size considering a mesh dimension of at least λ/10 in each
dimension, requires a huge number of mesh cells (i.e. triangles in MoM or volume elements in FDTD). Methods using surface discretization such as MoM have an advantage
over volume discretization based methods, because the large space comprising air does
not have to be discretized.
A particular characteristic which is rarely encountered in ordinary EM problems is the
varying geometry of the RC during operation: Since the stirrers inside an RC are rotating, the chosen simulation method must be able to accommodate a varying geometry
without introducing additional errors. This problem is primarily found in methods using volume discretization, where the mesh needs to be changed at least locally from one
stirrer step to another, which might create additional numerical artifacts (for FDTD
cf. chapters 10 and 11 in [41]). To achieve an optimal EM effectiveness, the stirrers
are usually very irregular and designed asymmetrically, which is particularly difficult to
consider 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 account. Otherwise (for a perfectly electrically conducting structure with κ → ∞) the RC
quality factor bandwidth ∆fQ [6] would be zero, i.e. at a given frequency f only a single
mode (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 the
field level within the cavity could reach unrealistically high values, only limited by the
chosen numerical method. Realistic RCs however can only achieve sufficient statistical
field 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 appropriate
for the simulation of an RC can only be determined through a validation of the simulated
results by measurements.
From the computational point of view, RCs represent an entirely bounded domain, which
does not radiate to the exterior. Several simulation techniques are numerically problematic 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 have
difficulties to achieve numerical convergence [41] so that e.g. “artificial” losses within the
air volume have to be introduced or the wall conductivity has to be modeled unrealistically 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 equations
or 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, making
adaptive (i.e. resonance-dependent) frequency sampling and interpolation in-between
awkward. This problem has been considerably reduced with the introduction of adaptive
frequency sampling algorithms that set simulation frequencies according to the presence
of resonances. For the simulation of an RC, however, adaptive frequency sampling only
makes sense up to or slightly beyond fLUF where the number of modes above cutoff is still
small (60. . . 100 modes [6]). For frequencies f fLUF , too many discrete frequencies
would need to be sampled to compute a truly broadband response. Highly resonant
structures pose additional problems for a simulation method: in time-domain codes such
as FDTD the correct derivation of the quality factor Q of an RC can be difficult, since
the underlying algorithm introduces phase errors due to numerical dispersion resulting
in shifts of resonance frequencies (cf. chapter 4 in [41]). In addition, identifying narrow
resonance 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 at
least on multiple planes within the chamber to analyze e.g. the effect of a stirrer or an
EUT. In a method such as FDTD, the user must decide before launching a simulation, at
which positions the near field is to be monitored and stored. If near field data throughout
the entire RC is to be monitored, memory and computational requirements for FDTD
increase significantly, especially for a fine frequency resolution.
For MoM the computationally expensive part is the inversion of a full matrix to obtain
the surface and line currents. Once however this system of equations is solved, the
near field at any arbitrary point can be computed a posteriori from the stored currents
without 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 electromagnetic (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, the
stirrers, the TX antenna itself, the RX antenna, and an EUT. The resulting fields are
defined by Maxwell’s equations which are given in differential form by (2.1)–(2.4). These
equations are further simplified by assuming that there are no free charges inside the
computational volume V , i.e. = 0 in (2.3). In addition, the properties of the utilized
materials are assumed to be linear, homogeneous, isotropic, and without memory. This
led to the simpler equations (2.8)–(2.11), which are used in the following sections.
3 NUMERICAL PROCEDURE
40
3.2.1 Incident and scattered field
i are incident
i and magnetic field strength H
If EM waves of electric field strength E
onto a scattering structure, electric currents I on metallic wires and surface currents J
along metallic surfaces will be excited. These currents are in return responsible for the
s . The total field consists of the superposition of the
s and H
resulting scattered fields E
incident and the scattered field
tot (r) =
E
tot (r) =
H
i (r) + E
s (r)
E
i (r) + H
s (r)
H
s, H
s can be calculated from
The scattered field E
s (r) = L
r , r ) + L
E J(
E
E
I(
r
,
r
)
I
J
H
J
(
r
,
r
Hs (r) = LH
)
+
L
)
I(
r
,
r
I
J
(3.1)
(3.2)
(3.3)
(3.4)
E , L
E , L
H
H
where L
, and L
I
I are linear integro-differential operators relating to the line
J
J
and surface currents I and J [45, 46]. For the sake of simplicity, (3.3) and (3.4) do
not take into account electric volume currents (there is no current flowing within the
computational 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 dielectric
materials. An extension of (3.3) and (3.4) to dielectric materials including these currents
can 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
j E
· J(
r ) · G(r, r ) dA
∇
∇
LJ J(r, r )
= −
A
4πεω
A
µ
r ) · G(r, r ) dA
J(
−jω
4π
A
1 H
r ) · G(r, r ) dA
L
J(
J(r, r )
∇×
=
J
4π
A
∂I(r )
j
E I(r, r )
L
∇
=
−
· G(r, r ) dl
I
4πεω
∂l
L
µ
−jω
I(r ) · el · G(r, r ) dl
4π
L
1
H
LI I(r, r )
∇ × I(r ) · el · G(r, r ) dl
=
4π
L
(3.5)
(3.6)
(3.7)
(3.8)
3.2 COMPUTATION OF ELECTROMAGNETIC FIELDS
41
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 of
the current density over the primed coordinate system is given by
· J(
r ) = ∇
· J(
r ) − en · ∂ J(r )
∇
A
∂n
with the normal unit vector en and the partial derivative
surface. G(r, r ) is the free space Green’s function
∂ J(
r )
∂n
(3.9)
taken normal to the
e−jk|r−r |
G(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 resources. Special formulations have been developed rendering these computations more
feasible [48, 42, 45].
3.2.3 Solution of integral equations
r ) on the surface dA and the
In order to solve (3.5)-(3.8), the surface current density J(
line current I(r ) on the wire of length dl must be known. A direct solution of these
equations is however not possible in most practical cases – this includes the simulation
r ) and I(r ) are either not known at all or very complicated over a large
of RCs – as J(
area dA and a long length dl . The solution to this problem is to divide the computational domain into small (compared to the operating wavelength λ) parts, wherein the
r ) and the line current I(r ) can be described by rather simple apcurrent density J(
proximations. An expansion of the current distribution within these small parts allows
the computation of (3.5)-(3.8) and hence the calculation of the electric and magnetic
field (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 the
form of a series of a-priori known basis (sometimes also called expansion) functions. In
general, one chooses as basis functions a set that has the ability to accurately represent
and resemble the anticipated unknown current density, while minimizing at the same
time the computational effort required to employ it. The sets of basis functions may be
divided into two general classes: the subdomain basis functions, which are nonzero only
over a part of the structure (with a subdomain being either a small part of a larger wire
structure or a metallic surface); and the entire domain basis functions that exist over the
entire structure to be simulated. For the latter basis functions, there is no discretization
of the structure under consideration involved, but there is a priori knowledge of the
anticipated current distribution to be modeled required (which is e.g. feasible for the
computation of the current distribution on a dipole, but not for more complex structures
such as an RC). Of these two classes, subdomain functions are the most common [49].
3 NUMERICAL PROCEDURE
42
Unlike entire domain functions, they may be used without prior knowledge of the nature
of the current density function they must approximate. The subdomains are created
by discretizing the overall geometric structure: wires are subdivided into smaller parts
(“segments” of length dl ), surfaces into smaller elements (“patches” of area dA ) such
as triangles or quadrangles. It is obvious that more sophisticated basis functions can
approximate arbitrary current distributions more accurately, resulting in a smoother
representation – this however comes at the cost of increased computational complexity.
Basis functions for wire structures
An example for a set of subdomain basis functions g used to approximate the line current
In on wire segments is given by
⎧
1
⎪
r − a+
r ∈ Sn+
+ · |
⎪
2,n | for l
⎪
n
⎪
⎨
1
gIn (r ) =
(3.11)
· |r − a−
r ∈ Sn−
2,n | for l−
n
⎪
⎪
⎪
⎪
⎩
0
for r ∈ Sn+ , Sn−
a−
wherein ln+ , ln− are the lengths of the segments Sn+ , Sn− . a+
2,n and 2,n refer to the
+
−
end points of the two segments Sn , Sn 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 wire
approximation”, assuming that the segments are infinitely thin; for “thicker” segments
a more accurate formulation can be used [48].
Using a superposition of the basis functions for wire segments (3.11), the actual current
in a segment will be expanded by
I(r ) =
NI
In · gIn (r )
(3.12)
n=1
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-linear
continuous manner. The currents In are the unknown coefficients which need to be
computed.
Basis functions for metallic surfaces
It is possible to model metallic surfaces through tightly spaced wire segments by creating
a 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 is
not very satisfying, vectorial basis functions for an approximation of the current density
JS on surface patches were introduced. As patch elements, often triangles are used since
they have the ability to conform to any geometrical surface. Due to the derivatives and
the kernel in the integral equations of the EFIE, there are complications in using vectorial basis functions for patches: In order to achieve physically consisting results, basis
functions must be constructed such that the normal component of the current is continuous across surface boundaries. The vectorial basis functions, which eliminate these
problems (and are used by FEKO ), are the so-called “Rao-Wilson-Glisson (RWG)” basis functions [51]. The RWG basis functions are not simply coincident with each triangle
3.2 COMPUTATION OF ELECTROMAGNETIC FIELDS
43
face, but consist of pairs of triangular faces along an adjacent edge, similar to “roof top”
functions. RWG functions are defined as
⎧
ln
ln
⎪
for r ∈ Tn+
rn+ = 2A
r − a+
+
+ · ⎪
3,n
2A
⎪
n
n
⎪
⎨
ln
ln
gJn (r ) =
(3.13)
rn− = − 2A
r − a−
for r ∈ Tn−
+ · 3,n
2A−
n
n
⎪
⎪
⎪
⎪
⎩
0
for r ∈ Tn+ , Tn−
and are associated to the n-th edge that two adjacent triangles have in common. ln is
−
+
−
the length of the n-th “inner” edge, A+
n , An resemble the areas of triangles Tn , Tn , and
+ −
rn , rn are the position vectors with respect to the vertices opposite from the n-th inner
edge [48]. The position vector rn+ is directed from the free vertex of Tn+ towards points in
a−
Tn+ , whereas rn− is directed towards Tn− . a+
3,n and 3,n denote the position vectors of the
two free triangle vertices which are not part of the inner edge. These triangle-pair basis
functions 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 surface
is a common edge between the two triangular faces of a subdomain. Hence, up to three
basis functions will be superimposed within each face of the triangulated surface, allowing
a 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 current
on a triangle will be expanded by
r ) =
J(
NJ
Jn · gJn (r )
(3.14)
n=1
where NJ 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. The
current densities Jn are the unknown coefficients which need to be computed.
Special basis functions are used to model the transitions between wire segments and
triangular surface elements [48]. Considerable advantages in computation time and reduction of approximation errors can be gained by a careful choice of the basis functions.
3.2.5 Computation of line and surface current coefficients
r ), the line
In order to approximate the actual line currents I(r ) and surface currents J(
and surface current coefficients In and Jn of (3.12) and (3.14) must be computed. These
coefficients can be determined by considering the boundary conditions for the EM field
obtained from Maxwell’s equations (2.1)-(2.4) which can be derived for the tangential
and 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 and
Jn 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 conditions
on 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
3 NUMERICAL PROCEDURE
44
(2.17). Also a combination of the EFIE and magnetic field integral equation (MFIE) is
used and known as the coupled field integral equation (CFIE). The CFIE uses both the
boundary conditions for the electric (2.16) and the magnetic (2.17) field.
For the example of a PEC surface, (2.16) simplifies to
tot = 0
n12 × E
(3.15)
For non-PEC structures, (3.15) needs to be reformulated so that the boundary conditions
imposed by (2.16) are satisfied. Substituting (3.3) into (3.1), (3.15) can be rewritten as
r , r ) + L
i (r) + L
E J(
E I(r, r )
= 0
(3.16)
n12 × E
I
J
Inserting the approximations for the surface current density (3.14) and the line currents
(3.12) in (3.16) yields
⎛
#
#⎞
N
N
J
I
i (r) + L
E
E
Jn · gJn (r ) + L
In · gIn (r ) ⎠ = 0
(3.17)
n12 × ⎝E
I
J
n=1
n=1
i (r) is on the righthand side results
Rearranging (3.17) so that the known incident field E
in
⎛ #
#⎞
N
NJ
I
E
i (r)
LE
Jn · g (r ) + L
In · gIn (r ) ⎠ = −n12 × E
(3.18)
n12 × ⎝
J
Jn
n=1
I
n=1
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 software
FEKO . It has to be outlined that the actual EM solution method as described in
Section 3.2.5 is commonly incorrectly referred to as “the MoM”. The MoM is merely
used 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 find
the 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 NJ + NI
unknowns Jn and In . To solve for the NJ + NI unknowns it is necessary to have NJ + NI
linearly independent equations. The MoM transforms the EFIE (3.18) containing linear
operators into a system of linear equations. It owes its name to the process of taking
moments by multiplying (3.18) with appropriate weighting functions and integrating [53,
46, 45].
3.3.1 Point matching and weighting functions
Having NJ +NI linearly independent equations can be accomplished by evaluating (3.18)
– i.e. applying the boundary conditions – at NJ + NI different locations on surface
3.3 METHOD OF MOMENTS
45
patches and wire segments. This is usually referred to as the collocation method or
point 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 these
points 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 geometric
structure approaches zero, the method of weighted residuals is utilized in conjunction
with a so-called inner scalar product
ν
ν
... ·w
m
dA
(3.19)
< ...,w
m >=
∂V =A
for the surface current density and
ν
< ...,w
m
>=
ν
...· w
m
dl
(3.20)
∂A=L
ν
are suitable weighting functions, which are defined on the
for the line currents [48]. w
m
NJ surfaces and NI line segments. This technique does not lead to a vanishing residual at
every point on a PEC surface, but it forces the boundary conditions (2.16) and (2.17) to
be satisfied in an average sense over the entire surface. The choice of weighting functions
ν
must be linearly independent, so that the NJ +NI
is important in that the elements of w
m
equations in (3.19) and (3.20) will be linearly independent, too. Furthermore, it will be
advantageous to choose weighting functions that minimize the computations required
to evaluate the inner products. Since the linear independence between elements and
the computational efficiency are also important requirements of basis functions, similar
types of functions are often used for both weighting and basis functions.
A particular choice of functions in the literature commonly referred to as “Galerkin’s
ν
method” is to let the basis and the weighting function be the same. Other choices for w
m
include Dirac δ-functions, which reduce the “average” boundary condition matching back
to point-wise matching. FEKO uses a computationally advantageous Quasi-Galerkin
approach with an adaptive integration technique, which distinguishes between spatially
near and far boundary condition matching points [46].
3.3.2 Matrix formulation
Evaluating (3.18) at NJ + 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 NJ + NI linear equations, which can be expressed in the commonly used
MoM matrix formulation
Z · I = V
(3.21)
where I is an N = (NJ + NI )-column vector which contains the expansion coefficients
In and Jn of the MoM basis functions. Z is the N × N system matrix, containing the
is an N -column vector relating to
coupling between basis and weighting functions; V
3 NUMERICAL PROCEDURE
46
the impressed fields originating from sources such as an incident wave radiated by a TX
antenna. 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 this
current approximation. In order to calculate the inverse of the system matrix Z−1
it is necessary to evaluate N 2 terms in (3.22), with each term requiring two or more
integrations. When these integrations are to be done numerically, depending on the size
of 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 Symmetry
If a complete structure or parts of a structure exhibit geometric symmetry in one or
more planes, the time for the computation of the elements of the system matrix Z can be
reduced. The source, however, is not symmetric, thus a symmetric current distribution
does not exist. This asymmetric current distribution leads to asymmetric electric and
magnetic fields. The order of the system matrix therefore remains the same, which
means that the computation time or the amount of memory needed for the solution of
the equation system cannot be reduced [13].
Electric or Magnetic Symmetry
If – in addition to symmetry of the geometric structure – also the excitation is symmetric with respect to one or more planes, then these planes can be considered as being
planes of electric (i.e. perfectly electrically conducting (PEC)) or magnetic (i.e. perfectly
magnetically conducting) symmetry. An electric (magnetic) symmetry plane is a plane
which can be replaced by an ideal electrically (magnetically) conducting wall without
changing the field distribution. For this situation, the corresponding coefficients of the
current 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 memory [46]. Each electric or magnetic symmetry reduces the number of unknowns in the
system matrix by a factor of two, that is the required amount of memory by a factor of
four.
Partial Symmetry
Normally, a structure must be perfectly symmetric for FEKO in order to speed up the
solution by exploiting the symmetries mentioned above. Any asymmetric segments or
triangles, or ones that lie in a symmetry plane or on the axis of rotation, will destroy the
symmetry. 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 symmetry
in a structure. Partial symmetry can be exploited to reduce solution time by running a
simulation of the symmetric part of the model first and saving this data. The asymmetric
parts 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
tot = E
Z
n12 × E
(3.23)
at the boundary to a non-PEC structure. Finitely conducting surface materials are
assumed to have a thickness d, permeability µ, and conductivity κ (ωε κ < ∞). They
are modeled by a fictitious surface impedance [42]
ZS =
1
1−j
2κδs tan (1 − j)
d
2δs
(3.24)
where f denotes the simulation frequency and δs the skin depth of the metal as defined
by (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 be
calculated as outlined in [48, 42, 45].
3.4 Computational requirements
The computational requirements needed for the simulation of RCs are significant at
higher 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 complete
one 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 leads
to 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 with
three basis functions (3.13), provided that all edges of the triangle are connected
to other triangles [55].
• Dielectric structures need twice as many basis functions (to model electric and
magnetic currents) as metallic triangles.
3 NUMERICAL PROCEDURE
48
• At the outer parts of a structure where there are triangles only connected on less
than three edges to other triangles, one or two basis functions are used (depending
on 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 number of unknowns in the system matrix by a factor of two; hence the required storage
memory for the system matrix Z is reduced by a factor of four per electric/magnetic
symmetry.
• One segment node which connects NS segments needs NS − 1 basis functions (e.g. a
loop 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 matrix 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
&
'
2
3
2
NT + NS = 36 NT2 + 16 NS2 Byte
M ≈ 16 Byte ·
(3.25)
2
Discretization with the CAD-software HyperMesh of the surface area A into triangles
of area A can be represented by
A = NT · A
(3.26)
The “chordal deviation” discretization algorithm in HyperMesh is controlled via the
parameters
• 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 a
single user-adjustable parameter “element size” d [56]. The surface size of a triangle A
created with HyperMesh can be approximated to
A ≈
1 2
d sin γ
2
(3.27)
where γ accounts for the apex angle of the triangle. d is directly related to the discretization requirements of the EFIE-MoM as implemented in FEKO . Values for d
used throughout the RC simulations are
d=
λ
5 . . . 10
(3.28)
3.4 COMPUTATIONAL REQUIREMENTS
Memory M [GByte]
1000
49
l2/100
l2/70
150
200
l2/50
l2/30
900
800
700
600
500
400
300
200
100
0
50
100
250
300
f [MHz]
Figure 3.1: Theoretical memory requirements of an RC simulation with A = 50 m2 of discretized
surface and NS = 870 segments for four different triangular mesh resolutions A (3.25) in
the f = 50 . . . 300 MHz frequency range.
with λ denoting the free-space wavelength. This finally results in an estimate for the
number of triangles
2
2A 5 . . . 10 · f
(3.29)
NT ≈
sin γ
c
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 to
the frequency, i.e. NT ∼ f 2 . Using a similar derivation, it can be easily shown that the
number 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 system
matrix 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 in
Table 3.1 and agree well with the approximate estimates of (3.25) and (3.30). It can
be clearly seen that with the computational approach presented in Section 3.2, a higher
3 NUMERICAL PROCEDURE
50
100
Memory M [GByte]
l2/100
l2/70
l2/50
l2/30
10
1
0.1
0.01
100
200
300
400
500
600
f [MHz]
700
800
900
1000
Figure 3.2: Theoretical memory requirements of an RC simulation with A = 50 m2 of discretized
surface and NS = 870 segments for four different triangular mesh resolutions A (3.25) in
the f = 50 . . . 1000 MHz frequency range.
operating frequency requires a finer discretization of the geometric structure, which
results in a greater number of triangles NT and segments NS . Figs. 3.1 and 3.2 show
the 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, and
λ2 /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 lower
f = 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 elements
must be computed as described in Section 3.2. According to [46], the time tsetup for the
determination of the system matrix elements can be estimated for segments as tsetup ∼ f 2
and for metallic triangles as tsetup ∼ f 4 . The total determination of the system matrix
3.4 COMPUTATIONAL REQUIREMENTS
51
Frequency f
[MHz]
Number of
triangles NT
Memory M
[MByte]
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
CPU runtime t
[s (hh:mm)]
a in addition to the triangles, for the discretization of the biconical TX/RX antenna 162 segments were used
b — — of the logper TX/RX antenna 435 segments were used
c 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 single
frequency and one stirrer position. A typical RC consists of walls, door, one 6-paddle vertical
stirrer and one antenna. Simulations were run using the standard FEKO solver without
symmetries or approximative methods.
elements is therefore on the order of
tsetup ∼ O f 4 + O f 2
(3.31)
With the implementation of a direct solver (e.g. LU-decomposition), the time needed
for the solution of the system of linear equations tsolve is for segments on the order of
tsolve ∼ f 2...3 and for metallic triangles tsolve ∼ f 4...6 . Under the assumption that the
time for initialization, checking of geometry, near and far field computation are negligible,
the total simulation time can be estimated as
(3.32)
ttot = tsolve + tsetup ∼ O f 4...6 + O f 4
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 a
high-frequency RC simulation with a fine discretization however, the matrix equation
solution time tsolve clearly dominates over the setup period tsetup . Computation times
obtained in actual RC simulations with FEKO are listed in Table 3.1: Due to the strong
frequency dependence of ttot (3.32), the rate at which the CPU runtime increases with
growing frequencies is even more dramatic than the increase in required memory (3.30).
The strong frequency dependence of both computational memory and solution time emphasizes the importance to use as few triangles NT and segments NS as possible within
the numerical restrictions of the MoM for the discretization. Therefore in order to minimize the number of triangles needed to mesh the large RC surface, a frequency-adaptive
52
3 NUMERICAL PROCEDURE
and geometry-dependent discretization was used. Table 3.1 states also the typically required numbers of triangles at different frequencies and a fixed stirrer position using a
standard MoM approach without symmetries (Section 3.3.3) or approximative methods
(Section 3.5.2). Simulations were performed on a distributed Sun Blade 1000/2000
workstation cluster with up to three double-processor machines in parallel. Looking
at Table 3.1 it is clear that the excessive runtimes render RC simulations with today’s
available 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 in
the interior resonance case the system of equations can be “ill-conditioned” (which results
in a very high condition number of Z) [57]. Practically however, this problem only shows
up if the simulated structure is completely closed and in addition PEC. Furthermore, it
is 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 the
MFIE. The MFIE, however, and hence also the CFIE, can only be used for regions that
are mathematically “simply connected” (“1-connected”) [48]. A simple cavity would
satisfy this requirement, whereas an RC with stirrers, antennas, and EUT certainly does
not. In addition, MFIE solutions tend to diverge if applied to structures involving thin
metallic sheets [49]. Therefore integral equation methods based on MFIE or CFIE are
not applicable to the solution of most “real world” RC problems.
Instead of using the MFIE or CFIE, EFIE-related problems can be eliminated (for most
practical cases) through the usage of materials of finite conductivity κ < ∞ together with
modified forms of the EFIE which are still valid in the interior resonance case [59, 44]. In
addition it is good practice to monitor the condition number of the system matrix and
to test explicitly whether this matrix is singular. Both of these precautionary measures
are implemented in the FEKO -kernel, which warns the user if the matrix is singular or
the condition number reaches 1016 [54].
3.5.2 Iterative solution techniques
In the traditional subdomain-based MoM, it normally occurs that the resulting system
of equations (3.21) is described with a dense (i.e. full) matrix Z having complex-valued
elements . The matrix structure is usually optimized for specific numerical solution techniques by a preconditioner. Solving systems with full matrices is – as mentioned in Sections 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 combining 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 and
3.5.4.
3.5 EXTENSIONS TO THE METHOD OF MOMENTS
53
Standard iterative schemes
Reducing the number of mathematical operations can be accomplished by using iterative
schemes such as
• stationary iterative methods: Jacobi, Gauss-Seidel, Successive-over-relaxation
• non-stationary iterative methods: Conjugate-gradient-on-normal-equations, Biconjugate gradient stabilized (BiCGstab) [60] or the transpose-free quasi minimum
residual (TFQMR) method [61]
which need fewer operations for the decomposition of the MoM system matrix Z. Since
there were problems with the convergence of these numerical methods when applied to
the 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 rotation
of a stirrer) result in changes to one or more small areas of the overall system matrix
Z (3.21), leaving relatively large portions of Z unchanged [62]. Therefore it would be
helpful to use techniques which avoid the repeated inversions of the original large, dense
system matrix. A solution to this problem is presented by [63] and [64] on the basis
of the so-called “Sherman-Morrison-Woodbury expansion”. This approach is useful for
configurations 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. However, the small object(s) may move or change its orientation with respect to the
large object(s).
• same as above, but where the small object(s) may change in size, shape, electrical
characteristics, position, and/or orientation.
• where one of the objects is much larger and unchanging, and the other smaller
object is changing geometrically or electrically.
Obviously the RC fits well in these categories of problems, since the greater part of the
geometry (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’s
function (NGF)”. The main purpose of the NGF is to avoid the unnecessary repetition
of calculations when a part of a model, such as e.g. the rotational position of a stirrer in
a complex RC environment, will be modified one or more times while the remaining RC
environment 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 smaller
self-interaction matrix for the stirrer, the mutual stirrer-to-RC-environment interactions,
and matrix manipulations for a partitioned-matrix solution. The NGF has not been
implemented in FEKO due to considerable incompatibility issues with parallelization
and the out-of-core solver.
3 NUMERICAL PROCEDURE
54
3.5.3 MoM and physical optics (PO) hybridization
In addition to the iterative techniques (Section 3.5.2), two other methods were considered
to 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 from
an excitation and a scatterer are [65]
• dimensions and radii of curvature of the illuminated object (i.e. the scatterer) must
be large as compared to the wavelength.
• source point (i.e. excitation) is sufficiently far away (typically d > λ) so that an
incident plane wave can be assumed locally.
• metallic region must be perfectly conducting, κ → ∞ (this restriction applies to
the PO implementation in FEKO [54]).
For complex geometric structures such as an RC, a large number of secondary reflections
needs to be taken into account. In order to accurately model the effects of secondary
reflections (e.g. for dihedrals at least two reflections and for trihedrals at least three
reflections must be considered) it is essential to use the multiple reflection PO. There
are however three problems associated with the PO applied to RCs:
• the ray tracing needed for the PO multiple reflection approach slows down the
computation considerably.
• the PO is computationally very efficient only as long as PO and MoM regions are
decoupled – 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 the
pronounced effects of the stirrer and the TX/RX antennas on the field structure
inside the RC, decoupling was not feasible in RC simulations [66].
• the RC cannot be modeled with infinite conductivity κ → ∞; otherwise the RC
quality factor bandwidth ∆fQ [6] would be zero, i.e. at a given frequency f only a
single 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 was
first used in fluid dynamics. Its derivative to Maxwell and Helmholtz equations was
initiated by V. Rokhlin [68]. The FMM is highly beneficial from around 25 000 triangles on – where computation with a standard full-wave MoM approach is not feasible
w.r.t. computation time and memory requirements. Using the FMM, even structures
discretized with 500 000 triangles can be computed on a standard personal computer
(2 GB RAM, 3 GHz processor) [54].
Single level FMM
The basic idea behind the FMM is to split the MoM system matrix Z into two parts
Z = Znear + Zfar which describe separately near- and far-field interaction between segments 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 expansion, T (translation) accumulates the global multipole expansions in a local one, and
L (disaggregation) evaluates the local multipole expansions at the observation element.
Then the whole system of equations can be solved using iterative solution techniques
mentioned in Section 3.5.2. This approach is known as the single level FMM.
Multilevel FMM
The multilevel FMM extends the idea of the single level FMM. Initially the currents in
one region are grouped to act as a multipole, then several multipole regions are bundled
together to act as another multipole and so forth – the whole EM problem is essentially
subdivided into smaller “cells”; in each “cell” the above mentioned Zfar = LTG transformation is carried out. This hierarchical procedure leads to the name “multilevel”
FMM [68, 69]. Using the multilevel fast multipole method (MLFMM) approach, the
memory requirements for electromagnetically large problems with N unknowns can be
drastically 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-medium
frequencies (large-medium wavelengths), the matrix Znear is much greater than the matrix Zfar . Consequently, the FMM does not exhibit a computational advantage over
the MoM at these frequencies. The frequencies at which the matrix Znear turns out to
be much smaller than Zfar are so high that an RC simulation is not reasonable due to
validation problems (cf. Section 7.4).
3.6 Conclusion
It was outlined that a numerical method suitable for reverberation chamber (RC) simulations must be able to compute the electromagnetic (EM) field over a broad range of
frequencies, handle a large, varying, and irregular geometry, and model finite conductivity. In addition, it must be able to cope with a highly resonant structure and should
allow the field calculation at a large number of spatial field points without introducing
too much computational overhead. In order to deliver broadband simulation data it
was shown that frequency-domain methods will need to compute the field at a lot of
discrete frequencies. Time-domain methods are at an advantage in this regard, since a
broadband frequency response can be calculated from only a single run. Methods using
unstructured grids (triangles, tetrahedra) are advantageous over structured grids (quadrangles, cubes) when it comes to modeling an irregular geometry.
After contrasting these requirements against advantages and disadvantages of several
state-of-the art methods, an electric field integral equation (EFIE)-based method-ofmoments (MoM) technique as implemented in the commercial field solver FEKO was
chosen. The MoM uses unstructured discretization elements and works in the frequencydomain. The basic concept of integral equations was outlined and the MoM solution
methodology introduced. Basis functions for wires and metallic surfaces used for the approximation of the currents in the RC simulations were described. It was explained how
the boundary conditions are satisfied in an average sense along a discretized structure
56
3 NUMERICAL PROCEDURE
through the usage of point matching combined with weighting functions. Computational
requirements regarding simulation time and memory were estimated; it was shown that
the MoM is computationally very expensive for high frequencies, often to the point of
being prohibitive – the simulation of the prototype RC at a frequency of e.g. 1 GHz
required more than 12 GByte of main memory and significant time expenditure. The
application of several types of symmetries was discussed and an outlook on MoM extensions and solver techniques (such as the NGF, PO and MLFMM) to reduce computation
time and memory was given. It was concluded that due to the geometric structure of
the RC prototype built for this thesis, symmetries and the PO could not be used. The
application of the MLFMM was found not to be appropriate as the interaction between
triangles (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 been
successfully used for a computationally efficient simulation of RCs by other authors, but
was 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 referenced 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 “distribute 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 Apparatus”. It already described some of the problems that are still among the main issues
addressed in today’s RC research and development:
• “. . . apparatus for producing a substantially uniform integrated radio-frequency
heat pattern. . . ” [71] (RC problem: field uniformity)
• “. . . cooking large volumes of food with the expenditure of a minimum amount
of input power . . . ” [71] (RC problem: large volume of uniform field, high field
strength with modest input power)
• “. . . means for producing periodic changes in the field distribution in a radiofrequency cavity, whereby the integrated heating effect of the field is made substantially uniform [71]” (RC problem: need for a device which is able to change
the field distribution efficiently and to provide a volume of uniform field)
• “. . . in which food masses, whose linear dimensions are large compared to the wavelength of the microwave energy used, may be cooked in a substantially uniform
manner [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 heating” (i.e. microwave cooking) were published – using such a device to test for EMC
simply 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 Standards (NBS), Boulder (CO), U.S. (now called National Institute of Standards and Technology (NIST)) which covers “Electromagnetic Field Measurements in Low Q Enclosures” [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 for
measurement of radiated power in the microwave range” [73].
John and Hall are among the first to describe a practical RC testing setup in their paper
dealing with “Electromagnetic susceptibility measurements using a mode-stirred chamber” from 1978 [74]. This paper is followed in 1980 by another publication of Corona et
al. dealing with the “Performance and analysis of a reverberating enclosure with variable
geometry” [75].
A historical overview on some parts of the above mentioned RC research is given in a
paper published in 2002 by Corona and Ladbury on the occasion of M. Kanda’s obituary [76].
4.2 Reverberation chamber standards
At the start of this thesis in December 2000, the general RC standard IEC 61000-421 was still in the stage of “Committee Draft for Vote (CDV)” [6]. The CDV stage
implies that both technical and editorial revisions can still be made to the preliminary
standard document as requested by organizations and countries participating in the
standardization process. In January 2001, the IEC 61000-4-21 passed the last objections
from the International Electrotechnical Commission (IEC) member states and entered
the “Final Draft for International Standard (FDIS)” phase. In the FDIS stage, final
editorial changes to the standard are done – changes regarding the technical content can
only be performed in a major revision at a later point of time once the standard was
published. After few editorial changes and voting the final standard was published in
August 2003 and is available since then from the IEC [6].
There is another IEC standard dealing with RCs available already since 1999, which
describes shielding efficiency measurements of components such as cable assemblies or
connectors: the IEC 61726 standard [77]. Contrary to the IEC 61000-4-21 mentioned
above, the IEC 61726 standard features a very detailed proposal of a suitable stirrer
design, almost in a “cookbook” style. Surprisingly, the IEC 61726 standard is rarely
cited 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 technical
reports published by Crawford, Koepke [9, 78], and Hill [8, 79, 80, 81, 20] of NIST
as well as original work by Hatfield [82, 83, 5], Freyer [84, 85], Ladbury [37], Lundén,
Bäckström [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 address
some of the questions outlined in Chapter 1. Because of the difficulties in the RC
simulations, 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 the
complexity of the stirrer design. The following Sections 4.3.1-4.3.3 summarize published
RC simulations performed in the time-domain, frequency-domain, and with statistical
models. Table 4.1 on page 61 provides a short overview and abstracts important key
facts on these simulation approaches: listed are the main author, whether the simulation
was carried out in 2-D or 3-D, which numerical method was used, and if it was a timeor a frequency-domain simulation. Furthermore, Table 4.1 shows at a glance whether a
commercial 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-domain
techniques, viz. the finite-difference time-domain (FDTD) method and the transmissionline-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 used
for the simulation, i.e. staircasing, errors introduced by the Fourier transform, difficulties
with modeling of losses, etc. In [86] also measurements are presented, but unfortunately
none of them measures quantities that were simulated. The simulation itself is exclusively benchmarked against a statistical analysis of computed results. Excitation of the
EM field is accomplished with an infinitely small ideal dipole. It should be noted that
most of the numerical problems described in [86] are already solved in commercially
available field solvers, such as e.g. CST’s FIT-based Microwave Studio (using its “perfect 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 of
employing a numerical method which is well-suited to the irregular stirrer design, the
stirrer 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 also
pursued by other authors employing FDTD-based simulation codes: Using two planar
stirrers that were well adapted to the rectangular grid of non-conformal FDTD, Harima
and Yamanaka [87, 88] investigated the impact of the numerical reflection coefficient
of the wall surface on the field distribution. Bai et al. [89, 90] studied the influence
of different stirrer mounting positions inside an RC on the field uniformity. A similar
analysis 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 conjunction
with structural staircasing or modeling of losses. A 2-D FDTD simulation of an RC
investigating the loading effect of different periodic “rat-like” EUT-configurations was
presented 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 was
simulated as a rectangular “moving-wall” cavity, with two one-dimensionally, in a staggered manner oscillating walls. In [94] no realistic VIRC could be simulated as the FDTD
60
4 LITERATURE OVERVIEW
code of the authors was not capable of modeling slanted, irregularly shaped walls. This
somewhat questionable procedure is only seconded by making the oscillations of the walls
a priori in a random fashion and then concluding a posteriori that the simulations work
well, because the obtained field structure is also random. In other words: Kouveliotis et
al. use randomly changing boundary conditions and validate their simulation by finding
randomly changing EM fields – the simulations were not verified by measurements at
all, only a statistical analysis is presented. Also other authors used the “moving-wall”
approach to simulate standard RCs (non-VIRC) [97].
Ritter and Rothenhäusler presented in their paper the FDTD simulation of a mediumsized RC (9.4 m · 6.5 m · 5.3 m) [98]. Their intention was to extend this simulation to a
very large RC measuring 40 m · 20 m · 30 m suitable for full-size aircraft testing. Excitation of the RC was carried out using the impressed field of a logper antenna obtained
from earlier simulations of this antenna through a far-to-near-field transformation. This
RC simulation was not benchmarked against measurements, but only against statistical
results. In [98], the performance of several tuners was compared, the results however
exhibited 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 very
difficult for high-Q devices such as RCs [99, 43]. In his simulation, he needed to lower
the 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 air
volume within the RC to improve the convergence speed. Also Petit’s FDTD simulations make use of somewhat inappropriate conductivities on the order of κ = 100 S/m
for metallic surfaces [86].
A group at EADS Airbus used the commercial codes ASERIS-FD (FDTD) and ASERISBE (BEM) to model and simulate a 3.7 m · 5 m · 2.5 m RC [100, 101]. This RC is currently
being used for immunity and emission testing of avionic equipment. They employed the
BEM code to run an RC simulation for the computation of the near field, while their
FDTD code is used for the estimation of the chamber quality factor Q. While this
approach seems to be promising, the simulation results were not validated by measurements, but instead only against statistical data. The RC-related problem of achieving
sufficient statistical field uniformity within a predefined volume has been addressed also
in 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 attempt to optimize the stirrer inside the RC they concluded that this was computationally
not feasible (simulation runtime typically 4 days for one stirrer revolution) and therefore
proposed 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 the
change in the Poynting vector for a certain number of spherically distributed sample
points as the stirrer rotates. Although the genetic algorithm approach was not very
convincing, the general idea of optimizing a stirrer without the RC seems promising (cf.
Section 7.7.5).
Wu et al. [113]
Bunting et al. [114]–[117]
Harima et al. [87, 88]
Bai et al. [89, 90]
Petirsch et al. [23]
Zhang et al. [91]
2-D/3-D a
TD/FD b
Method c
H/C d
Excitation
Validation e
Year f
2-D
TD
TLM
H
HD g
None
‘89
H
HD
g
HD
g
None
‘98/‘99
HD
g
None
‘99
HD
g
None
‘99
HD
g
None
HD
g
2-D
3-D
3-D
3-D
3-D
FD
TD
TD
TD
TD
FEM
FDTD
FDTD
TLM
FDTD
H
H
H
C
Statistics
h
‘98/‘99/‘02
‘00
Hoijer et al. [118]
3-D
TD
FDTD
C
Hoëppe et al. [101]
3-D
TD
FDTD
C
HD g /Horn
Statistics h
‘01
Hoëppe 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
C
HD
g
HD
g
Coates et al. [106]
Petit et al. [86]
Laermans et al. [119]
Åsander et al. [120]
Kouveliotis et al. [94]
Moglie et al. [121, 99, 43]
Ritter et al. [98]
Lammers et al. [92, 93]
Weinzierl et al. [111]
a
e
3-D
3-D
2-D
3-D
3-D
3-D
3-D
2-D
3-D
TD
TD
FD
FD
TD
TD
TD
TD
TD
TLM
FDTD
MoM
BEM
FDTD
FDTD
FDTD
FDTD
TLM
H
H
H
H
H
C
H
H
Dipole
HD
g
HD
g
HD
g
logper
HD
g
HD
g
Resonances
i
‘00
Statistics
i
‘02
Statistics
h
‘02
Statistics
h
‘02
None
Statistics
‘02
h
None
‘03
‘03/‘04
Statistics
h
‘03
Statistics
h
‘04
None
4.3 PREVIOUS REVERBERATION CHAMBER SIMULATIONS
Authors
‘04
2-dimensional / 3-dimensional simulation b time-domain / frequency-domain c numerical method d “home-made” / commercial solver
validation of the simulated results f year published g Hertzian dipole h simulation vs. theoretical data i simulation vs. measurements
61
Table 4.1: Summary of previously published RC simulations.
62
4 LITERATURE OVERVIEW
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 software package) to simulate the effects of different vane heights of a simple stirrer on the
SR (cf. section 2.2.3). Their modeled RC is 5.0 m · 2.9 m · 2.4 m large. Both publications [106, 107] presented simulations and measurements and compared them against
each other, but only on the basis of the SR. As outlined in [108], a validation of RC
simulations using “highly processed” data such as the SR is not possible, as completely
different EM fields can have identical SR values [108]. Therefore it remains unclear
whether the aforementioned TLM simulations actually model the EM fields inside the
RC correctly.
Petirsch et al. [109] investigated the effect of diffusors placed inside an RC on the field
homogeneity using their own TLM code. Most of the results shown in [109] remain
questionable, since the utilized performance measures were based exclusively on nonvalidated simulation data. As noted correctly in a comment to [109], Arnaut criticizes
the results as somehow “random”, since diffusors were used completely beyond their
originally intended operation conditions [110]. The improvement in field homogeneity
reported in [109] is most probably merely due to increased loading of the chamber by
absorbing material of the diffusors.
Weinzierl et al. [111] carried out a TLM-based simulation to find out whether an idea
proposed 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 limited
in their operation to frequencies f > fLUF , i.e. a large number of modes must be above
cutoff in order to achieve sufficient field uniformity within the chamber. The propagation of quasi-TEM waves on a two-wire line within an RC was intended to remove this
low-frequency-limitation by exciting EM fields below the fundamental RC mode [112].
Unfortunately, [111] only shows that there are quasi-TEM waves excited between the
two wires – whether they serve as an appropriate RC excitation in providing sufficiently
uniform fields remains unclear. It is evident however, that below the cutoff frequency of
the fundamental RC mode, the field distribution and magnitude within an RC does not
have 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 Laermans
and De Zutter in [119]. In their work, an ideal line current source was used as excitation
and boundaries were modeled as PEC. Whether these results are of any practical use
is questionable, since the three-dimensional nature of EM fields in an RC was not taken
into 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 using
the finite-element method (FEM). His results could not be validated by measurements
of 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 Hertzian
dipole. The paper mentioned is based on earlier publications [116, 115, 117] and relies as
many other papers heavily on the validation by comparison against statistical or “highly
processed” data (cf. Section 7.4.3).
Boundary-Element Method (BEM)
Åsander 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 generated and analyzed than is possible if measurements are used instead” and “that surface
discretization has some advantages over methods relying on volume discretization when
it comes to model irregularly shaped, rotating stirrers”. A validation of the simulation
using measurements was not performed.
Using the commercial frequency-domain BEM simulation tool ASERIS-BE, Hoëppe et
al. [101, 100] investigated the RC working volume of statistical field uniformity as well
as the effect of chamber loading. ASERIS-BE features a technique similar to the NGF
technique, so that from one stirrer step to another only small parts of a large system of
equations 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 developed
to avoid the long computation runtimes and large memory requirements associated with
time- and frequency-domain simulation methods mentioned above. Most of them are
based on the assumption of local plane waves in the RC and try to facilitate the understanding 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 cavities can be found in [33], out of which some parts were condensed into a journal paper [125]. Furthermore, most books dealing with mobile communications serve as an
excellent source for a better understanding of the statistics observed in RCs, since the
statistical properties of the EM field within an RC have a lot in common with a mixed
direct-path/multi-path environment [126, 127, 32]. Vice versa there is also a number
of publications dealing with RCs which make use of the statistical analogies to mobile
communications 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 mechanical 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. The
more important aspect, however, is that the stirrer (or tuner) represents a somehow disturbing mechanical element in this otherwise purely electrical testing environment. The
feed-through bearing for the stirrer axle is difficult to shield properly, care must be taken
not to move the stirrer manually since otherwise the precision gear may be damaged,
64
4 LITERATURE OVERVIEW
and rotating the stirrer from one position to another is time-consuming (acceleration,
deceleration, decay of stirrer oscillations). The following sections summarize the most
important concepts presented in the open literature on how to eliminate the mechanical
stirrer.
4.4.1 Moving walls
One possibility to remove the mechanical stirrer inside the RC is to change the field
distribution by actually moving the walls of the RC. A method that uses this approach
was proposed by Leferink et al. [95]. The “VIRC” is essentially a “tent” which is made
out of electromagnetically conducting cloth. Small motors with eccentric drive fixtures
wobble the cloth sufficiently so that the field is changed and a certain field uniformity is
achieved within the “tent” [96]. This approach does not get rid of mechanical problems
completely, but it removes the stirrer from the RC’s interior. It was successfully applied
to 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 through
electromagnetically conducting cloth.
4.4.2 Electronic stirring
Methods of field stirring without any mechanical stirring devices within (or around) an
RC are commonly referred to as “electronic stirring”. One limiting factor of electronic
stirring is that until now it is only applicable – if at all – to immunity testing of EUTs
in RCs. The major problem arising for these methods in emission measurements is
that one has no control over the emitted spectrum of a test object. This renders it
very difficult to ensure proper interaction between the electronic stirring equipment
and the EUT. Furthermore, since changes of the EM field distribution occur virtually
instantly (compared to the very slow changes using mechanical stirrers), the response
time of an EUT in immunity testing must be known [31]. In addition, the advantage of
eliminating (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 be
used.
Frequency and Gaussian noise stirring
A 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 the
two-dimensional case in [129], this technique had been applied one year earlier already
in “real-world”, three-dimensional RCs [130]. The underlying principle of “frequency
stirring” is to acknowledge that the change of the resonance frequencies of the cavity
modes by a rotating mechanical stirrer has some similarity to the frequency modulation
of the source [113].
Instead of changing the frequency “monochromatically” by standard frequency modulation, another proposed method uses additive white Gaussian noise (AWGN) which is
mixed onto a periodically changing center frequency [131]. This approach claims that
the 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 stirring
is investigated in [133]. This procedure leads to a larger number of independent samples
and hence lower measurement uncertainties for tests in the RC.
Multiple sources and phase stirring
Instead of frequency stirring, the usage of multiple source antennas is advantageous for
single-frequency high-power excitation of RCs, because they eliminate the need for combining high-power signals through external RF components. In [129] it was investigated
whether multiple sources alone (without any stirring) would lead to sufficient field uniformity: [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 even
if the sources were incoherent or varied in phase (so called “phase stirring”) [134].
Three-dimensional TEM cell
One 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 to
their physical dimensions. An alternative to RCs at lower frequencies could be possibly
three-dimensional TEM cells. A 3-D TEM cell is a combination of three individual TEM
cells in one, in such a way that the TEM coupling planes created by each plate are not
parallel to each other in the center of the test volume. A particular case is when the
coupling planes are orthogonally arranged two-by-two: Each plate creates an electric
and magnetic coupling which defines its TEM coupling plane [135]. Problems with 3-D
TEM cells are mainly due to the field distribution and gradients in the central region of
the structure, strongly limiting the test volume to a much smaller volume as compared
to RCs. The operating range of 3-D TEM cells is limited to frequencies below the first
fundamental resonance of the metallic cavity-like structure they are built into [136]. If
a 3-D TEM cell and an RC are combined, one is left with an EMC test device which
can be operated in the low frequency and the high frequency region – using completely
different operating principles –, but which fails to function in the intermediate region
between the first fundamental RC resonance and the LUF of the RC.
Two-wire TEM excitation
In a paper by Perini and Cohen published in 2000, it is proposed to use several wires
inside an RC to excite quasi-TEM modes [137]. As shown in Section 2.2, RCs are limited
in their operation to frequencies f > fLUF , i.e. a large number of modes must be above
cutoff in order to achieve sufficient field uniformity within the chamber. The propagation
of quasi-TEM waves on a two-wire line within an RC was intended to remove this
low-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 (which
is undoubtedly true) and therefore a bundle of correctly placed wires within an RC will
extend the operating range down to f = 0 Hz. Unfortunately, [137] (and later [111]) only
show that there are quasi-TEM waves excited between the two wires – whether they
serve as an appropriate RC excitation in providing sufficiently uniform and isotropic
fields remains unclear. Since until now nobody was able to exploit the proposed effects
in practice, the two-wire TEM excitation is discussed very controversially among RC
experts [112].
4 LITERATURE OVERVIEW
66
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 not
radiate like a dipole or quadrupole with known broadside direction)
• the EUT is large compared to the wavelength (such as e.g. a personal computer
operating 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, automotive, or avionic equipment
These requirements are difficult to meet with established, widely accepted EMC test environments such as ACs, OATS, or Gigahertz transverse electromagnetic (GTEM) cells
using the current testing methods as laid out in the respective standards. The following sections summarize important applications of RC testing for EUTs meeting these
requirements. A qualitative (and by no means fully exhaustive) comparison highlighting
the 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]. The
paper describes the implementation and measurement of an RC suitable for susceptibility
testing of automotive electronics in the 200 . . . 1000 MHz range. The original problem
was that in this frequency range, required power levels have entailed prohibitive testing
costs if the test is carried out within an AC or a similar EMC testing environment. It is
shown that the RC test method can be substituted for the AC method thereby effectively
solving this problem [141].
For the past several years there has been an increasing interest in the possibility of
testing large items such as aircrafts in an RC. As the use of electronic systems to
perform critical flight functions steadily increases, the application of RCs to testing of
aircraft and avionics systems is discussed by Hatfield et. al. in [142]: data is presented on
the statistical characteristics of the EM environments in aircraft cavities and compared
with 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 related
concerns is the scalability of the operational characteristics of RCs. For this reason, the
Naval Surface Warfare Center Dahlgren Division (NSWCDD) maintains a database (DB)
on the performance of operating RCs worldwide [83]. This data consists mainly of SR
data at 1 GHz. The volumes of chambers in the DB range from less than 1 m3 to
about 200 m3 . It was found that the characteristics are scalable over this two orders of
magnitude variation in volume. To test a reasonably large aircraft, a chamber would
4.5 PRACTICAL REVERBERATION CHAMBER APPLICATIONS
Anechoic chamber
(AC)
67
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
H
phase relation
E,
Fixed
Not fixed
Direction of incidence
Known, fixed
All directions, “isotropic”
Field impedance
377 Ω
Unknown
EUT radiation pattern
Assumptions:
- “well-behaved”
- “dipole-like”
No assumptions made
Emission testing
- extensive scanning
needed to get peak
- one direction at a time
- “integral approach”
- omnidirectional testing
Immunity testing
- uncertainty about
EUT 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 extracted
from [139]).
have to have a volume of 5 · 104 m3 and greater [143]. Measurements of the RF energy
coupled to instrumented avionics boxes from a large transport aircraft and a simulated
avionics box were presented in [144]. These measurements were made at NSWCDD
when the avionics bay, cockpit, and passenger cabin were internally excited with swept
68
4 LITERATURE OVERVIEW
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 aircraft
and simulated avionics boxes measured in an RC constitute valid descriptions of the
same boxes when installed and operating in an aircraft.
Extensive measurements of radiated emissions in RCs are adressed in a Ph. D. thesis by
Kürner [22]. In addition, the SE of various EUTs is analyzed, general recommendations
for 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 antennas
were measured once in an RC and once in an AC [22, 23]. Measurements in an RC to
compute the radiation efficiency and return loss of an electrically small antenna were
carried out by Carlsson et. al. in [145]. Madsén et. al. investigated whether RCs are
suitable for mobile phone antenna tests [146]. They conclude that an RC could be
used for production line testing to check periodically mobile phones for compliance with
specific absorption rate requirements.
Since antenna diversity in mobile terminals is likely to be commonly used in future mobile
communication systems, methods for characterization of antenna diversity performance
are therefore of interest. A new characterization method using an RC, with several
advantages over existing techniques, was proposed in [147, 148]. It was found that
when performing diversity measurements in an RC, the correlation between the signals
received by the diversity antenna elements was similar to the correlation in a real multipath environment. Hence an RC could be used to effectively model the propagation
characteristics commonly encountered in mobile communication systems.
4.6 Conclusion
An extensive overview on the literature treating the early development of reverberation
chambers (RCs), RC standards, and the simulation of RCs was given. On the order
of 35 papers were published in the past dealing with RC simulations, prepared by approximately 20 different research groups. The earliest published RC simulation dates
back to 1989. Most groups used a self-made numerical code, with FDTD being clearly
the preferred method (probably because this technique is rather easy to understand
and comparatively simple to implement – the latter statement is certainly only true for
very basic, non-conformal, non-subgridding FDTD). With the notable exception of one
group, all published papers employed a Hertzian dipole as excitation. Whereas earlier
simulations were still carried out in two dimensions, more recent RC analyses made use
of three-dimensional simulation tools. It is important to note that until today all but
one publication did not use appropriate means of validation. Half of the groups chose
not to validate their simulated results at all, the remainder performed only a statistical
benchmark (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 vibrating intrinsic reverberation chamber (VIRC), which consists of conductive fabric forming
a 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 comparison between the RC and anechoic chamber (AC) EMC test environment was presented
highlighting advantages and disadvantages of each methodology.
70
5 Prototype and Measurement System Development
Abstract — This chapter describes the construction and setup of the reverberation chamber prototype including walls, door, stirrers, and auxiliary equipment. Details of the measurement system
developed for data acquisition are outlined. Measurement errors originating from field probes, antennas, 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 as
a 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 surface
amounts to approximately 50 m2 .
5.1.1 Walls and door
The chamber walls consist of a sandwich-type construction made of chip board wood
between two galvanized sheet steel layers. The wood provides for sufficient mechanical
stability of the RC construction. The steel layers are made of several separate panels of
overlapping flat stock, which are connected by I-profiles. The main material components
of commercially available galvanized steel sheets are typically iron (97. . . 99 wt.-%), zinc
coating (0.5. . . 2 wt.-%), copper (0.4 wt.-%), and manganese (0.4 wt.-%) [149]. These
material properties are important and will be needed in Chapter 6 to accurately model
the RC for a simulation. Within the RC up to three stirrers can be mounted and operated
simultaneously on different axes. A schematic overview of the basic RC structure is
shown 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 also
two different coordinate systems are indicated, 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 the other is given by
w
(5.1)
xm = x −
2
l
−y
(5.2)
ym =
2
h
(5.3)
zm = z −
2
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
2.86 m
Dx
DzI
Dz
8m
2.4
Dy
II
3.06 m
z
DxI
I
x
III
y
Door
zm
ym
xm
Figure 5.1: RC geometry and dimensions with three different stirrer axes (I, II, III) and chamber
door. 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 a
0.07 m recess of the door frame from the chamber wall. The doorstep is elevated 0.07 m
from 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 that
they affect the chamber fields significantly and thus cannot be neglected. Details on
modeling 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 the
stirrers built for the RC prototype. Furthermore, details on the mechanical and electronic
part of the stirrer drive and controller are given.
Stirrer design
For the analysis of the RC prototype shown in Fig. 5.2a), initially one vertically mounted
stirrer 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 between
each paddle and stirrer axis is 45◦ . The horizontal spacing between the paddle centers
is approx. 0.45 m, the distance from the lowest and the topmost paddle edge is 0.15 m
to the RC floor and the ceiling, respectively. The stirrer was built according to the
generally accepted design principles available at the time of the thesis’ start, i.e.: the
stirrer should be electrically large at the LUF (which is around f = 300 MHz as shown in
Section 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’s
axle with anti-flexing fixture mechanism; c) stirrer motor drive unit with copper-shielding
around mounting aperture and d) pyramidal absorber placed on top.
distribution within an RC sufficiently. The asymmetry-criterion provides for dissimilar
field distributions, i.e. field distributions which are statistically weakly correlated. Among
the stirrers developed in the past exhibiting a good performance were the so-called “Zfold” 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
74
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
Science and Technology Organisation (DSTO), Australia [7]. These stirrer designs follow
the 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 to
the requirement for stirrers being electrically large: What does electrically large mean for
a rotating object? In an ordinary sense, one would refer to an object being electrically
large if it extends in one or more dimensions to a length greater than a significant
fraction of the wavelength (such as d > λ/2). The stirrer used in this thesis has e.g. a
total geometrical height (distance from upper edge of the topmost paddle to lower edge of
lowest paddle) of 2.76 m – its electrical height is however lower, as the individual stirrer
paddles are not connected with each other. Whether the performance of this stirrer can
be enhanced by manufacturing it with all paddles being connected (in order to increase
its electrical size in the traditional sense), is discussed in detail in Section 7.7. Taking
into considerations published results, it can be stated already that both stirrer types
made out of one single piece (e.g. “Z-fold” stirrer [150]) as well as stirrers consisting of
several separate parts (e.g. “Rot-Z” stirrer [151, 109]) exhibit good performance.
Stirrer drive and controller
The stirrer paddles are mounted on a plastic rod with a diameter of 50 mm using plastic
spacers cut at an angle of 45◦ ; these spacers allow the paddles to be rotated around the
rod’s axis so that all paddles can be aligned at any arbitrary angle with respect to each
other. In order to check proper alignment of the paddles on the rod, a rotary angular
scale is mounted on each paddle’s spacer. The rod itself is attached on the upper end to
the stirrer motor gearbox and on the lower end supported by a ball bearing on the chamber 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 the
settling time between different angular steps for the stirrer paddles considerably, which
allows faster measurement cycles [152].
The stirrer drive depicted in Fig. 5.2c) is an electrical 24 V DC, 20 W brush servo motor equipped with a planetary gearhead and an electronic encoder. The gearhead has
a gear reduction ratio of 128 : 1, the encoder is used to provide feedback on the actual angular stirrer position to the motor controller [153]. A pulse-width-modulation
technique is used to control the motor drive. The motor controller is a fully digital
position, speed, and current control unit, which can receive high-level motion commands
and provide continuously updated status feedback to the measurement system [154] (see
Section 5.2). Parameters for acceleration and deceleration ramps are stored in the controller’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 stirrer
speed 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, which
directly affects comparisons of measurements with simulations (see Section 7.7). The
theoretically achievable angular resolution in the RC prototype can be calculated as
follows:
• 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 output
revolution (1:1)
5.1 REVERBERATION CHAMBER PROTOTYPE
75
• with the above-mentioned reduction ratio, one revolution on the gearhead load-side
requires 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 is
converted to four so-called “quadcounts” (4:1) in the controller [153]
Since one quadcount is the smallest detectable movement, the theoretically achievable
angular resolution amounts to 256 000:1 or ∆ϕ ≈ 0.0014◦. The practically achievable
angular resolution is mainly limited by the quality of the planetary gearhead, i.e. essentially 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◦. Considering the strong dependence between rotational stirrer position ϕ and EM field within
the RC, this needs to be taken into account when evaluating simulation vs. measurement
benchmarks.
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, two
ceiling-mounted lights, a power line filter, and cable ducts. The feed-through panel shown
in Fig. 5.3a) is used to connect the measurement equipment outside of the RC to the
TX/RX antennas placed within the chamber. This panel is equipped with two Type N
feed-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) – correspond
to the three axes I, II, and III depicted in Fig. 5.1 and allow for an installation of the
stirrer at any of these positions. The length of the stirrers was designed so that they can
be mounted on all axes using appropriate rods without changing the mechanical paddle
configurations.
Fig. 5.3c) shows the power line filter, which is used to transfer AC power to any electrical
equipment operated within the RC. This filter prevents transmission of RF energy coupled to power lines running within the RC to the outside environment. The honeycomb
ventilation duct depicted in Fig. 5.3d) acts as an array of small waveguides operated below cutoff, providing high RF attenuation and at the same time means for air exchange
between chamber interior and exterior.
In order to achieve high field strengths within an RC and to prevent electromagnetic
interference (EMI) with devices outside of the RC, it is necessary that the chamber
structure guarantees sufficient SE for the frequency range of operation. The overall SE
was measured with the two-antenna-method, i.e. one antenna is placed inside the empty
RC and a second antenna is used outside the chamber. With the second antenna, the
walls of the RC are scanned, such that the worst-case SE is found. Using this measurement method, all the apertures depicted in Fig. 5.3 exhibited an SE well above 100 dB
within 50 MHz. . . 4.2 GHz. Initially, the stirrer drive mounting aperture had some EM
leakage; as shown in Fig. 5.2c) and d), this problem was solved with a copper tape bypass
76
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
a)
b)
c)
d)
Figure 5.3: Apertures in the RC: a) feed-through panel with coaxial Type N connectors and
circular waveguide for optical fiber (black bundle); b) stirrer mounting plate; c) external
power line filter; d) honeycomb duct for ventilation and interior metal fixtures.
between chamber wall and mounting plate in addition to the placement of a pyramidal
absorber on top of the stirrer drive. After retrofitting the RC with special corner elements (the original ones exhibited poor RF shielding), the overall chamber is capable of
providing 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 acquisition
computer (left); RF amplifiers, signal generator, power meter, power supplies, and stirrer
motor controller (right).
5.2 Measurement system
The RC measurement system is used to acquire information on the electric field within
the chamber as well as the power being transmitted and received by the TX/RX antennas. One part of the system consists of the equipment used to generate and measure the
EM field in the RC (Section 5.2.1), the other part is needed to control this equipment as
well as to record and process the gathered data (Section 5.2.3). As in any measurement
system, it is important to keep the influence of the measurement sensors on the quantities to be measured as small as possible (Section 5.3). Fig. 5.4 gives an overview of
the RC measurement system setup: the control and data acquisition computer is shown
on the left side, the measurement equipment consisting of RF amplifiers, signal generator, power meter, power supplies for the amplifiers and the stirrer drive, and stirrer
motor controller on the right. Two effects (which are not limited to RCs only) must be
particularly taken into consideration when measurements are performed:
• by its physically finite size, the measurement equipment performs spatial averaging
of the EM field – i.e. the field is not measured at a single, infinitely small point in
space, but rather over the geometric extension of the field sensor. The nonzero field
sensor size may mask an actually poor reverberation performance [28, 27]. This
effect becomes very pronounced if conventional, spatially large RX antennas are
used 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 stirrer
continuously rotates), the EM field changes very rapidly over time. As today’s
field probe systems have a rather slow response time and are therefore not capable
of 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].
78
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
Both spatial- and time-averaging result in deviations of the measured vs. the actual EM
field and make the gathered measurement data appear smoother. In the mode-tuned
operation of an RC (the stirrer is stepped from one angle to the next with comparatively
long pauses in-between), the time-averaging effect does not occur since the field is only
measured once steady-state is reached. For this reason, the RC analysis presented in
this 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 the
actually 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 electric near field as well as the forward and reverse power at the TX and RX antennas. As
outlined in Section 5.2, these antennas were not used for direct near field measurements
as they provide insufficient spatial field resolution. Furthermore it is almost impossible 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 a
highly repeatable procedure and without changing the field distribution). The following
equipment was used for EM field generation and antenna-based measurements:
Signal generator
Marconi (now IFR Test Systems) generator, type 2024, output level adjustable between
−137 dBm. . . +13 dBm, frequency range f = 100 kHz. . . 2 GHz
Amplifiers
Three broadband power amplifiers with a maximum input level of 10 dBm:
• Mini Circuits LZY-1, frequency range f = 10 MHz. . . 512 MHz, minimum gain
39 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 separate power supply (28 V, 10 A) and an external input/output stage protection against
excessively large input signals or highly reflective loads.
Directional couplers
The (bi-)directional couplers are used to measure the forward and reflected power at
the TX/RX antenna terminals. Two different couplers were employed to cover the wide
operating frequency range:
• Werlatone C3946, 40 dB attenuation, f = 100 kHz. . . 1 GHz
• Hewlett-Packard 788B, 20 dB attenuation, f = 100 MHz. . . 3 GHz
5.2 MEASUREMENT SYSTEM
79
7
16
13
6
Reverberation Chamber
5
1
2
3
11
15
4
14
10
8
Data signal path
RF signal path
9
11
17
12
2
17
3
7 9
1
3
6
4
8
5
Figure 5.5: RC measurement setup and photo. 1: signal generator; 2, 4: attenuators; 3: RF
amplifier; 5: bi-directional coupler; 6-9: power meter with probe heads; 10: electric field
probe 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 meter
In order to quantify the forward and reflected power provided by the bidirectional couplers, a Rohde & Schwarz NRVD dual-channel power meter is used [158]. The two
channels are completely independent from each other and can measure power simultaneously. In combination with a bidirectional coupler, the net power delivered to the RC
80
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
can be calculated from these measurements. The specified frequency range of the NRVD
unit is f = 0 Hz. . . 26.5 GHz, the actual operating frequency range is however defined by
the external probe heads. To measure the power, two Rohde & Schwarz insertion units
URV5-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 with
settling times on the order of 10 s) shunted on the output side with a 50 Ω resistor. Due
to their minimum insertion loss, these units leave the 50 Ω-line connected to the bidirectional coupler virtually unaffected. The NRVD power meter is fed with the current
operating frequency by the measurement system, so that the probe heads can provide
calibration data feedback to the NRVD using a correction factor lookup table.
TX/RX antennas
To cover the broad operational frequency range of the RC, several different antenna
types were used:
• One precision conical dipole antenna, manufactured by Austrian Research Center
Seibersdorf, 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 without
stirrers to validate the simulations in the lower frequency range, the other antennas
were utilized for all further TX/RX measurements in the RC. As shown in Section 5.3
and Fig. 5.10, the antennas were specially mounted to reduce the influence on the field
distribution within the RC.
Spectrum analyzer
For scattering parameter measurements and EUT emission tests, a Rohde & Schwarz
ESI 40 spectrum analyzer (f = 20 Hz. . . 40 GHz) was used.
Attenuators, cables, and adaptors
A 10 dB attenuator was inserted between the output of the signal generator and the
amplifier’s input to protect the input stage of the amplifier. Against excessively reflective
loads connected to the amplifier’s output, a 3 dB attenuator was used between the output
and the directional coupler’s input to protect the amplifier’s output stage. All coaxial
cables 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, a
field probe is not necessarily needed, but still recommended [6]. Since the calibration of
5.2 MEASUREMENT SYSTEM
81
Figure 5.6: Field probe (sensor head and processing unit) with non-conductive stand, suspended
TX/RX antenna, and measurement grid on the chamber floor.
an RC requires recording of the electric field in eight or respectively nine points (these
points form the so-called “volume of uniform field”), it is highly advantageous to employ
a 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 timeconsuming, 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 newer
systems also optically [160]), the probe’s processing unit (the “electronics box”) which
reads out and converts this data, and the probe’s mechanical support (this can be a simple
stand, 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 Solutions
Systems, 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 capable of providing simultaneously three-component electric field data [161]. Each of the
measured electric field components is rectified inside the sensor head using diodes and
82
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
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 the
probe system [155]. The distance between the sensor head and the processing unit is
kept constant at 30 cm (see Fig. 5.6), which allows a repeatable calibration of the whole
system. The Type-8 probe head can be operated between f = 100 kHz and 3 GHz.
Field probe processing unit
The rectified voltages of the three short dipoles are transferred to the field probe’s main
processing unit EMR-20 [156] (the box with the six buttons in Fig. 5.6). The EMR-20
performs all data processing which includes auto-zeroing (needed to correct for offset
voltages and temperature drift of the impedance transformers and the analog-to-digital
converters), sending/receiving of data to the equipment control PC, and power management. The dynamic range of the sensor head in combination with the processing unit is
greater than 60 dB, allowing to detect field strengths ranging from 0.6 . . . 800 V/m without range-switching. This is accomplished using precision analog-to-digital converters
with 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 data
is transmitted to the measurement PC with a bidirectional serial RS232 optical fiber
connection.
Optical fiber feed-through waveguide
In order to provide sufficient SE of the RC, a cylindrical waveguide is used to feed the
above-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 be
calculated for the TEmn mode from
fcmn =
J
√ mn
πd µ0 ε0 εr
(5.4)
wherein Jmn
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 propagation
constant for f < fcmn is given by
(
2 2
2
2πf
2πf
2π
2
2
2
βz = β0 − βmn =
− βmn = ±j
−
(5.5)
√ πd
co
co
εr J mn
The fundamental mode (i.e. the mode with the lowest cutoff frequency) of a circular
waveguide is TE11 which results in
(
2
2 2πf
2J βz = ±j
−
(5.6)
√ 11
εr d
co
Using J11
≈ 1.84, the attenuation of a circular waveguide of length l for the fundamental
TE11 mode is therefore equal to
(
(
2
2
2 2 2πf
2πf
2J11
2 · 1.84
α·l=
−
·l ≈
−
·l
(5.7)
√
√
εr d
co
εr d
co
5.2 MEASUREMENT SYSTEM
83
Geometry
Modeling
&
Simulation
Preprocessing
Field Solver
&
Graphical User
Interface
Data
Extraction
Simulation
Interface
Postprocessing
&
Measurement
Data Acquisition Measurement
System
Compliance®
Database
System
MS Access®
Statistics &
Benchmarks
MATLAB
Interface®
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 circular
waveguide with a diameter of d = 0.013 m was used, resulting in a cutoff frequency
for the fundamental TE11 mode of fc11 ≈ 13.5 GHz and a theoretical attenuation of
α · l ≥ 200 dB for frequencies f ≤ 6 GHz – this is considerably more attenuation than
the 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) are
remote controlled from a computer via the GPIB and RS232 bus. The RC control and
data acquisition programs were written using the Schaffner Compliance C3i software
suite [154] . Compliance provides the necessary equipment drivers enabling a high-level
control of measurement functions. The developed RC programs are specially adapted to
the requirements of measurements used for simulation validation: they allow to define a
rectangular test grid for the field probe system by setting spatial (x, y, z) start and stop
positions within the RC, measurement grid spacing, the frequency range of interest, and
the desired rotational stirrer angles.
The acquired measurement data is initially stored in the Compliance database (DB)
84
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
system. In order to perform extended measurement-vs.-simulation benchmarks and to
use advanced analysis tools, the procedure shown in Fig. 5.7 is needed to transfer the
data from Compliance : To facilitate data handling, measurement (and also simulation) results are gathered through a MATLAB -based data extraction interface from
Compliance (and respectively FEKO ) and fed directly into a common MS Access
DB. With this procedure, benchmarks, statistical analyses, and 2-D/3-D visualizations
of simulated and measured data are performed by retrieving the results from the DB
without the need to manipulate (e.g. scale, offset, etc.) underlying data – this in turn
significantly reduces the probability of accidentally introducing errors. Data transfers are
accomplished using SQL expressions from MATLAB via the ODBC application programming 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 the
simulation does not reproduce the physical phenomena inside the RC accurately or the
measured data is incorrect and additionally varying in repeated measurement trials. The
first issue is extensively discussed in Chapter 7, reasons for measurement errors are addressed in this section. It is essential to distinguish the term “error” (in a measurement
result) from the term “uncertainty”: error is the measurement result minus the true
value of the measurand [163, 164]. Whenever possible a correction equal and of opposite
sign 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 reported
result. Uncertainty can be characterized in terms of the spread of the probability distribution for the residual error; further references concerning measurement uncertainties
can be found in Section 5.4.
The sources of measurement errors can be attributed to the field probe system, the
TX/RX antennas, and the chamber prototype itself. A general problem for the determination of errors related to near-field measurements is the strongly frequency-dependent
impact on the results – if e.g. the field probe system is capable of a spatial resolution of
0.05 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 first
case measured field values are not going to change much as the probe is displaced by
0.05 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, quasiinherent, and non-inherent errors. Inherent errors are understood to be fundamental
errors that – with today’s available techniques – cannot be eliminated as they are coupled
e.g. to the physical measurement principle. Although one “has to live with” inherent
errors, it is important to be aware of these elementary limitations. Errors are referred
5.3 MEASUREMENT ERRORS
|E | [V/m]
60
0°
45°
85
90°
135°
180°
225°
270°
315°
50
40
30
20
10
0
50
100
150
200
Frequency f [MHz]
250
300
Figure 5.8: Isotropy of the electric field probe between f = 50 . . . 300 MHz for eight different
probe 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 equipment
being used. Differently designed equipment may not exhibit a particular effect to the
same 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 are
inaccuracies 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 field
distribution, 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 sufficiently
large 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 small
point in space. Practically this is not possible, instead the sensor “spatially averages” the EM field over its finite geometric extent. The nonzero field probe sensor
size may mask an actually poor reverberation performance [28]. This effect of
“spatial averaging” becomes extremely pronounced if antennas are used as sensors
instead of field probes to quantify the EM fields. Usage of antennas for near field
measurements is therefore strongly discouraged.
86
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
• offset between x-, y-, and z-dipole loops: theoretically all three field components
should be measured at the same location. As mentioned in Section 5.2.2, the
dipoles are mounted on a prism and therefore measure the field at slightly different
positions.
• mutual coupling between x-, y-, and z-dipole loops: ideally, all loops should measure the individual field components separately. In all practical field probes, there
is however mutual coupling by physical design constraints.
• response time/sampling rate of the field probe: an ideal field probe would measure
the 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 are
common [156, 167].
All the errors mentioned above cannot be eliminated and will be more or less pronounced
depending on the particular field probe system used. Compared with the quasi-inherent
and the non-inherent errors outlined below, however, they can be classified to a first
approximation as negligible in the EMR-20 field probe system. The probe’s sensor
Type-8 head exhibited a good isotropy, one exemplary result where the probe head is
rotated in 45◦ steps over 360◦ is shown in Fig. 5.8. The lower detectable electric field
magnitude of the field probe system is specified as 0.6 V/m, actually measured values
used for further analysis were always greater than 10 V/m. As in this thesis only the
mode-tuned RC operation is investigated with non-transient, steady-state fields, the
response 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 and
processing unit, probe stand) the EM field is disturbed. Several field probe manufacturers e.g. recommend to measure the field without placing the processing unit
nearby. Instead it is suggested to separate the probe’s sensor part from the electronic processing unit and to transfer the sensor data via a fiber optic link. The
processing unit is to be placed outside of the RC.
• general application of field probes in near field conditions with possibly strong field
gradients: several field probes are not suitable for use in near field conditions, they
may only be employed sufficiently far away from an excitation antenna, e.g. only
once plane wave propagation is predominant [168, 169].
The quasi-inherent errors can be a significant contribution to the overall measurement
error budget, especially if field probe systems are used beyond their intended range of
operation (cf. far field limitation). On the contrary, the EMR-20 system with the Type-8
sensor head is explicitly recommended for use in near field conditions such as the ones
encountered 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 probe
stand attached to it. At least the mechanical probe stand is made out of non-conductive,
“electromagnetically transparent” material.
-1.43
-1.23
-1.03
-0.63
-0.83
-0.43
-0.23
-0.03
0.17
0.37
87
0.57
0.77
0.97
1.17
1.37
5.3 MEASUREMENT ERRORS
-1.16
2.4
-0.96
2.2
-0.76
2
-0.56
1.8
-0.36
1.6
-0.16
z
x
0.04
1.4
1.2
1
0.24
y
0.8
0.84
1.04
0.4
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
1.24
0.6
xm
ym
0.6
zm
0.2
0.4
0
0
0.64
0.2
0.44
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 is
1.1 V/m) is not equal to the actual physical field strength (e.g. 1 V/m). Through
a proper broadband calibration, this effect can be reduced.
• accuracy of the x, y, z measurement position: a difference from the measured to the
simulated results can occur simply because the spatial measurement and simulation
position are not identical. This problem already starts with the difficulty of clearly
defining spatial measurement positions within a prototype RC, since the walls are
not 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 biggest
impact on an agreement between simulation and measurement, especially at higher frequencies. In order to position the field probe system correctly within the RC, two
methods were used: first of all, a fine measurement grid (spatial resolution 0.05 m) was
printed on the chamber bottom (see Fig. 5.9 for a schematic overview) for a coarse prepositioning 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.
88
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
a)
b)
Figure 5.10: Biconical TX antenna a) tripod-mounted and b) suspended from the ceiling with
plastic 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 measured
results can be summarized by essentially three issues:
• accuracy of the antenna position (x, y, z) and alignment angle θ: experiments in
the prototype RC revealed that the EM field distribution is very sensitive to the
excitation antenna position. The argument here is very similar to the one discussed
above 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 RC
measurements need to be somehow mechanically supported, in the simulations
they are “floating in the air”.
• coaxial cable feed: the simulated antennas are excited by an impressed voltage
across 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 however
always routed as close as possible to the RC’s walls.
Antennas in the prototype RC were carefully positioned using the measurement grid
printed on the chamber bottom, a plummet was attached to the antenna, and the laser
5.3 MEASUREMENT ERRORS
|E | [V/m]
80
89
Tripod
Plastic wires and Velcro
70
60
50
40
30
20
10
0
50
100
150
200
Frequency f [MHz]
250
300
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 with
plastic ropes and Velcro from the chamber ceiling (f = 50 . . . 300 MHz).
range distance metering device employed which had been introduced for the field probe
system. This ensures the greatest possible agreement between TX/RX antenna position
and angular alignment in the measurement setup and the simulation.
Initial comparisons between simulations and measurements were carried out with the
excitation antenna supported by a tripod. With this setup, at higher frequencies considerable differences between measured and simulated results occurred. A closer examination revealed that the tripod had metallic feet, “legs” made out of wood, a plastic
mounting rod, head, and antenna clamp. It was found that especially the highly absorbing wooden “legs” caused a significant distortion of the field distribution within
the prototype RC. In order to avoid further unintentional loading and distortion of
the EM field, the tripod was removed and the TX/RX antennas either suspended from
the chamber ceiling using nylon ropes and Velcro or placed onto styrofoam blocks (see
Fig. 5.10). A comparison between the 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 the tripod against the same antenna suspended by plastic ropes and Velcro from the
chamber ceiling (f = 50 . . . 300 MHz, ∆f = 1 MHz frequency resolution) is depicted in
Fig. 5.11. It can be seen that the measured field values are quite dissimilar concerning
their magnitudes and that – contrary to common believe – without the presence of the
absorbing 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 the
RC, rather than at a fixed position only: Fig. 5.12 depicts the magnitude of the electric
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
90
|E | [V/m]
50
Plastic wires and Velcro
Tripod
150 MHz
200 MHz
40
30
20
10
0
-1.2
-0.8
-0.4
0
y [m]
0.4
0.8
1.2
a)
|E | [V/m]
50
Tripod
Plastic wires and Velcro
250 MHz
300 MHz
40
30
20
10
0
-1.2
-0.8
-0.4
0
y [m]
0.4
0.8
1.2
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. suspended
with 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 biconical antenna supported by a tripod vs. suspended with plastic ropes and Velcro from the
chamber 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) the
electric field, it also distorts the field distribution. This implies that a peak measured at
a certain point with the tripod in the chamber will be shifted spatially once the tripod
is removed, which explains the behavior shown in Fig. 5.12. The impact of the tripod
can be seen at other spatial locations in Fig. B.1, B.2 and for additional frequencies in
Appendix 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 measurement
prototype 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 the
field probe system can be applied. Worthwhile mentioning are the following issues with
respect to the stirrer and chamber for modeling of the RC:
• the RC door (frame, door handle, gasket) exhibited a strong impact at certain
frequencies and needed to be taken into consideration in the simulation model (see
Section 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 was
found 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), ventilation ducts, lighting, and the RF feed-through panel proved to have an only
minor effect on the measured EM field in the lower-to-medium frequency range
and were therefore neglected in the RC simulation model
It should be emphasized that the sensitivity of both simulations and measurements on the
above-mentioned issues (e.g. the door or chamber imperfections) aggravates substantially
with 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 EMC
tests of EUTs inside an RC. An introduction to the field strength uncertainties to be
expected for a given number of stirrer steps within an RC was given in Section 2.5. In
order to establish an uncertainty budget, the “Guide to the expression of uncertainty
in measurement (GUM)” can be advantageously used [170]. This standard evaluation
approach is in the form of a cookbook and comes with the widely used “GUM Workbench” software. Further general application notes for uncertainty budgets are provided
92
5 PROTOTYPE AND MEASUREMENT SYSTEM DEVELOPMENT
in e.g. [163, 171, 164]. Very detailed information on the assessment of special RC-related
measurement uncertainties for chamber calibration and EUT emission and immunity
testing 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 inner
dimensions 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 mean
for a rotating object such as a stirrer in an RC?” versus “What does electrically large
mean in the traditional electromagnetic sense?” was brought up and will be addressed in
Section 7.7. Specifications of the measurement system (consisting of the transmit/receive
equipment and the field probe unit) developed for data acquisition were outlined. The
field probe system with its probe head and processing unit was analyzed in detail, since
it forms the most important part of the measurement system for validation of simulation
results. A system capable of providing simultaneously three-component electric field
data without distorting the field distribution significantly was chosen.
Measurement errors originating from field probes, antennas, and stirrers were discussed
and assessed for their impact on deviations between simulated and measured results.
The biggest deviations were found to result from the antenna tripods and position inaccuracies 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 ropes
and Velcro from the ceiling. A positioning and alignment system comprised of an optical
grid and a laser range distance metering device was proposed. Other measurement errors
resulting from anisotropy, sensitivity, mutual coupling and finite extension of the field
probe were found to be negligible. The fundamental limitations concerning the measurement system addressed in this chapter were taken into consideration when benchmarking
simulated 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 appropriate 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 to
find out which geometric features would be the most important for an optimally designed chamber. One particular challenge in RC simulations is the generation of a large
number (on the order of several hundreds) of chamber models where the only change in
the geometric structure is the step-by-step rotation of the stirrer. Following the requirements outlined in Section 2.5, at least 50 rotational stirrer angles need to be simulated
per frequency to achieve sufficiently low statistical uncertainty. The EM simulation software FEKO requires for each simulation involving a change in the RC geometry a new
PreFEKO input file. To speed up the modeling phase and to reduce the number of modeling errors, these input files were generated automatically with minimum user input
employing 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 different
CAD models. To set up an RC simulation, parameters such as linear/logarithmic frequency stepping and triangle discretization area are generated with the MATLAB tool
and passed on to the mesh generator and the preprocessor (Fig. 6.1). The mesh generator uses these input parameters together with the rules laid out in Section 3.4.1 to
compute an estimate of the appropriate discretization for all structures. All surfaces
are discretized with triangles and all wires with segments which facilitate an accurate
representation of arbitrarily shaped structures and are well suited for the EFIE-based
MoM technique (see Chapter 3). Furthermore, the stirrer positions are set up as well
as the position of the EUT, the TX and RX antennas and the location at which the
near field is to be computed. The procedure mentioned above allows the reliable, automated setup of a large number of similar simulations, where only certain parts of the
RC geometry are varying. The final RC geometry along with discretization-related data
is transferred back to the CAD system, which finally creates a mesh by generating a
mixed triangle/segment discretization [56]. The discretization process’ output data is
combined with the simulation settings in the preprocessor, which checks the validity
93
6 MODELING OF THE REVERBERATION CHAMBER
94
Geometry
Modeling
&
Simulation
Preprocessing
Data
Parameter
Extraction
Generator
3D CAD
Mesher
MATLAB
Interface®
HyperMesh
Simulation
Data Process
Geometry
Conversion
PreFEKO
Interface
®
Field Solver
&
Graphical User
Interface
Simulation
Postprocessing
&
Measurement
Data Acquisition
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 the
simulation 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 simulation
or to distribute the computational load equally across several machines using the parallel
FEKO solver. A scheduler is utilized to run all computations belonging to the same
base 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 were
obtained from initial simulations considering a simple RC model without stirrers, doors
or other geometrical details – i.e. a cavity. Triple geometric symmetries were used for
cavity modeling and during the computations (see Section 3.3.3). Also stirrers in the
prototype RC (see Section 5.1) were removed during measurements to match the simulation model. Furthermore, these initial simulations and measurements were used to
identify chamber 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 through the stirrer motor bearings). The reason not to include the stirrer in this
6.1 CHAMBER MODELS
95
z
Hertzian dipole
excitation antenna
Chamber walls
x
xy
xz-symmetryy
-s
ym
m
et
ry
V-stirrer axis
y
yz-
sym
me
try
6-paddle V-stirrer
Figure 6.2: Partly symmetric simulation model of the RC. Three geometrical symmetries are
utilized to construct the RC model by mirror imaging one eighth of the chamber walls (depicted in dark grey). The stirrer is moved to its designated position by a mathematical
translate/rotate operation. Note that this chamber (contrary to the RC in Fig. 6.3) does not
feature a door.
first comparison was to eliminate the possibility of deviations between simulations and
measurements resulting merely from the rotational positioning accuracy of the stirrer
paddles. Results of the cavity simulations are shown in Section 7.2. The cavity simulations and measurements were also used to investigate the loading effect of the TX/RX
antenna tripods mentioned in Section B.1.
6.1.3 Prototype reverberation chamber
Due to less powerful computational resources available in the past and to facilitate
modeling, simulations of RCs were commonly restricted to two dimensions – full-wave
3-D simulations of RCs have only become feasible in the most recent years. Although this
is a significant step forward compared to the 2-D models used earlier, 3-D simulations
are computationally still very slow (cf. Table 3.1). Therefore significant simplifications
are usually made in RC simulation models: Double or triple symmetry (such as in the
cavity simulations above) is often utilized and small geometrical details are neglected in
order to speed up simulations.
Chamber model without door
As a starting point, a partly symmetric RC was modeled using similar simplifications: As
depicted in Fig. 6.2, the walls are constructed by designing only one eighth of the chamber
6 MODELING OF THE REVERBERATION CHAMBER
96
Chamber walls
Receive
antenna
H
-s
Excitation
antenna
xis
1
a
er
r
tir
tir
re
ra
xi
s
2
-s
H
V-stirrer axis
y
Chamber door
z
x
6-paddle V-stirrer
Figure 6.3: Detailed fully asymmetric simulation model of the RC. Excitation source is a
logper antenna located in front of the door. Shown is its 3-D free space radiation pattern
superimposed for illustrative purposes. The logper antenna depicted in the right corner is
used for emission testing setups in the simulation model. E and H fields can be calculated
at any arbitrary point, as an example one of the near field computation planes is shown. All
metallic parts are of finite conductivity.
and mirror imaging it at the xy-, xz-, and yz-plane. The geometry mirroring process is
done by the simulation preprocessor (Fig. 6.1). To exploit triple geometric symmetry, the
vertical 6-paddle stirrer is discretized separately and subsequently moved to its intended
position using a mathematical shift/rotate operation. Using this approach, there is a
reduction in computation time possible when the elements of the MoM system matrix
are determined – however, the time and memory needed for solution of this system of
equations is not reduced as the RC is only partly symmetric (cf. Section 3.3.3). This
basic chamber has dimensions of 3.06 m · 2.86 m · 2.48 m (height h · width w · length l).
To minimize the number of triangles needed to mesh the large RC surface, a frequencyadaptive and geometry-dependent discretization was used (see Table 6.1 for details). For
ease of modeling, all walls and stirrers are assumed to be of single-layered metal. As
opposed to the RC model below, this chamber does not feature any apertures.
6.1 CHAMBER MODELS
97
Figure 6.4: Photo of the RC door with gasket on frame and contact finger strips.
Chamber model including door
Two RC models were designed and simulated, the “basic” one with flat walls described
above and a more “detailed” one including the chamber door as shown in Fig. 6.3. The
dimensions of the door are 2 m in height and 0.92 m in width with a 0.07 m recess of the
door frame from the chamber wall (see Fig. 6.5a)). Although, at first glance, these details
seem to be small, it is shown in Sections 7.2.1 and 7.3 that they affect the chamber fields
significantly and thus cannot be neglected. Since the RC model including the door does
not exhibit any symmetries, no reduction at all in memory or computation time can be
achieved (cf. Section 3.3.3).
As mentioned in Section 5.1.1, the prototype RC features beryllium-copper contact finger
strips which are used as gaskets to prevent EM leakage from the door (Fig. 6.3). Leakage
through the chamber door in the simulation would require the modeling of extremely
tiny gaps between the door and the chamber, which would introduce numerical artifacts
into the RC simulation and increase considerably the number of triangles (which would
98
6 MODELING OF THE REVERBERATION CHAMBER
need to have edge lengths on the order of the gap’s width). Modeling of the imperfect
door seal in FEKO can be carried out with two methods:
• Using a distributed series impedance between the edge of the chamber door cutout
and the chamber door itself (“LE-loading”): with this technique, the current distribution 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 be
modeled in the RC model in FEKO between the door cutout and the door. The
problem is that this tiny gap introduces numerical artifacts in the simulation due
to the scale of dimensions (size of the door vs. size of the gap). This prohibits a
meaningful inclusion of the door gap in the RC simulation model.
Therefore, modeling of the door gasket was accomplished by introducing a distributed series 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 simulated using different stirrers (Section 6.2) and antenna models (Section 6.4) and assuming
finite conductivity (Section 6.3) for all structures. This chamber is used as a benchmark
model 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 typical
RC parameters (stirring efficiency, field uniformity, correlation, etc.) corrugated, cubic,
and offset-wall chambers were modeled and simulated for performance comparisons.
Corrugated RC
Corrugations were introduced for use in RCs in analogy with acoustic ray theory and
were expected to exhibit similarly beneficial effects as in e.g. corrugated horns. The
driving force behind corrugations applied to RCs was the anticipated improvement of
field uniformity at lower frequencies close to the LUF. In [173] the corrugations have an
amplitude of 1.5 inch (i.e. 0.0254 m) and a valley-to-valley distance of 2 inch. Their RC is
rather compact and has dimensions of 1.2 m · 0.8 m ·1.8 m (h·w·l). To investigate whether
corrugations are indeed beneficial, the dimensions of the corrugations from [173] were
scaled to fit the size of the prototype RC used in this thesis. The scaled corrugations
have 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 ceiling
is 0.165 m, and between the walls and the outermost stirrer part 0.24 m. Employing
the corrugations also on the ceiling and the side walls of the RC would reduce the top
space 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 the
same door as the one in Section 6.1.3. Simulations of the corrugated RC are discussed
in Section 7.5.
Cubic and offset-wall RC
As shown in Section 2.2.1 and Fig. 2.4, cubic chambers suffer from mode degeneration so
that the usually required “∂N/∂f = 1.5 modes/MHz above cutoff”-criterion is reached
6.1 CHAMBER MODELS
99
a)
b)
c)
d)
Figure 6.5: Different simulation models of the RC: a) basic RC resembling the prototype
chamber (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 generate
a more uniform field than standard rectangular RCs. For this reason, a cubic chamber
was 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,
6 MODELING OF THE REVERBERATION CHAMBER
100
Number of triangles for discretization
30. . . 250
MHz a
30. . . 400
MHz b
30. . . 600
MHz c
30. . . 1000
MHz 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
Chamber type
a
b
for this frequency range the HyperMesh element edge size was set to 0.200
— — 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 in
Section 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 investigate
performance scaling:
• IEH chamber at the Universität Karlsruhe (Germany): this RC has dimensions of
2.34 m · 5.24 m · 2.39 m (h · w · l) and was initially simulated at frequencies lower
than 500 MHz to compare field uniformity levels of similar “Rot-Z” stirrers.
• Medium-sized (SMART200) and large (SMART80) ETS Lindgren chambers: these
chambers 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 MHz
and, respectively, 80 MHz on. Simulation requirements (time and memory) especially for the larger chamber grow quickly to astronomical levels, since for a
proper discretization of the chamber walls at frequencies beyond 500 MHz more
than 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 discretization levels to cover both the lower and higher frequency range. Discretization data
6.2 STIRRER MODELS
a)
101
b)
c)
d)
Figure 6.6: Vertical stirrer models with triangular discretization: a) 6-paddle stirrer, b) crossplate 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 m
away from the right side wall (designated as “position III” in Figure 5.1). Alternative
mounting 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 stirrer
design procedure are outlined in Section 5.1.2.
6.2.1 Vertical stirrers
In order to facilitate relative performance comparisons, all simulated vertical stirrers
can be circumscribed by 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 “rotational volume” of roughly 2 m3 . Vertical stirrers were always mounted in
position III.
Vertical paddle-type stirrers
• 6-paddle stirrer: simulation replica of the stirrer physically existing in the prototype RC (see Section 5.1.2 and Fig. 5.2). Consists of six rectangular paddles of
size 0.60 m · 0.60 m, rotationally offset by 60◦ with a slanting angle of 45◦ for each
paddle. Horizontal spacing between the paddle centers is approx. 0.46 m. Distance
from 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.
102
6 MODELING OF THE REVERBERATION CHAMBER
The stirrer rod (height 3.06 m), which supports the paddles, is not simulated as
it 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 connected
with spline surfaces, so that the paddles become one single structure. This stirrer
was used to analyze the impact of the electrical stirrer size on the field uniformity
and is shown in Fig. 6.6c).
• 4-paddle stirrer: essentially the same as the conventional vertical 6-paddle stirrer, 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 implies
that the rotational “volume” is the same as for all other stirrers.
Vertical single- and cross-plate stirrers
These stirrers were designed to compare “fancy, irregularly shaped” stirrers with very
rudimentary ones.
• cross-plate stirrer: two rectangular plates of equal size that intersect at an angle
of 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 of
the 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◦ . Each
plate measures 1.33 m · 0.735 m (h · w). The gap between the two stirrer plates is
0.09 m. This stirrer is shown in Fig. 6.6b).
• single-plate stirrer: the most basic stirrer, consisting of a single rectangular plate
measuring 2.745 m · 0.735 m (h · w).
Vertical upset Z-fold stirrers
The so-called Z-fold stirrers used in the prototype RC simulations are upset versions of
the original stirrers built by ETS Lindgren (described in Section 6.2.3).
• upset Z-fold stirrer: this stirrer is used for performance benchmarks against the
standard 6-paddle stirrers mentioned in Section 6.2.1. The upset version of the
original Z-fold stirrer was scaled so that it fits into an identical rotational volume
as 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 stirrer
mentioned above, however the metal “Z-fold”-part is not a single piece, but broken
into three separate parts with two gaps of approx. 0.09 m between each other.
6.2 STIRRER MODELS
103
Number of triangles
for discretization
Stirrer type
30. . . 600 MHz a 30. . . 1000 MHz c
6-paddle standard
443
599
6-paddle without gaps
695
926
4-paddle standard
399
481
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 )
for this frequency range the HyperMesh element edge size was set to 0.100
— — to 0.085
c supporting sidewall structure (electromagnetically irrelevant)
a
b
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), rotationally offset by 90◦ with a slanting angle of 45◦ . The nearest edge-to-edge distance between two paddles is 0.02 m. This stirrer is similar to the one described in
Section 6.2.3 and is shown in Fig. 6.7a).
• 5-paddle stirrer: features five rectangular paddles of 0.60 m · 0.60 m, rotationally
offset by 72◦ with a slanting angle of 45◦ . The nearest edge-to-edge distance
between 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 Section 6.2.1, but mounted horizontally at position II (see Fig. 5.1). This stirrer
is shown in Fig. 6.7c).
6 MODELING OF THE REVERBERATION CHAMBER
104
a)
b)
c)
Figure 6.7: Horizontal stirrer models with triangular discretization:
b) 5-paddle stirrer, c) 6-paddle stirrer.
a) 4-paddle stirrer,
6.2.3 Stirrers used in other reverberation chambers
The following stirrers are employed in the ETS Lindgren and IEH RCs mentioned in
Section 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 modified
version of the original one used in a small RC at the NSWCDD [173]. This stirrer
consists of four rectangular paddles of size 0.80 m · 0.80 m, rotationally offset by
90◦ with a slanting angle of 45◦ . The nearest edge-to-edge distance between two
paddles is 0.04 m.
• original vertical and horizontal “Z-fold” stirrer: stirrers employed in ETS Lindgren’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 and
Slocum [174], and measure 4.40 m · 1.52 m · 1.21 m (h · w · d). They consist of a
folded 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 not
included in the simulation (see Section A.1). It is used for illustrative purposes
however in the preprocessor PreFEKO.
6.3 WALL AND STIRRER CONDUCTIVITIES
105
Number of triangles
for discretization
Stirrer type
30. . . 600 MHz a
30. . . 1000 MHz b
6-paddle standard
431
611
5-paddle standard
347
487
4-paddle standard
513
671
a
b
for this frequency range the HyperMesh element edge size was set to 0.100
— — 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 conductivity
values for all metallic structures were assumed to be equal to DC conductivities and to
remain constant over the 50 MHz. . . 1 GHz range (according to NIST, DC conductivity
values can be safely used for metals at frequencies up to 5 GHz [175]). For the aluminum
structure 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 shielded
room construction, i.e. walls consisting of several interconnected sheets with intermediate overlapping flat stock.
The main material components of commercially available galvanized steel sheets are typically 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.-%). As
the 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 surface
impedance ZS (3.24) of the chamber walls tends to be increasingly defined by the conductivity of the zinc coating alone rather than the conductivity of the zinc combined
with the underlying iron. Commercially available iron has conductivities on the order of
κ = 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 conductivity
of 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 in
Table 6.4. Measurements of the chamber quality factor Q however always exhibit one
repeating pattern: the theoretically predicted Q (2.51) is much higher than the measured one (difference of a factor 10. . . 500 depending on the frequency) [24, 9]. This in
6 MODELING OF THE REVERBERATION CHAMBER
106
Material
Conductivity
κ 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
pure material at T = 300 K [149, 175, 176]
commercially available material at T = 300 K [149, 177]
c normalized to the conductivity of pure silver
a
b
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 high
or, vice versa, the chamber loading too low. In any case, assuming overall conductivity
values of zinc in an RC simulation, results in unrealistically high field strengths within
the chamber.
Taking into account interconnections between the different metal sheet panels, screws
and cutouts as well as dirt and occasional oxidation spots, the best agreement between
simulations 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 the
surface 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 simulations 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
Conductivity
κ 106 S/m
Conductivity
κ/κGS (relative b )
“High”
1.1
0.37
“Medium”a
0.09
0.03
“Low”
0.05
0.017
Simulation
Material
a
b
“Medium” is the default conductivity used in the RC simulations
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 measurements. Depending on the testing scenario in the RC (emission, immunity, benchmark
simulation-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 TX
and RX operation is depicted in Fig. 6.3. An overview of all utilized antennas along with
their respective 3-D far field radiation pattern is given in Fig. 6.8. Before employing the
TX and RX antennas in the RC simulation, their far field patterns were simulated separately and validated by measured or analytical results. All antennas can be positioned
in 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 the
horn antenna were modeled with triangles of finite conductivity σ = 1.1 · 106 S/m. Since
inside a typical RC with well-conducting walls there is EM field generated with virtually
zero 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 or
in some cases even slightly negative [178]. The reactive power at the antenna ports is
however fairly large, and therefore it is not possible to compute scattering parameter
data 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 frequency
range 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 analytically in the numerical code. Secondly, at the start of this project in June 2001, all (but
one) published RC simulations had used Hertzian dipoles as an excitation source (see
Table 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).
6 MODELING OF THE REVERBERATION CHAMBER
108
a)
b)
c)
d)
Figure 6.8: Antenna models with corresponding 3-D free-space far field radiation patterns used
in 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 unrealistic) RC excitation, a λ/2-dipole was modeled and simulated. This half-wavelength
dipole (discretized with 11 wire segments) was used as a model for the adjustable precision 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 range
TX/RX
antenna
Number of
segments / triangles
Arbitrary
Hertzian dipole
0/0
50 MHz. . . 3 GHz
λ
2 -dipole
11 a / 0
50 MHz. . . 350 MHz
biconical
162 b / 0
logper
435 a / 0
300 MHz. . . 3 GHz c
2.2 GHz. . . 3.3 GHz
a
c
1 a / 2 486
horn
excitation source was modeled with one segment b — — with two segments
discretization limits the maximum frequency of the logper antenna to 3 GHz
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 other
with a 0.1 m long feed line in-between. The largest overall dimension of the biconical
antenna is 1.32 m. The biconical antenna was modeled by 162 wire segments (with two
source segments for symmetry reasons), its free-space far field pattern (simulated gain
1.5. . . 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 type
USLP 9143) were also modeled for the simulations. Although these logper antennas are in
practice useable up to f = 5.2 GHz, the discretization limits their maximum operational
frequency to 3 GHz. This is due to the very short, but comparatively “thick” segments
utilized for the tip dipoles of the antenna. Given a certain wavelength λ and geometric
wire radius r, the conditions imposed on the discretized segments of length ∆l using the
EFIE 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 currents on a wire structure, the segment length ∆l must be smaller than λ/10. Furthermore
as (6.1) suggests, ∆l has at the same time a lower limit of r (Section 3.4). Large radii r
therefore limit ∆l to a certain minimum length greater than r. Since higher frequencies
f correspond to smaller wavelengths λ (c = λ · f ), and for small λ a finer discretization
is needed (i.e. segments with sufficiently small ∆l). This implies that the wire radius
r defines a maximum simulation frequency. With the “thick” segments of the logper
antenna, this maximum frequency is at around 3 GHz, even though the physical antenna
110
6 MODELING OF THE REVERBERATION CHAMBER
is specified to work up to 5.2 GHz. In total 435 wire segments (including one source
segment) are needed to model the logper antenna (Table 6.6). Its free-space far field
pattern (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 measurements (with the exception of a few special RC/horn simulations used for a coupling
analysis). Since the frequency range of most of the simulations in this thesis was limited
to 50 MHz. . . 1 GHz, horn antennas are not considered in the simulation vs. measurement comparisons discussed in Chapter 7. Reasons for the restriction to frequencies
f ≤ 1 GHz were the prohibitive computational runtimes and memory requirements
(cf. Section 3.4), along with the electromagnetically extremely sensitive RC structure
at higher frequencies (discussed in Section 7.4.2). The horn antenna has an aperture
of 0.21 m · 0.28 m with a rectangular waveguide feeding section of 0.088 m · 0.044 m and
an overall length (waveguide shorting back-plate to horn aperture) of 0.445 m. It was
modeled 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 is
shown 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 in
various testing environments [138]. The CEUT serves as a “standardized” EUT which
exhibits a consistent and reproducible radiation pattern. For simple CEUTs, the emission/immunity test response can be calculated even analytically. Comparisons are usually made between different laboratories (“inter-laboratory comparison”) where the tests
are made in either similar (e.g. different ACs against each other) or dissimilar testing
environments (e.g. OATS against AC). CEUTs are commonly also referred to as “imitated equipment” [180], “tightly specified test device” [138], “reference radiator” [181],
“representative EUT” [182], or “artificial EUT emitter/receiver” [183]. A typical EMC
test 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 practical
emission and immunity EMC tests. Details on CEUTs modeled in the course of the RC
simulations are given in Section 6.5.2.
Emission CEUT
In the emission case, the CEUT is typically a battery-powered device, so that feeding
cables are not needed. With this approach, the EM field is not influenced by the layout
of cables attached to the CEUT and their strong effect on the radiation characteristics is
eliminated. CEUTs can be operated either in an autonomous or in an interactive mode,
6.5 CANONICAL EQUIPMENT UNDER TEST
Chamber walls
111
EUT
RX/TX
antenna 2
RX/TX
antenna 1
co N
m ea
pu r
ta fie
tio ld
n
pl
an
e
V-stirrer axis
y
Chamber door
z
x
6-paddle V-stirrer
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 (immunity/emission). A canonical box-type EUT is measured in the RC.
which allows certain settings to be controlled remotely by the user. Remote control
capability is in practice usually provided through a fiber optic system (Section 5.2.2) with
which 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 components
inside a CEUT, where the first generate a very narrow and the latter a broad frequency
spectrum (e.g. 5 MHz spacing with 30 . . . 2000 MHz detectable output). Other possible
broadband excitation sources are comparison noise emitters (CNEs) [184]. As explained
in detail below in Section 6.5.2, EM energy is radiated through one or more simple wire
antennas (dipoles, loops) or one or more slots and gaps [182].
Immunity CEUT
CEUTs for immunity testing are significantly less widely used and “standardized” than
the ones for emissions mentioned above. Often devices for radiated immunity tests
are purpose-built for a very specific setup. Examples of special CEUTs for reference
immunity measurements are integrated circuit timers and comparators; combined with
some simple circuitry it is possible to monitor from which field level threshold on they are
6 MODELING OF THE REVERBERATION CHAMBER
f = 600 MHz
f = 300 MHz
Models
112
Figure 6.10: Different canonical EUTs models (top) with their corresponding simulated 3-D
free space radiation patterns at f = 300 MHz (middle) and f = 600 MHz (bottom). Shown
are (from left to right) a realistic dipole (finite length), a loop, and a box EUT operated in
slot mode (left) as well as in gap mode (right).
“disturbed” in their normal operation [185, 186, 187]. There are also box-type immunity
CEUTs where EM energy is coupled into the box through slots or gaps and picked up
by a metal rod. Voltage across the ends of the rod is measured and modulates inside the
CEUT an optical signal. This signal is output through a similar optic system as in the
emission EUT [188]. The optical signal is detected outside the testing environment and
serves as a measure of the field strength that the canonical immunity EUT is exposed
to [84].
6.5.2 CEUT modeling
For both emission and immunity CEUT simulations, test objects were used which were
already successfully employed in several round-robin tests (such as the FAR project of
the European Union [189, 138, 183] and the RC, GTEM, FAR, and OATS comparisons
carried out by the FCC in the U.S. [182, 190]). These box EUTs are made of thin brass
sheets soldered together at the edges. The following CEUTs were modeled and simulated
in 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 m
with 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 vice
versa) is possible through either the slot or the gap between the side panels and the top
or through a combination of both slot and gap. Simulation models of the CEUTEs and
CEUTIs along with their corresponding 3-D free space radiation patterns at 300 MHz
and 600 MHz are shown in Fig. 6.10.
Common requirements for EMC tests in RCs are that the EUT should take up less
than 8%. . . 10% of the chamber volume [6]. This recommendation aims at preventing
excessive RC loading and hence deterioration of field uniformity within the chamber. To
reduce direct coupling between EUT walls and RC, the EUT must be located between
λ/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 of
1.7. . . 2.2m3 and a minimum spacing towards any metallic object of 0.25. . . 0.5m in the
prototype RC. The actual volume of the CEUT is significantly below these limits, its
position was always adjusted to meet the requirements outlined above. All geometrical
specifications and discretization details of these CEUTs are summarized in Table 6.7.
Slot Mode
In the so-called “slot mode”, the CEUT features a slot of 0.12 m · 0.04 m in one of the side
walls (defined as the “front” of the CEUT, see Figure 6.10), through which radiation from
the inside of the box to the outside is possible. The EM field is excited by connecting the
Frequency range
50 MHz. . . 3 GHz
Realistic dipole (0.4 m)
50 MHz. . . 3 GHz
Loop (0.3 · 0.3 m2 )
50 MHz. . . 1.2 GHz
50 MHz. . . 1.2 GHz
50 MHz. . . 1.2 GHz
a
c
Canonical emission
EUT type
Box with slot c
(0.48 · 0.48 · 0.12m3 )
Box with gap c
(0.48 · 0.48 · 0.16m3 )
Box with slot and gap c
(0.48 · 0.48 · 0.16m3 )
Number of
segments / triangles
38 a / 0
120 a / 0
1 b / 922
1 b / 900
1 b / 906
excitation source was modeled with two segments b — — one segment
height of the box EUT is either 0.12 m (slot mode) or 0.16 m (gap mode)
Table 6.7: Discretization of the canonical emission EUTs (CEUTEs).
114
6 MODELING OF THE REVERBERATION CHAMBER
outer conductor of a coaxial feeding cable to the center of the lower slot edge, running
the inner conductor across the slot, and connecting it to the center of the upper slot
edge. Measured and simulated values for the gain in this mode of operation are on the
order of 6. . . 8 dBi in the 600. . . 1200 MHz frequency range [22, 186].
Gap Mode
By closing the aforementioned slot and mounting the top side of the CEUT 0.04 m spaced
away from the side walls, a gap is created which permits radiation of EM energy from
circuitry placed inside the box CEUT to the outside. The coaxial cable that was used
for the “slot mode” excitation is mounted in a similar fashion for the “gap mode”: the
outer 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 gap
edge, i.e. to the top side. Measured and simulated values for the directivity of the CEUT
are on the order of 2. . . 3 dBi in the 600. . . 1200 MHz frequency range [22, 186].
Slot and Gap Mode
The “slot and gap mode” CEUT combines the two apertures mentioned above. Excitation 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 introduced along with a tool automating the generation of RC simulation input data. This
tool allows the repeated, accurate, and consistent creation of a large number (on the
order of several hundreds) of RC geometries where the only change in the structure is
the step-by-step rotation of the stirrer. Once the stage of simple startup simulations had
passed, the usage of this automated tool turned out to be an absolute necessity.
Starting with a basic cavity, a comprehensive chamber model resembling the prototype
RC was elaborated. Discretization of the structure was performed using triangular surface patches and wire segments which were chosen according to the frequency range
of interest to minimize the computational effort. Obtaining and assigning reasonable
values for materials used in RCs proved to be rather cumbersome: the problem was to
find 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 for
the RC walls and stirrers in the simulations.
A prototype RC with and without door, cubic and corrugated chambers, and an offsetwall RC were modeled. Furthermore, eleven vertical and three horizontal stirrers as
well as various transmit and receive antennas (Hertzian dipole, λ/2-dipole, biconical antenna, logper antenna, horn) were designed. The concept of a canonical equipment under
test (CEUT) was introduced and three different CEUTs (dipole, loop, and box) were
modeled in the style of the EUTs employed in several international round-robin tests.
All structures designed and modeled in this chapter were used extensively throughout
the 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 measurements 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 reverberation 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 measurement 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 simulate
RCs 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 is
then employed to set the simulation frequencies and compute the data for the geometry
discretization. Furthermore, the stirrer positions are selected as well as the position of
the EUT, the TX and RX antennas and the location at which the near field is to be
computed. This generator allows the automated setup of a large number of similar simulations, where e.g. only the rotational stirrer angle ϕ is varying. The final RC geometry
along with discretization-related data is transferred back to the CAD system, which creates a mesh by generating a mixed triangle/segment discretization complying with the
requirements laid out in Section 3.4. Discretization data is combined with the simulation
settings in the preprocessor and passed on to the simulation kernel. After evaluating the
pros and cons of different numerical methods as discussed in Section 3.1, an EFIE-based
frequency-domain MoM field solver was chosen as the simulation kernel [54]. Once the
surface and line currents are computed and stored (see Section 3.2), near field data and
scattering parameters are calculated from the currents.
Simulations without any indication for validity are inherently problematic: at best they
are by chance correct, more often they are flawed in some way, and in the worst case
the results represent expensive and possibly utter nonsense. A thorough validation of
simulations 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 either
not to use any means of validation at all or to perform a validation utilizing unsuitable
quantities. Especially the latter approach will be addressed in detail in Section 7.4.3
in this chapter. In general, there are three possible validation options for simulation
results:
• benchmark against analytically calculated results; this includes also an evaluation
whether a simulated result “makes any sense” from a purely theoretical point of
view
• comparison with the results obtained with other EM solvers (preferably utilizing
a 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 hence
speed. For the validation of RC simulations, however the problematic aspects of analytical results dominate: first of all, the EM fields can only be computed in an ideal
cavity without stirrers, antennas or an EUT inside. Secondly, analytic approaches usually assume infinite conductivity for the cavity (PEC); this implies that the quality factor
bandwidth ∆fQ (Section 2.1.2) would be zero, i.e. at a given frequency f only a single
mode (and possibly respective degenerate modes) could be excited and coupling between
different non-degenerate modes does not exist. Realistic RCs however can only achieve
sufficient statistical field uniformity if ∆fQ > 0 so that several modes are excited at
a given frequency. Also lumped-elements circuit theory formulations cannot reproduce
realistic 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 numerical technique and with one solver operating in the frequency- and the other in the
time-domain (e.g. MoM vs. FDTD). There are however also several problems associated
with solver-to-solver benchmarks: a particular numerical method might be well-adapted
to simulate an RC whereas an other may not. Some methods might not be able to
simulate an RC simply because of the numerical size of the problem. In addition, RC
simulations are time-consuming to run, expensive software needs to be bought, and there
is a lot of experience required by the end-user (high-end EM field simulations still tend
to 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 addition, expensive equipment is needed (cf. Section 5.2), they provide an useful insight into
the “reality of the chamber physics”. This allows to identify critical parameters which
are important when performing RC tests in practice, but which would be neglected otherwise in RC simulations. Certainly, as outlined in Section 5.3, special care needs to be
taken of additional errors associated with measurements.
During the course of this thesis, measurements performed in the prototype RC usually
included the electric near field as well as forward and reflected power on all antennas.
7.2 CAVITY SIMULATION
Geometry
Modeling
&
Simulation
Preprocessing
Field Solver
&
Graphical User
Interface
Simulation
Postprocessing
&
Measurement
Data Acquisition
117
Data
Parameter
Extraction
Generator
3D CAD
Mesher
MATLAB
Interface®
HyperMesh
Simulation
Data Process
Geometry
Conversion
PreFEKO
Interface
®
FEKO®
r r r r
J, E, H, S
Visualization
WinFEKO
GraphFEKO
Data
Extraction
Database
System
Interface
MS Access
Measurement
System
Data &
Statistics
Extraction
Benchmarks
MoM Kernel
MLFMM / PO
Compliance
®
®
MATLAB
Interface®
Figure 7.1: Schematic simulation, analysis, measurement, and benchmark procedure.
Extensive details on the measurement system with a comprehensive error analysis can
be found in Chapter 5. To facilitate data handling, both simulation and measurement
results 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 measured
data are performed by retrieving the results from the database without the need to manipulate (e.g. scale, offset, etc.) underlying data – this in turn significantly reduces the
probability of accidentally introducing errors.
7.2 Cavity simulation
Initially, simulations considering a simple RC model without stirrers, doors or other
geometrical details (in other words: a cavity) were performed to obtain reasonable conductivity values for the RC walls and stirrers as listed in Table 6.4. The cavity model
used in the simulations is described in Section 6.1.2. For this purpose also stirrers in the
118
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
prototype RC were removed during measurements to match the simulation model. Furthermore, these preliminary simulations and measurements were used to identify chamber 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 through
the stirrer motor bearings). The reason not to include the stirrer in this first comparison
was to eliminate the possibility of deviations between simulations and measurements
resulting 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 same
frequency using greatly different mesh discretizations of the walls: decreasing the triangular mesh size from 2 498 triangles to 13 236 triangles did not change the field distribution within the cavity at all [191]. None of the changes in discretization introduced
any “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 view
was verified by examining if and how well boundary conditions for the EM fields were
met at the cavity walls. The computational region was extended beyond the cavity walls
so 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]. This
basic analysis did not bring up any major surprises, therefore the cavity model was used
as 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 excellent agreement with simulations, whereas for several higher frequencies considerable
differences started to appear. Therefore the cavity was closely examined to identify the
geometrical details causing the differences between the simulated and the real cavity.
The prototype RC without stirrer was modified with different coaxial cable and power
line routing, the simulated cavity was modeled with several modifications (walls with different conductivities, inclusion of the stirrer motor mount, etc.). This analysis revealed
that by far the biggest perturbance seemed to be caused by the door. To facilitate
modeling of the cavity, the door (among with other apertures) was neglected in the first
simulation models. In subsequent simulations, the cavity door was included in a new
model and proved to have a significant effect on the simulated results. Figure 7.2 clearly
shows the strong, very frequency selective impact of the door on the simulated EM near
field results: whereas at f = 200 MHz the field in the cavity with and without door is
virtually identical, at f = 250 MHz a significant difference can be seen. As shown below
in Section 7.3.1, a similar effect was noticed in later comparisons of RC measurements
and simulations; for this reason, the door was also included in the final simulation model
resembling the RC prototype.
7.2.2 Insertion of a stirrer
After initial “door-modified” cavity simulations proved to make sense, a stirrer was modeled for the simulations and installed into the prototype RC (see Section 6.2). Figure 7.3
depicts a comparison between an RC without (left) and including (right) a six-paddle
7.2 CAVITY SIMULATION
119
y
Without
chamber door
z
x
a)
c)
z
x
b)
y
Without
chamber door
y
With
chamber door
z
x
y
With
chamber door
z
x
d)
Figure 7.2: Effect of the chamber door on the magnitude of the electric field |E| within a cavity
at a) and b) f = 200 MHz and c) and d) at f = 250 MHz. Whereas at f = 200 MHz the
field in the cavity with and without door is virtually identical, at f = 250 MHz a significant
difference can be seen. Excitation is a biconical antenna.
120
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
y
z
x
a)
y
z
x
b)
Figure 7.3: Comparison of a) an empty cavity-like chamber against b) a reverberation chamber
with a single vertical six-paddle stirrer. Depicted is the magnitude of the electric field |E| in
the 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 readily
available, the empty cavity simulations were used to verify the simulation by benchmarking 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 theoretical cavity has PEC walls contrary to the simulated one, this benchmark is only an
estimate for the correctness of the numerical full-wave simulation. Rigorous measurements as presented from Section 7.4 onwards needed to be performed to clarify if the
simulations can pass a benchmark test satisfactorily. Figure 7.3 clearly illustrates that
the insertion of the large (in terms of wavelength λ) RC stirrer significantly changes the
field 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 of
stirrer structures and different chambers, any thorough RC simulation will generate large
amounts of data. In the following sections only the most important results are illustrated
and 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 chamber door already observed in the cavity simulations could be reproduced also in the
RC simulations: the simulated and measured results were in good agreement in the low
frequency range f ≤ 250 MHz, which is still below fLUF . Similar as in the cavity simulations at certain frequencies f ≥ 300 MHz however considerable differences between
RC measurements and simulations appeared. The difference in the magnitude of the
simulated electric field strength in the RC model with and without the door for a fixed
rotational 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 without and Figs. 7.4b) and 7.4f) including the door. Whereas the effect at a frequency
of f = 250 MHz (Figs. 7.4a) vs. 7.4b)) is just a slight distortion of |E| in the nearest
vicinity of the door, at f = 300 MHz in Figs. 7.4e) vs. 7.4f) a considerable change of
the 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 detailed
model (cf. Section 6.1). The influence of the door along with other small geometric
details increases with rising frequency (see also Section 7.4).
Along with the necessity to consider small structural details it is essential to utilize
correct inner dimensions in the RC simulations: The simulated chambers depicted in
Figs. 7.4c) (f = 250 MHz) and 7.4g) (f = 300 MHz) are ∆w = 0.1 m wider than the
other simulation models and the RC prototype. The length l and height h were kept
identical to the dimensions in the chamber prototype. Compared with the RC simulation
models 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 corresponding free-space wavelength λ0 is much larger than the small geometrical modification ∆w
of the RC.
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
122
a)
b)
e)
f)
y
z
x
c)
d)
g)
h)
Figure 7.4: Magnitude of the electric field strength |E| simulated in the xy-plane at z = 2 m above the chamber bottom at f = 250 MHz
in a). . . d) and f = 300 MHz in e). . . h). Analyzed is the effect of different RC geometries on the electric field: a) and e) simple
partly symmetric RC without door, b) and f) detailed asymmetric RC with door, c) and g) RC which is ∆W = 0.1 m wider than
the other RCs, d) and h) RC walls are of conductivity “High” (see Table 6.4). Excitation is a biconical antenna. Table 7.1 shows
the differences between the various geometries.
7.3 PROTOTYPE REVERBERATION CHAMBER ANALYSIS
123
To investigate the numerical stability of the simulations, the RCs were modeled with
various conductivity values. A comparison for two different conductivities is shown in
Fig. 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 results
are shown in Figs. 7.4b) and 7.4f). As expected, for a higher conductivity value the
magnitude of the electric field increases whereas the overall field pattern shape stays
the same. Similar results are obtained for other conductivities, i.e. higher conductivity
values consistently lead to higher, and lower conductivities to lower field magnitudes.
Stability problems reported by other authors (e.g. [43, 86]) using different numerical
techniques 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 a
rectangular cavity changes significantly the field distribution. In an RC analysis it is of
particular 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 sixpaddle 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 xyplane at a frequency of f = 400 MHz. It can be clearly seen how the magnitude of
the electric field varies. The excitation is a logper antenna positioned in front of the
chamber door and pointing towards the stirrer. Large E-field variations over one stirrer
revolution 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 stirrers
operated within the simulated prototype RC.
7.3.3 Different reverberation chamber excitations
An important issue for reliable RC simulations is the implementation of the excitation
source in the numerical code, as mentioned before in Section 4.3. To facilitate the
discretization of the RC structure and the simulation setup, often an ideal Hertzian
dipole is used, implicitly assuming that the effect on the actual field distribution inside
the chamber will be rather small. Whereas the assumption that different antennas will
lead to similar results in a statistical sense (i.e. a large number of samples taken from
a large number of stirrer positions, see Section 2.4) is certainly true, the results for a
given, 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 at
a frequency of f = 300 MHz using an ideal Hertzian dipole compared with a broadband
logper antenna in Fig. 7.6b). Both the Hertzian dipole and the center of the active region
of the logper antenna at f = 300 MHz were positioned at the same location inside the
RC. The difference in the field pattern is quite remarkable and can be attributed to the
strong 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 the
antenna setup in the measurement prototype) positioning and alignment of an excitation
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
124
j = 0°
j = 30°
j = 60°
y
j = 120°
j = 150°
z
j = 90°
x
j = 180°
j = 210°
Figure 7.5: Magnitude of the electric field strength |E| at f = 400 MHz simulated in the xy-plane at a height of z = 2 m above the
chamber bottom. The vertical six-paddle stirrer is rotated from one position to the next by ∆ϕ = 30◦ . Excitation is a logper
antenna in front of the chamber door.
7.3 PROTOTYPE REVERBERATION CHAMBER ANALYSIS
125
antenna are important for the simulation: As a reference configuration, the setup in
Fig. 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 at
an alignment angle of 0◦ . The rotational stirrer angle in Fig. 7.6 remained fixed for all
simulations at ϕ = 45◦ . Compared to the reference position and alignment of the logper
antenna in Fig. 7.6b), Fig. 7.6c) reveals that the field pattern changes significantly if the
excitation antenna is moved by ∆x = 0.2 m into +x-direction. Instead of e.g. four peaks
in 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 remains
at (x0 , y0 , z0 ), but is aligned at an angle of 90◦ perpendicularly instead of parallel to
the RC side walls: similar as in Fig. 7.6b) four peaks of the electric field magnitude in
the xz-cut plane exist, they are however slightly shifted in Fig. 7.6d). In addition, the
overall 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 the
reference configuration depicted in Fig. 7.6b).
The results shown in Figs. 7.4, 7.5, and 7.6 clearly indicate that a correct validation of
the simulated results through measured field data is only possible with an RC simulation
model accounting for seemingly insignificant small geometric details (indentations such
as a door or protrusions from stirrer mounts) and utilizing appropriate conductivity
values for the chamber materials. Furthermore, realistic as well as correctly positioned
and aligned TX/RX antennas, which resemble the actual antennas employed in the
prototype chamber, must be considered in any meaningful RC simulation model.
Excitation Antenna
Figure
Type
Position
Angle
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◦
Fig. 7.2
Table 7.2: Simulation parameter overview for the different types of reverberation chamber
excitations shown in Figs. 7.3, 7.6 and 7.10.
126
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
a)
c)
y
z
x
b)
d)
Figure 7.6: Magnitude of the electric field strength |E| simulated in the reverberation chamber
at f = 300 MHz for different excitations: a) Hertzian dipole, b) logper antenna, c) logper
antenna 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 a
glance 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 the
total amount of comparison data is presented in this section. Due to unavailability of
measurement equipment no validation has been carried out based on the magnetic field
H.
7.4.1 Measurement setup
Fig. 5.7 shows the equipment setup used for measurements of the electric near field as well
as 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 insufficient
spatial field resolution. Measurements of the near field were taken using diode-equipped
field probes (Section 5.2.2) measuring simultaneously |Ex |, |Ey |, and |Ez | components
of the electric field. Comparisons between simulated and measured results in this thesis
are 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 between
measured and simulated results:
• position and alignment of the field probe in the measurement is not the same as
in the simulation
• small geometric details of the prototype RC are not accurately modeled in the
simulation (Section 7.3.1)
• the excitation antenna used in the simulation is different from the actual antenna
in 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 probes
inside the RC using a coarse optical measurement grid of 0.1 m×0.1 m on the chamber
floor in combination with laser range distance metering for final precision alignment. By
including the RC door into the simulation model, a better agreement between simulated
and measured results was achieved. To avoid unintentional loading and distortion of
the EM field, tripods were removed and antennas either suspended from the chamber
ceiling with plastic ropes and Velcro or placed onto styrofoam blocks (see Fig. 5.10). An
encoder-controlled servo-motor and a special anti-backlash gearbox facilitate an angular
positioning accuracy of the vertical 6-paddle stirrer of ∆ϕ < 1◦ . Coaxial RF cables were
routed as close as possible to the walls of the prototype RC to reduce field distortion,
unnecessary power lines and cable ducts were removed.
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
128
|E | [V/m]
100
90
80
j=
|E | [V/m]
100
90°
f = 300 MHz
Simulation
Measurement
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
-1.2
-0.8
0.4
0
y [m]
-0.4
0.8
0
-1.2
1.2
j=
-0.8
0
y [m]
-0.4
a)
|E | [V/m]
100
90
80
j=
|E | [V/m]
100
90°
f = 500 MHz
Simulation
Measurement
90
80
70
60
60
50
50
40
40
30
30
20
20
10
10
-0.8
-0.4
0
y [m]
0.4
0.8
1.2
0.4
0.8
1.2
b)
70
0
-1.2
135°
f = 300 MHz
Simulation
Measurement
0.4
0.8
1.2
c)
0
-1.2
j=
135°
f = 500 MHz
Simulation
Measurement
-0.8
-0.4
0
y [m]
d)
Figure 7.7: Comparison between measurement and simulation of the magnitude of the electric
field 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◦ at
frequencies of a), b) f = 300 MHz and c), d) f = 500 MHz. Excitation source is a biconical
antenna at f = 300 MHz and a logper antenna at f = 500 MHz. Note the good agreement
between 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 actual
measured near field data. The laborious procedure of near field measurements is only
carried out for the purpose of chamber calibration and the gathered data is immediately “processed” to compute specific performance measures of the RC such as statistical field homogeneity (Section 2.7), randomness of polarization, correlation coefficients
7.4 MEASUREMENT VERSUS SIMULATION
|E | [V/m]
100
90
80
j=
|E | [V/m]
100
90°
f = 700 MHz
Simulation
Measurement
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
-1.2
-0.8
0
y [m]
-0.4
129
0.4
0.8
1.2
j=
0
-1.2
-0.8
0
y [m]
-0.4
a)
|E | [V/m]
100
90
80
j=
|E | [V/m]
100
90°
f = 1000 MHz
Simulation
Measurement
90
80
70
60
60
50
50
40
40
30
30
20
20
10
10
-0.8
-0.4
0
y [m]
c)
0.4
0.8
1.2
0.4
0.8
1.2
b)
70
0
-1.2
135°
f = 700 MHz
Simulation
Measurement
0.4
0.8
1.2
0
-1.2
j=
135°
f = 1000 MHz
Simulation
Measurement
-0.8
-0.4
0
y [m]
d)
Figure 7.8: Comparison between measurement and simulation of the magnitude of the electric
field 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◦
at frequencies of a), b) f = 700 MHz and c), d) f = 1000 MHz. Excitation source is a
logper antenna. Compared with Fig. 7.7, note how the agreement between measurements
and simulations progressively deteriorates as the frequency increases.
(Section 2.5), standard deviations (Section 2.7), or field anisotropy and inhomogeneity
coefficients (Section 2.3) [6]. These parameters are perfectly suited to analyze the performances of different RCs – they are however not suitable at all for the comparison
of RC simulation results against measurements: Since completely different EM fields
can still generate an identical field uniformity, the same correlation coefficient or equal
anisotropy coefficients, a thorough validation of RC simulation results cannot be accomplished using this type of “processed data”. “Processed data” can be classified as any
130
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
result obtained from large sets of values (e.g. electric field values measured or simulated
in several points over one stirrer revolution) to create one single metric (such as a correlation coefficient). Generally, any kind of data which cannot be uniquely “mapped” to
a corresponding near field or current distribution, should not be used for the validation
of an RC simulation. Therefore it was decided to use “raw”, “unprocessed” near field
data to benchmark simulated results.
Fig. 7.7 and Fig. 7.8 depict the mangitude of the electric field |E| measured along a line
in the y-direction (x = 0.77 m, y = −1.2 . . . 1.2 m, z = 2 m) within the RC compared
against 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. In
addition, Fig. 7.9 exhibits a comparison between measurement and simulation, where
the RC simulation was carried out once with and once without door. Excitation source
is a biconical antenna for f = 200 MHz, f = 250 MHz, f = 300 MHz and a logper antenna 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 values taken next to the RC walls, which might be due to limitations in the simulation
method 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 agreement
between measurements and simulations progressively deteriorates: Whereas some measured peaks in Fig. 7.8 can still be reproduced by the simulation, others are shifted in
their 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 agreement” [12]. The spatial p-norm in its most general form is denoted by
xp =
)
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) expressed
by a normalized spatial 2-norm ∆E as defined by (7.3).
7.4 MEASUREMENT VERSUS SIMULATION
131
Adapted to EM fields, the 2-norm (also called Euclidean vector norm) is defined as
E2 =
)
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
(
M
2
(|Es (xi , yi , zi )| − |Em (xi , yi , zi )|)
i=1
∆E
√
=
(7.3)
∆E = √
M
M
can be used, where M denotes the total number of spatial positions (xi , yi , zi ) used for
comparison. 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 was
computed for both stirrer angles ϕ = 90◦ and ϕ = 135◦ using the data shown in Fig. 7.7
and Fig. 7.8 and confirms the good agreement between simulation and measurement for
lower 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 the
fact that the EM field inside an RC becomes extremely sensitive to even tiny geometric
details. 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 a
simulation practically not feasible at higher frequencies f fLUF – unless one is willing
to undertake the challenge to model and discretize virtually every nut and bolt of the
chamber [67].
|E | [V/m]
RC without door
100
f = 200 MHz
90 j = 225°
Biconical
Simulation
80
feed
antenna
Measurement
70
RC with door
|E | [V/m]
RC without door
100
f = 250 MHz
90 j = 225°
Biconical
Simulation
80
feed
antenna
Measurement
70
60
60
50
50
40
40
30
30
20
20
10
10
0
-1.2
-0.8
-0.4
0
y [m]
a)
0.4
0.8
1.2
0
-1.2
-0.8
-0.4
0
y [m]
RC with door
0.4
0.8
1.2
b)
Figure 7.9: Influence of the chamber door: comparison between measurement and simulation of
the magnitude of the electric field strength |E|. |E| is shown at a rotational stirrer position
of ϕ = 225◦ and frequencies of a) f = 200 MHz and b) f = 250 MHz.
132
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
7.4.3 Statistical benchmarks
As the EM field within an RC is completely deterministically defined for a fixed rotational
stirrer angle ϕ = ϕ0 and at a given position (x0 , y0 , z0 ), near field based validations of a
simulation can be carried out as presented in Section 7.4 as well as Fig. 7.7, Fig. 7.8, and
Fig. 7.9. Since RCs are usually regarded as a “statistical” EMC test environment, most
papers 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 agreement
between statistics of simulated and theoretical or measured data even if there is total
disagreement between the field simulation and the measurement. Fig. 7.10 shows the
statistical distribution of the simulated electric field strength |E|/|Emax | sampled in a
rectangular 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◦ . Although the near field excited in the RC by the three antenna configurations has only
very little in common (cf. Fig. 7.6), all electric field histograms exhibit a fairly good
agreement with the theoretically expected χ(6) distribution for the field magnitude. This
is especially true for the logper versus the pivoted logper antenna. The Hertzian dipole
exhibits from a visual point of view a slight deviation from the analytical χ(6) distribution (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 obtained
with a Hertzian dipole excitation as shown in Fig. 7.10 will be rejected. For the two
logper antenna orientations however, a goodness-of-fit test accepts the hypothesis of a
χ(6) distribution. Looking only at the simulated EM fields processed to a statistical distribution, one would accept the two RC simulation results using the logper TX antenna
as “correct”, possibly also the one with the Hertzian dipole.
Contrary to this, the only simulated near field that matches measurements in the RC
prototype, results from the logper antenna in its standard position as indicated in Fig. 7.7
and Fig. 7.8. Therefore one cannot conclude that a simulation showing the χ(6) (respectively χ2(6) ) behavior is correct in the sense that it approximates the actual field inside
an RC. Validations exclusively based on statistics allow only the conclusion that the EM
simulator works well as a rather sophisticated random number generator producing 2-D
Gaussian-distributed Ex , Ey , and Ez field components – they do not reveal how well
reality 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 on
typical RC parameters (field uniformity, correlation, etc.), two other RCs were modeled
in 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 account the height of the corrugations (see Section 6.1.4)
7.5 CORRUGATED AND CUBIC REVERBERATION CHAMBER
Analytical c(6)-distribution
Number of samples
500
133
Distribution of simulated fields
Hertzian
dipole
Logper
antenna
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
0
a)
0.2
0.4
0.6
0.8
|E |
1 |Emax|
b)
Number of samples
500
Logper
90° pivoted
400
300
200
100
0
0
0.2
0.4
0.6
0.8
|E |
|
E
1 max|
c)
Figure 7.10: Statistical distribution of the normalized magnitude of the simulated electric
field strength |E|/|Emax | within the RC. Results shown were calculated at a frequency
f = 500 MHz and cumulated from angular stirrer positions of ϕ = 0◦ . . . 355◦ with 5◦ step
angle 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 near
field 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 against
the other RCs. In all RCs the same vertical 6-paddle stirrer is utilized, consisting of six
square 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 to
use identical stirrers was made to facilitate the comparison between different RCs, with
the shape and dimensions of the chamber walls being the only variables. The wedges in
the corrugated RC have an amplitude of 0.1 m and a valley-to-valley distance of 0.15 m.
134
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
7.5.1 Simulated near field distribution
For the prototype versus cubic versus corrugated RC comparisons, the three chambers
were simulated over a frequency range of 50 . . . 500 MHz (frequency 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 was computed in ten equally spaced
planes parallel to the xy-plane with a spatial resolution of 0.05 m, i.e. near field data
is available at 27 440 points throughout the chamber. With Fig. 7.11 it is possible to
compare qualitatively the different chambers: depicted is the magnitude of the simulated
electric 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 positions
of ϕ = 0◦ , 10◦ , 20◦ . The chamber performance can be qualitatively analyzed by looking
at the change of the overall field distribution and by examining how the field varies in
the cut planes at x = −1.45 m and y = −1.25 m from one stirrer step to the next. The
simulated EM near field within the RCs is utilized to investigate further the following
two controversially discussed statements:
• The idea to decrease the LUF by using corrugated chamber walls to enhance the
field uniformity was suggested by several authors (e.g. [173]) – however the approach presented in [173] is somewhat questionable since the “supporting data”
was achieved by changing twice antenna locations, shifting the frequency twice
and neglecting the worst 25% of the totally obtained data set. In addition there
is literature stating that great surface irregularities, such as corrugations, tend
to increase the difficulty of providing uniform EM fields within devices similar to
RCs. Already in e.g. [71] (see Section 4.1) it is outlined that for the application of
microwave food heating “it has been found that great surface irregularities [. . . ],
for example deep corrugations, tend to increase the difficulty of providing uniform
heating”. This in turn supports the theory that corrugations do not enhance EM
field uniformity within an RC.
• As shown in the modal analysis in Section 2.2, when choosing a rectangular room
as a basis for building an RC, ideally the dimensions should not be simple multiples or rational fractions of each other. The idea governing this statement
is that this choice will result in the largest number of (non-degenerate) modes
with different resonance frequencies, which is usually thought to improve chamber 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 usually required “∂N/∂f = 1.5 modes/MHz above cutoff”-criterion is reached consistently only at much higher frequencies compared to a cavity of similar rectangular,
but non-cubic shape. Nevertheless several authors claim that cubic chambers may
generate a more uniform field than standard rectangular RCs (e.g. [75]).
Visually inspecting the near field however does not yield a final conclusive (and especially quantitative) answer – it is necessary to take a closer look at the two key RC
parameters: 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 the
corrugations or the cubic shape should show the biggest impact.
7.5 CORRUGATED AND CUBIC REVERBERATION CHAMBER
10°
20°
Prototype RC
0°
135
y
z
x
a)
y
z
x
b)
c)
10°
20°
Cubic RC
0°
y
z
x
d)
y
z
e)
f)
10°
Corrugated RC
0°
y
g)
x
z
20°
x
y
h)
z
x
i)
Figure 7.11: Magnitude of the simulated electric field strength |E| computed in the xy-plane at a
height z = 2 m 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 a
logper antenna in all chambers, see Fig. 6.8c).
136
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
7.5.2 Correlation analysis
As opposed to the qualitative field-pattern-based analysis of different stirrers shown before, 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 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 in Fig. 7.12 corresponds to the absolute value of the
correlation |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 for
all RCs indicates that the stirrer is still rather ineffective in providing a large number
of sufficiently uncorrelated samples at low frequencies. The cubic RC shows the best
correlation-based performance at these frequencies. At f = 250 MHz and f = 300 MHz
however the corrugated RC exhibits a good performance whereas in the cubic RC only
relatively high correlation values are obtained (Fig. 7.12). |ρ(ϕj )| of the prototype chamber reaches intermediate levels at both frequencies. From a correlation point of view,
neither the cubic nor the corrugated RC exhibit convincing results across all frequencies
which 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 terms
of the combined three-axis standard deviation σxyz and the single-axis standard deviations σx , σy , and σz as proposed in [6]. These quantities are calculated from the three
components of the electric field Ex (xi , yi , zi ), Ey (xi , yi , zi ), and Ez (xi , yi , zi ). For both
the per-axis standard deviations σ̃ξ as well as the combined standard deviation σ̃xyz , the
IEC 61000-4-21 standard [6] requires for a “well operating” RC with sufficient statistical
field uniformity and a given uncertainty within all frequencies 80 MHz ≤ f ≤ 100 MHz
σ̃ξ ≤ 4 dB and σ̃xyz ≤ 4 dB
(7.4)
For frequencies 100 MHz ≤ f ≤ 400 MHz, the limits for σ̃ξ and σ̃xyz decrease linearly
from 4 dB to 3 dB. Finally,
σ̃ξ ≤ 3 dB and σ̃xyz ≤ 3 dB
(7.5)
is required for all frequencies f ≥ 400 MHz. Fig. 7.13 shows the statistical field uniformity envelopes of σ̃ξ and σ̃xyz for all three RCs. As expected, at frequencies in the
50 . . . 250 MHz range, the field uniformity is clearly insufficient in all chambers, whereas
the cubic RC performs the worst. Starting from f > 300 MHz however, sufficient statistical field uniformity is achieved, indicated by electrical field standard deviations on the
order of σ̃ ≤ 3 dB – with the exception of the cubic RC exhibiting significantly higher
values at some frequencies, e.g. at f ≈ 400 MHz. Detailed per-component field uniformities 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 exhibits consistently superior or inferior field uniformity performance. Especially the cubic
7.5 CORRUGATED AND CUBIC REVERBERATION CHAMBER
Correlation |r(j)|
1
Correlation |r(j)|
1
f = 100 MHz
0.9
0.6
0.6
Cubic RC
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
60
120
Correlation |r(j)|
1
180
240
Stirrer angle j [°]
300
360
Prototype RC
Corrugated RC
0.8
0
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
120
120
180
240
Stirrer angle j [°]
300
360
0
180
240
Stirrer angle j [°]
300
360
f = 300 MHz
Prototype RC
Corrugated RC
0.8
Cubic RC
60
60
0.9
0.7
0
0
Correlation |r(j)|
1
f = 200 MHz
0.9
Cubic RC
0.7
0.5
0
Prototype RC
Corrugated RC
0.8
Prototype RC
Corrugated RC
0.7
0
f = 150 MHz
0.9
0.8
137
Cubic RC
0
60
120
180
240
Stirrer angle j [°]
300
360
Figure 7.12: Absolute value of the correlation coefficient |ρ(ϕ)| as a function of stirrer angle
ϕ = 0◦ . . . 355◦ in the prototype RC, the corrugated RC, and the cubic RC at a frequency of
f = 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 field
distribution within a cubic RC (including a stirrer) does not have anything in common
with the fields observed in a cubic cavity. The presence of a stirring device shifts the
modes in frequency depending on their respective field distribution away from the analytically calculated resonance frequencies [19]. Therefore the usually observed problem
of degenerate modes does not come into play within a cubic RC and contrary to the
widely accepted RC design guidelines [6] a cubic RC will not exhibit worse (or better)
performance than other rectangular RCs.
138
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
Standard deviation $
s [dB]
6
IEC limit line
Prototype RC
Corrugated RC
5
Cubic RC
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure 7.13: Envelopes of the statistical field uniformities σ̃xyz and σ̃ξ in the prototype RC, the
corrugated 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 in
Fig. 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 used
along with the canonical EUTs (CEUTs) modeled in Section 6.5.2.
7.6.1 Simulated near field distribution
Near field simulations were performed over a frequency range of 50. . . 1000 MHz (frequency 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 was
computed in ten equally spaced planes parallel to the xy-plane with a spatial resolution
of 0.05 m, i.e. near field data is available at 27 440 points throughout the chamber. With
Fig. 7.14 it is possible to investigate the effects caused by placing different EUTs within
the RC: depicted is the magnitude of the simulated electric field strength |E| computed
at a height of z = 2 m above the chamber bottom. The chamber is operated in the
immunity testing mode. Results shown were calculated at a frequency of f = 400 MHz
and a fixed stirrer position of ϕ = 210◦. The chamber loading can be qualitatively analyzed by looking at the change of the overall field distribution and by examining how
the field varies in the cut planes at x = −1.45 m and y = −1.25 m: shown is the empty
RC without EUT in Fig. 7.14a), the RC with the loop EUT in Fig. 7.14b), and the
box EUT operated in gap mode configuration in Fig. 7.14c). These CEUTs are tested
for immunity, i.e. they are “passive” and the excitation is the logper antenna in front
7.6 EQUIPMENT UNDER TEST SIMULATION
y
z
x
a)
b)
y
c)
139
z
x
d)
Figure 7.14: Magnitude of the electric field strength |E| simulated in the reverberation chamber
at f = 400 MHz with different canonical EUTs: a) RC without CEUT, b) RC with loop
CEUT (immunity), c) RC with box CEUT operated in gap mode (immunity), d) RC with
box CEUT operated in gap mode (emission).
the RC door. When comparing Fig. 7.14a) and Fig. 7.14b) (i.e. empty and loop EUT
loaded RC) it is immediately apparent that the loading introduced by the loop EUT is
very small at f = 400 MHz: the change of the electric field pattern and magnitude is
almost negligible, which is due to the small size of the loop EUT together with its high
conductivity 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 RC
depicted in Fig. 7.14a). The latter is an indication that the box EUT (conductivity also
1.1 · 106 S/m) loads the chamber only moderately - simulations performed during the
140
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
Standard deviation $
s [dB]
6
IEC limit line
No EUT
Loop EUT
5
Box EUT
4
3
2
1
0
0
100
200
300
Frequency
f [MHz]
No EUT
400
500
600
Figure 7.15: Envelopes of the statistical field uniformities σ̃xyz and σ̃ξ without an EUT, with
the canonical loop EUTs, and the box EUT, obtained according to the procedure outlined in
Section 2.7 (- - - IEC limit line).
course of this thesis involving larger EUTs and hence higher losses showed that strong
loading of the chamber is exhibited by a reduction in field magnitude and at the same
time a significant change in the field distribution, starting from locally around the position of the EUT to globally throughout the entire chamber with increasing EUT-induced
loading [194]. Fig. 7.14d) exhibits the electric field distribution obtained when the box
CEUT is tested for emissions, so that the logper antenna serves as a receiving antenna
picking up EM fields radiated by the “active” CEUT. It can be seen that the fields close
to the box CEUT show some similarity for the emission and the immunity configuration;
further away from the CEUT however, this resemblance gradually diminishes. As the
frequency is increased, this effect becomes more and more pronounced [194].
7.6.2 Field uniformity
The field uniformity with the CEUTs placed inside the RC was computed as outlined in
Section 2.7 and [6]. Field uniformity computations are based on the simulated electric
near field with the respective stirrer in operation in the prototype RC and are shown
in 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 uniformity is insufficient. This is due to the chamber size as well as geometry (the “at least
1.5 modes/MHz” and “more than 100 modes above cutoff” criteria are passed around
f = 270 MHz) and the stirrer effectiveness.
Starting from 350 MHz, sufficient statistical field uniformity is achieved for all CEUTs
within the testing volume, indicated by electrical field standard deviations of σ̂ ≤ 3 dB.
7.6 EQUIPMENT UNDER TEST SIMULATION
141
Without any EUT in the RC the field uniformity exhibits the best performance; as the
loading is increased from the simple loop EUT to the box EUT, field uniformity values slightly worse than in the “empty RC” setup are obtained in the simulation. For
the whole frequency range of f = 350 . . . 600 MHz (and also higher), however, the percomponent and the combined field uniformity σ̂ remains below 3 dB, confirming that
the loading of the RC even with the relatively large box EUT is still within an acceptable level. With the near field distribution investigated in Section 7.6.1 and shown in
Fig. 7.14, this result is a logical implication. Detailed per-component and the combined field uniformity σ̂ 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 much
larger 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 utilized
or the requirements on the field uniformity and hence uncertainty of the results must be
relaxed.
7.6.3 TX/RX antenna coupling
As mentioned in Section 7.3.3, often analytical point or line sources and ideal Hertzian
dipoles are employed for RC simulations, because they are easy to implement in a numerical code. This is problematic due to undesirable coupling effects between an EUT and
the TX/RX antenna setup. Generally, the following classification can be made for typical RC operation modes and the resulting magnitude of the electric field |E| (it is always
assumed that the underlying EM field ensembles are statistically independent) [126]
• Strong dominant direct (i.e. deterministic) coupling path and comparatively small
multi-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 distributed (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 to
the highly undesirable result of strong coupling between an EUT within the chamber
and the excitation: Due to its very low directivity (1.76 dB), the |E|/|Emax | distribution resembles almost a Gaussian distribution with nonzero mean, which is a clear
indication of a dominant coupling path in an EM environment with only little multipath propagation [195] – the exact opposite of a “well-behaved” RC, where for a proper
operation implicitly “pure” multi-path propagation is assumed. Through the usage of
antennas with higher directivity, this unwanted direct coupling can be considerably reduced: Fig. 7.16b) and Fig. 7.16c) show the respective |E|/|Emax | distributions obtained
with 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 resemble more a Rice distribution where direct, non-dominant coupling paths still exist,
but multi-path propagation is clearly dominant [195, 126].
142
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
M /Mmax
Hertzian dipole
Biconical antenna
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
a)
M /Mmax
1
0
|E |/|Emax|
Logper antenna
(toward EUT)
0.2
0.4
0.6
0.8
1
b)
Logper antenna
(toward corner)
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
c)
0.6
0.8
1
0
|E |/|Emax|
0.2
0.4
0.6
0.8
1
d)
Figure 7.16: Statistical distribution (normalized number of samples M/Mmax ) of the magnitude of the simulated electric field strength |E|/|Emax | within the RC. Results shown were
calculated at a frequency f = 300 MHz and cumulated from angular stirrer positions of
ϕ = 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 test
volume V ; d) logper antenna pointing toward an RC corner opposite the vertical stirrer. In
total, 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, the
orientation of this antenna was changed such that it points into a corner opposite to the
stirrer (as shown in Appendix D, Fig. D.2). According to theoretical considerations, this
represents the situation of a multi-path environment without any dominant direct coupling path, which results in a central χ(6) distribution for the magnitude of the electric
field 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 comparing
Fig. 7.16c) with Fig. 7.16d): In Fig. 7.16c) the |E|/|Emax | distribution appears to be
slightly offset from the origin for the logper antenna oriented towards the rectangular
test volume. Once however the antenna points away from this test volume towards a
chamber corner, the offset vanishes and Fig. 7.16d) shows a very good agreement with
the theoretical χ(6) distribution. The latter antenna orientation is also recommended
in the IEC 61000-4-21 standard [6] in order to prevent direct “illumination” of the volume where an EUT will be placed during RC testing. It is important to note that in
the graphical representation of Fig. 7.16, the number of samples M per class has been
normalized 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 (as
mentioned in Section 2.8), this normalization together with a much larger of samples
are the main reason why the visual appearance of Fig. 7.16a) differs considerably from
Fig. 7.10a).
7.7 Comparison of different stirrers
In this section the simulation and performance analysis of various types of EM stirrers
inside the RC prototype is presented. Several different stirrer designs and sizes are
compared against each other, and the influence of the stirrer axis orientation (vertical
vs. 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 in
detail in Section 6.2. It should be noted that all simulated stirrers can be circumscribed
by 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 “rotational
volume”. This was done in order to make the comparison “even-handed”, since from
a basic understanding of electromagnetics it is obvious that a much larger stirrer will
introduce much greater changes of the field distribution as it is rotated and therefore
exhibits a better performance than a smaller one.
7.7.1 Simulated near field distribution
Performance comparisons between all stirrer types were carried out based on the simulated electric near field with the respective stirrer in operation in the prototype RC. The
same RC was simulated with the six stirrers mentioned above over a frequency range of
50 . . . 500 MHz (frequency 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
j = 200°
b)
e)
f)
z
x
c)
d)
g)
h)
j = 210°
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
144
a)
y
Figure 7.17: Magnitude of the simulated electric field strength |E| computed in the xy-plane at a height z = 2 m above the chamber
bottom. 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-paddle
connected V-stirrer; d), h) upset Z-fold V-stirrer. Excitation source is a logper antenna.
7.7 COMPARISON OF DIFFERENT STIRRERS
145
near field was computed in ten equally spaced planes parallel to the xy-plane with a spatial resolution of 0.05 m in x- and y-direction, i.e. near field data is available at 27 440
points throughout the chamber. In order to compare more intuitively the impact of the
different stirrers on the field distribution, results data shown here was chosen from the
lower to medium frequency range (close to the LOF/LUF), where it is easier to visualize
how effective a particular stirrer is.
Fig. 7.17 depicts the magnitude of the simulated electric field strength |E| computed at
a height of z = 2 m above the chamber bottom for four different stirrers (the horizontal
and the upset Z-fold without gaps are not shown in this figure). These results were calculated at a frequency of f = 300 MHz and for a fixed angular position of ϕ = 200◦ (top
row) and ϕ = 210◦ (bottom row). The stirrer performance can be qualitatively analyzed
by looking at the change of the overall field distribution and by examining how the field
varies in the cut planes at x = −1.45 m and y = −1.25 m from one stirrer step to the
next but one. Immediately apparent is that the resulting chamber field is quite different
among all stirrers. The only moderate similarity can be seen between the field pattern
excited 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) and
Fig. 7.17g). This similarity however does not prevail at other frequencies, which suggests that very similar stirrer designs (here in terms of the larger electrical length of
the “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 rotation of 10◦ occurs for the “stacked cross-plate V-stirrer” in Fig. 7.17b) and f): Both
the spatial field distribution as well as the field in the two cut planes at x = −1.45 m
and y = −1.25 m change only marginally, which is due to the rather simple geometric
structure and symmetry of this particular stirrer.
A near field analysis at very low frequencies (around f = 100 MHz – these field distributions are not shown in this thesis) revealed that all stirrers are equally (in-)effective
far below the LUF: shape, orientation, and electrical size do not matter [193]. As shown
in Fig. 7.17, at f = 300 MHz all stirrers are “somewhat similarly effective” and by looking at the near field distribution it is difficult to judge whether one stirrer outperforms
another. Furthermore the impact of the gaps (two models of the 6-paddle and upset
Z-fold stirrer) remains unclear. Summing up, evaluating directly the near field without
processing it to a more useful metric does not help in classifying stirrers with respect to
their performance.
7.7.2 Correlation analysis
As opposed to the qualitative field-pattern-based analysis of different stirrers shown
before, the computation of the correlation allows a quantitative comparison of the stirrer
performance. The correlation coefficient ρ(ϕ) is calculated from the magnitude of the
electric 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 in
Figs. 7.18 and 7.19 corresponds to the correlation between the simulated |E| for the
reference stirrer angle ϕ = 0◦ and |E| for ϕ = 0◦ . . . 355◦ . Fig. 7.18 compares |ρ(ϕ)| for
four stirrers (out of which three have completely different designs), but mounted in the
146
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
|r(j)|
6-paddle V
Cross-plate V
Z-fold V
Z-fold V (gaps)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
60
120
180
j [°]
240
300
360
Figure 7.18: Absolute value of the correlation coefficient |ρ(ϕ)| as a function of stirrer angle
ϕ 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 rapidly
for all stirrers (except for the Z-fold with gaps), which indicates that they are electrically
sufficiently large in order to achieve substantial changes of the field pattern within the
RC. For both the 6-paddle and the Z-fold stirrer without gaps |ρ(ϕ)| remains relatively
small (with some oscillations), whereas the cross-plate stirrer reaches |ρ(ϕ) = 1| again at
ϕ = 180◦, which is due to its rotational symmetry. Clearly, this stirrer is not suitable for
the application within an RC and is outperformed by the much better 6-paddle stirrer
and the very well operating Z-fold stirrer without gaps. It is interesting to note that the
small “gap modification” of the Z-fold stirrer changes completely the correlation. The
(in the traditional sense) electrically much larger Z-fold stirrer without gaps performs
significantly better than the stirrer with the artificially introduced gaps.
When designing an RC, typical questions often are: In which orientation is a stirrer to
be mounted and does it matter whether the stirrer is made from one piece or consists
out of several separate parts? Fig. 7.19 depicts the effect of the standard 6-paddle stirrer
mounted in two orientations in the RC: For this particular RC test volume it can be
seen that the vertically mounted stirrer works slightly better than the horizontal one
by exhibiting a lower |ρ(ϕ)| throughout the ϕ = 0◦ . . . 355◦ range. Fig. 7.19 also shows
that surprisingly the 6-paddle connected stirrer (although electrically larger than the
standard 6-paddle stirrer) performs significantly worse than the 6-paddle stirrer made
7.7 COMPARISON OF DIFFERENT STIRRERS
|r(j)|
1
6-paddle V
147
6-paddle connected V
6-paddle H
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
60
120
180
j [°]
240
300
360
Figure 7.19: Absolute value of the correlation coefficient |ρ(ϕ)| as a function of stirrer angle ϕ
at a frequency f = 300 MHz: vertical 6-paddle stirrer vs. vertical 6-paddle connected stirrer
vs. horizontal 6-paddle stirrer.
of separate, not-connected parts. This effect needs further analysis in order to clarify
whether the considerable performance difference appears consistently for a wide range
of frequencies or if it is only a narrow band phenomenon. Contrary to the frequencydomain EFIE-based MoM method utilized in this thesis, a numerical technique providing
broadband 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 any
intentional 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 than
a 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 all
stirrers the Z-fold stirrer however exhibits the best performance in the lower frequency
region, followed by the cross-plate and the vertical/horizontal 6-paddle stirrers. As the
frequency is increased, the overall correlation tends to be low (with the exception of the
cross-plate stirrer; this effect is due to its structural symmetry), the vertical 6-paddle
and the Z-fold stirrer perform the best, followed by the horizontally mounted 6-paddle
stirrer. With respect to the impact of gaps between individual stirrer elements (to
change the electrical size of a stirrer), their effect remains still unclear: whereas gaps are
148
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
Standard deviation $
s [dB]
6
IEC limit line
6-paddle V
6-paddle V (no gaps)
5
6-paddle H
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure 7.20: Envelopes of the statistical field uniformities σ̃xyz and σ̃ξ for the vertical 6-paddle
stirrer, the vertical 6-paddle connected stirrer, and the horizontal 6-paddle stirrer, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
beneficial in the 6-paddle stirrer (lower correlation than the connected version), they are
not beneficial for the upset Z-fold stirrer (the Z-fold stirrer made out of one single piece
performs better in terms of correlation than the model with gaps). It seems impossible
to clearly state that a large stirrer made out of one piece of metal performs significantly
better or worse than a replica version with small gaps in-between individual plates. For
this reason, unfortunately, a case-by-case analysis has to be carried out.
7.7.3 Field uniformity
Evaluating the field uniformity within the RC leads to a similar conclusion with respect
to the stirrer gaps as the correlation analysis: as shown in Fig. 7.20 and Fig. 7.21 for some
stirrers, gaps have a positive effect on the field uniformity (e.g. 6-paddle stirrer), whereas
for 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 perform
in a similar way over a broad range of frequencies. It is interesting to note that the stirrer
orientation plays an important role for the field uniformity, as the horizontal 6-paddle
stirrer performs significantly worse than the vertically mounted 6-paddle stirrer. This
effect might be due to the TX antenna orientation and strong, orientation-dependent
coupling between the antenna and the 6-paddle stirrers. Detailed per-component field
uniformities can be found in Appendix D, Fig. D.6. . . Fig. D.11.
7.7 COMPARISON OF DIFFERENT STIRRERS
149
Standard deviation $
s [dB]
6
IEC limit line
Cross-plate V
Z-fold V
5
Z-fold (gaps) V
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure 7.21: Envelopes of the statistical field uniformities σ̃xyz and σ̃ξ for the vertical 6-paddle
stirrer, vertical cross-plate stirrer, and vertical Z-fold stirrer, obtained according to the procedure 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 for
the RC from which the performance data originated, especially at frequencies around
the LOF/LUF. Generally, a stirrer exhibiting superior performance over other stirrers
(in terms of statistical field uniformity or isotropy, correlation, and especially the unfortunate SR), may perform also better in another chamber with a different geometry. As
shown here, there is however no obvious reason, why it must mandatorily outperform
these “other stirrers” in any other chamber, as the stirrer performance depends strongly
on parameters such as the position or the orientation of the stirrer within the chamber
and on the field distribution (which is strongly influenced by the chamber geometry
itself). As a conclusion resulting from the near field, correlation, and field uniformity
analysis of the stirrers spanning 50 . . . 500 MHz, the best performance in the prototype
RC is exhibited by the two Z-fold stirrers (with and without gaps), closely followed by
the vertically mounted 6-paddle stirrers (with and without gaps). These four stirrers
perform significantly better than the symmetric cross-plate stirrer, which is still better than the horizontal six-paddle stirrer. Table 7.4 summarizes the gathered stirrer
performance 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 simulation runtimes (on the order of several weeks to months). Therefore an idea originally
published in [104, 105] was taken up, proposing to do a stirrer analysis and optimization
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
150
100 MHz
300 MHz
Stirrer
performance
NF a
|ρ(ϕ)|
σ̃
NF a
|ρ(ϕ)|
σ̃
Best
Z b (gaps)
Zb
Z b (gaps)
Z b (gaps)
6V d
6V d
↑
–
X-plate c
X-plate c
Zb
Zb
Z b (gaps)
–
Z b (gaps)
Zb
6V d
6H e
Zb
–
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 the
field uniformity standard deviation σ̃ at 100 MHz and 300 MHz.
7.7 COMPARISON OF DIFFERENT STIRRERS
f
i
=
60
°
r
Ei
r
ki
151
f
i
=
60
°
r
Ei
r
ki
Figure 7.22: Stirrer RCS calculations using circularly polarized plane wave excitation. The
direction 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 field
during 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 direction
of the radar receiver, i.e. it is a measure of the ratio of backscatter power per unit solid
angle in the direction of the incident wave (from the target) to the power density that
is 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 directivity
of the target. The RCS was calculated by using the plane wave excitation (circularly
polarized) mentioned above and by varying the direction of the plane wave incidence
from 0 . . . 180◦ with increments of 20◦ (i.e. 10 directions, see Fig. 7.22). The stirrer was
rotated in angular steps of 5◦ . Far field calculations utilized for the RCS are based on
37×73 = 2701 data points and take only the scattered field into consideration (resolution
in both azimuthal and elevational direction 5◦ ).
Based on the RCS results, it was tried to compare the individual stirrer performances
against 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,
152
7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
|r(j)|
1
6-paddle V
6-paddle V (RCS)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
60
120
180
j [°]
240
300
360
Figure 7.23: Absolute value of the correlation coefficient |ρ(ϕ)| as a function of stirrer angle ϕ
at a frequency f = 300 MHz computed for the vertical 6-paddle stirrer within the RC and in
free-space using the RCS as defined by (7.6).
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 the
EM field (compare e.g. with Fig. 7.5)
• consequently, also processing the gathered RCS data further and computing the
correlation did not help much, as the correlation coefficient is almost unity regardless of the stirrer shape and the angle of incidence for the RCS over a broad
range of frequencies (see Fig. 7.23 for a comparison of the correlation coefficient
computed for the 6-paddle stirrer within the prototype RC and the same stirrer in
free-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 further
pursued.
7.8 Simulation and measurement time budget
Measurements of the electric field only provide limited information on the complex physical 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 it
takes to compute one single metric such as correlation or field uniformity over a broader
7.8 SIMULATION AND MEASUREMENT TIME BUDGET
E-field at. . .
1 spatial position
1 stirrer angle
10 frequencies
Simulation time
Measurement time
1 hour. . . 7 days a
60 s
10 s b
4 hours
1 hour. . . 7 days a
180 s
100 spatial positions
1 stirrer angle
1 frequency
1 spatial position
10 stirrer angles
1 frequency
a
153
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 electric
field in RC simulations based on the EFIE MoM versus measurements.
frequency range [178]. Table 7.5 gives an overview on the average time expenditure
required to simulate or measure the electric field at a given number of spatial positions
within the prototype (or any similarly sized) RC over several frequencies with a certain
number of rotational stirrer angles. It is interesting to note how much a particular EM
field analysis differs in time between simulation versus measurement: computing e.g. the
electric field within the prototype RC at ten frequencies and one stirrer position takes
between one hour up to several days depending on the actual frequency (see Section 3.4
and Table 3.1). Performing the same task in a measurement setup takes only about one
minute at one spatial position, irrespective of the frequency. Measuring however the
electric field throughout the entire chamber at 10 000 spatial positions for one frequency
can easily take two or three weeks (depending on the actual field probe system and the
number of field probes used in the measurement setup). Contrary to this, in the simulation – compared to the total simulation time – once the currents are computed it does
not 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 measurements yields a reciprocal behavior for the time expenditure: analysis tasks which
can be quickly performed in a simulation require a lot of time in measurements and vice
versa (see Table 7.5). Especially frequency sweeps and a rotation of the stirrer are timeconsuming tasks in a frequency-domain MoM simulation compared with measurements.
This is a problematic issue for several typical RC performance parameters: if, for example, the correlation coefficient or field uniformity is of interest at 20 frequencies and with
a stirrer angle resolution of 5◦ (i.e. 72 rotational stirrer positions), this translates into
1440 simulations which need to be performed. Generating plots such as e.g. Fig. 7.21
or Fig. 7.19 therefore easily adds up to combined CPU simulation run times of several
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7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
months. As long as the required number of spatial field points remains small and the
chamber 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 be
visualized at a large number of spatial positions in order to get a deeper understanding
of the physical processes within an RC, or the RC geometry has to be changed (e.g. from
a standard rectangular chamber to a corrugated RC), EM simulations are advantageous
and 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 particular
advantages and drawbacks of a comparison with analytically calculated results, results
obtained by a different numerical technique, and measurements were described. Measurements were utilized in this thesis to validate the simulation results, as they provide
an additional insight into the “reality of the chamber physics”.
Cavity simulations were performed to investigate the influence of the chamber door and
to derive suitable conductivity values. This analysis revealed that by far the biggest perturbance seemed to be caused by the door. To facilitate modeling of the cavity and the
RC, 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. The
simulation allowed a thorough analysis of the change in the field distribution with and
without 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 the
chamber geometry was investigated. As expected, the choice of a reasonable conductivity is important if the absolute field values in the chamber are of interest; if only the
relative field distribution needs to be known, absolute conductivity values do not play
a 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 the
geometry (for example the recess of the door) can have a significant impact on the field
within the RC.
The effect of Hertzian dipoles, λ/2-dipoles, biconical, logper, and horn antennas was
examined in the RC simulations. It was found that the actual antenna type employed in
an RC strongly influences the field distribution. Whereas the assumption that different
antennas will lead to similar results in a statistical sense (i.e. a large number of samples
taken 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 a
good agreement between measurements and simulations is desired, the correct modeling
of the TX/RX antennas as used in the measurements is of prime importance. This does
not 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 the
actual prototype RC, simulations were validated by extensive measurements. It was
pointed 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 using
this type of “processed data”. For lower frequencies (f = 50 MHz. . . 500 MHz) simulation and measurement were found to agree well, however as the frequency was increased
to f = 700 MHz. . . 1000 MHz, the agreement between measurements and simulations
progressively deteriorated. It was concluded that this “breakdown” of the simulation
starting from f > 600 . . . 700 MHz is related to the fact that the EM field inside an RC
becomes extremely sensitive to even tiny geometric details. This (from an RC application’s point of view desirable) phenomenon renders a simulation practically not feasible
at frequencies much greater than the lowest usable frequency – unless one is willing
to undertake the challenge to model and discretize virtually every nut and bolt of the
chamber.
The simulated near field distribution within the RC was used as a basis to compute the
correlation coefficients, statistical field uniformity, and coupling paths for the analysis of
different RCs, stirrers, and canonical equipment under tests (EUTs). The unprocessed
near field was useful to get a rough overview on how the field distribution (e.g. local
maxima and minima) changes as the stirrer rotates. Otherwise, raw near field data did
not provide very conclusive insights into whether an RC is particularly good or bad and
was therefore of little practical use.
To investigate the influence of special chamber designs on the RC performance, a corrugated and a cubic RC were modeled in addition to the prototype RC. From a correlation
point of view, neither the cubic nor the corrugated RC exhibited convincing results across
all frequencies which would clearly outclass the standard rectangular prototype RC. The
field uniformity at frequencies in the lower range was insufficient in all chambers, with
the cubic RC performing the worst. Starting from f > 300 MHz, however, sufficient
statistical field uniformity was achieved – with the exception of the cubic RC exhibiting
significantly higher values at e.g. f ≈ 400 MHz. As a summary, neither a cubic chamber
nor 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 perform
as bad as always alleged, mainly due to the fact that the field distribution within a cubic
RC (including a stirrer) does not have anything in common with the fields observed in
a cubic cavity. The presence of a stirring device shifts the modes in frequency depending on their respective field distribution away from the analytically calculated resonance
frequencies. Therefore the usually observed problem of degenerate modes does not come
into play within a cubic RC and contrary to the widely accepted RC design guidelines a
cubic 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 loading, field uniformity, and coupling path analysis. As the loading of the chamber was
gradually increased from a dipole via a simple loop EUT to the box EUT, field uniformity 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 loading of the RC even with the relatively large box EUT was still within an acceptable
level. The investigation of different coupling paths revealed that the usage of a Hertzian
dipole in an RC simulation leads to the highly undesirable result of strong direct coupling between an EUT and the excitation. Through the usage of realistic antennas with
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7 REVERBERATION CHAMBER SIMULATION AND MEASUREMENT
higher directivity, this unwanted direct coupling can be considerably reduced. Three
special multipath/direct path scenarios were simulated resulting in Gaussian, Rice, and
Rayleigh statistical distributions.
Several different stirrer designs and sizes were compared against each other, and the
influence of the stirrer axis orientation within the chamber was shown. In a correlation
analysis of all stirrers, the Z-fold stirrer exhibited the best performance in the lower frequency 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 performed
the best, followed by the horizontally mounted 6-paddle stirrer. By using only correlation data it was however impossible to clearly state that a large stirrer made out of one
piece of metal performed significantly better or worse than a replica version with small
gaps in-between individual parts. This was also true for the field uniformity: in some
stirrers, gaps had a positive effect, whereas in others their effect was negative. It can
be concluded that “complicated” stirrers exhibit consistently superior performance than
stirrers 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 frequencies. It is interesting to note that the stirrer orientation played an important role for the
field uniformity, as the 6-paddle stirrer performed significantly worse in the horizontal
than in the vertical position. Summing up, it can be stated that stirrer performance
comparisons are valid to a great extent only for the RC from which the performance
data originated. This makes the design of a universal “high performance” stirrer very
difficult, if not impossible. The much faster stirrer performance analysis in free space
proved to be unsuccessful and was therefore not further pursued.
8 Conclusion
The three-dimensional simulation of a reverberation chamber (RC) was presented in
this thesis. In the beginning, fundamental concepts and key parameters of an RC were
introduced in Chapter 2. These included the mode distribution, mode density, modal
gaps, and the quality factor. Furthermore, the RC was described as a statistical electromagnetic test environment and characterized by distribution functions, correlation,
uncertainty, and field uniformity. In Chapter 3 it was shown that it is crucial to select a
suitable numerical method to perform meaningful RC simulations. A chosen numerical
technique must be able to deliver results over a wide frequency range without using
excessive computational resources; the method must be able to handle large, irregular structures, and a varying geometry without introducing errors. Furthermore, there
must be a possibility to account for finite metal conductivity as well as highly resonant structures. The computation of near fields at an arbitrary number of chamber
locations should be possible without adding too much computational overhead. For
this thesis, a frequency-domain electric field integral equation (EFIE)-based method-ofmoments (MoM) technique was chosen. Chapter 4 put the work accomplished during
this thesis into perspective with previous publications on RC simulations. One striking
result of this literature survey was that in the majority of the published material a thorough simulation validation tends to be neglected. Chapter 5 described the construction
of a prototype RC used later on for simulation validations and explained the setup of the
measurement system. Measurement errors originating from field probes, antennas, and
stirrers were discussed and assessed for their impact on deviations between simulated
and measured results. The biggest deviations were found to result from the antenna
tripods and position inaccuracies of the field probe head or the antennas. Chapter 6
outlined how the practical prototype RC including the door, a stirrer, several different
antennas, and EUTs was modeled for the electromagnetic simulation. Electrical conductivity 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 flat
stock. In addition to the prototype RC, cubic and corrugated chambers, an offset-wall
RC 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 measurements and exhibited a good agreement in the lower-to-medium frequency range (at
frequencies less or equal to twice the lowest usable frequency). It was shown that a
proper validation of the simulation must be performed with direct comparisons against
measured near fields without further data processing or statistical analysis. Furthermore,
a deeper analysis of different chamber geometries, TX/RX antennas, various stirrer designs, and EUTs was performed. The importance of small geometric details and the
agreement between actual prototype and simulated RC dimensions was discussed. It
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8 CONCLUSION
was shown that the type, position, and alignment of the excitation source in the simulation model change the field pattern significantly. In addition, the effect of various
stirrers on the fields, correlation, and uniformity inside the chamber was visualized. The
6-paddle stirrer developed for this thesis and the commercially available Z-fold stirrer
exhibited the best performance. The insertion of gaps between parts of a stirrer did
not show a consistently “good” or “bad” effect – some stirrers performed better with,
some better without geometrical gaps. The stirrer analysis using a simulation approach
only confirmed the well-known conclusion that “complicated, asymmetric” stirrers will
outperform “simple, symmetric” stirrers over a broad frequency range. In addition, it
was found that the performance of a stirrer is linked to a particular RC, which renders
the 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 chamber revealed that the two latter chamber geometries do not offer significant advantages
concerning correlation and field uniformity. On the other hand, the cubic RC does not
perform as bad as always alleged, mainly due to the fact that the field distribution within
a cubic RC (including a stirrer) does not have anything in common with the fields observed in a cubic cavity. The presence of a stirring device shifts the modes in frequency
depending on their respective field distribution away from the analytically calculated
resonance frequencies. Therefore the usually observed problem of degenerate modes
does not come into play within a cubic RC and contrary to the widely accepted RC
design guidelines a cubic RC will not exhibit worse (or better) performance than other
rectangular RCs. Three special multipath/direct path coupling scenarios were simulated
(Gaussian, Rice, and Rayleigh statistical distributions). This investigation revealed that
the usage of a Hertzian dipole in an RC simulation leads to undesirable strong direct
coupling between an EUT and the excitation. Through the usage of realistic antennas
with 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 the
luxury of having the knowledge about the electromagnetic field at any arbitrary position
within the RC and being able to visualize the field distribution is very instructive, but
just one part of the whole story. For most, if not all, important RC quantities such as
correlation, anisotropy coefficients, or uniformity, knowledge of the field throughout the
chamber at only one stirrer step is essentially useless. In order to calculate these quantities 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 stirrer
positions. To complicate matters more, knowledge of the correlation coefficient or field
uniformity at only a single frequency is of not much use either. Essentially needed is
knowledge 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 frequencies
with a fine frequency resolution. Starting with these requirements, as shown at the
end of Chapter 7, the time it takes to compute one single metric such as correlation or
field uniformity is at minimum substantial, possibly even prohibitive. If, for example, a
frequency-domain solver is used and the correlation is of interest over a bandwidth of
500 MHz with 5 MHz steps at an angular stirrer resolution of 5◦ (i.e. 72 rotational stirrer
positions), 7200 simulations need to be performed, assuming that no interpolations can
be 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 used
in a simulation, generating plots as the ones presented in Chapter 7 easily adds up to
combined run times of several months – especially if two or more chambers or stirrers
are to be compared with each other.
For frequencies much smaller than the lowest usable frequency, the simulation of an RC
is possible, the chamber however becomes electrically too small compared to the operational wavelength, which prevents sufficient statistical field uniformity – the laws of
physics do not permit an optimization of RCs. Conversely, at frequencies much 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 optimization needed).
In addition, as shown in this thesis, with increasing frequency the field within an RC
becomes more and more sensitive to even small geometric details, which makes proper
modeling numerically not feasible at high frequencies. The possibilities for RC design
optimizations significantly below or above the lowest usable frequency are therefore limited. At frequencies around the lowest usable frequency, however, stirrer shapes or wall
geometries can be optimized with an electromagnetic simulation and the effect of multiple stirrers or other means for an improvement of field uniformity can be investigated
in order to extend the operating frequency for a given RC to lower frequencies. For a
successful optimization it is therefore crucial to have a numerical tool at hand, which can
accurately simulate an RC around the lowest usable frequency, accommodate a rotating
stirrer, and provide broadband simulation data with as few computations as possible.
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9 Outlook
In this thesis a simulation model of a reverberation chamber (RC) was developed, which
allows to accurately reproduce the electromagnetic fields within a prototype chamber in
the lower to medium frequency range. This model of the RC can be used for further
research. The biggest problem with frequency-domain RC simulations are the notoriously long computation runtimes; although time-domain simulation codes are known to
be problematic with highly resonant structures, it would be advisable to compare the results obtained in the frequency-domain simulations of this thesis with broadband results
from an electromagnetic time-domain solver. In order not to “reinvent the wheel”, for
this purpose preferably a commercial, state-of-the-art solver (such as for example CST’s
Microwave Studio ) should be utilized.
Two parameters were identified as crucial for the performance of an RC in terms of
correlation and field uniformity: the stirrer effectiveness and the quality factor of the
chamber. With respect to the first issue, it was shown in this thesis that a comprehensive stirrer analysis with the stirrer operating inside the RC is computationally extremely
expensive and hence time-consuming. A stirrer analysis and optimization in free-space
using multiple-angle plane wave illumination was attempted, but proved to be rather
unsuccessful 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 is
tempting and should be pursued further with other more suitable metrics.
Secondly, an extension to this thesis could deal with a rigorous analysis of the quality
factor of an RC. This is a challenging topic, as the usually applied analytical formulas for
multimode 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 and
the chamber. Furthermore, the electromagnetic modes will be shifted and widened in
their bandwidth by the stirrer rotation. Once the quality factor in a multimode RC is
derived with the help of a simulation tool, it needs to be validated by measurements in
a practical chamber. However, quality factor measurements tend to be inherently difficult in highly resonant structures. Conductivity and leakage in the RC model must be
adjusted accordingly in order to achieve a good agreement between quality factor values
of the measurement setup and computed from the simulation. Using the simulated and
measured results, suitable techniques can be developed allowing to influence the quality
factor which facilitates a control of the field uniformity within an RC.
An evaluation of the quality factor should also take into consideration the additional
loading of the chamber introduced through the presence of different EUTs. The CEUT
results shown in this thesis were mostly benchmarked against theoretical or previously
published measurement results. To gain more insight into the effects caused by an EUT
within an RC, a CEUT should be built (or borrowed from a round robin test) and
tested for immunity and emissions in a prototype RC. Results from these tests should
be compared with the immunity and emission simulations presented in this thesis. To
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9 OUTLOOK
measure the electric near field in the prototype RC, efficient methods for field measurements should be investigated allowing to sample the field distribution without distorting
it significantly (e.g. by electro-optical techniques). The measurement time needed for a
simulation validation could be considerably reduced by developing an automated near
field scanning system with multiple sensors and high positioning accuracy, thus facilitating fast sampling of the electric field at a large number of measurement positions. As
proposed in this thesis, it is advisable to feed both measured and simulated data into a
common database for fast, error-free, and convenient access.
Finally, an extension to this thesis could deal with novel TX/RX systems in an RC. This
could include research on electronic stirring methods aiming at complete removal of the
stirring device (this issue was briefly touched in Section 4.4) as well as spatially efficient
antennas resulting in larger maximum EUT volumes. Reproducing the electromagnetic
effect of a rotating stirrer by electronic means could lead to a significant reduction in
overall RC test time for EMC. Electronic stirring techniques together with the ability
to effectively control the quality factor would also help in reducing the lowest usable
frequency 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 the
preprocessor PreFEKO to generate the simulation model from the file filename (if it
does not already exist) and launches ‘x’ parallel FEKO processes on the machines listed
in the file machname. Load distribution and communication between these machines is
managed by FEKO .
Model geometry validation
To quickly check the geometry for errors such as inhomogeneous discretization, duplicate, badly connected, or degenerated elements, and label cross-referencing, FEKO can
be invoked with the option runfeko...--feko-options --check-only. FEKO will
check simply the geometry and try to allocate and de-allocate the required memory. It
will 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 processes
with 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 which
sets the execution priority on all parallel processes to 10. Note that adjusting process
priorities 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 added
for mechanical stability reasons only. An example is a styrofoam block used to serve as
a “table” onto which an EUT is placed. Simulating a styrofoam table would require a
significant number of dielectric triangles or cubes, although it is a priori known that the
EM effect of this table is negligible (since in practice exactly for this reason styrofoam
has been chosen).
Nevertheless, the presence of the styrofoam table is needed for illustrative purposes in
the 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 include
this capability by itself, a small trick has to be applied: The desired behavior can be
achieved 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 it
will appear to be “false”. When displaying the simulated geometry, currents, and fields
in WinFEKO the “Non-matching MD5 checksums” warning can be ignored.
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APPENDIX A ELECTROMAGNETIC SIMULATION SOFTWARE FEKO
A.2 Memory considerations and bugs
32 bit versus 64 bit FEKO version
At 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 which
unfortunately shows an error that could not be resolved by EMSS until the end of this
thesis. If run in 64 bit mode, FEKO crashes in a fully repeatable manner after the LU
decomposition 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 invoked
in 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 process
for the solution of the system equations. Sun however implemented a special memory
allocation mechanism in Solaris 8 which allows a single process to allocate and manage
up to 4 GByte of memory. This is referred to as “XMEM” (extended memory) support.
MLFMM versus MoM
The solutions for the near field obtained by using the MLFMM (introduced in Section 3.5.4) and MoM differ by a factor of approx. 4.5. The current distribution and
consequently the near field patterns are however completely identical for both solution
methods, i.e. the MLFMM solution is a scaled version of the MoM solution and vice
versa. This discrepancy could not be resolved with EMSS during the course of this thesis [198]. Therefore all results presented in this thesis were calculated with the reliable
full 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 plastics
on the electric field within an RC without a stirrer (i.e. a cavity). Each measurement
of the magnitude of the electric field |E| was performed once with the TX/excitation
antenna mounted on the tripod and once without the tripod and the antenna suspended
from 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 high
frequency 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, discrete
frequencies (Fig. B.3. . . Fig. B.5)
Broadband effect at lower frequencies
|E | [V/m]
80
Tripod
Plastic wires and Velcro
70
60
50
40
30
20
10
0
50
100
150
200
Frequency f [MHz]
250
300
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 with
plastic ropes and Velcro from the chamber ceiling (f = 50 . . . 300 MHz).
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APPENDIX B REVERBERATION CHAMBER MEASUREMENT SYSTEM
From Fig. B.1. . . Fig. B.5 it is obvious that the electric field with the tripod installed
in the chamber is not just a linearly scaled version of the field measured without the
tripod. In addition to being scaled, the EM field is also strongly distorted, as apparent
from Fig. B.3. . . Fig. B.5.
|E | [V/m]
80
Tripod
Plastic wires and Velcro
70
60
50
40
30
20
10
0
50
100
150
200
Frequency f [MHz]
250
300
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 with
plastic ropes and Velcro from the chamber ceiling (f = 50 . . . 300 MHz).
APPENDIX B REVERBERATION CHAMBER MEASUREMENT SYSTEM
167
Spatial effect at lower frequencies
|E | [V/m]
50
Tripod
Plastic wires and Velcro
50 MHz
100 MHz
40
30
20
10
0
-1.2
-0.8
-0.4
0
y [m]
0.4
0.8
1.2
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 with
plastic ropes and Velcro from the chamber ceiling (50 MHz and 100 MHz).
|E | [V/m]
50
Tripod
Plastic wires and Velcro
150 MHz
200 MHz
40
30
20
10
0
-1.2
-0.8
-0.4
0
y [m]
0.4
0.8
1.2
Figure B.4: 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 with
plastic ropes and Velcro from the chamber ceiling (150 MHz and 200 MHz).
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APPENDIX B REVERBERATION CHAMBER MEASUREMENT SYSTEM
|E | [V/m]
50
Tripod
Plastic wires and Velcro
250 MHz
300 MHz
40
30
20
10
0
-1.2
-0.8
-0.4
0
y [m]
0.4
0.8
1.2
Figure B.5: 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 with
plastic ropes and Velcro from the chamber ceiling (250 MHz and 300 MHz).
B.2 Data acquisition and interfacing
Simulation data can be transferred automatically from FEKO into a DB system via a
MATLAB -based tool. Due to the lack of an import/export interface of the measurement software Compliance C3i (this problem has been fixed for a short time), measurement 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 original
Compliance measurement data must be copied and pasted manually into Excel and
then preprocessed. The columns in the Excel worksheet must have exactly the same
names as the field names in Access; it is not enough that the Excel columns bear the
same name as the Access captions (which are essentially just to pretty-print the data
tables). To further complicate matters, problems arise if list boxes together with underlying lookup tables are used in the Access DB: it is not sufficient to have the values in
Excel as displayed by Access – one has to back-reference the displayed list box values
to the primary keys of the underlying lookup table. If e.g. the list box in the DB has
possible options “Measurement” and “Simulation” and the corresponding lookup table
primary keys are “1” and “2”, in the Excel table the value “2” must be entered in order
to get “Measurement” in the DB. Entering “Measurement” will not import the value
from the Excel worksheet into the Access DB. If alternatively the “Paste append” command 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 estimator
accuracy 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 the
field data to be estimated (usually 1 or 3) and N is the required number of statistically
independent stirrer positions. If the field probe responds to only one dimension of the
field in this case b = 1. Solving for the required number of statistically independent
stirrer positions N results in
2
k 2 10d̃/10 + 1
(C.1)
N=
b 10d̃/10 − 1
Equation (2.93) is plotted for different confidence levels in Fig. C.1. . . Fig. C.3. If, for
example, the uncertainty interval should be d˜ = ±1 dB and the desired level of confidence
is 90% (corresponding to k ≈ ±1.65σ), then one would obtain N ≈ 69 or N ≈ 207 for
b = 3 and b = 1 dimensions, respectively (see Fig. C.1 and Fig. C.3).
6
Confidence level p
68%
75%
95%
99%
~
Uncertainty d [dB]
4
2
0
-2
-4
-6
0
50
100
150
200
250
300
350
400
Number of stirrer positions N
Figure C.1: Number of statistically independent stirrer positions N required to achieve the
uncertainty interval ±d˜ for one EM field component at a confidence of p (see Fig. 2.7 for
corresponding standard deviation multiples).
169
APPENDIX C REVERBERATION CHAMBER STATISTICS
170
6
Confidence level p
68%
75%
95%
99%
~
Uncertainty d [dB]
4
2
0
-2
-4
-6
0
50
100
150
200
250
300
350
400
Number of stirrer positions N
Figure C.2: Number of statistically independent stirrer positions N required to achieve the
uncertainty interval ±d˜ for two EM field components at a confidence of p (see Fig. 2.7 for
corresponding standard deviation multiples).
6
Confidence level p
68%
75%
95%
99%
~
Uncertainty d [dB]
4
2
0
-2
-4
-6
0
50
100
150
200
250
300
350
400
Number of stirrer positions N
Figure C.3: Number of statistically independent stirrer positions N required to achieve the
uncertainty interval ±d˜ for three EM field components at a confidence of p (see Fig. 2.7 for
corresponding 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 commonly encountered in RC analysis.
Gaussian distribution
The PDF of a Gaussian distributed random variable X is given by
f (X | µ, σ) =
(X−µ)2
1
√ · e− 2σ2
σ 2π
(C.2)
with the mean µ and standard deviation σ. Its corresponding CDF is given by
X
F (X | µ, σ) =
f (u) du =
−∞
*
+
X −µ
1
1 + erf √
2
2σ
(C.3)
with erf(·) denoting the error function as given by [12, 195].
Chi-square distribution
A Chi-square (χ2 ) distributed random variable is related to a Gaussian-distributed random 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 = X 2 , then Y has a Chi-square distribution.
Two types of Chi-square distributions are distinguished: the central and the non-central
Chi-square distribution.
The central Chi-square distribution is obtained if the underlying Gaussian distribution
has zero mean µ = 0. The PDF of the central Chi-square distribution is given by
Y
1
· e− 2σ2
f (Y | σ) = √
2πY σ
Y ≥0
(C.4)
If, however, X is Gaussian distributed with non-zero mean µ = 0 and variance σ, then
the PDF of the non-central Chi-square distribution resulting from Y = X 2 is given by
√
+µ2
Yµ
1
− Y2σ
2
·e
cosh
Y ≥0
(C.5)
f (Y | µ, σ) = √
σ2
2πY σ
The corresponding CDF for the central Chi-square distribution is given by
Y
F (Y | σ) =
0
1
f (u) du = √
2πY σ
Y
u
1
√ · e− 2σ2 du
u
(C.6)
0
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 distribution
The 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
APPENDIX C REVERBERATION CHAMBER STATISTICS
172
variance σ 2 , then Y = X12 + X22 is central Chi-square distributed with two degrees of
freedom as given by (C.4). By introducing the new random variable
√
(C.7)
R = X12 + X22 = Y
the Rayleigh PDF
R − R22
· e 2σ
R≥0
(C.8)
σ2
is obtained. The corresponding CDF for the central Rayleigh distribution is given by
f (R | σ) =
R
F (R | σ) =
R2
f (u) du = 1 − e− 2σ2
(C.9)
0
Rice distribution
Just as the Rayleigh distribution is related to the central Chi-square distribution, the
Rice distribution is closely related to the non-central Chi-square distribution. If X1 and
X2 are two non-zero-mean (with means µ1 and µ2 ) i.i.d. Gaussian random variables, each
having the common variance σ 2 , then Y = X12 +X22 is non-central Chi-square distributed
with two degrees of freedom as given by (C.5). By introducing the new random variable
√
R = X12 + X22 = Y
(C.10)
the Rice PDF
f (R | µ1 , µ2 , σ) =
R − R2 +µ212+µ22
2σ
·e
J0
σ2
√
R µ1 µ2
σ2
R≥0
(C.11)
as given by [195] is obtained with J0 being the 0-th order Bessel function of the first
kind. The corresponding CDF for (C.11) can be found in [195].
Gamma distribution
The Gamma (Γ) distribution is a generalization of the Chi-square distribution (C.4) to
ν degrees of freedom. The PDF of the Gamma distribution is given by
f (X | σ, ν) =
X
ν
1
− 2σ
2
2 −1 e
ν ν · X
ν
Γ 2 σ 22
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 can
be found in [195].
Nakagami-m and multivariate Gaussian distribution
For the PDF and, respectively, CDF of the multivariate Gaussian distribution and the
Nakagami-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 are
indicated, 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 the
other is given by (5.1)-(5.3) by letting xs = x, ys = y, and zs = z. Spatial coordinates
of the field points used for the measurement and simulation 8-point field uniformity
analysis complying with IEC 61000-4-21 [6] are listed in Table D.1.
Measurement
points
ym
xm
zm
[m]
[m]
[m]
Simulation
points
ys
xs
zs
[m] [m] [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
Field point
number
Table D.1: Field points used in measurement and simulation for uniformity analysis (default
values). 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 MHz
excited by a) a 1 V 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 scenarios: 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 the
prototype RC (Fig. D.3), the corrugated RC (Fig. D.4), and the cubic RC (Fig. D.5).
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.3: Statistical field uniformities σ̃xyz and σ̃ξ in the prototype RC obtained according
to the procedure outlined in Section 2.7 (- - - IEC limit line).
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.4: Statistical field uniformities σ̃xyz and σ̃ξ in the corrugated RC obtained according
to the procedure outlined in Section 2.7 (- - - IEC limit line).
176
APPENDIX D REVERBERATION CHAMBER SIMULATION DATA
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.5: Statistical field uniformities σ̃xyz and σ̃ξ in the cubic RC obtained according to the
procedure 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 the
different RC stirrers introduced in Section 7.7 (see Fig. D.6. . . Fig. D.11).
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.6: Statistical field uniformities σ̃xyz and σ̃ξ for the vertical 6-paddle stirrer, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
APPENDIX D REVERBERATION CHAMBER SIMULATION DATA
Standard deviation $
s [dB]
6
177
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.7: Statistical field uniformities σ̃xyz and σ̃ξ for the vertical 6-paddle connected stirrer,
obtained according to the procedure outlined in Section 2.7 (- - - IEC limit line).
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.8: Statistical field uniformities σ̃xyz and σ̃ξ for the horizontal 6-paddle stirrer, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
178
APPENDIX D REVERBERATION CHAMBER SIMULATION DATA
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.9: Statistical field uniformities σ̃xyz and σ̃ξ for the vertical cross-plate stirrer, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.10: Statistical field uniformities σ̃xyz and σ̃ξ for the vertical Z-fold stirrer, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
APPENDIX D REVERBERATION CHAMBER SIMULATION DATA
Standard deviation $
s [dB]
6
179
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
sabs
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
Figure D.11: Statistical field uniformities σ̃xyz and σ̃ξ 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 different canonical EUTs placed inside the RC (see Fig. D.12. . . Fig. D.15).
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
600
Figure D.12: Statistical field uniformities σ̃xyz and σ̃ξ without an EUT, obtained according to
the procedure outlined in Section 2.7 (- - - IEC limit line).
180
APPENDIX D REVERBERATION CHAMBER SIMULATION DATA
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
600
Figure D.13: Statistical field uniformities σ̃xyz and σ̃ξ with the canonical loop EUT, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
Standard deviation $
s [dB]
6
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
600
Figure D.14: Statistical field uniformities σ̃xyz and σ̃ξ with the canonical box EUT, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
APPENDIX D REVERBERATION CHAMBER SIMULATION DATA
Standard deviation $
s [dB]
6
181
IEC limit line
sxyz
$
sx
$
sy
$
sz
$
5
4
3
2
1
0
0
100
200
300
Frequency f [MHz]
400
500
600
Figure D.15: Statistical field uniformities σ̃xyz and σ̃ξ with a large canonical box EUT, obtained
according to the procedure outlined in Section 2.7 (- - - IEC limit line).
182
Bibliography
[1] C. R. Paul, Introduction to Electromagnetic Compatibility.
Wiley & Sons, 1992.
New York, NY: John
[2] G. J. Freyer, “Distribution of responses for limited aspect angle EMC tests of
equipment with structured directional directivity,” in Proc. 2003 Reverberation
Chamber, Anechoic Chamber and OATS Users Meeting, Austin, TX, 2003.
[3] G. J. Freyer and M. Bäckström, “Comparison of anechoic & reverberation chamber
coupling data as a function of directivity pattern,” in Proc. IEEE Int. Symp. on
Electromagnetic Compatibility, vol. 2. Piscataway, NJ: IEEE, 2000, pp. 615–620.
[4] L. Jansson and M. Bäckström, “Directivity of equipment and its effect on testing in
mode-stirred and anechoic chamber,” in Proc. IEEE Int. Symp. on Electromagnetic
Compatibility, vol. 1. Piscataway, NJ: IEEE, 1999, pp. 17–22.
[5] M. O. Hatfield, J. L. Bean, G. J. Freyer, and D. M. Johnson, “Repeatability of
mode-stirred chamber measurements,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway, NJ: IEEE, 1994, pp. 485–490.
[6] CISPR/A and IEC SC 77B, IEC 61000-4-21 – Electromagnetic Compatibility
(EMC) - Part 4-21: Testing and Measurement Techniques - Reverberation Chamber Test Methods, International Electrotechnical Commission (IEC) International
standard, Aug. 2003.
[7] F. Weeks and K. R. Goldsmith, “Design philosophy and material choice for a tuner
in an electromagnetic reverberation chamber,” Department of Defense, Australia,
Tech. Rep., 2000.
[8] D. A. Hill, “Electromagnetic theory of reverberation chambers,” National Institute
of Standards and Technology (NIST),” Tech. Note 1506, 1998.
[9] M. L. Crawford and G. H. Koepke, “Design, evaluation, and use of a reverberation chamber for performing electromagnetic susceptibility/vulnerability measurements,” National Bureau of Standards (NBS),” Tech. Note 1092, 1986.
[10] R. E. Collin, Foundations for Microwave Engineering. New York, NY: McGrawHill, 1966.
[11] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, ImpedanceMatching Networks, and Coupling Structures. Boston, MA: Artech House, 1980.
183
184
BIBLIOGRAPHY
[12] I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der
Mathematik [in German]. Frankfurt am Main, Germany: Verlag Harri Deutsch,
1993.
[13] C. Bruns, P. Leuchtmann, and R. Vahldieck, “Introduction to reverberation chamber simulation,” in Proc. 2nd NPL FREEMET meeting at MIRA, Nuneaton. Teddington, UK: National Physical Laboratory (NPL), 2002, [Electronic].
[14] M. O. Hatfield, “Calibration of reverberation chambers,” in Proc. 2001 Reverberation Chamber, Anechoic Chamber and OATS Users Meeting, Seattle, WA, June
2001.
[15] L. R. Arnaut, “Operation of electromagnetic reverberation chambers with wave diffractors at relatively low frequencies,” IEEE Trans. Electromagn. Compat., vol. 43,
no. 4, pp. 635–653, Nov. 2001.
[16] B. H. Liu, D. C. Chang, and M. T. Ma, “Eigenmodes and the composite quality
factor of a reverberating chamber,” National Bureau of Standards (NBS), Tech.
Rep., 1983.
[17] P. Corona, G. Ferrara, and M. Migliaccio, “Reverberating chamber electromagnetic
field in presence of an unstirred component,” IEEE Trans. Electromagn. Compat.,
vol. 42, no. 2, pp. 111–115, 2000.
[18] C. Bruns, P. Leuchtmann, and R. Vahldieck, “Comparison of various reverberation
chamber geometries and excitations using a frequency domain method of moments
simulation,” in Proc. 17th Int. Wroclaw Symp. and Exhibition on Electromagnetic
Compatibility. Wroclaw, Poland: Politechniki Wroclawskiej, 2004, pp. 97–102.
[19] ——, “Cubic and corrugated reverberation chambers: mode distribution, correlation, and field uniformity,” in Proc. 16th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst.
Technol. Zurich, 2005, pp. 539–542.
[20] D. A. Hill, M. T. Ma, A. R. Ondrejka, B. F. Riddle, M. L. Crawford, and R. T.
Johnk, “Aperture excitation of electrically large, lossy cavities,” IEEE Trans. Electromagn. Compat., vol. 36, no. 3, pp. 169–178, 1994.
[21] D. A. Hill, “Linear dipole response in a reverberation chamber,” in Proc. IEEE Int.
Symp. on Electromagnetic Compatibility, vol. 41, no. 4. Piscataway, NJ: IEEE,
1999, pp. 365–368.
[22] W. Kürner, “Messung gestrahlter Emissionen und Gehäuseschirmdämpfungen in
Modenverwirbelungskammern [in German],” Ph.D. dissertation, Universität Karlsruhe, Karlsruhe, Germany, 2002.
[23] M. Petirsch, “Untersuchung zur Optimierung der Feldverteilung in Mode-Stirred
Chambers [in German],” Ph.D. dissertation, Universität Karlsruhe, Karlsruhe,
Germany, 1999.
BIBLIOGRAPHY
185
[24] J. G. Kostas and B. Boverie, “Statistical model for a mode-stirred chamber,” IEEE
Trans. Electromagn. Compat., vol. 33, no. 4, pp. 366–370, 1991.
[25] J.-F. Rosnarho, “Criteria of choice of mode stirred reverberation chamber,” in
Proc. 17th Int. Wroclaw Symp. and Exhibition on Electromagnetic Compatibility.
Wroclaw, Poland: Politechniki Wroclawskiej, 2004, pp. 274–277.
[26] M. O. Hatfield and M. B. Slocum, “Frequency characterization of reverberation
chambers,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway, NJ: IEEE, 1996, pp. 190–193.
[27] L. R. Arnaut and P. D. West, “Effect of antenna aperture, EUT and stirrer step
size on measurements in mode-stirred reverberation chambers,” in Proc. IEEE Int.
Symp. on Electromagnetic Compatibility. Piscataway, NJ: IEEE, 2000, pp. 29–34.
[28] L. R. Arnaut, “Effect of local stir and spatial averaging on measurement and
testing in mode-tuned and mode-stirred reverberation chambers,” IEEE Trans.
Electromagn. Compat., vol. 43, no. 3, pp. 305–325, Aug. 2001.
[29] ——, “Power normalization in field anisotropy and inhomogeneity coefficients,”
Private communication, Oct. 2003.
[30] L. R. Arnaut and P. D. West, “Electric field probe measurements in the NPL
untuned stadium reverberation chamber,” National Physical Laboratory (NPL),
Tech. Rep., 1999.
[31] ——, “Evaluation of the NPL untuned stadium reverberation chamber using mechanical and electronic stirring techniques,” National Physical Laboratory (NPL),
Tech. Rep., 1998.
[32] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York,
NY: McGraw-Hill, 1984.
[33] R. Holland and R. St. John, Statistical Electromagnetics. Philadelphia, PA: Taylor
and Francis, 1999.
[34] W. A. Stahel, Statistische Datenanalyse [in German].
Vieweg Verlag, 1995.
Braunschweig, Germany:
[35] J. R. Taylor, An introduction to error analysis: the study of uncertainties in physical measurements. Sausalito, CA: University Science Books, 1997.
[36] O. Lundén and M. Bäckström, “Stirrer efficiency in FOA reverberation chambers:
Evaluation of correlation coefficients and chi-squared test,” in Proc. IEEE Int.
Symp. on Electromagnetic Compatibility, vol. 1. Piscataway, NJ: IEEE, 2000, pp.
11–16.
186
BIBLIOGRAPHY
[37] J. M. Ladbury and K. R. Goldsmith, “Reverberation chamber verification procedures, or, how to check if your chamber ain’t broke and suggestions on how to
fix it if it is,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, vol. 1.
Piscataway, NJ: IEEE, 2000, pp. 17–22.
[38] H. Weber, Einführung in die Wahrscheinlichkeitsrechnung und Statistik für Ingenieure [in German]. Stuttgart, Germany: B. G. Teubner Verlag, 1992.
[39] M. O. Hatfield, “Background of reverberation chamber (RC) characterization using
eight field points,” Private communication, Apr. 2004.
[40] F. Beichelt, Stochastik für Ingenieure [in German].
Teubner Verlag, 1995.
Stuttgart, Germany: B. G.
[41] A. Taflove and S. C. Hagness, Computational Electrodynamics: The FiniteDifference Time-Domain Method. Boston, MA: Artech House, 2000.
[42] E. K. Miller, Ed., Computational Electromagnetics: Frequency-Domain Method of
Moments. Piscataway, NJ: IEEE Press, 1992.
[43] F. Moglie, “Finite difference, time domain analysis convergence of reverberation
chambers,” in Proc. 15th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst. Technol. Zurich,
2003, pp. 223–228.
[44] F. X. Canning, “Singular value decomposition of integral equations of EM and
applications to the cavity resonance problem,” IEEE Trans. Antennas Propagat.,
vol. 37, no. 9, pp. 1156–1163, Mar. 1989.
[45] J. Moore and R. Pizer, Eds., Moment Methods in Electromagnetics.
NY: John Wiley and Sons, 1984.
New York,
[46] U. Jakobus, “Erweiterte Momentenmethode zur Behandlung kompliziert aufgebauter und elektrisch grosser elektromagnetischer Streuprobleme [in German],”
Ph.D. dissertation, Universität Stuttgart, Stuttgart, Germany, 1995.
[47] C.-C. Lu, “A fast algorithm based on volume integral equation for analysis of
arbitrarily shaped dielectric radomes,” IEEE Trans. Antennas Propagat., vol. 51,
no. 3, pp. 606–612, Mar. 2003.
[48] U. Jakobus and F. M. Landstorfer, “Improved PO-MM hybrid formulation for
scattering from three-dimensional perfectly conducting bodies of arbitrary shape,”
IEEE Trans. Antennas Propagat., vol. 43, no. 2, pp. 162–169, Feb. 1995.
[49] C. A. Balanis, Advanced Engineering Electromagnetics. New York, NY: John
Wiley & Sons, 1989, ch. Integral equations and the moment method, pp. 670–732.
[50] G. J. Burke and A. J. Poggio, “Numerical Electromagnetics Code (NEC) – Method
of Moments,” Lawrence Livermore National Laboratory, Tech. Rep., 1981.
BIBLIOGRAPHY
187
[51] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by
surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol. 30, no. 3, pp.
409–418, May 1982.
[52] R. F. Harrington, Field Computation by Moment Methods.
Macmillan, 1968.
New York, NY:
[53] B. H. Jung, T. K. Sarkar, and Y.-S. Chung, Progress in Electromagnetics Research
PIER. EMW Publishing, 2002, no. 36, ch. A Survey of Various Frequency Domain
Integral Equations for the Analysis of Scattering from Three-dimensional Dielectric
Objects, pp. 193–246.
[54] U. Jakobus et al. (2004) FEKO – field computations involving objects of
arbitrary shape. EMSS Ltd. Stellenbosch, South Africa. [Online]. Available:
http://www.feko.info
[55] U. Jakobus, “Memory allocation in FEKO ,” Private communication, Dec. 2001.
[56] Altair Engineering Inc. (2004) HyperMesh . Altair Engineering Inc. Troy, MI.
[Online]. Available: http://www.altair.com
[57] U. Jakobus, “Internal resonance problems with the electric field integral equation
EFIE solved by the method-of-moments (MoM),” Private communication, Oct.
2002.
[58] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antennas Propagat., vol. 52, no. 2, pp. 362–372, Feb.
2004.
[59] F. X. Canning, “Protecting electric field integral equation (EFIE)-based scattering
computations from effects of interior resonances,” IEEE Trans. Antennas Propagat., vol. 39, no. 11, pp. 1545–1552, Nov. 1991.
[60] P. Sonneveld, “CGS, a fast Lanczos-type solver for nonsymmetric linear systems,”
SIAM J. Sci. Statist. Comput., no. 10, pp. 36–52, 1989.
[61] R. W. Freund, “A transpose-free quasi-minimal residual algorithm for nonhermitian linear systems,” SIAM J. Sci. Statist. Comput., no. 14, pp. 470–482,
1993.
[62] M. L. Waller and T. H. Shumpert, “Series-expansion representation of the reducedmatrix technique for TE and TM induced currents on coupled two-dimensional
scatterers,” Microwave and Optical Technology Letters, vol. 27, no. 4, pp. 238–245,
Nov. 2000.
[63] S. M. Rao and M. L. Waller, “Development and application of adaptive basis functions to generate a diagonal moment matrix for electromagnetic field problems,”
Microwave and Optical Technology Letters, vol. 28, no. 5, pp. 357–361, Mar. 2001.
BIBLIOGRAPHY
188
[64] S. M. Rao and G. K. Gothard, “A new technique to generate a sparse matrix using
the method of moments for electromagnetic scattering problems,” Microwave and
Optical Technology Letters, vol. 19, no. 4, pp. 271–274, Nov. 1998.
[65] M. Sabielny and H.-D. Brüns, “Practical aspects of the physical optics-moment
method hybrid method,” in Proc. 15th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst.
Technol. Zurich, 2003, pp. 251–256.
[66] C. Bruns, P. Leuchtmann, and R. Vahldieck, “Challenges and results of realistic reverberation chamber simulations and measurements,” in Proc. 2003 Reverberation
Chamber, Anechoic Chamber and OATS Users Meeting, Austin, TX, 2003.
[67] C. Bruns and R. Vahldieck, “A closer look at reverberation chambers – 3-D simulation and experimental verification,” accepted for publication in IEEE Trans.
Electromagn. Compat., Aug. 2005.
[68] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for
the wave equation: A pedestrian prescription,” IEEE Antennas Propagat. Mag.,
vol. 35, no. 3, pp. 7–12, June 1993.
[69] W. C. Chew, J.-M. Jin, C.-C. Lu, E. Michielssen, and J. M. Song, “Fast solution
methods in electromagnetics,” IEEE Trans. Antennas Propagat., vol. 45, no. 3,
pp. 533–543, Mar. 1997.
[70] D. Bohlen and K. Kessler, Hinter den Kulissen.
gruppe Random House GmbH, 2003.
München, Germany: Verlags-
[71] W. M. Hall, “Heating apparatus,” U.S. Patent 2,618,735, 1947.
[72] M. L. Crawford, “Electromagnetic field measurements in low Q enclosures,” National Bureau of Standards (NBS), Tech. Rep., 1972.
[73] P. Corona, G. Latmiral, E. Paolini, and L. Piccioli, “Use of a reverberating enclosure for measurement of radiated power in the microwave range,” IEEE Trans.
Electromagn. Compat., vol. 18, no. 2, pp. 54–59, 1976.
[74] L. B. John and R. A. Hall, “Electromagnetic susceptibility measurements using a
mode-stirred chamber,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, Atlanta, GA, 1978.
[75] P. Corona, G. Latmiral, and E. Paolini, “Performance and analysis of a reverberating enclosure with variable geometry,” IEEE Trans. Electromagn. Compat.,
vol. 22, pp. 2–5, 1980.
[76] P. Corona, J. Ladbury, and G. Latmiral, “Reverberation-chamber research – then
and now: a review of early work and comparison with current understanding,”
IEEE Trans. Electromagn. Compat., vol. 44, no. 1, pp. 87–94, Feb. 2002.
BIBLIOGRAPHY
189
[77] IEC SC 46A, IEC 61726 – Cable assemblies, cables, connectors and passive microwave components – Screening attenuation measurement by the reverberation
chamber method, International Electrotechnical Commission (IEC) International
standard, Nov. 1999.
[78] M. L. Crawford and G. H. Koepke, “Electromagnetic radiation test facilities: Evaluation of reverberation chambers located at NSWCDD,” National Bureau of Standards (NBS), Tech. Rep., 1986.
[79] D. A. Hill, “Spatial correlation function for fields in reverberation chambers,” IEEE
Trans. Electromagn. Compat., vol. 37, pp. 138–143, 1995.
[80] ——, “A reflection coefficient derivation for the Q of a reverberation chamber,”
IEEE Trans. Electromagn. Compat., vol. 38, no. 4, pp. 591–592, 1996.
[81] ——, “Plane wave integral representation for fields in reverberation chambers,”
IEEE Trans. Electromagn. Compat., vol. 40, no. 3, pp. 209–217, 1998.
[82] M. O. Hatfield, G. J. Freyer, and M. B. Slocum, “Reverberation characteristics of
a large welded steel shielded enclosure,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway, NJ USA: IEEE, 1997, pp. 38–43.
[83] M. O. Hatfield, G. J. Freyer, D. M. Johnson, and C. Farthing, “Demonstration test
of the electromagnetic reverberation characteristics of a transport size aircraft,”
Naval Surface Warfare Center Dahlgren Division (NSWCDD), Tech. Rep., 1994.
[84] G. J. Freyer, T. H. Lehman, J. M. Ladbury, G. H. Koepke, and M. O. Hatfield,
“Verification of fields applied to an EUT in a reverberation chamber using statistical theory,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, vol. 1.
Piscataway, NJ: IEEE, 1998, pp. 34–38.
[85] G. J. Freyer, M. O. Hatfield, D. M. Johnson, and M. B. Slocum, “Comparison
of measured and theoretical statistical parameters of complex cavities,” in Proc.
IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway NJ: IEEE, 1996,
pp. 250–253.
[86] F. Petit, “Modélisation et simulation d’une chambre réverbérante à brassage de
modes à l’aide de la méthode des différences finies dans le domaine temporel
[in French],” Ph.D. dissertation, Université de Marne-La-Vallée, Marne-La-Vallée,
France, 2002.
[87] K. Harima and Y. Yamanaka, “FDTD analysis on the effect of stirrers in a reverberation chamber,” in Proc. Int. Symp. on Electromagnetic Compatibility. Tokyo,
Japan: IEICE, 1999, pp. 223–229.
[88] K. Harima, “FDTD analysis of electromagnetic fields in a reverberation chamber,”
IEICE Trans. Commun., vol. E81-B, no. 10, pp. 1946–1950, Oct. 1998.
190
BIBLIOGRAPHY
[89] L. Bai, L. Wang, B. Wang, and J. Song, “Reverberation chamber modeling using
FDTD,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway,
NJ: IEEE, 1999, pp. 7–11.
[90] ——, “Effects of paddle configurations on the uniformity of the reverberation chamber,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway,
NJ: IEEE, 1999, pp. 12–16.
[91] D. Zhang and J. Song, “Impact of stirrers’ position on the properties of a reverberation chamber with two stirrers,” in Proc. IEEE Int. Symp. on Electromagnetic
Compatibility, vol. 1. Piscataway, NJ: IEEE, 2000, pp. 7–10.
[92] T. M. Lammers, “Numerical analysis of mode stirred chambers and their loaded
and unloaded configurations,” M.Sc. thesis, University of Colorado, Boulder, CO,
2004.
[93] T. M. Lammers, C. L. Holloway, and J. Ladbury, “The effects of loading configurations on the performance of reverberation chambers,” in Proc. Int. Symp. on
Electromagnetic Compatibility. Eindhoven, The Netherlands: Technische Universiteit Eindhoven, 2004, pp. 727–732.
[94] N. K. Kouveliotis, P. T. Trakadas, and C. N. Capsalis, Progress in Electromagnetics
Research (PIER). EMW Publishing, 2003, no. 39, ch. FDTD modeling of a
vibrating intrinsic reverberation chamber, pp. 47–59.
[95] F. Leferink et al., “Test chamber,” The Netherlands Patent WO 00/34 795, 1999.
[96] F. Leferink, D. Boerle, F. Sogtoen, G. Heideman, and W. van Etten, “In-situ EMI
measurements using a vibrating intrinsic reverberation chamber,” in Proc. 14th Int.
Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich,
Switzerland: Swiss Federal Inst. Technol. Zurich, 2001, pp. 653–658.
[97] Y. Huang and D. J. Edwards, “Investigation of electromagnetic field inside a moving wall mode-stirred chamber,” in Proc. IEE Electromagnetics, vol. 362. IEE
Conference Publication, 1992, pp. 115–119.
[98] J. Ritter and M. Rothenhäusler, “Mode stirring chambers for full size aircraft
tests: Concept- and design-studies,” in Proc. European Microwave Conference
2003. London, UK: Horizon House Publ. Ltd., 2003.
[99] F. Moglie and A. Pastore, “FDTD analysis of reverberating chambers,” in Proc.
Int. Symp. on Electromagnetic Compatibility. Eindhoven, The Netherlands: Technische Universiteit Eindhoven, 2004, pp. 6–11.
[100] F. Hoëppe, P.-N. Gineste, and B. Demoulin, “Numerical modelling for mode-stirred
reverberation chambers,” in Proc. 14th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst.
Technol. Zurich, 2001, pp. 635–640.
BIBLIOGRAPHY
191
[101] F. Hoëppe, P.-N. Gineste, B. Demoulin, L. Kone, and F. Flourens, “Numerical
predictions applied to mode stirred reverberation chambers,” in Proc. 2001 Reverberation Chamber, Anechoic Chamber and OATS Users Meeting, Seattle, WA,
June 2001.
[102] L. Ma, D. Paul, N. Pothecary, C. Railton, J. Bows, L. Barratt, J. Mullin, and
D. Simons, “Experimental validation of a combined electromagnetic and thermal
FDTD model of a microwave heating process,” IEEE Trans. Microwave Theory
Tech., vol. 43, no. 11, pp. 2565–2572, Nov. 1995.
[103] M. F. Iskander, R. L. Smith, A. O. M. Andrade, H. Kimrey, and L. M. Walsh,
“FDTD simulation of microwave sintering of ceramics in multimode cavities,”
IEEE Trans. Microwave Theory Tech., vol. 42, no. 5, pp. 793–800, May 1994.
[104] A. C. Marvin, J. F. Dawson, and J. Clegg, “Stirrer optimisation for reverberation
chambers,” in Proc. Int. Symp. on Electromagnetic Compatibility. Eindhoven,
The Netherlands: Technische Universiteit Eindhoven, 2004, pp. 330–335.
[105] J. Clegg, A. C. Marvin, J. F. Dawson, S. J. Porter, and M. Bruenger-Koch, “Optimisation of stirrer designs in a mode stirred chamber using TLM,” in Proc. 2002
URSI XXVIIth general assembly, Maastricht (NL). Ghent, Belgium: URSI, 2002.
[106] A. R. Coates, A. P. Duffy, K. G. Hodge, and A. J. Willis, “Validation of modestirred reverberation chamber modelling,” in Proc. Int. Symp. on Electromagnetic
Compatibility, Sorrento. Milano, Italy: AEI, 2002, pp. 35–40.
[107] A. P. Duffy and A. J. M. Williams, “Optimising mode stirred chambers,” in Proc.
13th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility.
Zurich, Switzerland: Swiss Federal Inst. Technol. Zurich, 1999, pp. 685–688.
[108] P. Leuchtmann, C. Bruns, and R. Vahldieck, “On the validation of simulated
fields in a reverberation chamber,” in Proc. European Microwave Conference 2003.
London, UK: Horizon House Publ. Ltd., 2003, [Electronic].
[109] M. Petirsch and A. J. Schwab, “Investigation of the field uniformity of a modestirred chamber using diffusors based on acoustic theory,” IEEE Trans. Electromagn. Compat., vol. 41, no. 4, pp. 446–451, Nov. 1999.
[110] L. R. Arnaut, “Comments on ‘Investigation of the field uniformity of a mode-stirred
chamber using diffusors based on acoustic theory’,” IEEE Trans. Electromagn.
Compat., vol. 45, no. 1, pp. 146–147, Feb. 2003.
[111] D. Weinzierl, A. Raizer, and A. Kost, “Investigation of exciting fields in an alternative mode stirred chamber,” in Proc. Int. Symp. on Electromagnetic Compatibility.
Eindhoven, The Netherlands: Technische Universiteit Eindhoven, 2004, pp. 723–
727.
192
BIBLIOGRAPHY
[112] J. Perini, “Why two-wire TEM excitation in RCs will not work,” Open discussion
at the 2001 Reverberation Chamber, Anechoic Chamber and OATS Users Meeting,
June 2001.
[113] D. Wu and D. Chang, “The effect of an electrically large stirrer in a mode-stirred
chamber,” IEEE Trans. Electromagn. Compat., vol. 31, no. 2, pp. 111–118, 1989.
[114] C. F. Bunting, “Statistical characterization and the simulation of a reverberation
chamber using finite element techniques,” IEEE Trans. Electromagn. Compat.,
vol. 44, no. 1, pp. 214–221, Feb. 2002.
[115] ——, “Two-dimensional finite element analysis of reverberation chambers: the
inclusion of a source and additional aspects of analysis,” in Proc. IEEE Int. Symp.
on Electromagnetic Compatibility, vol. 1. Piscataway, NJ: IEEE, 1999, pp. 219–
224.
[116] C. F. Bunting, K. J. Moeller, C. J. Reddy, and S. A. Scearce, “A two-dimensional finite element analysis of reverberation chambers,” IEEE Trans. Electromagn. Compat., vol. 41, no. 4, pp. 280–289, Nov. 1999.
[117] ——, “Finite element analysis of reverberation chambers: a two-dimensional study
at cutoff,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, vol. 1.
Piscataway, NJ: IEEE, 1998, pp. 208–212.
[118] M. Hoijer, A. M. Andersson, O. Lundén, and M. Bäckström, “Numerical simulations as a tool for optimizing the geometrical design of reverberation chambers,”
in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, vol. 1. Piscataway,
NJ: IEEE, 2000, pp. 1–6.
[119] E. Laermans and D. De Zutter, “Modelled field statistics in two-dimensional reverberation chambers,” in Proc. Int. Symp. on Electromagnetic Compatibility, Sorrento. Milano, Italy: AEI, 2002, pp. 41–44.
[120] H.-J. Åsander, G. Eriksson, L. Jansson, and H. Åkermark, “Field uniformity analysis of a mode stirred reverberation chamber using high resolution computational
modeling,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, vol. 1.
Piscataway, NJ: IEEE, 2002, pp. 285–290.
[121] F. Moglie, “Convergence of the reverberation chambers to the equilibrium analyzed with the finite-difference time-domain algorithm,” IEEE Trans. Electromagn.
Compat., vol. 46, no. 3, pp. 469–476, 2004.
[122] L. Musso, V. Berat, F. Canavero, and B. Demoulin, “A plane wave Monte Carlo
simulation method for reverberation chambers,” in Proc. Int. Symp. on Electromagnetic Compatibility, Sorrento. Milano, Italy: AEI, 2002, pp. 45–50.
[123] L. Cappetta, M. Feo, V. Fiumara, V. Pierro, and I. M. Pinto, “Electromagnetic
chaos in mode-stirred reverberation enclosures,” IEEE Trans. Electromagn. Compat., vol. 40, no. 3, pp. 185–192, 1998.
BIBLIOGRAPHY
193
[124] J. M. Dunn, “Local, high-frequency analysis of the fields in a mode-stirred chamber,” IEEE Trans. Electromagn. Compat., vol. 32, no. 1, pp. 53–58, 1990.
[125] R. Holland and R. St. John, “Statistical EM field models in an externally illuminated, overmoded cavity,” IEEE Trans. Electromagn. Compat., vol. 43, no. 1, pp.
56–66, 2001.
[126] N. Geng and W. Wiesbeck, Planungsmethoden für die Mobilkommunikation Funknetzplanung unter realen physikalischen Ausbreitungsbedingungen [in German]. Berlin, Germany: Springer Verlag, 1999.
[127] S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in Communication
Electronics, 3rd ed. New York, NY: John Wiley & Sons, 1994.
[128] M. Otterskog and K. Madsén, “Cell phone performance testing and propagation
environment modelling in a reverberation chamber,” in Proc. 2003 Reverberation
Chamber, Anechoic Chamber and OATS Users Meeting, Austin, TX, Apr. 2003.
[129] D. A. Hill, “Electronic mode stirring for reverberation chambers,” IEEE Trans.
Electromagn. Compat., vol. 36, no. 4, pp. 294–299, 1994.
[130] J. S. Hong, “Multimode chamber excited by an array of antennas,” Electronics
Letters, vol. 29, no. 19, pp. 1679–1680, 1993.
[131] M. L. Crawford, T. A. Loughry, M. O. Hatfield, and G. J. Freyer, “Band limited,
white Gaussian noise excitation for reverberation chambers and applications to
radiated susceptibility testing,” National Bureau of Standards (NBS), Tech. Rep.,
1996.
[132] T. Loughry, “Electronic mode stirring,” U.S. Patent 5,327,091, 1993.
[133] K. Madsén, P. Hallbjörner, and C. Orlenius, “Models for the number of independent samples in reverberation chamber measurements with mechanical, frequency,
and combined stirring,” IEEE Antennas Wireless Propagat. Lett., vol. 3, no. 3, pp.
48–51, 2004.
[134] J. S. Hong, “Effect of a modulated source on a multimode cavity,” IEEE Microwave
Guided Wave Lett., vol. 4, no. 2, pp. 43–44, 1994.
[135] M. Klingler, S. Egot, J.-P. Ghys, and J. Rioult, “On the use of 3-D TEM cells for
total radiated power measurements,” IEEE Trans. Electromagn. Compat., vol. 44,
no. 2, pp. 364–372, May 2002.
[136] M. Klingler, V. Deniau, L. Kone, B. Kolundzija, and B. Demoulin, “Characterization of direct electromagnetic coupling occurring in the vicinity of the lower
modes in reverberation chambers,” in Proc. 14th Int. Zurich Symp. and Technical
Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal
Inst. Technol. Zurich, 2001, pp. 641–646.
194
BIBLIOGRAPHY
[137] J. Perini and L. S. Cohen, “An alternative way to stir the fields in a mode stirred
chamber,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, vol. 2.
Piscataway, NJ: IEEE, 2000, pp. 633–637.
[138] P. A. Beeckman and J. J. Goedbloed, “Results of the CISPR/A radiated emission
round robin test,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility.
Piscataway, NJ: IEEE, 2001, pp. 475–480.
[139] H. G. Krauthäuser, “Mode-stirred chambers for EMC measurements,” Universität
Magdeburg, Magdeburg, Germany, Tech. Rep., 1999.
[140] K. Slattery, J. Neal, and S. V. Smith, “Characterization of a reverberation chamber for automotive susceptibility,” in Proc. IEEE Int. Symp. on Electromagnetic
Compatibility, vol. 1. Piscataway, NJ: IEEE, 1998, pp. 265–269.
[141] E. L. Bronaugh, J. J. Polonis, and I. Martinez, “Whole-vehicle EMC testing in
a reverberation chamber,” in Proc. 12th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst.
Technol. Zurich, 1997, pp. 469–474.
[142] M. O. Hatfield, G. J. Freyer, and M. B. Slocum, “NASA Boeing 757 cavity field
variability based on Boeing 757 and Boeing 707 test data,” Naval Surface Warfare
Center Dahlgren Division (NSWCDD), Tech. Rep., 1997.
[143] D. A. Hill, “Evaluation of the NASA langley research center mode-stirred chamber
facility,” National Institute of Standards and Technology (NIST),” Tech. Note
1508, 1999.
[144] M. O. Hatfield et al., “Means and methods for performing shielding effectiveness
measurements using mode-stirred chambers,” United States Statutory Invention
Registration H821, 2000.
[145] J. Carlsson, A. Wolfgang, and P.-S. Kildal, “Numerical FDTD simulations of a
validation case for small antenna measurements in a reverberation chamber,” in
Proc. IEEE Antennas and Propagation Society Int. Symp., vol. 2. Piscataway,
NJ: IEEE, 2002, pp. 482–485.
[146] K. Madsén and P. Hallbjörner, “Reverberation chamber for mobile phone antenna
tests,” in Proc. 2001 Reverberation Chamber, Anechoic Chamber and OATS Users
Meeting, Seattle, WA, June 2001.
[147] J. Byun, D. Kim, and P.-S. Kildal, “Actual diversity gain measured in the reverberation chamber,” in Proc. IEEE Antennas and Propagation Society Int. Symp.,
vol. 3. Piscataway, NJ: IEEE, 2002, pp. 718–721.
[148] P. Hallbjörner and K. Madsén, “Terminal antenna diversity characterisation using
mode stirred chamber,” Electronics Letters, vol. 37, no. 5, pp. 273–274, 2001.
BIBLIOGRAPHY
195
[149] CRC. (2004) Handbook of chemistry and physics. CRC Press Ltd. Boca Raton,
FL. [Online]. Available: http://www.hbcpnetbase.com
[150] Curran Company and U.S. Navy, “Reverberation chamber tuner and shaft with
electromagnetic radiation leakage device,” U.S. Patent WO 00/54 365, 2000.
[151] N. Eulig and A. Enders, “Reverberation Chamber: Eine preiswerte Alternative zur
Absorberhalle? [in German],” Technisches Messen, vol. 69, no. 2, pp. 85–89, Feb.
2002.
[152] F. Weeks and G. Philp, “A feasibility study into increasing the rotational speed
of the tuner in the DSTO electromagnetic reverberation chamber,” Department of
Defense, Australia, Tech. Rep., 2000.
[153] Maxon Motor AG, “MIP-10 position controller reference data sheet,” Maxon Motor
AG, Tech. Rep., 2001.
[154] Schaffner EMV AG, “EMC Compliance 3 – test-house software for emission &
immunity EMC testing,” Schaffner EMV AG, Tech. Rep., 2004.
[155] IEEE, IEEE standard methods for measuring electromagnetic field strength of sinusoidal continuous waves, 30 Hz to 30 GHz, IEEE Standards Board Std., 1991.
[156] H. Keller and R. Bitzer, “EMR-20/EMR-30 field probes,” Narda Safety Test Solutions GmbH, Tech. Rep., 2002.
[157] J. R. E. Richardson, “Mode-stirred chamber calibration factor, relaxation time,
and scaling laws,” IEEE Trans. Instrum. Meas., vol. 34, no. 4, pp. 573–580, 1985.
[158] Rohde & Schwarz, “Dual-channel power meter NRVD,” Rohde & Schwarz GmbH
& Co. KG, Tech. Rep., 1998.
[159] ——, “Power sensors NRV-Z,” Rohde & Schwarz GmbH & Co. KG, Tech. Rep.,
2001.
[160] E. Suzuki, T. Miyakawa, H. Ota, K. I. Arai, and R. Sato, “Characteristics of an optical magnetic probe consisting of a loop antenna element and a bulk electro-optic
crystal,” in Proc. 15th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst. Technol. Zurich,
2003, pp. 61–64.
[161] R. Bitzer, “Orientation of field probe sensors,” Narda Safety Test Solutions GmbH,
Tech. Rep., 2003.
[162] D. M. Pozar, Microwave Engineering.
Reading, MA: Addison-Wesley, 1993.
[163] NIST. (2003) Uncertainty of measurement results. National Institute of
Standards and Technology (NIST). Boulder, CO. [Online]. Available:
http://physics.nist.gov/cuu/uncertainty/basic.html
196
BIBLIOGRAPHY
[164] United Kingdom Accreditation Service (UKAS), “The expression of uncertainty
and confidence in measurement,” United Kingdom Accreditation Service (UKAS),
Tech. Rep., Dec. 1997.
[165] H. Hart, W. Lotze, and E.-G. Woschni, Messgenauigkeit [in German], 3rd ed.
München, Germany: Oldenbourg, 1997.
[166] R. H. Dieck, Measurement Uncertainty Methods and Applications. Research Triangle Park, NC: Instrument Society of America, 1992.
[167] Amplifier Research, “FP6001 field probes,” Amplifier Research Inc., Tech. Rep.,
2002.
[168] H. Trzaska, Electromagnetic Field Measurements in the Near Field. Atlanta, GA:
Noble Publishing Corp., 2001.
[169] M. Kanda, “Standard probes for electromagnetic field measurements,” IEEE
Trans. Antennas Propagat., vol. 41, no. 10, pp. 1349–1364, Oct. 1993.
[170] ISO, “Guide to the expression of uncertainty in measurement (GUM),” International Standardization Organization (ISO), Tech. Rep., 1995.
[171] Schaffner EMC Systems, “EMC measurement uncertainty – a handy guide,”
Schaffner EMC Systems Ltd., Tech. Rep., 2002.
[172] D. Bohlen and K. Kessler, Nichts als die Wahrheit.
Heyne Verlag, 2002.
München, Germany: W.
[173] E. A. Godfrey, “Effects of corrugated walls on the field uniformity of reverberation chambers at low frequencies,” in Proc. IEEE Int. Symp. on Electromagnetic
Compatibility, vol. 1. Piscataway, NJ: IEEE, 1999, pp. 23–28.
[174] D. Svetanoff, J. Weibler, R. Cooney, M. Squire, S. Zielinski, M. O. Hatfield, and
M. Slocum, “Development of high performance tuners for mode-stirring and modetuning applications,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility,
vol. 1. Piscataway, NJ: IEEE, 1999, pp. 29–34.
[175] NIST. (2003) Materials database. National Institute of Standards and
Technology (NIST). Boulder, CO. [Online]. Available: http://www.nist.gov
[176] American Society for Metals (ASM), ASM metals reference book, 3rd ed., M. Bauccio, Ed. Metals Park, OH: ASM International, 1993.
[177] AK Steel. (2003) Electrical steel specifications. AK Steel. Middletown, OH.
[Online]. Available: http://www.aksteel.com
[178] C. Bruns, P. Leuchtmann, and R. Vahldieck, “FEKO -simulation and measurement of a reverberation chamber,” in Proc. 2003 German FEKO Users Meeting.
Böblingen, Germany: EMSS GmbH, 2003, [Electronic].
BIBLIOGRAPHY
197
[179] U. Jakobus, “FEKO computes negative input power at antenna ports for reverberation chamber excitations,” Private communication, Feb. 2002.
[180] T. Kawashima, J. C. Aquino, and M. Tokuda, “Evaluation of anechoic chamber
characteristics using an optically driven imitated equipment,” in Proc. IEEE Int.
Symp. on Electromagnetic Compatibility. Tokyo, Japan: IEICE of Japan, 1999,
pp. 244–247.
[181] ETS Lindgren, “REFRAD - Reference Radiator Manual,” ETS Lindgren L.P.,
Tech. Rep., 2003.
[182] T. E. Harrington, “Update on informal interlaboratory comparison of reverb,
GTEM, FAR, and OATS sites using Refrad and European Union FAR project simple EUT,” in Proc. 2003 Reverberation Chamber, Anechoic Chamber and OATS
Users Meeting, Austin, TX, Apr. 2003.
[183] D. G. Rueda and M. J. Alexander, “EUT measurement comparison between
different EM environments: FAR, OATS, and GTEM cell,” in Proc. 14th Int.
Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich,
Switzerland: Swiss Federal Inst. Technol. Zurich, 2001, pp. 347–352.
[184] D. Bozec, “YES contribution to the emission and immunity measurements and
production of best practice guide on GTEM cells used for EMC measurements,”
York EMC Services (YES) Ltd., Tech. Rep., 2003.
[185] L. Musso, B. Demoulin, F. Canavero, and V. Berat, “Radiated immunity testing
of a device with an external wire: repeatability of reverberation chamber results
and correlation with anechoic chamber results,” in Proc. IEEE Int. Symp. on
Electromagnetic Compatibility, vol. 2. Piscataway, NJ: IEEE, 2003, pp. 828–833.
[186] T. E. Harrington, “Total-radiated-power-based OATS-equivalent emissions testing in reverberation chambers and GTEM cells,” in Proc. IEEE Int. Symp. on
Electromagnetic Compatibility, vol. 1. Piscataway, NJ: IEEE, 2000, pp. 23–28.
[187] D. A. Hill, D. G. Camell, K. H. Cavcey, and G. H. Koepke, “Radiated emissions
and immunity of microstrip transmission lines: Theory and reverberation chamber measurements,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility,
vol. 38, no. 2. Piscataway, NJ: IEEE, 1996, pp. 165–172.
[188] A. Nothofer, M. J. Alexander, D. Bozec, D. Welsh, L. Dawson, L. McCormack,
and A. C. Marvin, “A GTEM best practice guide – applying IEC 61000-4-20 to the
use of GTEM cells,” in Proc. 15th Int. Zurich Symp. and Technical Exhibition on
Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst. Technol.
Zurich, 2003, pp. 207–212.
[189] M. Spitzner, K. Münter, J. Glimm, and L. Dallwitz, “EMV-Ringvergleich der DATech/RegTP/PTB – Ergebnisse, Erkenntnisse und Schlussfolgerungen [in German],” Technisches Messen, vol. 70, no. 3, pp. 151–162, Mar. 2003.
198
BIBLIOGRAPHY
[190] M. B. Slocum and M. O. Hatfield, “Evaluation of proposed IEC reverberation
chamber methodology for radiated emissions measurements using a reference radiator,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway,
NJ: IEEE, 2001, pp. 734–739.
[191] C. Bruns, P. Leuchtmann, and R. Vahldieck, “Three-dimensional method of moments simulation of a reverberation chamber in the frequency domain,” in Proc.
15th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility.
Zurich, Switzerland: Swiss Federal Inst. Technol. Zurich, 2003, pp. 229–232.
[192] ——, “Broadband method of moment simulation and measurement of a medium
sized reverberation chamber,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway, NJ: IEEE, 2003, pp. 844–849.
[193] ——, “Simulation and comparison of different stirrer types inside a reverberation
chamber,” in Proc. IEEE Int. Symp. on Electromagnetic Compatibility. Piscataway, NJ: IEEE, 2004, pp. 241–244.
[194] ——, “Modeling and simulation of a canonical equipment under test inside a
medium-sized reverberation chamber,” in Proc. Int. Symp. on Electromagnetic
Compatibility. Eindhoven, The Netherlands: Technische Universiteit Eindhoven,
2004, pp. 744–749.
[195] J. G. Proakis, Digital communications.
New York, NY: McGraw-Hill, 1995.
[196] NAWCWD. (2004) Electronic warfare and radar systems engineering handbook.
Naval Air Warfare Center Weapons Division (NAWCWD). Point Mugu, CA.
[Online]. Available: https://ewhdbks.mugu.navy.mil
[197] U. Jakobus, “64 bit Sun version of FEKO crashes during LU decomposition,”
Private communication, Sept. 2003.
[198] ——, “Different results for computations with the method-of-moments (MoM)
versus the multilevel fast multipole method (MLFMM),” Private communication,
Aug. 2003.
Acknowledgments
Work for this thesis was done at the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH), ETH Zurich, Switzerland and Schaffner EMV AG, Luterbach, Switzerland.
First of all I would like to thank Prof. Rüdiger Vahldieck of IFH, ETH Zurich who gave
me the opportunity to carry out my Ph.D. thesis in his research group. He offered me
a highly interesting topic, bridging the gap between academia and industry, and gave
me 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 related
to 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 unique
possibility to attend virtually any conference related to my thesis topic and of interest
to 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 the
Politecnico di Torino, Torino, Italy, for the thorough review of this thesis, his very constructive 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 helpful
during this project and contributed to the results presented in this thesis.
Several big “Merci viiielmool” are due to the IFH measurement guru Hansruedi Benedickter, Aldo Rossi of the IFH electronics workshop, Ray Ballisti for help in computer matters
as 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 Dearing, Dan Hamblin, Richard Davy, David Riley, Uwe Karsten, and Michael Rehfeldt of
Schaffner EMV AG, who co-initiated this project, helped making it progress through
several stages, spent considerable time on the prototype measurements and software development, discussed new ideas, and allowed me to participate in the day-to-day business
of the company.
Dr. Ulrich Jakobus of EMSS GmbH, Böblingen, Germany, assisted me a lot with the
electromagnetic 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 Ladbury of NIST, Boulder (CO), USA, Dr. Luk Arnaut and Martin Alexander, both of NPL,
Teddington, UK, Dr. Wolfgang Kürner of EADS AG, Hamburg, Germany, Dr. Hans
Georg Krauthäuser of the Universität Magdeburg, Magdeburg, Germany, Dr. Nils Eulig
of the Universität Braunschweig, Braunschweig, Germany, Albin Maridet and Frédéric
199
200
ACKNOWLEDGMENTS
Hoëppe, both of EADS CCR S.A., Suresnes, France, and Gérard Orjubin of the Université Marne-La-Valleé, Marne-La-Valleé, France.
During the course of this thesis and at several conferences I had the pleasure to share the
results of my research with Dr. Mats Bäckström, Magnus Hoijer, and Olof Lundén, all
of the Swedish Defence Research Agency (FOI), Linköping, Sweden, Magnus Otterskog
from the Örebro University, Örebro, Sweden, Tim Harrington of the FCC, Laurel (MD),
USA, Gus Freyer, Monument (CO), USA, Michael Windler of Underwriters Laboratories
Inc., Northbrook (IL), USA, Peter Landgren of Saab Bofors Dynamics AB, Kent Madsén
of Flextronics International AB, Linköping, Sweden as well as Nico van Dijk of Philips
Research B.V., Eindhoven, The Netherlands.
Special thanks to all members of the IFH laboratory who made work, lunch and coffee
breaks 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 to
the gym to work out, skiing, hiking, biking or discussing political, economic, cultural or
scientific topics – and who became true friends instead of just colleagues and thus made
my time at ETH Zurich a great experience.
Apart from the scientific results presented in this thesis, what did I learn personally from
pursuing a Ph.D.? “If you can’t do it better – why bother doing it at all?” is an excellent
guideline to select what you do for your thesis and how you do it. If it is clear from the
beginning 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 will
be simply a repetition of someone else’s work. On top of this, you will waste yours and
other people’s time. Establish the starting point of your thesis by performing a thorough
search on what has been done up to now by others around the world – not just in the
offices next door. This approach will help you a lot when it comes to publishing your
results later on. Usage of readily available proven tools, perfectly suited to solve a certain
problem of your thesis, will keep you from reinventing the wheel, thus greatly accelerate
your progress. Identify your strongest scientific competitors – again worldwide, not just
locally – and learn from their achievements instead of ignoring them. Benchmark your
work against theirs and publish in a language which is universally understood. Choose
not to do simply another “Me-too” Ph.D. thesis. Cheers!
c 2003 Scott Adams Inc., distributed by United Feature Syndicate, Inc.
Copyright List of Publications
Publications related to this thesis
Journal papers
P1
C. Bruns and R. Vahldieck, “A closer look at reverberation chambers – 3-D
simulation and experimental verification,” accepted for publication in IEEE
Trans. Electromagn. Compat., Aug. 2005.
Conference papers
P2
C. Bruns, P. Leuchtmann, and R. Vahldieck, “Introduction to reverberation chamber simulation,” in Proc. 2nd NPL FREEMET meeting at MIRA,
Nuneaton. Teddington, UK: National Physical Laboratory (NPL), 2002, [Electronic].
P3
——, “Three-dimensional method of moments simulation of a reverberation
chamber in the frequency domain,” in Proc. 15th Int. Zurich Symp. and Technical Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss
Federal Inst. Technol. Zurich, 2003, pp. 229–232.
P4
——, “Challenges and results of realistic reverberation chamber simulations and
measurements,” in Proc. 2003 Reverberation Chamber, Anechoic Chamber and
OATS Users Meeting, Austin, TX, 2003.
P5
——, “Broadband method of moment simulation and measurement of a medium
sized reverberation chamber,” in Proc. IEEE Int. Symp. on Electromagnetic
Compatibility. Piscataway, NJ: IEEE, 2003, pp. 844–849.
P6
P. Leuchtmann, C. Bruns, and R. Vahldieck, “On the validation of simulated
fields in a reverberation chamber,” in Proc. European Microwave Conference
2003. London, UK: Horizon House Publ. Ltd., 2003, [Electronic].
P7
C. Bruns, P. Leuchtmann, and R. Vahldieck, “Comparison of various reverberation chamber geometries and excitations using a frequency domain method
of moments simulation,” in Proc. 17th Int. Wroclaw Symp. and Exhibition on
Electromagnetic Compatibility. Wroclaw, Poland: Politechniki Wroclawskiej,
2004, pp. 97–102.
201
LIST OF PUBLICATIONS
202
P8
C. Bruns, P. Leuchtmann, and R. Vahldieck, “Simulation and comparison of
different 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 a
medium-sized reverberation chamber,” in Proc. Int. Symp. on Electromagnetic
Compatibility. Eindhoven, The Netherlands: Technische Universiteit Eindhoven, 2004, pp. 744–749.
P10 ——, “Cubic and corrugated reverberation chambers: mode distribution, correlation, and field uniformity,” in Proc. 16th Int. Zurich Symp. and Technical
Exhibition on Electromagnetic Compatibility. Zurich, Switzerland: Swiss Federal Inst. Technol. Zurich, 2005, pp. 539–542.
P11 R. Vahldieck and C. Bruns, “Statistical characterization of reverberation chambers,” accepted for publication in Proc. 9th Int. Conference on Electromagnetics
in 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 simulation of a 1-18 GHz broadband parabolic reflector horn antenna system,” IEEE
Trans. Antennas Propagat., vol. 51, no. 6, pp. 1418–1422, June 2003.
P13 ——, “Analysis and simulation of a 1–18-GHz broadband double-ridged horn
antenna,” 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 experimental verification of a broadband ridged horn antenna system with parabolic
reflector,” 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,” in
Proc. URSI Int. Symp. on Electromagn. Theory. Ghent, Belgium: Int. Union
of Radio Science (URSI), 2001, pp. 621–623.
Curriculum Vitae
Personal data
Name:
Nationality:
Christian Bruns
Date of birth:
December 19, 1973
E-mail:
christian@bruns.com
German
Professional experience
11/04 – 03/05: Huber + Suhner AG, Herisau, Switzerland
Strategic evaluation of IEEE 802.16/WiMAX standard (diploma thesis)
11/00 – 05/05: ETH Zürich, Zurich, Switzerland
Laboratory for Electromagnetic Fields and Microwave Electronics
Doctorate in Electrical Engineering
Project with Schaffner EMV AG, Luterbach, Switzerland
05/97 – 10/97: Robert Bosch GmbH, Stuttgart, Germany
Gasoline engine management systems group (trainee)
01/96 – 08/96: Energy and High Voltage Systems Institute, Karlsruhe, Germany
Electromagnetic compatibility group (consultant, research assistant)
01/93 – 12/95: DATEC Elektroanlagen GmbH, Karlsruhe, Germany
Planning and setup of IT networks (trainee, freelance)
01/92 – 06/96: Bruns + Gerst GbR, Karlsruhe, Germany
Professional light and audio systems (founder and co-owner)
04/90 – 12/93: Radio Badenia GmbH, Karlsruhe, Germany
Interviews, data mining, reports, pre-/post-production (freelance)
Education
10/01 – 02/05: ETH Zürich, Zurich, Switzerland
Post-graduate studies in business, economics, and finance
08/99 – 04/00: ETH Zürich, Zurich, Switzerland
Analysis and simulation of a broadband horn antenna (diploma thesis)
08/96 – 05/97: Massachusetts Institute of Technology, Cambridge (MA), USA
University of Massachusetts, Dartmouth (MA), USA
Graduate study of Electrical Engineering (scholarship)
10/93 – 04/00: Universität Karlsruhe (TH), Karlsruhe, Germany
Dipl.-Ing. Electrical Engineering
09/84 – 05/93: High School Max-Planck-Gymnasium, Karlsruhe, Germany
203