Résumé

Study of rowing involves hydrodynamics twice : for the hull and for the oar blades. However, describe the unsteady 3D flow around blades moving with 6 D.O.F. at the end of a flexible shaft and including violent free surface deformation, is the most challenging task. To gain knowledge about this flow, both experimental and numerical studies have been started for a decade at the Laboratory of Fluid Mechanics (LMF) of Ecole Centrale Nantes. The experimental activity involves in-situ measurements using instrumented boats and tests in towing tank at LMF using a dedicated device that reproduces simplified rowing strokes but giving accurate and repeatable measurements. The numerical studies are based on unsteady CFD simulations carried out using an in-house RANS solver with advanced methods such as automatic grid refinement to capture accurately the free surface evolution. The pursued objective is to use experimental results as a reference database to validate the numerical simulations. When validated, simulations become a useful and trusty tool to undertake parametric studies in real situations easier than with experimentation. They also permit to improve the parametric models used for the fluid forces in the simulators of rowing systems. So, empirical indications could be replaced by objective criteria for the choice of oars and settings and also, in the future, for optimization of the rowing style.

Introduction

Due to the growing power of the computational ressources and the development of advanced numerical methods, CFD simulations become more and more popular and are used for many complex applications in naval hydrodynamics. During the last decade, free-surface capturing methods (volume of fluid, level-set formulation) have attracted an increasing interest because these approaches are more robust than those based on a free-surface fitting methodology since no regridding is necessary ( [START_REF] Carrica | Ship motions using single-phase level set with dynamic overset grids[END_REF][START_REF] Pm Carrica | An unsteady single-phase level set method for viscous free surface flows[END_REF][START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF]). Moreover, the merging or breakup of the interface is handled in a natural way. Here, the approach is used to compute the flow around a rowing blade, which is a far more violent test-case than usual. The blade moves with 6 D.O.F. close to the free sur-face, leading to an unsteady flow with a very complex shape of free surface varying in time and including breakup. It also involves fluid-structure interaction through the flexibility of the shaft, considering the motion of the oar is imposed by the rower. In reality, the "boat-oar-rower" global system with the rower as a human actuator is even more complex, since the fluid force acting on the blade will be linked to the kinematics of the rower, since the latter activates a sensory motor control of its own motion in order to adapt it to the current position and external loads. The study of this global system is an issue which is currently adressed at the Laboratory as well ( [START_REF] Rongère | Dynamic modeling and simulation of rowing with a robotics formalism[END_REF][START_REF] Rongère | Simulation dynamique des systèmes Bateau-Avirons-Rameur (s)[END_REF]), but it will not be addressed here since it is outside the scope of the current communication.

To conduct properly this ongoing research work, the experimental approach is used to provide a reference database. Thus, the article describes both aspects of the research carried out at the LHEEA (ex-LMF) Laboratory of Centrale Nantes to study the flow around rowing blades. First the experimental approaches are presented, including both towing tank and on-board measurements. Then, the numerical methodology will be briefly described prior to the different steps of the validation process we have adopted. In the next section, the complementarity of experimental and numerical approaches when dealing with validation will be highlighted. To conclude, the next steps of this ongoing work will be revealed.

Experimental approach

The Laboratory has carried out experimental studies on rowing boats and oars using specific devices in the towing tank, as well as on-board measurements with an instrumented boat ([6, 7, 8]). Towing tank tests, on one hand, are focusing only on one part of the system (the boat or the oar), the settings are necessarily simplified with respect to the real case. The main advantage is that the environment and the parameters are well controlled, with good accuracy and repeatability. This technique permits the separation of the propulsive device (the oar) from the motor (the oarsman) and from the boat, as it is done for propellers. On the other hand, on-board measurement works directly on the real system (boat-oar-rower). However, the effective reali-sation of the procedure involves numerous constraints :

-technological constraints : the instrumentation has to be light and as unintrusive as possible, -environmental constraints : weather conditions (wind, wave,..) are hard to measure, impossible to control and non-repeatable, -organizational and human constraints : elite rowers are not always free to spend time for such a test. It is also difficult to impose an experimental planning and to have a regular operation. The real phenomena get closer, but the price to pay is a larger number of parameters, which are sometimes not precisely controlled. Furthermore, only a limited number of measured quantities can be recorded. A loss of accuracy is then expected compared with towing tank experiments.

Towing tank experiment

Given the intermittent and unsteady nature of the propulsion of rowing boats, it is difficult to plan trials of selfpropulsion of the entire system as it is done for conventional vessels. However, it is possible to achieve separately testing on boats or on oars alone. Here, only the tests involving oars and using a specific device designed and build by the laboratory are described ( [START_REF] Barré | Etude expérimentale des systèmes de propulsion instationnaire[END_REF]). This is a six-component dynamometer equiped with a servomotor and a mechanism which reproduces accurately and repeatably the rotation of the oar blade in the water (Fig. 1). Two arms can be fixed to the system. One of them is dedicated to receive 0.7 scale blades with a small rigid shaft (Fig. 1). Another has been designed to test real oars. The device is fixed to the carriage of the towing tank (long=140 m, width=4m, depth=3m) and the speed of translation is constant during the stroke. The variable rotation velocity combined with the forward velocity of the carriage allow us to reproduce a simplified motion, but one that is representative of the physics of a real stroke (see the section 4.1 and [START_REF] Leroyer | Experimental and numerical investigations of the flow around an oar blade[END_REF] for more details). In any way, the generated flow reproduces more completely the reality than those obtained by [START_REF] Kinoshita | Improving rower motion and rowing equipment by using rowing velocity prediction program with estimating hydrodynamic load acting on an oar blade[END_REF], where a blade is only turning around its center in a uniform flow. However, this is still better than tests with constant incidence in uniform flow, where the crucial phenomenon of unsteadiness is not taken into account.

Here, the forces acting on the blade are directly measured in the axes of the dynamometer without having to apply friction correction, since the rotation part and the motor are fixed on the sensitive part. Inertia and aerodynamic contributions are deduced from some specific tests. The measurement of the velocity of the carriage and the rotation of the blade complete the recorded data. With this system, uncertainties on the blade forces are estimated to be 0.5 N with a gap of repeatability less than 2 %. The uncertainty on the rotation velocity is 0.1 % and 0.3% on the translation velocity. This material enabled to carry out systematic campaign tests to deduce the influence of kinematic parameters on hydrodynamic forces. This database of reference is also very useful to validate CFD simulations on such a complex physics. More details about the experimental material and procedure can be found in [START_REF] Leroyer | Experimental and numerical investigations of the flow around an oar blade[END_REF].

On-board measurements

Measurements with instrumented boats allow the direct analysis of the rowing stroke and the boat behaviour. Their use by the coaching staff becomes more and more common due to the commercial availability of light and efficient equipment. We have such material, but the present goal is to describe specific tests carried out to validate CFD computations on rowing blades. Such tests can not be used on a daily basis. They need development of specific sensors and the adjustment of specific procedures to obtain enough accuracy to reach the objective.

Measuring the boat dynamics

Commercial sensors specifically designed for rowers give a rather accurate mean velocity. However, inertia and friction of the turbine bias the measure of the instantaneous velocity. In addition, accurate GPS receivers with high sampling rate and post-treatment or Real-Time-Kinematic (RTK) are too heavy devices and measure the velocity with respect to the Earth. All these observations lead us to develop a specific sensor to measure the velocity of the boat. This is done through a small Schiltnecht ducted turbine fixed below the fin of the skiff. The turbine contains a Baumer magnetic sensor, which counts the passing of each blade. The position ensures that the turbine is outside the boundary layer of the hull. It has been calibrated fixed on the boat in the towing tank : the resolution is 0.7 mm of displacement per pulse with an accuracy of 2 cm/s.

Acceleration is measured at the middle of the boat and vertically at the bow and the stern through three piezoresistive +/-5g ENTRAN EGCSY sensors. These sensors are calibrated with the gravity through a half-turn. Vertical acceleration is integrated twice to give the pitch and the heave, considering the pitch is small (about 1˚of amplitude). To remove the drift after integration, we use the fact that displacements and their derivatives have zero averages for cyclic motions.

Measurements for the oars

To measure the trajectory of the blade, the oarlocks are equiped with three accurate potentiometers (Fig. 2) to measure the orientation of the oar : the horizontal sweep angle, the immersion angle and the spin angle (rotation around its own axis). Considering the dynamics of the oar, the following forces are involved : the action of the rower's hand on the handle, the reaction of the oarlock, the hydrodynamic and aerodynamic forces and the inertia and gravity forces. We do not know where the resultant of the hydrodynamic force is applied. To retrieve this information, all the forces and their moments should be measured, which is a arduous task. Considering the main objective which is to validate the numerical simulation, it is enough to measure bending and torsion moments in a section of the outside shaft and to compute the numerical results at the same location. If experiment and simulation match well, the numerical simulations can be used to obtain all the components of the hydrodynamic forces. For the measurements of these moments, the oars have been equiped with three strain gauges : two for the bending in orthogonal directions and the third one for the torsion. The pure hydrodynamic contribution is obtained by subtracting the inertia effects computed at the gauge positions. The inertial properties have been estimated from a model of the oar structure and the procedure has been globally checked for the position of the centre of mass, the global mass and the measure of inertia through the technique of bifilar pendulum. These dynamometric oars are calibrated in pair under static loads with a test bench specifically build for this procedure. This test bench is also used to measure the linear and angular deflection stiffness (Fig. 3), which are two essential parameters for the CFD simulations that take into account fluidstructure interaction.

Procedure to detect the contact of the blade with the water during the catch

The measure of the dynamic roll of the boat has not been integrated to the instrumented boat yet. The main reason is that this motion is neither of zero average over a cycle nor cyclic, but controlled by the rowers. Without this information, the position of the blade with respect to the water

(1) (3)
(2) (4) The head is fitted with three small wheels, of which one is mounted on an elastic piece to tighten and maintain the no-slip roll of the wheels. The wheel ( 5) is linked to a potentiometer to measure the rotation of the shaft along its own axis.

is not accurately known. A procedure to detect the instant where the blade is touching the water is thus required, to trim the immersion angle. In addition to that, we have to assume that the trim remains valid during the whole drive phase, i.e that the roll does not change, which is not totally true. The procedure of blade/water contact detection is mainly based on the analysis of the Z-bending and Xtorsion moment : both are approximatively null before the blade touches the water and increase suddenly when the contact is made. This detection can be checked by some kinematic signs : for elite rowers, the water entry is ob- tained after reaching the minimum sweep angle and when the spin rotation is stabilized. The analysis allows us to locate the sweep angle of the blade water entry with a accuracy of around 1˚(for the sweep angle).

Kinematic data to validate CFD simulations

Once the water entry of the blade is located, the angles of the oars and the motion of the boat are transformed into input data in a geometric rowing model used for the simulation of the global system ( [START_REF] Rongère | Dynamic modeling and simulation of rowing with a robotics formalism[END_REF]). As an output, the model gives the motion of the blade with respect to the water frame of reference (Fig. 4). All the details of the real oar are taken into account, in particular the shift between the three axis of rotation at the oarlock. Up till now, the spin of the shaft is the only property which is not activated in the simulation since it is not crucial during the propulsive phase (where the blade remains vertical).

Numerical approach

The flow around a rowing blade presents two main features, i.e. unsteadiness and a very complex free surface deformation, which made it difficult to investigate for a long time. For performance analysis of rowing crews (even for optimization), rough models ( [START_REF] Jf Wellicome | Some hydrodynamic aspects of rowing[END_REF][START_REF] Pope | On the dynamics of men and boats and oars[END_REF]) have been used to estimate the propulsive forces for lack of anything better, but with a limited accuracy. Under these conditions, the relevance of results is always debatable.

To investigate such a flow, only an experimental approach seemed possible around ten years ago. But nowadays, advances in computing power offer the possibility to investigate this flow using CFD. And the experimental investigations already available provide a useful and valuable reference database to validate CFD computations. In fact, CFD tools are no longer limited to simple physical problems and their field of applications has been significantly enlarged during the last years.

Description of the RANS solver

The numerical tool used for these studies is the ISIS-CFD solver developed in the DSPM group of LHEEA. The solver is based on the Finite-Volume method to build a second-order accurate discretisation of the RANS equations. The flow equations are constructed face by face, which means that cells having an arbitrary number of arbitrarily shaped faces can be accepted. This enables the simulation of flows around complex geometries. The velocity field is obtained from the momentum conservation equations and the pressure field is extracted from the mass conservation constraint, or continuity equation, transformed into a pressure equation. In the case of turbulent flows, additional transport equations for modeled turbulent parameters are solved in a form similar to the momentum equations : they are discretised and solved using the same principles. Here, the two-equation kω SST closure [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] is used to take into account the turbulence phenomena.

Free-surface flows are computed through an interface capturing method. The flow phases are modelled with a transport equation for the volume fraction of water c w in each cell : c w = 1 means that the cell is completely filled with water, c w = 0 means that only air is present in the considered cell. The interface between air and water is represented by the numerical discontinuity in the solution for c w . The effective flow physical properties (dynamic viscosity µ and density ρ) are obtained from the properties for each phase, (µ w ,µ a ) and (ρ w ,ρ a ) respectively for water and air, with the following constitutive relations :

µ = c w µ w + (1 -c w )µ a and ρ = c w ρ w + (1 -c w )ρ a .
Special attention has to be paid to preserving the sharpness of the interface, as well as the boundedness of the variable, when solving the transport equation for c w . These requirements are fulfilled by introducing downwind differencing since compressive characteristics are required for an accurate interface capturing. This implies a constraint on the local Courant number (the Courant number is an adimensional parameter roughly defined by : ∆t V /∆x , where V is the velocity through the considered cell, ∆x the size of the cell, and ∆t the global time step of the temporal discretisation). Here, the Blended Reconstructed Interface Capturing Scheme (BRICS) scheme is used ( [START_REF] Wackers | Free-surface viscous flow solution methods for ship hydrodynamics[END_REF][START_REF] Queutey | An interface capturing method for free-surface hydrodynamic flows[END_REF]).

An Arbitrary Lagrangian Eulerian (ALE) approach is used to deal with moving bodies ( [START_REF] Hirt | An arbitrary lagrangian-eulerian computing method for all flow speeds[END_REF][START_REF] Donea | Arbitrary Lagrangian-Eulerian methods[END_REF][START_REF] Leroyer | Etude du couplage écoulement/mouvement pour des corps solides ou à déformation imposée par résolution des équations de Navier-Stokes. Contribution à la modélisation numérique de la cavitation[END_REF]). All configurations of motion (up to 6 solved or imposed D.O.F.) can be applied. Analytical weighted deformation techniques have been developed to preserve a mesh fitted to the body during its motion [START_REF] Leroyer | Etude du couplage écoulement/mouvement pour des corps solides ou à déformation imposée par résolution des équations de Navier-Stokes. Contribution à la modélisation numérique de la cavitation[END_REF][START_REF] Leroyer | Numerical methods for RANSE simulations of a self-propelled fish-like body[END_REF].

The temporal discretisation scheme is the second order implicit Euler scheme when dealing with unsteady configurations. For each time step, an inner loop (denoted by nonlinear loop) associated to a Picard linearisation is used to solve the non-linearities of the system.

When the flexibility of the shaft is activated, the position of the blade is no longer fully imposed but depends on the fluid loads on the blade. Even if the position of the blade in the non-deformed configuration remains imposed, this fluid-structure interaction needs to be addressed with care. Since inertial effects of the oar are small compared to the fluid forces acting on the blade, the resolution of a static equilibrium of the structure is suitable. Due to the added mass effect, a part of the fluid loads on the blade depends on the acceleration and may lead to divergence of the numerical coupling ( [START_REF] Söding | How to integrate free motions of solids in fluids[END_REF]). To keep a stable fluid-structure coupling, the position of the blade is updated at each nonlinear iterations through a quasi-Newton method using a rough estimate of the added mass for the blade. It was evaluated from [START_REF] Lewandowski | The dynamics of marine craft : maneuvering and seakeeping[END_REF]) .

For the computations reproducing the real rowing stroke, we also benefit from the automatic grid refinement included in the solver ISIS-CFD, which is described in more detail by [START_REF] Wackers | Adaptive grid refinement applied to rans ship flow computation[END_REF] and [START_REF] Wackers | Adaptive grid refinement for hydrodynamic flows[END_REF], to capture accurately the evolution of the free surface. This technique enables to follow the free surface keeping always fine cells around it, and thus limiting the diffusion process of the volume fraction. This avoids to use a large area with fine cells surrounding all the region where the free surface takes place during the stroke.

Simulations and comparisons 4.1 Simulations of a planar 0.7 scale blade

This first validation reproduces one of the systematic tests carried out in the towing tank with a 0.7 scale oar (Fig. 1). In order to simplify as much as possible the first validations, we chose the reference blade, which is to say, a planar rectangular blade (Fig. 5) with a rigid shaft, which makes the meshing process very easy. Hence, it avoids a digitalization of a specific tested geometry and a fluid-structure coupling due to the interaction between the fluid force and the flexibility of the shaft. To define blade movements for systematic tests, the reproduction of real measured oar movements is not suitable because these are performed with oarsmen having different rowing styles and are not linked with reliable data to the instantaneous boat speed. Furthermore, the high number of parameters that define this kind of movement makes the exploitation of test results difficult. However, even if simplified kinematics have to be used in a laboratory, it is crucial that the main physical phenomena of the real conditions are reproduced. Refering to kinematics and energy considerations ( [START_REF] Leroyer | Experimental and numerical investigations of the flow around an oar blade[END_REF][START_REF] Barré | Comparison between common models of forces on oar blades and forces measured by towing tank tests[END_REF]), systematic tests were performed by imposing a significant parameter η 0 defined by equation (1) as a constant during the whole stroke.

η 0 = V b sin θ/(L e θ) , (1)
with V b velocity of the carriage, θ angle between the forward direction of the carriage and the shaft and L e the outer level (distance between the oar-lock and the middle of the blade). This parameter can be viewed as an advance parameter or a characteristic efficiency. To obtain a similar flow around the blade at reduced scale and around the blade at full scale, the advance parameter and the Froude number have therefore been respected. Under these conditions, the imposed blade trajectory is shown in Fig. 6. We can remark that the typical "l" path of the real motion in Fig. 4 is reproduced.

Using this configuration, we demonstrate in [START_REF] Leroyer | Experimental and numerical investigations of the flow around an oar blade[END_REF] that unsteadiness and free surface effect were two crucial phenomenum to accurately capture the physics of such a flow. On the contrary, we also show that viscosity (and turbulence induced) has no critical physical features for application of that kind, at least in the central part of the rowing stroke. For small θ angle, the blade is nearly aligned with the forward direction, and in this situation, the friction drag becomes an important effect. But in a real stroke, this situation never happens since the angle of attack is about 30˚.

Simulation of a real rowing stroke

From the experimental work described in section 2, a preliminary study has been launched to reproduce a real rowing stroke, including the fluid-structure interaction due to the flexible shaft. The only simplification is to keep the blade vertical (the rotation around its own axis has been removed). Except for that point, the 6 D.O.F. imposed kinematics is the one reproduced in Fig. 4, That was done to avoid resorting to sliding grids for the moment and starting with a simpler configuration, knowing that this motion is not crucial for the propulsive phase. A moderate adaptative grid refinement around the free surface has been activated too. A coarse mesh contains initialy 240,000 cells (before refinement around the free surface) to reach about 1.6 million cells during the computation (Fig. 7). The first results, FIGURE 7 -Global view of the mesh. in the absence of full validation, show the feasibility of such a configuration. On the one hand, the evolution of the free surface is realistic Fig. 8. On the other hand, the Z-axis moment, which is the highest load during the drive and where the flexibility of the shaft acts, is quite well reproduced with respect to the experimental data (Fig. 9), except a more oscillating signal. In this figure and in the two figures after (Fig. 10 and Fig. 11), the reference stroke corresponds to the stoke which was chosen to build the imposed kinematics among all the stokes of the recorded sequence (plotted with thin lines). The whole sequence highlights the variations obtained between different strokes of the sequence due to minor changes of the kinematics of the rower and of external parameters. It is an indicator of the reproducibility using on-board measurements. The other bending component (Y-Axis) matches quite well with the experiments too (Fig. 10). The observed gap before the blade enters the water (and where the simulation gives a value close to zero whereas the experimental signal is about -7N.m ) is due to the weight of the oar which is not caught by M y where the blade is horizontal (the zero value of the sensor was calibrated with a vertical blade). As for the torsion component (Fig. 11), the evolution is quite well caught, even if the relative differences are much larger. Note that the order of magnitude for this component is quite weak compared to the others. A difference setting of the longitudinal pitch of the oarlock (which was not measured) may be the cause of this gap.

Conclusions

This article aims at reporting progress about the approach, both experimental and numerical, pursued at LHEEA to gain knowlegde about the flow around rowing blades. In particular, it shows the complementarity of these two ap- proaches for this kind of research work. Accurate and well documented towing tank experiments dealing with simplified configurations, but reproducing the major physical features with respect to a real stroke are very useful because they enable to validate numerical models with care. This step has been initiated using the simplest case. This case has also been used to quantify the influence of physical parameters. As it was done for swimming ( [START_REF] Toussaint | An alternative fluid dynamic explanation for propulsion in front crawl swimming[END_REF]) and for insect flight ( [START_REF] Dickinson | Unsteady mechanisms of force generation in aquatic and aerial locomotion[END_REF]), the crucial role of unsteadiness is demonstrated, invalidating works based on a series of "equivalent" steady flows to study such a configuration. And unsurprisingly, as the blade motion occurs just below the free surface, we show that this physical feature is also a sine qua non parameter to investigate such a flow.

Then, a preliminary study involving a real measured kinematic of a real oar has been started. Even if this work is at its first stage, it has reinforced the potential of the numerical methods which are used, especially the adaptive grid refinement to limit the diffusion of the interface, and procedure to stabilize numerically the interaction between the flexible shaft and the fluid. However, experimental uncertainties, hard to estimate and resulting from nonmeasured parameters (such as the roll of the boat), make the validation task for this case difficult. As a consequence, we planned to come back to towing tank tests involving a full-scale oar to pursue the validation process with more controlled experimental data. As for the experimental part, a new on-board campaign should be considered. The objective will be to add a synchronized video acquisition as well as a measure of the roll of the boat to make up the missing data. This should improve the accuracy of the path of the blade with respect to the free surface and thus becomes reference data to validate a real configuration.

Once the validation process is fully completed, numerical investigations will be started to better understand the influence of some parameters, such as the flexibility of the shaft, the immersion of the blade, the setting of the oar (pitch of the oarlock, lever length, etc.). The final goal is to improve the simplified models which are used in the simulators of the global system boat-oar-rower (which corresponds to the Velocity Prediction Programs in sailing yachts).

  Towards a better knowledge of the flow around rowing blade Alban Leroyer, Sophie Barré, Jeroen Wackers, Patrick Queutey 18 juin 2012

FIGURE 1 -

 1 FIGURE 1 -Motion device fixed to the carriage of the towing tank for testing 0.7 scale blades.

FIGURE 2 -

 2 FIGURE 2 -Sensors to capture oar orientation : the horizontal sweep angle is directly measured with a potentiometer (1) fixed on the axis of the oarlock. An oscillating arm (2), with the head (3) clipped on the shaft, operates the potentiometer (4) which measures the immersion angle.The head is fitted with three small wheels, of which one is mounted on an elastic piece to tighten and maintain the no-slip roll of the wheels. The wheel (5) is linked to a potentiometer to measure the rotation of the shaft along its own axis.

FIGURE 3 -

 3 FIGURE 3 -Representation of the linear and angular deviation due to the flexible shaft.

FIGURE 4 -

 4 FIGURE 4 -Motion of the blade built through on-board measurement.

FIGURE 5 -

 5 FIGURE 5 -Surface mesh of the planar rectangular blade (the shaft is not taken into account in the simulation).

FIGURE 6 -

 6 FIGURE 6 -Example of prescribed motion with the simplified kinematics used in the towing tank.

FIGURE 8 -

 8 FIGURE 8 -Two views of the free surface at different time.

FIGURE 9 -

 9 FIGURE 9 -Z-axis moment (oar frame of reference).

FIGURE 10 -

 10 FIGURE 10 -Y-axis moment (oar frame of reference).

FIGURE 11 -

 11 FIGURE 11 -X-axis moment (oar frame of reference).

Acknowledgments

This work was granted access to the HPC resources of CINES and IDRIS under the allocation 20129 made by GENCI (Grand Equipement National de Calcul Intensif).


A Nakahata Medrado

J M Guichon

P O Jeannin

Senior Member, IEEE J L Schanen

E Batista

G Desportes

Determining the minimal decoupling capacitor in a railway traction inverter using optimization

This paper aims determining the minimum decoupling capacitor Cdec for a railway traction inverter. It is defined when any increase of its value does not decrease the overvoltage at turn off anymore. A simple generic model is deduced from a full understanding of the switching cell behavior, including the effect of all stray inductances and all interactions within the switching cell. This model is used in an optimization process in order to obtain directly the minimal value of Cdec. The methodology is validated in comparison with time simulation, as well as using experimental results. It has been used to analyze the impact of the switching cell parameters on the minimum value of Cdec: switching speed, stray inductances and additional resistance have been considered.

I. INTRODUCTION

The railway industry is focusing on the advantages of using the SiC wide bandgap devices in traction applications. Within the European project "Shift2rail" [START_REF]Shift2Rail Project[END_REF], the potential benefits of using SiC MOSFETs in future converters are investigated, but also the potential impacts on module design. With the high speed commutation capability of these new switches, there is a strong need of properly designing a decoupling capacitor very close to the switching cell: therefore a small value is desired. In addition, this allows decreasing volume, weight and cost of the global system.

The goal of a decoupling capacitor is to reduce undesired events during the commutation, originated by stray inductance such as voltage overshoot. Additionally, at turn off, the output capacitance of the MOSFET oscillates with the stray inductance, generating high frequency ringing, inducing disadvantages such as Common Mode currents or radiated fields. The use of a decoupling capacitor Cdec allows thus compensating the impact of the stray inductive elements between the DC voltage source and the switching cell, and reduces the semiconductor peak voltage [START_REF] Liang | Research on an improved DC-side snubber for suppressing the turn-off overvoltage and oscillation in high speed SiC MOSFET application[END_REF].

An empirical method to define the minimal decoupling capacitance is presented in [START_REF] Chen | Electrical Integration of SiC Power Devices for High-Density Applications[END_REF]. An effective Cdec that achieves a fully decoupled condition without interfering the resonance between the power module stray inductance Lpack and the output capacitor Coss is obtained using a rule defined as: Cdec must be 50 to 100 times larger than Coss to achieve a good decoupled condition. However, this approach has no link with the minimal voltage overshoot and neglects the parasitic impedance of the decoupling capacitor.

If for a small power design, the value of decoupling capacitance is not really impacting its volume, this is not the case in high power traction inverters, where a specific bus bar has to be carefully designed to associate several capacitors in parallel, together with the power modules. We aim therefore minimizing this value, taking also into account the effective stray inductance of the association of all decoupling capacitors. Fig. 1 presents a switching cell, where the input filter capacitor Cfilter is replaced by an ideal voltage source due to its large value. The DC source (catenary) is not supposed to participate to the switching events, given the high value of the line inductance. Lfilter in Fig. 1 represents the stray inductance between the filter capacitor and the switching cell, composed of a power module and the decoupling capacitor. It also takes into account the esl of the filter capacitor.

As clearly explained in [START_REF] Ren | Voltage Suppression in Wire-Bond-Based Multichip Phase-Leg SiC MOSFET Module Using Adjacent Decoupling Concept[END_REF], if Cdec is infinite, the overvoltage is generated by Lstray_pack + esl, Lstray_pack being the inductance of the semiconductor package and esl the stray inductance of Cdec. If Cdec is null then the overvoltage is generated by Lfilter + Lstray_pack, and is usually higher, otherwise Cdec would not be necessary. Defining a value of Cdec that ensures that the voltage overshoot generated by this mix of stray inductances is limited to a maximum value is the objective of this paper. Other design criteria as the behavior under short circuit conditions can be used [START_REF] Timms | New Boundary Condition for Decoupling Capacitance Selection in SiC Phase Legs Considering Short-Circuit Events[END_REF], but this paper is focused on the impact of switching speed and stray elements on the overvoltage as a function of the decoupling capacitor.

It should be noticed that in railway traction inverters, the value of the stray inductance between the DC bus capacitor tank and the decoupling capacitors (Lfilter) might be quite low. The capacitor tank is indeed composed of a large number of paralleled components, thus leading to a low value of Lfilter, since all Esl (Equivalent series Inductance) of electrolytic capacitances are paralleled. In addition, the link between this capacitor tank and the decoupling capacitors is very compact and as low inductive as possible. Therefore, interactions between filter and decoupling capacitors cannot be neglected.

The time domain waveform of the voltage on the semiconductor for different values of Cdec is illustrated in Fig. 2, based on simple simulations. The resonances between Coss, Cdec and the parasitic inductances impact both semiconductor overshoot and the ringing during switching. A mix of low and high frequency ringing can be observed. In some case, the maximum voltage overshoot is even not reached at the first oscillation, but later (see the example of 500nF in Fig. 2 where the maximum voltage does not occur at /2). The explanation of these phenomena comes from a simple current divider between the filter capacitors and the decoupling capacitor, including also the semiconductor's capacitance Coss. The switching waveform excites an infinite number of frequencies, and furthermore resonances between all stray inductances, Cdec and the output capacitance Coss of the semiconductors occur, as clearly illustrated in [START_REF] Chen | A frequency-domain study on the effect of DClink decoupling capacitors[END_REF]. Therefore, the evaluation of the voltage overshoot cannot be obtained using the classical second order results.

Using the impedance comparison between the two branches (Cdec and Cfilter) to determine a good decoupling condition is not obvious, since it has to be carried out for each individual frequency, each one contributing to the final result. It has thus been decided in this paper to use a time domain analysis, via a simplified representation to be considered in an optimization process. For this study, the representation of parasitic inductances is essential and must be studied carefully. The paper will start with introducing a rigorous electromagnetic model of a switching cell to account for all stray inductances of the layout. After simplifications, this equivalent circuit will be used into an optimization process to obtain the optimal value of the decoupling capacitance Cdec_optimal. It is defined as the minimal value necessary to obtain the minimal voltage overshoot i.e. when there is no longer interest in increasing Cdec. This value depends on several parameters as stray elements, switching speed or devices characteristics. Therefore, it is possible to have a direct link between Cdec_optimal and all these parameters. The influence of these parameters on Cdec_optimal will be discussed. All results will be validated using precise simulations and some measurements.

II. SWITCHING CELL ELECTROMAGNETIC MODEL

A. Defining the model

The electrical circuit representation of an industrial power inverter has to be carefully addressed. In traction inverters, the 3D integration is so complex that it is not possible to use distributed models of the interconnects, simply translating each part of the layout to R-L circuits as in [START_REF] Chen | A frequency-domain study on the effect of DClink decoupling capacitors[END_REF]. Indeed the decoupling capacitors are plugged on a DC bus bar, also linked to the power modules.

A generic way to take into account all the parasitic inductances of a converter from its geometry is proposed in this section. Fig. 3 presents the "ONIX 671" converter, from Alstom, which will be used as a case study for optimization in section III. It is a three-phase inverter with a dissipative chopper leg. The maximum rms current per phase is 1020 A, and the rated DC bus voltage is 750-900 V. The maximum switching frequency is 1.5 kHz.

Each phase is composed of 3 dual pack IGBT 1.7 kV / 1200A power modules in parallel (Top IGBT + Bottom IGBT) [START_REF]Mitsubishi High voltage insulated gate bipolar transistor[END_REF].The electrical schematics of phase W and the dissipative leg of ONIX 671 is displayed in Fig. 4. This structure will be considered to illustrate the generic representation in terms of electrical circuit, and will be validated by simulation and measurements. As already stated, the transposition of a complex industrial 3D geometry to an electric scheme is not straightforward. Indeed, the electrical connections are made using bus bars, which connect several components. It is not a simple inductance between the input and the output of a wire or a PCB track. In this context, the circuit representation requires a generic approach.

In order to reproduce the inductive effects due to the layout, a possible option is to use the PEEC method [START_REF] Daroui | Bus Bar Simulations Using the PEEC Method[END_REF], which means usually replacing each conductor by a stray inductance and couplings with the rest of the structure. However, this solution generates as many equivalent circuits as studied layouts, and does not allow a generic study of the phenomena involved in a commutation. Furthermore, the use of laminated bus bar structures does not facilitate identifying separate portions of the circuits which can be replaced by lumped elements, since the current flow in a plate is not really known. Therefore, we propose a global approach based on the number of Input/Outputs of a conductor. If a conductive link is only associating two points, it will be replaced by a single inductor (with coupling with the rest of the structure). However, if it is used to connect N_port > 2, the electrical circuit will be composed of N_port-1 inductors and resistors. One point among the I/O ports will be considered as a reference and linked with the others with R-L circuits (+couplings). It is worth noting that the full PEEC matrix in this representation exhibits both real and imaginary part. In other words, not only the inductors but also the resistors are coupled. Physically, this corresponds to common resistive path within the same conductor and eddy currents induced in the rest of the geometry. This generic representation has already been used and validated in several previous papers [10] [11].

Fig. 5 presents a simple example of this representation. First, we identify the system inputs / outputs that correspond to the interface points between the cabling model (bus bar) and other system components. Then, all points are connected electrically in groups. Among each group, a reference point is chosen arbitrarily. With this representation is possible to account for the impedances (and couplings) between any of the points connected together.

To illustrate the approach, let us consider the Alstom ONIX 671 power converter of Fig. 3. The DC bus is composed of 3x18 Electrolytic capacitors, and 2 power bus bars, each one linking the power modules to 8 decoupling capacitors (Fig. 6).

Even in this complex geometry, the proposed method can be used to obtain an equivalent circuit presented in Fig. 7. In this figure, "Mod1+" and "Mod1-" on the power bus bar have been chosen the reference points.

Many conductors are involved during the switching event: the positive and the negative bus bars and one bus bar for each phase that links the semiconductors. The positive conductor (Bus bar+) is connecting the positive pins of Cfilter, Cdec and the Collectors of the top IGBTs, and the negative conductor (bus bar-) is linking the negative pins of Cfilter, Cdec and the Emitters of the bottom IGBTs. Of course, we account for the intrinsic stray resistive and inductive behavior of the capacitors, the well-known esl (equivalent series inductance) and esr (equivalent series resistance) [START_REF] Salcone | Selecting Film Bus Link Capacitors for High Performance Inverter Applications[END_REF].

To simplify the representation of the 3x18 electrolytic capacitors, a global equivalent device has been considered, including the equivalent Esl of all capacitors, as well as all associated interconnects (Capacitor tank is displayed on the top of Fig. 6). For the 8 decoupling capacitors, the inductive link between the reference points mod+ and the plus of the capacitors pins, and mod-and the minus of the capacitors pins, is composed of three main parts: a main busbar (in blue and red in Fig. 6), mechanically supporting all 8 Cdec in parallel, and two phase busbars (dark red and magenta in Fig. 6) to close the loop.

After simulations, the impact of the main busbar, supporting the 8 Cdec was found very small in comparison with the two phase busbars, which have no overlap between plus and minus. Therefore, its impact has been neglect in comparison with the two lateral phase bus. This allowed simplifying the electrical representation, by replacing 8 Cdec with an equivalent circuit with Esr/8and Esl/8. The contribution of the phase busbars is included in the model of Fig. 7, with the equivalent inductances connected to the reference points Mod1+ and Mod1-.

This circuit of Fig. 7 can be compacted by associating the stray impedances in series (and re-computing the couplings), what leads to the generic circuit of Fig. 8. In this circuit, Rfilter and Lfilter are the stray resistance and inductance of all paralleled Capacitors of the DC bus tank (Cfilter), in series with the parasitic behavior of the bus bar; Rdec and Ldec are the stray resistance and inductance of all paralleled Cdec, in series with the parasitic behavior of the bus bar; Rpack and Lpack are the stray resistance and inductance of the device package [START_REF]Application Note AN2011-05 Industrial IGBT Modules Explanation of Technical Information[END_REF], in series with the parasitic behavior of the bus bar.

Equations ( 1) to [START_REF] Timms | New Boundary Condition for Decoupling Capacitance Selection in SiC Phase Legs Considering Short-Circuit Events[END_REF] provide the link between the elements of Fig. 7 and those of Fig. 8. The expressions of the couplings are given by ( 7)-(18). They are not displayed on these figures for clarity. To be noticed that couplings between capacitors and the rest of the structure have been neglected. All resistances of the interconnects have been evaluated from Ansys Q3D at 1 MHz. This overestimates the resistive effects, which depend on skin effect. However, using complex equivalent circuits to represent frequency dependent resistance in a time domain simulation was not considered necessary.

𝐿𝑓𝑖𝑙𝑡𝑒𝑟 = 𝑒𝑠𝑙 + 𝐿1 + 𝐿2 + 2. 𝑀12 (1)
𝑅𝑓𝑖𝑙𝑡𝑒𝑟 = 𝑒𝑠𝑟 + 𝑅1 + 𝑅2 + 2. 𝑅12 (2)
𝐿𝑑𝑒𝑐 = 𝑒𝑠𝑟2 + 𝐿3 + 𝐿4 + 2. 𝑀34 (3)
𝑅𝑑𝑒𝑐 = 𝑒𝑠𝑟2 + 𝑅3 + 𝑅4 + 2. 𝑅34 (

B. Switching effect representation

To avoid time simulation with semiconductor models, which are sometimes originating convergence issues, a simple model reproducing the effects of switching has been used. This model replaces the power switches by a simple current source Iex, with a parallel impedance (Norton model, Fig. 9). This impedance is defined by the output capacitance of the semiconductor, Coss, to keep a physical link between the proposed model and the real switch. This approach has already been used for instance in [START_REF] Chen | A frequency-domain study on the effect of DClink decoupling capacitors[END_REF], [START_REF] Liu | EMI Suppression in Voltage Source Converters by Utilizing dc-link Decoupling Capacitors[END_REF]. The current source is roughly a linear current slope with the same turn off speed (dI/dt) as the power switch. Therefore the number of parameters is reduced to the minimum: the switching speed (dI/dt) and Coss. These two elements, together with stray inductances and decoupling capacitance are the main responsible of switching events, overshoot and ringing [START_REF]Design Considerations for Designing with Cree SiC Modules Part 1. Understanding the Effects of Parasitic Inductance[END_REF].

The circuit model corresponding to the power converter ONIX 671 is presented in Fig. 10. The DC operating point is represented with current and voltage sources: Cfilter is large enough to keep a constant voltage during commutation, so it is replaced by a DC voltage source, Vbus (800V in the following). Of course, the corresponding stray elements are kept.

Iex is a square waveform with an arbitrary duty cycle, swinging from 0 to desired current (1000A in the following). Ibus provides the average value of Iex, so if duty cycle is 0.5, Ibus = 500A. Note that this choice of duty cycle and Ibus does not fix the switching conditions and is completely arbitrary.

As explained before, all values of the stray elements displayed in Fig. 10 have been obtained with a 3D simulation using Q3D. The mutual coefficients between all stray inductances and resistances have also been computed, but in the particular case of this geometry, they were so small that they have been neglected.

The esl and esr values of the filter and decoupling capacitors have been identified through impedance measurements: esr = 1m, esl = 25nH for Cdec. The low esr /low loss is one of the reason for choosing this technology for railway, as well as robustness.

III. DEFINING THE OPTIMAL DECOUPLING CAPACITANCE

A. Optimization problem statement

We are looking for the minimal decoupling capacitance value since it will correspond to the minimal volume and therefore the best possible integration. This optimum decoupling capacitance is obtained above a specific value Cdec_opt which provides no more benefit on the voltage overshoot V = VMOS peak -Vbus. The minimum voltage overshoot Vmin is obtained for an "infinite" Cdec value, actually Cdec = 10mF. Cdec_opt is deduced when the overvoltage V is 5% more than this minimum value, i.e. V = 1.05 x Vmin.

Using the proposed model to determine the minimal decoupling capacitor is easier than controlling time domain simulations, since the number of parameters is reduced (especially regarding the MOSFET and diode model), simulation time is much lower and no convergence issue can be expected.

B. Simple model for optimization

To validate the model of Fig. 10, experimental measurements have been carried out in a simple chopper configuration for the switching of one leg, using a double pulse test [START_REF]Design Considerations for Designing with Cree SiC Modules Part 1. Understanding the Effects of Parasitic Inductance[END_REF].

The setup is displayed in Fig. 11 and Fig. 12. It uses only one IGBT module and one phase. The objective is to measure the voltage overshoot on the top side IGBT for different values of decoupling capacitance and different switching speeds. The switched current was 1 kA and the voltage applied to the DC bus was 800 V. For voltage measurement, a 50 MHz bandwidth differential probe was used; the current measurement was performed with a Rowgoski probe.

The double pulse test was carried out for two switching speeds (2.5 kA/µs, 4.4 kA/µs) and the following decoupling capacitor values: 8x47uF; 4x47uF; 2x47uF; 1x47uF. The various dI/dt were obtained by modifying the gate resistor Rg of the driver.

The overvoltage as a function of the decoupling capacitance is displayed in Fig. 13. It is worth noting that when the amount of decoupling capacitors in parallel changes, the inductance and resistance of the switching loop also change. If Cdec increases (more capacitors in parallel) then Ldec decreases, and therefore the voltage overshoot Ldec . dI/dt also decreases. This phenomenon will be considered in the equivalent circuit, by changing the value of Rdec and Ldec according to Cdec change (Table I). In experiment, it was only possible to use multiple values of 47µF (due to capacitor availability and bus bar structure), so a simulation approach was also considered to get results with other capacitor values. The circuit of Fig. 10 has therefore been used on Ansys Simplorer and results compared with the measurements, to validate this precise model. A dedicated IGBT model, mainly based on the MOSFET behavior, was used and its parameters extracted from datasheet (Mitsubishi CM1200DC-34S [START_REF]Mitsubishi High voltage insulated gate bipolar transistor[END_REF]). The switching speed in simulation was controlled by varying the value of the gate resistance Rg, as in experiments. Cdec, Rdec and Ldec have been changed according to the number of paralleled Capacitors, as indicated in Table I. The results are presented in Fig. 14. Despite some differences in the absolute values, the evolution of the curves is very similar. Even if the voltage measurement was carried out with care, using a very low inductance path for the voltage probe, the voltage measurement may have been disturbed by external field, what might explains the difference between the experiment and the simulation. An interesting property is to be noticed, especially for dI/dt = 2.5 kA/µs: the voltage overshoot is minimal for Cdec = 4 x 47µF, and increases for 8 x 47µF. This results from the interaction of resonances between all elements of the circuit (Coss, Cdec and stray inductances), which originate a minimal voltage spike for this specific value of Cdec. Unfortunately this property cannot be used to obtain an optimal value for Cdec, due to too high sensitivity to all stray elements.

Once validated in comparison with experiment, this circuit has been used to extend the capacitance variation Cdec. For each time domain simulation, the values of Cdec, Ldec, Rdec change according to ( 13)-( 15 The simplified model of Fig. 10 (for 8 x 47µF) has then been compared with the precise simulation of Simplorer. Fig. 16 shows the overvoltage on the IGBT terminals as a function of the decoupling capacitance (from 10µF up to 10mF i.e. "infinity value"), for a dI/dt of 4,4 A/µs. Even if there is a difference, the general trend is the same, what is important for optimization purpose. The simple model is thus able to reproduce the impact of a change in Cdec. The differences can be explained since the current source in the model uses a linear behavior, which is not the case in reality and in Simplorer simulations. In addition, changing the value of Cdec, Ldec and Rdec elements of the switching loop has an impact on the switching speed (variation of ± 200 A/µs noticed during experiments).

Since the simplified model gives the correct trend, it has been implemented in an optimization process to obtain the optimal value of the decoupling capacitance. This is described in section C.

C. Optimization process

The optimal value of Cdec is obtained using the function 'fzero' from Matlab®. It solves the circuit presented in Fig. 10 in Simulink®, calculating the semiconductor voltage in the time domain using trapezoidal resolution method. Values of Cdec, Rdec, Ldec are changed according to ( 13)- [START_REF]Design Considerations for Designing with Cree SiC Modules Part 1. Understanding the Effects of Parasitic Inductance[END_REF]. The current commutation speed is defined in Iex source. The Matlab function returns the optimal value of Cdec that minimizes the objective function defined as (16):

𝐹 𝑜𝑏𝑗 = (𝑉 𝑀𝑂𝑆 𝑝𝑒𝑎𝑘 -𝑉 𝑏𝑢𝑠 ) -1.05. (𝑉 𝑀𝑂𝑆 𝑝𝑒𝑎𝑘 𝑚𝑖𝑛 -𝑉 𝑏𝑢𝑠 ) (16)
VMOS peak is the semiconductor peak voltage obtained with Cdec, whereas VMOS peak min is the one obtained with an "infinite" Cdec (actually 10 mF).

Fig. 16 shows the evolution of the voltage overshoot ∆Vmos as a function of Cdec, for 3 different commutation speed. Cdec_opt is indicated with a star on the curves. The voltage overshoot obtained at Cdec_opt is 5% more than the minimal possible value, according to our optimization strategy. To be noticed that the minimum overvoltage obviously increases with dI/dt.

By comparing the three test cases from 2.5kA/µs to 5.2A/µs, (Fig. 16) it can be seen that the optimal decoupling capacitance for a faster system (373 μF) is lower than for a slower system (410 μF) using our condition regarding the relative overvoltage definition. This is quite intuitive since higher dI/dt implies higher equivalent frequency, and therefore a lower value of capacitance is needed for reaching the same impedance. However, this is much higher than the empirical method, which suggests using 50-100 times Coss (i.e. 400 nF-800 nF in the considered application) [START_REF] Chen | Electrical Integration of SiC Power Devices for High-Density Applications[END_REF] [START_REF] Chen | A frequency-domain study on the effect of DClink decoupling capacitors[END_REF]. It shows that changing the criterion of Cdec selection has a huge impact on its value. This optimal result is actually really close from the original value of Cdec (8 x 47 μF = 376 μF) in the ONIX 671 converter, but it is also linked to the choice of technology (Esl and Esr of the capacitor chosen as a reference, busbar, semiconductor package).

To check the impact of the objective function, it has been changed from 5% to 10% of the overshoot obtained for an infinite Cdec. The optimal value Cdec_opt changes from 410 μF to 181 μF for dI/dt = 4,4k A/μs and from 380 μF to 197 μF for dI/dt = 2,5 kA/μs. This is quite interesting for the designer: accepting a voltage overshoot a bit larger may help in reducing the value of decoupling capacitor, and therefore facilitate its integration. Since all stray elements and switching speed impact the optimal decoupling capacitance, a sensitivity study has been carried out in the next section.

IV. SENSIBILITY STUDY

The first parameter to be changed was the switching speed. In the previous section, it was already noticed that a change of a factor 2 in the switching speed was only originating a variation of 10% in the optimal value of decoupling capacitance (410 µF to 373 µF). However, higher change has been investigated, corresponding to a change from Silicon IGBT to Silicon Carbide MOSFET. The results (without changing the other elements of the circuit) is a huge decrease of the optimal decoupling capacitor, as displayed in Table II. Unfortunately, even if the gain in term of Cdec_opt is impressive, one has to take care the overvoltage which will occur on the device: without changing the package (Lpack = 41nH) the situation of dI/dt as high as 33 or 52 kA/µs is not possible, leading to voltage overshoots of some kilovolts. Of course the full interconnect topology has to be redesigned when changing from IGBTs to SiC MOSFET. The proposed optimization method will be useful to determine the optimal decoupling capacitance corresponding to the new low inductive layout, as it will be illustrated in the conclusion.

To help in understanding the effect of other parameters, few additional sensibility studies have been carried out.

Lpack obviously directly influences the overvoltage. However, during the sensibility study, it was noticed that increasing Lpack allows reducing the value of Cdec_opt. This was not intuitive at all, but it is useless since the aim is not to increase the voltage overshoot.

The influence of Ldec (which originates from both Esl and bus bar stray inductive behavior) has also been studied. Once again, increasing Ldec results in higher voltage overshoot. Overall, Ldec and Lpack have the same role as they are both part of the switching loop. From a design point of view, there is no interest in increasing the value of Lpack and/or Ldec to reduce Cdec as this would increase the overvoltage.

Another remark is that for low values of the parasitic inductance of the capacitor tank, Lfilter, the electrolytic capacitors participate in decoupling and the voltage overshoot is reduced. Obviously, it is not possible to reduce Lfilter too much, otherwise Cdec would not be necessary at all.

As noticed in previous section, increasing the switching speed (dI/dt) decreases the need for a high Cdec value, and at the same time increases the voltage overshoot. The overall impact of dI/dt on Cdec is actually limited by the values of the parasitic inductances.

By applying the same approach for different parasitic resistance values of the decoupling capacitor, Rdec, it has been observed that the Cdec_opt changes considerably (Fig. 17). This is explained by an increased damping of the MOSFET voltage. Due to the large number of circuit equations, it was not easy to predict this with classic second-order models. For Rdec between 2 mΩ and 10 mΩ, the optimal decoupling capacitor remains around 250 µF, but if Rdec = 50m Ω, the optimal decoupling capacitor is divided by two, and even by 10 for Rdec = 100 m. In order to verify the impact of a resistor in series with the decoupling capacitor, the ONIX 671 structure has been simulated using Simplorer, with dI/dt = 5.2A/µs and an additional resistance of 100 m (Fig. 18). The value of Cdec_opt is displayed on this figure. The losses in the additional resistance have been evaluated around 130W in a real situation of PWM inverter, which for this power range may be acceptable, if it can be evacuated. However the gain on the integration has to be carefully evaluated, since this solution is clearly not easy to implement.

All previous results are summarized in Table III, which illustrates the impact of all stray elements on both overvoltage and optimal value of decoupling capacitor.

V. CONCLUSION

The aim of this paper was to propose a methodology to size the minimal decoupling capacitor for high power traction inverters used in railway. The proposed condition was the voltage overshoot in the time domain, which results from complex interactions between Coss, Cdec and all stray inductances brought by the 3D layout and components. The chosen condition was to define the "optimal" decoupling capacitance when the voltage overshoot is 5% more than using an infinite Cdec.

First, a generic method to account for the stray inductance of the geometry has been introduced. This is more than roughly evaluating the contribution of a stray inductive behavior with some order of magnitude, but really provides a close link with the layout and the number of devices in parallel used to realize the decoupling function.

To be implemented in an optimization process, the time domain simulation using precise semiconductor models is not very convenient, due to potential convergence issues and amount of time. Therefore, a simple equivalent model using a linear slope for dI/dt associated with the device's capacitance has been used. This model has been validated in comparison with precise simulations, these ones being validated themselves with measurements. This simple model used in optimization has allowed obtaining the minimal value of decoupling capacitance according to the chosen criterion, in the example of the traction inverter ONIX 671. The value is much higher than the one obtained in previous work using different criterion and lower power.

Sensibility study has been carried out to evaluate the most impacting factors on Cdec_opt, and it has been shown that the commutation speed may help in decreasing this value, but also that this should be combined with a clear reduction of the stray elements, to avoid large voltage overshoots. Indeed, the proposed criterion checks the relative voltage overshoot, but the absolute voltage has also to be limited. Therefore, the proposed design method should be integrated in a full design process, as illustrated in Fig. 19.

Finally, the possibility of further reduction of Cdec_opt using an additional series resistance has been proposed, but should be considered with care, regarding the amount of losses generated and heat dissipation issues for high power structures.

Fig. 1 .

 1 Fig. 1. Simplified switching cell and resonant circuits.

Fig. 2 .

 2 Fig. 2. Time domain simulation of the semiconductor voltage for different values of Cdec. Different behaviors and instant of maximum voltage.

Fig. 3 .

 3 Fig. 3. Alstom converter ONIX 671. Top: 3x18 Electrolytic capacitor association (filter capacitor). Bottom: 4 power bus bars (2 used for 2 legs), 8 decoupling capacitors and 12 IGBT dual power modules 1.7kV / 1200A (3 per phase).

Fig. 4 .

 4 Fig. 4. Electrical schematics of phase W and rheostatic leg of power converter ONIX 671 from Alstom.

Fig. 5 .

 5 Fig. 5. Example of a circuit representation considering its input/output ports and reference port (real and imaginary couplings not displayed in this figure).

Fig. 6 .

 6 Fig. 6. Identification of the common points Mod1 + and Mod1-on the bus bars that connect the filter capacitors (top) and the bus bars that connect the decoupling capacitors and the semiconductors (bottom).

4 ) 5 )

 45 𝐿𝑝𝑎𝑐𝑘 = 𝐿𝑚 + 𝐿5 + 𝐿6 + 2. 𝑀56 + 2. 𝑀6𝑚 + 2. 𝑀5𝑚 (𝑅𝑝𝑎𝑐𝑘 = 𝑅𝑚 + 𝑅5 + 𝑅6 + 2. 𝑅56 + 2. 𝑅6𝑚 + 2. 𝑅5𝑚 + 𝑀16 + 𝑀1𝑚 + 𝑀25 + 𝑀26 + 𝑀2𝑚 (9) 𝑅𝑓𝑖𝑙𝑡𝑒𝑟_𝑝𝑎𝑐𝑘 = 𝑅15 + 𝑅16 + 𝑅1𝑚 + 𝑅25 + 𝑅26 + 𝑅2𝑚 (10) 𝑀𝑑𝑒𝑐_𝑝𝑎𝑐𝑘 = 𝑀35 + 𝑀36 + 𝑀3𝑚 + 𝑀45 + 𝑀46 + 𝑀4𝑚 (11) 𝑅𝑑𝑒𝑐_𝑝𝑎𝑐𝑘 = 𝑅35 + 𝑅36 + 𝑅3𝑚 + 𝑅45 + 𝑅46 + 𝑅4𝑚 (12)

Fig. 7 .

 7 Fig. 7. ONIX 671 equivalent circuit (before simplification). Coupling not displayed for clarity

Fig. 9 .

 9 Fig. 9. Proposed model to simulate the MOSFET voltage overshoot.

Fig. 10 .

 10 Fig. 10. Complete model circuit of power converter ONIX 671 for Cdec = 8 x 47µF. The operating point is Vbus = 800V, Iex is a square waveform with 0.5 duty cycle swinging from 0 to 1000A, and Ibus = 500A to provide the average value of Iex..

Fig. 11 .

 11 Fig. 11. Experimental bench using ONIX 671 power converter.

Fig. 12 .

 12 Fig. 12. Simplified equivalent circuit of the test bench from Fig. 11.

Fig. 13 .

 13 Fig. 13. IGBT overvoltage from Alstom test bench, in function of the decoupling capacitor value for two switching speeds (4.4kA/µs and 2.5kA/µs).

  the constant terms in (14) and (15) being the contribution of the busbars.

Fig. 14 .

 14 Fig. 14. Simulation results of the model in Simplorer simulation (♦) and test bench measurements (■) for two switching speeds.

Fig. 15 .

 15 Fig. 15. Overvoltage on the IGBTs in function of decoupling capacitance (Simplorer and simple models comparison).

Fig. 16 .

 16 Fig. 16. Voltage overshoot vs Cdec and Cdec_opt.

Fig. 17 .

 17 Fig. 17. MOSFET peak voltage as a function of the decoupling capacitor (Cdec) for different parasitic resistance values of Rdec (bus bar and esr of the decoupling capacitors).

Fig. 18 .

 18 Fig. 18. Overvoltage on the IGBT vs decoupling capacitance with and without a 100 m series resistance Rsupp (Simplorer simulations).

Fig. 19 .

 19 Fig.19.Illustration of the proposed optimal sizing method inside a full 3D layout design process.

TABLE I

 I

	.	SUMMARY OF DIFFERENT TESTS CONDITIONS
	Test Condition	Cdec [µF]	Ldec [nH]	Rdec m
	1 x 47µF	47	35.3	2.9
	2 x 47µF	94	22.8	2.4
	4 x 47µF	188	16.5	2.1
	8 x 47µF	373	13.4	2.0

TABLE II .

 II IMPACT OF CURRECT COMMUTATION SPEED dI/dt [A/µs] Cdec_opt [µF]

	5.2	373
	33	2.6
	52	1.6

TABLE III

 III

	.	SUMMARY OF SENSIBILITY STUDY
	Parameter	Cdec_Opt	Overvoltage
	Lpack	+	++
	Ldec	+	++
	Lfilter	+	+
	dI/dt	++*	++
	Rdec	++	+**

* limited by maximum allowed overvoltage and stray inductances ** for reasonable values of Rdec