telice.univ-lille1.frtelice.univ-lille1.fr/fileadmin/general/theses/PhD_LM_LILLE.pdf · N°...

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 Rapporteur A. C. MARVIN Rapporteur S. PIGNARI Rapporteur

W. TABBARA Rapporteur F. CANAVERO Directeur de thèse

B. DEMOULIN Directeur de thèse V. BERAT Co-Directeur de thèse F. FLOURENS Examinateur

I

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 not-yet 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

Summary

II

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.

III

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)

Acknowledgements

IV

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) 4 “Union Technique de l’Automobile du motocycle et du Cycle” (Technical Union of Car,

motorcycle and Cycle)

V

List of acronyms

AC Anechoic Chamber CLT Central Limit Theorem DUT Device Under Test EM Electromagnetic EMC Electromagnetic Compatibility RC Reverberation Chamber

VI

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

Table of contents

VII

for RC EM fields.......................................................................................37 3.3 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 3.4 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

Table of contents

VIII

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

1

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 sub-systems.

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 – General introduction

2

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


3

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


4

characterized by statistical criteria, based on fields characterization. Such a formulation of uncertainty is furthermore independent from the particular used test-facility, 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


5

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.

6

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

2 – Renault RC facility

7

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 non-dissipative 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


8

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.


9

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 Probability density

function f(x)

Mean value xx =E

Variance E(x- x )2

zyxEe ,,ℜ ,

zyxEm ,,ℑ Normal

(Gaussian)

− 2

2

2exp

21

σσπx

0

2σ

zyxE ,, 2χ (Rayleigh)

− 2

2

2 2exp

σσxx σπ

2

−

222 πσ

2,, zyxE

22χ

(Exponential)

− 22 2exp

21

σσx 22σ 44σ

TotE 6χ

− 2

2

6

5

2exp

8 σσxx πσ 2

1615

1282256

22 πσσ −

2TotE 2

6χ

− 26

2

2exp

16 σσxx 26σ 412σ

All distributions reported in Table 2.1 are one-parameter distributions, depending

only on the variance 2σ 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, 2σ 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


10

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:

2

22

η

TotTot

EH = (2.1)

aTot

rec mE

P ηπλ

η 421 2

2

= (2.2)

where εµη /= is the wave impedance, ( )fµελ /1= is the wavelength, m is

the antenna impedance mismatch and aη is the antenna efficiency (m and aη vary

from 0 to 1).

Thus, by the estimation of 2σ 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 aη 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


11

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


12

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 nx , and the maximum sample value as nx ; 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 2xσ is the

variance of the measured sample, the variance of the Normal distribution

associated to the sample mean value is nx /22 σσ = .

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

nx /22 σσ = (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.


13

Table 2.2 95% confidence intervals of mean estimated values

Original distribution Confidence interval

Rayleigh

+⋅

−⋅

nx

nx nn

02.11,02.11

Exponential

+⋅

−⋅

nx

nx nn

96.11,96.11

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 1=nx (0 dB) for each distribution.

10 50 100 150 200 250 300 3500.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Sample size n

95%

con

fiden

ce in

terv

als

95% confidence intervals of mean values (⟨x⟩n=1)

Rayleigh Exponential

Figure 2.1 Mean values confidence intervals in linear units


14

10 50 100 150 200 250 300 350-5

-4

-3

-2

-1

0

1

2

3

Sample size n

95%

con

fiden

ce in

terv

als

(dB

)

95% confidence intervals of mean values (⟨x⟩n=1)


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 12≥n and lower than 0.1 dB for

50≥n , 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,


15

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 5≥n . For Exponential distributed samples, 150≥n 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:

( )

( )

++

−

−⋅

++

−

−⋅

nn

x

nn

x

n

n

n

n

21ln577.0

975.01ln1,

21ln577.0

025.01ln1

11

(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 1=nx for each distribution.

3 Anderson-Darling normality test was used for testing the normality of the sample.


16

10 50 100 150 200 250 300

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Sample size n

95%

con

fiden

ce in

terv

als

95% confidence intervals of maximum values (xn=1)


Figure 2.3 Maximum values confidence intervals in linear units

10 50 100 150 200 250 300-5

-4

-3

-2

-1

0

1

2

3

4

Sample size n

95%

con

fiden

ce in

terv

als

(dB

)

95% confidence intervals of maximum values (xn=1)


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


17

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


18

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 sub-section 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

mSe /54.4=σ , 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.


19

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 21S 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 21S . 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 221S


20

from the network analyser) over one stirrer rotation with the ideal 22χ (see Table

2.1) by the help of the 2χ 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, 22χ 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

iipK

1ln2 (2.4)

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 22χ distributed at each frequency, it can be shown that K is 2

2Nχ

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 22χ 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.


21

0 50 100 150 200 250 300 350 400 4504800.080.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

No. of stirrer positions

Sta

rt fre

quen

cy (G

Hz)

Cumulative χ2 test for number of stirrer positions vs. frequency

Accepted

Rejected

accept level > 5%accept level > 1%

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 22χ 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 22χ 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 22χ

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.


22

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

iri

n

ii

n

irii

PPPP

PPPPr

1

2

1

2

1ρ (2.5)

In (2.5), ( )rρ 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 ( )rρ 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 ( ) 1=rρ , which means complete correlation. If the stirrer is able to actually stir the cavity modes, increasing r will decrease ( )rρ ; an ideal ( )rρ = 0 points out complete

uncorrelation. Several upper limits for ( )rρ can be adopted to define uncorrelation

for practical purposes. The empirical choice of ( ) 37.0/1 =≤ erρ 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 ( )rρ . The

significance of the correlation coefficient ( )rρ is evaluated by calculating the probability that n measurements of two uncorrelated variables would give a result ρ , as large as or greater than ( )rρ . This probability is given by [21]:

( )( ) ( )[ ]( )[ ]

( )( )

( )∫

−−⋅

−Γ⋅−Γ⋅

=≥1 2/421

2/22/12Prob

r

nn d

nnr

ρρρ

πρρ (2.6)


23

where ( )Γ is the gamma function. An experimental ( )rρ , for which the probability in (2.6) is small, points out a correlation. In particular, in [21], if

( )( ) %5Prob ≤≥ rn ρρ 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 ( ) 37.0/1 =≤ erρ .

0 50 100 150 200 2500.080.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

No. of stirrer positions

Freq

uenc

y (G

Hz)

Number of uncorrelated stirrer positions vs. frequency

Non correlated

Correlated

significant correl. ρ=0.0895 highly sign. correl. ρ=0.1175correlation ρ=1/e

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 e/1 . Figure 2.6 evidences that the use of

e/1 as an empirical limit may lead to consider as uncorrelated those stirrer


24

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 22χ 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 22χ 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].


25

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, meanσ , and of

the maximum values distribution, Maxσ , can be computed (see Table 2.2 and sub-

section 2.3.1 for mean values distribution and Annex A.3 for maximum value distribution):

7390.02

22

50 =

−⋅

== NNmean

ππ

σ (2.7)

1373.050 ==NMaxσ (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:


26

+⋅=

xx

dBσ

σ 10log20 (2.9)

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 10

1

2

3

4

5

6

7

Frequency (GHz)

σ (d

B)

27 measurements of |Er|, n=50 stirrer positions

σmeanmeas.σmeanth. σMaxmeas. σMaxth.

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


27

used it with theoretical value in (2.7) to calculate:

dBU theormeas 83.09980.022 ≈≈−= σσ (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 sub-section, 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.


28

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 10

1

2

3

4

5

6

7

Frequency (GHz)

σ (d

B)

IEC 61000-4-21 - σMax

fmin = 117 MHz

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.


29

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 10

1

2

3

4

5

6

7

Frequency (GHz)

σ (d

B)

27 measurements of |Er|n, n=50 and n=12 stirrer positions

σ meas. n=50σ th. n=50 σ meas. n=12σ th. n=12

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


30

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]:

( )21)(

38

3

3+++−⋅⋅=

cfdba

cfVfN π (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 QfBW /= , 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:

Qc

fVMDQBW 3

38 ⋅⋅=

π (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


31

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 No. of frequencies (log-spaced)

80 MHz – 240 MHz 50 20 240 MHz – 480 MHz 18 15 480 MHz – 800 MHz 12 10

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

1

2

3

4

5

6

7

Frequency (GHz)

σ (d

B)

IEC 61000-4-21 - σMax

fmin = 95 MHz

Figure 2.11 IEC 61000-4-21 field uniformity validation – chamber loaded with a

VelSatis car


32

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.

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-14

-12

-10

-8

-6

-4

-2

0

2

4

6

Frequency (GHz)

PV

elS

atis

/PE

mpt

y (dB

)

Normalized mean received power ratio: VelSatis / Empty chamber

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:

ChamberEmptyVelSatis

ff min3

1

min 31

⋅

≅ (2.13)

If we take the lowest usable frequency for the empty chamber obtained in Figure


33

2.9, that is MHzfChamberEmpty

117min = , (2.13) gives MHzfVelSatis

81min = . 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


34

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


35

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.

36

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.

3 – RC EM fields coupling with el. objects: a statistical plane wave approach

37

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 wave electric field in one point of the space rr as an integral over the solid angle of the plane wave spectrum:

∫∫ Ω⋅Ω=π4

d)jexp()()( rkFrE rrrrr (3.1)

In equation (3.1) Ω is the solid angle, kr

is the wave propagation vector and )(ΩFr

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 field as generated in a RC during one stirrer rotation, the angular spectrum )(ΩF

r 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 and in-quadrature real and imaginary parts of the plane wave spectrum )(ΩF

r.

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

iii rkErE

1)j(exp)( rrrrr (3.2)

The i-th plane wave geometry is shown in Figure 3.1.


38

z

y

x iϕ

iθ

piθ

iθ− ˆ

ikiθ

iϕ

iϕ

ie

iEr

O

Figure 3.1 i-th contributing plane wave geometry

In (3.2), n is the number of contributing plane waves, ikr

is the propagation vector

of the i-th plane wave given by:

)cosˆsinsinˆcossinˆ( iiiiii zyxkk θϕθϕθ ++−=r

(3.3)

where 00εµω=k is the wave number in free space. iEr

is the electric field

vector which lays on the phase plane defined by the unit vectors iθ and iϕ . iEr

, iθ

and iϕ are defined by:

iiiiii zyx θϕθϕθθ sinˆsincosˆcoscosˆˆ −+= (3.4)

iii yx ϕϕϕ cosˆsinˆˆ +−= (3.5)

( ) ( ) ]ˆ)sin(ˆ)cos([expˆexp ip

iip

iiiiiii jEejEE ϕθθθφφ +−⋅⋅=⋅⋅=r

(3.6)

In (3.6), iE is the i-th plane wave amplitude, and ie and iφ 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


39

preferred field polarisation. This means that uniform distributions are chosen for propagation direction and polarization angle, over the solid angle and over

π2 , 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 0E .

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: ( )ii ϕθ ,Ω Uniform: [ ]π4,0U

Polarisation: piθ Uniform: [ ]π2,0U

Phase : iφ Uniform: [ ]π2,0U

Amplitude: iE Dirac delta: ( )0E−iEδ

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:


40

∑ ∑= =

=⋅=⋅=n

i

n

iiziz EzEzOEOE

1 1,ˆˆ)()(

rr (3.7)

Where )(OEz is the resulting field z-component in the origin, and izE , is the field

z-component given by the i-th plane wave in the origin. Statistical properties will be derived for izE , first. According to (3.6), izE ,

can be written as:

( )

( )

4444 34444 214444 34444 21

r

iziz Em

iip

ii

Ee

iip

iiiiz EjEzEE

,,

)sin(sin)cos()cos(sin)cos(ˆ,

ℑℜ

+=⋅= φθθφθθ (3.8)

When random plane wave parameters are distributed according to Table 3.1, mean value and variance for izE , in (3.8) can be analytically computed. Details of

calculation are reported in Annex B, and the results are:

( ) ( ) 0,, =ℑ=ℜ iziz EmmeanEemean (3.9)

( ) ( )6

Evarvar

20

,, =ℑ=ℜ iziz EmEe (3.10)

where iEi ∀= ,E0 , 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, )(OEe zℜ and )(OEm zℑ 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]:

⋅==≈ℑℜ

6E

,0)(,)(2

02 nNOEmOEe zz σµ (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 in-phase and in-quadrature field components as well as for the three field rectangular


41

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 0E 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 mean value

zyxE ,, 12

E0π

⋅⋅ n

2,, zyxE

3E 2

0⋅n

totE 316

15E0π

⋅⋅⋅ n

2totE 2

0E⋅n

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.


42

the field mean amplitude measured in a real chamber can be obtained by using Table 3.2 by adjusting the value of 0E .

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

i

pwixx

1 (3.12)

where n is the number of contributing plane waves and pwix 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 pwix 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 pwix . 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).


43

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 m , and mean

values in RC over the stirrer rotation will be simply indicated by .

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:

m

n

i

pwixx ∑

==

1 (3.13a)

m

n

i

pwixx

2

1

2 ∑=

= (3.13b)

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 right-hand side in (3.13). As shown in Annex B, under specific conditions, the right-hand side in (3.13b) can be re-written as:

n

pwi

m

n

i

pwi xnx

22

1⋅=∑

= (3.14)


44

For generic independent complex random variables pwix , 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 iPWx , 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).

2

,20

1E RCtotEn

= (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

pwi

n

pwi

RCtotERCtotEn

xxn22

2,

20E2

,12

0E ===⋅ (3.16)

In the left hand side of (3.16) the coupled quantity pwix corresponds to a plane

wave of amplitude 2,

1RCtotEn , while in the right hand side pw

ix corresponds to a

plane wave of amplitude 2,RCtotE . From results in (3.13b), (3.14) and (3.16), it

is possible to write:


45

2

,20

22 E RCtotn

pwi Exx =⇔= (3.17)

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 2

,0E RCtotE= . 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):

( )npwi

m

n

i

pwi xeStdnx ℜ=∑

= 21

π (3.18)

n

pwi

m

n

i

pwi xnx ⋅=∑

=1 (3.19)

where Std stands for standard deviation. As reported in Annex B, (3.18) is valid for

any complex random variable pwix 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

pwix 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 pwix , 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


46

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:

m

n

i

pwixx

= ∑

=1 (3.20a)

m

n

i

pwixx

= ∑

=

2

1

2 (3.20b)

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 pwix 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


47

RC maximum coupling.

On the other hand, since x and 2x 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:

( ) ( )22χπ

NnpwixeStdnx ↑⋅ℜ= (3.21a)

( )22

22 χNn

pwixnx ↑⋅⋅= (3.21b)

In (3.21) “ N↑ ” stands for the maximum to average ratio of N values of a given

distribution, in our case a 2χ in (3.21a) and a 22χ 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


48

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):

⋅+⋅− −−−− n

tn

t nNnNσµσµ αα 2/1,12/1,1 , (3.22)

where Nµ is the estimated mean value, σ is the estimated standard deviation of

the coupling sample, 2/1,1 α−−nt is the inverse of the Student’s T cumulative density

function with n-1 degrees of freedom at the value in 2/1 α− , 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):

−⋅−⋅

−−−2

1,2/12

1,2/

1,1

nn

nn

αα χ

σ

χ

σ (3.23)

where σ is the estimated standard deviation of the real parts over random plane

waves, 21,2/ −nαχ is the inverse of the 2χ cumulative density function with n-1


49

degrees of freedom evaluated in 2/α , 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


50

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.

51

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

4 – EM fields coupling to wires in a RC

52

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.


53

z

y

x

Z 0 Z L

piθ

iϕ

iθ

iθ− ˆ

iϕ

h

Lik

iEr

Er

(*)

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 piθ 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

pwi LILI

1)()( (4.1)

where )(LI pwi 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 )(LI pwi , can be expressed analytically as:


54

+−+

−−

+−

+−

+

+−

+++

⋅=

−−+

Ljk

ci

pi

ii

Ljk

cii

Ljk

c

ip

iiip

iii

i

c

LcL

jipw

i

ii

iiii

i

eZZ

LL

jke

ZZ

jke

ZZ

jkkh

hkZZZ

ZLZZL

eEhLI

ϕθ

ϕθγϕθγ

φ

γγθθ

ϕθγϕθγ

ϕθϕθθθθθ

γγ

sinsin0

)sinsin(0

)sinsin(0

00

)sinh()cosh(1sinsin

sinsin11

21

sinsin11

21

)coscossincossin(coscos

)cossin(

)sinh())(cosh(

2)(

(4.2)

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]:

( )( )cjgljr ωωγ ++= (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 λ and is generally


55

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 mm2 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 5/λ and 3/λ (see e.g. [24]).


56

NetworkAnalyser Port 1Port 2NetworkAnalyser Port 1Port 2

(a)

(b)

(c)(c)(d)

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 50Ω 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 50Ω . 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 0== gr in (4.3) and clZc /= 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


57

the coupled current amplitude obtained in conjunction between (4.2) and the application of (3.13a), (3.20a), which become for coupled current:

m

n

i

pwiRC

LILI ∑=

=1

)()( (4.4)

m

n

i

pwiRC LILI

= ∑

=1)()( (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)).

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 1

-110

-100

-90

-80

-70

-60

Frequency (GHz)

|I L| (dB

A)

L=50 cm, h=3 cm, Z0=ZL=50 Ω

IL144

⟨IL⟩144

measurementprediction

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


58

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 ( ) dB7.0,8.0 +− , and the 95%

confidence interval for maximum values is ( ) dB0.2,7.1 +− . 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 22χ

distribution.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|IL|2 referenced to mean

Cum

ulat

ive

dens

ity fu

nctio

n

L=50 cm, h=3 cm, Z0=ZL=50 Ω

f=1 GHz

meas: χ2 test (r.s.l.=0.05%) passed

pred: χ2 test (r.s.l.=0.05%) passed

χ22

measurementprediction

Figure 4.4 Numerical prediction and measurements versus ideal 22χ – 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 22χ . This is confirmed


59

by the 2χ goodness of fit test between experimental and prediction distributions

from one side and a 22χ 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 22χ 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:

n

pwi

RCLInLI

22 )()( ⋅= (4.6)

( )npwiRC

LIeStdnLI )(2

)( ℜ=π (4.7)

n

pwiRC

LInLI )()( ⋅= (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


60

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)).

0.08 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-165

-160

-155

-150

-145

-140

Frequency (GHz)

⟨ |IL|2 ⟩

(dB

A2 )

L=50 cm, h=3 cm, Z0=Z

L=50 Ω

reduced complete

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.


61

0.08 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

-82

-80

-78

-76

-74

-72

Frequency (GHz)

⟨|IL| ⟩

(dB

A)

L=50 cm, h=3 cm, Z0=ZL=50 Ω

reduced - (4.7)reduced - (4.8)complete

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 )(LI pwi . 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

)(LI pwi 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


62

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.

0.08 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.5

1

1.5

2

2.5

3

3.5

Frequency (GHz)

95%

con

dife

nce

inte

rval

s up

per b

ound

s (d

B)

L=50 cm, h=3 cm, Z0=Z

L=50 Ω

⟨ |IL|⟩

n, n=20

⟨ |IL|2⟩

n, n=20

⟨ |IL|⟩

n, n=1500

⟨ |IL|2⟩

n, n=1500

Figure 4.7 Upper bounds of 95 % confidence interval for the estimation of 2)(LI , and )(LI , 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


63

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.

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.9 1-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

⟨|IL| ⟩

(dB

mA

)

L=1 m, h=5 cm, Z0=ZL=50 Ω

Straight lineU-line

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 ( ) dB7.0,8.0 +− . 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


64

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 50Ω 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.


65

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.

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-110

-100

-90

-80

-70

-60

-50

-40

Frequency (GHz)

⟨|IL| ⟩/

⟨|Eto

t| ⟩ (d

B (1

A /

1V/m

))

Wire bundle total current (ZL=C//(R+L), R=50 Ω L=0.1µH C=1pF)

MeasurementPrediction

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.


66

For obtaining Figure 4.10 prediction results, the 50Ω 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 ( LI 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 ( ) dB2.1,4.1 +− .

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 50Ω 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.


67

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-85

-80

-75

-70

-65

-60

-55

-50

Frequency (GHz)

⟨|IL| ⟩/

⟨|Ei| ⟩

(dB

(1A

/ 1V

/m))

Total current on terminal load c1t8 (network on ground plane)

MeasurementPrediction

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 ( ) dB2.1,4.1 +− .

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


68

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 dB3± .

In order to evaluate the applicability of prediction methods for automotive

2 Mean value, maximum value and statistical distribution over stirrer rotation.


69

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.

70

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

5 – Radiated immunity test of electronic devices in RC

71

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:

inCALin

Test PP

calEE ⋅−

=−

RC- (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.


72

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” fully-AC 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 ACE 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 failP received at two identified DUT electric terminals. A

DUT failure is present if the received power recP , due to external EM fields, is

failrec PP ≥ . 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:

2

RC-

2AC

3 rRCrec

ACrec

E

EPP

⋅=

−

− (5.2)

where:

• AC−recP is the mean received power by the DUT in AC with respect to

inspection angle and field polarization;


73

• RC−recP is the mean received power in RC with respect to the stirrer rotation;

• 2ACE is the square amplitude of the AC plane waves, as defined above;

• 2RC-rE 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 AC−recP , with respect to the inspection angle and the field

polarization, and the maximum received power in RC nrecP RC− , with respect to

n stirrer positions, can be expressed as a function of the relative mean values according to:

( ) ACAC 2 −− ⋅⋅= recDUTrec PfDP (5.3)

( ) RCRC −− ⋅↑= recrecnnrec PPP (5.4)

where:

• ( )fDDUT is the frequency dependent maximum directivity of the DUT, as

defined in antenna theory; • ( )recn P↑ 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

RC-

2AC

RC

AC

RC

AC322

1rrecn

DUT

rec

rec

recn

DUT

nrec

rec

E

EP

fDPP

PfD

PP

⋅⋅↑

=⋅↑⋅

==−

−

−

− (5.5)

which can be re-written as:


74

( )

( )fDP

E

E

DUT

recn

r

↑⋅=

23

2RC-

2AC (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 ACE 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 2RC-rE of

(5.6). This can be done by the following relation, obtained by using results in Table 2.1:

( )[ ]2RC-

2RC-

2RC- 14

rNnr

r

EE

E

↑=

π (5.7)

where ( )RC-rN E↑ 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:

( )

( )( )RC-RC-

AC-

2112

rn

recn

DUTTest

TestE

PfDE

E↑

↑⋅

⋅⋅=

π (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 In deriving (5.7), we used the asymptotic ( ∞→n ) relation

2RC-

2RC-

4rr EE

π= from Table 2.1.


75

the maximum directivity ( )fDDUT 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 non-intentional 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.


76

External wire(50 cm)

Optical fibre outputBattery

16 cm

10 cm

Figure 5.1 Printed circuit board of the test device

The DUT is composed of two commercial integrated circuits, and a 9 Volt on-board 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).


77

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

15

20

25

30

35

40

45

50

55

Frequency (GHz)

Ete

st-R

C (d

B V

/m)

Device threshold values (E-field)

RC1 test-1RC1 test-2RC1 test-3RC1 test-4

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.


78

0.5 0.6 0.7 0.8 0.9 110

15

20

25

30

35

40

45

50

55

Frequency (GHz)

Ete

st-R

C (d

B V

/m)


UTAC RC TELICE RC RENAULT RC

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.


79

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.

0.6 0.7 0.8 0.9 110

15

20

25

30

35

40

45

50

55

Frequency (GHz)

Ete

st-R

C (d

B V

/m)


UTAC RC TELICE RC RENAULT RCUTAC AC

Figure 5.5 Radiated immunity results for the benchtest obtained in different test-facilities: UTAC AC, Renault RC, TELICE RC and UTAC RC.


80

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


81

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.

82

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

6 - Conclusions

83

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

6 - Conclusions

84

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 dB3± . 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

6 - Conclusions

85

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.

6 - Conclusions

86

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 key-issue of the future automotive EMC.

87

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[17] J. M. Ladbury, G. H. Koepke and D. Camell, “Evaluation of the NASA Langley Research Center Mode-Stirred Chamber Facility”, United States Department of Commerce, Technology Administration, National Institute of Standards and Technology, NIST Technical Note 1508, 1999.

[18] A. Papoulis, Probability, Random Variables, and Stochastic Processes, Third Ed., McGraw-Hill International Editions, 1991.

[19] N. Wellander, O. Lunden, M. Bäckström, “The maximum value distribution in a reverberation chamber”, Proceedings of 2001 IEEE International Symposium on Electromagnetic Compatibility, August 2001, Montreal, Canada, pp. 1227 -1232 vol.2.

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[20] M. Hatfield, Reverberation chamber theory / experiment short course, Naval Sea System Command (NSWC) Dahlgren Division, 1998.

[21] O. Lundén, M. Bäckström, N. Wellander, “Evaluation of stirrer efficiency in FOI mode-stirred reverberation chambers”, Swedish Defence Research agency, Division of Sensor Technology, SE-581 11 Linköping, Sweden, FOI Scientific report FOI-R-0250-SE, November 2001.

[22] D. I. Wu and D. C. Chang, “The effect of an Electrically Large Stirrer in a Mode-Stirred Chamber”, IEEE Trans. on Electromagn. Compat., vol.31, No. 2, 1989, pp.164-169.

[23] P. Corona, G. Ferrara, M. Migliaccio, “Reverberating chambers and absorbers”, in Proc. 14th Int. Zurich Symp. on Electromag. Compat., Zurich, CH, Feb. 2001, pp. 631-634.

[24] F. Hoëppe, “Analyse du comportement électromagnétique des chambres réverbérantes à brassage de modes par l’utilisation de simulations numériques”, Ph.D. Thesis, University of Lille, France, December 2001.

[25] S. Silverskiöld, M. Bäckström, J. Lorén, “Microwave field-to-wire coupling measurements in anechoic and reverberation chambers”, IEEE Trans. on Electromagn. Compat., vol.44, No. 1, 2002, pp.222-232.

[26] J. M. Ladbury and F. H. Koepke, “Reverberation chamber relationships: corrections and improvements or three wrongs can (almost) make a right”, in Proc. IEEE 1999 Int. Symp. on EMC, Seattle, WA, Aug. 2-6, 1999, p. 1.

[27] M. H. Kalos, Monte Carlo methods, Volume 1, Basics, Wiley-Interscience, 1986.

[28] D. Bellan, S. Pignari, “A Probabilistic model for the response of an electrically short two-conductor transmission line driven by a random plane wave field”, IEEE Trans. on Electromagn. Compat., vol. 43, pp. 130-139, May 2001.

[29] R. Holland, R . H. St. John, “Statistical response of EM-driven cables inside an overmoded enclosure”, IEEE Trans.on Electromagn. Compat., vol.40, No. 1, Nov. 1998, pp.311-324.

[30] V. Rannou, F. Brouaye, M. Hélier, W. Tabbara, “Coupling of the field radiated by a mobile phone to a transmission line: a simple statistical and probabilistic approach”, in Proc. 14th Int. Zurich Symp. on Electromag. Compat., Zurich, CH, Feb. 2001, pp. 130-139.

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[31] D. Bellan, S. Pignari, “Susceptibility analysis of wiring harness in a reverberation chamber environment”, Proc. Of the 2001 IEEE Int. Symp. on Electromag. Compat., August 13-17, 2001, Montreal, Canada, pp. 746-750.

[32] C. R. Paul, Analysis of multiconductor transmission lines, Wiley series in microwave and optical engineering, New York, 1994.

[33] L. Musso, B. Demoulin, F. Canavero, V. Berat, “Susceptibility of a Transmission Line in two Reverberation Chambers”, Proceedings of the 2001 Reverberation Chambers, Anechoic Chambers and OATS Users Meeting, Seattle, WA (USA), June 4-6, 2001.

[34] S. Salio, F. Canavero, J. Lefèbvre and W. Tabbara, “Statistical description of random propagation on random bundles of wires”, in Proc. 13th Int. Zurich Symp. on Electromag. Compat. , Zurich, CH, Feb. 1999, pp. 499-504.

[35] G. Koepke, D. Hill, J. Ladbury, “Directivity of the test device in EMC measurements”, Proceedings of 2000 IEEE International Symposium on Electromagnetic Compatibility, August 21-25, 2001, Washington DC, USA, pp. 535-539.

[36] M. Bäckström, J. Lorén, G. Eriksson and H.J. Åsander, “Microwave coupling into a generic object. Properties of measured angular receiving pattern and its significance for testing”, Proc. Of the 2001 IEEE Int. Symp. on Electromag. Compat., August 13-17, 2001, Montreal, Canada, pp. 1227-1232.

[37] G .J. Freyer, M. G. Bäckström, “Impact of equipment response characteristics on absorber lined and reverberation chamber test results”, Proceedings of EMC Europe 2002, September 9-13, 2002, Sorrento, Italy, pp. 51-55.

[38] T. H. Lehman, G. J. Freyer, “Characterization of the maximum test level in a revereration chamber“, Proc. Of the 1997 IEEE Int. Symp. on Electromag. Compat., 1997, Austin, USA, pp. 44-47.

[39] J. M. Ladbury, "Monte Carlo simulation of reverberation chambers", National Institute of Standards and Technology, Boulder, CO, Internal Note, Oct. 1999.

[40] E. D. Cashwell and C. J. Everett, Monte Carlo Method for random walk problems, London – New York – Paris – Los Angeles: Pergamon Press, 1959.

[41] H. B. Dwight, Tables of integrals and other mathematical data, Fourth Edition, Mac Millan Publishing Co., Inc., Nez York, 1961.

91

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 2χ 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,

Annex A – Probabilistic and statistical tools

92

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 recP is between 1recP and 2recP ? 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 dxx + , for each real value x . For instance, the power recP 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:

( )

<

≥

−=

0,0

0,2

exp2

122

rec

recrec

recP

P

PP

Pfrec σσ (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.

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

f(x)

Exponential distribution - probability density function

Figure A.1 Probability density function for the Exponential distribution, σ =1

The probability that, for a particular stirrer position, recP is between 1recP and

2recP is given by the integral:


93

( ) rec

P

P

recrec

P

PrecPrecrecrec dPPdPPfPPP

rec

rec

rec

recrec ∫∫

−==≤≤2

1

2

12221

2exp

21P

σσ

(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:

∫+∞

∞−

⋅= dxxfxx x )()(E (A.3)

( )∫+∞

∞−

⋅−= dxxfxxx )()(E)var( 2 (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 recP in (A.1), we found the received power expected or mean

value and variance, given respectively by:

222 2

2exp

21)(E σ

σσ=

−= ∫+∞

∞−rec

recrec dP

PP (A.5)

422

22 42

exp2

1)2()var( σσσ

σ =

−−= ∫+∞

∞−rec

recrec dP

PxP (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


94

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 )(xFx of a random

variable, which is the integral of the above defined probability density function )(xf x , according to:

ξξξ∫∞−

=x

x dfxF )()( (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:

( ) )(x xFdxxfxP x

x

x ==≤ ∫∞−

(A.8)

If we dispose of two random variables x1 and x2, both characterised by the same cumulative density function )(xFx , 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:

[ ]22121 )(xx)x,max(x xFxPxPxP x=≤⋅≤=≤ (A.9)

If we generalize (A.9) to n random variables, the probability that the maximum of n random variables nx is less than a value x is given by:

[ ]nxn xFxP )(x =≤ (A.10)

In other words, the cumulative density function )(x xFn

of the maximum value of

n identical independent random variables x is:

[ ]nx xFxFn

)()(x = (A.11)

where )(xFx is the cumulative density function of one random variable x. The

probability density function )(x xfn

of the maximum of n random variables x, can


95

be obtained by deriving (A.11):

[ ] )()()( 1x xfxFnxf x

nxn

⋅⋅= − (A.12)

where )(xf 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 ∑ =⋅= ni inrec PnP 1/1 .

The interval estimate is an interval of the form ul ≤≤ µ , with the associated probability:

αµ −=≤≤ 1ulP (A.13)

In summary, when a point estimate nrecP 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 α−1 (for instance, α =0.05 for 95% probability). It is important to underline that the real mean


96

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

2σ , and if the sample mean is nx , the confidence interval for the unknown mean

value µ is given by:

ασσαα −=

+<<− −− 1xxP 2/12/1 n

zµn

z nn (A.14)

Where 2/1 α−z is the inverse of the standardized1 Normal cumulative density

function, evaluated in 2/1 α− . 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 2σ 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]:

ααα −=

+<<− −−−− 1xxP 2/1,12/1,1 n

stµ

ns

t nnn

nnn (A.15)

1 The standardized Normal distribution has mean value µ =0 and variance σ =1


97

where 2/1,1 α−−nt is the inverse of the Student’s T cumulative density function

with n-1 degrees of freedom evaluated in 2/1 α− , and sn is the estimated sample standard deviation, computed as:

[ ]∑=

−−

=n

inin xx

ns

1

2

11 (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:

αχ

σχ αα

−=

−⋅≤≤

−⋅

−−−

111

21,2/

21,2/1 n

n

n

n nsnsP (A.17)

where 21,2/1 −− nαχ ( 2

1,2/ −nαχ ) is the inverse of the 2χ cumulative density function

with n-1 degrees of freedom evaluated in 2/1 α− ( 2/α ), and ns 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:

( )

( )

ααα

−=

++

−−

−⋅≤≤

++

−

−⋅ 1

21ln577.0

)2/1(1ln1M

21ln577.0

)2/(1ln1

11

nn

x

nn

xP

n

n

n

n

(A.18)


98

where M is the real unknown maximum value and nx 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 35010 ≤≤ n .

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 2χ 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 2χ goodness-of-fit computation, the sample is divided into k ordered sub-sets, called bins, and a test statistic is defined as:

∑=

−=

k

i i

iio E

EO

1

22 )(

χ (A.19)

where iO is the number of sample measurements contained in the bin i and iE is

the number of data coming from the supposed distribution contained in the bin i. The expected number of data iE is calculated by:

( ))()( lFuFnEi −⋅= (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


99

distribution, 2oχ follows approximately a 2χ distribution with 1−−mk 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 2χ 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 21,1

2−−−> mkpo χχ , where 2

1,1 −−− mkpχ is the

inverse of the 2χ cumulative density function with 1−−mk degrees of freedom evaluated in p−1 , 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 2χ 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 ),( ϕθΩ , the field polarization angle pθ , and the phase φ . Such parameters must follow the probabilistic distributions detailed in Table 3.1, reported below in Table A.1.


100

Table A.1 Random plane waves parameters probabilistic distributions

Plane wave parameter Probabilistic Distribution

Propagation direction: ( )ϕθ ,Ω Uniform: [ ]π4,0U

Polarisation: pθ Uniform: [ ]π2,0U

Phase : φ Uniform: [ ]π2,0U 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 )(xFx of the desired distribution (see [18]). It can be

shown that if u is a random number produced by an uniform generator, and if

)(1 xFx− is the inverse of the cumulative density function of the desired

distribution, then:

)(1 uFy x−= (A.21)

is a random number following the distribution given by )(xFx .

This technique can be directly applied to the generation of the polarization

angle pθ and of the phase angle φ . In the case of uniformly distributed angles

between 0 and π2 , the probability density function is given by:

ππ

20,21)( ≤≤= xxf x (A.22)

Thus pθ and φ can be generated, starting from a uniform generated number u and

according to (A.7) and (A.21), by:

uup

⋅=⋅=

πφπθ

22 (A.23)


101

Concerning the plane wave propagation angles, it can be shown that to have the solid angle ( )ϕθ ,Ω uniformly distributed over π4 , ϕ must be uniformly distributed over π2 , while θ must be distributed according to the density function [40]:

πθθθθ ≤≤= 0)sin(21)(f (A.24)

As a result, by applying (A.21) the propagation angles can be generated, starting from a uniform generated number u, by:

u

u⋅=

−= −

πϕθ

2)21(cos 1

(A.25)

102

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 z-component of the electric field in the origin resulting from an uniform linearly polarised electromagnetic plane wave is given by (3.8), that is:

4444 34444 214444 34444 21

r

iziz Em

iip

ii

Ee

iip

iiiiz EjEzEE

,,

sinsincoscossincosˆ,

ℑℜ

⋅⋅⋅+⋅⋅⋅=⋅= φθθφθθ (B.1)

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 piii θϕθ ,, and iφ are independent from

each other. Starting from the pdfs in Table B.1, we derive now the pdf of izEe ,ℜ in

(B.1). As a first step, we express izEe ,ℜ as:

444 3444 214434421

321321321

zw

yi

x

pi

viizEe θθφ sincoscosE0, ⋅⋅⋅=ℜ

(B.2)

Annex B – Computation details of the statistical properties of the field resulting from one random plane wave

103

Table B.1 Random plane waves parameters pdfs

Plane wave parameter Probability density function

Propagation direction: ( )ii ϕθ ,Ω

πϕπ

ϕ

πθθθ

ϕ

θ

20,21)(

0),sin(21)(

≤≤=

≤≤=

ii

iii

i

i

f

f

Polarisation: piθ πθ

πθθ 20,

21)( ≤≤= p

ip

ipi

f

Phase : iφ πφπ

φφ 20,21)( ≤≤= iii

f

Amplitude: iE ( )0E)( −= iiE EEfi

δ

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:

10,1

)(

1,1

1)(

1,1

1)(

2

2

2

≤≤−

=

≤−⋅

=

≤−⋅

=

yy

yyf

xx

xf

vv

vf

y

x

v

π

π

(B.3)

Given the pdfs in (B.3), and assuming independent random variables, according to [18] (section 6: Two random variables) the pdf of yxw ⋅= can be written as:

∫∫∞

∞−

∞

∞−

≤⋅=

≤≤=

−−=

=

1/,10,

,/1

11

1

1,1)(222 uwyxw

uyudu

uwu

uu

duuwuf

uwf xyw π

(B.4)

which results in:


104

1,21)( ≤= wwfw

(B.5)

Proceeding with the same method used in (B.4), the pdf of wvz ⋅= can be derived by using the pdfs in (B.3) and (B.4); the result is:

( ) 1,log11log1)( 2 ≤

−

−+= zzzzf z π

(B.6)

(B.6) is the probability density function for the real part of the electric field in (B.1) when considering 1E0 = . The corresponding cumulative density function (cdf)

can be found by integrating (B.6):

( ) [ ]

( ) [ ]

≤<

⋅−

−+⋅++

≤≤−

⋅+

−+⋅−−+

== ∫∞− 10,log11logarcsin1

21

01,log11logarcsin121

)()(2

2

zzzzzz

zzzzzzdxxfzF

z

zz

π

π

(B.7)

The pdf and the cdf of zEet iz ⋅=ℜ= 0, E , for any 0E , can be expressed in

function of (B.6) and (B.7) [18]:

1E

,E

logE

11logE

1)(00

2

00≤

−

−+

⋅=

ttttft π

(B.8)

≤<

⋅−

−+⋅+

+

≤≤−

⋅+

−+⋅−

−+

=

000

2

000

0

00

2

000

E0,E

logEE

11logEE

arcsin121

0E

,E

logEE

11logEE

arcsin121

)(

tttttt

t

ttttt

tFt

π

π

(B.9)


105

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

ReEz,i/E0

ReEz,i - pdf

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

ReEz,i/E0

ReEz,i - cdf

Figure B.1 pdf and cdf for izEe ,ℜ

The pdf and the cdf for izEe ,ℜ

, in (B.8) and (B.9) respectively, are shown in

Figure B.1. The mean value and variance of izEe ,ℜ

can be easily derived by the mean

values and variances of v, and w (see (B.2)). Mean values and variances of v

( 2, vv σµ ) and w ( 2, ww σµ ) can be directly computed by using the definitions in

Annex A, and the pdfs in (B.3) and (B.5):

3/1,0

2/1,02

2

==

==

ww

vv

σµ

σµ

(B.10)

Since v and w are independent, mean value and variance for izEe ,ℜ

in (B.2) can

be computed as:

[ ]6

EE

0E2

022222220

2

0

,

,

=⋅+⋅+⋅⋅=

=⋅⋅=

ℜ

ℜ

vwwvwvEe

wvEe

iz

iz

σµσµσσσ

µµµ

(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).

106

Annex C

Details of Monte Carlo mean values estimation

We consider the following equality:

22

1EE i

n

ii xnx ⋅=

∑=

(C.1)

where iii bjax ⋅+= 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 0EE == ii ba .

The left hand-side of (C.1) can be decomposed as follows:

434214434421

444 3444 21

C

1,

B

1,A

1

2

1

2

1,1,1

2

1

2

2

1

2

1

2

11

2

1

EEE

E

EEE

+

+

+=

=

+++=

=

+

=

⋅+=

∑∑∑∑

∑∑∑∑

∑∑∑∑∑

≠=

≠===

≠=

≠===

=====

n

kiki

kin

kiki

kin

ii

n

ii

n

kiki

kin

kiki

kin

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

bbaaba

bbaaba

babjax

(C.2)

In (C.2), A can be expressed as:

Annex C – Details of Monte Carlo mean values estimation

107

2

1

2

1

2

1

22

1

2

1

2 EEE)(EE in

ii

n

ii

n

iii

n

ii

n

ii xnxxbaba ⋅==

=

+=

+ ∑∑∑∑∑=====

(C.3)

If we make the hypothesis that 0EE == ii ba , we can write, for the terms B and

C of (C.2):

0EEE1,1,

==

∑∑≠=

≠=

n

kiki

kin

kiki

ki aaaa

(C.4)

0EEE1,1,

==

∑∑≠=

≠=

n

kiki

kin

kiki

ki bbbb

(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

ii

n

ii

n

ii bjax

111EE

(C.6)

under the hypothesis that iii bjax ⋅+= are independent identically distributed

complex random variables, 0EE == ii ba and 2varvar σ== ii ba . In this

case, according to [18], p. 140, for large values of n it can be shown that ∑ =ni ix1

follows a Rayleigh distribution, with mean value:

nx

n

ii 2

E1

πσ=

∑=

(C.7)

On the other hand, in the case where ia and ib are Normally distributed with zero

mean value, by the same argument reported above, it can be shown that:

2E π

σ=ix

(C.8)

Annex C – Details of Monte Carlo mean values estimation

108

Thus, in this case, from (C.7) and (C.8) is follows:

nxx i

n

ii EE

1=

∑=

(C.9)

109

Annex D

Measurement set-up

Renault chamber qualification measurements, proposed in Chapter 2, were performed with the following instrumentation.

Table D.1 Antennas

Model Frequency range Transmitting antenna

EMC test systems model n° 3144 serial no. 9906-1055

80MHz - 2 GHz

Receiving antenna

Amplifier Research AT1100 serial no. 11543

80 MHz - 1 GHz

Table D.2 Electric field probe

Model Specifications

Electric field probe Thomson E1602 N°5

EfARV e ⋅⋅⋅⋅πε= 02

Ω= 50R

00113.0=eA

cut-off frequency 600 MHz

Table D.3 Network analyser

Model Frequency range Vector network analyser HP 8753 C 300 KHz - 3 GHz S-parameters test set HP 85046 A 300 KHz - 3 GHz

Annex D – Measurement set-up

110

The measurement set-up is shown in Fig. D.1.

NetworkAnalyser Port 1Port 2

(d)

(a)(b)

(c) PinPrec

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 ijS ,

defined with respect to the incident and reflected waves ia and ib , at the reference

planes, by:

2221212

2121111aSaSb

aSaSb+=+=

(D.1)

The power recP , received at the Port 2 reference plane, can be expressed in terms

of the incident power on the transmitting antenna inP and of the scattering

parameter 21S , by:

221SPP inrec = (D.2)

The electric field measured by the electric probe, can be obtained, according to Table D.2, by:

Annex D – Measurement set-up

111

fARπε

VE=e ⋅⋅⋅02

(D.3)

where V is the voltage measured at the probe input, which can be expressed in terms of the received power recP , according to:

0RPV rec= (D.4)

where Ω= 500R is the input impedance of the network analyser. From (D.2)-

(D.4), the measured electric field is:

fRA

RPSE

e

in

0

021 2πε

= (D.5)

112

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

Annex E – Noise reduction

113

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.

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