F Sanchez

M Budinger
email: marc.budinger@insa-toulouse.fr

S Delbecq
email: scott.delbecq@safrangroup.com

I Hazyuk
email: ion.hazyuk@insa-toulouse.fr

MODELLING AND DESIGN APPROACHES FOR THE PRELIMINARY DESIGN OF POWER ELECTRONIC CONVERTERS

Keywords: Power converters, system engineering, surrogate models, model based design, multidisciplinary optimization

This paper presents different modelling approaches and design tools initially developed for the preliminary design of mechatronic system and applied here to power electronic converters. Different type of models, components level and system level, are needed and exemplify with a DC/DC converter. A dedicated framework is used to manipulate and associate the models. An optimization of the converter is finally realized.

INTRODUCTİON ON DESİGN TOOLS

The increase of computation capabilities of computers and the development of numerical tools boosted modelling & simulation (M&S) usage during product design process [START_REF]Systems Engineering Fundamentals[END_REF]. Similarities exist between mechatronic system design and power converter design, especially in early design stages [START_REF] Capella | Capella Modelling Tool[END_REF] [START_REF] Qamar | A Domain-Specific Language for Dependency Management in Model-Based Systems Engineering[END_REF]. Two levels of knowledge are necessary in order to achieve a mechatronic or power electronics system design [START_REF] Hehenberger | Advances in Model-Based Mechatronic Design[END_REF]. First, a component level knowledge which includes sizing laws, design drivers and technological limits of a component. Secondly, a system level knowledge which concerns sizing scenarios and requirements of the system and how to link them with component sizing. This paper illustrates how methodologies and models developed initially for the design of mechatronic systems can also be applied to the power electronic domain. In addition, different levels of knowledge and their uses are introduced in order to create an optimal sizing procedure. A DC/DC converter with filtering and dissipation components is used to illustrate the concepts introduced.

THE PRELİMİNARY DESİGN OF POWER CONVERTERS

SİZİNG PROBLEM OF POWER CONVERTER

A lot of new applications require the use of power converters to manage power sources, such as battery, supercapacitor, for embedded systems. The objectives of this case study are:

 The definition of the sizing procedure and the optimization problem of a DC/DC converter for an ultra-capacitor.  The sizing of the main components and the determination of optimal design variables such as the Pulse Wide Modulation (PWM) frequency.

ARCHİTECTURE, COMPONENTS

The studied DC/DC converter (Fig. 1 and Fig. 2) is composed by one film capacitor, two IGBT transistors, and one inductance. A heatsink is used to cool the IGBT modules.

Components design drivers

Inductance

The inductance have to store magnetic energy for the maximum current. This will have an influence on the section of iron and the value of the airgap.

For thermal aspects Joules and iron losses have to be evaulated and the winding hot spot temperature have to be estimated through a thermal model. Moreover, we will check that the size of the winding remains acceptable. Generally during the sizing the value of airgap is limited to ensure good precision for the magnetic analytical model. In this study, it is proposed to build a model which allows the estimation of the influences of airgap for widest range of variation [START_REF] Mühlethaler | Optimal Design of Inductive Components Based on Accurate Loss and Thermal Models[END_REF].

Capacitor

The capacitor has to limit the amplitude of the voltage ripple. For the thermal aspects, it is assumed that the losses are dissipated from the RMS current in the electrodes and the dielectric losses in the dielectric. A thermal model of the capacitor will be used here to estimate the hot spot temperature.

IGBT

In order to select the IGBT modules, the junction temperature will be estimated. This criteria requires to choose the size of the heatsink and to make compromise for the value of the switching frequency (filter / loss compromise).

Requirements

The DC/DC converter specifications used for numerical applications are the followings:

 DC bus voltage: E = 300 V  Ripple voltage max. of the DC bus: 1%  DC current of the load (supercapacitor): 140 A  Current ripple max. of the load: 35 %  DC voltage of the load: 125 V

SYSTEM LEVEL AND COMPONENTS LEVEL MODELS

Models support Systems Engineering processes. Models also facilitate capitalization especially if their built or used are generic. Different types of models can be used such as distributed parameters (3D FEM, 3D CFD [START_REF] Sanchez | Surrogate modeling technique for the conceptual and preliminary design of embedded systems and components[END_REF]) for local level, lumped parameters (1D/OD, ODE/ADE) [START_REF] Maré | Simulation Modelling Practice and Theory Requirement-based system-level simulation of mechanical transmissions with special consideration of friction , backlash and preload[END_REF] and state machine for global level. In order to capture and structure knowledge, Knowledge-Based Engineering [START_REF] Pfennig | A knowledgebased approach for design and modelling of high lift actuation systems[END_REF] can be used. This tends to use component database and model libraries [START_REF] Pfennig | A knowledgebased approach for design and modelling of high lift actuation systems[END_REF].

SYSTEM LEVEL MODELS

Knowledge at system level enables the evaluation of sizing variables for system' components for several sizing scenario. The simple layout of the considered converter allows to express these models with algebraic equations. Table I represents the current and voltage filtering equations. Table II shows the variables required for the thermal evaluation of the components. For more complex architectures, these models can be obtained using regression process on PWM circuit simulations [START_REF] Giraud | Methodologies for the optimal design of the Integrated Modular Power Electronics Cabinet[END_REF] or using directly system models [START_REF] Delbecq | A framework for the conceptual and preliminary design of embedded mechatronic systems[END_REF].

I Current and voltage filtering equations

Component

RMS current

Inductor In the design of mechatronic systems scaling laws allow estimation models [START_REF] Budinger | Estimation models for the preliminary design of electromechanical actuators[END_REF] to be obtained from a single reference component by using three main modeling assumptions: a. Material similarity: all material and physical properties are assumed to be identical to those of the component as the reference; b. Geometric similarity: the ratio of all the lengths of the component under consideration to all the lengths of the reference component is constant; c. Uniqueness of the design driver: only one main dominant physical phenomenon drives the evolution of the secondary characteristic 𝑦. The mathematical form of scaling laws is a power law:

∆𝐼 𝐿 = 𝐸(1 -𝛼). 𝛼 𝐿 𝑓 Capacitor ∆𝑉 = 𝐼 𝑚𝑎𝑥 (1-∝) ∝
𝑦 = 𝑘𝑋 𝑎 (1)
With y is the secondary characteristic to be estimated, 𝑋 is a definition parameter of the component, and k and a are numerical constants calculated from a reference component.

Their simple form makes them easy to manipulate and customize as they require only one reference to determine the coefficients k and a. They have a monotonous progression valid over a wide range of sizes (several orders of magnitude) which avoids the risk of possible mathematical aberrations of metamodels used outside their construction bounds [START_REF] Budinger | Scaling-law-based metamodels for the sizing of mechatronic systems[END_REF].

Scaling laws for ferrite cores

Geometrical similarities, i.e. conservation of geometrical ratio during scale changes, enable to obtain interesting characteristics by reducing considerably the number of inputs [START_REF] Forest | Design and comparison of inductors and intercell transformers for filtering of PWM inverter output[END_REF]. With the notation proposed by M. Jufer [START_REF] Jufer | Design and Losses -Scaling Law Approach[END_REF] where 𝑥 * = 𝑥/𝑥 𝑟𝑒𝑓 is the scaling ratio of a given parameter.

Scaling laws for IGBT modules

The table IV introduces the different scaling laws used to estimate the IGBT characteristics where I express the current rating of the module.

IV : Scaling laws established for the IGBT modules

Scaling laws Reference

Definition parameter : Current Scaling laws have assets that make them attractive for the design of mechatronic systems. However they have some limitations. Although the similarity of the materials can be easily verified for a given technology of electronic components, the geometric similarity is not necessarily verified or sought. Surrogate modelling technique can be used to overcome these limitations [START_REF] Hazyuk | Scaling laws based metamodels for the selection of the cooling strategy of electromechanical actuators in the early design stages[END_REF]. Surrogate modelling technique consists in replacing high detail models (FEM, CAD models) by analytical models which are explicit and easily handle-able. It is proposed here to use a specific surrogate modelling technique, the VPLM methodology [START_REF] Sanchez | Dimensional analysis and surrogate models for the thermal modeling of Multiphysics systems[END_REF], to build the magnetic and thermal models of inductance. Similar thermal models of capacitor and heatsink can be found in another Electrimacs contribution submitted to the special session: Thermal Management of power electronics and electrical machines.

I 80 A Maximum voltage V max * = 1 900 V Voltage drop V 0 * =1 1 V Dynamic resistance R 0 * =I *-1

Surrogate thermal model for inductance

The thermal resistance of inductance component is required to estimate the hot spot temperature of its winding. Inductance using ferrites pot cores is considered for this study. Fig. 3 introduces the simplified model of the inductance considering the following assumptions:  Geometrical simplification: axi-symmetry hypothesis.  Ferrites cores follow geometrical similarities except for the airgap.  Material simplification: the winding is composed by copper wires and potting resin. The winding coefficient drives the thermal conductivity of the equivalent material. The equivalent thermal conductivity is calculated using the formula given in [START_REF] Laïd | Thermal Model With Winding Homogenization and FIT Discretization for Stator Slot[END_REF].  Natural convection cooling, conductive and radiative heat transfers are considered. According to the VPLM methodology [START_REF] Sanchez | Dimensional analysis and surrogate models for the thermal modeling of Multiphysics systems[END_REF], the thermal resistance of the inductance is given by [START_REF] Capella | Capella Modelling Tool[END_REF] and has a maximum relative error less than 8% with the FEM simulations (Fig. 4). is the radiative dimensionless number.

Surrogate magnetic model for inductance

The objective is to represent the magnetic circuit of an inductance. The model has to take in account the magnetic field in the airgap for small and big values, and using finite elements simulations the proportion of the magnetic field outside the magnetic circuit is also considered. These effects are not well modelled by classical analytic models.

The inductance is represented by the reluctance of the magnetic circuit 𝑅 𝐿 :

𝐿 = 𝑛²/𝑅 𝐿 ( 3
)
Where 𝐿 is the inductance, 𝑛 the number of turns.

Two assumptions are considered for the study:

 The reluctance of the ferrite magnetic circuit is considered during the FEM simulation but neglected for the final expression.

 The saturation effects are not considered.

According to the VPLM methodology [START_REF] Sanchez | Dimensional analysis and surrogate models for the thermal modeling of Multiphysics systems[END_REF], the reluctance is given by ( 4) and the model has a maximum relative error less than 1% with the FEM model (Fig. 4).

Where 𝜋 𝑒 = 𝑒/𝐷.

MODELS MANİPULATİON AND DESİGN PROCESSUS

ASSOCİATİON OF ACAUSAL MODELS

In order to facilitate capitalization in models libraries [START_REF] Mattsson | Physical system modeling with Modelica[END_REF] and reuse of the models for different applications, acausal models are generally preferred [START_REF] Capella | Capella Modelling Tool[END_REF], [START_REF] Allain | Automatic modelling of Power Electronic Converter, Average model construction and Modelica model generation[END_REF].

FROM ACAUSAL TO CAUSAL MODELS

Moving from acausal non-oriented model and unusable for sizing or optimization tasks to causal model which has inputs and outputs, requires to solve several problems to be sure to obtain an explicit and ''balanced'' code (no complex numerical solving and equality between number of inputs and number of outputs).

Three types of problem must be solved:

Set of equations underconstrained

The set of equations introduces too much variables to be determined. It is required assuming ones of them known. During optimization procedures, it will be possible to includes them into the set of optimization variables. For the DC/DC converter, the explicit evaluation of diode and IGBT losses and inductance and capacitance value require assuming the switching frequency known.

Set of equations overcontrained

A set of equations is over-constraint if the number of equations is greater than the number of variable to be determined. This sizing problem appears for the selection of a component which has to fulfill two constraints. For example, the capacitor has to ensure a filtering function (limitation of voltage ripple) without overheating. During optimization procedure, it is possible to adapt the set of equations with an oversizing coefficient used for one equation and the other equation is added to the constraints of the optimization problem.

Algebraic loop

This problem is classical for coupling equations like: 𝑥 = 𝑓(𝑦) and 𝑦 = 𝑔(𝑥). Solving these equations requires the use of numerical solver inside the optimization procedure.

This problem occurs when a criterion used to select a component depends on the internal characteristics of this component. For example, the selection of IGBT module with thermal criterion: the silicon temperature of the transistor depends on its thermal resistance junction to case. It is possible to transform the problem in an explicit way by breaking the algebraic loop with the use of an oversizing coefficient. It is used on a simplified selection criterion independent of internal characteristics of the IGBT. The silicon temperature will be verified by the optimization algorithm trough the problem constraints.

CAUSAL MODELLİNG OF THE SİZİNG PROCEDURE

Acausal models presented before can generate causal models for the problem introduced before thanks to ordering and matching algorithms [START_REF] Reysset | Computer-aided definition of sizing procedures and optimization problems of mechatronic systems[END_REF]. Causal model were generated component by component and they are associated in a causal modelling [START_REF] Allain | Capitalisation et traitement des modèles pour la conception en génie électrique[END_REF] environment win order to represent system level sizing procedure. Their connections uses N 2 diagram [START_REF]NASA Systems Engineering Handbook[END_REF] topology. This way, a sizing procedure and the related optimization problem architecture can easily be implemented. In addition to component level knowledge, the choices made during sizing must be validated by operational scenarios. These sizing scenarios can be represented by different model types such as algebraic equations, time response simulations [START_REF] Roos | Optimal selection of motor and gearhead in mechatronic applications[END_REF] harmonic response simulations [START_REF] Malmquist | Holistic design methodology for mechatronic systems[END_REF] or surrogate models [START_REF] Budinger | Optimal preliminary design of electromechanical actuators[END_REF] like response surfaces for instance.

Usually, these kinds of model have imposed inputs and outputs. Therefore, information flows are causal at system level design and causal model must be used.

OPTİMİZATİON AND RESULTS

OPENMDAO FRAMEWORK

In order to optimize the system design, it is possible to associate optimization algorithms to the sizing procedure [START_REF] Budinger | Optimal preliminary design of electromechanical actuators[END_REF]. Multidisciplinary Design Optimization (MDO) is relevant for mechatronic systems design [START_REF] Friedl | A design optimization framework for multidisciplinary mechatronic systems[END_REF] and by the way for converters design. MDO frameworks such as openMDAO [START_REF] Gray | OpenMDAO: An Open Source Framework for Multidisciplinary Analysis and Optimization[END_REF] enable building global system models by associating elementary models and offer optimization algorithms libraries. Due to the number of requirements and design drivers, MDO of power converters could require constrained (linear or non-linear) optimization.

The preliminary design framework used in this paper aims to satisfy in the best manner the needs and problems while designing a system by using the interesting concepts introduced earlier. More details about this framework are described in [START_REF] Delbecq | A framework for the conceptual and preliminary design of embedded mechatronic systems[END_REF].

OPTİMİZATİON

The optimization procedure conducts on the problem defined in 2 gives a minimum mass of 9.78 kg. The final PWM frequency, around 10 kHz, is a compromise between cooling mass (heat sink) and filtering mass (inductor and capacitor). The airgap inductor, 7.2 mm, find thanks to FEM surrogate models, is bigger than that would be find with an over simple analytical model.

CONCLUSİON

In this paper several design approaches to generate models dedicated to the preliminary design of power electronic converters were introduced. The framework presented enables to make connections between all these models and to conduct optimization tasks.

The preliminary sizing of a DC/DC converter has been done using optimization and different types of models such as scaling laws and surrogate models. These models permit to evaluate components characteristics such as hot spot temperature, air gap and mass. Optimization enable to realize the compromise between cooling mass and filtering mass.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Power converter prototype (ARCEL)

Fig. 3 :

 3 Fig. 3 : Simplified model of inductance Two types of losses are considered for the thermal modelling: Joules losses dissipated in the equivalent material for the winding and cores losses dissipated in ferrite cores.

  in term of heat flux) which defined the natural convection phenomena;

  -0.226 log(𝜋 𝑒 )-0.0355 log(𝜋 𝑒 )

Fig. 4 :

 4 Fig. 4 : COMSOL simulation results for inductance models (right: thermal, left: magnetic)


Dr Marlène Rio
email: marlene.rio@nck.aphp.fr

Valérie Malan

Sarah Boissel

Annick Toutain

Ghislaine Royer

Stéphanie Gobin

Nicole Morichon-Delvallez

Catherine Turleau

Jean-Paul Bonnefont

Arnold Munnich

Marle ` Ne Rio

Vale ´rie Malan

Ste ´phanie Gobin

Michel Vekemans

Laurence Colleaux

Familial interstitial Xq27.3q28 duplication encompassing the FMR1 gene but not the MECP2 gene causes a new syndromic mental retardation condition

Keywords: X-linked mental retardation, chromosome duplication, FMR1

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

X-linked mental retardation (XLMR) is a heterogeneous group of syndromic and nonsyndromic disorders. Although the most common cause of XLMR is an expansion of an unstable trinucleotide repeat in the 5¢-UTR of the FMR1 gene that causes the fragile X syndrome, [START_REF] Verkerk | Identification of a gene (FMR-1) containing a CGG repeat coincident with a breakpoint cluster region exhibiting length variation in fragile X syndrome[END_REF] mutations in more than 80 other genes have also been involved in XLMR. [START_REF] Chiurazzi | XLMR genes: update 2007[END_REF] Yet, linkage data and mutation screening of known XLMR genes strongly suggest that many more are to be discovered. [START_REF] Ropers | X-linked mental retardation: many genes for a complex disorder[END_REF] Moreover, recent data suggest that novel mechanisms likely account for unknown genetic defects. This is why the search for novel XLMR mutations was recently shifted toward a high-throughput sequence analysis of brain-expressed X-chromosomal miRNA genes, [START_REF] Chen | Mutation screening of brain-expressed X-chromosomal miRNA genes in 464 patients with nonsyndromic X-linked mental retardation[END_REF] or all genes on the X chromosome, [START_REF] Raymond | The genetics of mental retardation[END_REF] as well as screening of submicroscopic imbalances by array-comparative genome hybridization (array-CGH). [START_REF] Bauters | X chromosome array-CGH for the identification of novel X-linked mental retardation genes[END_REF][START_REF] Froyen | Detection of genomic copy number changes in patients with idiopathic mental retardation by high-resolution X-array-CGH: important role for increased gene dosage of XLMR genes[END_REF][START_REF] Madrigal | X-chromosome tiling path array detection of copy number variants in patients with chromosome X-linked mental retardation[END_REF] In this study, we report on the molecular and clinical characterization of a small interstitial Xq27.3q28 duplication encompassing the FMR1 gene in a large X-linked syndromic mental retardation family.

MATERIALS AND METHODS Patients

Patients were referred to the Genetic Department of the Necker-Enfants Malades hospital for investigating their developmental delay. Blood samples were obtained from the probands and their parents after obtaining informed consent, and genomic DNA was isolated from blood leukocytes using the Nucleon kit (Amersham, UK) according to the manufacturer's instructions.

Genotyping

Fluorescent genotyping was carried out as previously described. [START_REF] Colleaux | Fluorescence genotyping for screening cryptic telomeric rearrangements[END_REF] Polymerase chain reaction (PCR) amplification of genomic DNA from parents and patients (100 ng) was undertaken separately. PCR products were then pooled according to their size and labeling, and were analyzed on a capillary electrophoresis system (ABI 3130 Applied Biosystems), and alleles were identified using GeneScan and Genotyper softwares.

Standard and molecular karyotyping

Standard karyotyping using GTG and RHG banding analysis and the highresolution R-banding technique were performed on patients and parental peripheral blood lymphocytes according to standard procedures.

Array CGH analysis was performed using the CytochipTM V2 Microarray (BlueGnome, AmpliTech) with a resolution of 1 Mb according to the manufacturer's instructions.

Fluorescence in situ hybridization

BACs were selected from several databases (UCSC, University of California, Santa Cruz http://genome.ucsc.edu/ and NCBI, the National Centre for Biotechnology Information http://www.ncbi.nlm.nih.gov/). FISH experiments were performed on chromosome preparations as described previously. [START_REF] Romana | In situ hybridization to interphase nuclei in acute leukemia[END_REF] Metaphase spreads were prepared from patient-derived lymphocytes using standard procedures. The human genomic clones from RPCI BAC and PAC libraries were obtained from BACPAC resources (http://bacpac.chori.org/).

X-inactivation analysis

X-chromosome-inactivation patterns in carrier females were performed on genomic DNA extracted from white blood cells at the FMR1 locus as previously described. [START_REF] Carrel | an assay for X inactivation based on differential methylation at the fragile X locus, FMR 1[END_REF]

RESULTS

Clinical presentation

We had the opportunity to ascertain a French family in which three male members presented with similar clinical findings of mild mental retardation, hypogonadism and short stature (Figures 12and Table 1). Individual III.1 was born at 35 weeks of gestation after a normal pregnancy and delivery. Birth weight was 1500 g (o 5th percentile), birth length was 43 cm (10th percentile) and head circumference was 30 cm (10th percentile). The neonatal period was complicated by hypotonia, failure to thrive and regurgitations, which resolved by 3 months of age. He walked alone at 16 months, was toilet trained at 2 years and had no speech delay. At 5 years of age, poor motor performances were noted. He had mild learning difficulties and left school at 12 years. He was noted to have undescended testis in childhood and underwent bilateral orchidopexy at 7 years. Subsequently, primary hypogonadism was diagnosed and testosterone therapy was prescribed. On examination at 35 years, his height was 149 cm (À4 SD), whereas his predicted final height was 168 cm. His weight was 46 kg (+1 SD) and his head circumference was 54.5 cm (À1 SD). He had small hands and feet, and showed mildly dysmorphic facial features with deep-set eyes, a bulbous nasal tip and thin lips (Figure 2). He also had a high-pitched voice, sparse body hair, abdominal adiposity and bilateral small testes ascribed to primary testicular failure confirmed by a subsequent endocrine investigation (Table 1). He was able to read and write. He was highly social and communicated verbally with good sentence structure. He had no abnormal behavior. He was unemployed, lived semi-independently, was independent for tasks of daily life but needed help to prepare food and he was unable to look after his financial affairs. No abnormalities were noted on brain CT-scan and brain magnetic-resonance imaging performed at 35 years. Complete skeletal radiographs did not show any abnormalities. Blood karyotype was normal. Fragile X was ruled out with the detection of one trinucleotide repeat expansion of normal size (44) and no abnormal methylation.

His maternal uncle (II.3), was born at term with a low birth weight (2400 g, 5th percentile). His clinical features were similar to those of his nephew, as shown in Table 1 andFigure 2. He had mild learning difficulties, was able to read and write and to communicate fairly well. He worked 12 years as a worker in a factory. When examined at 59 years, his height was 150 cm (À4 SD), his weight was 63 kg (+3 SD) and his head circumference was normal (56 cm). Neurological examination showed mildly increased reflexes. He had a high-pitched voice, sparse body hair, overweight with truncal obesity (BMI 28), gynecomastia and small testis. He also had small hands and feet and mildly dysmorphic features (Figure 2).

Individual III.4 was born at 38 weeks with intra-uterine growth retardation (weight 2400 g (5th percentile), length 43 cm (o5th percentile) and head circumference 32 cm (5th percentile)). His clinical features were similar to those of his cousin and his uncle (Table 1 and Figure 2). At 7 years of age, he had orchidopexy for undescended left testis. He was in good health as a child, but was small, with height at À4 SD (1 m at 7 years) and weight at 16.4 kg (À2 SD). Endocrine testing at this time revealed a markedly delayed bone age (2½ years) but no growth hormone deficiency. He was said to have idiopathic short stature, but was nevertheless given Growth Hormone. Subsequently, he was found to have testosterone deficiency because of primary testicular failure and received testosterone treatment. Early psychomotor development was normal but he always had poor motor performances, was slow in his movements, ran with difficulty and was described as having amyotrophy of the lower legs. He had mild learning difficulties at school, was able to read and write and to communicate, but was unable to work in adulthood. At the age of 29 years, his height was 157.5 cm (À2.5 SD), his weight was 81.2 kg (+2.5 SD) and he had a normal head circumference (55 cm). He had small hands and feet, truncal obesity (BMI 32) and mildly dysmorphic facial features (Figure 2). He also had a high-pitched voice and bilateral small testis. He was unable to look after his financial affairs and to live independently, but he successfully passed his driving test. Mutation screening in the genes SOX3, PHF6 and CUL4B was negative.

Obligate carrier females (individuals I.1, II.1, II.2) had neither abnormal findings nor learning difficulties, but all of them presented with a short stature (148, 148 and 145 cm, respectively) and early menopause between 39 and 40 years of age.

Molecular analyses

Because the pedigree of the family was consistent with X-linked recessive inheritance, linkage analysis was performed on three affected males and obligate carrier females. A panel of 25 polymorphic microsatellite markers spanning the X chromosome was genotyped and showed putative linkage to a 50-Mb-long region between DXS1106 (in Xq22.2) and DXS8091 (Figure 1).

To rule out the hypothesis of a chromosomal rearrangement, we performed array-CGH on DNA from patient III.1 using the Cytochip V2 Microarray (BlueGnome, AmpliTech). We identified a small duplication of the long arm of chromosome X. A total of 15 BACs (RP3-433G13, RP6-224C24, RP11-37P24, RP11-161L9, RP11-75A14, RP11-64L19, RP11-242E3, RP5-937E21, RP13-406D21, RP11-413L16, RP5-892C22, RP11-45D17, RP11-329E24, RP11-274C1 and RP1-14D6) spanning a region of 5.1 Mb from Mb positions 146 479 822 (Xq27.3) to 151 603 959 (Xq28) had an enhanced log2 intensity ratio, whereas both RP11-226B15 and RP11-76K17 clones had a normal ratio. Metaphase and interphase FISH analyses on leukocytes confirmed the presence of the intrachromosomal duplication. The duplication segregated with the disease in the family, was identified in obligate carriers and was also identified in individuals III.3 and III.5. This duplication was not visible on routine chromosome analysis, even when the karyotype was reexamined with particular attention to this region. In addition, no other significant chromosomal imbalance was detected.

Finally, we investigated the X-inactivation profile in all female carriers. Three out of five (I.1, III.3 and III.5) showed a completely skewed inactivation. Moreover, methyl-sensitive PCR analysis indicated that 100% of their cells had inactivated the duplicated X chromosome. The two remaining carriers (II.1 and II.2) showed a random X-inactivation profile (60/40 and 70/30, respectively).

DISCUSSION

Partial duplications of the long arm of the X chromosome are rare and have been mainly described in males with severe mental retardation/ developmental delay. [START_REF] Novelli | Disomy of distal Xq in males: case report and overview[END_REF][START_REF] Cheng | Xq chromosome duplication in males: clinical, cytogenetic and array CGH characterization of a new case and review[END_REF] Usually, they are inherited from heterozygous mothers in whom preferential inactivation of the dup(X) chromosome leads to a skewed X-inactivation pattern and results in a normal or mildly abnormal phenotype. [START_REF] Heard | Recent advances in X-chromosome inactivation[END_REF] By contrast, structural X duplication in XY males results in functional disomy, and excess gene dosage leads to an abnormal phenotype. Until recently, most cases of functional disomy have been detected using standard cytogenetic methods because of intrachromosomal duplications, an unbalanced X/Y, X autosome translocation or X pericentric inversion. The availability of the array-comparative genomic hybridization (array-CGH) technique has tremendously increased the ability to detect cryptic interstitial chromosome X imbalances, as illustrated by duplications of the MECP2 gene causing severe mental retardation and progressive neurological symptoms in males. [START_REF] Van Esch | Duplication of the MECP2 region is a frequent cause of severe mental retardation and progressive neurological symptoms in males[END_REF] In this study, we report an Xq27.3q28 5.1 Mb-long interstitial duplication segregating in a large family with syndromic mental retardation. The three affected males present with mild mental retardation, short stature, bilateral testicular ectopia and hypogonadism, as well as phenotypic features likely caused by testosterone deficiency (that is, gynecomastia, decreased pilosity on the body and high-pitched voice). Hypogonadism, one of the striking clinical features observed in our patients, is peripheral in origin, with elevated gonadotrophins and low testosterone levels. Hypogonadism has been described in several known syndromic forms of XLMR, especially in the Borjeson-Forssman-Lehmann syndrome [START_REF] Turner | The clinical picture of the Bo ¨rjeson-Forssman-Lehmann syndrome in males and heterozygous females with PHF6 mutations[END_REF] (MIM:301900), Cabezas syndrome [START_REF] Tarpey | Mutations in CUL4B, which encodes a ubiquitin E3 ligase subunit, cause an X-linked mental retardation syndrome associated with aggressive outbursts, seizures, relative macrocephaly, central obesity, hypogonadism, pes cavus, and tremor[END_REF] (MIM:300354) and the Wilson Turner syndrome [START_REF] Wilson | New X-linked syndrome of mental retardation, gynecomastia, and obesity is linked to DXS255[END_REF] (MIM:309585), which have been mapped, respectively, to Xq26.3, Xq23 and Xp21.1-q22. However, our patients do not fulfill the clinical criteria of any of these conditions. First, none of them has the large ears, fleshy earlobes and small penis usually observed in the Borjeson-Forssman-Lehmann syndrome. Second, the occurrence of short stature with small testes, obesity and normal head circumference raises the issue of CUL4B mutations. Yet, neither tremor, aggressive outbursts nor seizures was reported in our family. Third, the Wilson Turner syndrome could also have been discussed as Wilson Turner male patients often present with mild-to-moderate mental retardation, gynecomastia, small or undescended testes and obesity. However, endocrinological investigations in these patients suggest secondary rather than primary gonadal insufficiency.

Largely, 20 male patients with interstitial duplications of the long arm of chromosome X have been reported in literature. [START_REF] Steinbach | Tandem duplication dup(X)(q13q22) in a male proband inherited from the mother showing mosaicism of X-inactivation[END_REF][START_REF] Cremers | An interstitial duplication of the X chromosome in a male allows physical fine mapping of probes from the Xq13q22 region[END_REF][START_REF] Vejerslev | Inherited tandem duplication dup(X) (q131-q212) in a male proband[END_REF][START_REF] Schwartz | Inherited X-chromosome inverted tandem duplication in a male traced to a grandparental mitotic error[END_REF][START_REF] Thode | A new syndrome with mental retardation, short stature and an Xq duplication[END_REF][START_REF] Yokoyama | Growth hormone deficiency and empty sella syndrome in a boy with dup(X) (q13.3-q21.2)[END_REF][START_REF] Apacik | Two brothers with multiple congenital anomalies and mental retardation due to disomy (X)(q12-4q13.3) inherited from the mother[END_REF][START_REF] Shapira | Inherited inverted duplication of X chromosome in a male: report of a patient and review of the literature[END_REF][START_REF] Monaghan | Prader-Willi-like syndrome in a patient with an Xq23q25 duplication[END_REF][START_REF] Hol | Identification and characterisation of an Xq26-q27duplication in a family with spina bifida and panhypopituitarism suggests the involvement of two distinct genes[END_REF][START_REF] Solomon | Increased gene dosage at Xq26-q27 is associated with X-Linked hypopituitarism[END_REF][START_REF] Solomon | Array comparative genomic hybridisation analysis of boys with X linked hypopituitarism identifies a 3.9 Mb duplicated critical region at Xq27 containing SOX3[END_REF][START_REF] Jehee | An Xq22.3 duplication detected by comparative genomic hybridization microarray (Array-CGH) defines a new locus (FGS5) for FG syndrome[END_REF][START_REF] Bedeschi | Association of syndromic mental retardation with an Xq12q13.1 duplication encompassing the oligophrenin 1 gene[END_REF][START_REF] Gabbett | Prader-Willi syndrome phenocopy due to duplication of Xq21.1-q21.31 with array CGH of the critical region[END_REF][START_REF] Tzschach | Characterization of interstitial Xp duplications in two families by tiling path array CGH[END_REF] Yet, interstitial duplication encompassing the Xq27.3-Xq28 region has been reported in only three patients. [START_REF] Gecz | Fibroblast growth factor homologous factor 2 (FHF2): gene structure, expression and mapping to the Bo ¨rjeson-Forssman-Lehmann syndrome region in Xq26 delineated by a duplication breakpoint in a BFLS-like patient[END_REF][START_REF] Lachlan | Functional disomy resulting from duplications of distal Xq in four unrelated patients[END_REF] The probands were males and duplication was detected using routine cytogenetic analysis. Unfortunately, a detailed clinical description of these patients is not included in the reports. The described clinical features are summarized in Table 2. Moderate-to-severe MR was observed in all cases. In addition, as observed in our patients, two cases were obese and had cryptorchidism, but no mention of primary testicular failure, one of the most relevant findings in our patients, was made. These three patients presented with microcephaly, a feature not observed in our family. Differences in size and position of individual duplications, disruption of various genes or position effects might explain this wide phenotypic variability.

Recently, an Xq28 duplication encompassing the MECP2 gene was found to cause severe mental retardation associated with progressive spasticity, axial and facial hypotonia, seizures and recurrent infections in males. [START_REF] Van Esch | Duplication of the MECP2 region is a frequent cause of severe mental retardation and progressive neurological symptoms in males[END_REF] By contrast, the duplicated segment in our patients encompasses 28 genes but does not include the MECP2 gene and causes a completely different clinical phenotype. Among the duplicated genes, several are of particular interest (Figure 3). Loss-offunction mutations of FMR1 and FMR2 genes are the cause of FRAXA syndrome (MIM:309550) and FRAXE syndrome (MIM:309548), respectively. The aneusomic region also contains the IDS gene responsible for the Hunter Syndrome (MIM309900) and the MTM1 gene responsible for X-linked myotubular myopathy (MIM310400). Whether functional disomy of any of these genes, disruption of genes at the break points of duplication or position effects are the major causative factor for mental retardation in our patients remains speculative. Yet, a twofold higher iduronate sulfatase enzyme activity was observed in peripheral leukocytes from individual III.4 when compared with control samples (data not shown), supporting the hypothesis of a functional disomy. As FMR1 status analysis in the probands revealed a stable expansion of the CGG repeat without abnormal methylation, we suggest that the duplication may also result in increased FMR1 transcript expression. Interestingly, several features such as short stature or small testis are contrary to those observed in the fragile X syndrome. [START_REF] Mattei | X-linked mental retardation with the fragile X. A study of 15 families[END_REF] Similarly, although fragile X cases have severe speech delay, our patients had good verbal performance. Interestingly, some of these findings are reminiscent of what is observed in transgenic mice. [START_REF] Peier | Over) correction of FMR1 deficiency with YAC transgenics: behavioral and physical features[END_REF] Indeed, although Fmr1 knockout mice showed increased locomotor activity suggesting increased exploratory behavior, FMR1 YAC transgenic mice showed low levels of locomotor activity suggesting decreased exploratory behavior.

Similarly, the presence of the FMR1 transgene not only rescues the macroorchidism phenotype in the knockout animals, but FMRP overexpression results in reduced testis weight in wild type.

It is worth noting that all female carriers fulfill the diagnostic criteria of premature ovarian failure (POF) defined by hypergonadotropic ovarian failure before 40 years of age. [START_REF] Coulam | Incidence of premature ovarian failure[END_REF][START_REF] Aittoma ¨ki | Mutation in the follicle-stimulating hormone receptor gene causes hereditary hypergonadotropic ovarian failure[END_REF][START_REF] Pasquale | Hypergonadotropic ovarian failure associated with an inherited mutation of human bone morphogenetic protein-15 (BMP15) gene[END_REF] This could be a direct consequence of chromosome rearrangement. X-chromosome rearrangements are known to lead to POF because of the presence of unsynapsed regions that might increase apoptosis and reduce the number of ovarian follicles. [START_REF] Toniolo | X-linked premature ovarian failure: a complex disease[END_REF] Alternatively, early menopause could also be a consequence of FMR1 overexpression. Indeed, normal carriers of FMR1 premutation are known to have a significantly higher frequency of POF. Moreover, expansion of the FMR1 CGG repeat represents the most common risk factors causing ovarian dysfunction. [START_REF] Mcconkie-Rosell | Evidence that methylation of the FMR-I locus is responsible for variable phenotypic expression of the fragile X syndrome[END_REF] The risk increases with increasing length of the repeat, suggesting a toxic effect of the premutated mRNA itself, which might sequester CGG binding proteins that are important for RNA processing. Similarly, FMR1 gene overexpression because of Xq27.3q28 duplication could result in ovarian dysfunction.

In conclusion, this study reports the first familial case of an Xq27.3q28 interstitial duplication encompassing FMR1 and FMR2 genes. This chromosomal imbalance is responsible for a novel clinical phenotype characterized by mild mental retardation, short stature and hypogonadism because of primary gonadal failure. On the basis of these findings, we suggest that this chromosomal imbalance is responsible for a novel clinically recognizable syndrome that may represent the fragile X syndrome 'contre-type' .

Figure 1

 1 Figure 1 Two-generation genealogy of the family. Black symbols indicate the affected individuals related through their clinically normal mothers. Genotyped markers are from centromere to telomere.

Figure 2

 2 Figure 2 Photographies and facial features of affected individuals. a, d, g: Individual III.1, b, e, h: individual II.3, c, f, i: individual III.4. Note truncal obesity and sparse body hair. Similar facial features include deep-set eyes, bulbous nasal tip and thin lips.

Figure 3

 3 Figure 3 Schematic representation of chromosome X indicating the duplicated region.

Table 1

 1 Clinical findings of affected individuals

		III.1	II.3	III.4
	Developmental history
	IUGR	+	+	+
	Failure to thrive	+	+	+
	Walking (months)	16	24	22
	Mild learning difficulties	+	+	+
	Good verbal skills	+	+	+
	Orchidopexy for	+	+	+
	undescended testis
	Age at examination	35 years	59 years	29 years
	Height in cm (SD)	149 (À4 SD) 150 (À4 SD) 157.5 (À2.5 SD)
	Head circumference	54	56	55
	in cm (SD)
	Small hands and feet	+	+	+
	Dysmorphic features	+	+	+
	Small testes	+	+	+
	Testicular volume	o 1 ml	o 0,5 ml	o4 ml
	Endocrine investigation
	FSH (1,5-14 mUI/ml)	65.2	16.4	21.7
	LH (1,2-10 mUI/ml)	23.68	11.94	3.7
	Testosterone (2,5-10 ng/ml)	2.67	0.54	1.30
	Inhibine B (80-400 pg/ml)	o15	o15	o10
	IGF1 (100-247 ug/l)	182	97	118

Table 2

 2 Summary of clinical details of patients previously reported with interstitial Xq duplications encompassing the Xq27.3-Xq28 region

		Lachlan et al 2004 36	Gecz et al 1999 35
		Patient 1	Patient 2
	Duplication Xq region	Xq27.2-Xq28	Xq27.1-Xq28	Xq26-Xq28
	Inheritance	De novo	Maternal	Maternal
	Duplication size	7.7-8.8 Mb	9.1-10.2 Mb
	Birth weight	2830 g (o9th)	2950 g (9th)	2300 g
	Developmental delay/MR	Severe	Moderate	Severe
	Growth failure	?	À	+
	Microcephaly	+	+	+
	Cryptorchidism	À	Unilateral	+
	Obesity	?	+	+

European Journal of Human Genetics

ACKNOWLEDGEMENTS

We express our deepest gratitude to the family for its participation in this study. This study was supported in part by the Centre National de la Recherche Scientifique (CNRS), the Agence Nationale de la Recherche (ANR), the Fondation LEJEUNE and by the Region Ile de France.

CONFLICT OF INTEREST

The authors declare no conflict of interest.


F Sanchez

M Budinger
email: marc.budinger@insa-toulouse.fr

I Hazyuk
email: ion.hazyuk@insa-toulouse.fr

DIMENSIONAL ANALYSIS AND SURROGATE MODELS FOR THERMAL MODELLING OF POWER ELECTRONIC COMPONENTS

Keywords: Thermal resistance, Capacitor, Heatsink, Heat transfer, Design, Power converter 1. INTRODUCTİON

This paper presents the use of a surrogate modelling technique, called VPLM (Variable Power Law Meta-model) to conduct the thermal modelling of power converter electronic components. The proposed methodology combines dimensional analysis and surrogate modelling technique to build analytical models from finite element simulations. The VPLM methodology is illustrated with the building of the thermal models of a film capacitor and a heatsink. Finally typical utilizations of the thermal models are presented.

Since lot of embedded power systems shift to electrical solutions and become more complex, the amount of electronic components used within is growing up. The thermal environment of such systems is generally critical for the electronic components. Thus thermal modelling of power electronic components is crucial for the design of more electrical systems. The design space exploration and the optimization of these systems needs component models adapted to the level of the considered design:  Design and optimization of a component for given specifications implies the determination of a large number of geometric dimensions and a set of possible physical properties. However, at this level the external geometrical dimensions vary relatively little and the main objective is the optimization of the internal shape of the component [START_REF] Cova | Microelectronics Reliability Thermal modeling of high frequency DC -DC switching modules : Electromagnetic and thermal simulation of magnetic components[END_REF];  Design and optimization of a global system (assembly of components from different fields) involves determining the best compromise between the various components [START_REF] Cova | Thermal modeling and design of power converters with tight thermal constraints[END_REF]. The external dimensions (geometric size) of each component can vary on a wide range while the technological choices and internal shapes of each component will follow some given rules determined at the previous design level. This article focuses on power electronic components models suitable at system design level where useful characteristics of the components should be estimated with a limited number of input variables but which may vary on wide ranges. Therefore, the following features are crucial for the models used at this stage: simpleto gain rapidity and easy handling, i.e. simple algebraic or ordinary differential equations that can be easily implemented in worksheets or optimization loops; explicitto enable analytic manipulations that provide direct access to the main integration parameters (mainly weight and geometrical parameters) disposing of very little input information (usually functional requirements); continuousat least in the range of conventional parameter variation for the given application, which is especially useful for the optimization; broad validity rangemodels have to be usable and reusable on large validity ranges of variables to allow the capitalization of knowledge and optimization without a priori. The approaches to generate such models may take different forms. A classical way of work is to solve analytically physical equations after simplification of the geometry or neglecting some physical phenomena. The validation of these analytical models is usually done on more detailed models (FEM and/or nodal network [START_REF] Boglietti | Evolution and Modern Approaches for Thermal Analysis of Electrical Machines[END_REF]) or on experimental or statistical data [START_REF] Verstraten | Modeling and design of geared DC motors for energy efficiency: Comparison between theory and experiments[END_REF]. This approach requires expertise and becomes difficult for multiphysics systems. A second approach uses empirical or semiempirical models that may use response surface approximations and occasionally can be supported by dimensional analysis [START_REF] Gogu | Dimensionality Reduction Approach for Response Surface Approximations: Application to Thermal Design[END_REF]. The last approach uses surrogate models or meta-models of computer experiments [START_REF] Forrester | Engineering Design via Surrogate Modelling[END_REF]. Surrogate modeling can be combined with dimensional analysis to build surrogate models that relies more on physical reasoning than on mathematical approaches [START_REF] Vignaux | Simplifying regression models using dimensional analysis[END_REF]. The VPLM (Variable Power Law Metamodel) methodology [START_REF] Sanchez | Dimensional analysis and surrogate models for the thermal modeling of Multiphysics systems[END_REF] uses this approach and the used mathematical form is well adapted especially for heat transfer problems. Section 2 recalls this methodology and the different steps for the construction of surrogate models adapted for sizing tasks. Section 3 illustrates the process with the generation of thermal models of some power converter electronic components. The last section shows examples of utilization of these thermal models.

THE VPLM METHODOLOGY

INTRODUCTİON TO DİMENSİONAL ANALYSİS

Dimensional analysis is a powerful way to reduce the dimension of the model and is mainly based on the Buckingham theorem [START_REF] Buckingham | On physically similar systems: illustration of the use of dimensional equations[END_REF]. A classical mathematical representation of a component model is:

𝑦 = 𝑓 (𝑑 1 , 𝑑 2 , … , 𝑝 1 , 𝑝 2 , … ⏟ 𝑛-1 ) (1)
where:

 𝑦 is the physical characteristic of a component which is useful for its selection in a system.

 𝑑 𝑖 are the geometrical dimensions of the component. These dimensions may vary on big intervals during global system design.

 𝑝 𝑖 are boundary conditions or material properties used in the design of the component.

Note that 𝑑 𝑖 and 𝑝 𝑖 are 𝑛 -1 independent variables.

Using dimensional analysis principle, equation (1) can be rewritten in the form:

𝜋 0 = 𝐹 (𝜋 1 , 𝜋 2 , … , 𝜋 𝑞-1 ⏟ 𝑞-1 ) (2)
where 𝜋 𝑖 are dimensionless variables, also called dimensionless numbers, and they are defined as linear combinations of 𝑑 𝑖 and/or 𝑝 𝑖 . 𝑞 is the number of dimensionless variables, which is function of the 𝑚 independent physical units (e.g. m, kg, s, etc.) and the 𝑛 variables involved in the problem. It is calculated as: 𝑞 = 𝑛 -𝑚.

The use of dimensional analysis enables to reduce the number of variables of a problem by grouping them into dimensionless numbers without loss of information. The readers can follow the reference [START_REF] Szirtes | Applied Dimensional Analysis and Modeling[END_REF] to become more familiar with dimensional analysis.

THE VPLM METHODOLOGY

The methodology presented here is called Variable Power Law Meta-model (VPLM). It is applied in three main steps: data generation, surrogate model definition and surrogate model selection. Fig. 1 shows the global process of the methodology by presenting all the activities involved in each main step. A detailed description of the methodology is given in [START_REF] Sanchez | Dimensional analysis and surrogate models for the thermal modeling of Multiphysics systems[END_REF]. This methodology aims at facilitating the design of multi-physics systems and components by building surrogate models based on dimensional analysis and finite elements simulations.

Fig. 1: Procedure of the VPLM methodology

Data generation

This step deals with the identification of the influent physical variables and the construction of the corresponding dimensionless numbers. Then, a design of experiments is used to define the configurations which have to be simulated by a finite element simulation software. The readers can follow the reference [START_REF] Montgomery | Design and Analysis of Experiments[END_REF] for more details on design of experiments.

Surrogate model definition

This second step deals with the definition of the surrogate model, or in other words the algebraic expression of the model. As states the name of the methodology (VPLM), the chosen mathematical form of the model is a variable power law.

π 0 = 𝑘π 1 𝑎 1 (𝜋 1 ,𝜋 2 ,…,𝜋 𝑞-1 ) π 2 𝑎 2 (𝜋 1 ,𝜋 2 ,…,𝜋 𝑞-1 ) … π 𝑞-1 𝑎 𝑞-1 (𝜋 1 ,𝜋 2 ,…,𝜋 𝑞-1 ) (3)
where 𝑎 𝑖 (𝜋 𝑖 , … , 𝜋 𝑞-1 ) are polynomial functions of log (𝜋 𝑖 ). In order to define the functions 𝑎 𝑖 (𝜋 𝑖 , … , 𝜋 𝑞-1 ), a sensitivity analysis is conducted using the simulation data gathered during the first step. Several candidate models are built for which the relative errors and/or the standard deviation can be evaluated. In order to appreciate and compare the accuracy of the surrogate models built during the second step, the evolution of the relative errors of each model (differing in the number of selected higher order terms) are shown in Fig. 2.

Fig. 2: Example of relative errors comparison between different models

Surrogate model selection

This third step of the VPLM methodology concerns the selection of the appropriate model and its validation regarding its accuracy and complexity.

The engineer can make a compromise between accuracy and simplicity of the expected surrogate model. Thus, by looking at Fig. 2, it can be seen that the prediction accuracy increases with the number of selected terms, but the model becomes more bulky as well.

THERMAL MODELLİNG OF POWER CONVERTER ELECTRONİC COMPONENTS

PROBLEM STATEMENT

The preliminary design of power converters requires mastering the thermal performances of somes electronic components. Thus it is necessary to have accurate thermal models since early design stage. The methodology described in this paper enables to generate thermal models in accordance with the requirements. The studied system is based on a prototype designed by ARCEL (Fig. 3).

THERMAL MODEL OF A FİLM CAPACİTOR

The life expectancy of capacitors highly depends upon their operating temperature [START_REF] Parler | Thermal Modeling of Aluminum Electrolytic Capacitor[END_REF]. Thus the evaluation of its thermal resistance is required. Due to its geometrical integration on the converter, the capacitor dissipates the generated losses through all the three heat transfer modes (conduction, convection and radiation). Indeed, for this geometry, semi-empirical laws [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF] are available for each heat transfer mode. But, in order to use them we need to establish a nodal model including at least three nodes, i.e. the hot-spot, the case surface and the surrounding air. Generally, for the preliminary design of capacitors a straight forward relation that estimates the equivalent thermal resistance between the hot-spot and the ambient temperature is preferable. The VPLM methodology enables to build such model from finite element simulations.

It is considered that the capacitor is fixed on the printed circuit board through its terminals and the heat losses dissipated through them can be neglected. Natural convection cooling is considered and there is two assumptions (Fig. 4):

 Geometrical simplification: axi-symmetry hypothesis.

 Material simplification: an equivalent homogenized material (function of physical and geometrical characteristics of the layers) is modeling all the dielectric and electrode layers. This material has different physical properties for radial 𝜆 𝑟 and axial 𝜆 𝑧 directions. The thermal model of the capacitor is built by following the steps of the VPLM methodology:

Data generation

The dimensional analysis shows that the problem depends on eleven physical quantities (4) and can be reduced to seven dimensionless numbers (5):

𝑅 𝑡ℎ = 𝑓(𝑔βΔθ, λ, μ, C p , 𝐷, 𝐿, 𝜆 𝑟 , 𝜆 𝑧 , 𝜌, 𝜀𝜎𝑇 𝑎 ) (4)

𝜋 0 = 𝐹(𝐺𝑟, 𝜋 2 , 𝑃𝑟, 𝜋 4 , 𝜋 5 , 𝜋 6 )

where the dimensionless numbers are: the nondimensional thermal resistance, 𝜋 0 = 𝜆𝐷𝑅 𝑡ℎ ; the modified Grashof number (expressed in term of .The air properties are assumed to be constant and evaluated at the mean temperature. In addition, the radial thermal conductivity 𝜆 𝑟 does not change with the geometrical characteristics of the layers. Thus the Prandtl number and the dimensionless number 𝜋 4 are constants and the problem depends only on five dimensionless numbers.

A design of experiments of 81 points is built to define the configurations which have to be simulated by a finite elements model. The finite element model of the capacitor is built in COMSOL Multiphysics software.

Surrogate model definition and selection

From the simulations results all the possible VPLM models are built and the evolution of the relative errors is represented in function of the complexity of the considered model (Fig. 6). During preliminary design stage, a model with less than 15% of maximum relative error remains acceptable. Thus the model ( 6) with ten higher order terms represents the best compromise between the expected accuracy and the complexity of the mathematical expression.

Surrogate model validation

The prediction capability of the selected model is assessed by cross validation, which is illustrated on a scatter plot in Fig. 7. In the previous section was built a thermal model of a film capacitor cooled by natural convection. This cooling configuration is often insufficient for high power electronic components such as IGBT modules. A solution to improve the cooling performances of IGBT is to use a heatsink (Fig. 3).

For the selection of this type of heatsink two physical parameters are important: the pressure drop through the heatsink and its thermal resistance.

In this study a rectangular heatsink is considered cooled by a single fan.

Data generation

The dimensional analysis shows that the problem depends on twelve physical quantities [START_REF] Vignaux | Simplifying regression models using dimensional analysis[END_REF] and can be reduced to eight dimensionless numbers:

𝑅 𝑡ℎ = 𝑓(ΔP, 𝜇, 𝜌, 𝜆, 𝐶 𝑝 , 𝑊 ℎ , 𝐿 ℎ , 𝑊 𝑝 , 𝑒 𝑓 , 𝐻 𝑏 , 𝑊 𝑓 )

𝜋 0 = 𝐹(𝐵𝑒, 𝑃𝑟, 𝜋 3 , 𝜋 4 , 𝜋 5 , 𝜋 6 , 𝜋 7 )

where the dimensionless numbers are: the nondimensional thermal resistance, 𝜋 0 = 𝜆𝐷𝑅 𝑡ℎ ; the As previously, the air properties are assumed to be constant and are evaluated at the mean temperature implying constant Prandtl number. Additionally, in order to reduce the number of parameters, two geometrical simplifications are made: the geometrical ratio between the height of the heatsink base (𝐻 𝑏 ) and its length (𝐿 ℎ ) is kept constant; the thickness of the fins (𝑊 𝑓 ) is equal to the space between the fins (𝑒 𝑓 ). Thus, the dimensionless numbers 𝜋 7 is constant and the dimensionless numbers

Surrogate model definition and selection

The results of the VPLM methodology (Fig. 9) show that the model with four higher order (9) terms seems to be the best compromise between accuracy and complexity.

Surrogate model validation

The prediction capability of the selected VPLM model is assessed by cross validation, which is illustrated on a scatter plot in Fig. 10.

POSSİBLE UTİLİZATİON OF THE MODELS

The use of VPLM enables to reduce nodal network models composed of non-linear resistances (depending on the temperature, for example) of a capacitor (Fig. 11). Thus it allows linking directly the hot-spot temperature of capacitor to the ambient temperature without estimating the case temperature. Another possible use of these models is their association to conduct the preliminary design of a system (the models are function of the geometry and some physical properties) or to investigate other cooling configurations for a component. For example, it is possible to evaluate the cooling capabilities of the capacitor if a heatsink is used (Fig. 12).

Fig. 12 : Nodal network representation for the association of models

CONCLUSİON

This paper presented and applied a methodology for building algebraic thermal models of electronic components from finite elements simulations.The models of the components obtained by this method are modular and re-usable for different engineering problems and different systems involving these components. Finally it is shown that this type of models permits to simplify nodal network approach by reducing the numbers of required thermal resistances.

I

Fig. 3 :

 3 Fig. 3 : DC/AC (one phase) converter prototype It is proposed here to generate the thermal model of a film capacitor used for filtering and the thermal model of the heatsink used to cool the IGBT modules.

Fig. 4 :

 4 Fig. 4 : Simplifying assumptions for film capacitor detailed thermal modeling Two types of losses are considered for the thermal modelling: Joules losses dissipated in the conductors and dielectric losses dissipated in the dielectric layers. For the finite elements simulations these losses are generated in the equivalent material.

Fig. 5 :

 5 Fig. 5 : Finite elements simulations results for diffferent capacitor geometry (left: 𝐷/𝐿 < 1, right: 𝐷/𝐿 > 1, temperature scale in °K)

Fig. 6 : 2 𝑎 2

 622 Fig. 6 : Evolution of the relative errors for different models of the capacitor 𝜋 0 = 0.039𝐺𝑟 𝑎 1 (𝐺𝑟,𝜋 2 ,𝜋 5 ) 𝜋 2 𝑎 2 (𝐺𝑟,𝜋 2 ,𝜋 5 )

Fig. 7 :

 7 Fig. 7 : Validation of the capacitor thermal model 3.3. THERMAL MODELLİNG OF A HEATSİNK

  𝜋 4 and 𝜋 6 are equal. Considering all these assumptions, the problem depends only on five dimensionless numbers: 𝐵𝑒, 𝜋 3 , 𝜋 4 and 𝜋 5 . A design of experiments of 81 points is built to define the configurations which have to be simulated by finite elements model in COMSOL Multiphysics software. The boundary conditions used for the simulations are:  A pressure drop (Δ𝑃) is applied between the inlet and the outlet of the heatsink.  A heat flux density of 2 • 10 5 𝑊/𝑚 2 is dissipated on a square surface (𝑆 = 𝑊 𝑝 2 ) representing the IGBT module.  The inlet temperature is equal to 300°𝐾 and the heatsink does not exchange heat through its external surfaces.

Fig. 8 :

 8 Fig. 8 : Finite elements simulation results for the heatsink model (Temperature scale in °K)

3 - 1 .Fig. 9 :

 319 Fig. 9 : Evolution of relative errors for different models of the heatsink

Fig. 10 :

 10 Fig. 10 : Validation of the heatsink thermal model

Fig. 11 :

 11 Fig. 11 : Nodal network representations of the thermal model of a capacitor: classical (top) and proposed representations (bottom)