Léonard Turpin
email: leonard.turpin@ens-paris-saclay.fr

Stéphane Roux

Olivier Caty

Sébastien Denneulin

Coupling tomographic and thermographic measurements for in-situ thermo-mechanical tests

Keywords: Pin-hole projective model, Integrated-DIC, Infra-red imaging, In-situ thermo-mechanical test, Full-field measurement, Computed tomography

In order to determine the thermo-mechanical properties of a complex 3D woven ceramic composite material, an experiment at high and inhomogeneous temperature and its dedicated full-field measurement procedure is developed. 3D tomographic images of the tested sample are captured at different stages of loading in a synchrotron beamline, and an infrared camera captures a side view of the sample as it rotates in the X-ray beam. A pin-hole projective model of the thermographic camera allows one to map the thermal field measured under numerous orientations onto a 3D mesh of the sample built from an initial tomographic image or a model. The projective model has to be calibrated, and an original procedure is proposed thanks to an integrated digital image correlation algorithm based on the "silhouette" of the sample (as only the protruding edges outlining the sample shape can be seen clearly). This procedure is illustrated with an experimental case study.

Introduction

The use of in-situ mechanical tests in material science has substantially increased during past three decades. It consists in performing experimental tests (such as a mechanical test) inside a sophisticated piece of equipment delivering full field measurements, e.g. Transmission or Scanning Electron Microscopy (TEM or SEM) [START_REF] Legros | Quantitative In Situ Mechanical Testing in Electron Microscopes[END_REF], Electron BackScatter Diffraction (EBSD) [START_REF] Wright | A review of strain analysis using electron backscatter diffraction[END_REF] or tomograph [START_REF] Merrifield | A novel technique combining high-resolution synchrotron X-ray microtomography and X-ray diffraction for characterization of micro particulates[END_REF][START_REF] Maire | In-situ X-ray tomographic study of materials[END_REF]. Such experiments are quite complex to set up. Most of the time, a dedicated experimental device has to be developed. Providing exceptionally rich and highly resolved in space or time information about evolutions of the sample, they are among the most invaluable tools to understand and describe quantitatively material behaviour when subjected to a variety of different loadings including extreme conditions. The very last improvement consists in using multi-modality to further broaden in-situ measurement [START_REF] Tudisco | An extension of digital volume correlation for multimodality image registration[END_REF].

This work is focused on thermo-mechanical behaviour of thermo-structural composite materials [START_REF] Narottam | Handbook of ceramic composites[END_REF]. Those materials are used at very high temperatures (more than 1000 • C), under severe thermal gradients, and yet, they are expected to preserve their structural qualities. Their very complex micro-structure produces anisotropic 3D thermal and mechanical fields calling for full-field measurements. Testing materials at such high temperatures is challenging and development of new experimental methodologies is needed [START_REF] Mazars | Damage investigation and modeling of 3D woven ceramic matrix composites from X-ray tomography in-situ tensile tests[END_REF][START_REF] Berny | On the Analysis of heat haze effects with spacetime DIC[END_REF].

The originality of our experiments is that they are carried out using tomography and thermography. An in-situ loading device was designed to perform multi-modal experiments on complex-shaped samples [START_REF] Turpin | Design of an in-situ multi-axial test to characterise the thermo-mechanical behavior of CMC parts[END_REF]. The sample is heated externally in a range from 1000 • C to 1200 • C with a non-homogenous thermal field to induce elastic stress from kinematic incompatibilities so that damage induced by extreme thermal and/or mechanical loadings can be studied.

Tomography provides volume images of the sample. It is particularly relevant to study micro-structured materials or samples submitted to complex localized loading [START_REF] Baruchel | X-Ray Tomography in Material Science[END_REF]. In-situ tomographic observation has first been developed to study damage and crack propagation [START_REF] Guvenilir | Direct observation of crack opening as a function of applied load in the interior of a notched tensile sample of Al-Li 2090[END_REF]. It is the focus of most of those tests including the present study where the occurrence of damage is investigated.

Due to acquisition time, a set of tomograms is generally acquired at pre-defined loading steps. The acquisitions were done on steady states. After the thermal and mechanical loadings are applied, few minutes are waited for to let the temperature and damage fields reach a stationary state before the acquisition is ran. The comparison between two steps provides information on the evolution of the bulk of the sample. The bulk displacement field taking place between two acquisition steps can be computed from tomography using Digital Volume Correlation (DVC) [START_REF] Bay | Digital volume correlation: Threedimensional strain mapping using x-ray tomography[END_REF][START_REF] Buljac | Digital volume correlation: Review of progress and challenges[END_REF]. In the global approach, DVC computation may be based on a Finite-Element (FE) mesh which can be built from a tomogram to reproduce accurately the sample geometry.

Infra-red (IR) thermography imaging is used in parallel to tomography. The thermal field emitted by sample surfaces is measured as the sample is rotated for the tomographic scan. Emissivity of the surfaces is uniform, and hence digital levels of the IR camera can be converted into temperatures. Moreover, because the tomographic scan is fast, the thermal inertia of the sample ensures that the temperature field is steady. Thus the temperature of the sample surface over most of the sample height can be estimated safely.

In order to exploit tomography and thermography jointly, it is necessary to associate the acquired raw data in a common framework. This requires a geometrical calibration of the IR camera that will account quantitatively for the sample geometry, described in 3D with a finite element (FE) mesh built from an initial tomogram, and captured at any rotation angle as a 2D image with the IR camera.

Such a geometrical calibration appears to share some similarities with that used in stereo-vision. The latter, a modern evolution of photogrammetry [START_REF] Wrobel | The evolution of digital photogrammetry from analytical photogrammetry[END_REF], consists in determining the 3D geometry of a scene from two (or more) 2D images, usually photographs, taken under several angles [START_REF] Faugeras | Real time correlation-based stereo: algorithm, implementations and applications[END_REF]. Calibration is a needed first step that aims to determine the optical parameters of the acquisition devices and their position on the scene. Calibration is typically performed thanks to a dedicated Digital Image Correlation (DIC) procedure using a textured (speckled) sample [START_REF] Chen | Calibrating stereo-digital image correlation system using synthetic speckle-pattern calibration target[END_REF]. If the sample cannot be speckled, the calibration can be performed projecting a pattern on the object (e.g. fringe pattern [START_REF] Gai | A flexible multi-view calibration and 3D measurement method based on digital fringe projection[END_REF]).

One specific difficulty of our experiment is that very few details of the sample surface can be tracked. Indeed, no speckle was used as a uniform emissivity allows an easy evaluation of temperature. Moreover, even edges between two visible surfaces cannot be clearly distinguished on the IR images. Hence only the sample protruding edges, namely the sample silhouette, can be trusted to calibrate the projective model. This very severe constraint leads to the development of an original methodology designed to be robust and accurate, to perform the geometrical calibration based solely on the sample silhouette.

Once the camera parameters are known, the projection operation is standard and can be used backward to transport the measured temperature field onto the mesh surface. Care has to be exercised to identify the visible surfaces -and to discard hidden ones -so that the thermal field can be mapped onto the visible parts of the sample surface.

An original procedure is proposed to couple those two full-field measurement modalities. After a brief presentation of the experimental setup, Section 2 details the formulation of the projective model problem and its calibration based only on the information available in thermographic images. In Section 3, this method is implemented and illustrated in an actual experimental case study. The filtering of the hidden sample surfaces is explained. The application is discussed in Section 4 together with the evaluation of the uncertainty of the thermal field. Finally, Section 5 sums up the main results and presents some future perspectives.

Methods: Calibration of projection matrix

Experimental setup

A loading device was developed to perform tests on T-shaped or L-shaped samples of ceramic matrix 3D-woven composite materials [START_REF] Turpin | Design of an in-situ multi-axial test to characterise the thermo-mechanical behavior of CMC parts[END_REF]. It can be used both in laboratory or synchrotron tomography. The thermograms presented here were acquired during an experimental campaign at PSICHE beamline at synchrotron SOLEIL [START_REF] King | Tomography and imaging at the PSICHE beam line of the SOLEIL synchrotron[END_REF]. The mechanical quasi-static loading is applied step by step, alternating loading and tomographic acquisition (which takes about 15 min). The geometry of the experimental configuration is schematized in Figure 1. The (O, X t ) axis is the optical axis of the tomograph. The loading device and the sample are fixed on the tomograph rotating plate. They rotate with respect to the tomograph about (O, Z t ) axis. During a tomographic acquisition, a large number of radiographs at different angles θ are taken. The angular step is sufficiently small so that the rotation of the plate can be continuous without degradation of the radiograph quality (no motion blur).

During the tomography, a series of thermograms is acquired with an IR camera which is positioned so that its optical axis (T, Z c ) is oriented perpendicularly to (O, X t ). This procedure provides a large set of view angles. The objective is to use the information acquired by the IR camera as boundary conditions of a 3D FE model of the sample. The thermal field will be "reprojected" onto the 3D mesh surface using a pin-hole projective model. However, prior to this exploitation, it is necessary to calibrate the projective geometry, as detailed below.

Stereo-correlation matrix decomposition

The projection from the real space to the image space can be written as a projection matrix [M ] which links the 3D homogeneous spatial coordinates of a point, {X}, to the 2D homogeneous coordinates of its corresponding point, {x}, in the image [START_REF] Sutton | Image Correlation for Shape, Motion and Deformation Measurements[END_REF].

     sx sy s      = [M ]          X Y Z 1          (1)
This formulation, where s is a scale factor (which is the expression of the perspective effect), enables writing the projection as a linear problem. [M ] is an a priori unknown 3 × 4 matrix. Due to s, [M ] is defined up to a global magnification factor, so that eleven parameters remain to be determined.

This matrix has to be identified for the whole set of n images. Without assumptions on the respective displacement of the object and the camera, n projection matrices have to be computed. During the test, the camera if fixed and the sample, clamped to the tomograph rotating stage, undergoes only a rotation about the vertical axis. The stage movement is assumed to be accurate enough to rewrite the projection matrix [M i ] 1 i n for each angular position θ i , as

M i = [M ] R i (2)
where [R i ] is the rotation matrix of angle θ i around the tomograph axis.

[M ] has 11 parameters. The position and the orientation of the rotation axis of [R i ] is fully set with 5 parameters. The angular position of the first image can be arbitrarily set to

θ 1 = 0 (thus [M 1 ] = [M ]
), only (n -1) unknown angular positions remain. For n images, n + 15 parameters are to be determined. Any 3×4 matrix does not represent an admissible projection matrix. A way to take into account the constraints obeyed by [M i ] consists in decomposing it in elementary transformations (Figure 2) which can easily be written as linear transformations, and thus as matrices. Only R(θ i ) depends on the considered image. P gathers intrinsic parameters which stand for camera optics. Extrinsic ones (i.e. related to the position of the object with respect to the camera) are decomposed to match the geometry of the considered experimental set-up. This decomposition increases the number of parameters of the problem. Transformation T 1 and T 2 have 6 degrees of freedom (d.o.f.) (3 rotations and 3 translations). For T 1 , as it just consists in aligning one of the basis axis with the tomograph rotation axis (O, Z t ), 1 translation is irrelevant. R(θ i ) has only one d.o.f. θ i . And, without further assumption on the camera optics, P has 11 parameters. Finally 5 + 6 + n + 11 = 22 + n parameters have to be evaluated.

{X} {X p } {X t } {X c } {sx} T 1 R(θ i ) T 2 P [R i ] [M ]
As all the transformations T 1 , R(θ i ), T 2 and P do not change during the experiment, calibration can be performed only once. Moreover, because a very stiff ceramic material was tested, the displacement of the sample under the mechanical load turned out to be very small (lower than thermogram resolution). Would the deformation of the sample be significant, it could be computed by DVC on tomographic images. The positions of 3D mesh (surface) nodes would then account for the deformed geometry. In fact, the only case which may be troublesome is when the sample slips out of its grip. If the sample requires a pre-loading for a good positioning, it is recommended to perform calibration only after the pre-loading step.

A gross projection matrix is first computed assessing the values of parameters. This matrix is used as an initialization for the digital image correlation procedure presented below.

Shared information between 3D mesh and thermograms

Calibration consists in computing the projection matrix parameters in a given configuration (cameras and their respective positions and orientations). Usually this procedure operates on couples of images acquired by two different cameras [START_REF] Faugeras | Real time correlation-based stereo: algorithm, implementations and applications[END_REF][START_REF] Gai | A novel dual-camera calibration method for 3D optical measurement[END_REF]. When images are not sufficiently textured for digital image correlation (DIC), a preliminary step using a dedicated calibration target is used. Such an approach determines the transformation between two images (1 and 2), [M 2 ][M 1 ] -1 which is not sensitive to some parameters which have to be set arbitrary as the 3D basis axes and a scale factor.

Here the exact transformations from the 3D frame used for the sample mesh and the image planes are sought. To make the correlation possible, data have to be pre-processed to be comparable (2D images). Moreover, thermography implies a homogeneous emissivity and grey levels are susceptible to vary in time so conventional calibration methods (based on sample texture) reveal improper [START_REF] Archer | Multi-instrumentation of very high temperature tests[END_REF]. The common information between the 3D mesh and the thermograms is thus the sample silhouette geometry. The projection of the protruding 3D mesh edges on 2D images (the silhouette) is compared to edges of the sample on the IR images. To do so, for each image, Integrated-DIC (IDIC) is performed between the silhouette of the mesh on the image plane and the binarized thermograms (Figure 3).

As the sample temperature is very high, the contrast between sample and surrounding is quite high, and binarization is easy. The grey level threshold is determined with an Otsu filter [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF] but it could equally well be determined manually over a significant range without noticeable changes. This simple and classical procedure revealed satisfactory. In more complicated cases, images can be pre-processed to apply any needed correction (e.g. mask, more than one threshold, etc.).

The information supported by the edges of the silhouette of the sample is very poor for DIC, yet suitable for global DIC. Not using directly grey level field and hence being compatible with grey level changes expected for the thermographic images, it implies an enhanced robustness. 2.4. IDIC procedure to determine matrix parameters DIC, is a full-field measurement procedure consisting in computing a displacement field u(x) between two images: the reference image, f , and a deformed one, g. Assuming grey level conservation, u(x) minimizes the residual field, ρ(x),

ρ (x, u) = f (x) -g (x + u (x)) (3)
The global formulation consists in finding the field u minimizing the quadratic norm of the residual on the whole Region of Interest (ROI) [START_REF] Gao | A global approach for Digital Speckle Correlation[END_REF].

Γ (u) = ROI ρ(x, u) 2 dx ( 4
)
To do so, the displacement field u is progressively corrected by elementary corrections δu which minimize the linearization of Γ(u) about the current point.

The main interest of such a formulation lies in the fact that u can be decomposed within any kinematic basis ϕ (x) (u (x) = ϕ i (x) a i ) -commonly, ϕ (x) is a FE basis [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF]. Finding a is a non-linear problem which is classically solved from iterative corrections, computed for the tangent (and hence linear) problem. The incremental correction δa minimizes Γ

Γ {δa} = 1 2 {δa} [H] {δa} -{δa} {b} (5)
where [H] is the Hessian matrix

     [H] = ϕ ∇f ∇f ϕ dx {b} = ϕ ∇f ρ dx (6)
The above minimization corresponds to solving the following linear system

[H] {δa} = {b} (7)
and the correction {δa} is to be combined with the current determination of the displacement {a}.

The displacement field u can depend on several parameters {p}. For each of them, p i , the sensitivity field S i is defined as the ratio of the variation of displacement induced by an infinitesimal variation of p i .

S i = ∂u ∂p i (8)
Integrated-DIC consists in directly computing a vector containing the sought unknowns {p}. Instead of solving Eq. 7, the problem is reformulated as

[S] [H] [S] {δp} = [S] {b} (9)
where [S] is the matrix concatenating all sensitivity fields of {a} with respect to the parameters {p} of [M ]. Passing through the intermediate basis {a} is not mandatory but it simplifies the writing of the problem.

In the considered case, the unknowns are the geometrical parameters of [M i ] decomposition (Figure 2). The links between u and the parameters are explicitly known and the derivatives can easily be computed analytically. As an example, using the decomposition of Figure 2

, [S T 2 ] is [S T 2 {X t }] = ∂ {X c } ∂ {p T 2 } = ∂ {T 2 X t } ∂ {p T 2 } (10)
The sensitivity of [M i ] w.r.t. the set of parameters of elementary matrix is written as a composition of elementary sensitivity matrix

[S M i {X}] = PT 2 RS T 1 {X} PT 2 S R {T 1 X} PS T 2 {RT 1 X} S P {T 2 RT 1 X} (11)
and the global sensitivity matrix, considering the full set of images is naturally

[S M ] =    S M 1 . . . S M N    (12)
2.4.1. Support of sensitivity fields As explained in section 2.3, the relevant information for DIC is located at the edges of the sample: in the area where the gradient of images (see Figure 3c-3d) is not null. So only the edges which lines up the silhouette (outline of the indicator function) are to be exploited. Moreover, to compute the sensitivity matrix, the coordinates of the corresponding points in the 3D domain have to be known. Thus only the projection of the protruding edges of the 3D mesh is to be considered.

Pyramidal approach to DIC

The registration of the silhouette onto the thermographic image using DIC cannot be performed directly with binary images. Indeed, as seen in Eqs. 6-7, images are used only through their gradients, which are sparse and ill-behaved. The abrupt variations of grey levels at boundaries induce a "short-sightedness" of the algorithm. Intermediary steps to fix it consist in first blurring images using a convolution with a Gaussian of width σ (Figure 4). In a pyramidal approach, several decreasing lengths, σ, allows one to solve Eq. 9 even if the initialization is quite far from the solution. Using first a large width σ makes registration robust with respect to large displacements and then, at the end of the process, exactness is secured by not using any filter (σ = 0 px). It speeds up the resolution without prejudice to accuracy.

Figure 4: Blurred binarized thermograms used in the pyramidal approach, σ = [3, 2, 1, 0 px] from left to right Two different convergence criteria are defined. The first one is based on the ∞norm of incremental nodal displacement vector. When this criterion is met, the value of σ is decreased to sharper gradients (and hence better accuracy) once the current registration is close to the sought solution. The second is the 2-norm of the residual decrease rate and is only used at the end of the computation, at the lower floor of the pyramidal approach, where σ = 0.

Conditionning improvement

With a single projection, it is obviously not possible to determine the parameters of the projective model. The problem is ill-posed. As for stereo-vision, one has to consider at least two different points of view. But even so, in our case because of the poor information content of the silhouette, the problem defined by Eq. 9 appears as illconditioned.

The definition of parameters {p} involves the choice of parameters expressed in a specific frame. Although a mere translation and/or rotation does not change the nature of the problem, and hence not the conditioning of the Hessien, the inherent singularity of the Euler angles for an axis oriented close to the "north" pole is susceptible to make the problem artificially singular (the azimuth direction is undefined at the north pole). This is especially true in the present problem where the rotation axis of the tomograph appears to be a natural choice for the frame of reference. Hence it revealed convenient to rotate the frame within which the sample is described by a large angle to keep safe from this singularity.

Most of the parameters are rotations and translations. A second step is to normalize all rotation parameters (angles) with a characteristic length so that all variations of those parameters have about the same magnitude. Typically, this characteristics length is chosen equal to half the sample diameter.

Finally, when the above procedures are not sufficient, a Thikonov regularization is used [START_REF] Tikhonov | Solutions of Ill-Posed Problems[END_REF]. It consists in adding the identity matrix scaled by a very small parameter to the Hessian. Here, this parameter is chosen to truncate the condition number (i.e. the ratio of maximal and minimal eigenvalues) at 10 6 .

Application

Simplifying assumptions

Additional assumptions on the problem geometry may help reducing the number of d.o.f. with no loss of accuracy. First, as the rotation speed of tomograph plate and the IR camera acquisition frequency are kept constant, the angular step between images is accurately known and so is θ i , for all i. The set of rotation transformations R(θ i ) is fully determined. With this assumption, all parameters can be determined using two images. More images can be used to improve accuracy, computation time and the conditioning of the problem.

T 1 and T 2 cannot be simplified further and require respectively 5 and 6 d.o.f. At the resolution of thermograms, optical distortions are assumed to be negligible. Thus P depends only on 3 parameters: the focal length, λ, and a possible translation offset if the centre of the camera sensor is not centered on the optical axis.

P =    λ 0 0 t x 0 λ 0 t y 0 0 1 0    ( 13
)
Optical distortions are linked to intrinsic parameters of the camera. If they were not negligible, they could be determined once and for all before experiments with a dedicated calibration setup (as in [START_REF] Yang | Precision calibration method for binocular vision measurement systems based on arbitrary translations and 3D-connection information[END_REF] for example).

Eventually, 14 parameters are to be determined. In the hereafter discussed case study, eight images are used to perform the calibration.

Mesh projection

On the thermograms, only well-oriented surfaces (i.e. facing the camera) are visible. Building the projected mesh, the scalar product between the outgoing normal of each 3D surface element and the vector connecting the barycentre of the considered element to the focal point of the camera is computed (Figure 5b). Only elements for which this product is positive are considered (the other ones being obviously hidden). Because the considered sample is not convex, this criterion is not sufficient to decide whether a surface element is visible or not because parts of the object can hide some well-oriented elements. All elements whose barycentre is hidden are considered to be hidden themselves. If the line segment connecting a barycentre N of the 3D mesh element to its projection P on the image plane crosses a 3D surface element then it is hidden. Efficient algorithms to compute the intersection of a segment line and a triangle have been developed for ray tracing. The approach followed here is a simplified version of the algorithm proposed by Moller et al [START_REF] Möller | Fast, minimum storage ray/triangle intersection[END_REF].

Let ABC be a 3D surface element and I the intersection point of a ray (P N ) with the plane defined by ABC (Figure 6), the vector -→ P I can be decomposed as

-→ P I = λ --→ P N = -→ P A + ξ -→ AB + χ -→ AC (14)
Using Cramer's rule

                     λ = -→ P A • n --→ P N • n ξ = - -→ P A ∧ -→ AC • --→ P N --→ P N • n χ = -→ P A ∧ -→ AB • --→ P N --→ P N • n (15) where n = -→ AB ∧ -→ AC. B A C N I P -→ P A χ -→ AC ξ -→ AB
     0 ≤ ξ and 0 ≤ χ ξ + χ ≤ 1 0 < λ < 1 (16)
Not to be sensitive to computational errors, a tolerance ε is integrated such that

0 < ε 1      ε < ξ and ε < χ ξ + χ < 1 -ε ε < λ < 1 -ε (17)
This system has to be solved for each couple node-element, its crude implementation needs two nested loops. To easily bypass this problem and to drastically reduce computation time, the projection on the focal plane of the camera is considered. The projection of every node coincides with the focal point: ∀N, P = F . Decoupling the vectors linked to the nodes and the ones linked to the element, the computation of coefficients λ, ξ and χ is reduced to matrix products. The three matrices (nb. of nodes× nb. of el.), corresponding to the three coefficients enable to find all the nodes which respect the conditions given in Eq. 17. Figure 5c shows remaining elements.

Thermal field mapping

Using a prior black body calibration, the thermogram grey levels are converted to an equivalent temperature. This temperature field can then be re-projected onto the visible surfaces of the 3D mesh, and affected to the mesh nodes. Assuming the temperature field to be stationary in time, each nodal temperature value is averaged in time (over the frames where the node is visible).

Assuming further the temperature field to be spatially smooth enough to be faithfully described by a 3D polynomial field, the previous determination is projected onto such a polynomial field thanks to a least squares regression,

T (X) = argmin T (X) -θ δ(θ, X)f (x(θ)) θ δ(θ, X) 2 (18)
where {x(θ)} = [M (θ)]{X}, δ(θ, X) is equal to 1 or 0, respectively when the node X is visible in the image under the angle θ or not. This additional step further filters out small fluctuations that resulted from the previous mapping. The difference between the two temperature fields allows one to assess the unphysical nature of the filtered fluctuations, and its very small magnitude.

Result and discussion

"Re-projecting" thermal field

Figures 7a and 7b present the projection of the 3D mesh in a thermogram with, respectively, the initial estimation of parameters and their fine calibration. In this example, the initialization of parameters is estimated from the nominal position of the sample with respect to tomograph rotating stage (i.e. the sample is vertical and centered on the rotation axis), and the position of the camera (as in Figure 1). The scale of the sample is rather well-estimated. The calibration algorithm efficiently corrects angular and positioning errors. After calibration, projected (2D) meshes fit accurately the sample in the IR images.

Camera calibration problem is ill-conditioned and the convergence is typically very slow (a fortiori when the calibration is based only on few reference points, as in manual procedure). So the number of iterations at convergence depends much on the value of convergence criterion. Here the conditioning improvement (part 2.5) and the relevant amount of data provided using the full edges of the sample and 8 images for calibration drastically reduce the number of iterations (22 iterations for the presented case). Each iteration is more complex and more time-consuming but the overall procedure remains fast without any computational optimisation. The temperature field is expressed at the nodes of the 2D mesh (Figure 7c). Due to interpolation, the temperature close to the protruding edges is distorted. During the computation of 3D field, the nodes of those edges are omitted. However, the information provided by the whole set of images is redundant enough to allow for the determination of a 3D polynomial fit (here a degree 2 revealed sufficient) of the temperature data to be performed accurately, and hence the full 3D surface temperature map can be obtained (Figure 8a).

After calibration, the temperature was followed in time over the observed surfaces. Even if the temperature field is not strictly stationary -as the sample is rotated on the tomographic turn-table -the temperature change revealed hardly measurable because of the thermal inertia of the sample. Regarding the uncertainty on the measured field, the thermal field can be considered stationary.

This 3D field can effortlessly be mapped from a mesh to another. As an illustration, the mesh of Figure 8b corresponds to the part of the sample where the interpolation is relevant (that is the part which is visible on IR images). Setting the thermal field re-projection procedure within the wider framework of inverse measurement, the field should be mapped on the mesh used for DVC.

Considerations about errors and uncertainties

Diffuse emissivity assumption

The thermal field is diffuse, indeed the edges between two visible faces of the sample cannot be distinguished in any IR image. Moreover the grey level value of a physical point of the sample does not vary from one frame to the next. The emissivity of the sample seems not to depend on the view

Uncertainty induced by thermograms noise

The uncertainty on the re-projected thermal field induced by IR images noise can be estimated empirically. IR-image noise level η is computed as one half of the standard deviation of the difference between two images of the same scene. (I i ) i=1...n denotes the original set of IR images. A new set of images (J i ) i=1...n is built adding a Gaussian white noise of intensity η to each images I i .

The two 3D thermal fields, T I and T J , corresponding respectively to set (I i ) and (J i ) are computed. The error field E is the difference

E = T J -T I (19)
The uncertainty is characterized by the standard deviation of E. It is dependent on the artificial noise realizations. A large number of samples has to be used to estimate the average standard deviation e.

In our images, the noise level is η = 0.25% (of the full range of grey level). The 3D fields are computed onto the visible (meshed) surface of the sample (Figure 8b). Figure 9a shows an example of error field whose intensity remains low. It is quite smooth because it is the difference between two polynomial fields. The uncertainty e is computed over 100 realizations. The graph (Figure 9b) presents the influence of the number of images used for the computation on the average uncertainty e (the minimum and maximum uncertainties on the samplings are plotted in dotted lines). As expected, e decreases with the number of images used. When the full set (48 images) is used, the uncertainty, induced by the noise affecting the IR images, is about 0.01%.

Residual fields

To evaluate the relevance of the 3D polynomial interpolation, residual fields are computed. They are a set of 2D images defined as the difference between the projection of the 3D polynomial field and the IR images (Figure 10). Their intensities are low (a few percent of image dynamic). As expected, the polynomial approximation smoothens the field and some differences are visible, namely near the edges of the sample. The luminance data on the thermograms can be affected by the environment of the sample. Taking into account the whole set of images, this effect is averaged out in the 3D field. Residuals were computed for several polynomial fields of different order. All three have the same order of magnitude. For degree 2, 3 and 4 their norms are respectively 1.20%, 1.22% and 1.12%. The assumption on the smoothness of the temperature field to be described by a low order polynomial approximation is deemed validated.

Conclusion

To draw an intimate connection between two full-field measurement modalities, a new calibration procedure is introduced. In order to allow a direct use of the IR image intensity to infer the temperature value, the sample is not speckled. Hence, the calibration has to rely only on the silhouette of the sample. The proposed procedure is based on the correspondence of the protruding edges of the 3D mesh projected on the IR-camera image plane, and the outline of the sample extracted from the thermograms. The use of the calibrated projective model allows the temperature field to be mapped onto the surface of the 3D mesh of the sample, and to be readily accessible for any further thermo-mechanical modelling.

Up to now, in-situ thermo-mechanical tests within a tomograph, were carried out either at room or at high temperatures, but most of the times with a spatially uniform temperature field. However, for ceramic materials and especially ceramic composites it is very important to assess the material resistance to extreme temperature gradients potentially combined to mechanical loads. Moreover samples with a complex shape (such as T-or L-shaped woven composites) is required to obtain meaningful results, representative of extreme conditions that industrial parts have to sustain in service.

The proposed procedure allows to quantify thermal fields on the actual sample 3D geometry. Coupled with tomography and DVC, both temperature and displacement field can now be accessed with a consistent description, that of a 3D mesh of the sample, directly amenable to quantitative comparison with a numerical modelling. This opens up the possibility to validate complex thermo-mechanical model of complex microstructures with local highly anisotropic properties.

The presented procedure is quite generic and can be extended to other acquisition modalities. It can provide different fields, mapped onto the same frame, such as the temperature field in the present case, but it is also applicable to multiscale or multiresolution acquisitions.

Figure 1 :

 1 Figure 1: Scheme of the experimental configuration. The sample and the rotating stage are in red and the camera, in blue. S is the X-ray source, O, the center of rotation and C, the center of the scintillator. (O, X t , Y t , Z t ) is the coordinate system of the tomograph frame of reference. (O, X p , Y p , Z t ) is the rotating plate frame. Finally (F, X c , Y c , Z c ) denotes the frame of reference of the IR camera.

Figure 2 :

 2 Figure 2: Decomposition of [M i ] into elementary transformations. T 1 and T 2 correspond respectively to the transformation from the 3D mesh frame to the tomograph rotating stage frame and from the tomograph frame to the camera frame. R(θ i ) corresponds to the rotation of the stage. And P represents the projection onto the image plane.

  (a) Thermograph (b) Projection of 3D mesh outlines (c) Silhouette from the thermograph (d) Silhouette from the 3D mesh

Figure 3 :

 3 Figure 3: Silhouettes of the sample are derived from either the thermograph (a) or a 3D mesh (b). From the thermograph, binarization is used to determine the silhouette from the thermograph (c), and edges extraction provides that of the mesh (d).

Figure 5 :

 5 Figure 5: 3D Mesh projection (a) before filtering, (b) only well-oriented elements, (c) after filtering of hidden elements

Figure 6 :

 6 Figure 6: Decomposition of P I in the basis defined by the triangular element ABC

Figure 7 :

 7 Figure 7: Result of calibration and affectation of temperature values onto the surface mesh (a) before calibration, (b) after calibration, (c) temperature mapping

Figure 8 :

 8 Figure 8: Registration of thermal field in 3D mesh (digital level) (a) on calibration mesh, (b) on another (finer) mesh

Figure 9 :

 9 Figure 9: Estimate of the uncertainty of the re-projected field: (a) an error field; (b) average uncertainty e is shown as a solid curve, whereas minimum and maximum uncertainties on the 100 independent samplings are indicated as dotted curves.

Figure 10 :

 10 Figure 10: Residual (%): Difference between the projection of the 3D polynomial field and infra-red images evaluated for polynomials of degrees 2, 3 and 4, respectively for (a), (b) and (c)

Acknowledgments

L. Turpin PhD is supported by a convention CIFRE implying Association Nationale de la Recherche et de la Technologie and Safran Ceramics. Safran group and Direction Générale de l'Aviation Civile are acknowledged for funding this work through PRC MECACOMP. In-situ experiments were carried out at PSICHE beamline at synchrotron SOLEIL (Saclay, France) with the valuable support of Andrew King and were part of proposal 20181639. It is a pleasure to acknowledge David Damiani for the numerous discussions about infra-red thermography and his help during experiments.


Fatima Zahra Boudara

Marc Rivaletto

Laurent Pecastaing
email: laurent.pecastaing@univ-pau.fr

Antoine Silvestre De Ferron

Sylvain Paquet

Jean-Pierre Brasile

Journal of Electromagnetic Analysis and Applications Design and Development of High Efficiency 150 kW Very Compact PLA Core Electromagnetic Coupler for Highly Resonant Power Transfer Technology

Keywords: PLA Core, Toroidal Coupler, Resonant System, Low Stray Emission, Smart Grid

Highly Resonant Power Transfer (HRPT) technology is currently receiving very significant attention from the industry and the smart power grid distribution community in particular. This technology ensures electrical power transmission between two points while controlling the level of transmitted power and ensures the immediate shutdown of the transmitted power in the event of a problem. This paper reviews the inductive power transfer method and describes the design of an ultra-compact PLA core electromagnetic coupler. The proposed architecture confines the magnetic field in a toroidal PLA core transformer, and by avoiding the use of heavy and bulky shielding plates, reduces magnetic losses and avoids the Curie point. As a result, the overall unit has a weight of 5 kg and a volume of only 0.013 m 3 . The electromagnetic coupler is capable of transferring a peak power of 150 kW with an operating frequency of 193 kHz, giving a satisfactory efficiency of 95%. The proposed novel system was first investigated through CST 3D numerical modelling to determine the electrical parameters of the coupler's equivalent circuit and its efficiency, to verify its compatibility with the ICNIRP 2010 standard and to evaluate its temperature rise with an air-cooling system. Afterwards, the designed coupler was built with a 3D printing device and finally tested experimentally. Simulation and experimental results are compared and show a good agreement.

Introduction

In toroidal coupler, the winding occupies the full periphery of the core, resulting in reduced leakage inductance and stray electromagnetic field. Besides, it has advantages related to size and volume. Thanks to those merits, it is widely used in low power applications but is still not very available in high power ones and less studied compared to the conventional transformer. Technically, the aim of this paper is to demonstrate that toroidal electromagnetic coupler can be used in modern high power applications, such as smart grid technology.

One of the main components of this technology is the transformer, which must ensure efficient power transmission with high-performance control and communication systems. So, in this present, we will show the possibility of replacing a conventional transformer by a galvanically isolated PLA core toroidal electromagnetic coupler that will address the main challenges of smart grid technology using the inductive transmission method.

Figure 1 shows a diagram of the power grid distribution. The power plant produces electricity. Through the step-up voltage transformer, the voltage is increased and power is directed to transmission systems that supply this power to customers from the distribution substation. Smart grid technology enables better coordination of local needs and capabilities while maximizing system reliability, resilience and stability [START_REF] Nicholas | Lessons Learned on Early Electric Vehicle Fast-Charging Deployments[END_REF].

In this context, the goal of this paper is to study the development of a very compact PLA core electromagnetic coupler that can be integrated in a voltage converter ensuring a galvanic isolation between the input and the output of the system. The design of a lightweight structure and a high-efficiency system are the key elements taken into account.

First the power transfer system is presented. Secondly, the electrical characteristics of the designed coupler are discussed in detail through the simulation results. Next, the prototype of the toroidal electromagnetic coupler and the experimental results are presented in the fourth section. Finally, in order to comply with international guidelines, electromagnetic leakage fields (EMLF) around the coupler are simulated, measured and discussed in the last section.

Power Transfer System Overview

A power transmission system block diagram is shown in Figure 2. It consists of a DC/AC power converter, two resonant compensation networks for the primary and secondary coils and a rectifier converter on the receiver side. In a conventional transformer, the primary and secondary coils share the same magnetic core. Thus, the magnetic flux generated by the primary moves forward to the secondary, allowing a high coupling factor and good efficiency. On the other hand, in an electromagnetic PLA core coupler, the primary and secondary coils are not physically connected. Therefore, a small amount of magnetic flux generated by the primary flows through the secondary. This results in a poor coupling factor, in the range of 0.1 to 0.3. This coefficient decreases as the air gap between the primary and secondary coils increases.

The differential Equation (1) and Equation ( 2) for the coupler are presented by the main components of the equivalent circuit in Figure 3.

1 2 1 1 1 1 d d d d i i U L M R i t t = ⋅ + ⋅ + ⋅ (1) 2 1 2 2 2 2 d d d d i i U L M R i t t = ⋅ + ⋅ + ⋅ (2)
The inductances L 1 and L 2 are the primary and secondary self-inductances and M is the mutual inductance between these two coils. R 1 and R 2 are the primary and secondary resistances of the wires that characterize the Joule losses. The coupler voltages on the primary and secondary sides are denoted respectively U 1 and U 2 , while i 1 and i 2 are the coupler currents.

For the ideal transformer, the two windings are crossed by the same magnetic flux, therefore M L L = ⋅ . However, for the PLA core electromagnetic coup- ler, the magnetic leakage inductance L σ leads to

1 2 M L L < ⋅ .
That is why the coupling factor k is defined by: The weak value of the coupling factor in a PLA core electromagnetic coupler results in an increase in the leakage inductance value, so the voltage drop is increased and the magnetizing inductance is decreased. In this way, the magnetizing current is higher. Thus, additional reactive power must be supplied from the voltage source to the load for the same amount of active power, resulting in a significant reduction in efficiency. To compensate for the inductive effect and to ensure efficient power transfer, resonance capacitors are connected in series or parallel with the primary and secondary coils giving rise to four possible configurations (Series-Series, Series-Parallel, Parallel-Series, Parallel-Parallel). This double compensation is essential because the compensation on one side does not take into account the reactive energy linked to the other side. Besides, the selection of the compensation topology should mainly depend on the proposed application and the topology of the power converter chosen [START_REF] Steigerwals | Analysis and Design of a Contactless Rotary Power Transfer System[END_REF] [3]. To ensure an efficient energy transfer from the transmitter to the receiver, the capacitances of the primary (C 1 ) and secondary (C 2 ) compensations must be chosen such that the resonance frequency indicated by Equation ( 4) is equal to the switching frequency of the electronic power converter.

1 2 M k L L = ⋅ (3)
0 1 1 2 2 1 1 1 1 2 2 f L C L C π π = ⋅ = ⋅ ⋅ ⋅ (4)
So, as a first step in the design of the Power Transmission System, a coil design as well as a suitable compensation topology should be selected. This will be discussed in the next section.

3D Numerical Simulation Results

The values of the primary and secondary currents provide the initial information for the selection of the characteristics of the wires of the coils. The cross-section of the conductor is chosen as a function of the current density. In our case, a density of 10 A/mm 2 is considered an acceptable benchmark. The conductor cross-section also influences the coil resistances and the self-inductances. In addition, the selection of the section must deal with aspects of cost and mechanical realization.

The maximum RMS current supplied by the Mosfets of the power converter is 400 A, so four Litz wires in parallel with a cross-section of 10 mm 2 are mandatory in order to be able to withstand this value of current without destroying the Litz wire. Therefore, a Litz wire consisting of 2500 × 71 μm strands and an outer diameter of 5 mm with a cross-section of 9.98 mm 2 was selected for the primary and secondary windings. The diameter of the strands was chosen based on the skin depth at the operating frequency. The maximum current value also allowed us to estimate the minimum allowed value of the self-inductances of the electromagnetic coupler and therefore the length of the Litz wire.

The compactness of the system is the key element considered when sizing the toroidal coupler; for this reason, a spacing of 1 cm between the loops of the toroidal coupler was chosen. This distance corresponds to the minimum possible distance between loops without mechanical contacts and allows us to obtain a maximum coupling coefficient. This project does not only target the civilian market, but also the military one.

To meet its needs in terms of system compactness, a resonant frequency of around 200 kHz was selected. This frequency differs from the usual working frequency found in the literature, which generally does not exceed 85 kHz. The parameters of the selected design are listed in Table 1.

Using CST 3D numerical software, the toroidal coupler was simulated to determine the electrical parameters of the coupler's equivalent circuit. The diameter of a single strand of the Litz wire in high frequency applications is very small (a few tens of micrometers). Taking into account the depth of the skin effect requires substantial meshing and is time-consuming. Therefore, we used the method proposed by [START_REF] Etemadrezaei | Equivalent Complex Permeability and Conductivity of Litz Wire in Wireless Power Transfer Systems[END_REF] to simulate the Litz wire.

The modeled coupler with EMLF distribution is presented in Figure 4. The primary and secondary windings wound over the entire surface of the toroid confine the magnetic field lines, offering the toroidal coupler very reduced electromagnetic radiation. The circuit parameters obtained from the CST software are presented in Table 2 in the next section.

As mentioned earlier, to overcome the issue of large leakage inductance, resonant capacitors are used. Numerous studies have been conducted to compare the four basic topologies [START_REF] Ryu | Analysis of the Contactless Power Transfer System Using Modelling and Analysis of the Contactless Transformer[END_REF] [6] [START_REF] Bhuyan | Effect of Design Parameters on Resonant Wireless Energy Transfer System[END_REF]. It has been found that the Series-Series (SS) topology induces the lowest current in power electronics systems and prevents the destruction of components in the event of undesirable overvoltage. Besides, according to [START_REF] Lee | Practical Bifurcation Criteria Considering Coil Losses and Compensation Topologies in Inductive Power Transfer System[END_REF] [9] [10] the SS topology is the main compensation method often used in power transmission systems because it is the only configuration that does not vary the resonant frequency with coupling factor or load. Given these main advantages over other compensation methods, the SS topology was found to be the most suitable for our application and will therefore be used.

The overall system including the 3D model of the coupler was simulated using CST Design Studio with an equivalent load of 1.85 Ω, as shown in Figure 5.

The maximum input power obtained is 152 kW and the maximum output power is 147 kW with an approximate efficiency of 96.4% using the SS compensation topology.

For the thermal management of the power losses in the electromagnetic coupler, a forced-air cooling system is considered. The heat transfer coefficient needs to be determined for the Litz wire and for the PLA material. This coefficient empirically reflects the heat exchange from the inside (Litz wire) to the outside (forced air) and its value depends essentially on the type of material and the fluid flow.

So, a theoretical thermal modelling of the hollow torus was carried out and the heat equation is solved in a torus coordinate system for the appropriate boundary conditions. The steady-state thermal problem is solved first and then the simulation shows that the temperature is not too high for the Litz wire and for the PLA support. The maximum temperature reached respects the maximum thermal limits supported by the Litz wire, which is 120˚C, and the PLA, which is 60˚C.

The results predicted by the 3D modeling are very encouraging. A prototype of the electromagnetic coupler was then developed in order to experimentally verify the modeling results.

Experimental Tests and Validation

Considering all the criteria discussed in the third section, a 150 kW HRPT system was designed and built. The operating frequency of 193 kHz was determined and the SS topology was chosen as the compensation technique. To verify the efficiency of the proposed system, experimental tests were carried out. The experimental setup is shown in Figure 7.

DC/AC Converter

A 400 DC power supply is used for the measurements. Additionally, the energy storage is provided by (1ER300UAVT8)300 μF/1.2 kV type capacitors from SCR.

Thanks to a capacitor bank composed of three of these capacitors, the energy delivered by the DC power supply is stored and then transmitted to the DC/AC converter. This system relieves the DC power supply during high power demands by preventing an excessive drop in the voltage it delivers. The DC/AC converter was built with 1.2 kV/300 A rms SiC-MOSFETs (R DS,on = 4.2 mΩ at 25˚C) with a switching frequency of up to 200 kHz. The MOSFET high blocking voltage will allow future experiments with a higher DC voltage up to 1 kV.

SS Compensation

SS topology was chosen for the reasons discussed above. According to (4), capacitors of 350 nF in the primary and secondary sides are required to ensure the highest efficiency of power transmission. As result, high-power resonant capacitors from Illinois Capacitor (104HC1102K2SM6) were used. According to the datasheet, the equivalent series resistance of the component at 200 kHz is 1.5 mΩ.

Rectifier Circuit

A rectifier diode circuit is located at the output of the system. Fairchild RHRP30120 diodes 1.2 kV, 300 A rms were chosen. power is 155 kW (420 V, 368 A) and the output peak power is measured to be 146 kW (424 V, 344 A), thus demonstrating an efficiency of 94.4%.

Electromagnetic Coupler's Efficiency

Human Safety Concerns and International Regulations

To enable the application of any electromagnetic coupler and to comply with international guidelines, the magnitude of EMLF must be checked. For this reason, several solutions have been proposed in the literature in order to limit the emission of the electromagnetic field [START_REF] Knecht | Optimization of Transcutaneous Energy Transfer Coils for High Power Medical Applications[END_REF] [13] [START_REF] Bosshard | Multi-Objective Optimization of Inductive Power Transfer Systems for EV Charging[END_REF]. Most of these studies use the active shielding method in order to better confine EMLF noise. In the OLEV system [START_REF] Kim | Coil Design and Shielding Methods for a Magnetic Resonant Wireless Power Transfer System[END_REF], a shielding method using a reactive resonant current loop in which the original magnetic field is cancelled by the magnetic field generated by the resonant loop current is adopted. These solutions have proven to be effective in ensuring low EMLF levels. But this makes the system more cumbersome and adds complications to the design of the coupler. The unshielded toroidal structure of the PLA coupler has the merits of confining the EMLF inside the torus and keeping the system lightweight, as will be demonstrate in this paragraph.

Two international groups have set guidelines for human exposure to variations in flux density. The first is the International Committee on Electromagnetic Safety (ICES) under the leadership of the Institute of Electrical Electronic Engineers (IEEE) [START_REF]IEEE Standard for Safety Levels with Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz. IEEE C95[END_REF]. The second is the International Commission on Non-Ionizing Radiation Protection (ICNIRP) [START_REF]ICNIRP Guidelines for Limiting Exposure to Time-Varying Electric, Magnetic and Electromagnetic Fields up 300GHz[END_REF]. These regulations have been drawn up to limit human exposure to variations in flux density in order to prevent adverse effects on human health.

In the IEEE C95.1-2005 standards, the maximum permissible exposure of the head and torso is 205 μT for public exposure and 615 μT for occupational exposure. The limit value is even higher for the limbs, with 1130 μT. The latest published version of the ICNIRP [START_REF]ICNIRP Guidelines for Limiting Exposure to Time-Varying Electric, Magnetic and Electromagnetic Fields up 300GHz[END_REF] states that for a frequency range from 3 kHz to 10 MHz, people should not be exposed to a flux density greater than 27 μT for public exposure and 100 μT for occupational exposure.

Compared with IEEE C95.1-2005, the ICNIRP 2010 standards are more stringent. Moreover, as mentioned in [START_REF] Budhia | Design and Optimization of Circular Magnetic Structures for Lumped Inductive Power Transfer Systems[END_REF], the standards do not explicitly describe the measurement techniques for determining whether systems comply with the guidelines, thus making the development of the coupler's prototype difficult.

In order to verify compliance with international standards, simulations and measurements were undertaken to analyze leakage EMLF around the system. Simulations were carried out with the CST software and measurements were performed with a SMP2 WAVE CONTROL three-axis probe.

The evolution of the stray field during transmission of 150 kW at 193 kHz, together with the X and Z axes used (Figure 10 (a)) for the reference simulation and measurement, is shown in Figure 10. The probe is moved along the X horizontal axis in Figure 10

Figure 1 .

 1 Figure 1. Schematic of the power grid distribution [1].

Figure 2 .

 2 Figure 2. Block diagram of the power transmission system.

Figure 3 .

 3 Figure 3. The coupler equivalent circuit.

DOI: 10

 10 .4236/***.2020.***** 5 Journal of Electromagnetic Analysis and ApplicationsTaking all this into account, a toroidal electromagnetic coupler was designed after a long optimization process carried out by our industrial partner. The outer diameter of the electromagnetic coupler is 248 mm. The primary and secondary windings have the same loop diameter of 110 mm. They consist of nine turns.

F

  . Z. Boudara et al. DOI: 10.4236/***.2020.***** 6 Journal of Electromagnetic Analysis and Applications transient one. The calculated values for the surface heat transfer coefficients are h Litz_wire = 245 W/(m 2 •K) for the Litz wire and h PLA = 150 W/(m 2 •K) for the PLA support with air flow of 25 m/s. Figure 6 shows the temperature distribution of the electromagnetic coupler in steady state at 150 kW. The electromagnetic 3D

Figure 4 .

 4 Figure 4. The toroidal coupler, the primary winding is represented by the red color and the secondary is represented by the blue (a) the coupler structure, (b) the EMLF distribution.

Figure 5 .

 5 Figure 5. The simulation circuit combining the 3D model with circuit component.

Figure 6 .

 6 Figure 6. Temperature distribution of the electromagnetic coupler in steady state at 150 kW.

Figure 9 Figure 8 .

 98 Figure9shows the waveforms of the DC/AC converter output voltage U 1 and current i 1 (yellow and blue curves respectively) as well as the system output voltage U 2 and current i 2 (purple and green curves respectively). Currents were measured with Pearson 3972 current probes and voltages were measured with

Figure 9 .

 9 Figure 9. Waveforms at the transmission at 150 kW.

  (b) and the Z vertical axis in Figure 10(c). The origin of the measurement is located in the center of the electromagnetic coupler. DOI: 10.4236/***.2020.***** 11 Journal of Electromagnetic Analysis and Applications

Figure 10 .

 10 Figure 10. Evolution of unshielded induction B, (a): The axes used for the reference measurement (b): along X axis, (c): along Z axis.

Table 1 .

 1 Parameters of the selected design.

	Parameter	Variable	Value
	Coupler diameter	D coil	248 mm
	Loop diameter	D loop	110 mm
	Number of turns	N p /N s	9/9
	Litz wire diameter	D wire	5 mm
	Copper cross-section	S cu	9.98 mm 2
	Resonant frequency	F res	193 kHz

Table 2 .

 2 The measured and simulated electrical parameters.

	Parameter	Experimental value	Simulation value	Error
	L 1 , L 2	1.50 μH	1.47 μH	2.0%
	M	1.05 μH	1.1 μH	4.8%
	L σ	0.72 μH	0.66 μH	8.4%
	k	0.72	0.74	2.8%
	(a)		(b)

Acknowledgements

The authors gratefully acknowledge the French Ministry for Defense (Direction Générale de l'Armement) for its financial support for this work (RAPID CEMPER N˚172906093).

The authors would also like to thank the support team of CST Dassault system for their useful help and discussions along the simulation process. F. Z. Boudara et al. DOI: 10.4236/***.2020.***** 13 Journal of Electromagnetic Analysis and Applications

The converter includes an electronic board to control the H-bridge power converter and a measurement system is set up to adjust the real-time control of the system.

Electromagnetic Coupler

Based on the parameters of the coil design shown in Table 1, a prototype of the electromagnetic coupler was built. The cable holder was manufactured using a 3D printer.

As shown in Figure 8, the printable volume did not allow us to print the cable support in one piece, so we had to make the coupler in three sections. To prevent a high additional eddy current loss, the use of metals for the cable holder is not permitted. The PLA material was thus chosen to support the 3D printed cable holder because of its low-loss tangent and high-volume resistivity [START_REF] Veselý | Study of Electrical Properties of 3D Printed Objects[END_REF].

Table 2 shows a comparison between the measured and simulated equivalent circuit parameters with a calculation error with respect to the measured values.

The measurements were performed using the RLC meter from HIOKI IM 3536 and demonstrate good agreement between the simulation and the practical implementation. The maximum difference between simulation and measurements for inductances is 4.8% and less than 2.8% for the coupling factor. The highest error appears for the leakage inductance because it is calculated according to ( )

, so the error in the self-inductance and the coupling factor are added up. Besides, the high value of the coupling factor leads to a reduction in the leakage inductance. Overall, the results obtained are fully consistent with the intended performance of the designed coupler. rimental measurement. This difference can be explained by the fact that the simulation is simplified and doesn't take into consideration all the parasitic elements found in the practical setup. In addition, the induction B is measured to be 27 μT at a distance of 26 cm for the X axis and 32 cm for the Z axis. By simulation, this value of EMLF is found at a distance of 28 cm for the X axis and 35 cm for the Z axis.

Despite this slight difference, in general, the results show a good match between simulation and measurement with a maximum related error of 25%. This error is due both to the measurement uncertainty of the probe, which is 5%, and to the accuracy of its position, which is estimated at +/-1 cm.

Those values are consistent with the safety guidelines recommended by the ICNIRP 2010 and IEEE C95.1-2005 for public exposure. Hence, the use of the representative electromagnetic coupler is safe.

Conclusions

In this paper, a very compact PLA core electromagnetic coupler which occupies an overall volume of 0.013 m 3 was presented, designed, simulated, built and tested. In order to achieve a high magnetic coupling factor and low emission of stray radiation, a toroidal geometry was chosen. Furthermore, to enhance the power transmission capability, a SS compensation topology was considered. Based on the 3D simulation and the experimental implementation, the electromagnetic coupler designed demonstrates high efficiency of 94.4% for transmitted peak power of 150 kW at 193 kHz.

The work presented also shows that the electromagnetic coupler is in line with the ICNIRP 2010 recommendations for electromagnetic field exposure of the general public at a horizontal distance of 26 cm and a vertical distance of 32 cm from the center of the coupler.

Actually, the electromagnetic coupler designed could transfer up to 1 MW peak power. However, the thermal limit of PLA, which is 60˚C does not allow the conduction of high-power tests. Working at this level of power requires more efficient cooling systems for the toroidal electromagnetic coupler or replacing the PLA material with other more heat resistant material such as TC poly which has a better temperature limit of 175˚C. The project team is still working on this aspect.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.


Jérôme Long

Ivan V Basalov

Natalia V Forosenko

Konstantin A Lyssenko

Ekaterina Mamontova

Anton V Cherkasov

Marko Damjanović

Liviu F Chibotaru

Yannick Guari

Joulia Larionova

Alexander A Trifonov

Dysprosium Single-Molecule Magnets with Bulky Schiff Base Ligands: Modification of the Slow Relaxation of the Magnetization by Substituent Change

come

Since their discovery fifteen years ago, lanthanide Single-Molecule Magnets (SMM) captivate the chemistry and physics communities due to their possible applications in data storage, spintronics or quantum computing. [1] In such coordination complexes, a slow relaxation of the magnetization, eventually associated with a magnetic bistability, originates from the appearance of an energy barrier, , that opposes two directions of the magnetization. The foundation of such remarkable property lies in the synergistic combination between the intrinsic nature of the lanthanide ions through their angular dependence of the 4f electronic density (oblate vs. prolate) [2] with the crystalfield generated by the surrounding ligands. In this sense, mononuclear lanthanide SMM with tremendous magnetic properties represent a challenge for synthetic chemists in order to design optimized complexes with targeted symmetry and coordination sphere (that includes the electronic density of the donor atoms) in order to maximize the magnetic anisotropy of the ± mJ states and reduce the Quantum Tunneling of the Magnetization (QTM). This latter produces direct transition between the ground state doublet, however the relaxation of the magnetization could also proceed through other spin-lattice processes (Raman and direct) that decreases the overall magnetic performances through underbarrier pathways.

Understanding these relaxation mechanisms appears essential to improve the magnetic features. In this sense, since 2016, major breakthroughs in the field have been achieved with examples of mononuclear SMM exhibiting exceptionally large anisotropic barriers of few thousands of cm 1 , [3] and a magnetic hysteresis that could almost reach liquid nitrogen's boiling point, [3d, 4] making such molecular systems particularly promising for future technological devices.

Among the possible synthetic strategies to design SMM with large anisotropic barriers, organometallic chemistry appears to be particularly efficient to accurately control the coordination sphere of the lanthanide ion. Such attribute has been efficiently employed to engineer axial crystal-field systems, which can be obtained in linear complexes exhibiting negatively charged ligands along an axis. [3d, 3e, 4-5] This arrangement maximizes the crystal-field splitting for oblate lanthanide ions, while suppressing off-diagonal matrix elements in order to shortcut the QTM. However, examples of such strong axiality in pseudo-linear SMM based on typical coordination chemistry ligands involving nitrogen or oxygen donor atoms are rather scarce [3a, 3f, 6] due to the difficulty to realize a linear arrangement and efforts have to be pursued in order to engineer such systems. For instance, Schiff base ligands involving phenoxide moieties have been largely used to obtain lanthanide SMM, [1b, 7] but the complexes displaying an enhanced axial-crystal field and, as an extent, high energy barriers are rarely met. [START_REF] Oyarzabal | [END_REF] In this sense, we report in this communication the synthesis and study of two dysprosium Schiff base complexes that display high temperature slow relaxation of their magnetization and luminescence.

Our strategy to obtain Schiff base complexes with a pronounced axial crystal-field relies on using unsymmetrical bulky tridentate ligands, either 2,4-di-tert-butyl-6-((quinolin-8ylimino)methyl)phenolate (L1) and 2,4-di-tert-butyl-6-(((pyridin-2ylmethyl)imino)methyl)phenolate (L2) containing the donor set N,N,O  in association with the oblate Dy 3+ ion. The steric hindrance imposed by tert-butyl groups is favourable to generate mononuclear complexes, while the presence of different pendant groups, such as pyridine or quinoline moieties, will allow tuning the emission wavelength. Thus the reaction of anhydrous DyCl3 with 2 equivalents of in situ prepared potassium phenolates KL1 or KL2 and 1 equivalent of NaBPh4 in THF (Scheme 1) affords the X-Ray diffraction on single crystals indicates that both, 1 and 2 are cationic mononuclear complexes that crystallize in the chiral C2221 and centrosymmetric P21/n space groups, respectively, with a unique crystallographic Dy 3+ ion (Figure 1, Table S1). The electroneutrality is ensured by tetraphenylborate anions. The coordination spheres of the Dy 3+ ions in 1 and 2 are composed of two deprotonated Schiff base ligands and two coordinated solvate molecules (THF and pyridine for 1 and 2, respectively) leading to a coordination number of eight. The coordination Dy-N bond lengths in 1 are ranging from 2.503(7) to 2.536( 6) Å. The Dy-O(phenoxide) bond lengths are equal to 2.190( 6) Å, while the distance involving THF moieties of 2.469( 6) Å are much longer. For 2, the two Dy-O(phenoxide) distances (2.193(2) and2.196(2) 2) and 121.20(4) ° for 1 and 2 respectively. The eight-coordinate polyhedrons were quantitatively analysed using the SHAPE [9] software indicating a biaugmented trigonal prism like geometry of the lanthanide site (Table S2). Thanks to the presence of tetraphenylborate anions, the shortest intermolecular Dy 3+ -Dy 3+ distances are relatively large (13.959 and 10.658 Å for 1 and 2 respectively) indicating that the complexes are well isolated (Figure S1).

The magnetic properties of complexes 1 and 2 were investigated in both, static and dynamic modes. The room temperature T values of 13.74 and 14.04 cm 3 .K.mol 1 are slightly lower than the value of 14.17 cm 3 .K.mol 1 expected for a single Dy 3+ ion, suggesting a significant crystal-field splitting (Figure S2). [10] For temperature lower than 10 K, a dramatic decrease of T is observed confirming the large crystal-field splitting. The field dependence of the magnetization at 1.8 K shows an inflection point at low field before reaching the values of 5.66 and 5.61 N under a 70 kOe field (Figure S2). Such sigmoidal shape has been previously observed and imputed to the occurrence of a high anisotropy or magnetic blocking. [3b, 3e] The occurrence of a slow relaxation of the magnetization was then investigated using alternate currents (ac) measurements. The frequency dependence of the out-ofphase susceptibilities under a zero dc-field shows a single frequency dependent peak for both compounds (Figure 2, Figure S3). The temperature dependence of the ac susceptibility indicates that out-of-phase signals could be observed up to 60 and 55 K for 1 and 2, respectively (Figure S4). For temperature higher than 10 K, fitting of the Cole-Cole plots (Figure S5) with a Debye model confirms the presence of a narrow distribution of relaxation processes with  lower than 0.2 (Table S3-S4). The relaxation dynamics can be further analyzed by studying the temperature dependence of the relaxation time, . The ln vs. T 1 plot (Figure 3) clearly shows a linear region corresponding to a thermally activated relaxation process. However, deviation from linearity is observed for lower temperatures indicating the occurrence of other relaxation processes. Therefore, the whole temperature dependence could be modeled with the following equation  1 = 0 1 exp(/kT) + CT m +  1 QTM (Eq. 1). [11] The first term accounts for an activated process, while the second and third ones stand for two-phonon Raman and QTM, respectively. To avoid over-parameterization, the m coefficient was successfully fixed to integer numbers (1 to 12). The best fitting correlation coefficient was obtained with m = 3, suggesting the presence of acoustic phonons. [12] The other parameters could be found in Table 1 and indicate that for both compounds,  is found nearly identical. Remarkably, it can be noticed that these values are among the highest reported for mononuclear lanthanide Schiff base complexes. In order to reduce the QTM, ac measurements were performed under an optimum field of 1000 Oe (Figure S6, Table S5) for both compounds and confirmed the strong decrease of the tunneling rate (Figures S7-9, Table S6-7). The temperature dependence of  can be fitted with  1 = 0 1 exp(/kT) + CT m + AT (Eq. 2) (Figure 3, Table S8) in which the third term stands for the direct relaxation process, which may become operative under dc magnetic field. [11] Figure 2. Frequency dependence of the out-of-phase susceptibility,", for 1 and 2 under a zero dc field.

An important feature for lanthanide SMM concerns also their ability to retain the magnetization. The zero-field cooled (ZFC)/field cooled (FC) curves corroborates the occurrence of a magnetic irreversibility with a maximum of the ZFC curves (corresponding to the blocking temperature TB [13] ) located at 3.1 K and 3.0 K for 1 and 2, respectively (Figure S10). However, a clear divergence in the ZFC/FC curves, equivalent to an irreversible temperature (Tirr), could be observed at 8.0 and 5. 5 K for 1 and 2 respectively which is also concomitant with a crossing between the curves. Such behavior has been previously evidenced in some lanthanide SMM and was interpreted as an out-of-equilibrium state due to the occurrence of competitive relaxation paths. [3d, 3e, 4, 6, 14] Moreover, opening in the hysteresis loops is observed up to 8.0 and 5.0 K for 1 and 2, respectively (Figure S11).

In order to get further insights into the mechanisms of slow relaxation, ab initio calculations of the CASSCF level were carried out to quantify the energies of the Kramers doublets, as well as the orientation of the anisotropic axes. The first excited Kramers doublets (mJ = ± 13/2) are found at 250 and 246 cm 1 (Table S9-S10) for 1 and 2, respectively (Figure 4). These values are in good agreement with those obtained by ac magnetometry suggesting a dominant Orbach relaxation process. This is further confirmed by looking at the g tensor values of the 1 st excited state (Table S9-S10) which clearly shows a moderate axiality, explaining the observed anisotropic barriers (Figure 4). In contrast, the g tensor values of the ground Kramers doublet (mJ = ± 15/2) point out weak transverse anisotropy components but the small difference observed in the gz values (19.85 for 1 vs. 19.78 for 2) explains the lowest tunneling rate (Figure 4) observed in 1 and therefore the presence of a magnetic bistability up to higher temperature for this latter. The orientations of the anisotropic axes were also evaluated by ab initio calculations. As expected from the structural analysis, they are slightly tilted with respect to the O-Dy-O sequence with an easy-axis-Dy-O(phenoxide) angle of 27.6° for 1 vs. 33.2 and 32.8 ° for 2 (Figure S12). The largest crystal-field parameter is observed for in both complexes (Table S11), but it is 11 % greater in magnitude for 1, confirming its stronger axiality. While the Dy-O distances are nearly similar for 1 and 2, the O-Dy-O angles show a significant difference of about 6 %. Remarkably, it could be noticed that the change in the nature of the coordinated solvates (THF vs. pyridine) does not dramatically influence the relaxation dynamic in these complexes. Hence, although THF molecules are stronger coordinated (shorter distances) than pyridine, a greater axiality is observed in 1. Hence, the smaller O-Dy-O angle observed in 2 explains its slightly enhanced tunneling rate with respect to 1. In addition, Schiff base ligands could also be promising to implement luminescence properties to obtain bifunctional lanthanide that have recently received considerable attention. [15] At room temperature, both, 1 and 2, exhibit a broad emission band located at 580 and 490 nm, respectively (Figures S13-14). Apart from some changes in the width, these bands resemble those of the HL1 and HL2 ligands (Figure S15) indicating a ligand-based luminescence for 1 and 2. It was however not possible to observe a dysprosium-centred luminescence, even at low temperature (Figure S16), indicating that these two ligands could not sensitize the Dy 3+ ion.

To summarize, we have shown that the use of bulky Schiff base ligands involving phenoxide groups could be viewed as an appealing synthetic strategy to design Dy 3+ -based SMM. The modulation of the substituents directly affects the O-Dy-O angle that in turn alters the QTM. Hence, two genuine SMM with out-of-phase signals up to 60 K and among the highest anisotropic barriers for Schiff base systems have been obtained. Magneto-structural correlations indicate that the control over the axiality of the excited doublets should be achieved to provide relaxation via higher excited states to obtain enhanced magnetic performances. In addition, the compounds exhibit a luminescence originating from the Schiff base ligands which can be tuned depending on substituents making such compounds particularly promising to design multifunctional materials.

  Scheme 1. Synthesis of and 2.

  Å) are comparable to those in 1. The Dy-N(phenoxide) distances in 2 (2.482(2)-2.548(2) Å) are somewhat longer compared to the Dy-N(py) distances (2.594(2), 2.622(2) Å). The O-Dy-O angles show the values of 128.5(

Figure 1 .

 1 Figure 1. Molecular structure of 1 (a) and 2 (b). Colour code: orange, Dy; blue, N; red, O; grey, C. Hydrogen atoms and [B(Ph)4]  moieties are omitted for clarity.

Figure 3 .

 3 Figure3. Temperature dependence of the relaxation time using the ac susceptibility data for 1 and 2. The solid lines represent the fit with Eq. 1 (0 Oe) or Eq. 2 (1000 Oe).

Figure 4 .

 4 Figure 4. Energy diagrams for the Kramers doublets for 1 (top) and 2(bottom). The states are placed according to their magnetic moments (bold blue lines). The horizontal black dashed lines show the tunnelling transitions within each doublet state, while the non-horizontal lines show the spin-phonon transition paths. The numbers at non-horizontal lines are averaged transition moments in µB connecting the states. The numbers at horizontal lines are tunnelling gaps. The red dashed arrows delineate the relaxation path.

Table 1 .

 1 : Fit parameters of the temperature dependence of the relaxation time for 1 and 2.

	Compound	 (cm 1 )	0 (s)	m [a]	C(s 1 .K m )	QTM (ms)
	1 (0 Oe)	263 ± 8	0.05 ± 0.01	3	0.015 ± 0.001	14 ± 1
	2 (0 Oe)	270 ± 8	0.015 ± 0.004	3	0.020 ± 0002	17 ± 1
	[a] fixed parameter

Acknowledgements

The authors thank the University of Montpellier, CNRS and PAC of ICGM. The Russian co-authors thank the Russian Science Foundation (grant 17-73-30036). The X-ray study has been carried out using an equipment of The Analytical Center of The G.A. Razuvaev IOMC RAS.

Conflict of Interest

The authors declare no conflict of interest.