text:
Carmen Riveiro Moreno

Marie Couliou

Nicolò Fabbiane

Olivier Marquet

Reynald Bur

Riveiro Moreno

Experimental characterization of synchronized interactions between a normal shock-wave and compliant wall

Keywords: Fluid structure interaction, Shock/Boundary Layer Interaction, compressible flow

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Introduction

Due to minimum weight requirements, the structure of high speed vehicles consists of thin panels or shells-like components that are subjected to large aerodynamic and aerothermal loads caused by supersonic and hypersonic flows. Their reduced thickness and thus flexibility enhance the receptivity of the structure to flow induced vibration which, in case of coupling, can be detrimental to the safety of the vehicle and thus should be taken into account during the design process [START_REF] Jordan | Structural Dynamic Experiences of the X-15 Airplane[END_REF]. The compressible flow around such high speed vehicles is characterized by the presence of shock-waves which interact with the incoming boundary layer, causing large adverse pressure and thermal gradients, as well as unsteady three dimensional separated flow [START_REF] Clemens | Low-frequency unsteadiness of shock wave/turbulent boundary layer interactions[END_REF]. Hence, better understanding the aeroelastic coupling between shock-waves and flexible walls is fundamental for the design of future high-speed flight vehicles.

Several research groups have conducted experimental campaigns with non intrusive measurement techniques such as Digital Image Correlation (DIC), Pressure Sensitive Paint (PSP) and Particle Image Velocimetry (PIV) among others to describe the fluid-structure interaction. At Imperial College of London, research has focused on the impact the static deformation of the thin panel and the relative position of the shock to it have on the interaction [START_REF] Gramola | Experimental FSI study of adaptive shock control bumps[END_REF][START_REF] Gramola | Off-design performance of 2D adaptive shock control bumps[END_REF][START_REF] Jinks | Wind tunnel experiments with flexible plates in transonic flow[END_REF][START_REF] Tan | Oblique shockwave boundary layer interaction on a flexible surface[END_REF][START_REF] Gomez-Vega | Oblique shock control with steady flexible panels[END_REF]. By varying actuators and cavity pressure, they achieved a panel deformation that recreates the effect of Shock Control Bumps (SCB) [START_REF] Bruce | Unsteady shock wave dynamics[END_REF], thus causing drag reduction and stabilization of the shock unsteadiness. The experimental campaigns conducted in Air Force Research Laboratory (AFRL) Research Cell 19 monitored the influence of thermal stress on the thin panel, cavity pressure, and shock intensity [START_REF] Spottswood | Exploring the response of a thin, flexible panel to shock-turbulent boundary-layer interactions[END_REF][START_REF] Brouwer | Experiments on a Thin Panel Excited by Turbulent Flow and Shock/Boundary-Layer Interactions[END_REF]. Brouwer et al. found that in the presence of a weak shock the panel presented limit cycle oscillations, and that these disappear and reemerge as the shock intensity was increased. The impact of temperature has also been studied in DLR at the H2K wind tunnel, where the interaction behaviour was found to be highly sensitive to small changes in temperature [START_REF] Daub | Experiments on aerothermoelastic fluid-structure interaction in hypersonic flow[END_REF]. Additionally, it was found that larger incident shock angles prompt the onset flutter when in a non-shock configuration it would not take place. At Florida Center for Advanced Aero-Propulsion, the impact of the compliant wall shape due to change of cavity pressure as well as shock impingement were studied [START_REF] Tripathi | Response of a compliant panel to shock boundary layer interaction at mach 2[END_REF]. In the work by Eitner et al. [START_REF] Eitner | Effect of Shock-Wave Boundary Layer Interaction on Vibratory Response of Compliant Panel[END_REF], several plate thicknesses were tested. Their results show that the shock oscillates at the first vibration frequency of the panel, being this the first bending mode. Both the panel and the shock-wave present larger oscillations for thinner plates. The tests conducted by A. D'Aguanno [START_REF] D'aguanno | Characterization of shock-induced panel flutter with simultaneous use of DIC and PIV[END_REF] present a shock that oscillates at the two first vibration frequencies of the plate, corresponding to the first and third bending modes. Despite the effort of the research community, the physics behind the coupling between a shock-wave and a compliant wall is still not fully understood. Moreover, most study cases involve an oblique shock caused by either a compression ramp [START_REF] Eitner | Effect of Shock-Wave Boundary Layer Interaction on Vibratory Response of Compliant Panel[END_REF] or a shock generator [START_REF] Tan | Oblique shockwave boundary layer interaction on a flexible surface[END_REF][START_REF] Gomez-Vega | Oblique shock control with steady flexible panels[END_REF][START_REF] Spottswood | Exploring the response of a thin, flexible panel to shock-turbulent boundary-layer interactions[END_REF][START_REF] Brouwer | Experiments on a Thin Panel Excited by Turbulent Flow and Shock/Boundary-Layer Interactions[END_REF][START_REF] Daub | Experiments on aerothermoelastic fluid-structure interaction in hypersonic flow[END_REF][START_REF] Tripathi | Response of a compliant panel to shock boundary layer interaction at mach 2[END_REF]. For normal transonic shocks [START_REF] Gramola | Experimental FSI study of adaptive shock control bumps[END_REF][START_REF] Gramola | Off-design performance of 2D adaptive shock control bumps[END_REF], the interaction has been less explored, and not focused on a strong dynamic coupling scenario. The aim of this study is to shed new light on the fluid structure interaction with the presence of a normal shock-wave (transonic regime). For this purpose, the present work first characterizes the natural unsteady normal shock oscillations over a rigid wall. Knowing its dynamics, a compliant wall is designed so that its natural frequency is of the same order of magnitude as the shock oscillations. Lastly, the fluid structure interaction of the oscillating shock and the compliant wall is studied by high speed schlieren imagery and an optical displacement sensor.

Experimental investigation of a shock wave on a rigid wall

Experimental facility and measurements

Experiments are conducted in S8Ch ONERA's transonic open circuit wind tunnel in Meudon (France), shown in Fig. 1a. This wind tunnel, has been largely used for experimental studies on passive and active (fluidic) control of shock boundary-layer interaction [START_REF] Bur | Study of passive control in a transonic shock wave/boundarylayer interaction[END_REF]. The stagnation temperature in the settling chamber is 300 ± 10 K and the stagnation pressure 96.0 ± 0.3 kPa. For this study, the wind tunnel is equipped with a half nozzle that generates an unsteady normal shock-wave with a nominal upstream Mach number of 1.4 [START_REF] Bur | Experimental and numerical study of forced shock-wave oscillations in a transonic channel[END_REF]. The origin of the cartesian coordinate system (O,x,y) is located at 370 mm from the end of the converging part, as indicated in Fig. 1a, close to the end of the half nozzle. The size of the test section is 100 mm in the wall-normal (y) direction and 120 mm in the spanwise (z) direction. The wind tunnel is equipped with optical accesses on the two lateral doors (white area) and on the top wall of the test section (green area). showing the cartesian coordinate system (0,x,y) and the optical access for the schlieren visualization (white area) and the wall-tracking (green area). On the top and bottom walls, static pressure probes are installed to retrieve (b) static wall-pressure as a function of the streamwise position x (bottom wall), here for the nominal shock configuration 4 (see table 1). The flow is "chocked" in the test section by means of a second throat located at the outlet which isolates the test section from downstream perturbations. Varying the height h of the second throat (see Fig. 1a) allows changing the (average) streamwise position xs of the shock in the test section. As the measurement of h is not accurate enough, we only report in Table 1 the shock position for various cases investigated in the present study.

Several measurements are performed to characterize the flow in the nominal shock configuration, number 4 in Tab. 1. The average wall-pressure is measured on the centerline (z = 0) of the bottom and top walls using 60 static pressure sensors. Their localisation is illustrated with blue squares in Fig. 2a and typical measurements are shown in Fig. 1b as a function of the streamwise position x, for the nominal configuration. Note that unsteady pressure sensors (Kulite transducers type XCQ-093-15A) are also used to measure the wall-pressure fluctuations at fewer spatial positions on the bottom wall, but results are not reported here. The interested readers are referred to [START_REF] Moreno | Interaction of shockwaves with a compliant wall[END_REF] for a discussion about these unsteady pressure measurements. The flow is also analysed by means of schlieren imagery. Schlieren visualizations are conducted with a Z-type Schlieren setup with a high speed Phantom camera and a blue LED source. A vertical knife orientation is used, which yields images whose light intensity is proportional to horizontal density gradients. This orientation enhances the visibility of the shock on the schlieren image. The schlieren videos are acquired at a sampling frequency of 7 kHz for a duration of 3.87 s, which used the maximum camera memory capacity. The schlieren images are used to track the location of the shock as shown in Fig. 2a. This was conducted on the normal part of the shock-wave (Fig. 2a, white rectangle). Therefore, for each snapshot, the region displaying the upper and lower shock feet is cut out of the images (Fig. 2b). The schlieren images are reported in grey scale, where an intensity value of 255 corresponds to white and 0 to black. In order to increase the brightness of the shock, all pixels whose intensity are above 180 in the grey scale are set to 255 (white) (Fig. 2b). For each y location, the location in x of the most downstream pixel whose intensity is 255 (white) is recorded. As a result, one obtains the temporal displacement of the shock for each y location on the normal shock. As similar displacement signals are retrieved for different locations along y, the middle position y = 50 mm will be used to represent the motion of the shock in this study (Fig. 2, blue dot).

Description of the nominal rigid configuration

We now detail the nominal configuration denoted 4, highlighted in bold in the table 1. The Mach number upstream of the shock is equal to 1.35. Fig. 1b shows the pressure along the lower wall of the test section for this configuration. This pressure distribution reveals the high adverse pressure gradient the normal shock imposes on the flow. At the shock location xs (mm) = -4.8, no pressure plateau is visible meaning that the shock does not induce a massive separation or recirculating flow. The schlieren image in Fig. 2a presents an instantaneous image of the flow in a zoom region around the shock. In it, it is possible to observe a normal shock (A) with its λ shaped shock foot, with the front leg (B) composed of a compression fan, and the rear leg (C) consisting of an oblique shock. The existence of a λ region confirms the presence of shock induced separation, although, for this configuration, it is small. Note that the apparent thickness of the shock is due to a spanwise integration of light.

In order to study the dynamics of the shock, x s , the location in x of a point at the shock at y = 50 mm (Fig. 2a, blue dot) was tracked in time. The fluctuating part, x ′ s = x s -xs , of the resulting signal is presented in Fig. 3a (grey). For the computation of the power spectrum density using the Welch method, 2048 samples per block were used with 50% overlap. The premultiplied spectrum of the shock displacement signal is presented in Fig. 3b (grey). The low frequency peak present in the spectrum, below 20 Hz, is generated by fluctuations of the stagnation pressure in the settling chamber. These fluctuations were found to be related to vibrations of the wind tunnel installation. On the other hand, the high frequency content, above 1kHz, is due to noise steaming from the tracking method to retrieve the shock location. Thus, the spectrum of the natural unsteady shock oscillation corresponds to the non shaded area in Fig. 3b (grey). Fig. 3b presents a zoom of the spectrum in this area. As it can be seen, the shock oscillations are low frequency and broadband in nature, in agreement with results reported in literature [START_REF] Sartor | Unsteadiness in transonic shock-wave/boundarylayer interactions: experimental investigation and global stability analysis[END_REF][START_REF] Erengil | Unsteady wave structure near separation in a Mach 5 compression rampinteraction[END_REF][START_REF] Dupont | Space and time organisation of a shock wave/turbulent boundary layer interaction[END_REF][START_REF] Clemens | Low-frequency unsteadiness of shock wave/turbulent boundary layer interactions[END_REF]. For the present study, the low frequency bump is centered around 100 Hz. In order to understand the contribution of the natural shock oscillations to the total shock displacement signal, the signal was filtered with a Butterworth filter of order 2, a low pass at 20 Hz and a high pass at 1000 Hz. The resulting signal is presented in Fig. 3a and the spectrum in Fig. 3b-c (black). As it can be seen in the spectrum, within the frequency range of interest (Fig. 3b-c), the filtered spectrum matches that of the raw data, whereas the forcing from the settling chamber and the spurious oscillations have been removed. Hence, the oscillations present in the filtered signal in Fig. 3a (black), correspond to the oscillations steaming from the interaction of the shock with the incoming boundary layer. Since the forcing from the settling chamber and the spurious noise coming from the shock tracking method are not within the frequency range of the natural shock oscillations (Fig. 3b-c), there is no need to filter the shock displacement signal. For a better comprehension, all the shock displacement spectra will be presented only from 20 Hz to 1000 Hz in this study.

0 1 2 3 t [s] -2 0 2 x ′ s [′′] (a) Shock displacement signal
3. Investigation of the shock-wave/compliant-wall interaction

Description of the compliant wall

The compliant wall is made of a polyurethane block that is inserted in an aluminum container. The streamwise (x) and spanwise (z) lengths of this container are respectively fixed to 160 and 85 mm, the latter being constrained by the spanwise length of the test section (120 mm). The thickness (y) of this container may be varied but will be fixed in the present study to 20 mm. As shown later, this thickness allows the lowest natural vibration modes of the compliant wall to be in the range of frequency 30Hz ≤ f ≤ 200Hz characterizing the shock oscillation. As shown in Fig. 4a, the container is placed on the lower wall of the test section. Its midpoint in the streamwise direction coincides with the origin of the cartesian coordinate system, i.e. x = 0. Therefore, for the nominal configuration (denoted 4) described before, the shock is almost located at the center of the compliant wall, as sketched in Fig. 4b. The polyurethane block is clamped to the four lateral surfaces of the container but is free to move at the bottom and top surfaces. The top surface is the interface with the flow and the pressure distribution therefore depends on the shock position. Under the bottom surface, the pressure is constant but not necessarily equal to the atmospheric pressure (leaks, gaps, ...). Its value is measured using a pressure probe so as to obtain the pressure difference between the top and bottom surface, and it was found to be equal to the average pressure on the test section.

Optical measurements of the wall displacement

The vertical displacement of the compliant wall in the symmetry plane (z = 0) of the test section is measured using two non-intrusive optical methods.

The first method is based on the schlieren images. As the compliant wall deforms upwards, it blocks the light projecting a shadow which translates into a black area on the schlieren image (Fig. 5a). Since the schlieren method provides a spanwise integration of light, this deformation, corresponds to the highest displacement of the material, which is found at the centerline (z = 0). Thus, the schlieren images provide access to the compliant wall's deformation at z = 0. For this purpose, the width of the images matches the length of the compliant wall. The first step to extract this data from the schlieren images is to crop the image so that only the lower test section wall and the lower shock foot are displayed. Then, a Sobel filter that computes the pixel intensity gradient on the y direction is applied to retrieve horizontal edges on the image, and thus the compliant wall edge. The resulting image, in grey scale, will display the edge of the compliant wall in black. Then, for each x position, the location of the compliant wall edge in y, pixel intensity equal to 0 (black), is retrieved. When this procedure is conducted on all the schlieren images, one obtains the temporal evolution of the compliant wall's displacement along x.

The videos are sampled at 7 kHz for a duration of 3.87 s. Vertical knife orientation is used so that horizontal density gradients are retrieved in order to enhance the visibility of the shock, which helps the tracking of the shock location as explained in §2.1. This knife orientation has also the advantage that it does not highlight the boundary layer, which would interfere with the wall tracking process. Nonetheless, this technique presents two disadvantages. Firstly, the lateral windows of the wind tunnel only allow optical access to the test section, meaning that any downward motion of the compliant wall is not captured (Fig. 5b). Secondly, the resolution of the displacement is linked to the image resolution in pixels, which for this study is 0.22 mm/pixel, which is troublesome when small displacements are at play. Another way the compliant wall's displacement was measured was through a non-

EXPERIMENTAL CHARACTERISATION OF SYNCHRONIZED INTERACTIONS BETWEEN A NORMAL SHOCK AND A COMPLIANT WALL

intrusive optical pointwise displacement sensor Keyence LK-G402. The resolution of this sensor is 0.2 µm, much higher than the one provided by the schlieren. The measurements were conducted through the optical access on the test section (Fig. 5, green rectangle), which allows downward displacement measurements (Fig. 5b), not available with the schlieren setup.

The sensor was mounted on a metallic structure outside of the wind tunnel facing the wind tunnel's upper wall, which is also equipped with a window. The sensor points at location x sensor = 27 mm and z sensor = 2 mm (Fig. 5, red dot). Due to restrictions on the optical window of the test section, it was not possible to install the sensor perpendicular to the compliant wall. Therefore the measured displacement is not along the y axis. Additionally, owing to reflection issues, data was not available for shock configurations 3 and 4. Nonetheless, for the reminding configurations, its average value can still be used qualitatively, and this does not entail any problem for the dynamic analysis of the compliant wall's motion. The signal for all shock configurations was acquired at 20 kHz and 60 s. The signal (grey) is superimposed with the filtered signal (black) which will be later explained. The amplitude of the oscillations is less than 0.1 mm, meaning that these fluctuations would not be captured by the schlieren images as their resolution is 0.22 mm per pixel. The power spectrum density of the entire signal is presented in Fig. 6b (grey), and was obtained through the Welch method using 8192 samples per block and 50% overlap. The grey areas presented in the spectrum in Fig. 6b, refer to the frequency range where spurious data has been found on the shock displacement spectrum on the rigid configuration (Fig. 3b). The spectrum of the wall displacement, presents no peaks below 20 Hz (grey area on the left), meaning that the low frequency forcing of the flow coming from the settling chamber (presented in §2.1), does not interact with the compliant wall. The same result was found in all configurations. On the grey area on the right, starting at 1000 Hz, there are frequency peaks, related though to electronic noise. This implies that also in the compliant wall dynamics, high frequency content on the spectrum is spurious. So as to mitigate the effect of the non-physical peaks, a high pass 6th order Butterworth filter at 600 Hz was applied to the displacement signal. The parameters of the Butterworth filter were chosen to include in the attenuation the first non-physical peak found already at 800 Hz. The resulting filtered signal and spectrum are represented in Fig. 6 in black. As shown in Fig. 6a the filtered (black) and raw (grey) signals are coincident, meaning that the contribution of the high frequency noise to the displacement signal is barely relevant. Fig. 6c presents a zoom on the frequency range of interest, where it is possible to observe frequency peaks that correspond to the different vibration frequencies of the compliant wall. For a better interpretation, the different displacement spectra presented in this study are filtered with the aforementioned Butterworth filter and displayed from 20 Hz to 1000 Hz to allow a comparison with the shock displacement spectrum. Clearly, the experimental measurements are too sparse to allow a complete description of the wall deformation. Therefore, we will propose in §3.4 a method to reconstruct the instantaneous displacement by combining a numerical model of the wall displacement with the experimental data. The numerical modeling is detailed in the next section.

Experimental and numerical characterisation of the compliant wall's displacement

The mechanical properties of polyurethane were characterised with shear torsion tests on a rheometer at Laboratoire de Mécanique des Solides at École Polytechnique. This provided the evolution of the complex shear modulus G * = G ′ + iG ′′ as a function of frequency for the temperature expected at the surface of the wind tunnel. For the frequency range of the shock oscillations, the real part of the shear modulus, G ′ , is practically constant and equal to 1.0095 MPa. Using the Lamé coefficients and taking into account the correspondence principle of viscoelasticity, the Young storage modulus E ′ can be deduced from the shear storage modulus G ′ . This yields a Young Modulus E ′ equal to 3.03 MPa. The density of the material was found to be ρ s ∼ 1000kg/m 3 .

To validate this value, we propose to compare the static deformations measured experimentally and predicted numerically. For the experiments, it is advantageous to consider configuration 8 where the shock position (x = 73.9 mm) is closer to the downstream side of the compliant wall (x = 80 mm). Indeed, the wall-normal displacement of the compliant wall is upward (y w > 0) for most of the streamwise positions and can thus be captured with schlieren vizualisations. The time-averaged wall-normal displacement of the compliant wall is shown in Fig. 7b with the solid curve in red. The maximal displacement is around 2 mm, i.e. 10 times smaller than the thickness of the material and is obtained around the middle point of the compliant wall x ∼ 0. For x > 60 mm, the wall displacement is downward (y w < 0) and thus not observable in the schlieren vizualisation. The red dot seen in Fig. 7b corresponds to the time-averaged displacement obtained from instantaneous pointwise measurements with the non-intrusive laser technique at the single point x sensor = 27 mm. The two techniques used to measure the wall displacement thus give coherent results although they do not perfectly match. The wall-normal static displacement of the compliant wall can also be estimated numerically as described in §A using the static aerodynamic wall-pressure difference, ∆P (x) = P (x) -P bottom , where P (x) is measured on the rigid wall and P bottom with a static pressure probe in the cavity. Fig. 7a presents the wall pressure difference for case 8. Compared to the nominal case (4, shown in Fig. 1b), the sharp streamwise variation of the pressure associated with the shock is located more upstream. In the present case, the pressure upstream of the shock is lower than the pressure measured under the compliant wall, and thus the pressure difference is negative. More specifically, for x < 60 mm, the pressure difference between the top and bottom interface induces an upward aerodynamic loading, while for x > 60 mm, a downward aerodynamic loading is applied. The numerical prediction of the static displacement induced by such aerodynamic loading is shown in Fig. 7b for the centerline z = 0. For the Young modulus E = 3.03 MPa measured with the rheometer, we obtain the displacement displayed with the dash-dotted curve. It clearly underestimates by a factor of 1.86 the displacement measured experimentally (solid curve red). By adjusting the value to E = 1.63 MPa, we obtain (dashed curve) a much better agreement with the experimental result. Therefore, we propose in the following to estimate the Young modulus of our material as E = 1.63 MPa. The dynamic of the compliant wall can also be estimated numerically by examining the natural vibration modes of the structures as described in §A. The wall-normal displacement of the top interface (compliant wall) is shown in Fig. 8 for the three eigenmodes corresponding to the lowest frequencies. The instantaneous displacements in the two-dimensional plane (x, z) are displayed in the top figures while the temporal evolution of the displacement at the centerline of the test section z = 0 is shown in the bottom figures. The first mode is characterized by maximal/minimal displacements at the center, i.e. (x, z) = (0, 0), oscillating at frequency f n 1 = 104.3 Hz. The second and third modes display a spatial oscillation in the streamwise direction. The second mode, oscillating at frequency f n 2 = 133.2 Hz, is characterized by one node (zero displacement) at the center of the compliant wall and maximal/minimal displacements in the centerline at x = ±40 mm. The third mode, oscillating at frequency f n 3 = 179Hz, is characterized by two nodes located at x = ±27 mm with maximal/minimal displacements in the centerline at z = ±53 mm. The natural frequencies of the first two modes are close to the range of frequency characterizing the oscillation of the shock on the rigid wall, as shown in Fig. 3b.

Reconstruction of the instantaneous wall displacement

For most cases investigated here, the downward displacement of the wall prevents extracting its positions from schlieren images. In these cases, the optical sensor gives access to only a pointwise measurement of the wall. To complete these experimental data, we propose to reconstruct the full displacement of the compliant wall as follow. This reconstruction method is based on the three dimensional linear equation model used to describe the behaviour of the material and it is presented in §A.

The instantaneous vertical displacement of the interface at the centerline z = 0, denoted y w (x, t), is modeled as the sum of a static component and a fluctuating component, i.e.

y w (x, t) = a 0 y 0 w (x) + N k=1 a c k ŷk w,r (x) cos(2πf k t) + a s k ŷk w,i (x) sin(2πf k t) . (1)
The first term is the static component with a constant amplitude a 0 and a spatial shape y 0 w (x). These amplitudes and shape can be estimated using the static numerical computation described before. Knowing the spatial shape, the amplitude can also be retrieved with the method described below. The second term in the above decomposition is the fluctuating component that is expanded as a Fourier series. The real coefficients a c k and a s k are the amplitudes of the modes vibrating at the frequency f k . The spatial shapes ŷk w,r and ŷk w,i are defined as the real and imaginary parts of the vertical component of the natural vibration mode

ŷk w (x) = ξ k n y (x, y = 0, z = 0) (2)
restricted to the centerline (z = 0) of the fluid/solid interface (y = 0). These natural vibration modes are eigenvectors from the eigenvalue problem 13 shown in §A and are associated with natural vibration frequencies f n k = ω n k /2π, ω n k being the square root of the corresponding eigenvalue. Note that in the proposed model ( 1), the modes rather vibrate at frequencies f k determined via a frequency analysis of the experimental wall-displacement signal: the identified peaks in frequency are associated to each mode in order of appearance. Indeed, in the coupled fluid-structure system, the structure does not necessarily oscillate at its natural frequencies and frequency-shift usually appears because of the coupling with the fluid, thus the need to identify the correct frequencies from the experimental data. The coefficients a c k and a s k are computed by means of a least-square fit of the experimental data. The data may come from the tracking of the wall position in the schlieren images or from the optical displacement sensor. The former is used when the deformed wall is visible in the schlieren image and the oscillation amplitude is significantly larger than the image resolution. In this case, the set of observations used to fit the model is defined as

O schlieren = {(x i , t j ) 1≤i≤Nx,1≤j≤Nt | y exp w (x i , t j ) > ϵ} (3)
where N x is the number of points (pixels) in a schlieren image and N t is the number of images. The positive real number ϵ is introduced to select wall-position visible in the image while accounting to the image resolution. As shown in §3.2, the schlieren resolution is 0.22 mm. Therefore, for the computation ϵ is set to 0.22 mm. The total number of observations used for the reconstruction is N schlieren = N t × N x = 704 × 1000 = 704000; the number of retained points in O schlieren is 227378.

On the other hand, the displacement sensor is able to measure the complete oscillation dynamic but only for one spatial location. The set of observations used for fitting the model is then defined as

O sensor = {(t j ) 1≤j≤Nsensor | y exp w (x sensor , t j )} (4)
where N sensor = 600000 is the number of temporal measurements obtained with the optical displacement sensor at point x sensor = 27 mm.

The model is then fitted to the observations. To that aim, we classically write that the wall-displacement model ( 1) is equal to the available measurements, for instance

y exp w (x i , t j ) = a 0 y 0 w (x i ) + N k=1 a c k ŷk w,r (x i ) cos(2πf k t j ) + a s k ŷk w,i (x i ) sin(2πf k t j )
for any points in the schlieren data-set O schlieren . This relation can be recast as a linear problem

y exp w = Υa (5)
where

y exp w = (y exp w (x 1 , t 1 ), • • • , y exp w (x Nx , t Nt )) T is a vector gathering experimental observations, a = (a 0 , a c 1 , a s 1 , • • • , a c N , a s N )
T is the vector gathering unknown amplitudes of model and Υ is the interpolation matrix defined as

Υ =    . . . . . . . . . y 0 w (x i ) . . . ŷk w,r (x i ) cos(2πf k t j ) ŷk w,i (x i ) sin(2πf k t j ) . . . . . . . . . . . .    (6)
This linear system is overdetermined since the number of equations (rows of Υ) is larger than the number of unknowns (coefficients of a). Still, it is possible to solve this type of system by means of a least-square approach. Left-multiplying ( 5) with the transpose of the interpolation matrix, we obtain

Υ T Υa = Υ T y exp w ( 7
)
where Υ T Υ is a square matrix of small dimension, equal to the number of unknowns. It can be inverted, yielding

a = Υ T Υ -1 Υ T Υ † y exp w (8)
where Υ † is known as the Moore-Penrose pseudo-inverse. From a computational point of view, the pseudo-inverse is computed via the Python library scipy, which implements a method based on the singular-value decomposition of the matrix.

An example of the reconstruction from wall-tracking data coming from the schlieren images is presented in Fig. 9a-b. This corresponds to the shock configuration 4. Fig. 9a shows the comparison between the reconstructed displacement signal (dashed lines) and the experimental one (solid lines) extracted at the wall streamwise location x = -43 mm. We recall that the data set used for fitting the model is not limited to this temporal signal, as explained before, but includes all positive instantaneous positions. The signal is periodic and the wall is not visible (y w < 0) in part of the period. Clearly, the algorithm allows us to reconstruct the complete signal. The vibration frequency f k of the first three modes used for the reconstruction and their associated amplitudes a k = (a s 2 k + a c 2 k ) are given in Tab. 2. Being a 2 the highest values, it is clear that the second mode vibrating at frequency f 2 = 153Hz is the dominant one. This is also shown in Fig. 9b that compares the spectra of the experimental (solid) and reconstructed (dashed) wall displacement signal (Fig. 9a). Note that in the experimental spectrum, we also observe peaks that are multiples of the leading frequencies f 2 . These are spurious peaks appearing in the spectrum because the experimental signal is clipped when y w < 0, and thus are not used in the reconstruction. This effect can be easily shown by clipping the reconstructed signal when y w < 0. The frequency spectrum of this synthetic signal is shown with the dotted line in Fig. 9b. Super-harmonic peaks appear while they do not exist in the original spectrum (dashed).

Fig. 9c-d reports also an example of reconstruction from data measured with the optical displacement sensor at point x sensor = 27 mm for the shock configuration 6 (Tab. 1). In this case, we recall that the data set is much more limited since only the temporal signal at this point is used to fit the model. Fig. 9c compares the reconstructed displacement signal (dashed line) with the experimental one (solid line). Even if the time evolution is not perfectly recovered, the spectral content at the identified frequency peaks is more than satisfactory (Fig. 9d). As in the previous example, further information that is retrieved by this analysis is the dominant modeshape of the structure. This case is of particular interest since it highlights how a single-point measurement can be misleading: if the signal spectrum identifies ŷ1 w as The amplitudes a k are defined as a s 2 k + a c 2 k , cf. (1). For each case, the modal shape associated with the largest amplitude is reported in bold.

Case f 1 [Hz] a 1 [mm] f 2 [Hz] a 2 [mm] f 3 [Hz] a 3 [mm]
the dominant mode at f 1 = 119 Hz, the true dominant mode is ŷ3 w at f 3 = 210 Hz (Tab. 2). This is due to the fact that the displacement sensor is close to a node of ŷ3 w (Fig 8c). The amplitude a k of a basis ŷk w vibrating at f k is linked to the wall displacement amplitude Y (f k , x) at a given streamwise position x by the relation

|Y (f k , x)| = a k |ŷ k w (x)|. If x is
positioned close to a node of ŷk w -i.e. ŷk w (x) ≈ 0 -, the apparent amplitude of the mode at this location will be biased. The algorithm is able to overcome this limitation by retrieving the real modal amplitudes a k and, thus, identifying the true dominant mode. Tab. 2 presents the modal amplitudes for a selection of the shock cases presented in this work. These cases are representative of the different shock oscillation regimes observed in the experimental data and that will be discussed in the following.

Description of various regimes of interaction.

Different regimes of interaction between the compliant wall and the shock wave are found depending on the shock position. First, we describe a regime of largeamplitude synchronized interaction where the amplitude of wall's vibration is of the same order as its static deviation. Then we focus on two regimes of small-amplitude synchronized interaction where the wall's vibration is of lower amplitude than its static deviation. A last regime exhibiting no synchronization between the wall and shock dynamics is described before classifying all of these regimes as a function of the shock positions.

Regime of large-amplitude synchronized interaction

To illustrate this regime of interaction characterized by large periodic synchronized oscillations of the compliant wall and the shock, we describe the results obtained for the shock configuration 4 for which the normal shock is located around xs = -4.8 mm as shown in Tab.1. Fig. 10a shows the frequency spectrum of the wall-displacement at x = -36.9 mm, located upstream of the shock. At this point the maximum upward deformation takes place. The spectrum shows a dominant peak f 2 at 153 Hz as well as two other minor peaks f 1 and f 3 , located at 119 Hz and 205 Hz (dashed lines). The frequency peaks above 205 Hz are multiples of the leading frequency f 2 , 153 Hz. These are the result of the lack of displacement information when y w < 0 as illustrated in §3. The spectrum of the shock displacement obtained at the center of the channel is presented in Fig. 10b. The dashed lines indicate the three vibration frequencies of the structure. Just as for the compliant wall, the flow spectrum is dominated by a peak at 153 Hz, accompanied by its first harmonic 2f . Therefore, both the structure and the flow present large synchronized periodic oscillations at the same frequency f 2 = 153 Hz, which is the second vibration frequency of the structure. This largeamplitude peak at a frequency characteristic of the compliant wall clearly suggests a lock-in of the shock oscillation with the second vibration mode of the structure. In this regime of interaction, the frequency spectrum of the shock-displacement is notably different from the rigid wall case, which is characterized by a broadband amplification of frequencies with much weaker amplitudes (see Fig. 3c). To further explore the interaction, the temporal evolution is shown in Fig. 11 by depicting instantaneous schlieren images for six different instants along a full oscillation cycle of period T = 1/f = 6.5 ms. The blue dots show the positions of the shock in the center of the channel, while the red (resp. green) dots highlight the position of the compliant wall upstream (resp. downstream) of the shock at x = -40 mm (resp. x = 40 mm). At the initial time t 0 of the oscillation period (Fig. 11a), an upward displacement of the wall is clearly visible for -80 mm ≤ x ≤ 0 mm with a maximum located around x = -40 mm. This is the maximum upward deformation of the compliant wall during the cycle. At this instant, the shock at the channel's center is located at x = 0 mm and presents a large λ shape on the lower foot. The front leg is located slightly downstream of the position of the maximal deformation. Downstream, a large separation region may occur and large structures are shed in the mixing layer. At the time t 1 = t 0 + T /6 (Fig. 11b), the upward deformation of the wall decreases while the shock moves upstream. At the time t 2 = t 0 + 2T /6, we do not see any upward deformation of the wall and the shock has continued to move upstream. Regarding the shock, the λ region has decreased, indicating a reduction in the separation under the shock foot. Likewise, there is a reduction in the size of the structures shed downstream of the shock foot. At the time t 3 = t 0 + T /2 corresponding to half-period of the cycle, there is no visible displacement of the compliant wall upstream of the shock, but an upward deformation is observed downstream of the shock with a maximum of around 40 mm. Note that this upward displacement is weaker downstream the shock than upstream the shock (compare with Fig. 11a). In the meantime (from t 2 to t 3 ), the streamwise position of the shock has barely changed but the size of the λ region is reduced and the shedding of the structures appears less intense. At time t 4 = t 0 + 4T /6, no upward deformation of the wall is visible downstream of the shock, while a very small deformation is visible upstream of the shock. The shock has moved downstream and its topology resembles the one found in the rigid case (Fig. 2a). Finally, at the last time of the cycle t 5 = t 0 + 5T /6, the upward deformation of the compliant wall is more pronounced upstream of the shock. This upward motion is accompanied by a downstream displacement of the shock. The features here observed resemble those present in t 0 .

In order to give an overall view of the compliant wall displacement, Fig. 12a shows isocontours of the vertical displacement in a spatio-temporal diagramm where x is the streamwise position and t/T is the time normalized by the oscillation period T . Since downward displacements of the wall are not visible in the schlieren images, negative values are not visible in this figure and are displayed here with grey color. The various instants of the schlieren images shown in Fig. 11 are marked here with horizontal dashed lines. This spatio-temporal evolution clearly confirms that the upward displacement of the compliant wall is of larger magnitude upstream of the shock, the position of the latter being shown with the black curve. To complete the analysis of the interaction between the shock wave and the compliant wall, we need to reconstruct the displacement of the latter as explained in §3.4. We briefly recall that the instantaneous displacement is the sum of a static component and a fluctuation. The static displacement, computed here using the spatial distribution of the wall-pressure for the rigid case, is shown in Fig. 12c, with the average position of the shock identified with the black line. This deformation is positive for x < 0, where the pressure on the upper surface is lower than on the bottom surface, and vice-versa for x > 0. The fluctuation is then modeled with the first three modes of the modal analysis vibrating at a frequency given by the experimental data (see Fig. 10). The amplitude of the three modes are then determined to best fit the experimental data shown in Fig. 12d. As shown in Tab. 2, the amplitude a 2 associated with the second mode of vibration is dominant compared to the first and third modes. Therefore, the pattern of the reconstructed fluctuation shown in Fig. 12d is very similar to that of the second mode. The sum of the static and fluctuation gives the reconstructed displacement of the compliant wall shown in Fig. 12b. We observe that the duration and amplitude of the negative displacement for x > 0 are longer and larger than for the positive displacement. The opposite pattern is found for x < 0. This is clearly the result of the existence of a the static deformation of the compliant wall caused by the pressure gradient imposed by the shock. We now investigate the phase difference between the shock and wall oscillations by looking at Fig. 12d. An upward movement of the material upstream of the shock coincides with a downstream displacement of the shock and vice-versa. When the downstream part of the material presents an upward deformation, the shock position (solid black line) barely changes, presenting almost a plateau, as it is for the case between t 2 and t 3 . This is consistent with the observation made in Fig. 11, where the shock seemed to remain in the same positions from t 2 till t 3 . When looking at .. Fig. 12d, it is possible to see that the oscillation of the material presents a lag with respect to that of the shock. This implies that the most downstream position of the shock happens before the material has reached its maximum deformation. In order to assess the lag, the cross spectral density of both signals was computed. The phase between both wall displacement signals and the shock signals is 28 • .

Regime of small-amplitude synchronized interaction This regime of interaction is characterized by periodic synchronized oscillations of the wall and the shock, whose amplitude is smaller than the static deformation of the wall. The particularities of this regime are illustrated by analyzing the interaction in shock configurations 1 and 6 (Tab. 1). The dynamics for both shock configurations were analyzed by computing the spectrum with the Welch method for both the shock and wall displacement. For the wall, the spectrum was computed using the displacement signal at x sensor = 27 mm retrieved from the displacement sensor as the schlieren does not provide enough resolution to capture the reduced wall oscillations. With respect to the flow, the spectrum was calculated with shock displacement at the center of the channel. The resulting spectra for both shock configurations are shown in Fig. 13 For shock configuration 1, the wall spectrum (Fig. 13a, red) displays several peaks. The first three peaks, f 1 , f 2 , and f 3 , are located at 119 Hz, 149 Hz and 205 Hz (dashed lines) respectively. Out of the three, f 1 is the dominant one. The spectrum of the flow (Fig. 13a, blue), is characterized by a dominant peak at a frequency 119 Hz (first dashed line). This suggests that the shock presents a periodic oscillation at the first vibration frequency of the structure. Nonetheless, note that with respect to the regime of large-amplitude synchronized interaction (Fig. 10b), the magnitude of the shock oscillations is three orders of magnitude smaller. Therefore, the motion of the shock and the compliant wall is synchronized at f 1 with small amplitude oscillations.

In shock configuration 6, the wall spectrum displays a dominant peak f 1 at 122 Hz (Fig. 13b,red). The two next peaks, f 2 and f 3 are located at frequencies 153 Hz and 209 Hz. On the flow spectrum however, the dominant peak appears at 209 Hz (third dashed line), which coincides with the third vibration frequency of the wall (Fig. 13b, blue). This suggests that the shock oscillates at the structure's third vibration frequency. Seeing this result, one would expect a dominant peak at f 3 , 209 Hz, on the compliant wall spectrum instead of at f 1 , 122 Hz. The explanation for this discrepancy lies on the location of the displacement sensor with respect to the nodes of the third vibration mode (Fig. 8c). Being one of the two nodes located at x = 27 mm, the displacement sensor, which measures the wall motion at x sensor = 27 mm, cannot see any contribution of the third mode to the wall motion, and thus its amplitude on the spectrum is small. In §3.4 the reconstructed data shows that the modal amplitude a 3 , representative of the contribution of the third mode to the total motion of the compliant wall, is the largest, indicating that the compliant wall is mostly oscillating at its third vibration frequency. Consequently, this suggests that the shock and the compliant wall present a synchronized motion at structure's third vibration frequency, f 3 . As in configuration 1, the amplitude of the shock oscillations is three orders of magnitude smaller than in the regime of large-amplitude synchronized interaction (Fig. 10b), meaning that the amplitude of the wall and shock motion is small. Given the reduced amplitude of the oscillations, there are no major visible changes on the flow and the wall during one cycle. Thus, only one instantaneous schlieren visualization is used to illustrate the flow and wall for configurations 1 and 6 in Fig. 14a-b. On the first configuration, the shock topology resembles that of the rigid wall (Fig. 2a), except for the fact that the upper shock foot is smaller than the lower shock foot. In this case, no deformation is present in the schlieren, suggesting that there is a negative deformation. In configuration 6, the shock is inclined with a strong difference between the upper and lower foot. For this case, the wall presents a large upward deformation with a maximum around x = -20 mm. Nonetheless, from x > 40 mm no information of the deformation is available. So as to complete the information on the wall's deformation obtained from the schlieren images, the static deformation of the compliant wall was computed assuming a Young modulus of 1.68 MPa. Fig. 14c-d presents the pressure difference f 2 , and third, f 3 , peaks located at 151 Hz, and 210 Hz respectively (dashed lines). When looking at the spectrum of the flow, this presents a low frequency broadband bump, similar to the one obtained in the rigid case Fig. 3b. Thus, the flow excites the compliant wall but the flow does not lock-in to any of its vibration frequencies, contrary to the previous exposed regimes. Additionally, the amplitude of the shock oscillations is four orders of magnitude smaller than the one found in the synchronized large-amplitude regime, and one order of magnitude smaller than in the synchronized small-amplitude regime.

Regarding the topology of the flow as well as the deformation of the compliant wall caused by the pressure gradient imposed by the shock, it resembles that of configuration 6, explained in §3.5. Therefore, it is interesting that despite having similar flow and static deformations of the compliant wall, the behaviour of the interaction in both configurations is completely different.

Summary Having presented the three different regimes of interaction, this section aims at analyzing the influence the shock position along the wall has on the dynamics of the interaction as well as on the deformation of the compliant wall. In order to analyse the effect on the compliant wall deformation, the static deformation fitted to the experimental data is presented as a function of the shock position in Fig. 16b. Fig. 16a presents the wall pressure difference used for the computation.

For the most upstream shock position, most of the compliant wall's upper surface is in contact with the high pressure associated with the subsonic regime downstream the shock. Therefore, the aerodynamic loading exerted on the wall is positive, causing a negative deformation. At this point, the compliant wall presents its maximum negative displacement. As the shock moves upstream, the region upstream the shock, in contact with the low pressure from the supersonic region, becomes larger. In this area, the compliant wall is subjected to a negative pressure difference causing a positive deformation. When the shock is close to the center, the regions with positive and negative deformations are of the same length. As the shock reaches its most downstream position, the compliant wall presents a positive displacement. It is at this point were the maximum positive displacement is found. The influence of the shock location on the interaction dynamics is displayed in Fig. 17. Fig. 17a presents the normalized premultiplied spectrum of the shock displacement for the different shock locations along the wall displayed in Tab. 1. Its value has been normalized by the maximum amplitude on the premultiplied spectrum found between 20 Hz and 1000 Hz. Along side with the spectrum in Fig. 17a, the average value of the vibration frequencies f 1 (orange), f 2 (blue) and f 3 (green), are reported. On the other hand, Fig. 17b shows the values used to normalized the premultiplied spectrum in Fig. 17a. For the first configuration, the dynamics of the most upstream location, x = -45.1 mm, corresponds to the synchronized small-amplitude regime (SA 1 ). At this location, the dominant peak matches the first vibration frequency of the structure (Fig. 17a, orange). As the shock moves downstream, in configuration 2, at x = -29.4 mm, it undergoes a transition regime (T ). At this point, the shock presents broadband oscillations as well as a peak coinciding with the third vibration frequency of the structure (Fig. 17a, green). When the shock is close to the middle of the compliant wall (locations 3 and 4, x = -7.3 mm and x = -4.8 mm respectively), it presents only one peak that coincides with the second vibration frequency of the structure (Fig. 17a, blue). Note that the amplitude of the oscillations for these configurations is at least two orders of magnitude higher than for SA 1 and T (Fig. 17b, blue). For this reason, these two locations are said to belong to the regime of synchronized large amplitude oscillations (LA 2 ). Further downstream, shock configurations 5 and 6, at locations x = 32.1 mm and x = 41.1 mm, present a dominant peak at the third vibration frequency of the structure (Fig. 17a, green). As the amplitude of the oscillations is smaller than for LA 2 (Fig. 17b), they are considered to be in the regime of the synchronized small amplitude vibrations (SA 3 ). At the two most downstream locations, configuration 7 and 8, positioned at x = 65.3 mm and x = 73.9 mm, they are unsynchronized (U ) with the vibration frequencies of the structure (Fig. 17a) and present the smallest shock oscillations amplitudes (Fig. 17b,U ).

From this last analysis it is clear that the shock location along the compliant wall has a paramount impact on the dynamics of the interaction. In order to find an explanation for this change in behavior, further analysis using the numerical tools should be conducted. For instance, it would be clarifying to retrieve the response of the compliant wall to an unsteady pressure field that could recreate the shock oscillations.

Conclusion

The interaction of a shock wave and a compliant wall has been experimentally characterized by means of schlieren visualization and a point displacement sensor. Due to the sparsity of the experimental data, a reconstruction method based on the three dimensional linear elastic equations is proposed to complete the experimental description of the interaction. The effect of the shock location on the compliant wall deformation was assessed. It was found that due to the pressure gradient imposed by the shock, the compliant wall presents a downward deformation downstream of the shock and an upward deformation upstream of the shock. When the shock is located close to the compliant wall's leading edge, it presents only a downward deformation. As the shock moves downstream, an upward deformation appears upstream of the shock and the downward deformation downstream of the shock is reduced. When the shock is located at the trailing edge, the compliant wall presents only an upward deformation.

Throughout the experiments, three different regimes of interaction have been identified. The first one consists of large amplitude synchronized interactions of the shock and the compliant wall. A second regime of synchronized motion has also been identified, but with lesser amplitude oscillations. The last regime presented unsynchronized small oscillations of the shock and compliant wall. Whether one regime or another took place, highly depended on the position of the shock along the compliant wall. It was observed that for shock location close to the middle of the compliant wall, the interaction corresponded to the large-amplitude synchronized regime, where a lock-in of the shock motion to the second structure vibration frequency took place. For shock locations upstream of the middle, the interaction corresponded to the small-amplitude synchronized regime, presenting a lock-in of the shock motion to the first vibration frequency of the structure. For downstream shock positions, the small-amplitude synchronized regime was also found but the lock-in occurred with the third vibration frequency of the structure. When the shock was located at its most downstream positions, close to the trailing edge of the compliant wall, no lock-in of the shock motion to the structure vibration frequencies was found. In this stage, the shock presented low frequency broadband oscillations. As to which is the cause for the existence of the three regimes of interaction, it is not clear to the authors. For a better understanding, the three dimensional linear elastic equation will be used to model the behaviour of the material and compute its response to an unsteady external forcing representative of the oscillatory motion of the shock. In this manner, it would be possible to discern if the observed lock-in at the different vibration frequencies of the structure is due to the excitation of a particular mode by the shock depending on its location with respect to the modal shapes. 3: Mesh convergence study. The size of the mesh is represented by the number of elements along the x, y, and z directions, as well as by the total number of elements (n el ).

Results from both dynamic and static analyses are monitored: the first natural frequency (f 1 n ) and the maximum static displacement along the z direction (ξ max z ) for the nominal shock configuration. The relative variation with respect to the results for the reference mesh (bold) is also reported. pressure P f is thus retrieved by a linear interpolation of the experimental measurements in the symmetry plane and a constant extrapolation in the spanwise (z) direction.

The unsteady displacement ξ ′ s and pressure p ′ s of the structure can be analyzed by assuming a modal decomposition as ξ ′ s (x, t) = ξ n (x) e iωnt + c.c. [START_REF] Gramola | Off-design performance of 2D adaptive shock control bumps[END_REF] where ω n is the (real) natural vibration frequency associated to the modal shape ξ n . Injecting the above modal decomposition in [START_REF] Eitner | Effect of Shock-Wave Boundary Layer Interaction on Vibratory Response of Compliant Panel[END_REF] gives the eigenvalue problem ρ s ω 2 n ξ n = -∇p n -µ s ∇ • ∇ξ n + ∇ξ T n , 0 = -λ s ∇ • ξ n + p n [START_REF] Gramola | Experimental FSI study of adaptive shock control bumps[END_REF] where the eigenvalues correspond to the square of the natural vibration frequencies ω 2 n and the eigenvectors are the corresponding modal shape. Similar Dirichlet conditions are applied on the lateral boundaries while stress-free conditions are applied on the top and bottom boundaries.

The static elasticity equation [START_REF] Erengil | Unsteady wave structure near separation in a Mach 5 compression rampinteraction[END_REF] and the eigenvalue problem (13) are discretized using a standard finite-element method. The open-source library FreeFEM [START_REF] Hecht | New development in freefem++[END_REF] is used to assemble the sparse matrices. The numerical interface with the open-source library MUMPS is used to perform sparse-LU factorization/inversion of these matrices. The interface with the open-source library ARPACK is used to determine the eigenvalues and eigenvectors.

In order to determine a suitable discretization of the domain, a convergence study was conducted. The number of elements along the three directionsn x , n y , and n z -has been varied and results from both dynamic and static analyses have been monitored. Secondorder langrangian elements are used for the displacement, while first order ones are for the pressure field. Tab. 3 presents the evolution of the prediction of the first natural frequency with the mesh spatial resolution, as well as of the maximum static deformation for the nominal shock configuration. The reference configurationn x = 24, n y = 18, and n z = 6 -was chosen as it is a good compromise between computation time and spatial resolution. An example of the static deformation computed for configuration 4 computed with the corrected Young modulus -as it is explained in §3.3 -can be found in Fig. 18 as well as the reference mesh.

Figure 1 :

 1 Figure1: (a) Picture of the high-speed wind tunnel S8Ch showing the cartesian coordinate system (0,x,y) and the optical access for the schlieren visualization (white area) and the wall-tracking (green area). On the top and bottom walls, static pressure probes are installed to retrieve (b) static wall-pressure as a function of the streamwise position x (bottom wall), here for the nominal shock configuration 4 (see table1).

Figure 2 :

 2 Figure 2: (a) Full view of an instantaneous schlieren image with (A) the normal shock and (B,C) the front and rear legs of λ-shaped shock-foot. The blue squares correspond to the localisation of static pressure sensors. The white rectangle delimits (b) the close-up view around the normal shock. The dot corresponds to a particular tracked position x s (t) of the normal shock in the symmetry plane y = 50 mm of the test section. Nominal shock configuration 4 (see table 1).

  f*PSD Shock displacement zoom

Figure 3 :

 3 Figure 3: Nominal configuration 4. (a) Fluctuation of the streamwise displacement of the shock x ′ s = x s -xs as a function of time, defined by substracting the average position from the instantaneous displacement x s obtained by tracking the shock position in the instantaneous schlieren. (b-c) Power spectral density (premultiplied by the frequency) as a function of the frequency for the shock displacement signal shown in (a). The grey areas in (b) correspond to spurious oscillations as explained in the text. (c) Zoom in the frequency range of interest. In grey the raw signal, in black the filtered signal as explained in the text.

  BETWEEN A NORMAL SHOCK AND A COMPLIANT WALL (a) Lower wall test section (b) Sketch of compliant wall setup

Figure 4 :

 4 Figure 4: (a) View of the compliant wall on the lower wall of the test section. (b) Sketch of a lateral view along the center line (white dashed line on (a)) of the compliant wall with the location of the shock.

Figure 5 :

 5 Figure 5: Sketch of the two optical techniques used to measure the wall displacement illustrated with cases (a) 6 and (b) 1. The dashed black line indicates y < 0 wall displacement that cannot be observed with schlieren. Green rectangles show the location of the optical access (on the upper wall of the test) for the light beam (red line) allowing us to measure the displacement of one point (red dot) on the compliant wall.

  Zoom on PSD of the wall displacement

Figure 6 :

 6 Figure 6: Fluctuating part of the compliant wall displacement signal x ′ s = x s -xs and its power spectrum density (PSD). In grey the raw data, in black the filtered data. Grey area in (b) referrers to the range of frequencies with spurious data on the shock displacement spectrum shown in Fig. 3b

Fig. 6a presents

  Fig. 6a presents an interval of 4 s of the total duration of the displacement fluctuation signal, x ′ s = x s -xs , measured with the optical sensor for configuration 7.The signal (grey) is superimposed with the filtered signal (black) which will be later explained. The amplitude of the oscillations is less than 0.1 mm, meaning that these fluctuations would not be captured by the schlieren images as their resolution is 0.22 mm per pixel. The power spectrum density of the entire signal is presented in Fig.6b(grey),

Figure 7 :

 7 Figure 7: Configuration 8 corresponding to the shock position x = 73.9 mm. (a) Averaged wall-pressure difference, ∆P (x) = P (x) -P bottom , where P (x) is measured on the rigid wall as a function of the streamwise coordinate x. (b) Wall-normal displacement y w of the compliant wall as a function of x.The solid red curve corresponds to the time-averaged displacement extracted from the schlieren vizualisation. The dashed and dashed-dotted curves are numerical predictions of the static displacement computed using the experimental wall-pressure as static aerodynamic loading at the interface for the Young modulus E = 3.03 MPa (dashed-dotted) and E = 1.63 MPa (dashed). The red dot is the averagedisplacement measured with the optical pointwise sensor.

  C. RIVEIRO MORENO ET AL. (a) f n 1 = 104.3 Hz (b) f n 2 = 133.2 Hz (c) f n 3 = 179 Hz

Figure 8 :

 8 Figure 8: Natural vibration modes of the structure (polyurethane inserted in the aluminium container) for the three lowest frequencies. Wall-normal displacement of the compliant wall (fluid/solid interface) shown (top) in the two-dimensional plane (x, z) at a given instant and (bottom) shown in the spatio-temporal diagram for the centerline z = 0 .

  (a) Case 4: time signals at x = -43 mm. (b) Case 4: spectra at x = -43 mm. (c) Case 6: time signals at x = 27 mm (d) Case 6: spectra at x = 27 mm.

Figure 9 :

 9 Figure 9: Wall-displacement reconstruction from wall-tracking data (Cases 4 and 6 in Tables 1-2). (a, c) Temporal signals at two streamwise positions. Solid and dashed lines correspond to experimental and reconstructed data, respectively. (b, d) Power Spectral Density as a function of frequency for the experimental (solid), reconstructed (dashed). The clipped reconstructed signal (dotted) is also reported in (b); see text for description.

  4. The three frequencies identified experimentally are close to the natural frequencies of the material (f n 1 = 104.3 Hz, f n 2 = 133.2 Hz, f n 3 = 179 Hz) determined numerically in §3.3.

Figure 10 :

 10 Figure 10: Regime of large-amplitude synchronized interaction illustrated with the shock position 4. (a) Power spectrum density of the wall-displacement extracted from schlieren images at point x = -36.9 mm and (b) premultiplied power spectrum density of the shockdisplacement tracked in the schlieren images. Dashed lines correspond to the first vibration frequencies of the compliant wall.

t 3 = 6 -

 36 t 0 + 3 T

Figure 11 :

 11 Figure 11: Instantaneous schlieren visualizations during an oscillation cycle of the compliant wall. Red, green and blue dots illustrate the displacement of the compliant wall and the shock.

  C. RIVEIRO MORENO ET AL. (a) Experimental data. (b) Reconstructed data. (c) Reconstructed static. (d) Reconstructed fluctuation.

Figure 12 :

 12 Figure 12: Spatio-temporal evolution of the compliant-wall displacement (color) and streamiwse shock position (black lines) (a) measured experimentally and (b) reconstructed by fitting a model. (c) Shows the reconstructed average displacement (color) and the average streamwise position of the shock (black line). Similarly, (d) reports the perturbation with respect to the time average of the wall displacement (color) and streamwise shock position (solid black line).

Figure 13 :

 13 Figure 13: Two regimes of small-amplitude synchronized interaction with large static deviation. (a) Case 1 for xs = -45.1 mm and (b) Case 6 for xs = 41.1 mm. In red the spectra of the compliant wall displacement measured with the displacement sensor at x sensor = 27 mm.In blue the premultiplied spectra of the shock displacement. Note that it is three orders of magnitude smaller than in (Fig.10b). The three dashed lines correspond to the first three vibration frequencies of the structure

Figure 14 :

 14 Figure 14: Two regimes of small-amplitude synchronized interaction with large static deviation. (a,c) Case 1 for xs = -45.1 mm and (b,d) Case 6 for xs = 41.1 mm. (a-b) Instantaneous schlieren images.(c-d) Wall-pressure difference, ∆P (x) = P (x) -P bottom , as a function of x and deviation of the compliant wall computed with the linear elastic model (dashed curve), extracted from schlieren visualization (red curve) and obtained with the optical pointwise sensor (red dot).

  (a) Pressure wall difference (b) Reconstructed static deformation

Figure 16 :

 16 Figure 16: Isocontours of the (a) wall-pressure difference and (b) reconstructed static deformation of the compliant-wall as a function of the streamwise position x for several position xs of the shock.

Figure 17 :

 17 Figure 17: (a) Normalized premultiplied spectrum of the shock displacement signal by the maximum amplitude between 20 -1000 Hz as a function of the shock location. Horizontal dashed lines indicate the average values for the first vibration frequencies of the structure, f 1 (orange), f 2 (blue), and f 3 (green). (b) The maximum amplitude of the premultiplied spectrum of the shock oscillations used to normalized the premultiplied spectrum is presented in (a). SA stands for small amplitude, LA for large amplitude and U for unsynchronized and T for transition.

C

  . RIVEIRO MORENO ET AL.

Figure 18 :

 18 Figure 18: Typical mesh and isocontours of the static deformation computed for the nominal shock configuration.

Table 1 :

 1 Cases investigated in the present study. Average streamwise position of the shock for various heights of the second throat. The nominal case (4) is highlighted in bold.

	Case	1	2	3	4	5	6	7	8
	xs (mm) -45.1 -29.4 -7.3 -4.8 32.1 41.1 65.3 73.9

Table 2 :

 2 Frequencies and reconstructed modal amplitudes (cf. Table1for case definition).

	Data source

Acknowledgements

This work is co-funded by the Direction Générale de l'Armement (AID Convention N°2020 65 0022) and ONERA. The authors would like to express their gratitude to Julie Diani for granting them access to the rheometer at Laboratoire de Mécanique des Solides at École Polytechnique.

distribution across the compliant wall, ∆P (x) = P (x) -P bottom , used on the computation. Fig. 14c-d shows as well the resulting static deformation with the average displacement value measured with the displacement sensor, and from the schlieren when available (red line). For shock configuration 1, up to x = -40 mm, the low pressure from the supersonic region on the upper wall is lower than that of the cavity under the wall (Fig. 14c), which causes a negative pressure difference that results in an upward deformation of the compliant wall (Fig. 14c,dashed). Downstream this point, the situation reverses as the upper surface of the compliant wall is in contact with the higher pressure values of the subsonic region downstream of the shock, causing a positive pressure difference and a downstream deformation of the wall (Fig. 14c). In this configuration, there is no information from the schlieren visualizations but there is from the displacement sensor. Although the values do not coincide, the displacement sensor measures a negative displacement (Fig. 14c, red dot). In shock configuration 6, up to x = 40 mm, the upper surface of the compliant wall is in contact with the low pressure coming from the supersonic flow. As this is lower than the pressure in the cavity, the pressure difference is negative causing an upward deformation of the compliant wall (Fig. 14d). The situation reverses downstream x = 40 mm, where the compliant wall presents a small downward deformation. In Fig. 14d, although there are some differences, both the average wall deformation from the schlieren images (red) and the displacement sensor (red dot) are in agreement with the computed static deformation (black dashed line).

Regime of Unsynchronized interaction This regime, illustrated by shock configuration 7 (Tab. 1), is characterized by small oscillations of the compliant wall and the shock which are not synchronized. The dynamics of the interaction are illustrated through the spectrum of the compliant wall displacement obtained from the displacement sensor as well as the spectrum of the shock displacement at the center of the channel (Fig. 15a-b). The spectrum from the compliant wall presents a dominant peak at 122 Hz, f 1 , with the second,

Appendix

A. Numerical modelling and discretization of the compliant structure

The three-dimensional instantaneous displacement ξ s (x, t) of the polyurethane inserted in the aluminum container is modelled with the linear elasticity equations formulated as

where the second equation defines the instantaneous solid pressure p s (x, t). These equations are satisfied in the domain Ω s that is a parallelepiped with rectangular faces extending as -80mm ≤ x ≤ 80mm in the streamwise direction, -42.5mm ≤ z ≤ 42.5mm in the spanwise direction and -L y ≤ y ≤ 0 in the vertical direction. The top surface (y = 0) corresponds to the interface with the high-speed flow in the test section of the wind tunnel. We recall that the Lame coefficients λ s = (Eν s )/(1 + ν s )/(1 -2ν s ) and µ s = 0.5E/(1 + ν s ) λ s and µ s are defined as a function of the Young modulus E and Poisson coefficient ν s .

The static displacement Ξ s (x) and pressure P s (x) of the structure are obtained by solving

with appropriate conditions on the boundaries ∂Ω s of the domain. At the four lateral boundaries with the aluminum container ( x = ±80mm and z = ±42.5mm), Dirichlet boundary conditions are applied on the displacement, i.e. Ξ s = 0. At the top (y = 0) and bottom (y = -L y ) boundaries, Dirichlet boundary conditions are applied for the static pressure. For the bottom boundary that is the interface with still air, we impose P s (x, y = -L y , z) = P meas where P meas = 45 kPa is the constant pressure measured with a static probe under the compliant structure during the experiments. For the top surface, we impose P s (x, y = 0, z) = P f (x, y = 0, z)

where P f is the static aerodynamic pressure. Note that it is partially determined in the experiments for the rigid wall configuration since it is measured at some streamwise positions in the symmetry plane z = 0 of the test section. The two-dimensional aerodynamic


Buttay Cyril

Bergogne Dominique

Morel Hervé

IEEE member, Ehlinger rené Allard Bruno

Bevilacqua Pascal

Towards a Sensorless Current and Temperature Monitoring in MOSFET-Based H-Bridge

Chip temperature in power MOSFETs is commonly obtained (e.g. for thermal characterisation purposes) by feeding a known, controlled low DC-current in the body diode and measuring the forward-voltage drop.

Drain current might be estimated by a voltage measurement across drain and source terminals, on-state resistance acting as a shunt, assuming that the temperature is known and constant.

Such voltage-based measurements are attractive, but dependencies limit their application to laboratory environment. In industrial converters, neither temperature nor current are known a priori.

Here we show that by combining the two techniques, previous limitations are overcome. Temperature and current come as two separate polynomial expressions.

The application to a low-cost monitoring system is presented and results are discussed. The validity of the system over the full temperature and current ranges is verified and a confidence map is given.

I. INTRODUCTION

Current and Temperature are critical parameters in power converters. The monitoring of their values is at a premium to ensure reliable operation. Extensive work and literature relate techniques based upon an estimator to avoid the cost and pitfalls of dedicated sensors [START_REF] Bergogne | An estimation method of the channel temperature of power MOS devices[END_REF][START_REF] Allard | Application of averaged Models to Real-Time Monitoring of Power Converters[END_REF][START_REF] Ammous | Transient Temperature Measurements and Modelling of IGBT's under Short Circuit[END_REF]. The method proposed in this paper is based on the measurement of the drain source voltage at the terminals of one MOSFET in an inverter.

Two operating states of the transistor are used to compute current and temperature values. When in the on-state, a MOSFET behaves as a temperature-dependent resistor. Drainto-source voltage is therefore related to the transistor temperature and drain current. When freewheeling on the body diode, the drain-to-source voltage is equal to the intrinsic body diode forward-voltage. This value is mainly governed by the forward (drain) current and the transistor temperature. Thanks to the proximity of the body diode and the channel (a few µm), the temperature can be assumed to be identical in both areas. Providing there is a relation between the two currents in the two states, it is possible to extract the temperature and current values.

In this paper, we consider that temperature is constant in the active area of the silicon die. It is referred to as "chip temperature". This is commonly admitted in all circuitsimulation models where device behaviour is described by a set of equations. Further more, the only observable variables in a converter are global such as drain current and drain to source voltage. Taking into account the cellular structure of MOSFET requires the knowledge of difficult to obtain technological parameters. The sophisticated modelisation that would be required is not compatible with a converter operating in real-time.

II. PRINCIPLE OF OPERATION

The two parameters used to compute the temperature and the current have opposite behaviour : RDSon presents a positive temperature coefficient whilst the body diode forward-voltage shows a negative temperature coefficient. This allows for a robust discrimination.

In a classical inverter, a deadtime is observed between the turn-off of a pair of switches (e.g. H1 and L2, as described on fig. 1) and the turn-on of the opposite pair (i.e. H2 and L1). Given the load is inductive, current flows through body diodes of H2 and L1, which will therefore act as freewheeling devices during this period. The forward-voltage of these diodes cannot however be measured at this time, due to the influence of transient phenomena (recovery, overvoltages caused by circuit stray inductances...). To achieve high efficiency, the deadtime must be as short as possible to reduce 0-9999-9999-0/09$17.00©2003 9999

DC bus voltage

the voltage drop across the body diode, so it doesn't last as long as the transients occurring at commutation. Forwardvoltage will therefore be measured by turning-off the transistor L1 a short lapse of time after the beginning of its conduction. This will not disturb the operation of the inverter, as it doesn't involve a high voltage step (less than 1V) nor high current gradient (the only change in the current path will be across the silicon die). This turning-off must be performed before the current ID becomes positive in the switch (with respect to the conventions above), otherwise it would flow through the diodes of H1 and L2.

The measurement of the drain-to-source voltage (in the resistive on-state) will be performed just before the end of the conduction, to ensure a good signal-to-noise ratio (the current is maximal at the end of the conduction cycle).

III. ANALYTICAL APPROACH

A. On-state resistance model

The effect of the temperature on the on-state MOSFET resistance, is quite well taken into account by classical models such as those implemented in SPICE.

For the sake of simplicity, the equations given here are those of the SPICE level 1 MOS model [START_REF] Massobrio | Semiconductor device modelling with SPICE[END_REF].

In the linear region, i.e. VGS > VTH and VDS < VGS-VTH,

I D =KP W L-2X jl V GS -V TH - V DS 2 V DS 1 V DS  (1)
Where KP is the transconductance parameter, W and L respectively the width and length of the channel, Xjl the metallurgical junction depth, VGS the gate to source voltage, VTH the gate to source threshold voltage, and λ is the channel length modulation.

For power MOSFETs, λ is considered as zero. KP can be supposed to be the only temperature dependent parameter,

KP T 2 =KP T 1   T 1 T 2  3 2 (2)
The influence of the temperature on the on-state resistance is given by replacing (2) into (1), and then deriving.

∂ I D ∂ V DS = KPT 1   T 1 T 2  3 2 W L-2X jl V GS -V TH -V DS  (3)
Notice that RDSon evolves as a power of 1.5 of the temperature. On a limited temperature range, (3) can therefore be approached by the 2 nd order polynomial below.

R DSon =C 0 C 1 TC 2 T 2 (4)

B. Intrinsic body diode forward-voltage model

As it can be seen in the extensive literature [START_REF] Profumo | instantaneous Junction Temperature Evaluation of High Power Diodes (Thyristors) During Current Transients[END_REF][START_REF] Maxim | Power PIN Diode Electro-Thermal SPICE Macromodel with Forward and Reverse Recovery[END_REF][START_REF] Muñoz-Yagüe | High Level Behavior of Power Rectifiers: A Quantitative Analysis of the Forward Voltage Drop[END_REF], there are few simple but accurate formulations of the diode forwardvoltage drop. This is due to the great number of phenomena occurring in a classical PIN diode. Ohmic contacts are of a major concern in this voltage drop, although they are hardly taken into account in numerous models. The temperature and doping gradients across the PN junction have also an influence, but are difficult to quantify.

In [START_REF] Profumo | instantaneous Junction Temperature Evaluation of High Power Diodes (Thyristors) During Current Transients[END_REF], an analytical expression is used to determine the junction temperature of a power diode from its forwardvoltage and current, but is far too complex for the targeted application.

Thought for monitoring purposes, the proposed method is not supposed to return very accurate estimations. Unavailable technological parameters of the diode prevent the building of a physical model. An empiric model is proposed in [START_REF] Profumo | instantaneous Junction Temperature Evaluation of High Power Diodes (Thyristors) During Current Transients[END_REF], its validity will be studied later in this paper. Notice that special care is taken in the fitting of such simple model in order to remain accurate where necessary.

V F =V S T K 2 I (5)
Vs(T) is the temperature dependant diode threshold voltage, and K1 a fixed coefficient.

V S = K 0  K 1 T (6) Therefore V F = K 0  K 1 T K 2 I (7)

C. Temperature and current expressions

From (4) it comes, when in the on-state:

V DS1 = R DSon I V DS1 =  C 0 C 1 TC 2 T 2  I 1 (8)
when in the "freewheeling" mode, (7) yields to

V DS2 =-V F =-K 0 -K 1 T-K 2 I 2 (9)
VDS1 is measured at the beginning of the conduction period, while VDS2 is the value of the drain-to-source voltage at the end of the conduction. I1 and I2 are therefore different, but T can be considered as constant, due to the high thermal time constant (Thermal constants are higher than 1ms while conduction period in our case is 25 µs).

When using a pure inductive load, I1 and I2 are linked by: 0-9999-9999-0/09$17.00©2003

I 2 =I 1 - U  t L load ( 10
)
where ∆t is the time interval between acquisition of VDSon and acquisition of VF, U is the DC-bus voltage, and Lload the load inductance. From (9), it comes:

I 2 =- V DS2  K 0  K 1 T K 2 (11)
Using ( 10) and ( 11) in ( 8) yields:

V DS1 =C 0 C 1 T C 2 T 2   U  t L load - V DS2  K 0 K 1 T K 2  (12)
As all parameters (C0, C1, C2, K.0, K1, K2, Lload) are constant and ∆t, VDS1, VDS2 are measured, T is obtained by solving the third order polynomial (12). Equation (11) gives the corresponding I2 value.

IV. PARAMETER ESTIMATION

A. Description of the test bench

In order to characterise the studied MOSFET (75V 100A), an H bridge is built using two industrial inverter legs. It is placed in a controlled temperature furnace. To achieve good thermal spreading, an insulated metal substrate (I.M.S) mounted on a heatsink was used. Burst square signals are used to drive the transistors, avoiding self-heating (see fig. 9). However, slight self-heating can be observed at the end of the conduction period (see fig. 8), with the increase in RDSon. It has little effects on the voltages measured during the parameter estimation, and is taken into account in the fitting.

A small load is required in order to reach the full current during one single period. This imposes an experimental condition where both measurements (VDS and VF) occur within a short period of time, thus avoiding temperature variations. Placing the inverter in the furnace has a detrimental effect on load current linearity (see fig. 7) as the electrolytic capacitors bank is kept out of the furnace (electrolytic capacitors cannot 0-9999-9999-0/09$17.00©2003 9999 Temperature is acquired as the average value given by two thermocouples put at both end of the bridge heatsink. Temperature value is therefore accurate within the ±1°C range. More accuracy would be very difficult to achieve, due to the size of the active area (active area of the silicon MOS die). Higher accuracy is not required because of the simplicity of the proposed model for the expression of the body diode forward voltage drop.

Fig. 3 represents the measured drain-to-source voltage when the MOSFET is in the on-state, for 6 different ambient temperatures. As can be seen on fig. 4 , the RDSon evolves as a 2 nd order polynomial:

R DSon =C 0 C 1 TC 2 T 2 (13)
When fitting data, such a polynomial yields to: C2=16×10 -8 Ω.T -2 C1=32×10 -6 Ω.T -1, C0=58×10 -4 Ω The body diode forward-voltage, shown on fig. 5 is more difficult to identify. Fitting is set to minimise error at high current, near SOA (Safe Operation Area) limits. Error is therefore not negligible at low current. Higher order modelling would be required to reduce this inaccuracy, but is beyond of the scope of the targeted application.

In our case, a simple expression gives the value of the body diode voltage drop:

V F =V S T K 2 I
The relation between Vs and temperature can be seen on fig. 6 , and is expressed by [START_REF] Maxim | Power PIN Diode Electro-Thermal SPICE Macromodel with Forward and Reverse Recovery[END_REF]. Body diode forward voltage can therefore be expressed as follow:

V F = K 0  K 1 T K 2 I
From the identification of data with (7) it comes: K2 =1.9×10 -3 Ω K1 = -1.5×10 -3 V.K -1 K0 = 0.786 V This set of parameters (C0, C1, C2, K0, K1, K2), allows therefore for the calculation of temperature and current across the MOSFET by only proceeding to two distinct drain-source voltage measurements.

Solving (12) for the measurements performed on fig. 7, where VDS1=1.08 V and VDS2=-0.743 V and the parameters computed above, gives T=133°C and I2=84A, while the temperature imposed by the furnace was 128°C and the current at the time of the acquisition is 88A.

V. APPLICATION AND DISCUSSION

It has been established in the previous section the possibility to compute the temperature and current in any MOSFET of the H-bridge by performing two voltage measurements across the device.

In this section, an implementation of the method is described, and the results are discussed.

A monitoring system is built around the H-Bridge: a differential amplifier senses the voltage across the MOSFET for a fast sample-and-hold stage. An analog-to-digital conversion and a data processing is devoted to a small outline microcontroller. One can notice that both analog data (diode forward-voltage drop and on-state MOSFET voltage) use the same acquisition path.

The previous equations are coded into the microcontroller which displays temperature and current values concerning the monitored MOSFET. 0-9999-9999-0/09$17.00©2003 9999 The analytical approach is chosen for its advantages: ease of calibration (device characteristics dispersion is taken into account by updating a six parameters only), small memory requirement (an equation set replaces a measurement look-up table ).

With respect to our application, some assumptions can be made: Load current is pure AC triangular-shaped. We also have identity of the duration between the beginning of L1 conduction and VF measurement, and duration between VDSon measurement and turning-off of L1. So we may assume I1=-I2. It comes from (12):

V DS1 =-C 0 C 1 T C 2 T 2  V DS2 K 0 K 1 T K 2 (14)

A. Results

To solve the previous equation, a dichotomy algorithm is used. It can be seen on fig. 10 that over the whole shaded area the Temperature is monotonous with respect to Vf and VDS, and will not therefore "trap" the algorithm. Current value is then directly calculated from (11).

As the current waveform has a fixed shape, the system can directly display the RMS current value. This allows for precise measurement using the oscilloscope, as it integrates current value over several periods. Peak and RMS current values are simply proportional in a 3 ratio. Temperature computation requires about 1ms with a general purpose low-outline 8-bit microcontroller running at 20MHz. This is due to the intense use of floating point 32-bits numbers, and the dichotomy algorithm which requires 8 solving iterations of (12) in the worst case (as temperature is coded as an 8-bits integer). Current computation is a lot faster (few tens of µs) as it only requires a single floating point multiply and two additions.

In order to measure the accuracy of the monitoring system, it is tested over the full current and temperature ranges of the MOSFETs, using previously described test bench where temperature is set using a furnace. These measurement points are chosen different from those of the parameters estimation (above) to ensure our method is reliable. Driving of the bridge is still made using burst square signals to avoid self-heating that would result in different chip and heatsink temperatures.

The current and temperature estimated by the monitoring system are compared to the values measured by the current probe and the thermometer. For the sake of brevity, a global error factor of merit, combining error values on current and temperature is proposed:

 I , T =  I 2  T 2 ( 15
)
where

 x = x measured -x estimated x measured (16)
Fig. 12 Shows the value of ∆I,T over the whole operating range of the MOSFET.

We can see that this monitoring system is reliable, as it computes accurate current and temperature values at high current or high temperature values (near SOA limits).

Error is relatively high at low current level, due to limitations of the estimator: current is calculated as an integer, resulting in high rounding errors under 10 amps. However, the error does never exceed 2 amps over the whole map, and is 0-9999-9999-0/09$17.00©2003 9999 fig. 12 error map mainly less than 1 amp. This accuracy is achieved thanks to the -relatively-low sensitivity of VDSon with respect to the temperature. Another issue at low current level is caused by the assumptions we make on body diode forward voltage drop. The hypothesis of proportionality between body diode forward voltage and drain current is accurate at high current level (see fig. 5), but as current value decreases, the error grows. Fitting of measurement curves is chosen to optimise estimation at high-current (to achieve high reliability), at the cost of worse results when switching low current. This does not conflict with the monitoring purposes, mainly thought for reducing security margins and therefore increasing MOSFET usage.

VI. FURTHER DEVELOPMENTS AND CONCLUSION

Implementation of the proposed method is described in the previous section. The application is based on two industrial inverter legs connected as a H-Bridge, operating over the full current and temperature range, as specified in device datasheets. However, driving signals (fixed duty cycle) and load (almost pure inductance, with a low value with respect to the switching frequency) are adapted to suit a demonstration purpose only, where the current in one of the switches changes its sign during every commutation cycle. This is not the case of a majority of converters.

Therefore further developments are required to adapt the method for "Real World Applications".

Thanks to current and temperature properties (long time constant vs. switching frequency, periodic behaviour, isothermal areas...) some assumptions can be made, allowing the implementation of the presented technique in operational converter.

Faster estimation of temperature is possible by replacing the 8-bit microcontroller used in our prototype by a low-end signal-oriented processor. Real time computation of the parameters can then be expected. On the other hand, one can use a data look-up table containing current and temperature values corresponding to given VF and VDS instead of the analytical approach described in this paper. The computing power and time required by this method is almost zero, at the extra cost of some kilo-bytes of EPROM.

In addition, it appears that temperature estimation is very sensitive to the forward voltage drop of the body diode. A high resolution measurement of this voltage (such as 14-bit A/D conversion) would be required to increase estimation accuracy.

A special interest is the monitoring of paralleled devices in inverters, where transistors are used very intensively, and current sensors are difficult to integrate.

An interesting feature is that the proposed method gives a reliable approximation of the MOSFET die temperature. Classical techniques using external sensor, such as thermistors only give substrate or ambient temperature, which is sometimes quite different from chip temperature, especially during load surges. Accurate knowledge of temperature allows for a better usage of active components by reducing security coefficients, and therefore decreasing overall system cost.

As a conclusion, it is confirmed that the proposed method, based on voltage measurements, allows for the estimation of two key variables in MOSFET-based converters : current and internal chip temperature.

fig. 1

 1 fig.1 conventions

fig. 5 .fig. 4 .

 54 fig. 5. body diode voltage drop for several ambient temperatures0 1 6 3 2 4 8 6 4 8 0 9 6 1 1 2 1 2 8

fig. 8 7

 87 fig. 8 Drain to source voltage during the conduction period.

9

 9 fig. 10. computed temperature over the whole Vds, Vf range.Shaded area corresponds to operating temperature.