N° d’ordre : 3275
THESE
présentée à
L’Université des Sciences et Technologies de Lille
pour obtenir le grade de
Docteur de l’Université
Spécialité : ELECTRONIQUE
Par
Luca Musso
Assessment of reverberation chamber testing
for automotive applications
soutenue le 21 Février 2003 devant la Commission d’Examen
JURY:
MM.
M. BÄCKSTRÖM
A. C. MARVIN
S. PIGNARI
W. TABBARA
F. CANAVERO
B. DEMOULIN
V. BERAT
F. FLOURENS
Rapporteur
Rapporteur
Rapporteur
Rapporteur
Directeur de thèse
Directeur de thèse
Co-Directeur de thèse
Examinateur
Summary
The use of electronics in automotive industry has increased at a phenomenal rate in
the last decades, leading to more than 70 electronically driven functions on most
recent cars. To ensure the reliability of each electronic function and device,
automotive manufacturers must face the problem of conceiving robust electronic
systems, with respect to possible internal and external electromagnetic
perturbations. In order to validate single devices and whole cars with respect to
such constraints, electromagnetic compatibility tests are performed, according to
standardized methods.
The contribution brought by this work is inherent to the radiated immunity
tests performed on automotive electronic devices and on cars. The classical test
methodology in fully– or semi–anechoic chambers proves to be unsatisfactory,
from the point of view of quality and cost–effectiveness, forcing to investigate for
alternative methods. We investigate in this work the reverberation chamber
methodology, for radiated immunity tests in automotive applications. Several notyet consolidated aspects concerning this methodology are analysed.
As a first aspect we address to the quantification of the measurement
uncertainty in a real chamber, and to the loading effects obtained by introducing a
car inside the chamber. The first aspect is at the basis of the evaluation of the
reliability of reverberation chamber tests, and the second one must be investigated
to evaluate the feasibility of car tests in chambers with reasonable sizes. To fulfil
these two tasks, we carry out both a theoretical analysis and statistical analysis of
experimental data obtained inside the Technocentre Renault reverberation
chamber.
Later on, we focus on the problem of modelling the electromagnetic fields
coupling to electrical objects. An original solution is proposed to this problem,
based on a discrete statistical plane wave approach for reverberation chambers
electromagnetic environment. The proposed approach has two potential
applications. The first one is the possibility of predicting reverberation chamber
coupled quantities by the numerical simulation of a set of random plane waves
I
Summary
coupling contributions. We investigate this possibility for the case of
electromagnetic fields coupling to wires, which is a relevant problem for
automotive electromagnetic compatibility. A second kind of application, concerns
the correlation between the coupling results obtained in reverberation chamber and
in anechoic chambers, which are supposed to operate in a plane wave environment.
This correlation is possible, according to the proposed approach, in the case where
the electrical object is inspected over different incidence angles and field
polarizations in anechoic chambers. The consequent possibility of correlating
radiated immunity results obtained in reverberation and anechoic chambers is
finally experimentally investigated for a device representative of automotive
devices.
II
Acknowledgements
Guyancourt, March 2003
This thesis is situated within the French context of a CIFRE1 agreement, which
implies both an industrial and an academic direction of the thesis research work.
The industrial direction has been ensured by the Research Direction of Renault
Automobile, where I worked as a research Engineer during the three years of the
thesis. A scientific co-direction agreement between the Electronics Department of
the Politecnico di Torino and the TELICE2 Laboratory of the University of Lille
ensured the academic direction of the thesis.
The thesis began thanks to the will of the Renault Engineer Jérôme Bossé,
who introduced me to Renault and to reverberation chamber testing. I wish to thank
him for the determination and the scientific competence that he transmitted to me
during the beginning of the thesis. Renault Engineer Vincent Berat was charged to
direct my thesis later on. I would like to thank him for constantly encouraging me
during these three years and for always pushing me to find the good compromise
between theoretical research and industrial requirements. It was a pleasure to work
with him. I wish also to take this opportunity to thank the whole Renault
electromagnetic compatibility team and its responsible Anselmo Soria, for the
kindness and friendship that they showed towards me, and for helping me each one
with his particular skills.
Special thanks are addressed to my thesis Director Prof. Flavio Canavero.
He supported my candidature in Renault and encouraged me to take this great
opportunity. His scientific methodology and valuable comments, as well as his
complete availability for useful discussions much improve the quality of this work.
I also particularly thank him for accurately revising this manuscript.
1
“Convention Industrielle de Formation par la Recherche” (Industrial Agreement of
Training by Research)
2
“Laboratoire de Télécommunications, Interférences et Compatibilité Electromagnétique”
(Laboratory of Telecommunications, Interferences and Electromagnetic Compatibility)
III
Acknowledgements
Particular thanks are also addressed to my thesis Director Prof. Bernard
Démoulin. His wide experience in reverberation chambers and his enthusiasm in
scientific research much helped me and encouraged me to constantly make
progress.
This manuscript was revised by four referees, who gave important
indications and valuable comments to improve the final version of the manuscript. I
would like to thank the referees Dr. Mats Bäckström (Swedish Defence Research
Agency FOI, Sweden), Prof. Andrew C. Marvin (University of York, UK), Prof.
Sergio Pignari (Politecnico di Milano, Italy) and Prof. Walid Tabbara (LSS3
SUPELEC, France) for accepting to revise the manuscript, for their effort and their
thorough revision work. I thank in particular Dr. Mats Bäckström who helped me
to better understand the tested devices directivity effects in correlating
reverberation chamber results and anechoic chamber results.
I would also like to thank Dr. Franck Flourens (AIRBUS, France), who
accepted to take part to the thesis jury as examiner. I thank him for his precious
comments and remarks made during the thesis defence.
I finally wish to thank also all those people who collaborated with me to
obtain the results of this thesis, and in particular Dr. Lamine Kone for helping me
during the measurement campaigns at the TELICE laboratory and Dr. Serge
Ficheux and his staff for measurement campaigns at the UTAC4 EMC laboratory.
The most important thanks go to those who allowed me to succeed in
making all this possible. I think to my family (my parents and my sister Elena),
which is close to me even when I’m far away. I learn from you the essential of life.
And I think to Cristina, who had the patience to follow me in this adventure
and the constancy to give me the strength to overcome every difficulty. I learn from
you the beauty of life.
3
“Laboratoire des Signaux et Systèmes ” (Laboratory of Signals and Systems)
“Union Technique de l’Automobile du motocycle et du Cycle” (Technical Union of Car,
motorcycle and Cycle)
4
IV
List of acronyms
AC
CLT
DUT
EM
EMC
RC
Anechoic Chamber
Central Limit Theorem
Device Under Test
Electromagnetic
Electromagnetic Compatibility
Reverberation Chamber
V
Table of contents
Summary
I
Acknowledgements
III
List of acronyms
V
1 General introduction
1
1.1 Automotive EMC
1
1.2 RC methodology for radiated immunity testing .........................................3
1.3 Thesis objectives and plan ..........................................................................4
2 Renault RC facility
6
2.1 Introduction.................................................................................................6
2.2 Ideal RC EM fields statistical model ..........................................................7
2.3 Measurement uncertainty statistical model ..............................................10
2.3.1 Mean values uncertainty ...............................................................12
2.3.2 Maximum values uncertainty........................................................15
2.4 Qualifications principles and techniques for a real chamber....................17
2.4.1 Statistical tests...............................................................................19
2.4.2 Measurement uncertainty in a real chamber .................................24
2.4.3 Calibration following standards....................................................27
2.4.4 Chamber loading ...........................................................................29
2.5 Conclusions...............................................................................................33
3 RC EM fields coupling with electrical objects: a statistical plane
wave approach
36
3.1 Introduction...............................................................................................36
3.2 From a plane wave integral model to a plane wave discrete model
VI
Table of contents
3.3
3.4
for RC EM fields.......................................................................................37
Statistical plane wave coupling approach for RC.....................................42
3.3.1 Coupled quantities mean values....................................................43
3.3.2 Coupled quantities maximum values ............................................46
3.3.3 Monte Carlo methods accuracy ....................................................47
Conclusions...............................................................................................49
4 EM fields coupling to wires in a RC
51
4.1 Introduction...............................................................................................51
4.2 RC EM fields coupling to single wire transmission lines: modelling
and experimental validation......................................................................52
4.2.1 Modelling......................................................................................52
4.2.2 Validation......................................................................................54
4.3 Extensions to wire bundles .......................................................................64
4.4 Conclusions...............................................................................................67
5 Radiated immunity test of electronic devices in RC
70
5.1 Introduction...............................................................................................70
5.2 Radiated immunity testing in RC..............................................................70
5.3 Directivity based approach for RC and AC radiated immunity
results comparison ....................................................................................71
5.4 Measurement results of a generic test device ...........................................75
5.4.1 Immunity test results in RC ..........................................................76
5.4.2 Immunity test results in AC ..........................................................78
5.5 Conclusions...............................................................................................80
6 Conclusions
82
Bibliography
87
A Probabilistic and statistical tools
91
A.1 Introduction...............................................................................................91
A.2 Probability theory fundamentals...............................................................91
A.3 Extreme order statistics.............................................................................93
A.4 Confidence intervals .................................................................................95
A.5 Goodness-of-fit tests .................................................................................98
VII
Table of contents
A.6 Monte Carlo method .................................................................................99
B Computation details of the statistical properties of the field
resulting from one random plane wave
102
C Details of Monte Carlo mean values estimation
106
D Measurement set-up
109
E Noise reduction
112
VIII
Chapter 1
General introduction
1.1 Automotive EMC
The use of electronics in automotive industry has increased at a phenomenal rate in
the last decades. Such a proliferation can be classified into three separate categories
[1]. The first one is tied to the basic security, driving-control and comfort
functions, such as airbags, ABS, stability control and electrical brake system. The
second one is related to the car body functions, such as thermal comfort, anti-theft
devices and remote key-less control. The third and most recent one, is the
communication world, which deals for instance with telephone, GPS and Bluetooth
systems. As a result, most recent cars have more than 70 electronically driven
functions. A first key issue of automotive EMC is thus to ensure the reliability of
such a complex electronic system, by eliminating the EM interactions between subsystems.
On the other hand, the car external environment presents many risks of
possible EM aggression, given, for instance, by broadcast and mobile
communication systems, or power lines and systems. Furthermore, the use of
electronic equipments by passengers presents an uncontrolled additional
contribution to the external EM pollution, and an increased threat to the intended
function of electronic devices.
As a consequence, automotive industries are faced to the problem of
conceiving robust electronic systems, with respect to internal and external
perturbations, with low EM emissions. To such constraints, typical automotive
constraints must be added, such as the use of low-cost and low-weight cables and
devices. In order to validate the single devices and the complete cars with respect
1
1 – General introduction
to such constraints, EMC tests are performed, according to standardized methods.
The qualities required for test methods can be identified in the representativity of
the test with respect to the identified risk, in a good repeatability of measurements,
and in the lack of pollution of the environment external to the test site. An
additional quality, is the capability of testing the devices with regard to the most
severe conditions to which they may be exposed in reality. This implies the
investigation of the most sensitive characteristics of the device.
The contribution brought by this work is inherent to the radiated immunity
tests performed on automotive electronic devices and on cars. The European
directive EC/95/54 addresses the problem of EMC for motor vehicles, by
specifying test levels as well as test methods. Classical radiated immunity test sites
in automotive industry are fully or semi-ACs. In the validation and homologation
process, in order to anticipate the risks of malfunctioning when operating on-board,
benchtests are first performed on single devices placed on conducting planes in
semi-ACs. Benchtests are also necessary face to the poor availability of car
prototypes, dictated by ever-shortened development and production processes of
cars, and to costs reduction.
The classical methodology shows several critical problems, in particular
when considering the measurement repeatability and the investigations of the most
sensitive zones of the devices. Firstly, measurement repeatability in AC is strongly
dependent on the measurement configuration, e.g. the device and wires distances
and positions with respect to the emitting antenna. For instance, at low frequencies,
where far-field conditions are barely verified, the distance from the device and the
antenna influences the results in a non-controlled way. Furthermore, the experience
shows that it is difficult to ensure the reproducibility of results in test-facilities with
different sizes and instrumentations.
Additionally, the investigation of the most sensitive device zones in ACs
implies the inspection of a large number of device orientations with respect to the
emitting antenna. In practice, this is done by rotating the device over a rotational
axis, and changing the antenna height. Nevertheless, for automotive applications,
only the front inspection angle is adopted in benchtests, and one or two incidences
(frontal and lateral) are adopted for cars, with a consequent strong potential risk of
missing critical directions.
In order to take into account such difficulties, automotive manufacturers
impose large safety margins to mandatory immunity levels, during validation tests.
If we add to these problems also the high costs of test-facilities, mainly due to
2
1 – General introduction
absorbers and amplifiers, such a process reveals to be unsatisfactory, from the point
of view of quality and cost-effectiveness.
Such practical and economic difficulties, have inspired the investigation of
alternative methods. Among the alternative methods, the use of RC is promising
because of their practical and economic advantages.
1.2 RC methodology for radiated immunity testing
The introduction of EM RC dates approximately to 25 years ago [2], but their use
is only now becoming accepted in EMC standards, starting with aeronautic [3] and
automotive applications ([4] and [5]). At the beginning, the basic idea was to
dispose of an emission test method for evaluating the total radiated power by an
electrical device, despite of its radiation pattern. Equivalently, for immunity
testing, the aim was to dispose of an uniform and isotropic EM environment,
allowing a homogeneous illumination of the tested device. Scientific works made
their appearance proposing at first experimental investigations in radiated
immunity tests [6] and in shielding effectiveness measurements [7]. After such an
experimental phase of about 15 years, in the 90s, the theoretical characterisation of
RCs has been developed. The major encountered difficulty was the theoretical
characterization of a complex-shaped EM cavity. Deterministic approaches were
abandoned, since they are not able to give a characterization of a cavity with a
complex general shape, and statistical characterizations made their appearance in
[8], [9] and [10]. The transition from complex shaped cavities to regular cavities
equipped with metallic rotating objects of complex shape, called stirrers, finally
took place, and an EM theory of RC was established in [11]. Since then, RCs made
their appearance in European research laboratories [12], universities [13],
industries [14] and, among the others, at the Technocentre Renault.
The RC methodology offers many potential advantages, which will be
briefly listed below and will be investigated in this thesis from the point of view of
industrial automotive radiated immunity tests. The statistical uniformity and
isotropy of the EM environment allow to have an omni-directional illumination of
the DUT, and to avoid the research of the most penalizing incidence direction. This
is a first advantage in using the reverberation testing methodology. At the same
time, the statistical fields uniformity eliminates the dependence of the results from
the measurement configuration, that is the wires and device positions and distances
from emitting antenna. Further, the measurement repeatability may in principle be
3
1 – General introduction
characterized by statistical criteria, based on fields characterization. Such a
formulation of uncertainty is furthermore independent from the particular used testfacility, and can, at least in principle, characterize measurement reproducibility in
different sites. Other interesting advantages are the possibility of generating high
field strengths with reduced power amplifiers, and the low realization costs due to
the lack of absorbing panels. An extensive validation of such properties could
make the RC testing a reliable and cost-effective methodology.
Nevertheless, several not yet consolidated aspects must be investigated.
Among these, the validation of the above exposed ideal properties for a real
chamber is primary, for defining the limits of acceptance of ideal models. A variety
of different qualification criteria has been proposed so far, with some difficulties of
correlation among them, as it is discussed in [15]. Secondly, the understanding of
the EM coupling of RC fields with electrical and electronic devices is necessary for
establishing testing methodologies. Finally, the correlation between RC immunity
results and classical tests of immunity is required for several reasons. On one hand,
the knowledge of the correlation is necessary when investigating not only the
reliability of a test method, but also when searching for a characterization of the
real response of the considered device. On the other hand, establishing a
correlation is necessary for wide acceptance of a new method.
Several of these aspects will be investigated and formalized in this thesis, as
discussed in more details in the next section.
1.3 Thesis objectives and plan
The final objective of this thesis is to provide a characterization of a reverberation
testing methodology for radiated immunity in automotive industry. A complete
analysis of the testing methodology will be conducted in this thesis work,
extending from the phase of the empty chamber qualification, to the EM coupling
to electrical objects, and to a study of the immunity test of a generic electronic
device and of the correlation with classical test methodologies.
In the first part of the work, the RC principles and operation will be shortly
recalled and, in the light of the theoretical models for ideal chambers, a rigorous
experimental analysis and qualification of a real chamber performances will be
proposed. In order to formalize the repeatability and reproducibility of
measurements, the statistical formulation of measurement uncertainty related to
fields statistics will be presented. The validity of the ideal uncertainty model will
4
1 – General introduction
be then experimentally investigated in the Renault test-facility. Several relevant
elements, such as the agreement of real chamber fields statistics with the ideal
model, the knowledge of the number of independent stirrer positions, the
evaluation of the residual uncertainty and the effect of introducing a big object
inside the empty chamber will be systematically investigated in Chapter 2.
The problem of the EM coupling of RC fields to electrical objects will be
then analysed in Chapter 3. An original contribution of this thesis, consisting in the
formulation of a model based on statistical plane wave coupling, will be
introduced. The theoretical basis and the detailed formulation of the proposed
approach will be presented. The feasibility of such approach in predicting the EM
coupling in RCs will be then investigated and validated for the practical cases,
relevant to automotive applications, of field coupling to wires and wire bundles. In
Chapter 4, the validation of the proposed coupling approach will be achieved by
comparing numerical coupling results, obtained by applying the proposed
approach, with measurement results obtained in the Renault chamber.
The last part of this work will consider radiated immunity tests on a generic
test device in a RC. In particular, we will present an experimental validation of the
advantages expected in using the RC methodology. In order to perform
experimental investigations, a simple electronic device was conceived and realized
during the thesis. The characteristics of such device are a low radiated
susceptibility level over a wide frequency range, a simple non-intrusive system of
detecting the failures, and a radiated susceptibility mostly due to an external wire.
The repeatability and reproducibility of RC measurements will be investigated in
Chapter 5, by repeating the tests in three chambers with different sizes, stirrers and
instrumentations. Finally, the problem of correlation between immunity results
obtained in RCs and in fully ACs, considered here as a reference for classical
testing methodologies, will be also presented in Chapter 5. Based on the statistical
plane wave coupling approach proposed in Chapter 3, a correlation methodology
will be presented, and the applicability conditions will be discussed and analysed.
The correlation methodology will be applied to correlate experimental results
obtained with the above described device in both kind of test-facilities.
The ensemble of results obtained in this work clarifies several not-yet
consolidated aspects of the RC methodology, building the ground for a robust and
reliable testing procedure.
5
Chapter 2
Renault RC facility
2.1 Introduction
An EM RC is a metallic enclosure or cavity that allows to create a statistically
uniform EM environment, once excited by an internal EM source. This is possible
at high frequencies, by redistributing the EM energy among the cavity resonant
modes, with the help of a mode tuning or a mode stirring technique (hence the
names of mode-tuned and mode-stirred RC, MTRC and MSRC respectively).
Mode tuning and mode stirring can be achieved either mechanically or
electronically. In the first case, mechanical, a complex shaped metallic stirrer is
inserted inside the chamber and turned around a rotation axis to change EM fields
boundary conditions during time. When the stirrer is turned continuously in time, a
mode stirring operation takes place, when the stirrer is rotated by fixed steps, a
mode tuning operation takes place. Under particular conditions, mode tuning and
mode stirring operations allow to obtain different independent boundary conditions
for EM fields during time, such that statistical uniformity of fields is ensured
during one stirrer rotation inside a limited portion of the chamber volume, far from
the stirrer and the chamber walls. The Renault RC is provided with a mechanical
stirrer that can be turned both continuously and by discrete steps. In the following
of this work only the mode-tuned operation will be considered. This means that
measurements of EM quantities are carried out by sampling for fixed stirrer
positions. We made this choice since a continuous stirrer rotation can not be used
for testing most of automotive devices which often have a long response time.
Furthermore, only single frequency continuous wave fields are considered.
This Chapter is intended to introduce the Renault RC EM properties. This is
6
2 – Renault RC facility
done by firstly recalling the theory of an ideal RC and then proposing the
qualification measurements with regard to the ideal model. By this approach, a
scrupulous statistical methodology of analysis for a real RC is proposed.
The Chapter is structured as follows. The theoretical properties of an ideal
RC environment will be recalled in section 2.2. Particular attention will be stressed
on probabilistic and statistical characterisations of EM quantities inside the
chamber. Then, based on this probabilistic model, the uncertainty model for mean
and maximum values measurement will be exposed in section 2.3. The
characterisation and qualification measurements of the Renault chamber will be
finally presented in section 2.4. The agreement of the real EM environment with
ideal models will be investigated by means of statistical tests applied to
measurements and by means of a standard calibration method for the empty
chamber. Additionally the chamber electrical loading effect will be investigated
and experimentally analysed. The different effects of a dissipative and nondissipative loading will be underlined by measurement results analysis. In fact, the
introduction of a complex object like a car into the chamber gives rise to both
effects, and must be taken into account.
2.2 Ideal RC EM fields statistical model
Several approaches have been adopted to reach a theoretical characterisation of an
ideal RC EM environment. Two classes of approaches can be distinguished.
According to the first kind of approach, a RC is considered as an EM cavity
characterised by quasi-stationary fields structures corresponding to the cavity
resonant modes. In [8] and [9] it has been pointed out that when a complex shaped
cavity is considered, resonant fields structures become too complicate for
analytical solutions. A probabilistic solution for fields is proposed in [9] and [10],
limited to high excitation frequencies.
The starting point of the second approach is the plane wave integral
representation for fields based on the angular plane wave spectrum [11]. According
to this approach, inside a limited portion of the interior volume of a RC the angular
spectrum can be modelled in a probabilistic way basing on simple correlation
assumptions. For this portion of the chamber’s volume, which is called the working
volume of the chamber and is far from the walls and the stirrer, the determination
of a probabilistic model for EM fields is thus possible. This model is also limited to
high excitation frequencies, such that random properties of the plane wave
7
2 – Renault RC facility
spectrum are assured.
Both kind of approaches conclude that for an ideal RC operating in the high
frequency domain a probabilistic description is suitable to characterise EM fields.
Furthermore, both approaches lead to the same field probabilistic description.
Three main results characterise the probability model: the ergodicity property, the
probabilistic distributions for EM quantities and the correlation functions. These
three properties are recalled in the following.
Ergodicity. It concerns three kinds of processes: spatial shift in the working
volume, stirrer rotation and frequency shift. This property can be explained by
considering a RC excited by an EM field in the three following different situations:
1) a single frequency EM field is excited and the stirrer is not in motion. We
identify with Sn a set of n values of an EM quantity measured in different
spatial points of the working volume;
2) a single frequency EM field is excited in the chamber, and the stirrer is in
motion. We identify with Rn a set of n values of an EM quantity measured in a
fixed spatial point for different position of the stirrer;
3) the stirrer is not in motion and a single spatial position is considered. We
identify with Fn a set of n values of an EM quantity measured for different
frequencies of excitation1.
For an ideal RC, ergodicity assures that statistical properties of the three samples
Sn, Rn and Fn are the same, when measurements within each sample are
independent. This means that when a spatial shift, a stirrer rotation or a frequency
shift are considered, the probabilistic distributions that characterise EM quantities
are the same. Such probabilistic distributions are recalled below.
Probabilistic distribution. In the working volume of a fully operating RC,
probability distributions for electric and magnetic field, power density and power
received by antennas have been derived in [11]. The starting point of the
probabilistic model is that real and imaginary parts of the three electric and
magnetic field rectangular components have a Normal (Gaussian) distribution and
are independent from each other. It is thus possible to derive distributions for
electric and magnetic fields amplitude and squared amplitude, power density and
1
At high frequencies, and with the hypothesis of a cavity quality factor constant over the
frequency band considered.
8
2 – Renault RC facility
power received by an antenna. Results for electric field are summarised in Table
2.1.
Table 2.1 Electric field probabilistic distributions inside a RC
EM
quantity
x
Distribution
{ }
ℑm{E x, y , z }
Probability density
function
f(x)
Normal
(Gaussian)
 x2 
exp −

2π σ
 2σ 2 
E x, y , z
χ2
ℜe E x , y , z ,
E x, y , z
(Rayleigh)
2
(Exponential)
χ6
ETot
ETot
χ 22
2
χ 62
1
Mean
value
Variance
E{x} = x
E{(x- x )2}
0
σ2
 x2 
exp
−

σ2
 2σ 2 
π
σ
σ 22 −
x 

exp
−


2σ 2
 2σ 2 
2σ 2
4σ 4
 x2 
exp
−
2
8σ 6
 2σ 
15
σ 2π
16
x
1
x5
x2
x 

exp −

16σ
 2σ 2 
6
2
6σ 2


6σ 2 −
π

2
225πσ 2
128
12σ 4
All distributions reported in Table 2.1 are one-parameter distributions, depending
only on the variance σ 2 of real and imaginary parts of one electric field rectangular
component. This means that, if this parameter is estimated from measurements, all
electric field quantities are completely characterised. As an example, σ 2 can be
estimated via the measurement of the mean value of any of the quantities in the
first column of Table 2.1, and by equating the result to the corresponding fourth
column.
Magnetic field quantities follow the same distribution of corresponding
electric field quantities, reported in Table 2.1. The current amplitude and the power
received by an impedance matched antenna follow the same distributions of the
amplitude and the squared amplitude, respectively, of one electric field rectangular
component (Table 2.1, second and third rows), that are Rayleigh and Exponential
distributions respectively. Furthermore, any test object that can be identified by
terminals with linear loads can be thought as an antenna with loss and impedance
9
2 – Renault RC facility
mismatch, thus received current and power follow Rayleigh and Exponential
distributions.
The relations between mean values of electric field, magnetic field and
power received by a test object are:
H Tot
Prec
2
=
1 ETot
=
2
η
ETot
2
(2.1)
η2
2
λ2
mη a
4π
(
(2.2)
)
where η = µ / ε is the wave impedance, λ = 1 / µε f is the wavelength, m is
the antenna impedance mismatch and η a is the antenna efficiency (m and η a vary
from 0 to 1).
Thus, by the estimation of σ 2 for the electric field (Table 2.1), and by
using equations (2.1) and (2.2), also the magnetic field and power received by the
test object are completely characterised, if m and η a are known. It is important to
remind that the statistical model in Table 2.1 is representative for RC measurement
samples constituted of independent measurements. How to obtain independent
measurements will be discussed in sub-section 2.4.1.
Correlation functions. For single-frequency continuous wave fields that
are mechanically stirred, the above probabilistic description characterises the fields
in a given spatial point during the stirrer rotation. To complete the probabilistic
description, knowledge of the spatial correlation for EM fields is necessary.
Starting from the plane wave integral representation, spatial correlation functions
for fields and energy density have been derived in [16], thus completing the
probabilistic description of EM fields for the working volume of a RC.
2.3 Measurement uncertainty statistical model
Measurement uncertainty is strictly related to the statistical concept of confidence
interval. Measurement uncertainty associates to the measurement result an interval
of possible values which characterises both the reliability and the repeatability of
measurement. Statistics help to associate a value to the probability of this interval
10
2 – Renault RC facility
of possible values, by introducing the concept of confidence interval, which is
characterised by the two interval bounds and the associated probability level. Such
a characterization is possible only if the probabilistic distribution of the
measurement is known.
According to the approach adopted in [17], the uncertainty that affects RC
measurements consists in three individual components, which are combined to
produce a total combined uncertainty:
1) Uncertainty due to the random nature of RC EM fields.
2) Residual, unexplained uncertainty (imperfections of the chamber).
3) Measurement instrumentation uncertainty.
Each RC measured parameter has an associated probabilistic distribution, as
reminded in section 2.2, and thus an associated uncertainty which can be quantified
by confidence intervals as discussed above. This is the first uncertainty component.
If the chamber and the instrumentation were perfect, it would be the only
measurement uncertainty observed.
The second uncertainty component is due to the imperfections of the
chamber, and the consequent bad agreement of EM measured quantities with ideal
RC probabilistic distributions. We include in this kind of uncertainty fields spatial
non-uniformity due to operation at low frequencies and/or close to chamber walls,
as well as other imperfections such as a non effective stirrer.
Uncertainty due to instrumentation is the last contribution to the total
measurement uncertainty. This contribution depends on several parameters, such as
measurement instruments and methods, cables, connectors and antennas.
Characterising this kind of uncertainty is out of the aim of this work, for more
information see [17].
The first uncertainty component is addressed in the following of this
section, and this component will be used to characterise the whole measurement
uncertainty in the rest of this work, neglecting thus the second and third uncertainty
components. An experimental quantification of the amount of the second and third
uncertainty components for a real chamber will be proposed in the next section.
We consider now the first uncertainty component, and we derive the
corresponding confidence intervals. Details about confidence intervals
determination for a given distribution are included in Annex A.4. If a measurement
is made for a single stirrer position, distributions outlined in Table 2.1 can be
directly used to determine confidence intervals. On the other hand, for practical
applications the most concerned measurements are mean values and maximum
11
2 – Renault RC facility
values during stirrer rotations, requiring thus the knowledge of mean and maximum
values distributions for determining the associated confidence intervals. The CLT
can be used to determine mean values distributions, and extreme order statistics
can be used to determine maximum values distributions. Mean and maximum
values of Rayleigh and Exponential distributions are only considered, as they cover
the most of practical measurement cases. Mean and maximum values confidence
intervals will be derived respectively in sub-sections 2.3.1 and 2.3.2.
In the following, when considering a measurement sample x = (x1, x2, …, xn)
taken for n positions over one stirrer rotation, the mean sample value will be
indicated as x
n
, and the maximum sample value as x  n ; the assumption will be
made that the n measurements are independent. Finally, all measurement
uncertainties reported in the following correspond to measurements expressed in
linear units (V/m, A, W, …), but results are reported both in linear units and in
decibel (dB). Of course, in the latter case results are computed for linear values,
and the dB conversion is made as a final step. Different distributions and thus
different uncertainties characterize dB measurements (for details about dB
distributions see [17]).
2.3.1 Mean values uncertainty
According to the CLT, the arithmetical mean value of a measurement sample
collected over a stirrer rotation follows a Normal distribution, provided that the
sample size n is sufficiently large and the n measurements are independent,
regardless of the particular measured quantity distribution [18]. If σ x2 is the
variance of the measured sample, the variance of the Normal distribution
associated to the sample mean value is σ 2 = σ x2 / n .
The 95% confidence interval for mean values can be thus obtained by the
0.025 and 0.975 quantiles of a Normal distribution whose variance is given by
σ 2 = σ x2 / n (see Annex A.4). The 95% confidence intervals for mean values of
Rayleigh and Exponential distributed samples are reported in Table 2.2.2
2
See reference [18] pag. 249, where the “known variance” technique is used to estimate
mean value of an Exponential distribution.
12
2 – Renault RC facility
Table 2.2 95% confidence intervals of mean estimated values
Original distribution
Confidence interval
Rayleigh

 x


Exponential

 x


n
n
 1.02 
, x
⋅ 1 −
n 

 1.96 
, x
⋅ 1 −
n 

n
n
 1.02  
 
⋅ 1 +
n  

 1.96  
 
⋅ 1 +
n  

Mean value confidence intervals as a function of the sample size n are shown in
Figures 2.1 and 2.2, respectively in linear units and in dB (20*log10 was used for
Rayleigh distribution and 10*log10 was used for the Exponential one). Figures refer
to measured mean values x
n
= 1 (0 dB) for each distribution.
95% confidence intervals of mean values (⟨ x⟩ =1)
n
1.8
Rayleigh
Exponential
95% confidence intervals
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
10
50
100
150
200
Sample size n
250
300
350
Figure 2.1 Mean values confidence intervals in linear units
13
2 – Renault RC facility
95% confidence intervals of mean values (⟨ x ⟩ =1)
n
3
Rayleigh
Exponential
95% confidence intervals (dB)
2
1
0
-1
-2
-3
-4
-5
10
50
100
150
200
Sample size n
250
300
350
Figure 2.2 Mean values confidence intervals in dB
Results in Figures. 2.1 and 2.2 show that confidence interval amplitudes decrease
as sample size increases both in linear units and dB, as expected. On the other
hand, one could expect to have the same confidence intervals, when plotted in dB,
for Rayleigh and Exponential distributions. Think for example to the measurement
of one electric field rectangular component. If the mean value is computed for the
amplitude of the measurement sample, and the mean value is computed for the
squared amplitude of the same sample, we expect the two mean values to be
characterised by the same confidence interval, as they come from the same
measurement data. The slight difference appearing in Figure 2.2 for low n values is
probably due to the fact that, for computing Table 2.2 results, asymptotic values (n
→ ∞) were used for mean values and variances of the underlying distribution.
However, differences are lower than 0.6 dB for n ≥ 12 and lower than 0.1 dB for
n ≥ 50 , thus it is very difficult to appreciate them in experimental results.
Furthermore, a discussion about the applicability of the above confidence
intervals must be done. Table 2 confidence intervals are based on the hypothesis of
large samples sizes, so that the CLT can be applied. The type of the adopted
statistical distribution for a measured quantity influences the minimum sample size
required for the CLT applicability [18]. Mean values of measurements coming
from distributions with smoothed density functions (e.g. Rayleigh) follow the
Normal distribution even with relatively small sample sizes n. On the contrary,
14
2 – Renault RC facility
mean values of measurements coming from distributions with sharper density
functions (e.g. Exponential) require larger sample size n to follow a Normal
distribution. Numerical tests with a random number generator showed us that in the
case of Rayleigh distributed samples, mean values follow a Normal distribution
with samples sizes n ≥ 5 . For Exponential distributed samples, n ≥ 150 was
required for mean values to follow a Normal distribution3. As a result, there is a
potential error in using Table 2.2 confidence intervals for mean values of
Exponential distributions when the sample size is small.
2.3.2 Maximum values uncertainty
Confidence intervals for maximum values over a stirrer rotation can be drawn from
maximum values distribution. Such distribution can be found from underlying
distributions of Table 2.1, with the help of extreme order statistics [17]. For the
Exponential distribution, analytical expressions for maximum values probability
density, as well as for 0.025 and 0.975 quantiles can be found according to Annex
A.3. The 95% confidence interval for maximum values of Exponential distributed
samples is given by:



 x  n



1  



ln1 − 0.025 n  


 , x
⋅ 1 −
 n
1 
(
)
 0.577 + ln n +

2n 

1  



ln1 − 0.975 n   


 
⋅ 1 −
1 
 0.577 + ln (n ) +

2n  

(2.3)
On the other hand, numerical solution is required for maximum values of the
Rayleigh distribution, as outlined in [17].
The confidence intervals for maximum values of Rayleigh and Exponential
distributions as a function of the sample size n are reported in Figures 2.3 and 2.4,
respectively in linear units and in dB. Results of Figures 2.3 and 2.4 refer to
measured maximum values x  n = 1 for each distribution.
3
Anderson-Darling normality test was used for testing the normality of the sample.
15
2 – Renault RC facility
95% confidence intervals of maximum values (xn=1)
2.2
Rayleigh
Exponential
2
95% confidence intervals
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
10
50
100
150
200
Sample size n
250
300
Figure 2.3 Maximum values confidence intervals in linear units
95% confidence intervals of maximum values (xn=1)
4
Rayleigh
Exponential
95% confidence intervals (dB)
3
2
1
0
-1
-2
-3
-4
-5
10
50
100
150
200
Sample size n
250
300
Figure 2.4 Maximum values confidence intervals in dB
Considerations similar to those made for mean values confidence intervals in
Figures 2.1 and 2.2 apply also to the results of Figures 2.3 and 2.4. The main
difference here is that for maximum values the confidence intervals are not
symmetric around measured maximum value neither in linear values nor in dB, due
16
2 – Renault RC facility
to the non symmetry in the shape of linear and dB maximum values distributions.
The reported confidence intervals are based on ideal maximum value
distribution for Rayleigh and Exponential distributions. However, it is worth to
mention that it was recently shown in [19] that such maximum value distributions
show a slight discrepancy with experimental results even for well operating
chambers, that is chambers where the statistics of Table 2.1 are verified. This is not
surprising, since taking the maximum values is a selective process, and it can point
out problems of the underlying model in the tails of the distributions. Table 2.1
model allow theoretical infinite maximum values over the stirrer rotation, which is
not possible in a real chamber. Thus, testing the goodness of fit of the derived
maximum distribution can evidence problems in the underlying model which are
not evident when testing the goodness of fit of the model itself. However, since the
discrepancy found in [19] between the experimental results and the actual model
for maximum values distributions is not too large, and in the lack of a better model,
the confidence intervals of Figures 2.3 and 2.4 will be used in this work for
estimating maximum values uncertainty.
2.4 Qualification principles and techniques for a
real chamber
The analysis of the performances of a real chamber, with regard to the ideal
statistical model of sections 2.2 and 2.3, will be introduced in this section.
Several approaches and standard chamber calibration methods have been
proposed in the past to determine good operating conditions for a RC. Some of
these criteria have been analysed and discussed in [15], where it has been pointed
out that the correlation among different approaches is somehow difficult. This is
due to the fact that many of the early proposed criteria had empirical basis. We take
as an example the so called “stirring ratio” introduced in [20] and [4]. This
parameter evaluates the ratio between the maximum and the minimum received
power by a fixed antenna during one stirrer rotation. Intuitively, it is clear that an
effective stirrer is responsible for a variation of the received power during its
rotation. Thus a stirring ratio greater than 1 is expected in a well operating RC. In
the early days, empirical thresholds were selected for characterising a good stirring
ratio. For instance the standard [4] points out an empirical minimum value of 20
dB. More recently, on the basis of the ideal RC statistical model, it has been shown
that the stirring ratio is a known statistical parameter, which is a function of the
17
2 – Renault RC facility
number of independent measurements taken during the stirrer rotation [17]. Thus,
statistical goodness-of-fit tests can be adopted to evaluate the agreement of a
measured stirring ratio with the stirring ratio of an ideal chamber, giving a
quantifiable criteria. In this section, only statistical tests will be considered, as they
have the advantage of quantifying the agreement of a given chamber EM
environment with ideal models, with a known confidence level.
In sub-section 2.4.1 the statistical tests which allow the qualification of a
RC are proposed, and such tests are applied to Renault RC measurement results.
Sub-section 2.4.2 considers the qualification procedure proposed by the standard
IEC [5], and discusses the correlation of this procedure with statistical tests of subsection 2.4.1. In sub-section 2.4.3, the measurement uncertainty of a real chamber
is investigated with regard to the ideal model proposed in section 2.3. Finally, the
effect of loading the chamber with a big object is investigated in sub-section 2.4.4.
The ensemble of the collected information allows a complete characterization of a
real RC.
Before to proceed, it is helpful to introduce the physical properties of the
considered chamber. The Renault facility is a parallelepiped cavity of dimensions
L1 = 9.40m (length), L2 = 5.00m (height), L3 = 4.70m (width), for a total volume V
of about 221m3, whose walls are made of zinc plated steel (conductivity
σ = 4.4e5 S / m , from [20]). The chamber is provided of a mechanical stirrer
placed horizontally along L3 at the top of the chamber and made of copper panels.
The stirrer has dimensions D1 = 3.7m (length), D2 = 1.5m (height), giving a total
rotational volume of about 6m3 (about 2.7% of chamber volume), and appears as a
hollow complex shaped solid with big apertures that help in breaking the rotational
symmetry. Figure 2.5 shows the interior of the chamber with the mechanical stirrer.
18
2 – Renault RC facility
Figure 2.5 Renault RC facility
2.4.1 Statistical tests
The aim of statistical tests applied to RC measurements is to define the good
operating domain of the chamber, characterized by a frequency domain, a volume
domain and the information about the number of independent stirrer positions. The
tests that we propose in this sub-section and the consequent results analysis are
mainly based on in [21].
We carried out measurements inside the Renault RC with the help of a
vector network analyser, an emitting log-periodic antenna connected to the port one
of the analyser and a receiving log-periodic antenna or alternatively a derivative
electric field probe connected to the port two of the analyser (see Annex D). The
measurement procedure consisted in collecting a measurement sample of the
parameter S 21 of the network analyser for a number n of linearly spaced angular
stirrer positions at each frequency. By using the relations in Annex D, the
measured electric field or received power can be derived for each stirrer position
from parameter S 21 . Measurements were carried out for 1601 frequency points
(log-spaced) between 80 MHz and 2 GHz and for 480 stirrer positions at each
frequency.
The agreement of the chamber EM environment statistics with ideal
probabilistic model introduced in section 2.2 will be first investigated. This is done
by testing the agreement of the measured normalized received power (that is S 21
19
2
2 – Renault RC facility
from the network analyser) over one stirrer rotation with the ideal χ 22 (see Table
2.1) by the help of the χ 2 goodness of fit test, described in Annex A.5. For a given
frequency and a given number n of stirrer positions measurements, the test result is
given as a probability, p, the rejection significance level (r.s.l.), yielding the risk
that the assumed distribution, even if correct, be rejected. Normally the hypothesis
of the assumed distribution, χ 22 in our case, is rejected if p is less than 5% or 1%.
A way to treat all the data for a given frequency range is to calculate a logarithmic
sum of all p-values, according to [21]:
N
K = −2 ⋅ ∑ ln pi
(2.4)
i =1
where i = 1 to N denotes the frequency points and pi the above calculated
probability of rejection at each frequency. Assuming that the normalized received
power is χ 22 distributed at each frequency, it can be shown that K is χ 22N
distributed (see [21], appendix C). Thus, for each value of K a corresponding
probability pk can be calculated that gives the significance level for rejecting the
result for the complete frequency range. The hypothesis that measurements follow
a χ 22 distribution is normally rejected if pk is less than 5% or 1%. Since p and pk
are sensitive both to the frequency range and to the number of stirrer positions n
used in measurements, we calculated pk for starting frequencies fstart from 80 MHz
to 2 GHz, and for a number of stirrer position from 1 to 480. The test result, in
terms of the pk values, is shown in Figure 2.6.
20
2 – Renault RC facility
Cumulative χ 2 test for number of stirrer positions vs. frequency
2
1.8
Start frequency (GHz)
1.6
Accepted
1.4
1.2
1
0.8
0.6
Rejected
0.4
0.2
0.08
accept level > 5%
accept level > 1%
0
50
100
150 200 250 300 350
No. of stirrer positions
400
450480
Figure 2.6 Cumulative rejection significance levels for complete frequency
intervals
Figure 2.6 has been processed with the noise reduction technique proposed in [21].
For completeness, the same results before noise removal are proposed in Annex E.
Two kind of information can be drawn from results in Figure 2.6. Firstly, the
maximum number of stirrer positions, for which the hypothesis of χ 22 distribution
is accepted, can be determined for each frequency interval. For example for fstart =
200 MHz, that is for the frequency interval from 200 MHz to 2 GHz, the
assumption of χ 22 distribution is accepted when number of stirrer positions is
lower than around 50. Equivalently, the lowest usable frequency for a desired
number of stirrer positions can be drawn form Figure 2.6. For instance, if one
wants to use 50 stirrer positions, staying in the region of acceptance of the χ 22
distribution, he has to work at frequencies f > 200 MHz.
According to the approach that we have adopted, Figure 2.6 defines the
validity domain of the ideal RC statistical model of section 2.2, and consequently
of the uncertainty model of section 2.3, for the Renault chamber. On the other
hand, as the uncertainty model in Figures 2.1 to 2.4 depends on the number of
independent measurements considered, it is important to know such number over
one stirrer rotation. Thus, the number of independent stirrer positions is
investigated next.
21
2 – Renault RC facility
Inside a real chamber, a too small stirrer rotation step is not able to excite a
new different (independent) field pattern, thus a limited number of independent
stirrer positions are available. In addition, for low frequencies the stirrer is small
when compared to the wavelength and not able to excite different field patterns
over its rotation [22]. In other words, at low frequencies a small number of
independent stirrer positions are available, and this number grows with frequency.
A technique to quantify the number of available independent stirrer positions is to
evaluate the linear correlation coefficient of the received power P by an antenna
over one complete stirrer rotation [21]:
n
ρ (r ) =
∑ (Pi −
P ) ⋅ (Pi + r − P

∑ (Pi − P
i =1
)2  ∑ (Pi+r −
i =1
n
n
 i =1
)
P
(2.5)
)2 

In (2.5), ρ (r ) is the linear correlation coefficient for an angular shift r of the
stirrer, Pi is the power measured for the ith position of the stirrer (from 1 to n) and
P is the mean received power over the complete stirrer rotation. In (2.5), if the
index i+r is greater than the total number of stirrer positions n, the index becomes
i+r-n. The linear correlation coefficient ρ (r ) gives information about the level of
correlation of the received power at the stirrer position r with the received power at
the position 0. For r = 0, that is with no stirrer rotation, we have ρ (r ) = 1 , which
means complete correlation. If the stirrer is able to actually stir the cavity modes,
increasing r will decrease ρ (r ) ; an ideal ρ (r ) = 0 points out complete
uncorrelation. Several upper limits for ρ(r ) can be adopted to define uncorrelation
for practical purposes. The empirical choice of ρ (r ) ≤ 1 / e = 0.37 was chosen at
first for RC applications in [5] and [20]. In [21] a solution is proposed which takes
into account also the size n of the sample used for evaluating ρ (r ) . The
significance of the correlation coefficient ρ (r ) is evaluated by calculating the
probability that n measurements of two uncorrelated variables would give a result
ρ , as large as or greater than ρ (r ) . This probability is given by [21]:
Prob n ( ρ ≥ ρ (r ) ) =
2 ⋅ Γ[(n − 1) / 2]
π
∫ (1 − ρ
⋅ Γ[(n − 2 ) / 2] ρ (r )
22
⋅
1
)
2 (n − 4 ) / 2
dρ
(2.6)
2 – Renault RC facility
where Γ(
)
is the gamma function. An experimental ρ (r ) , for which the
probability in (2.6) is small, points out a correlation. In particular, in [21], if
Prob n ( ρ ≥ ρ (r ) ) ≤ 5% the correlation is called significant, if it is less than 1% is
called highly significant. By calculating the linear correlation coefficient of (2.5),
and by using (2.6), the performances of the stirrer can be analysed, i.e. it is possible
to compute the number of uncorrelated stirrer positions at each frequency. The
results are shown in Figure 2.7, for measurement of the normalized received power
performed from 80 MHz to 2 GHz in 401 frequency points using 480 stirrer steps.
Each zone of the diagram shows whether the considered number of stirrer positions
at each frequency are correlated or not, basing on the values of 1% and 5%
significance level computed by (2.6) and on ρ (r ) ≤ 1 / e = 0.37 .
Number of uncorrelated stirrer positions vs. frequency
2
1.8
Frequency (GHz)
1.6
Non correlated
1.4
1.2
1
0.8
Correlated
0.6
significant correl. ρ =0.0895
highly sign. correl. ρ =0.1175
correlation ρ =1/e
0.4
0.2
0.08
0
50
100
150
No. of stirrer positions
200
250
Figure 2.7 Number of uncorrelated stirrer positions versus frequency (480 stirrer
steps used for qualification measurements)
As Figure 2.6, also Figure 2.7 has been processed with the noise reduction
technique proposed in [21]; the results before the noise reduction process are
proposed in Annex E. Results in Figure 2.7 show that, for instance, at 200 MHz, 42
uncorrelated stirrer positions are available according to the limit of 5% significance
level given by (2.6), 46 positions according to the limit of 1% significance level,
and 57 positions according to the limit 1 / e . Figure 2.6 evidences that the use of
1 / e as an empirical limit may lead to consider as uncorrelated those stirrer
23
2 – Renault RC facility
positions that are potentially correlated according to the 1% and 5% significance
defined by (2.6).
As in general uncorrelation doesn’t strictly imply independence4, in [21] a
discussion is proposed for relating uncorrelated stirrer positions with independent
field patterns. In this work, for simplicity, we refer to independent samples when
dealing with uncorrelated stirrer positions.
Figure 2.7 results can thus be used in conjunction with results in Figure 2.6
and confidence intervals derived in section 2.3 for defining the minimum
measurement uncertainty which can be achieved at each frequency. For instance, at
200 MHz, assuming a maximum of about 50 independent stirrer positions from
Figure 2.7, the lowest uncertainty level is about 2.5 dB for mean values and about
4.6 dB for maximum values (see Figures 2.2 and 2.4). If we want to reduce the
uncertainty and we need e.g. 100 stirrer steps, the lowest usable frequency, from
Figure 2.7, is between about 675 MHz and 1 GHz, depending on which limit is
adopted for defining uncorrelation.
Figures 2.6 and 2.7 give thus a complete statistical characterization of the
performances of real chamber. The question is discussed in [21], if it is possible to
have the complete statistical information just by the results of only one of the two
Figures. In our case, for instance, we see that starting from fstart ≈ 110 MHz, the
assumption of a χ 22 distribution is accepted if the underlying data come from
uncorrelated stirrer positions. Also, the area of rejection of Figure 2.6 falls within
the correlation region of Figure 2.7. On the other hand, correlated data can also
give “accepted” result for χ 22 distribution (see e.g. fstart = 1.8 GHz and number of
stirrer positions N > 200). Tests with different stirrers in [21] however lead to the
conclusion about the difficulty of generalizing such conclusions, and thus with the
necessity of using both kind of characterizations.
2.4.2 Measurement uncertainty in a real chamber
Based on information about the number of uncorrelated stirrer positions the
uncertainty due to ideal RC statistical fields can be computed. An experimental
evaluation of the amount of the other two uncertainty components, due to chamber
imperfections and to measurement instrumentation, is now given. To take into
account spatial non uniformity of fields due to chamber imperfections, we
4
Independence implies uncorrelation, but the vice versa is true only for Normal random
variables [18].
24
2 – Renault RC facility
performed measurements with the “derivative” electric field probe for three field
polarizations in 9 positions inside the chamber, having thus 27 independent
measurements of one electric field rectangular component. The 9 positions include
the 8 corners of a parallelepiped volume inside the chamber, 1 m far from walls,
and 1 central position inside this volume. It must be noticed that the measurements
are carried out with an intrusive method, since the electric field probe is directly
connected to a coaxial cable (see Annex D). As a result, by changing the position
of the field probe, we change the measurement conditions. However, since we are
interested in ensemble statistical properties, we neglect this problem.
Measurements were performed for 101 frequency points (log spaced) from 80 MHz
to 1 GHz, for 50 stirrer positions. From Figures 2.6 and 2.7 we see that, starting
from about 200 MHz and for 50 stirrer positions the ideal RC statistical model is
accepted, thus we are in the conditions of a good operating chamber.
For the 27 electric field measurements (each measurement is the amplitude
of one electric field rectangular component), we computed mean and maximum
values over 50 stirrer positions. We then evaluated the standard deviations of the
27 mean and maximum values providing an indication of the spatial uniformity of
mean and maximum electric field values. Such standard deviations, if normalized,
can be directly compared to standard deviations of ideal mean and maximum
values distributions, whose confidence intervals were introduced in sub-section
2.3.1 and 2.3.2, respectively. For normalized electric field rectangular components
amplitudes, the standard deviation of the mean values distribution, σ mean , and of
the maximum values distribution, σ Max , can be computed (see Table 2.2 and subsection 2.3.1 for mean values distribution and Annex A.3 for maximum value
distribution):
σ mean
N =50
=
σ Max
π
2 
⋅2 − 
π 
2
= 0.0739
N
N =50
= 0.1373
(2.7)
(2.8)
Result in (2.8) was obtained by numerical solution.
Theoretical values in (2.7) and (2.8) are compared to measured values in
Figure 2.8. Results are reported in dB, according to:
25
2 – Renault RC facility
 x +σ 


x


σ dB = 20 ⋅ log10 
(2.9)
27 measurements of |Er|, n=50 stirrer positions
7
σmeanmeas.
σmeanth.
σMaxmeas.
σMaxth.
6
σ (dB)
5
4
3
2
1
0
0.08
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.8 0.9 1
Frequency (GHz)
Figure 2.8 Standard deviation of mean and Max values of one electric field
rectangular component (27 measurements for 50 stirrer positions)
Results in Figure 2.8 show that, both for mean and maximum values, the
experimental standard deviation approaches the ideal one at high frequencies. If we
compare results of Figure 2.8 to results of Figures 2.6 and 2.7, we see that, for
frequencies from 400 MHz and 700 MHz, even if we have uncorrelated data which
follow the ideal statistical properties, fields spatial uniformity is considerably
worse than expected. We expect this to be mostly due to the fact that, for such
frequencies, statistical tests in Figures 2.6 and 2.7 are close to the threshold values.
Nevertheless, even at high frequencies (700 MHz – 1 GHz) experimental standard
deviation is slightly greater than would be expected under ideal conditions. This
difference points out the uncertainty components due to chamber imperfections and
to instrumentation. This uncertainty components constitute a sort of “noise floor”
for the experimental standard deviation, as it is evident especially for mean values
standard deviation. A rough estimate of the total amount of residual and
instrumentation uncertainty U can be derived by comparing experimental standard
deviation with the ideal one. To do this, we calculated an average value of the
measured standard deviation at high frequencies, between 700 MHz and 1 GHz and
26
2 – Renault RC facility
used it with theoretical value in (2.7) to calculate:
2
2
U = σ meas
− σ theor
≈ 0.0998 ≈ 0.83 dB
(2.10)
The value in (2.10) gives an estimate of the total amount of residual and
instrumentation uncertainty that we have even in the good operating region of the
chamber. For detailed discussion about this subject, see [17]. In the next subsection, we will see that the evaluation of experimental standard deviation is also at
the basis of the fields uniformity evaluation according to standard [5].
2.4.3 Calibration following standards
It is useful to briefly introduce the RC calibration method contained in the
proposed standard IEC 61000-4-21 [5], and to discuss the calibration results in the
light of results obtained in sub-sections 2.4.1 and 2.4.2. Standard [5] is already
used in industry for carrying out tests in RC both for immunity and emission, and a
very similar method is already adopted in the avionic standard [3].
Annex B of [5] contains the field uniformity validation procedure for mode
tuning operation. The aim of the procedure is to obtain both the chamber lowest
operation frequency, and the ratio between the field amplitude and the injected
power to be used for immunity tests. Field uniformity validation procedure is based
on the estimation of the variance of the maximum of the electric field rectangular
components amplitude for the 8 corner positions of the measurement volume (see
[5] for details). This procedure is very similar to what was done in the previous
sub-section (see blue traces in Figure 2.8). According to [5], the chamber passes
the field uniformity evaluation if the experimental variance fall within an imposed
tolerance.
By using the same measurement data used to obtain Figure 2.8, the results
according to [5] together with the imposed tolerance versus frequency are shown in
Figure 2.9.
27
2 – Renault RC facility
IEC 61000-4-21 - σ
7
Max
6
f min = 117 MHz
σ (dB)
5
4
3
2
1
0
0.08
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.80.9 1
Frequency (GHz)
Figure 2.9 IEC 61000-4-21 field uniformity validation – empty chamber
According to [5], the chamber passes the validation provided the tolerance in
Figure 2.9 (4 dB decreasing linearly to 3 dB from 100 MHz to 400 MHz) is
respected for all frequencies; a maximum of three frequencies per octave may
exceed the tolerance by an amount not to exceed 1 dB. From Figure 2.9, the
obtained lower frequency is thus about 117 MHz.
A risk associated with the procedure in [5] is to obtain large variations in
the low frequency result, by using different frequency points or by using different
number of stirrer steps for calibration measurements. This is due to the fact that [5]
specifies a minimum number of frequency points and stirrer steps for each
frequency band (but not maximum numbers), while the tolerance in Figure 2.9 is
not a function of the such quantities. On the contrary we have shown in sub-section
2.4.1 that the variance of maximum values is function of the number of stirrer
positions. As an example, the results of Figure 2.8 for maximum values over 50
stirrer positions are reported in Figure 2.10, together with the result obtained by
undersampling the same data to obtain 12 stirrer positions.
28
2 – Renault RC facility
27 measurements of |Er|n, n=50 and n=12 stirrer positions
7
σ meas. n=50
σ th. n=50
σ meas. n=12
σ th. n=12
6
σ (dB)
5
4
3
2
1
0
0.08
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.8 0.9 1
Frequency (GHz)
Figure 2.10 Standard deviation Max values of one electric field rectangular
component (27 measurements for 50 and 12 stirrer positions)
Figure 2.10 shows the influence of the number of stirrer positions on the variance
used for defining the field uniformity in [5]. The choice of the frequency points
also influence the minimum frequency result obtained by [5]. Results of Figure 2.9
were obtained by considering a number of frequency points which is almost the
double of the minimum recommended in [5]; by undersampling in frequency the
results in Figure 2.9, it is possible that the new minimum frequency were reduced
of about 10 MHz.
Such considerations show the importance of associating the result obtained
by [5] both to the considered chamber and to the considered frequency points and
number of stirrer positions used.
2.4.4 Chamber loading
When a non negligible (in size or in energy absorption coefficient) object is placed
into a RC, several effects can take place. As a first effect, the introduction of an
absorbing object will influence the chamber quality factor Q, and may lead to an
improvement of chamber performances at low frequencies. This is explained in the
following, by cavity resonant modes theory.
Fields inside a parallelepiped RC can be modelled as resonant fields of an
ideal lossless parallelepiped EM cavity. This would be exact for a chamber without
29
2 – Renault RC facility
a stirrer and for perfectly conducting walls, but can be used for a real RC as a first
order approximation. The number N of resonant modes with resonant frequency
lower that f for a parallelepiped EM cavity can be expressed as [22]:
N( f ) =
8π
f3
f 1
⋅ V ⋅ 3 − (a + b + d ) +
3
c 2
c
(2.11)
where V is the cavity volume, a, b and d are the cavity dimensions and c is the
speed of light. Deriving (2.11) with respect to f, the mode density D(f) at each
frequency can be obtained. Finally, by multiplying D(f) times the excitation
bandwidth of a cavity with losses, given at each frequency f by BW = f / Q , where
Q is the cavity quality factor, it is possible to obtain the number of the modes
actually excited at the frequency f. This number is given by:
MDBWQ =
8π ⋅ V ⋅ f 3
c 3Q
(2.12)
In [22] it is shown that the effect of the rotation of an electrically large stirrer
inside a RC is to shift the cavity resonance frequencies. As a result, at high
frequencies, there is a considerable number of modes which enter and go out of the
excitation bandwidth of the chamber (mode stirring). Thus, if MDBWQ in (2.12) is
large, it is likely for a large stirrer to be effective in stirring the cavity modes.
Therefore, a threshold value could be established for MDBWQ to define good
operating conditions for a given chamber with a given stirrer [15]. Without
attempting to do that, we simply notice that MDBWQ in (2.12) is inversely
proportional to the cavity factor Q, thus, for constant conditions, a chamber with a
lower cavity factor will have a better mode stirring. The technique of deliberately
electrically charging the chamber is indicated in [5] for improving chamber
performances. However, it is important to notice that experimental results reported
in [23] show that also a degradation of ideal chamber statistics may take place
when introducing a considerable number of absorbing objects, which is a function
of the positioning of the objects itself.
A second and different loading effect is investigated in [24], where it is
shown by numerical simulation that introducing a perfectly conducting regularly–
shaped object inside the RC volume leads to a deterioration of the statistical
properties of the fields. In particular, it is indicated the upper limit of 8% ratio
30
2 – Renault RC facility
between the object volume and the chamber volume, to maintain an acceptable
deterioration from ideal statistical properties. This effect may partly be explained
by the help of (2.12), where a decrease of the actual chamber volume, by the
introduction of a closed metallic object, results in a degradation of stirring
performances.
When a complex object, which has both conducting and absorbing parts, is
inserted inside a RC, the two above highlighted effects take place at the same time.
This is the case when inserting a car inside a RC. We conducted the experiment of
inserting a Renault VelSatis car (about 8% of the chamber volume) inside the
Renault RC, to investigate the effects on the EM environment of the chamber. We
used the [5] field uniformity procedure presented in sub-section 2.4.3 to evaluate
the effect on field uniformity. The number of frequency points and stirrer positions
adopted are outlined in Table 2.3 (according to [5]). Field uniformity validation
results are shown in Figure 2.11.
Table 2.3 Frequency points and stirrer position number
Frequency range
No. of stirrer positions
80 MHz – 240 MHz
240 MHz – 480 MHz
480 MHz – 800 MHz
50
18
12
IEC 61000-4-21 - σ
7
No. of frequencies
(log-spaced)
20
15
10
Max
6
f min = 95 MHz
σ (dB)
5
4
3
2
1
0
0.08
0.1
0.2
0.3
0.4
0.5
0.6 0.7 0.8
Frequency (GHz)
Figure 2.11 IEC 61000-4-21 field uniformity validation – chamber loaded with a
VelSatis car
31
2 – Renault RC facility
To explain the decrease of the lowest usable frequency in the loaded
chamber, we investigated the influence of the loading on the chamber quality factor
Q. The variation of the quality factor of the loaded chamber with respect to the
empty chamber can be evaluated by the ratio of the mean normalized received
power over one stirrer rotation, measured when the chamber is loaded and when
the chamber is empty, respectively (see [11] for quality factor measurement). The
ratio between the normalized powers is shown in Figure 2.12, where measurements
and smoothed measurements are shown versus frequency.
Normalized mean received power ratio: VelSatis / Empty chamber
6
4
2
P
VelSatis
/P
Empty
(dB)
0
-2
-4
-6
-8
-10
-12
-14
0.08
0.1
0.2
0.3
0.4
0.5
0.6 0.7 0.8
Frequency (GHz)
Figure 2.12 Normalized mean received power ratio between the case of a chamber
loaded by a VelSatis car and an empty chamber. The heavy line represents a sliding
window smoothing of the original measurement (thin line)
Figure 2.12 points out a decrease of the quality factor Q between 4 and 6 dB
between 100 MHz and 240 MHz. By assuming for simplicity a mean reduction of 5
dB, this means an increase of MDBWQ in (2.12) of about a factor of 3. The
corresponding expected decrease of the lower usable frequency can be computed
by (2.12) as:
f minVelSatis
1
≅ 
 3
1
3
⋅ f min Empty Chamber
(2.13)
If we take the lowest usable frequency for the empty chamber obtained in Figure
32
2 – Renault RC facility
2.9, that is f min Empty Chamber = 117 MHz , (2.13) gives f minVelSatis = 81 MHz . This
result, if compared with the one of 95 MHz found in Figure 2.11, points out that
the adopted approach can be used at least to give a first-order quantification of the
loading effect due to a car.
In summary, the effect of loading the chamber with a car appears to be
dominated by the electrical loading effect, which is reflected into an improvement
of chamber performances, while no apparent degradation of chamber performances
due to the volume of the car seems to take place.
2.5 Conclusions
The RC EM environment has promising qualities for performing EMC radiated
immunity tests. In particular, fields statistical homogeneity and isotropy are
encouraging for the reliability of the test methodology and for the test
exhaustiveness. In this Chapter, in order to have an experimental validation of such
hypothesis, we have investigated the performances of a real chamber installed in an
industrial context.
At first, the RC principles and operation were introduced, and the statistical
theory of an ideal RC at high frequencies was recalled. Among the different
possibilities of RC operating conditions, we focused on the mechanical mode
stirring, with a mode-tuned operation. Electronic mode stirring is in fact not
applicable to radiated immunity testing, while mechanical mode-stirring, which
implies a continuous stirrer rotation, can not be used for testing automotive devices
which have often long response times. On the basis of ideal fields statistics, we
have derived the formulation of confidence intervals for mean and maximum
values over independent stirrer positions. The dependence of measurement
uncertainty from the number of used stirrer positions has thus been quantified.
The performances of the Renault chamber were then analysed. We firstly
presented a statistical methodology, mainly based on the work presented in [21], to
evaluate the agreement between real chamber statistics and ideal statistics, and to
evaluate the number of available independent stirrer positions. The agreement with
ideal statistics as a function of frequency can be used for establishing the lowest
usable frequency of a real chamber. Results show that the lowest usable frequency
is a function of the number of stirrer positions used for the evaluation: in a real
chamber, operating at a given frequency, one can use a limited number of stirrer
positions, if he wants the measurements to agree with ideal statistical models. Such
33
2 – Renault RC facility
maximum number of stirrer positions increases with frequency. As a result, the
number of stirrer positions used in qualification measurements should be specified
when defining the lowest usable frequency of a given chamber. The number of
independent stirrer positions available at each frequency, regardless of the
underlying distribution, was also investigated by a statistical test. The results put in
evidence that this information is complementary to the previous one. We showed,
for instance, that correlated data can pass a goodness of fit test when evaluating the
agreement with ideal statistical models. Since the statistical uncertainty model can
be applied only if measured fields agree with ideal RC fields and if independent
stirrer positions are used, the information given by both tests is necessary, for
characterizing measurement uncertainty in a real chamber.
For an ideal RC, fields isotropy and spatial uniformity fall within the
bounds given by the ideal measurement uncertainty. We investigated the
limitations of this hypothesis when dealing with measurements in a real chamber.
The comparison of the measured fields spatial uniformity with the ideal standard
deviation, allowed us to have an estimation of the residual measurement
uncertainty. Residual uncertainty, which exists even at high frequencies where the
measured fields agree with ideal statistical model, is due to chamber imperfections
and instrumentation uncertainty. The residual uncertainty must be considered as a
lower limit of the RC uncertainty, and must be taken into account when trying to
reduce the measurement uncertainty by increasing the number of stirrer positions.
Such uncertainty is of the order of 1 dB for the Renault chamber.
The procedure used for estimating the residual uncertainty is at the basis of
RC calibration according to standard [5], as we have shown in section 2.4.3. We
also showed the influence of the number of stirrer positions used for the calibration
on the resulting lowest usable frequency as defined in [5].
We finally addressed the problem of chamber loading by a big object. The
understanding of loading effects is necessary if one wants to perform tests on cars.
Introducing a car inside the Renault RC (we used a Renault VelSatis, which is
about the 8 % of the chamber’s volume) seems to not degrade the fields uniformity.
On the contrary, the effect of decreasing the cavity quality factor Q, allows the
decrease the chamber lowest usable frequency. We gave a quantitative prediction
of the lowest frequency decrease based on the estimation of the chamber resonant
mode density.
By the results presented in this Chapter, we can conclude that for
developing a correct test procedure in RC, a deep understanding of different
34
2 – Renault RC facility
elements mandatory. First, the knowledge of ideal physical properties and ideal
chamber statistics is necessary for understanding the stochastic nature of RC
testing. Secondly, a complete qualification of chamber performances is required for
correct test operations and a correct definition of measurement uncertainty. Once
adopted a correct methodology, RC testing can make use of the expected
advantages in measurement repeatability and reproducibility, as it will be shown in
Chapter 5.
35
Chapter 3
RC EM fields coupling with electrical
objects: a statistical plane wave
approach
3.1 Introduction
The problem of EM fields coupling to electrical objects in RC is presented in this
Chapter, and an original solution to this problem is proposed with a statistical
formulation based on random plane waves coupling. The proposed approach is
based on Hill’s plane wave integral representation for RC fields in [11], which is
reformulated here in terms of the superposition of a finite number of contributing
plane waves. Such a fields representation is then used for modelling the RC EM
coupling as the superposition of a finite number of coupling effects due to random
plane waves. This formulation allows the prediction of RC coupling by a plane
wave coupling mechanism. The proposed approach will be applied in Chapter 4 for
predicting the RC EM coupling to distributed transmission lines.
The Chapter is organised as follows. Section 3.2 briefly recalls the plane
wave integral representation for RC fields and develops the reformulation of the
integral model into a discrete model which considers a finite number of random
plane waves contributions. Such a discrete model is then adopted in section 3.3 to
propose the statistical plane wave coupling for RC. The statistical estimation of
mean and maximum values of coupled quantities in RC will be then addressed in
sub-sections 3.3.1 and 3.3.2, and it will be shown how it can be obtained by Monte
Carlo trials of random plane wave couplings. The statistical accuracy of the Monte
Carlo methods will be finally formulated and analysed in sub-section 3.3.3.
36
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
3.2 From a plane wave integral model to a plane
wave discrete model for RC EM fields
The starting point for the plane wave integral representation for EM fields inside a
RC proposed by Hill in [11] is the expression of a single frequency continuous
r
wave electric field in one point of the space r as an integral over the solid angle of
the plane wave spectrum:
r r
E (r ) =
r r
r
∫∫ F (Ω) exp( jk ⋅ r )dΩ
(3.1)
4π
r
r
In equation (3.1) Ω is the solid angle, k is the wave propagation vector and F (Ω)
is the angular plane wave spectrum. Expression (3.1) is complete for a spherical
volume, and can be adopted for a spherical volume inside a RC. For a statistical
r
field as generated in a RC during one stirrer rotation, the angular spectrum F (Ω) is
taken to be a random variable depending on stirrer position. Thus, statistical
properties must be selected for the angular spectrum to make the expression (3.1)
representative of a well stirred field obtained in an electrically large, multimode
chamber. Such properties are chosen in [11] as correlation functions of in-phase
r
and in-quadrature real and imaginary parts of the plane wave spectrum F (Ω) .
Starting from (3.1) and from plane wave spectrum statistical assumptions,
statistical properties for resulting EM fields and power received by electrical
objects are derived in [11]. The resulting fields statistical characterisation was
recalled in Chapter 2 of this work.
The electric field representation in (3.1) is reformulated in this work as a
finite sum of plane waves contributions:
n r
r r
r r
E (r ) = ∑ Ei exp ( jki ⋅ r )
i =1
The i-th plane wave geometry is shown in Figure 3.1.
37
(3.2)
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
r
Ei
z
− θˆ i
ϕ̂i
êi
p
θi
θi
ϕ̂i
k̂i
θ̂i
O
y
ϕi
x
Figure 3.1 i-th contributing plane wave geometry
r
In (3.2), n is the number of contributing plane waves, k i is the propagation vector
of the i-th plane wave given by:
r
k i = − k ( xˆ sin θ i cos ϕ i + yˆ sin θ i sin ϕ i + zˆ cosθ i )
(3.3)
r
where k = ω µ 0ε 0 is the wave number in free space. Ei is the electric field
r
vector which lays on the phase plane defined by the unit vectors θˆi and ϕ̂ i . Ei , θˆi
and ϕ̂ i are defined by:
θˆi = xˆ cosθ i cos ϕ i + yˆ cosθ i sin ϕ i − zˆ sin θ i
(3.4)
ϕˆ i = − xˆ sin ϕ i + yˆ cos ϕ i
(3.5)
r
Ei = Ei ⋅ exp( jφ i ) ⋅ eˆi = Ei ⋅ exp( jφ i ) ⋅ [− cos(θ ip )θˆi + sin(θ ip )ϕˆ i ]
(3.6)
In (3.6), Ei is the i-th plane wave amplitude, and êi and φi are the unit vector
giving the field polarization and the plane wave phase in the origin O, respectively.
The following assumptions are now made about random plane waves
parameters that allow (3.2) to represent fields inside a RC volume:
•
in an isotropic environment, as in the case of the working volume of a RC,
plane waves are supposed to have no preferred propagation direction and no
38
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
preferred field polarisation. This means that uniform distributions are chosen
for propagation direction and polarization angle, over the solid angle and over
2π , respectively.
•
Additionally, multiple scattering phenomena inside a RC result in the fact that
the phases of plane waves have no preferred value; thus, a uniform distribution
is chosen for the phase angle.
•
Finally, constant amplitude is chosen for plane waves, equal to E 0 .
The first two assumptions are physically justified and agree with statistical
assumptions about angular spectrum in [11]. It would be more delicate to give a
physical characterisation of statistical properties of plane waves amplitudes, as
plane waves come from multiple scattering phenomena. The choice of a constant
amplitude (that is a Dirac delta probability density function – see [18]) is not
physical, but the utility of this choice will appear clear later on.
Probabilistic distributions for plane waves parameters are summarised in
Table 3.1.
Table 3.1 Random plane waves parameters probabilistic distributions
Plane wave parameter
Probabilistic Distribution
Propagation direction: Ω(θ i , ϕ i )
Uniform: U [0,4π ]
Polarisation: θ ip
Uniform: U [0,2π ]
Phase : φi
Uniform: U [0,2π ]
Amplitude: Ei
Dirac delta: δ ( Ei − E 0 )
Starting from (3.2) and Table 3.1 distributions, statistical properties for fields will
be now analytically derived. We will start to derive statistical properties for one
electric field rectangular component in one point of the space, and we will extend
the results to the three components in any point of the space in virtue of fields
isotropy and uniformity inside an ideal RC. This approach allows a simple
analytical calculation without loss of generality.
The z-component of the electric field in the origin will be considered.
According to (3.2), this field component can be expressed as:
39
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
n r
n
r
E z (O) = E (O) ⋅ zˆ = ∑ Ei ⋅ zˆ = ∑ E z ,i
i =1
(3.7)
i =1
Where E z (O) is the resulting field z-component in the origin, and E z ,i is the field
z-component given by the i-th plane wave in the origin.
Statistical properties will be derived for E z ,i first. According to (3.6), E z ,i
can be written as:
r
E z ,i = Ei ⋅ zˆ = Ei cos(θ ip ) sin (θ i ) cos(φ i ) + j Ei cos(θ ip ) sin (θ i )sin(φ i ) (3.8)
14444244443
14444244443
ℜe{E z ,i }
ℑm{E z ,i }
When random plane wave parameters are distributed according to Table 3.1, mean
value and variance for E z , i in (3.8) can be analytically computed. Details of
calculation are reported in Annex B, and the results are:
mean (ℜe{E z ,i }) = mean (ℑm{E z ,i }) = 0
var(ℜe{E z ,i }) = var(ℑm{E z ,i }) =
E02
6
(3.9)
(3.10)
where E 0 = Ei , ∀i , as outlined in Table 3.1. Results in (3.9) and (3.10) can be
inserted into (3.7), and with the help of the CLT, the statistical properties of the
field component in (3.7), for large values of n, can be finally obtained. According to
the CLT, for large values of n, ℜe{E z (O)} and ℑm{E z (O)} are distributed
according to a Normal distribution, with mean values and variances given by the
sum over n of (3.9) and (3.10) respectively [18]:

E 0 2 
2

ℜe{E z (O)}, ℑm{E z (O)} ≈ N µ = 0, σ = n ⋅

6 

(3.11)
Given the isotropy of random plane waves assumed in Table 3.1, the result in (3.11)
can be extended to any field rectangular component in O . Furthermore, as the
choice of the coordinate system origin O is arbitrary, results can be extended to
any point in the space. Finally, by supposing statistical independence for the inphase and in-quadrature field components as well as for the three field rectangular
40
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
components, from (3.11) the statistics for amplitudes and squared amplitudes of
field components and total field can be derived [18], leading to the same results
obtained by the integral approach in [11].
It is helpful now to shortly discuss the model proposed and the results
obtained. Drawing on Hill’s plane wave integral representation for RC fields, we
propose a discrete representation considering a finite number of contributing
random plane waves. Each plane wave has random propagation direction,
polarization and phase, but constant amplitude. When considering a sufficiently
large number of contributing plane waves, the CLT ensures a normal distribution
for real and imaginary parts of the electric field rectangular components resulting
from the superposition of such random plane waves. As a result, this model is
representative for ideal RC EM fields. Furthermore plane waves amplitude E 0 is a
free parameter to match the amplitude of fields resulting from this model with fields
amplitude measured in a real chamber1. To find the matching relation between the
model and measured field quantities, one has to derive the field amplitudes and
squared amplitudes mean values as a function of real and imaginary parts mean
value and variance in (3.11). Resulting mean values, derived according to [11] are
reported in Table 3.2.
Table 3.2 Amplitudes and squared amplitudes mean values of electric field
resulting from the sum of n random plane waves described in Table 3.1
RC electric field quantity
E x, y , z
E x, y , z
E0 ⋅ n ⋅
2
n⋅
π
12
E02
3
E0 ⋅ n ⋅
Etot
Etot
mean value
15 π
⋅
16 3
n ⋅ E02
2
The matching between the field in (3.2), resulting from the superposition of n
contributing plane waves whose random parameters are described in Table 3.1, and
1
The number of contributing plane waves n is also a free parameter, but n must be
sufficiently large for the CLT to be respected.
41
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
the field mean amplitude measured in a real chamber can be obtained by using
Table 3.2 by adjusting the value of E 0 .
3.3 Statistical plane wave coupling approach for RC
By referring to the field representation in (3.2), and invoking the superposition of
effects, an EM coupled quantity x into an electrical object inside a RC can be
expressed as a finite sum of plane waves coupling contributions:
n
x = ∑ xipw
(3.12)
i =1
where n is the number of contributing plane waves and xipw is the coupled quantity
corresponding to the i-th random plane wave, whose parameters distribution are
those in Table 3.1. In (3.12) x and xipw can be for instance complex electric
voltages or currents induced into an electrical device by RC fields and by a random
plane wave, respectively.
Equation (3.12) can be used to evaluate statistical properties of the coupled
quantity in RC. As a first result, by applying the CLT to (3.12), as it was done for
(3.7), real and imaginary parts of x follow a Normal distribution. Thus, x
amplitude and squared amplitude follow a Rayleigh and an Exponential
distribution respectively, regardless of the distribution of xipw . This result agrees
with the derivation by the plane wave integral representation in [11]. As the
Rayleigh and the Exponential distributions are one-parameter distributions2, the
knowledge of the estimated mean value is sufficient for completely characterize the
coupled quantity.
The estimation of mean and maximum values of coupled quantities in (3.12)
by a Monte Carlo method will be proposed in sub-sections 3.3.1 and 3.3.2.
Maximum coupling value is interesting when analysing the threshold level of the
immunity of an electrical device exposed to external EM fields (see Chapter 5).
Finally, it is useful to underline that the correlation between the statistical
plane wave coupling formulation in the right term of (3.12) and actual coupling
measurements in a given RC facility, is possible once the amplitude of plane waves
2
i.e. the probability density function has just one free parameter (see Table 2.1).
42
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
is chosen according to Table 3.2 to match electric field mean value measurement in
RC, for a given number n of contributing plane waves.
3.3.1 Coupled quantities mean values
Monte Carlo methods for estimating mean values of the amplitude and squared
amplitude of the right term in (3.12) will be investigated in this sub-section. In the
following, mean values over a set of m values will be indicated by
values in RC over the stirrer rotation will be simply indicated by
m
, and mean
.
Equation (3.12) expresses the coupled quantity in RC for a fixed stirrer
position. The effect of rotating the stirrer gives rise to different plane waves
patterns (whose parameters follow the statistical distributions of Table 3.1), thus
mean values over stirrer rotation can be estimated as Monte Carlo trials of the right
term of (3.12) over m independent plane waves patterns. Mean values of amplitude
and squared amplitude of (3.12) can thus be expressed as:
n
∑ xipw
x =
(3.13a)
i =1
x
2
n
∑
=
i =1
m
2
xipw
(3.13b)
m
where the mean values in (3.13a) have dimensions of voltage or current, and the
mean values in (3.13b) have dimensions of power. The right-hand side of (3.13)
must be interpreted as follows: m independent sums of plane-wave coupling
contributions must be available and the mean value must be computed over such m
values. From a practical point of view, this means that m × n independent plane
wave coupling contributions must be available, in order to compute the mean
values of (3.13).
Some simplifications are now investigated for the estimation of the righthand side in (3.13). As shown in Annex B, under specific conditions, the righthand side in (3.13b) can be re-written as:
n
∑
i =1
2
xipw
= n ⋅ xipw
m
43
2
(3.14)
n
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
For generic independent complex random variables xipw , equation (3.14) states that
the mean value of the squared amplitude of the sum of n variables is equal to n
times the mean value of the squared amplitude of a single variable. In Annex B it is
shown that (3.14) has a general validity, under the condition that real and
imaginary parts of x PW , i have zero mean value. Thus, in our case if the plane
wave coupled quantity has zero mean value for real and imaginary parts, (3.14) can
be used as a correct estimate of the right-hand side in (3.13b).
The practical interest of (3.14) is that only one set of n independent plane
wave coupling contributions is required to estimate the RC mean values. The
physical interpretation of (3.14) is that a direct relation exists between RC mean
coupling values and mean coupling values over random incident plane waves. For
instance, we consider the case when the contributing plane waves amplitudes are
chosen according to the last row of Table 3.2, i.e. related to the a real RC total field
squared amplitude as in (3.15).
E 02 =
1
E tot , RC
n
2
(3.15)
In the case when (3.15) is used for choosing the plane waves amplitudes, we
showed that (3.2) is representative of EM fields of a given RC facility whose
measured total field squared amplitude mean value is that of the right term of
(3.15). It follows that (3.12) and (3.13) are representative for the EM coupling
measurements in that facility. Now, if we consider the coupling with a linear
device, by the linearity property of the superposition of effects we can write for the
right-hand side in (3.14):
n ⋅ x ipw
2
1
2
Etot , RC
E2 =
0 n
=
n
x ipw
2
E 2 = Etot , RC
0
(3.16)
2
n
In the left hand side of (3.16) the coupled quantity xipw corresponds to a plane
wave of amplitude
1
n
Etot , RC
plane wave of amplitude
2
, while in the right hand side xipw corresponds to a
Etot ,RC
2
. From results in (3.13b), (3.14) and (3.16), it
is possible to write:
44
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
x
2
= xipw
2
2
⇔ E 02 = Etot , RC
(3.17)
n
where “ ⇔ ” stands for a necessary condition. Equation (3.17) states that the mean
value of squared coupled quantity in RC is equal to the mean value of the squared
coupled quantity over random plane wave incidence, provided that the plane waves
amplitude is chosen as E 0 = Etot , RC
2
. This important result was already proven
theoretically in [11] and experimentally in [25], by comparison of coupling
measurements in RC and in AC.
Concerning the right–hand side of (3.13a), things are a little more
complicated, as it is not correct to simply take the square root of (3.14). Two
possible solutions are proposed (see Annex C for calculation details):
n
∑ xipw
i =1
= n
m
n
∑ xipw
i =1
π
2
( { })
Std ℜe xipw
= n ⋅ xipw
m
n
n
(3.18)
(3.19)
where Std stands for standard deviation. As reported in Annex B, (3.18) is valid for
any complex random variable xipw which has zero mean value and equal variance
for real and imaginary parts. In this case, by substituting (3.14) in conjunction with
(3.13a) state that the mean value of the amplitude of the coupled quantity in RC
can be estimated by the standard deviation of real and imaginary parts plane wave
couplings. Finally, equation (3.19) is valid only when real and imaginary parts of
xipw are normally distributed, with zero mean values and equal variances. It is
important to underline the fact that in this case, Normal distribution is required for
xipw , which means that the coupled quantity must be Normally distributed over
random plane waves incidence, polarisation and phase. It is interesting to notice
that directly taking the square root of the expressions inside the operator of mean
value in (3.14) leads to (3.19), but the two expressions have different validity
domains.
It is useful to shortly discuss the hypothesis laying at the basis of Equations
(3.14), (3.18) and (3.19). In a general way, statistical properties of coupled
45
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
quantities in function of random plane waves parameters depend on the particular
electrical object considered. As an example, the variance of the coupled quantity in
function of plane wave incidence direction depends on the directivity of the
considered object: greater is the directivity, lower is the variance and vice versa.
On the other hand, even if no rigorous demonstration is available, the hypothesis of
zero mean value and equal variance for real and imaginary parts seems to be
reasonable for coupled quantities into linear devices. If these hypothesis are
verified, (3.14) and (3.18) can be used asymptotically (for large values of n). The
hypothesis of Normal distribution for real and imaginary parts of plane wave
coupling is on the contrary more restrictive and depending on the considered
object. The validity of (3.19) must thus be investigated case by case.
In Chapter 4, the validity domains of Monte Carlo prediction methods based
on (3.14), (3.18) and (3.19) will be investigated for the case coupled currents into
transmission lines
3.3.2 Coupled quantities maximum values
Monte Carlo methods can be used also to estimate the maximum values of the
amplitude and squared amplitude of the right-hand side of (3.12). If the maximum
value over m samples is indicated by   m and the maximum value in RC simply
by   , (3.10) gives rise to:
 n pw 
x
=
   ∑ xi 
 i =1
m
(3.20a)
 n
=  ∑ xipw
 i =1

(3.20b)
x 
2
2


m
In analogy with the previous sub-section, (3.20) allow to estimate maximum values
of coupled EM quantities in RC by a processing of m×n plane waves couplings.
Unfortunately, no simplification is possible for the right-hand sides of (3.20), as it
was the case for mean values in (3.13). The reason is that more detailed
information about the xipw distribution is required in order to proceed with a
simplification of (3.20). In Chapter 5, a different approach, based on the knowledge
of the device under test directivity, will be investigated to relate plane wave and
46
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
RC maximum coupling.
2
On the other hand, since x and x in RC follow respectively a Rayleigh
and an Exponential distribution, extreme order statistics can be used to estimate
maximum values in the left-hand side of (3.20) from the estimation of the mean
values in (3.18) and (3.14). According to the notation in [26], maximum values can
be expressed in terms of mean values of (3.18) and (3.14) by:
x  =
n
 
x
2
π
2
( { }) ⋅ ↑
Std ℜe xipw
= n ⋅ xipw
n
N
(χ 2 )
( )
2
⋅ ↑ N χ 22
(3.21a)
(3.21b)
n
In (3.21) “ ↑ N ” stands for the maximum to average ratio of N values of a given
distribution, in our case a χ 2 in (3.21a) and a χ 22 in (3.21b). The formulation of
↑ N functions can be found in [26].
As a result, also maximum coupling values in RC can be estimated by (3.21)
with just one set of random plane waves coupling contributions.
3.3.3 Monte Carlo methods accuracy
One advantage of using Monte Carlo methods for simulating complex physical
processes, is that one has both the estimation of the process result, and an
estimation of the accuracy of the result [27]. This means that it is possible to
associate a confidence interval to the RC coupled quantities estimation by random
plane wave contributions, as formulated in sub-sections 3.3.1 and 3.3.2. The
confidence interval associated to each of the different proposed Monte Carlo
methods is discussed in the following.
Mean coupling values in (3.13) are computed as mean values over a set of
couplings each one corresponding to the RC coupling for a fixed position of a
virtual stirrer. This means that the mean value over m Monte Carlo trials in (3.13)
corresponds to the mean value over m stirrer positions. As a result, the associated
statistical uncertainty is the same uncertainty of mean values measurements in RC,
which was exposed in section 2.3 of this work. Thus, if the plane waves number n
in (3.13) is sufficiently large for the CLT to be respected, the 95% confidence
intervals in Table 2.2 can be used for estimating the Monte Carlo prediction
47
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
accuracy. In this case, the sample size n in Table 2.2 is given by the number of
Monte Carlo trials m.
The same argument can be applied for maximum values estimation in
(3.20); in this case RC maximum values uncertainty exposed in sub-section 2.3.3
can be used as an estimation of the accuracy of the Monte Carlo prediction.
We consider now the uncertainty associated to Monte Carlo reduced
estimation methods of Equations (3.14) and (3.19). In (3.14) and (3.19) the RC
mean coupling values are estimated as mean values over n plane wave coupling
contributions. As discussed above, when considering a general device, we don’t
know the probabilistic distribution of the coupled quantities over random plane
waves. This means that we are estimating mean values of samples with unknown
distribution and variance. In this case, the lower and upper bounds of confidence
interval associated to (3.14) and (3.19) are given by (see Annex A.4):

σ
σ 

 µ N − t n −1,1−α / 2 ⋅
, µ N + t n −1,1−α / 2 ⋅
n
n

(3.22)
where µ N is the estimated mean value, σ is the estimated standard deviation of
the coupling sample, t n −1,1−α / 2 is the inverse of the Student’s T cumulative density
function with n-1 degrees of freedom at the value in 1 − α / 2 , and α is the
confidence degree ( α =0.05 for 95% confidence interval). Confidence interval in
(3.22) can be used for estimating the accuracy of Monte Carlo methods in (3.14)
and (3.19).
Finally, in (3.18) the standard deviation of the real part of the coupled
quantity over random plane waves is used for estimating RC coupled quantity
amplitude mean value. The associated uncertainty can thus be estimated as the
uncertainty associated to the estimation of the standard deviation of a sample. In
this case, the lower and upper bounds of the associated confidence interval are (see
Annex A.4):

 σ ⋅ n −1 σ ⋅ n −1
,

 χ α2 / 2,n −1
χ 12−α / 2,n−1






(3.23)
where σ is the estimated standard deviation of the real parts over random plane
waves, χ α2 / 2,n −1 is the inverse of the χ 2 cumulative density function with n-1
48
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
degrees of freedom evaluated in α / 2 , and α is the confidence degree ( α =0.05
for 95% confidence interval).
Maximum values in (3.21a) and (3.21b) are estimated via the estimation of
mean values in (3.18) and (3.14) respectively. Thus (3.23) can be used for the
uncertainty estimation in (3.21a) and (3.22) can be used for the uncertainty
estimation of (3.21b).
3.4 Conclusions
The characterization of a radiated immunity testing methodology requires the
determination of the reliability of results, which is identified with the repeatability
of measurements in the same test site and the reproducibility in different sites. The
representativity of the test, with regard to the DUT real world conditions, defines
the robustness of the test. Thus, the reliability of results and the representativity of
the test are the relevant parameters for a good testing procedure. An additional
property is however required in the cases where the actual response of the device is
important, and when the correlation with other test techniques is needed. In these
cases, the understanding of the mechanism of EM fields coupling to the device is
required. In this Chapter, we have investigated this aspect, trying to understand the
nature of EM field coupling to electrical objects in RC.
We have proposed a coupling model for RC based on plane wave coupling
mechanism. This approach is based on the hypothesis that the RC EM environment
is the result of the superposition of plane waves, as formalized in [11]. For an ideal
RC, at high frequencies, the superposition of a large number of plane waves with
random parameters, is supposed to take place. Starting from this hypothesis, and by
applying the superposition of effects, we modelled the RC coupling as the
superposition of random plane wave coupling contributions. In order to have an
operational method for predicting RC coupling, we considered a finite number of
plane wave contributions and we derived the relation between the amplitude and
number of contributing plane waves and the mean value of the resulting fields
amplitude. Based on this approach, we presented Monte Carlo statistical techniques
for estimating mean and maximum values of RC coupled quantities, by coupling
contributions due to random plane waves. The estimation of Monte Carlo methods
accuracy can be used to assess the feasibility of this approach when using low
numbers of plane wave contributions.
The proposed method has two potential applications. The first one is the
49
3 – RC EM fields coupling with el. objects: a statistical plane wave approach
prediction of RC coupled quantities by the numerical simulation of a set random
plane waves coupling contribution. This approach is particularly interesting when
treating the EM coupling to distributed transmission lines, since in this case the
coupling theory is well established, and numerical codes for the solution of
multiconductor lines are available. This kind of application, relevant for the
automotive domain, will be analysed in the next Chapter.
A second kind of application, concerns the correlation between the coupling
results obtained in RC and in classical radiated immunity test sites, like AC, which
are supposed to operate in a plane wave environment. This correlation is possible,
according to the proposed approach, in the case where we dispose of different
plane waves inspection angles and polarizations. The consequent possibility of
correlating radiated immunity results obtained in RC and AC will be investigated
in Chapter 5.
The approach proposed in this Chapter is valid for an ideal RC EM
environment, characterised by fields uniformity and isotropy and independent field
patterns for different stirrer positions. A possible extension to the proposed
approach would take into account non ideal reverberating conditions for fields
representation, such that low frequency poor field uniformity and isotropy, or
direct coupling between emitting antenna and tested device. A second possible
extension would consist in taking into account the dependence of the considered
device directivity on the number of contributing plane waves required for
estimating mean and maximum values of the EM coupling. Concerning this last
point, it should be noted that, for an omni-directional device a single plane–wave
coupling contribution is required for estimating the mean and maximum values of
coupling. On the contrary, for a strongly directive device and/or for complex
directivity patterns, a large number of plane wave incidences should be inspected
to have good estimates for mean and maximum coupling values. The effect of the
device directivity on the correlation between plane wave coupling and RC coupling
will be introduced in Chapter 5.
50
Chapter 4
EM fields coupling to wires in a RC
4.1 Introduction
Nowadays, car electrical networks, which are constituted of unscreened wire
bundles, reach an overall length on the order of kilometres. By experience, in the
frequency range of about 80 MHz – 1 GHz, EM energy picked up by wires is one
of the principal causes of car radiated immunity problems. As a consequence, in the
industrial automotive EMC process, all electric and electronic devices are
individually tested on table (benchtest), connected to the functional wire bundles of
a normalized length. Benchtests are performed in fully- or semi-ACs, with the
DUT placed over a conducting ground plane.
Before establishing benchtest procedures in RC, a parametric analysis of the
EM field coupling to wires is necessary to quantify the influence of relevant
parameters such as wires length, height from the ground plane, wires orientation
and paths, terminal loads. Modelling of RC fields coupling to wires can be helpful
to perform this task by numerical simulation.
The problem of modelling statistical EM fields coupling to wires has been
the subject of several recent works (see e.g. [28], [29] and [30]). More specifically,
the problem of RC fields coupling to wires has been experimentally analysed in
[25], where a comparison with AC coupling is also presented, and theoretically
analysed in [31], where the plane wave integral model for RCs is adopted for
deriving closed form expressions for statistical estimators of induced electrical
quantities in a wiring harness. Based on the statistical plane wave coupling
approach for RC introduced in Chapter 3, we present in this Chapter an original
solution for modelling the RC fields coupling to wires by using Monte Carlo
51
4 – EM fields coupling to wires in a RC
simulation methods. A validation of the proposed approach is performed by
comparison with experimental results obtained in RC.
The Chapter is structured as follows. Section 4.2 contains the application of
the statistical plane wave approach, introduced in Chapter 3, to EM fields coupling
to a single wire over a ground plane. Numerical modelling, based on plane wave
coupling to transmission lines theory, is proposed and validated by comparison
with measurements. In section 4.3 the analysis is extended to modelling wire
bundles representative for automotive applications. Concluding remarks are finally
contained in section 4.4.
4.2 RC EM fields coupling to single wire
transmission lines: modelling and experimental
validation
The statistical plane wave coupling approach for RC proposed in Chapter 3 is well
adapted for modelling RC field coupling to distributed transmission lines. In fact,
the theory of plane waves coupling to transmission lines is well established (see
e.g. [32]) and the plane wave coupling contributions can be computed in an
analytical way for simple cases or by numerical solution of transmission line
equations for complex cases. In order to validate the approach proposed in Chapter
3, we analyse the simple case of a single wire transmission line over a ground
plane. For a uniform wire over an infinite, perfectly conducting ground plane, the
analytical solution of plane wave coupling is possible, as recalled in sub-section
4.2.1. The validation of prediction results is then presented in sub-section 4.2.2, by
comparison with experimental results obtained for a transmission line running over
the Renault RC floor.
4.2.1 Modelling
The considered line, illuminated by an incident plane wave described by equations
(3.3) – (3.6), is shown in Figure 4.1.
52
4 – EM fields coupling to wires in a RC
− θˆ i
r
Ei
z
θi
θip
r
E
ϕ̂i
k̂i
Z0
(*)
L
h
Z
L
y
x
ϕi
Figure 4.1 Single wire transmission line over a ground plane illuminated by an
incident plane wave. Geometrical and electrical quantities are defined. The insert
“(*)” represents the wave front, where the polarization angle θ ip is defined.
According to (3.12), when the transmission line in Figure 4.1 is exposed to RC
fields, the induced current flowing into the terminal load ZL can be expressed as:
n
I ( L) = ∑ I ipw ( L)
(4.1)
i =1
where I ipw (L) is the current induced by the i-th incident plane wave described by
random parameters as in Table 3.1, and n is the number of considered plane wave
contributions.
For the case of a straight wire over a perfectly conducting ground plane,
when line parameters are indicated in Figure 4.1 and plane waves parameters are
described as in (3.3) – (3.6), the current I ipw (L) , can be expressed analytically as:
53
4 – EM fields coupling to wires in a RC
I ipw ( L)
2h Ei ⋅ e jφi
=

Z Z 
cosh(γ L)( Z 0 + Z L ) + sinh(γ L) Z c + 0 L 
Zc 

sin( k h cosθ i )
jk cosθ i (− sin θ ip cosθ i sin ϕ i + cosθ ip cos ϕ i )
kh cosθ i
{
(γ + jk sin θ i sin ϕ i ) L
(4.2)
−(γ − jk sin θ i sin ϕi ) L
1  Z0  e
− 1
−1 1  Z0  e


− 1 +
 1 +

2  Z c  γ − jk sin θ i sin ϕ i 
 2  Z c  γ + jk sin θ i sin ϕ i
 
 
Z 
+ sin θ ip sin θ i 1 −  cosh(γ L) + sinh(γ L) 0 e jk sin θi sin ϕi L  
Zc 
 
 
Expression (4.2) can be derived according to [32].
In (4.2), ZC is the characteristic impedance of the line and γ depends on the
per-unit-length parameters of the line [r l g c]:
γ =
(r + jω l )(g + jω c )
(4.3)
By the use of a random number generator, random plane parameters can be
generated according to Table 3.1 as discussed in Annex A.6, and inserted in (4.2).
Methods proposed in Chapter 3 can then be used for estimating mean and
maximum values of the coupled current in RC over stirrer rotation.
4.2.2 Validation
In order to validate the modelling approach of sub-section 4.2.1, a transmission line
experimental set-up has been tested in the Renault RC, which was analysed in
Chapter 2. In choosing the experimental set-up, we had to face the problem of
disposing of a device with characteristics respecting as far as possible the
hypothesis at the basis of (4.2). We are in fact seeking a validation of the statistical
coupling approach instead of an assessment of the model for individual plane wave
coupling contributions. With this aim, we investigated the possibility of using the
floor of the chamber as the ground plane of the line, allowing thus to approach the
ideal conditions of infinite ground plane and to eliminate EM diffractions at the
ground plane edges. At this point, it should be noticed that, even if the working
volume of a RC is defined far from walls and the chamber floor1, when considering
1
The minimum distance is defined in function of the wavelength
54
λ
and is generally
4 – EM fields coupling to wires in a RC
a device over a ground plane we are authorised to approach the chamber floor in
the case where we can neglect common mode induced currents on the ground
plane, and we are interested only in differential mode currents between the tested
device and the ground plane. In fact, in the case where we can neglect both
common mode current and border effects at the ground plane edges, only the EM
fields above the ground plane are relevant to the coupling, and the distance of the
ground plane from the chamber floor is not relevant. We carried out some
experiments to verify such hypothesis with an experimental set-up consisting in a
movable ground plane with a semi-rigid transmission line mounted on it. We tested
such set-up for different positions inside the chamber volume, including close to
chamber walls and directly on the chamber floor. Results in RC show no
dependence of the coupled current flowing in the terminal loads of the line from
the set-up position in the frequency range where the differential mode is prevailing
[33]. We have thus chosen to use the chamber floor as the ground plane of the line.
The experimental set-up consists in a commercial single wire having a 0.6
2
mm section and dielectric coating. Each wire end was soldered to the inner
conductor of a type N microwave connector, mounted on a supporting metallic
plate fixed on the floor of the chamber. The outer conductor of the N connector
was mechanically connected to the metallic plate and the electric contact was
assured with the chamber floor. The measurement of the coupled current was made
by a network analyser, as shown in Figure 4.2.
defined between
λ /5
and
λ / 3 (see e.g. [24]).
55
4 – EM fields coupling to wires in a RC
(d)
(c)
(c)
(a)
(b)
Port 2
Network
Analyser
Port 1
Figure 4.2 Measurement configuration. (a) is the single wire exposed to the EM
field inside the RC; (b) is the far end load; (c) are the supports allowing various
line heights; (d) are coaxial cables.
Port 1 of the network analyser is connected to the emitting antenna, and port 2 to
one end of the line. The measurement of the S21 parameter is used to derive the
coupled current I(L). The adopted configuration allows measurements for different
terminal loads at one line end, while the other end is always terminated by the
50 Ω of the network analyser. The supports allow different wire heights from 2 to
5 cm above the chamber floor.
We present results for measurements performed over the frequency range of
80 MHz – 1 GHz for 144 stirrer positions, with a 50 cm long line placed at 3 cm
from the floor, and for both terminal loads of 50 Ω . To match prediction results
with measurement results, the amplitude of plane waves to be inserted into (4.2)
was chosen according to measurement of electric field in the Renault chamber, as
outlined in Table 3.2. Furthermore we modelled the line at first approximation as a
lossless line in an homogeneous medium. This means that for the Monte Carlo
simulations we used r = g = 0 in (4.3) and Z c = l / c in (4.2), where [r l g c] are
the per-unit-length parameters of the line. With these assumptions we neglect the
effects of the dielectric coating of the wire and of Ohmic and radiation losses of the
line. It should be noticed that radiation losses may play an important role when
considering small values of the terminal loads.
We first propose the numerical prediction of mean and maximum values of
56
4 – EM fields coupling to wires in a RC
the coupled current amplitude obtained in conjunction between (4.2) and the
application of (3.13a), (3.20a), which become for coupled current:
I ( L)
RC
 I ( L) RC
=
n
∑ I ipw ( L)
i =1
(4.4)
m
 n

=  ∑ I ipw ( L) 
 i =1
m
(4.5)
In (4.4) and (4.5), left terms correspond to RC measurements and right terms
correspond to Monte Carlo prediction results. Eq. (4.4) and (4.5) can be either used
for current amplitude and squared amplitude. Prediction and measurement results
of current amplitudes are compared in Figure 4.3. For the prediction, 144 Monte
Carlo trials of 50 plane waves contributions were used (this means m = 144 and n =
50 in (4.4) and (4.5)).
L=50 cm, h=3 cm, Z0=ZL=50 Ω
-60
IL144
|IL| (dBA)
-70
-80
-90
⟨IL⟩ 144
-100
measurement
prediction
-110
0.08
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.8 0.9 1
Frequency (GHz)
Figure 4.3 Coupled current amplitude mean value – prediction: 144 Monte Carlo
trials of 50 plane waves contributions, measurement: 144 stirrer positions.
Results in Figure 4.3 show a good agreement between predictions and
measurements over the useful frequency range of the chamber (see Chapter 2), thus
validating the approach presented in Chapter 3. The length L of the line used for
prediction in Equation (4.2) was adjusted, to match measurement resonant
57
4 – EM fields coupling to wires in a RC
frequencies of the line, to 53 cm. The slight shifting in prediction and measurement
resonant frequencies before this adjustment is probably due to the fact that (4.2)
doesn’t take into account the additional length of the line vertical risers and that we
neglected to model the dielectric sheath of the line. The uncertainty associated to
measurement and prediction results in Figure 4.3 can be computed according to
Section 2.3.1 (for Monte Carlo simulation uncertainty see Section 3.3.3).
According to Figures 2.2 and 2.4, for 144 stirrer positions and Rayleigh distributed
data, the 95% confidence interval for mean values is (− 0.8,+0.7 ) dB , and the 95%
confidence interval for maximum values is (− 1.7,+2.0 ) dB .
A further validation consists in comparing the statistical distributions of the
coupled currents amplitudes. The prediction and experimental cumulative density
functions (c.d.f. in the following) of the induced current squared amplitude at the
frequency f = 1 GHz are compared to the cumulative density function of the χ 22
distribution.
L=50 cm, h=3 cm, Z0=ZL=50 Ω
1
2
χ2
Cumulative density function
0.9
measurement
prediction
0.8
0.7
0.6
0.5
0.4
f=1 GHz
0.3
2
meas: χ test (r.s.l.=0.05%) passed
0.2
2
pred: χ test (r.s.l.=0.05%) passed
0.1
0
0
1
2
3
4
5
6
7
8
2
|IL| referenced to mean
Figure 4.4 Numerical prediction and measurements versus ideal χ 22 – prediction:
144 Monte Carlo trials of 50 plane waves contributions, measurement: 144 stirrer
positions.
Results in Figure 4.4 show a very good visual agreement between the c.d.f. of
experimental and predicted squared current amplitude and a χ 22 . This is confirmed
58
4 – EM fields coupling to wires in a RC
by the χ 2 goodness of fit test between experimental and prediction distributions
from one side and a χ 22 from the other side; in both cases the test is passed for a
rejection significance level of 5 %. The agreement of experimental results with a
χ 22 distribution was already theoretically predicted in [11] and experimentally
proven in [25]. The important practical consequence is that the measurement
confidence intervals determined in section 2.3 can be extended to measurements of
field coupling to wires.
From results in Figures 4.3 and 4.4 we can conclude that the prediction
method based on statistical plane wave coupling in Eq. (4.2) is able to predict both
mean and maximum values and the statistical distribution of coupled current.
Nevertheless, large numbers of plane wave coupling contributions have been used
for Monte Carlo methods (m*n = 144*50 = 7200 plane wave coupling
contributions!). On the other hand, once proven that the statistical distribution of
the current belongs to the one-parameter distribution family χ , the prediction of
coupled mean value becomes sufficient to have a complete prediction. Reduced
Monte Carlo methods were discussed for predicting mean values with lower
numbers of plane wave coupling contributions in sub-section 3.3.1. Such methods,
given by Equations (3.14) (3.18) and (3.19), and their validity domains in the case
of coupled current will be now investigated. In the case of coupled current, (3.14),
(3.18) and (3.19) become:
I ( L)
I ( L)
2
RC
I ( L)
= n ⋅ I ipw ( L)
RC
= n
RC
π
2
2
(4.6)
n
( {
})
Std ℜe I ipw ( L)
= n ⋅ I ipw ( L)
n
n
(4.7)
(4.8)
The previous Equations should be interpreted in the sense that the mean value of an
electrical quantity in the RC (e.g. the current at one end of the line) is represented
by an average taken over the simulated effects due to n plane waves.
The validation of (4.6) is investigated at first, for the same line analysed in
Figures 4.3 and 4.4. Validation is made by comparison with results obtained with
(4.4) applied to squared current amplitude mean value. A plane wave amplitude of
59
4 – EM fields coupling to wires in a RC
1 V/m is used in (4.2), since in this case we do not refer to measurements.
Comparison between results obtained by applying (4.4) and (4.6) is shown in
Figure 4.5. A large number of coupling contributions is used at first in order to
reduce results uncertainty due to Monte Carlo methods (m = 1500 n = 20 for the
complete method in (4.4), n = 1500 for the reduced method in (4.6)).
L=50 cm, h=3 cm, Z =Z =50 Ω
0
-140
L
reduced
complete
-150
2
2
⟨ |I | ⟩ (dBA )
-145
L
-155
-160
-165
0.08
0.1
0.15
0.2
0.25
Frequency (GHz)
0.3 0.35 0.4 0.45
Figure 4.5 MC complete and reduced methods for the prediction of the mean
squared amplitude of the line coupled current. Complete method refers to the
evaluation of Eq. (4.4) with m = 1500, n = 20. Reduced method refers to Eq. (4.6)
with n = 1500.
The validity of (4.6) is based on the hypothesis of zero mean value of the real and
imaginary parts of the coupled current over random plane-wave incidence, phase
and polarization (see sub-section 3.3.1). Repeated numerical simulations of (4.2)
showed us that this hypothesis is verified, proving the validity of (4.6) and the good
agreement between results in Figure 4.5. The slight difference at low frequencies is
within Monte Carlo method accuracy, which is discussed in the following and
shown in Figure 4.7.
The validation of (4.7) and (4.8), with regard to results obtained by (4.4), is
proposed in Figure 4.6.
60
4 – EM fields coupling to wires in a RC
L=50 cm, h=3 cm, Z0=ZL=50 Ω
-72
reduced - (4.7)
reduced - (4.8)
complete
-74
⟨|IL|⟩ (dBA)
-76
-78
-80
-82
0.08
0.1
0.15
0.2
0.25
Frequency (GHz)
0.3 0.35 0.4 0.45
Figure 4.6 MC complete and reduced methods for the prediction of the mean
amplitude of the line coupled current. Complete method refers to the evaluation of
Eq. (4.4) with m = 1500, n = 20. Reduced methods refer to Eq. (4.7) and (4.8) with
n = 1500.
Reduced method in (4.7) is based on the same hypothesis of (4.6), validated above,
plus the hypothesis of equal variance for real and imaginary parts of I ipw (L) . Such
hypothesis has been numerically verified by repeated trials of (4.2), proving the
validity of (4.7) and explaining the good agreement with results obtained by the
complete method (see Figure 4.6 red and black traces). Reduced method in (4.8) is
based on the hypothesis of Normal distribution for real and imaginary parts of
I ipw (L) over random plane wave incidence, polarization and phase. By numerical
simulation, we noticed that real and imaginary parts have a frequency dependent
statistical distributions, approaching more or less a Gaussian in function of
frequency. This explains the frequency dependent agreement of reduced and
complete methods results in Figure 4.6 (blue and black traces). The study of the
distribution of coupled current into a transmission line by random plane wave
incidence and polarization as been theoretically studied in [28], and experimentally
studied in [25]. Both works conclude that the analytical distribution of coupled
currents is frequency dependent, and [28] concludes also that an analytical
expression of the distribution is possible only for simple cases at low frequencies.
Such results limit the validity domain of (4.8), which is based on Normal
61
4 – EM fields coupling to wires in a RC
distribution for real and imaginary parts of coupled current over all the frequency
range. As a conclusion, reduced methods in (4.6) and (4.7) have general validity
for predicting mean coupled current values, while (4.8) doesn’t have general
validity.
In order to evaluate the accuracy of reduced methods (4.6) and (4.7) when
considering lower number of plane waves contributions, Monte Carlo methods
accuracy as formulated in sub-section 3.3.3 can be used. The upper bounds of 95%
confidence intervals for results obtained by (4.6) and (4.7) are shown in Figure 4.7,
for contributing plane wave numbers n = 20 and n = 1500.
L=50 cm, h=3 cm, Z =Z =50 Ω
0
95% condifence intervals upper bounds (dB)
3.5
3
L
⟨ |I |⟩ , n=20
L n
2
L
n
⟨ |I | ⟩ , n=20
2.5
⟨ |I |⟩ , n=1500
L n
2
L
n
⟨ |I | ⟩ , n=1500
2
1.5
1
0.5
0
0.08
0.1
0.15
0.2
0.25
Frequency (GHz)
0.3 0.35 0.4 0.45
Figure 4.7 Upper bounds of 95 % confidence interval for the estimation of
I (L)
2
, and I (L) , according to (4.6) and (4.7), respectively
The uncertainty levels corresponding to n = 1500 plane waves contributions must
be associated to results of reduced methods in Figures 4.5 and 4.6. With n = 20
plane wave contributions, results in Figure 4.7 show that the uncertainty level is
limited to about ±3 dB. Furthermore, as this value depends only on the number of
considered plane waves contributions and not on the particular considered line (see
Chapter 3, Eq. (3.23)), the same accuracy characterises any coupling prediction
based on (4.7) with 20 plane wave contributions. This conclusion supports the
practical feasibility of Monte Carlo methods for predicting RC coupled quantities
with a good accuracy (to be compared to measurement uncertainty presented in
62
4 – EM fields coupling to wires in a RC
Chapter 2) even with low numbers of contributing plane waves.
For the case of one wire transmission line, we also investigated
experimentally the hypothesis of independence of the coupled current from the
wire position, inside the measurement volume, and configuration, when
considering a straight or a bent wire. The first hypothesis is supported by fields
homogeneity in the working volume of the chamber, and the second hypothesis
comes from the fact that all antennas that have the same internal losses (in this case
same wire and same loads) should theoretically have, apart from impedance
mismatch between the antenna and the load, the same receiving cross-section in an
isotropic environment [11]. The two hypothesis have been experimentally
investigated in [25] and [33]. Taken from results in [33], Figure 4.8 shows the
coupling results obtained for the same line in two different positions, the first one
for a straight line configuration and the second one for a U-bent configuration.
L=1 m, h=5 cm, Z0=ZL=50 Ω
0
-5
-10
-15
⟨|IL|⟩ (dBmA)
-20
-25
-30
-35
-40
Straight line
U-line
-45
-50
0.08
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.8 0.9 1
Frequency (GHz)
Figure 4.8 Line coupled current amplitude mean value for two positions inside the
chambers volume with two line configurations: straight and U-bent
Current amplitude mean values in Figure 4.8 correspond to 144 stirrer positions,
having thus an associated 95% confidence interval of (− 0.8,+0.7 ) dB . Results in
Figure 4.8 and other results in [33] validate the hypothesis of the independence of
coupling from wire position, inside the measurement volume, and configuration.
The practical interest of this result is that the position and configuration of cables
during radiated immunity tests in RC don’t influence measurement results, at least
63
4 – EM fields coupling to wires in a RC
as a first approximation.
4.3 Extensions to wire bundles
In the light of the results obtained in the previous section, we investigate now the
prediction of coupled currents into wire bundles representative for automotive
applications. When dealing with real world cables, the major difficulties are the
simulation of per-unit length parameters of non uniform lines and in the modelling
of non controlled terminal loads. Statistical approaches to overcome these
problems have been proposed in [34]. A statistical approach seems necessary when
dealing with automotive wire bundles, which can reach almost a hundred of wires
and are placed in a non uniform medium as the interior of a car.
Modelling of real world automotive bundles is out of the scope of this work.
It is however interesting to experimentally analyse the response of real wire
bundles in an isotropic environment such as a RC, and to evaluate the plane wave
statistical coupling approach for more realistic cases. We have thus chosen to
analyse real wire bundles and to model them by a commercial code, CRIPTE
(ONERA), which allows the modelling of uniform multiconductor lines over a
ground plane. The code, based on transmission line equations, allows the
introduction of dielectric coating and computation of coupled currents due to
incident plane waves. Single plane wave coupling contributions for real bundles
will thus be computed by CRIPTE, and treated via Monte Carlo methods for
obtaining RC coupled currents.
We first analyse the case of a 1 m long, 10 wires bundle. We realised an
experimental set-up, with each wire terminated to a 50 Ω carbon resistance and
connected to a metallic plate support fixed to the RC floor. The bundle is placed at
2 cm over the floor. The total coupled current is measured by a current probe
placed at a distance of 5-10cm from the loads. An example of measurement is
shown in Figure 4.9.
64
4 – EM fields coupling to wires in a RC
Figure 4.9: Measurement set-up for bulk coupled current on a wire bundle. The
current probe and the bundle termination with resistors are visible.
For numerical prediction, we modelled the wires bundle by CRIPTE as a uniform
lossless bundle. Mean value for the coupled current amplitude was measured in the
Renault RC over one stirrer rotation, and computed by the Monte Carlo reduced
method in (4.7), with 20 plane waves contributions. For frequencies between 80
MHz and 700 MHz, the results normalised to RC total electric field mean value are
shown in Figure 4.10.
Wire bundle total current (ZL=C//(R+L), R=50 Ω L=0.1µH C=1pF)
-40
⟨|IL|⟩/⟨|Etot|⟩ (dB (1A / 1V/m))
-50
-60
-70
-80
-90
-100
-110
0.08 0.1
Measurement
Prediction
0.2
0.3
Frequency (GHz)
0.4
0.5 0.6 0.7 0.8
Figure 4.10 Coupled total current amplitude mean value for the 10 wires bundle:
prediction by Monte Carlo reduced method in (4.7), with 20 plane waves
contributions.
65
4 – EM fields coupling to wires in a RC
For obtaining Figure 4.10 prediction results, the 50 Ω carbon resistances terminal
loads were modelled according to RLC resistor models, whose parallel
capacitances and series inductances values are given in Figure 4.10 title. As
previously discussed, the accuracy of prediction results in Figure 4.10 is the same
accuracy found for the case of one wire in Figure 4.7 ( I L
for n = 20), that is
about ±3 dB. The accuracy of measurement results is given by 95% confidence
intervals of mean values for data coming from a Rayleigh distribution over 50
stirrer positions, that is (− 1.4,+1.2 ) dB .
As a concluding case, a simplified electrical network of a car is considered.
The network is constituted of 11 bundles with a number of wires varying from 4 to
10, for an overall length of 10 m. The network has been extracted from the car and
laid at 2 cm above the chamber floor, as shown in Figure 4.11. Terminal loads are
50 Ω carbon resistances for all the wires, and the total current measurement has
been carried out at several terminal ends of the network, by measurement
procedure shown in Figure 4.8.
Figure 4.11 Simplified car electrical network laid at 2 cm from the chamber floor
The electrical network has been modelled by CRIPTE, and 20 random plane wave
coupling contributions were computed by using Monte Carlo method in (4.7).
Comparison of measurement and prediction results, normalized to the measurement
of one RC electric field component mean value, is shown in Figure 4.12.
66
4 – EM fields coupling to wires in a RC
-50
Total current on terminal load c1t8 (network on ground plane)
⟨|IL|⟩/⟨|Ei|⟩ (dB (1A / 1V/m))
-55
-60
-65
-70
-75
-80
-85
0.08 0.1
Measurement
Prediction
0.2
0.3
Frequency (GHz)
0.4
0.5 0.6 0.7 0.8
Figure 4.12 Coupled total current amplitude mean value for the simplified car
electrical network: prediction by Monte Carlo reduced method in (4.7), with 20
plane waves contributions.
The accuracy of Figure 4.12 prediction results is ±3 dB, and the accuracy of
measurement results (− 1.4,+1.2 ) dB .
The validity of prediction results in Figures 4.9 and 4.11 is limited by the
simplified modelling used for line parameters. In particular for the simplified car
electrical network, twisted bundles have been modelled as uniform, and bundles
interconnections have been modelled as ideal short circuit. However, results show
a good agreement between prediction and measurement in the case of the 10 wires
bundle, over almost the entire frequency range, and a certain agreement in the case
of the car electrical network, for limited frequency points. These results indicate
that the statistical plane wave coupling approach is a valid prediction approach
even for real lines. On the other hand, we can suppose that a statistical
characterisation of cables parameters used in conjunction with the statistical
coupling approach could give a complete statistical characterisation of the coupling
phenomena more representative for reality.
4.4 Conclusions
EM fields coupling to wires is one of the key-issues of automotive EMC. The
67
4 – EM fields coupling to wires in a RC
understanding the coupling mechanism in RC is necessary to define the
representativity of the RC radiated immunity test, and to correlate immunity results
with those obtained in classical test sites like AC. Furthermore, a numerical
prediction tool could be useful to performe parametrical studies in the early stages
of the automotive EMC process. In this Chapter we have analysed the application
of the plane wave statistical model for RC, exposed in Chapter 3, to the problem of
coupling to wires. In particular, we investigated the possibility of RC coupling
numerical prediction via Monte Carlo trials of coupling contributions due to plane
waves.
We first focused on the simple case of a one-wire line over an ideal ground
plane. In this case we dispose of an analytical closed form to compute the coupled
current due to an incident plane wave. In order to validate the prediction model,
measurements were carried out with a one-wire transmission line placed over the
Renault chamber floor. We measured the coupled current flowing in the far-end
load with the help of a network analyser. For the prediction, we used two kinds of
Monte Carlo methods. The first method, called “complete”, allows the complete
characterization of the coupled current, namely mean values, maximum values and
statistical distributions2. Prediction results obtained with the “complete” method
for the current amplitude and squared amplitude show a very good agreement with
measurements. The drawback of this method lays in the computational cost, given
the large number of plane wave coupling contributions required for characterizing
the current statistics. In fact, n plane wave coupling contributions are required for
each of the m stirrer positions used in measurement. This means that n*m
independent plane wave coupling contributions must be numerically computed for
predicting the RC coupled current. We thus focused on “reduced” methods, which
require reduced numbers of plane wave coupling contributions, and allow the
prediction only of mean coupled current values. We showed that with a single set
of n plane wave coupling contributions it is possible to predict mean values of
coupled current amplitude and squared amplitude which agree with the “complete”
method results, and thus with measurements. The accuracy of “reduced” methods
as a function of the number of considered plane waves was also addressed, and we
showed that, with n = 20 plane waves coupling contributions, the accuracy is of the
order of ± 3 dB .
In order to evaluate the applicability of prediction methods for automotive
2
Mean value, maximum value and statistical distribution over stirrer rotation.
68
4 – EM fields coupling to wires in a RC
applications, we turned to consider more complex line structures. We analysed a 10
wires bundle and a simplified electrical car network, composed of 11
interconnected bundles. Plane wave coupling contributions were computed with the
help of a commercial transmission line code. Real bundles were modelled as
uniform bundles (i.e. with uniform transversal section), and bundles
interconnections were modelled as ideal short circuits. Despite of such
simplifications, the prediction method proves its validity for predicting coupled
currents into real structures. Furthermore, we showed that the implementation of
the prediction method in an industrial context is possible by using commercial
codes which have a good computational time effectiveness.
69
Chapter 5
Radiated immunity test of electronic
devices in RC
5.1 Introduction
As a final task, in this Chapter we investigate the feasibility of radiated immunity
test in RC. In section 5.2, a summary of the test procedure in RC will be presented,
as it is proposed in [5]. Secondly, in section 5.3, we will focus our attention on the
correlation between RC radiated immunity test and AC test results. The aim is that
of defining the conditions, with regard to the tested device characteristics and to
the testing methodology adopted, which allow to establish a correlation. The
statistical plane wave approach introduced in Chapter 3 will be an essential
element for establishing a comparison between the two environments. In section
5.4 we will finally present and analyse the immunity results obtained in RC and
AC by means of an electronic device representative of automotive devices. A
description of the device, built on purpose, is proposed, and the repeatability of RC
results as well as correlations between RC and AC results will be analysed.
5.2 Radiated immunity testing in RC
Radiated immunity testing in RC is a stochastic process in which the DUT is
exposed to different field patterns obtained by rotating the stirrer. In mode-tuned
operation, if the tuner is rotated by n fixed independent positions, the DUT is
exposed to n independent field patterns. According to [5], the test procedure is
structured in two phases.
In analogy with AC calibration, a RC calibration is performed in the first
70
5 – Radiated immunity test of electronic devices in RC
phase of the procedure, with a double objective. The first one is to verify the field
uniformity in the test volume, as it was described in sub-section 2.4.3. The second
one is to predetermine the field strength inside the chamber for a given injected
power Pin-CAL. This is done by measuring the ratio between the electric field and the
square root of the injected power for the 8 corner points of the test volume.
Maximum values of the three electric field rectangular components are retained for
each position. The average of the 24 electric field maximum values, E-cal in the
following, is the calibrated chamber field strength1. As a result, the calibration of
the chamber is performed with regard to the expected maximum value of one
electric field rectangular component.
During the second phase of the testing procedure, the DUT is placed inside
the test volume, the power Pin is injected to have the desired field strength,
according to the calibration results, and the stirrer is rotated by fixed steps. If the
number of stirrer positions used during this phase is the same of that used in the
calibration phase, we expect that the maximum electric field rectangular
component ETest-RC, to which the DUT is exposed during one stirrer rotation, be
equal to:
ETest - RC =
E − cal
⋅ Pin
Pin − CAL
(5.1)
Pin is adjusted to have ETest-RC as the threshold level for the DUT over one stirrer
rotation.
5.3 Directivity based approach for RC and AC
radiated immunity results comparison
The differences between the RC radiated immunity testing, presented in the
previous section, and the classical fully or semi-AC test are evident. In RC the
DUT is exposed to an omni-directional un-polarized stochastic perturbation, while
in AC the DUT is exposed to a directional perturbation and the failure field level
corresponds to a deterministic field predetermined by a calibration procedure. The
DUT directivity pattern and the incidence direction of the perturbation influence
1
Variations of the E-field due to the introduction of the DUT inside the chamber, are taken
into account in [5] by a loaded chamber calibration procedure to be conducted before testing.
71
5 – Radiated immunity test of electronic devices in RC
thus AC results, but not RC results, at least in principle. Thus, it makes no physical
sense to search for a correlation between RC results and AC results obtained for a
single incidence direction and polarization. On the other hand, it is possible to
investigate for a statistical correlation, when testing several inspection angles and
polarizations in AC. In this section, we consider for simplicity a “complete” fullyAC test, that is with a large number of incidence directions and field polarizations.
The outcome of such test is the field strength ETest-AC corresponding to the
threshold level for the DUT, for the worst-case inspection angle and field
polarization.
The possibility of correlating the above defined ETest-RC and ETest-AC will be
investigated in the following. The correlation approach proposed in the following is
mainly based on [26] and [35], and is presented here as a further development of
the statistical plane wave coupling approach for RC introduced in Chapter 3.
Some simplifying hypothesis are at the basis of this approach. First, we
assume to be using an ideal RC, as defined in Chapter 2 of this work. Secondly, we
suppose that inside the fully-AC the DUT is exposed to an ideal plane wave EM
environment. The latter hypothesis is verified only in far-field regions and with no
scattering phenomena in the chamber. We identify in this case with EAC the plane
wave amplitude, that is the polarised electric field strength to which the DUT is
exposed for each inspection angle and polarization.
We consider then a linear DUT, and we characterise failures with regard to
the electrical power level Pfail received at two identified DUT electric terminals. A
DUT failure is present if the received power Prec , due to external EM fields, is
Prec ≥ Pfail . In the case where the AC test incidence directions and polarization
angles can be assumed as uniformly distributed (over the solid angle and over the
plane angle, respectively), according to the plane wave statistical approach for RC
in Chapter 3 (see (3.16) and (3.17)), we can derive the following equation:
Prec − AC
Prec − RC
=
EAC
2
3 ⋅ Er - RC
2
(5.2)
where:
•
Prec − AC
is the mean received power by the DUT in AC with respect to
inspection angle and field polarization;
72
5 – Radiated immunity test of electronic devices in RC
•
Prec − RC is the mean received power in RC with respect to the stirrer rotation;
•
E AC
•
2
Er - RC
is the square amplitude of the AC plane waves, as defined above;
2
is the mean value of the square amplitude of one electric field
rectangular component in RC.
Equation (5.2) has also been derived by a different approach in [26]. The maximum
received power in AC Prec − AC  , with respect to the inspection angle and the field
polarization, and the maximum received power in RC Prec − RC  n , with respect to
n stirrer positions, can be expressed as a function of the relative mean values
according to:
Prec−AC  = 2 ⋅ DDUT ( f ) ⋅ Prec−AC
(5.3)
Prec − RC n =↑ n (Prec ) ⋅ Prec − RC
(5.4)
where:
•
DDUT ( f ) is the frequency dependent maximum directivity of the DUT, as
defined in antenna theory;
•
↑ n (Prec ) is the maximum-to-average function of the received power in RC,
which is a function of the number n of stirrer positions and is independent of
the specific DUT. Complete formulation of this function can be found in [17],
and point estimations can be found in [26].
When considering a “complete” fully AC test, Eq. (5.3) is derived directly from the
antenna theory definition of the DUT directivity, as outlined in [26] and [36].
Starting from relations (5.2)-(5.4), it is possible to derive a relation between
the above defined ETest-AC and ETest-RC. As a first step, we equate the maximum
received power by the DUT in the two facilities, by taking the ratio of (5.3) and
(5.4) and equating it to 1. This is done in (5.5), by using the result in (5.2):
2
E AC
Prec−AC  2 ⋅ DDUT ( f ) Prec−AC
DDUT ( f ) 2

=
⋅
=
⋅ ⋅
1=
2
Prec−RC
↑ n (Prec )
↑ n (Prec ) 3 E
Prec−RC  n
r -RC
which can be re-written as:
73
(5.5)
5 – Radiated immunity test of electronic devices in RC
E AC
E r -RC
2
=
2
3 ↑ n (Prec )
⋅
2 DDUT ( f )
(5.6)
Equation (5.6) relates thus the fields amplitudes used in AC and RC tests which are
responsible of a DUT failure in each test-facility. This means that EAC in (5.6) is
the above defined ETest-AC. On the other hand, to obtain the final expression for test
correlation, the RC immunity result ETest-RC, expressed as the expected maximum
value of one electric field rectangular component, must be related to
Er - RC
2
of
(5.6). This can be done by the following relation, obtained by using results in Table
2.1:
E r -RC
2
Er -RC  n
2
=
4
1
2
π [↑ ( E
N
r -RC )]
(5.7)
where ↑ N ( Er - RC ) is the maximum-to-average function of one electric field
rectangular component in RC (reported in [17])2.
By combining (5.6) and (5.7), we finally obtain:
↑ n (Prec )
ETest -AC
12
1
=
⋅
⋅
ETest -RC
π
2 ⋅ DDUT ( f ) ↑ n ( E r -RC )
(5.8)
Relation (5.8) expresses the ratio between the DUT immunity field level for a DUT
when tested in AC (“complete” fully AC test) and in RC (testing procedure
described in section 5.2). In the right-hand side of (5.8), the second factor
expresses the dependence from the DUT characteristics, as a function of the DUT
directivity. The first and the third factors express the dependence from the choice
of the “equivalent test conditions” and the number of stirrer positions used in RC.
Different choices of “equivalent test conditions” are discussed in [26].
The major difficulty in the practical use of (5.8), is tied to the estimation of
2
E r -RC
In deriving (5.7), we used the asymptotic ( n → ∞ ) relation
2
=
4
π
E r -RC
2
from Table 2.1.
74
5 – Radiated immunity test of electronic devices in RC
the maximum directivity DDUT ( f ) of the tested device. For a general device, the
directivity is not known a-priori. Thus, in most cases, estimated values have to be
inserted in (5.8). Estimation methods for the directivity of intentional and nonintentional emitters are discussed in [35]. However, the validity of such methods is
not general. For instance in [37], it is shown how, for shielded enclosure with
apertures, given the great variability of the DUT directivity versus frequency, it is
difficult to obtain a frequency by frequency correlation between AC and RC results
based on directivity estimation. In the next section, we will experimentally
investigate the applicability of (5.8) to the case of a typical automotive device,
whose primary source of radiated susceptibility are external wires.
It must be finally underlined that (5.8) is valid for a “complete” AC test, that
is when a large number of incidence directions and polarization are inspected in
order to find the real maximum received power. In most cases, for practical
applications, only a few inspection angles are tested, for one or two orthogonal
field polarization. This means that there is a potential risk of missing the real worst
case angle. Thus, for experimental results, the equality in (5.8) should be replaced
by “larger than”.
5.4 Measurement results of a generic test device
In this section, we will experimentally investigate the RC radiated immunity tests
repeatability (within the same facility) and reproducibility (in different facilities),
as well as the validation of the approach presented in section 5.3 for correlating RC
and AC immunity results. With the aim of disposing, for our investigations, of a
simple device both representative of automotive devices and susceptible over a
wide frequency range, we realized the test device which is shown in Figure 5.1.
75
5 – Radiated immunity test of electronic devices in RC
Battery
Optical fibre
output
External wire
(50 cm)
10 cm
16 cm
Figure 5.1 Printed circuit board of the test device
The DUT is composed of two commercial integrated circuits, and a 9 Volt onboard battery. Since most of the times the energy picked-up by external wire
bundles is the responsible of the immunity level of automotive devices, we
connected a 50 cm long external wire to a critical input of the device (a voltage
comparator input). A continuous wave EM signal coupling to the external wire
allows thus to trigger a device failure, and the failure information is extracted from
the chamber via an optical fibre.
Radiated immunity benchtests have been performed for the considered DUT
in three different RCs and in one AC. Results are analysed in sub-sections 5.4.1
and 5.4.2, respectively.
5.4.1 Immunity test results in RC
We performed the tests in the Renault chamber (Vol.=221 m3), analysed in Section
2, in the TELICE chamber (Vol.=14m3), and in the UTAC chamber (Vol.= 48m3),
using different instrumentations in each facility. We adopted the testing procedure
described in section 5.2 working in the good operating conditions for each
chamber, and the DUT was placed over a non conducting support.
We first present, in Figure 5.2, the measurement repeatability results
obtained in the Renault RC, for different positions of the DUT inside the testing
volume. We used 50 stirrer positions over the frequency range 200 MHZ – 1 GHz.
The results, for 4 measurement positions, are expressed as the electric field
strength giving a DUT failure (ETest-RC as defined in section 5.2).
76
5 – Radiated immunity test of electronic devices in RC
Device threshold values (E-field)
55
RC1
RC1
RC1
RC1
50
E
test-RC
(dB V/m)
45
test-1
test-2
test-3
test-4
40
35
30
25
20
15
10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
Figure 5.2 Radiated immunity results for the benchtest in the Renault RC:
measurements repeated in 4 different positions inside the test volume, for 50 stirrer
positions.
Analysis of Figure 5.2 results will be proposed later on.
Figure 5.3 shows the reproducibility results in the three different RC, over
the frequency range of 500 MHz – 1 GHz, being the lowest usable frequency of
TELICE chamber about 500 MHz, and the lowest usable frequency of UTAC
chamber about 400 MHz.
77
5 – Radiated immunity test of electronic devices in RC
Device threshold values (E-field)
55
UTAC RC
TELICE RC
RENAULT RC
50
E
test-RC
(dB V/m)
45
40
35
30
25
20
15
10
0.5
0.6
0.7
0.8
Frequency (GHz)
0.9
1
Figure 5.3 Radiated immunity results for the benchtest in three RC: Renault,
TELICE and UTAC chambers, 50 stirrer positions.
Since the ETest-RC is expressed in terms of maximum values of one electric field
rectangular component, measurement repeatability and reproducibility can be
related to the maximum values uncertainty for electric field rectangular
components, which was presented in section 2.3. As discussed in section 2.3, for an
ideal RC with an ideal instrumentation, maximum value uncertainty derived from
ideal statistical model is the only contribution to overall uncertainty3. From Figure
2.4, with 50 independent stirrer positions we expect a 95% confidence interval for
maximum values of 4.6 dB. The experimental results show that 73% of
susceptibility frequency points in Figure 5.2 and 76 % in Figure 5.3 fall within a 5
dB range. We can thus conclude that, provided that we are working with RC in
good operating conditions, the uncertainty given by ideal RC statistics proves to
be a good estimation both of the measurement repeatability and of the
measurement reproducibility, regardless of the measurement configuration and
instrumentation.
5.4.2 Immunity test results in AC
We finally address the correlation between RC and AC susceptibility results, with
regard to the approach presented in section 5.3. We tested the above described
3
An estimation of the residual uncertainty was also given in sub-section 2.4.2.
78
5 – Radiated immunity test of electronic devices in RC
DUT inside the UTAC semi-AC, where absorbing panels were placed on the floor
to obtain a fully-anechoic effect. The DUT was placed over a wooden table, and
the emitting antenna was placed at 1m from the DUT. A one-point electric field
calibration was performed prior to the test. A picture of the measurement set-up is
shown in Figure 5.4.
Figure 5.4 AC measurement set-up for benchtest radiated immunity testing
We tested the DUT for 10 inspection angles and 2 polarizations over one principal
plane. For each tested frequency, we retained the worst susceptibility case over the
20 measurements, giving the ETest-AC defined in section 5.3. AC results are shown
in Figure 5.5, together with RC results of Figure 5.3, over the frequency range 600
MHz – 1 GHz. The lowest usable frequency is imposed here by the ability of
available power amplifiers to generate adequate field strengths.
Device threshold values (E-field)
55
UTAC RC
TELICE RC
RENAULT RC
UTAC AC
50
E
test-RC
(dB V/m)
45
40
35
30
25
20
15
10
0.6
0.7
0.8
Frequency (GHz)
0.9
1
Figure 5.5 Radiated immunity results for the benchtest obtained in different testfacilities: UTAC AC, Renault RC, TELICE RC and UTAC RC.
79
5 – Radiated immunity test of electronic devices in RC
In order to correlate AC results with RC results by relation (5.8), we approximated
the unknown device directivity by the external wire directivity. By this
approximation, we suppose that the only responsible of the DUT susceptibility is
the EM energy picked-up by the wire. This hypothesis was verified by testing the
DUT with and without the external wire. The directivity of the external wire was
estimated in an analytical way as the directivity of a perfectly conducting 50 cm
long wire, loaded with an open circuit at one end. In the 600 MHz – 1 GHz range,
the estimated directivity is a slowing varying function of frequency ranging from
1.8 to 2.4.
By inserting the above estimated directivity, and the number of stirrer
positions n = 50, used for RC measurements, into relation (5.8), we obtain an
expected ratio between AC and RC results which is a slowly varying function of
frequency ranging from -0.7 dB to –2.0 dB.
These results are consistent with the results of Figure 5.5, pointing out the
validity of the approach for correlating immunity results obtained in AC and RC.
Moreover, results in Figure 5.5 suggest that when dealing with non directive
devices (over the considered frequency range) an almost direct correlation between
AC and RC immunity results is possible, with AC results which are a few dB more
severe than RC results.
Finally, since a non-directive object has a smoothed radiation pattern, for a
given number of inspection angles there is a large probability of inspecting the
worst case coupling direction. In other words, a few incidence directions have to be
inspected in order to find the worst susceptibility case for non-directive devices.
This is the case of the analysed device, for which, given also the rotational
symmetry, with only 10 inspection angles, we obtain a good correlation with RC
results. Missing the worst incidence or polarization case in AC would have resulted
in higher fields threshold values in Figure 5.5.
Such considerations support the feasibility of correlating AC and RC
immunity results for non directive devices.
5.5 Conclusions
One of the potential advantages in using RC for immunity testing lays in the
possibility of a statistical control of uncertainty, which makes the test results
independent from the measurement configuration and from the specific test-facility
and instrumentations used. The validation of this hypothesis, concerning the
80
5 – Radiated immunity test of electronic devices in RC
measurement of EM quantities inside the empty chamber, was discussed in Chapter
2, where the uncertainty statistical model was introduced and the real measurement
uncertainty was evaluated. In this Chapter, we extended this analysis to the
measurement uncertainty of radiated immunity testing of electronic devices. This
was done by repeated immunity benchtests for a specially-conceived electronic
device for several configurations within the same chamber and in different
chambers with different instrumentations. Experimental results show that both the
measurement repeatability (within the same facility for different configurations)
and the measurement reproducibility (in different facilities) agree pretty well with
the confidence intervals of an ideal RC. The only hypothesis we used, is that of
working with RC in well operating conditions, hypothesis that was evaluated in
this Chapter by the calibration procedure of [5]. However, for a better
characterization of the measurement uncertainty, we recommend to use rigorous
chamber evaluation statistical tests as discussed in Chapter 2. This would avoid the
risk, for instance, of using correlated stirrer positions and thus underestimating the
real uncertainty.
Correlation of RC and AC radiated immunity results was also addressed in
this Chapter. The two testing approaches have difference nature, and while the AC
test is strongly dependent on the radiation pattern of the DUT, the omni-directional
RC testing is transparent to the radiation pattern. As a matter of fact, a correlation
between single direction tests in AC and RC test is not possible. On the other hand,
a correlation is possible when considering a “complete” fully AC test, which
considers multiple inspection angles and field polarizations. Basing on the plane
wave statistical approach for RC, proposed in Chapter 3, a statistical correlation
between the two methods has been proposed, which has a practical interest under
particular conditions. In particular, we showed the applicability of this method
when dealing with non-directional devices, as for instance automotive devices with
external wires. In this case, a direct correlation between worst case susceptibility
with regard to inspection angle in AC and worst case susceptibility over stirrer
rotation in RC, is possible, provided a correct definition of the equivalent test
conditions.
81
Chapter 6
Conclusions
The contribution of this work is inherent to the radiated immunity tests performed
on automotive electronic devices and on cars. The classical test methodology in
AC reveals to be unsatisfactory from a quality and cost-effectiveness point of view,
and there is a need to investigate alternative methods. We have investigated in this
thesis the RC methodology, with mechanical mode-tuning operation, in order to
assess the feasibility of radiated immunity tests and the possible advantages that
can be drawn in an industrial context. The thesis work has been structured into four
complementary parts, corresponding to Chapters 2–5, each one analysing a
different element necessary for assessing the testing methodology.
As a first relevant aspect, we have experimentally analysed the
performances of the Renault RC, in order to verify the coherence with the
statistical theory of an ideal RC. Several past works have proposed different
criteria to achieve this task, and we made in Chapter 2 a critical choice from the
principal criteria, providing a complete set of techniques for characterizing a real
chamber. The final goals for us were to obtain a quantification of the real
measurement uncertainty and an evaluation of the loading effects due to the
introduction of a car inside the chamber. The analysis of the first aspect is
necessary for evaluating the reliability of test results, while the second aspect must
be investigated to evaluate the feasibility of car tests in chambers with reasonable
sizes.
The ideal uncertainty statistical model for RC would allow, if validated, a
measurement uncertainty and the consequent repeatability of test results,
independent from the measurement configuration and from the specific used
facility. The two elements necessary for a statistical characterization of uncertainty
were experimentally investigated for the Renault chamber. The first one is the
82
6 - Conclusions
agreement of real fields statistics with ideal RC statistics, and the second one is the
number of independent stirrer positions available in a real chamber. This analysis
allowed us to evidence that the two elements are related, and the critical parameter
is the choice of the number of stirrer positions. In particular, we have shown that
the technique of increasing the number of stirrer positions for decreasing the
measurement uncertainty has three limitations. The first limitation is imposed by
the necessity of using independent stirrer positions. The second limitation rises
from the need of having a good agreement of measured fields with the ideal RC
fields statistical model. We have shown that such two limitations, which are based
on different requirements, lead to two different upper bounds for the number of
usable stirrer positions, and that both bounds increase with frequency. The third
limitation to arbitrarily increase the number of stirrer positions is due to the
residual uncertainty, which is the uncertainty existing even in good operating
chambers at high frequencies, and due to chamber imperfections. We have
estimated that the residual uncertainty for the Renault chamber is of the order of 1
dB. If the three upper bounds for the number of used stirrer positions are respected,
at the frequency of interest, the fields statistics agree with the ideal RC statistical
model, and the ideal statistical uncertainty model is applicable.
As a final remark, we notice that, for industrial applications the choice of
the number of stirrer position will be a compromise between the maximum number
allowed by the two above exposed criteria, and the time/cost-effectiveness of the
test.
Loading the chamber with a big object has, in principle, two different
effects. The first effect is an electrical loading of the cavity, with a consequent
decrease of the quality factor Q. The second effect is a decrease of the actual
chamber volume, since a portion of the empty chamber volume is taken by the
object. The first effect is prevailing when dealing with absorbing objects, while the
second one is prevailing for conducting objects. It has been shown in literature that
the two effects have contrasting results on chamber performances. The electrical
loading seems to extend fields uniformity towards low frequencies, while the
“volume” loading seems to deteriorate fields statistical properties. Introducing a car
inside the chamber will probably give rise to both effects, and an evaluation of total
effect on the chamber performances is necessary. Experimental results point out
that the electrical loading effect is prevailing when considering a car which is about
the 8% of the chamber volume, and that the chamber performances are improved at
low frequencies. A theoretical criterion, based on the measurement of the chamber
83
6 - Conclusions
quality factor Q and on the estimation of number of excited modes inside the
chamber, has been proposed in Chapter 3 to predict the decrease of the chamber
lowest usable frequency when inserting an electrical load inside the chamber. This
criterion is of course limited by the lowest chamber frequency determined by the
chamber dimensions.
In Chapter 3 we focused on the problem of the modelling of EM fields
coupling to electrical objects inside RC. The understanding of the coupling
mechanism is necessary when one wants to characterise the actual response of the
tested device, or if one wants to correlate the immunity results with those obtained
by different methodologies. The original contribution of this work consists in the
formulation of the RC coupling as a discrete statistical plane-wave coupling. We
modelled the RC EM environment as a superposition of a finite number of plane
waves with random parameters, and we derived the relation between the number
and amplitude of contributing plane waves, and the fields amplitude in RC. Based
on the theoretical formulation of this approach, the EM coupling inside a given RC
can be predicted by Monte Carlo trials of random plane waves contributions. The
number n of contributing plane waves is a free parameter, and, by statistical
inference techniques, it is possible to quantify the relation between n and the
prediction method accuracy.
The application of the statistical plane wave coupling approach to the
numerical prediction of RC coupling is interesting in cases where the plane wave
coupling with the considered electrical object can be easily computed. This is the
case of distributed transmission lines, since in this case the plane wave coupling
theory is well established. In Chapter 4, we have thus applied such coupling
approach to the numerical prediction fields coupling to wires, which is a relevant
problem in automotive EMC. The feasibility of the method was evaluated first for
the simple case of one wire transmission line over a ground plane, and validated by
comparison with measurement results. The principal result obtained is the
possibility to predict the RC coupling by using a relatively low number of plane
waves, with a good accuracy. For instance, by using 20 plane waves we have an
uncertainty of the order of ± 3 dB . Later on, we applied this method to the
prediction of coupled current in more realistic automotive wire bundles. In this
case, we used a commercial code based on transmission lines theory, to compute
plane wave coupling contributions. Also in this case, the prediction method proves
to be valid for predicting measurement results obtained on real bundles, with a low
number of plane wave contributions. The method has thus a strong application
84
6 - Conclusions
interest in the automotive domain, since it allows the prediction of RC fields
coupling to wire bundles by using commercial transmission line codes.
In order to fully exploit the meaning of the measured data, we analysed the
repeatability and reproducibility of RC radiated immunity tests, and the correlation
with the classical test methodology in AC. With the aim of carrying out repeated
tests in different facilities we realized a specially-conceived electronic device,
susceptible over a wide frequency range and representative for automotive devices.
Radiated immunity tests were repeated in three different test facilities with
different sizes and instrumentations. An automatic test procedure for RC radiated
immunity testing was established and validated during such measurements
campaign. The obtained results show that, when using RC in the good operating
conditions defined in Chapter 2, repeatability and reproducibility of radiated
immunity tests agree well with the predicted statistical measurement uncertainty of
ideal RC. In particular, by using 50 stirrer positions, we obtained a reproducibility
of results in different facilities which lays statistically in a 5 dB range. This is a
good result if compared to classical automotive tests reproducibility.
Finally, we analysed the possibility of correlating AC and RC radiated
immunity results. Based on the plane wave statistical coupling approach proposed
in Chapter 3 and on the current state of the art, we have proposed a statistical
correlation approach, applicable only in the case of a “complete” fully-AC test, that
is when multiple inspection angles and field polarizations are tested in AC. A
critical aspect of this approach is given by the dependence of the correlation factor
from the tested device maximum directivity. Since this parameter is generally not
known a-priori, estimated values must be used for establishing the correlation. It
has been shown in literature that this approach is difficult to be used with directive
devices, whose directivity is also strongly frequency dependent. However, this is
not the case of typical automotive devices, whose radiated susceptibility is
generally given by external functional wire bundles, which have in general a low
directivity, with a low frequency dependence. In this case, we showed the
theoretical feasibility of a direct correlation between AC and RC results, if a
correct definition of the equivalent test conditions is used. Experimental results
obtained with the special test device confirm the validity of this approach, and
show that for non-directive devices the correlation is possible even for a low
number of inspection angles in AC. In particular we obtained a correlation between
AC and RC results laying in a 5 dB range over a 400 MHz frequency span, by
using 10 inspection angles with 2 fields polarizations in AC.
85
6 - Conclusions
In conclusion, several aspects of RC testing relevant for automotive
applications have been assessed in this work. The obtained results promote the RC
testing as a reliable and robust methodology. Furthermore, this work brings an
original contribution to the understanding of RC EM coupling with electrical
objects by an approach which allows a statistical correlation between RC coupling
and plane wave coupling.
Several aspects and other possible advantages of RC immunity testing must
still be investigated and consolidated. Some of them are suggested by automotive
applications. With the aim of establishing Renault/Nissan EMC specifications for
RC radiated immunity benchtests, the analysis performed in Chapter 5 should be
extended to all the categories of automotive devices. This will impose to deal also
with devices with a non-uniform radiation pattern, and thus to face the problem of
a difficult correlation with AC tests. Nowadays, benchtests are carried out in AC
with standardised incident field strengths, which are independent on the particular
device directivity. The different nature of RC testing will require the investigation
of a different approach.
These questions should be moreover extended to the radiated immunity tests
of whole cars. Finally, it would be interesting to evaluate the contribution of RC to
the problem of correlating benchtests with whole-cars test results, which is a keyissue of the future automotive EMC.
86
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90
Annex A
Probabilistic and statistical tools
A.1 Introduction
Given the statistical nature of the RC EM environment, one has to be familiar with
several probabilistic and statistical concepts to correctly use the RC for radiated
immunity testing. It is thus useful to report in this Annex the principal probabilistic
and statistical tools used in this thesis and necessary when dealing with RC
measurements.
A brief introduction to random variables definition and probability theory is
presented in section A.2. The principles of extreme order statistics for determining
the distribution of the maximum of one random variable are contained in section
A.3. Statistical theory of inference and confidence intervals determination is
recalled in section A.4. The χ 2 goodness-of-fit test, used in Chapter 2 for
evaluating the agreement between measured EM fields with ideal RC statistical
model, is introduced in section A.5. Finally, an introduction to Monte Carlo
methods, focusing on the generation of random EM plane waves parameters used
in Chapter 3 ad 4, are discussed in section A.6.
A.2 Probability theory fundamentals
According to [18], a random variable is “a number x (ζ ) assigned to every
outcome ζ of an experiment. This number could be the gain in a game of chance,
the voltage of a random source, the cost of a random component, or any other
numerical quantity that is of interest in the performance of an experiment”.
Dealing with RC measurements, ζ can be associated to the position of the stirrer,
91
Annex A – Probabilistic and statistical tools
and x (ζ ) to any measurement of EM quantities. The basic question asked when
dealing with a random variable is: what is the probability that the random variable
x is less than a given value x, or what is the probability that the random variable x
is between the values x1 and x2? In the case of RC measurements, the question
could be: what is the probability that for a given stirrer position the power received
by an antenna Prec is between Prec1 and Prec 2 ? Probability theory answers to such
questions by associating a probability distribution to random variables. For a
continuous random variable, a continuous probability density function is defined,
giving the probability that the random variable x is between x and x + dx , for each
real value x . For instance, the power Prec received in a RC by an antenna for a
fixed stirrer position is distributed according to an Exponential distribution (see
Chapter 2), whose probability density function is:
 1
 Prec 
 2 exp −
 , Prec ≥ 0
f Prec (Prec ) =  2σ
 2σ 2 
0
, Prec < 0

(A.1)
where σ is a parameter depending on the particular experiment. The probability
density function in (A.1), is shown in Figure A.1, for σ =1.
Exponential distribution - probability density function
0.5
0.45
0.4
0.35
f(x)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
14
x
Figure A.1 Probability density function for the Exponential distribution, σ =1
The probability that, for a particular stirrer position, Prec is between Prec1 and
Prec 2 is given by the integral:
92
Annex A – Probabilistic and statistical tools
P{Prec1 ≤ Prec ≤ Prec 2 } =
Prec 2
Prec 2
 P 
exp − rec2  dPrec
 2σ 
Prec1 2σ
∫ f Prec (Prec ) dPrec = ∫
Prec1
1
2
(A.2)
The expected or mean value of a random variable E(x) , which is the most probable
value of x, and the variance var(x) , which gives an information about the spread of
the density function, are respectively defined by:
E ( x) =
+∞
∫ x ⋅ f x ( x) dx
(A.3)
−∞
var( x) =
+∞
∫ (x − E( x) )
2
⋅ f ( x) dx
(A.4)
−∞
The information about the variance is often given in terms of the standard deviation
“std”, which is the square root of the variance.
When applying (A.3) and (A.4) to the probability density function of the
received power Prec in (A.1), we found the received power expected or mean
value and variance, given respectively by:
E( Prec ) =
var( Prec ) =
+∞
∫
−∞
+∞
 P 
exp − rec2  dPrec = 2σ 2
 2σ 
−∞ 2σ
∫
1
2
( x − 2σ 2 ) 2
 Prec 
exp
dPrec = 4σ 4
−
2
2
2σ
 2σ 
1
(A.5)
(A.6)
The probability density functions for RC EM quantities were presented, together
with the relative mean values and variances in Chapter 2, Table 2.1.
A.3 Extreme order statistics
When dealing with RC for radiated immunity testing, the threshold level of a
device is defined by the maximum field value over one stirrer rotation, as discussed
in Chapter 5. It is important to underline that the maximum value is a random
variable itself, and must be characterised by an appropriate distribution. For
instance, if we look at the smooth cut-off at the high end of the density function in
Figure A.1, we see that it is difficult to quantify the maximum value of the received
93
Annex A – Probabilistic and statistical tools
power basing on its density function. Extreme order statistics can be used to
examine the probability distributions of maximum values in RC [38]. The starting
point is the definition of the cumulative density function Fx (x) of a random
variable, which is the integral of the above defined probability density function
f x (x) , according to:
Fx ( x) =
x
∫ fξ (ξ ) dξ
(A.7)
−∞
The probability that the random variable x is less than a given value x, is thus
simply the cumulative density function evaluated in x:
P{x ≤ x} =
x
∫ f x (x ) dx = Fx ( x)
(A.8)
−∞
If we dispose of two random variables x1 and x2, both characterised by the same
cumulative density function Fx (x) , the probability that the maximum between the
two variables is less than a given value x, is the joint probability that x1 is less than
x, and that x2 is less than x. If x1 and x2 are independent, the joint probability is
simply the product of the two probabilities, that is:
P{max(x1 , x 2 ) ≤ x} = P{x1 ≤ x}⋅ P{x 2 ≤ x} = [Fx ( x)]2
(A.9)
If we generalize (A.9) to n random variables, the probability that the maximum of n
random variables x  n is less than a value x is given by:
P{x  n ≤ x} = [Fx ( x)]n
(A.10)
In other words, the cumulative density function Fx  ( x) of the maximum value of
n
n identical independent random variables x is:
Fx  ( x) = [Fx ( x)]n
n
(A.11)
where Fx (x) is the cumulative density function of one random variable x. The
probability density function f x  ( x) of the maximum of n random variables x, can
n
94
Annex A – Probabilistic and statistical tools
be obtained by deriving (A.11):
f x  ( x) = n ⋅ [Fx ( x)]n−1 ⋅ f x ( x)
(A.12)
n
where f x (x) is the probability density function of a single variable x. By applying
this technique, the probability density functions for the maximum values of EM
quantities in RC over n independent stirrer positions can be obtained using the
corresponding single values distributions (see Chapter 2, Table 2.1).
The complete derivation of maximum values distributions for fields and
power in a RC can be found in [17].
A.4 Confidence intervals
Given an abstract probabilistic model, statistics is the discipline which deals with
the applications of the model to real problems, making inferences and drawing
conclusions based on experimental observations.
Statistical inference techniques allow to draw conclusions concerning
general properties of physical phenomena, based on the information contained in a
set of collected observations or measurements, which is called experimental
sample. Inference techniques are used in RC, for instance, to draw conclusions
about mean or maximum values of EM quantities over one stirrer rotation, based
on one measurement sample collected for different stirrer positions. The inference
consists of a point estimation and of an interval estimation of the unknown
parameter. For instance, if we are estimating the mean value µ of the power
received by an antenna over n stirrer positions, the point estimation that we use is
the arithmetical mean value of the measurement sample Prec
n
n
= 1 / n ⋅ ∑i =1 Pi .
The interval estimate is an interval of the form l ≤ µ ≤ u , with the associated
probability:
P{l ≤ µ ≤ u} = 1 − α
In summary, when a point estimate Prec
n
(A.13)
of the mean received power is given,
the associated confidence interval in (A.13) ensures that the real, and unknown,
mean received power is between l and u with a probability 1 − α (for instance,
α =0.05 for 95% probability). It is important to underline that the real mean
95
Annex A – Probabilistic and statistical tools
received power µ , which would ideally consists in the mean over all the possible
stirrer positions and all the possible antenna positions, remains unknown.
A confidence interval in the form of (A.13) can be drawn from the
probabilistic distribution of the estimated parameter. We present in the following
the confidence intervals used in this work: for mean values, standard deviations
and maximum values.
According to the CLT, mean values are supposed to follow Normal (or
Gaussian) distributions. Two cases must be distinguished when determining
confidence intervals for mean values: the first case is when the variance of the
measurement sample is known, and the second one is when the variance is
unknown. According to [18], if we have a sample of size n with a known variance
σ 2 , and if the sample mean is x n , the confidence interval for the unknown mean
value µ is given by:

P x

n
− z1−α / 2
σ
n
<µ< x
n
+ z1−α / 2
σ 
 = 1−α
n
(A.14)
Where z1−α / 2 is the inverse of the standardized1 Normal cumulative density
function, evaluated in 1 − α / 2 . The confidence intervals for mean values, in the
case where the sample variance is known, are useful in RC applications. In fact, all
EM quantities in RC have one-parameter distributions, and the variances can be
derived from the estimated mean values (see Chapter 2, Table 2.1), and thus
considered as known. This technique is used in [18], p. 249, to estimate the mean
value of an exponential distribution. For our applications, mean values for
Exponential and Rayleigh distributions, presented in Chapter 2, can be obtained by
(A.14) simply determining the variance σ 2 by the mean estimated value,
according to Table 2.1, and inserting this value in (A.14).
In Chapter 3, dealing with Monte Carlo methods, the confidence intervals
for mean values of samples with unknown variances were required. In this case, the
following confidence interval can be found in [18]:

P x

1
n
− t n−1,1−α / 2
sn
n
<µ< x
n
+ t n−1,1−α / 2
The standardized Normal distribution has mean value
96
sn 
 = 1−α
n
µ =0 and variance σ
(A.15)
=1
Annex A – Probabilistic and statistical tools
where t n −1,1−α / 2 is the inverse of the Student’s T cumulative density function
with n-1 degrees of freedom evaluated in 1 − α / 2 , and sn is the estimated sample
standard deviation, computed as:
sn =
[
1 n
∑ xi − x
n − 1 i =1
n
]2
(A.16)
In Chapter 3, we also needed the confidence interval for the standard deviation of a
sample with unknown distribution. To determine such confidence interval, an
hypothesis must be made about the probabilistic distribution of the considered
sample. Usually, the hypothesis is made that the sample follows a Normal
distribution [18]. In this case, we can use the following confidence interval for the
standard deviation:


sn ⋅ n − 1
sn ⋅ n − 1 

= 1−α
P
≤σ ≤

 χ2
2
χ
α / 2,n −1 
 1−α / 2,n−1
(A.17)
where χ12−α / 2, n −1 ( χ α2 / 2, n −1 ) is the inverse of the χ 2 cumulative density function
with n-1 degrees of freedom evaluated in 1 − α / 2 ( α / 2 ), and s n is the estimated
standard deviation.
Finally, confidence intervals for maximum values were used in Chapter 2
for maximum values measurements in RC. Extreme order statistics must be used in
this case to determine the maximum values distributions. In [17] it is shown that an
analytical closed form expression for maximum value density function, found by
applying (A.12), is possible for the Exponential distribution, while is not possible
for the Rayleigh distribution. For the exponential distribution, the following
confidence interval is derived in [17] for maximum values:
1  
1  






 ln1 − (1 − α / 2) n  
ln1 − (α / 2) n  

  ≤ M ≤ x ⋅ 1 − 
  = 1 − α
P  x  n ⋅ 1 − 


n



1
1

0.577 + ln (n ) +
 0.577 + ln (n ) +




2n 
2n 



(A.18)
97
Annex A – Probabilistic and statistical tools
where M is the real unknown maximum value and x  n is the estimated maximum
value.
The determination of confidence intervals for the Rayleigh distribution
maximum values, requires the numerical integration of (A.12). Point values of the
95% confidence intervals are reported in [17], and numerical results for lower and
upper bounds of 95% confidence intervals are reported in Chapter 2, Figures 2.3
and 2.4 of this work, for 10 ≤ n ≤ 350 .
A.5 Goodness-of-fit tests
Goodness-of-fit tests are used to test the hypothesis that a probabilistic distribution
is suitable to model the statistical distribution of experimental data. Operating with
RC, goodness-of-fit tests can be used to establish whether the chamber respect or
not the ideal probabilistic model. The test that we used in this work is the χ 2
goodness-of-fit test, which can be easily found in statistical literature, and is briefly
described in the following.
The test procedure requires a sample of size n, which is supposed to follow
the probabilistic distribution with cumulative density function F(x). For the χ 2
goodness-of-fit computation, the sample is divided into k ordered sub-sets, called
bins, and a test statistic is defined as:
χ o2
(Oi − Ei ) 2
=∑
Ei
i =1
k
(A.19)
where Oi is the number of sample measurements contained in the bin i and Ei is
the number of data coming from the supposed distribution contained in the bin i.
The expected number of data Ei is calculated by:
Ei = n ⋅ (F (u ) − F (l ) )
(A.20)
where n is the sample size, F(u) is the cumulative density function of the
hypothesized probabilistic distribution evaluated in the upper limit u of the bin i,
and F(i) is the cumulative density function of the hypothesized probabilistic
distribution evaluated in the lower limit l of the bin i.
It can be shown that, if the measurement sample follows the hypothesized
98
Annex A – Probabilistic and statistical tools
distribution, χ o2 follows approximately a χ 2 distribution with k − m − 1 degrees
of freedom, where k is the number of considered bins and m is the number of
parameters of the hypothesized distribution estimated by the sample. In the case of
hypothesized distributions belonging to the χ 2 family (as it is for RC
applications), we have m = 1, since we deal with one-parameter distributions. We
would reject the hypothesis that the distribution of the sample is the hypothesized
distribution if the computed value is χ o2 > χ12− p, k − m −1 , where χ12− p, k − m −1 is the
inverse of the χ 2 cumulative density function with k − m − 1 degrees of freedom
evaluated in 1 − p , and p is the probability of rejecting the assumed distribution
even if it is correct. The probability p is called rejection significance level of the
test.
This test is sensitive to the choice of bins. There is no optimal choice for the
bins width (since the optimal bin width depends on the distribution). Most
reasonable choices should produce similar results. For the χ 2 test to be valid, the
expected number of experimental measurements contained in each bin should be at
least 5. This is the choice we used to obtain results in Chapter 2.
A.6 Monte Carlo method
Numerical methods that are known as Monte Carlo methods can be described as
statistical simulation methods using sequences of random numbers to perform the
simulation. Monte Carlo methods are well adapted for simulating random EM
fields in RC, as illustrated for instance in [39]. We used in this work Monte Carlo
simulations to simulate RC EM quantities as superposition of random plane waves.
The application contained in Chapter 4 concerns the simulation of the current
induced in a transmission line by RC EM fields, by the superposition of random
plane wave contributions. Some details of the random plane wave parameters
generation will be given here.
Plane waves geometry was shown in Chapter 3, Figure 3.1. Plane wave
parameters which have to be randomly generated are the propagation direction,
given by the solid angle Ω(θ , ϕ ) , the field polarization angle θ p , and the phase φ .
Such parameters must follow the probabilistic distributions detailed in Table 3.1,
reported below in Table A.1.
99
Annex A – Probabilistic and statistical tools
Table A.1 Random plane waves parameters probabilistic distributions
Plane wave parameter
Probabilistic Distribution
Propagation direction: Ω(θ , ϕ )
Uniform: U [0,4π ]
Polarisation: θ p
Uniform: U [0,2π ]
Phase : φ
Uniform: U [0,2π ]
At the basis of Monte Carlo methods there is commonly an uniform generator,
which produces numbers which are independent and uniformly distributed between
0 and 1. In particular we used the uniform generator provided with the commercial
code MATLAB, version 5. Starting from a uniform generator, it is possible to
generate random numbers with other distributions. Several methods are available,
and for our applications we used a direct method based on the inversion of the
cumulative density function Fx ( x) of the desired distribution (see [18]). It can be
shown that if u is a random number produced by an uniform generator, and if
Fx−1 ( x) is the inverse of the cumulative density function of the desired
distribution, then:
y = Fx−1 (u )
(A.21)
is a random number following the distribution given by Fx ( x) .
This technique can be directly applied to the generation of the polarization
angle θ p and of the phase angle φ . In the case of uniformly distributed angles
between 0 and 2π , the probability density function is given by:
f x ( x) =
1
, 0 ≤ x ≤ 2π
2π
(A.22)
Thus θ p and φ can be generated, starting from a uniform generated number u and
according to (A.7) and (A.21), by:
θ p = 2π ⋅ u
φ = 2π ⋅ u
100
(A.23)
Annex A – Probabilistic and statistical tools
Concerning the plane wave propagation angles, it can be shown that to have the
solid angle Ω(θ , ϕ ) uniformly distributed over 4π , ϕ must be uniformly
distributed over 2π , while θ must be distributed according to the density function
[40]:
fθ (θ ) =
1
sin(θ ) 0 ≤ θ ≤ π
2
(A.24)
As a result, by applying (A.21) the propagation angles can be generated, starting
from a uniform generated number u, by:
θ = cos −1 (1 − 2u )
ϕ = 2π ⋅ u
101
(A.25)
Annex B
Computation details of the statistical
properties of the field resulting from
one random plane wave
In a polar coordinate system, as shown in Chapter 3, Figure 3.1, the Cartesian zcomponent of the electric field in the origin resulting from an uniform linearly
polarised electromagnetic plane wave is given by (3.8), that is:
r
E z ,i = E i ⋅ zˆ = E i ⋅ cos θ ip ⋅ sin θ i ⋅ cos φ i + j E i ⋅ cos θ ip ⋅ sin θ i ⋅ sin φ i (B.1)
14444244443
14444244443
ℜe{E z ,i }
ℑm{E z ,i }
where all the parameters have been defined in section 3.2. We consider random
plane wave parameters whose probabilistic distributions are shown in Table 3.1.
The corresponding probability density functions (pdfs) are reported in Table B.1.
We assume that the random variables θ i , ϕ i , θ ip and φ i are independent from
each other. Starting from the pdfs in Table B.1, we derive now the pdf of
(B.1). As a first step, we express
ℜe{E z ,i }
ℜe{E z ,i }
in
as:
ℜe{E z ,i } = E 0 ⋅ cos φ i ⋅ cos θ ip ⋅ sin θ i
123 123 123
y
v
x
14
4244
3
w
144424443
z
102
(B.2)
Annex B – Computation details of the statistical properties of the field
resulting from one random plane wave
Table B.1 Random plane waves parameters pdfs
Plane wave parameter
Probability density function
Propagation direction: Ω(θ i , ϕ i )
1
fθi (θ i ) = sin(θ i ), 0 ≤ θ i ≤ π
2
1
fϕi (ϕ i ) =
, 0 ≤ ϕ i ≤ 2π
2π
Polarisation: θ ip
1
, 0 ≤ θ ip ≤ 2π
2π
fθ p (θ ip ) =
i
Phase : φi
f φi (φ i ) =
1
, 0 ≤ φi ≤ 2π
2π
f Ei ( Ei ) = δ ( Ei − E 0 )
Amplitude: Ei
According to [18] (section 5: Function of one random variable) and to the pdfs in
Table B.1, the pdfs for v, x, and y in (B.2) can be directly derived:
f v (v ) =
f x ( x) =
f y ( y) =
1
π ⋅ 1− v
2
1
π ⋅ 1− x2
y
1− y
, v ≤1
, x ≤1
(B.3)
, 0 ≤ y ≤1
2
Given the pdfs in (B.3), and assuming independent random variables, according to
[18] (section 6: Two random variables) the pdf of w = x ⋅ y can be written as:
f w ( w) =
∞
∫
−∞
∞
1
1
 w
f xy  u, du = ∫
u
u
 u
−∞
u
1
1− u2 π
u = y, 0 ≤ u ≤ 1
du, 
w = x ⋅ y, w / u ≤ 1
1 − w2 / u 2
1
(B.4)
which results in:
103
Annex B – Computation details of the statistical properties of the field
resulting from one random plane wave
f w ( w) =
1
,
2
w ≤1
(B.5)
Proceeding with the same method used in (B.4), the pdf of z = v ⋅ w can be derived
by using the pdfs in (B.3) and (B.4); the result is:
f z ( z) =
1 
log1 + 1 − z 2  − log( z ),


π 

z ≤1
(B.6)
(B.6) is the probability density function for the real part of the electric field in (B.1)
when considering E 0 = 1 . The corresponding cumulative density function (cdf)
can be found by integrating (B.6):
1 1
 2 + π
Fz ( z ) = ∫ f z ( x)dx = 
1 + 1
−∞
 2 π
z
− arcsin( z ) − z ⋅ log 1 + 1 − z 2  + z ⋅ log[ z ], − 1 ≤ z ≤ 0






arcsin(z ) + z ⋅ log 1 + 1 − z 2  − z ⋅ log[z ], 0 < z ≤ 1






(B.7)
The pdf and the cdf of t = ℜe{E z ,i } = E 0 ⋅ z , for any E 0 , can be expressed in
function of (B.6) and (B.7) [18]:
 

2
 t 
 t  
1  


 − log
f t (t ) =
log 1 + 1 − 
 E ,
π ⋅ E 0  
E 0  

 0 
 


t
≤1
E0
(B.8)



2

 1 + 1 − arcsin t  − t ⋅ log 1 + 1 −  t   + t ⋅ log  t  ,


E   E
2 π 
E  E

E 0  

0 
0
0
0










− E0 ≤ t ≤ 0
Ft (t ) = 


2

 1 1 
 t 
 t  
 t 
t
t

 +
 −
 + arcsin
⋅ log 1 + 1 − 
⋅ log   , 0 < t ≤ E 0

E 0   E 0
2 π 
 E0  E0

 E 0 





(B.9)
104
Annex B – Computation details of the statistical properties of the field
resulting from one random plane wave
4
Re{Ez,i} - pdf
3
2
1
0
-1
-0.8
-0.6
-0.4
1
-0.2
0
0.2
Re{Ez,i}/E0
0.4
0.6
0.8
1
0.5
Re{Ez,i} - cdf
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Re{Ez,i}/E0
0.4
0.6
0.8
1
Figure B.1 pdf and cdf for ℜe{E z ,i }
The pdf and the cdf for ℜe{E z ,i } , in (B.8) and (B.9) respectively, are shown in
Figure B.1.
The mean value and variance of ℜe{E z ,i } can be easily derived by the mean
values and variances of v, and w (see (B.2)). Mean values and variances of v
( µ v , σ v2 ) and w ( µ w , σ w2 ) can be directly computed by using the definitions in
Annex A, and the pdfs in (B.3) and (B.5):
µ v = 0 , σ v2 = 1 / 2
(B.10)
µ w = 0 , σ w2 = 1 / 3
Since v and w are independent, mean value and variance for ℜe{E z ,i } in (B.2) can
be computed as:
µ ℜe{E z ,i } = E 0 ⋅ µ v ⋅ µ w = 0
σ ℜ2 e{E z ,i } =
2
E0 ⋅
[
σ v2
⋅ σ w2
+ µ v2
⋅ σ w2
+ µ w2 ⋅ σ v2
]
E02
=
6
(B.11)
Similar computations can be made for the imaginary part in (B.1), leading to the
same results in (B.8), (B.9) and (B.11).
105
Annex C
Details of Monte Carlo mean values
estimation
We consider the following equality:
 n

E  ∑ xi
 i =1
{ }
2

2
 = n ⋅ E xi

(C.1)
where xi = ai + j ⋅ bi are independent identically distributed complex random
variables, and E stands for expected or mean value. We will show below the
validity of (C.1) in the case where E{ai } = E{bi } = 0 .
The left hand-side of (C.1) can be decomposed as follows:
 n

E  ∑ xi
 i =1
2
 n
 n  2  n  2 
n





 = E  ∑ ai + j ⋅ ∑ bi  = E  ∑ ai  +  ∑ bi   =
i =1
 i =1  

 i =1

 i =1 


n
n
 n 2 n 2

= E  ∑ ai + ∑ bi + ∑ ai a k + ∑ bi bk  =
i , k =1
i , k =1
i =1
i =1



i≠k
i≠k




n
n
n
n









= E  ∑ ai2 + ∑ bi2  + E  ∑ ai a k  + E  ∑ bi bk 
14
i =4
i =14
i, k =1

i, k =1 
14
424
3
i ≠ k

i ≠ k

A
142
4 43
4 14243
2
B
In (C.2), A can be expressed as:
106
C
(C.2)
Annex C – Details of Monte Carlo mean values estimation
n

n

n
n
E  ∑ ai2 + ∑ bi2  = E  ∑ (ai2 + bi2 ) = E  ∑ xi
i =1 

i =1
i =1
i =1
2
n
{ }= n ⋅ E{x }
 = ∑ E xi
 i =1
2
i
2
(C.3)
If we make the hypothesis that E{ai } = E{bi } = 0 , we can write, for the terms B and
C of (C.2):


n
 n

E  ∑ ai a k  = ∑ E{ai }E{a k } = 0
i, k =1
 i, k =1
 i ≠ k
i ≠ k
(C.4)


n
 n

E  ∑ bi bk  = ∑ E{bi }E{bk } = 0
i, k =1  i, k =1
i ≠ k
 i ≠ k
(C.5)
By inserting (C.3)-(C.5) in (C.2), we finally validate the equality (C.1).
We consider now the following quantity:
 n
E  ∑ xi
 i =1
n

 n
=
+
⋅
E
a
j
∑ bi

∑ i

 i =1
i =1



(C.6)
under the hypothesis that xi = ai + j ⋅ bi are independent identically distributed
complex random variables, E{ai } = E{bi } = 0 and var{ai } = var{bi } = σ 2 . In this
case, according to [18], p. 140, for large values of n it can be shown that ∑ in=1 xi
follows a Rayleigh distribution, with mean value:
 n
E  ∑ xi
 i =1

π
n
=σ
2

(C.7)
On the other hand, in the case where ai and bi are Normally distributed with zero
mean value, by the same argument reported above, it can be shown that:
E{ xi } = σ
π
2
107
(C.8)
Annex C – Details of Monte Carlo mean values estimation
Thus, in this case, from (C.7) and (C.8) is follows:
 n
E  ∑ xi
 i =1

 = E{ x i } n

108
(C.9)
Annex D
Measurement set-up
Renault chamber qualification measurements, proposed in Chapter 2, were
performed with the following instrumentation.
Table D.1 Antennas
Transmitting antenna
Receiving antenna
Model
EMC test systems
model n° 3144
serial no. 9906-1055
Amplifier Research
AT1100
serial no. 11543
Frequency range
80MHz - 2 GHz
80 MHz - 1 GHz
Table D.2 Electric field probe
Model
Specifications
V = 2πε 0 ⋅ R ⋅ Ae ⋅ f ⋅ E
Electric field probe
Thomson
E1602 N°5
R = 50 Ω
Ae = 0.00113
cut-off frequency 600 MHz
Table D.3 Network analyser
Vector network analyser
S-parameters test set
Model
Frequency range
HP 8753 C 300 KHz - 3 GHz
HP 85046 A 300 KHz - 3 GHz
109
Annex D – Measurement set-up
The measurement set-up is shown in Fig. D.1.
(b)
Prec
(a)
Pin
(c)
(d)
Port 2
Network
Analyser
Port 1
Figure D.1 Network analyser measurement set-up. (a) is the emitting antenna; (b)
is the receiving antenna; (c) is the electric field probe; (d) are coaxial cables. O are
the network analyser reference planes.
The network analyser allows the measurement of the scattering parameters S ij ,
defined with respect to the incident and reflected waves ai and bi , at the reference
planes, by:
b1 = S11a1 + S12 a 2
b2 = S 21a1 + S 22 a 2
(D.1)
The power Prec , received at the Port 2 reference plane, can be expressed in terms
of the incident power on the transmitting antenna Pin and of the scattering
parameter S 21 , by:
Prec = Pin S 21
2
(D.2)
The electric field measured by the electric probe, can be obtained, according to
Table D.2, by:
110
Annex D – Measurement set-up
E=
V
2πε 0 ⋅ R ⋅ Ae ⋅ f
(D.3)
where V is the voltage measured at the probe input, which can be expressed in
terms of the received power Prec , according to:
V = Prec R0
(D.4)
where R0 = 50 Ω is the input impedance of the network analyser. From (D.2)(D.4), the measured electric field is:
E = S 21
Pin R0
2πε 0 RAe f
111
(D.5)
Annex E
Noise reduction
In Chapter 2, Figures 2.6 and 2.7 have been processed with the noise reduction
technique reported in [21]. We report here the same Figures before the noise
removal.
Figure E.1 Figure 2.6 before noise removal
112
Annex E – Noise reduction
Figure E.2 Figure 2.7 before noise removal
The noise reduction technique can be visualized on Figure E.2, as explained in the
following. For each frequency, we start from the highest number of stirrer positions
(250, right-end) and we move to the lowest one (0, left-end). The first time that we
encounter a non-correlation value, corresponding to a number of stirrer positions n,
we mark as non-correlated all the stirrer positions from n to 0. Then we repeat the
same procedure for each fixed number of stirrer positions, starting from the lowest
frequency (80 MHz) and moving to the highest one (2 GHz). The results obtained
after this procedure are reported in Chapter 2, Figure 2.7.
A similar procedure is used for obtaining Figure 2.6, starting from Figure
E.1.
113