Julie Uhl

Aurelien Doitrand
email: aurelien.doitrand@insa-lyon.fr

Sylvain Meille

Variability in porous ceramic fracture : influence of apparent density and critical pores

Keywords: Strength, porosity/flaw, Four-point bending, Coupled criterion

Gypsum specimen failure under four-point bending is investigated both experimentally and numerically. The analysis accounts for the variation in specimen thickness (in a large range from 10 to 90 mm), in pore volume fraction (related to a variation in elastic and fracture properties) and for the presence of macropores that are likely to trigger failure. Examination of specimen fracture surfaces reveals several isolated macropores near the specimen face loaded under tension. The critical pore size does not depend on the specimen volume. The pore criticity does not only depends on its size but also on its location with respect to the specimen face sollicitated under tension. Both pore volume fraction variations between samples and the presence of critical pores induce failure stress scattering for a given specimen width, which is well reproduced numerically using the coupled criterion.

Introduction

The influence of the characteristic structure size on failure stress is generally referred to as "size effect". Two main points of view enable describing this size effect. The first explanation relies on the weakest link theory which is a statistical description of failure [START_REF] Weibull | The phenomenon of rupture in solids[END_REF][START_REF] Weibull | A statistical representation of fatigue failures in solids[END_REF][START_REF] Weibull | A statistical distribution function of wide applicability[END_REF]. It is based on the assumption that failure occurs due to a critical flaw in an area sollicitated in tension. The second assumption states that the larger the specimen, the larger the probability to encounter a large critical flaw. Based on these assumptions, Weibull's approach describes the failure probability as a function of the maximum local stress level and sample volume sollicitated under tensile loading. Even if these assumptions rely on mechanical considerations, this is an empirical statistical approach that is not based on a mechanical description of failure. This approach is frequently used for ceramic failure study [START_REF] Lu | Scaling of fracture strength in ZnO: Effects of pore/grain size interaction and porosity[END_REF][START_REF] Danzer | Fracture statistics of ceramics -Weibull statistics and deviations from Weibull statistics[END_REF][START_REF] Bermejo | Influence of measurement uncertainties on the determination of the Weibull distribution[END_REF] including porous ceramic failure assessment [START_REF] Genet | A two-scale weibull approach to the failure of porous ceramic structures made by robocasting: Possibilities and limits[END_REF][START_REF] Lei | Statistics of ceramic strength: Use ordinary weibull distribution function or weibull statistical fracture theory?[END_REF].

The alternative viewpoint to explain size effects is based on a mechanical description of failure and the interaction between a characteristic length of the material and the structure characteristic dimensions. It was introduced by Bazant et al. [START_REF] Bažant | Size effect on structural strength: a review[END_REF][START_REF] Bažant | Size effect in blunt fracture: concrete, rock, metal[END_REF][START_REF] Bažant | Determination of fracture energy from size effect and brittleness number[END_REF][START_REF] Bažant | Statistical size effect in quasi-brittle structures: II. nonlocal theory[END_REF] who combined Linear Elastic Fracture Mechanics (LEFM) and the theory of plasticity to define the material characteristic length EG c /((1 -ν 2 )σ 2 y ), where σ y is the yield limit, E the material Young's modulus, ν the Poisson's ratio and G c the critical energy release rate. LEFM studies the propagation of a pre-existing crack and thus cannot assess the initiation of a crack. This shortfall was overcome by the Coupled Criterion (CC) introduced by Leguillon [START_REF] Leguillon | Strength or toughness? a criterion for crack onset at a notch[END_REF]. The CC is based on the simultaneous fulfillment of stress and energy conditions which enables the determination of the initiation load and crack length [START_REF] Weißgraeber | A review of finite fracture mechanics: crack initiation at singular and non-singular stress raisers[END_REF][START_REF] Doitrand | Numerical implementation of the coupled criterion: Matched asymptotic and full finite element approaches[END_REF].

The CC is able to account for size effects such as, for instance, in composite materials [START_REF] Leguillon | Strength or toughness? a criterion for crack onset at a notch[END_REF][START_REF] García | A numerical study of transverse cracking in cross-ply laminates by 3D finite fracture mechanics[END_REF][START_REF] García | The effect of residual thermal stresses on transverse cracking in cross-ply laminates: an application of the coupled criterion of the finite fracture mechanics[END_REF][START_REF] García | Experimental study of the size effect on transverse cracking in cross-ply laminates and comparison with the main theoretical models[END_REF], in specimens with notches [START_REF] Cornetti | Finite fracture mechanics: A coupled stress and energy failure criterion[END_REF][START_REF] Leguillon | Size effects for crack initiation at blunt notches or cavities[END_REF][START_REF] Carpinteri | Brittle failures at rounded V-notches: a finite fracture mechanics approach[END_REF][START_REF] Cornetti | Mode mixity and size effect in V-notched structures[END_REF] or holes [START_REF] Leguillon | Size effects for crack initiation at blunt notches or cavities[END_REF][START_REF] Cornetti | Mode mixity and size effect in V-notched structures[END_REF][START_REF] Doitrand | Experimental characterization and numerical modeling of crack initiation in rhombus hole pmma specimens under compression[END_REF][START_REF] Martin | A coupled strength and toughness criterion for the prediction of the open hole tensile strength of a composite plate[END_REF][START_REF] Cornetti | Penny-shaped cracks by finite fracture mechanics[END_REF][START_REF] Doitrand | Nonlinear implementation of Finite Fracture Mechanics: A case study on notched Brazilian disk samples[END_REF]. It was also recently extended to explain the large apparent failure stress obtained when studying nanoor micro-scale specimens [START_REF] Doitrand | Revisiting the strength of micron-scale ceramic platelets[END_REF][START_REF] Gallo | Brittle failure of nanoscale notched silicon cantilevers: a finite fracture mechanics approach[END_REF][START_REF] Doitrand | UO 2 micron scale specimen fracture : Parameter identification and influence of porosities[END_REF][START_REF] Alfaro | Finite fracture mechanics at the micro-scale. Application to bending tests of micro cantilever beams[END_REF]. The CC was used in previous works first to study the bending strength variation as a function of the specimen width in elastic brittle materials such as gypsum [START_REF] Doitrand | Strength and fracture toughness estimation from fourpoint bending test[END_REF]. The bending strength tends towards the tensile strength σ c only if the specimen is large enough compared to the material characteristic length l mat = EG c /((1 -

ν 2 )σ 2 c
), otherwise it increases as the specimen size decreases. Regardless of the material elastic and fracture properties, we highlighted the existence of a master curve describing the variation of the ratio of maximum flexural stress to tensile strength as a function of the specimen height to material characteristic length ratio. An analytical expression was derived which enabled the set-up of inverse identification procedure of both tensile strength and fracture toughness from bending tests. The influence of the specimen size on the failure stress was thus highlighted based on a mechanical analysis. The analysis presented in [START_REF] Doitrand | Strength and fracture toughness estimation from fourpoint bending test[END_REF] enabled the determination of the failure force and therefore maximum stress under bending of specimens that do not contain any flaw. Then, we implemented the CC to evaluate the influence of a critical flaw (i.e. a pore located in the area under tension) in gypsum specimen under bending [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF]. It was shown that depending on the pore size and location within the specimen, crack initation may occur beyond the pore or in the pore to free surface ligament sollicitated under tension (starting either from the pore or from the free surface).

For any pore size and location configurations, the CC enabled determining the failure force decrease compared to the reference configuration without pore. It was finally shown that several configurations may lead to the same failure force decrease, a larger pore further from the free surface being equivalent to a smaller pore closer to the free surface. These results questionned the assumption of Weibull's increasing critical flaw size with increasing specimen size since a smaller flaw may be more critical depending on its location. From this analysis, we determined for a given failure force decrease, all the admissible corresponding pore size and location couples. From experiments on gypsum specimens taken from [START_REF] Doitrand | Strength and fracture toughness estimation from fourpoint bending test[END_REF], based on the admissible pore size/location couples, we inversely estimated the range of critical pore diameters, which lied around 50-250 µm. Interestingly, the pore size was found to be rather constant whatever the specimen size rather than decreasing as expected in Weibull's approach assumption. This result was obtained from inverse estimate based on failure forces and the CC, but no microscope observations were available to confirm these data.

The objectif of the present work is to study the influence of porosity on failure of gypsum specimen with different thicknesses under four-point bending. On one hand, it aims at evaluating the influence of the overall porosity fraction on failure due to mechanical property variations. On the other hand, it is dedicated to the experimental characterization of critical spherical macropores always present in gypsum samples and their influence on failure. Experiments are presented in Section 2. Macropore size and location characterization as well as fracture surface analysis are presented in Section 3. In Section 4, failure of four-point bending specimen is assessed using the CC considering the influence of critical pores.

Experiments

Experiment design

Previous works were dedicated to the comparison between Weibull's approach and the CC ability to reproduce experimentally observed size effects [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF][START_REF] Leguillon | Flexural vs. tensile strength in brittle materials[END_REF]. The failure stress variation as a function of specimen size could be studied and compared to experimental measurements, but no experiments with direct observations of the fracture surfaces were available. Weibull's approach assumes that the larger the specimen, the larger the critical flaw that causes failure whereas the CC is based on a mechanical reasoning that accounts for the actual size, location and shape of the flaw. We thus establish experiments in order to assess the assumption and results predicted by both models. We propose to test under four-point bending specimens having the same length and width, but with several thicknesses. From a theoretical point of view, it should enable assessing both approach assumptions and predictions. Indeed, since the specimen volume varies due to the thickness variation, Weibull's approach predicts that the critical flaw size should increase with increasing specimen volume, and therefore failure stress should decrease with increasing specimen volume. On the contrary, the CC will predict that in the absence of flaws, the same failure stress should be obtained whatever the specimen thickness since the same tensile stress gradient and energy release will be obtained whatever the specimen thickness. Of course, gypsum is a porous material thus we expect large pores in the specimens corresponding to entrapped air bubbles [START_REF] Meille | Mechanisms of crack propagation in dry plaster[END_REF] which may act as critical flaws.

Therefore, we need to quantify the pores shape, size and location to take it into account in the CC approach and to evaluate both model assumptions and predictions.

4-points bending tests

The material under investigation is commercial β-gypsum. β-gypsum samples are prepared by hydration of β-hemihydrate at W/P=0.75 water to hemihydrate powder weight ratio. Rectangular specimens are prepared according to the following process: molds are cleaned and covered with demolding oil, water and hemihydrate powder are precisely weighted to respect the W/P ratio, powder is regularly added in water and followed by 30 s break, water and powder are spirally mixed until the paste is homogeneous, the paste is put into the mold and the paste surplus is removed by leveling, the specimen are unmolded after 30 minutes (the setting time is 15 minutes) and stored at 25°C during one week before the tests.

The temperature at which the tests are performed is 21+1 °C at 50+ 15% relative humidity. All tests are done in two consecutive days. The specimens all exhibit similar length (L = 100 + 0.5 mm) and width (h = 9.7+0.7 mm) but different thicknesses: t=10 mm, 30 mm, 50 mm, 70 mm, or 90 mm. The specimen dimensions are summarized in Tab. 1. Four-points bending tests are performed on a Instron 8862 (USA) machine equipped with a 5 kN load cell at 0.5 mm/min displacement rate. The distance between lower supports is 75 mm and the distance between the upper supports is 29.5 mm. The specimen top face which was in contact with air during manufacturing in the mold is the most likely to present flaw or irregularities. Therefore, the specimen is placed in the four-point bending apparatus so that this face is sollicitated under compression. It is worth noting that the other faces do not present visible surface flaws. For all the specimens, a linear elastic behavior is recorded until brittle failure as typically noted for dry gypsum samples. The measured failure force F c allows determining the corresponding failure stress σ max from Euler-Bernoulli's beam theory (Tab. 1).

Specimen

Results

Apparent density and mechanical property variation

Gypsum specimen apparent density depends on the amount of water and hemihydrate powder that is directly linked to the pore volume fraction inside the material. Fig. 1 depicts the apparent density variation as a function of specimen thickness. Relative variations up to 5% are noticed between the different specimens related to very small variations in W/P ratio. Gypsum specimen apparent density has a first order influence on the mechanical properties [START_REF] Meille | Linear elastic properties of 2D and 3D models of porous materials made from elongated objects[END_REF][START_REF] Sanahuja | Micromechanical explanation of elasticity and strength of gypsum: from elongated anisotropic crystals to isotropic porous polycrystals[END_REF]. Actually, it is directly linked to the pore content in the specimen since the larger the pore volume fraction, the smaller the density. The relation between the pore volume fraction ϕ and the sample density ρ is given by:

ρ = ρ 0 (1 -ϕ), (1)
where ρ 0 = 2.32 is the pure gypsum specific gravity with no porosity. The variation in apparent density corresponds to pore volume fractions varying between 52 % and 58 % for the tested specimens. Pore volume fraction influence on gypsum mechanical properties (Young's modulus, tensile strength and critical stress intensity factor) was for instance studied by Meille [START_REF] Meille | Etude du comportement mécanique du plâtre pris en relation avec sa microstructure[END_REF], Meille and Garboczi [START_REF] Meille | Linear elastic properties of 2D and 3D models of porous materials made from elongated objects[END_REF], Sanahuja et al. [START_REF] Sanahuja | Micromechanical explanation of elasticity and strength of gypsum: from elongated anisotropic crystals to isotropic porous polycrystals[END_REF] and Liu et al. [START_REF] Liu | Towards understanding the influence of porosity on mechanical and fracture behaviour of quasi-brittle materials: experiments and modelling[END_REF]. From these works, we can derive the variation of the Young's modulus, tensile strength and critical energy release rate as a function of the pore volume fraction (Fig. 2). It was also shown in [START_REF] Meille | Linear elastic properties of 2D and 3D models of porous materials made from elongated objects[END_REF][START_REF] Meille | Etude du comportement mécanique du plâtre pris en relation avec sa microstructure[END_REF] that the pore content variation induces a moderate variation of Poisson's ratio. From the density variations measured experimentally, we can derive variations on the pore volume fraction and thus deduce variations on the mechanical and fracture properties based on Fig.

It yields the following variations:

Young's modulus between 2.8 GPa and 4.3 GPa, The influence of such property variations on numerical predictions of the failure stress under bending is studied in Section 4.2.

Fracture surfaces

Gypsum is obtained by mixing hemihydrate powder and water which results in entangled needle-shaped crystals formed by hydratation reaction. The porosity in the material consists in interconnected pores (micrometer size) between the gypsum crystals as well as some macropores (tens to hundreds of micrometers in diameter). The latter are due to the presence of entrapped air in the slurry, that will lead to macropores in the material after hydration and drying [START_REF] Meille | Mechanisms of crack propagation in dry plaster[END_REF][START_REF] Devillard | Highlighting the role of heterogeneity on the indentation hardness of foamed gypsum[END_REF]. Gypsum specimen fracture surfaces are observed using a magnifier in order to characterize the macropores (Fig. 3). Fracture surfaces were observed for all the tested specimens but only two specimens for each thickness are shown in Fig. 3 for the sake of clarity. Whatever the specimen thickness, spherical macropores are present on the fracture surface. Their diameter is in the order of magnitude of hundreds of micrometers. For most of the observed fracture surfaces, these macropores seem relatively far from one another. It seems that there is no preferential location for the pores on the fracture surface. A cluster of two or three macropores is also observed on few fracture surfaces. Almost no pores emerging at the surface under tension are observed on the fracture surfaces. In particular, no edge pores are observed on the specimen faces sollicitated under tension. These few edge pores are located on the specimen surface sollicitated under compression. Hydratation of hemihydrate powder with water is processed in a mold which means that the specimen top face is in contact with air. Edge pores are thus more likely to be observed on this face rather than other specimen faces since air bubbles entrapped in the slurry rather tend to move up to the top surface before setting. Macropores are likely to induce prematurate fracture provided they are large enough and located near the specimen surface sollicitated under tension [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF].

For all tested specimens, we identify these critical pores and measure their diameter d and their distance to the specimen free surface h d (Fig. 4). If several of these pores are observed on one fracture surface, we select the most critical based on the analysis given in [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF]. Critical pore diameter and distance to the face under tension measurements for each sample are summarized in Table 2.

Porosity characterization and influence on failure stress

Fig. 5a shows the critical pore size as a function of specimen thickness. The mean critical pore size for a given specimen thickness lies between 0.3 mm and 0.4 mm for all the tested specimens without exhibiting an increasing trend with increasing specimen volume, as assumed in Weibull's approach. Indeed, it rather appears that the critical pore size lies in the same order of magnitude whatever the specimen thickness and thus the specimen volume. This result is consistent with pore size variation as a function of specimen volume obtained by inverse identification from other experiments on gypsum specimens [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF]. This result can be explained by considering the processing, with dissolution of hemihydrate powder and subsequent precipitation of gypsum crystals. Macropores originate from air initially entrapped in the dry hemihydrate powder before mixing with water. The characteristics of these macropores in the final material are likely to be related to the rheology of the slurry during processing. Since the slurry rheology is controlled by the W/P ratio, similar for all samples, the characteristics of the macro porosity (number and diameter of pores) can be expected to be relatively similar for all the tested specimens. The pore distance to the specimen face under tension is shown in Fig. 5b. Most of the critical pores are located at less than 1.5 mm from the edge (i.e. approximately 1/7 of the specimen width), except for the smaller specimen series (thickness: 10 mm) for which three critical pores are located between 1.5 mm and 2.2 mm. One critical pore is also located between 1.5 mm and 3.5 mm away from the specimen face under tension for three other specimen series (with respective thickness: 50 mm, 70 mm and 90 mm). It can be noted that the criticity of a pore is not only linked to its size but also to its location with respect to the face under tension [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF].

Fig. 6 shows the critical pore size and distance to free surface as a function of specimen pore volume fraction. First, it seems that there is no direct relation between the critical pore size and the total pore volume fraction in the specimen (Fig. 6a). Indeed, the macropores only represent a very small fraction of the total porosity, the pore volume fraction is mainly linked to the microporosity which does not vary much between all the samples. For a given specimen thickness, we observe a slightly decreasing trend of the critical pore distance to free surface as a function of the pore volume fraction even though the trend is not clearly observed for all the specimen sets. Fig. 7 shows the variation of failure stress as a function of the specimen thickness.

Seven specimens of each size may not be sufficient to observe a statistical effect and conduct Weibull's model analysis, nevertheless it is sufficient to observe a failure stress scattering for a given specimen size. Indeed, for a given specimen thickness, stress variations are observed which may be explained by the specimen having different densities, widths and critical pore size and location. Apart from this source of variability, the failure stress is almost constant as a function of the specimen thickness. The mean failure stress value slightly increases for thicknesses smaller than 40 mm and remain constant for larger specimen thicknesses. Fig. 8 shows the failure stress variation as a function of the critical pore size and distance to the free surface. It is not straightforward to establish a clear trend between the failure stress and one of these parameters. From a theoretical point of view [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF][START_REF] Doitrand | Asymptotic analysis of pore crack initiation near a free edge[END_REF], for a given distance to the specimen face under tension, a larger pore is expected to trigger failure at a smaller force and conversely, for a given pore size, the smaller the pore distance to free surface, the smaller the failure stress. Nevertheless, it is not made evident to observe this result from Fig. 8 since the data for a given pore size may not correspond to the same pore distance to free surface. Moreover, the different specimens also exhibit slightly different apparent densities and widths, both parameters having an influence on failure stresses.

The failure stress variation as a function of the specimen pore volume fraction is shown in Fig. 9. For a given specimen thickness, the failure stress exhibits a decreasing trend as a function of the pore volume fraction. This variation can only partly be explained by higher mechanical and fracture properties corresponding to lower pore volume fraction (see Fig. 2)

as the scatter for a given pore volume fraction seems related to the macropore characteristics.

Pore-induced failure modeling

In this section, we assess pore-induced failure using the numerical approach presented in [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF], considering in addition the mechanical property variation as a function of the pore content. A brief recall of the CC is first presented. A more detailed presentation of the methodology so as to determine pore-induced crack initiation is also provided in [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF].

The Coupled Criterion

Crack initiation assessment with the CC [START_REF] Leguillon | Strength or toughness? a criterion for crack onset at a notch[END_REF] consists in simultaneously fulfilling stress and energy conditions. The energy condition states that the potential energy variation before and after crack initiation is larger than the energy needed to nucleate the crack G c × S (where G c is the material critical energy release rate and S the crack surface). The stress condition requires that prior to crack nucleation, the stress along the crack path is larger than the tensile strength σ c . Under small deformation framework and linear elasticity assumption, the potential energy and the stress are respectively proportional to the square applied load and the applied load, which allows combining both conditions into a single equation that has to be solved in order to determine the initiation load level and crack extension [START_REF] Doitrand | Numerical implementation of the coupled criterion: Matched asymptotic and full finite element approaches[END_REF]. The load level for crack initiation corresponds to the lowest value for which both energy and stress conditions are fulfilled. In practice, the stress and the potential energy are obtained by means of FE calculations corresponding to four-point bending specimen containing a critical pore [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF] or not [START_REF] Doitrand | Strength and fracture toughness estimation from fourpoint bending test[END_REF]. The proposed approach requires several input parameters, namely the specimen dimensions, the critical pore size and location in the specimen, the material elastic (Young's modulus, Poisson's ratio) and fracture (critical energy release rate, tensile strength)

properties. Given these input parameters, the CC yields as ouptput the crack initiation force which also corresponds to the failure force for the problem under investigation.

Influence of input parameter variations

A first source of scattering is due to the specimens not having the exact same height, which varies from 9 mm to 10.4 mm. The influence of height variation on bending failure stress was already quantified in [START_REF] Doitrand | Strength and fracture toughness estimation from fourpoint bending test[END_REF], it results in a +1 % relative variation in failure stress for given elastic and fracture properties. Another difficulty arising when studying gypsum is the dependency on the elastic and fracture properties to the apparent density (or equivalently to the pore volume fraction). The studied specimens exhibit around +5 % variations in the pore volume fraction, which induces variations in elastic and fracture properties (see Fig. 2). We study the influence of such variations on failure stress variation in specimen without pore based on the analytical approach proposed in [START_REF] Doitrand | Strength and fracture toughness estimation from fourpoint bending test[END_REF]. Fig. 10a shows the bending failure stress variation of gypsum specimens as a function of the pore volume fraction for a given h = 10 mm specimen width. Note that this result is obtained without considering the presence of a critical macropore, the variation is only due to elastic and fracture properties dependency on pore volume fraction. The pore volume fraction in the specimens under investigation varies between 52 % and 58 %, the parameters used in the CC are determined based on the specimen pore volume fraction using the data presented in Fig. 2. This variation in pore volume fraction and thus in elastic and fracture properties results in up to +10 % relative variation in failure stress. Therefore, the influence of the pore volume fraction on failure stress is much more pronounced than the influence of the specimen height. For a given specimen thickness, the obtained scattering lies between 26 % and 50 % of the scattering observed experimentally.

Pore-induced failure

We now compare the numerical predictions obtained with the coupled criterion to experimental results considering both apparent density variation and critical pore influence.

Fig. 11 shows the failure stress variation as a function of the specimen thickness measured experimentally and obtained numerically with the CC considering the presence of a critical pore. The presence of a pore induces a decreases in the failure stress which can be calculated for all the specimens (Table 3). The failure force decrease induced by the presence of a pore varies up to 15.5% for all the studied specimens. It depends both on the pore diameter and location with respect to the specimen face under tension [START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF][START_REF] Doitrand | Asymptotic analysis of pore crack initiation near a free edge[END_REF]. The failure stress variation obtained using the coupled criterion is almost constant as a function of the specimen thickness. It enables reproducing similar scattering as obtained experimentally, except for 30 mm thickness specimen series for which the scattering obtained numerically represents around 30 % of the experimental scattering. The comparison between experimental and numerical results can be deepened by comparing the failure forces obtained with both approaches for each specimen (Fig. 12). An overall good agreement is obtained since most of the points are close to the bisector (which corresponds to a perfect match) in Fig. 12. Table 3: Decrease (in %) in the failure stress due to the presence of a pore with respect to the same configuration without pore.

Discussion

The experiments set-up in this work were initially motivated by a study of size effect in brittle fracture. They finally provide several conclusions regarding the use of the CC or Weibull's approach to assess such size effect.

First, it is worth noting that the variability in the failure stress is not only due to the presence of critical pores that induce prematurate failure. Indeed, for the specimen under investigation another source of variability is the variation in the pore content, which has a first order influence on macroscopic mechanical properties and thus on failure stress. This is an essential data that should be evaluated when using Weibull's approach for failure assessment. Since Weibull's modulus calculation mainly relies on failure stress scattering, all sources of variability are taken into account in Weibull's approach. Therefore it is of primary importance to correctly identify the source of failure stress scattering in order to draw consistent conclusions and not directly relate it to variations in larger pore size. This result could explain some limitations about Weibull's model interpretation that were already noticed previously especially on porous materials [START_REF] Lu | Scaling of fracture strength in ZnO: Effects of pore/grain size interaction and porosity[END_REF][START_REF] Doitrand | Size effect assessment by weibull's approach and the coupled criterion[END_REF].

The CC enables considering the influence of the different sources of variability on failure stress, such as the variation in mechanical properties which yields failure stress variation range similar to those measured experimentally. The CC also enables the quantification of the influence of a critical pore on failure. For the specimen under investigation in this work, the failure stress decrease associated to the presence of a macropore lies around the same order of magnitude as the variation due to the influence of pore volume fraction. Therefore it is not straigthforward to identify the influence of such critical pores without considering the influence of the variation in the total pore content on the failure stress scattering. The method developped here could be applied to dense ceramics containing individual pores, thus avoiding the influence of fluctuation in total pore volume fraction. For instance, a prospective study would focus on dense ceramics with controled macro-defects possibly obtained using additive manufacturing. Finally, a limitation of the proposed approach applied to porous ceramics is the case where failure is induced by a cluster of pores, which has not be taken into account in the model and would require further developments.

Conclusion

Examination of the fracture surfaces of gypsum specimens tested in bending indicate the presence of several isolated critical macropores that are likely to trigger prematurate fracture.

The critical pore size does not depend on specimen size whereas it is more likely to be located near the specimen face under tension for larger pore volume fractions. Pore criticity does not only depend on pore size but also on pore location with respect to the specimen face under tension. Two reasons mainly explain the failure stress scattering measured experimentally for a given specimen thickness, namely the variation in specimen pore volume fraction and the presence of critical pores triggering failure. Another source of variability comes from the variation in the specimen width. It may have a strong influence on the failure stress when varying in a significant range and yield a mechanically-based explanation for experimentally observed size-effects [START_REF] Doitrand | Strength and fracture toughness estimation from fourpoint bending test[END_REF]. However, in the present analysis, the specimen width variation is smaller than around 5% and induces an almost negligible variability on the failure stress in comparison to the two above-mentioned variability sources. Despite the failure stress scattering originating from both critical macropores and pore volume fraction variation, almost no size effect is observed when varying the specimen thickness (and thus increasing the specimen volume). The variation in pore volume fraction directly influences the material elastic and fracture properties. The presence of a pore induces a decrease in the failure stress depending on its location and size. Considering both pore volume fraction, critical pore presence and width variations, the CC enables the prediction of failure stress as a function of specimen thickness and allows accounting for the scattering measured experimentally. A good agreement is obtained between failure forces predicted by the CC and that measured experimentally.

Figure 1 :

 1 Figure 1: Specimen density variation as a function of specimen thickness.

Figure 2 :

 2 Figure 2: Young's modulus, tensile strength and critical energy release rate variations as a function of gypsum pore volume fraction [35, 36, 37].

Figure 3 :

 3 Figure 3: Fracture surfaces for specimen with different thicknesses (from two different specimens for a given thickness). Some pores are highlighted on the top left fracture surface.

Figure 4 :

 4 Figure 4: (Left) Focus on a critical pore and definition of its characteristics: distance to the specimen face under tension h d and pore diameter d. (Right) SEM observation of a critical pore.

Figure 5 :

 5 Figure 5: Pore a) diameter and b) distance to the specimen face under tension as a function of specimen thickness.

Figure 6 :

 6 Figure 6: Pore a) diameter and b) distance to the specimen face under tension as a function of pore volume fraction.

Figure 7 :

 7 Figure 7: Failure stress as a function of specimen thickness obtained experimentally.

Figure 8 :

 8 Figure 8: Failure stress as a function of pore a) diameter and b) distance to specimen face under tension obtained experimentally.

Figure 9 :

 9 Figure 9: Failure stress as a function of pore volume fraction obtained experimentally.

Fig. 10b shows

  Fig. 10b shows the failure stress as a function of the specimen thickness considering both actual pore volume fractions and sample widths. The variation in pore volume fraction and specimen width allows partly explaning the failure stress scattering measured experimentally.

Figure 10 :

 10 Figure 10: a) Failure stress variation of four-point bending specimen without critical pore as a function of pore volume fraction obtained with the CC for w=10 mm width. b) Failure stress as a function of specimen thickness obtained numerically (symbols) without considering the presence of a critical pore (range of values measured experimentally are also indicated as error bars).

Figure 11 :

 11 Figure 11: Failure stress as a function of specimen thickness obtained experimentally (error bars) and numerically using the CC accounting for density and width variation as well as the presence of a critical pore.

Figure 12 :

 12 Figure 12: Failure force obtained numerically with the CC as a function of failure force measured experimentally.

Table 1 :

 1 Dimensions, failure force F c and bending stress σ max .

	type	1	2	3	4	5
	t (mm)	10.2 + 0.1	30.4 + 0.3	50.4 + 0.1	70.3 + 0.2	90.4 + 0.2
	Number of specimens	7	7	7	7	6
	h (mm)	9.65 + 0.6	9.6 + 0.5	9.6 + 0.3	9.7 + 0.4	9.8 + 0.6
	ρ (g/cm 3 )	1.01 + 0.04 1.05 + 0.06	1.05 + 0.02	1.06 + 0.04	1.03 + 0.03
	F c (N)	61.9 + 13.0 161.1 + 39.5 214.6 + 40.0 303.6 + 76.0 426.6 + 83.0
	σ max (MPa)	4.1 + 0.7	3.9 + 0.7	3.1 + 0.4	3.3 + 0.9	3.2 + 0.4


Innovative Fault-Tolerant Three-Phase SPMSM Drive without Split Capacitors, Auxiliary Legs or TRIACs Xiaokang Zhang, Student Member, IEEE, Jean-Yves Gauthier, and Xuefang Lin-Shi

Abstract-The existing fault-tolerant three-phase AC drives need split capacitors, auxiliary legs and triode for alternating current (TRIACs) upon a standard three-phase two-level voltage source inverter (VSI) to obtain the self-healing ability towards open-phase faults (OPFs). In order to reduce the cost, volume and weight, this paper proposes a novel fault-tolerant threephase surface-mounted permanent magnet synchronous machine (SPMSM) drive without split capacitors, auxiliary legs or TRI-ACs. The proposed drive is still built upon a three-phase two-level VSI, while its DC-source is directly connected to the neutral point of the motor. First, mathematical model and equivalent circuit are established to reveal the existence of an equivalent DC/DC boost converter in the drive system. Second, the brand-new d-0 axes current references are proposed in order to preserve the motor's performance when an OPF occurs. In this case, both i d and i0 are non-constant. Third, an overall control strategy including the healthy and fault-tolerant modes is presented. In the fault-tolerant mode, a deadbeat current controller is developed to track the non-constant current references. Finally, experiments are carried out on a SPMSM test-bench to verify the effectiveness of the proposed scheme.

Index Terms-Permanent magnet synchronous machine, openphase fault, fault-tolerant control, DC/DC boost converter, equivalent circuit, deadbeat current controller.

I. INTRODUCTION

D UE to their merits of high torque and power density, high efficiency and structure simplicity, surface-mounted permanent magnet synchronous machines (SPMSMs) are preferred in various applications [START_REF] Han | Initial Rotor Position Detection Method of SPMSM Based on New High Frequency Voltage Injection Method[END_REF]. With the development of power electronic devices, AC drives have occupied a dominant position in industrial fields, such as air conditioning/heat pumps, wind turbines, robotics and electric vehicles (EVs) [2] [3]. Nevertheless, the failure rate of AC drives increases with the use of these devices. According to [START_REF] Wang | A Fault-Tolerant Permanent-Magnet Traction Module for Subway Applications[END_REF], open-circuit and short-circuit failures occurring in switching devices or one of motor phases are the most common faults at the power level. As the demand for high reliability drives grows, researchers and engineers have put extensive efforts into developing faulttolerant AC drives. Although multi-phase motors, such as five-phase [START_REF] Tao | Simplified Fault-Tolerant Model Predictive Control for a Five-Phase Permanent-Magnet Motor With Reduced Computation Burden[END_REF], six-phase [START_REF] Munim | A Unified Analysis of the Fault Tolerance Capability in Six-Phase Induction Motor Drives[END_REF] and nine-phase [START_REF] Abdel-Khalik | Postfault Operation of a Nine-Phase Six-Terminal Induction Machine Under Single Open-Line Fault[END_REF] motors, have gained special attentions during the past few years due to their inherent fault-tolerant capability, three-phase machines are still the workhorses in industrial areas owing to low cost and The authors are with the loboratory Ampère (CNRS UMR5005), the University of Lyon, Institut National des Sciences Appliquées de Lyon, 69621 Villeurbanne, France (e-mail: xiaokang.zhang@insa-lyon.fr; jeanyves.gauthier@insa-lyon.fr; xuefang.shi@insa-lyon.fr).

simple structure. This paper focuses on fault-tolerant threephase SPMSM drives.

For a three-phase two-level voltage source inverter (VSI) fed AC motor with Wye-connection (as shown in Fig. 1), there are four typical fault-tolerant topologies [START_REF] Beltrao De Rossiter Correa | An induction motor drive system with improved fault tolerance[END_REF]- [START_REF] De Araujo Ribeiro | Fault-tolerant voltage-fed PWM inverter AC motor drive systems[END_REF], as shown in Fig. 2. Thanks to the fast-acting fuses (F j and F j , j = A, B, C), the short-circuit fault will eventually lead to one phase being open-circuit. Therefore, the open-phase fault (OPF) is an usual fault. To be specific, if the fault occurs in inverter legs, remedial strategies can be achieved by moving the motor terminal of the faulty leg to the midpoint of split capacitors or to an auxiliary inverter leg via triode for alternating current (TRIACs) [START_REF] Errabelli | Fault-Tolerant Voltage Source Inverter for Permanent Magnet Drives[END_REF]. These topologies are known respectively as 3 legs with phase to capacitor midpoint (3L-PCM) and (3+1) legs with phase to an auxiliary leg ((3+1)L-PAL), as shown respectively in Fig. 2(a) and Fig. 2(b). The two topologies still allow the motor to operate with three phases under post-fault operations. As for the post-fault control strategy, the 3L-PCM should adopt the control approach of a three-phase four-switch VSI [START_REF] Zhu | Comprehensive Analysis and Reduction of Torque Ripples in Three-Phase Four-Switch Inverter-Fed PMSM Drives Using Space Vector Pulse-Width Modulation[END_REF] while the (3+1)L-PAL can continue using the same control as that in healthy operations. In addition, both the 3L-PCM and (3+1)L-PAL are able to output the rated torque under post-fault operations. However, the motor speed in the 3L-PCM is limited to around half of the base speed because of the voltage limitation. Besides, the 3L-PCM is exposed to a risk of performance degradation caused by the oscillation of the capacitor midpoint voltage [START_REF] Zhu | Comprehensive Analysis and Reduction of Torque Ripples in Three-Phase Four-Switch Inverter-Fed PMSM Drives Using Space Vector Pulse-Width Modulation[END_REF].

If an OPF occurs in motor windings, fault-tolerance can be achieved by connecting the motor's neutral point either to the midpoint of split capacitors or to an auxiliary inverter leg [START_REF] Zhao | An approach to modeling and field-oriented control of a three phase induction machine with structural imbalance[END_REF]- [START_REF] Yang | On-Line Stator Open-Phase Fault Detection and Tolerant Control for Permanent Magnet Machines Using the Neutral Point Voltage[END_REF]. These topologies are known respectively as 3 legs with neutral point to capacitor midpoint (3L-NCM) and (3+1) legs with neutral point to an auxiliary leg ((3+1)L-NAL), as shown

A B + - u in + - u bus N C S A S B S C S' A S' B S' C

Motor

Fig. 1. VSI-fed three-phase AC motor drive.

respectively in Fig. 2(c) and Fig. 2(d). For the two topologies under post-fault operations, the motor must operate with two active phases and the control method needs to be modified appropriately. Generally speaking, the 3L-NCM and (3+1)L-NAL are more popular because they can tolerate the faulty cases appearing both in inverter and motor sides. Numerous fault-tolerant strategies based on the 3L-NCM and (3+1)L-NAL were presented in [START_REF] Liu | A strategy for improving reliability of field-oriented controlled induction motor drives[END_REF]- [START_REF] Tousizadeh | Performance Comparison of Fault-Tolerant Three-Phase Induction Motor Drives Considering Current and Voltage Limits[END_REF].

The first effective example of the 3L-NCM applied to a three-phase induction machine was presented in [START_REF] Liu | A strategy for improving reliability of field-oriented controlled induction motor drives[END_REF], in which a crucial conclusion indicated that it is still possible to generate the same rotating magnetic motive force by imposing a 60 • phase-shift between the remaining phase-currents (the current amplitude should increase to √ 3 times) when an OPF occurs. Based on this conclusion, a simple fault-tolerant implementation is to use hysteresis controllers [START_REF] Hoang | Comparative study of current vector control performance of alternate fault tolerant inverter topologies for three-phase PM brushless ac machine with one phase open -circuit fault[END_REF]. However, hysteresis controllers suffer from the problem of inconstant switching frequency. Afterwards, [START_REF] Gaeta | Modeling and Control of Three-Phase PMSMs Under Open-Phase Fault[END_REF] proposed a field-oriented control method by exploiting a suitable reference frame, in which the pulse-width modulation (PWM) with constant switching frequency can still be used. It needs to note that the 3L-NCM also gets involved in the trouble of drifting capacitor midpoint. To address this problem, a hardware-based solution by using a pair of resistors to clamp the capacitor midpoint was presented in [START_REF] Liu | A strategy for improving reliability of field-oriented controlled induction motor drives[END_REF].

A B + - u in o + - u bus N C S A S B S C S' A S' B S' C A B + - u in o + - u bus C S A S B S C S' A S' B S' C (a) (b) (c) (d) A B + - u in + - u bus C S A S B S C S' A S' B S' C S D S' D A B + - u in u bus C S A S B S C S' A S' B S' C S D S' D + - Motor Motor Motor Motor F A F C F B F' A F' C F' B F A F C F B F' A F' C F' B F A F C F B F' A F' C F' B F A F C F B F' A F' C F' B
Compared to the 3L-NCM, the (3+1)L-NAL can avoid the problem of drifting capacitor midpoint due to an auxiliary leg provided to the motor's neutral point. The reformulating current references (60 • phase-shift) for the remaining phases are consistent with that in the 3L-NCM. Therefore, hysteresis controller based fault-tolerant control is also applicable to the (3+1)L-NAL. [START_REF] Bolognani | Experimental fault-tolerant control of a PMSM drive[END_REF] and [START_REF] Bianchi | Innovative remedial strategies for inverter faults in IPM synchronous motor drives[END_REF] proposed a voltage feed-forward compensation method whose effectiveness was verified on a PMSM. Moreover, an unified voltage feed-forward compensation method by injecting suitable zero-sequence voltages was developed in [START_REF] Tousizadeh | Fault-Tolerant Field-Oriented Control of Three-Phase Induction Motor Based on Unified Feedforward Method[END_REF], which is general and applicable both in the 3L-NCM and (3+1)L-NAL. Nevertheless, voltage feedforward compensation method is sensitive to motor's parameters so that it is unfavorable in some high-precision applications. Recently, [START_REF] Zhou | High Performance Three-Phase PMSM Open-Phase Fault-Tolerant Method Based on Reference Frame Transformation[END_REF] [START_REF] Zhou | PMSM Open-Phase Fault-Tolerant Control Strategy Based on Four-Leg Inverter[END_REF] proposed a new fault-tolerant control method for the (3+1)L-NAL, in which the neutral current is proactively controlled due to an additional control degree of freedom provided by the auxiliary leg. Meanwhile, a novel reference frame was introduced, in which the neutral current reference remains constant so that a classic proportionalintegral (PI) regulator is sufficient to track the reference.

A detailed comparison of all these existing fault-tolerant topologies was presented in [START_REF] Tousizadeh | Performance Comparison of Fault-Tolerant Three-Phase Induction Motor Drives Considering Current and Voltage Limits[END_REF]. In a word, compared to the standard three-phase two-level VSI, the existing fault-tolerant topologies need redundant components (split capacitors or auxiliary legs) to obtain the remedial ability when an OPF occurs. These redundant components definitely increase the cost, volume and weight of the drive system. Besides, the circuit structure changes by triggering TRIACs. This actually increases the complexity and reduces the reliability of the whole system.

In order to achieve the objectives of lower cost and simpler structure, this paper aims to propose a novel fault-tolerant three-phase SPMSM drive without using split capacitors, auxiliary legs or TRIACs. The rest of this paper is organized as follows: In Section II, a brief description of the proposed drive is presented. Then, mathematical model and equivalent circuit are established. In order to maintain the torque performance when an OPF occurs, the novel fault-tolerant scheme is proposed in Section III. In this case, the d-0 axes current references are non-constant. An overall control strategy including the healthy and the fault-tolerant modes is presented in Section IV, in which a deadbeat current controller is developed to track the post-fault current references. Experiments and assessments are presented in Section V. Finally, conclusions are given in Section VI.

A B o + - u bus N C S A S B S C S' A S' B S' C SPMSM F A F C F B F' A F' C F' B C + - u in i A i B i C i N i bus (α A ) (α B ) (α C )

II. TOPOLOGY DESCRIPTION AND MODELING UNDER THE HEALTHY OPERATION

The proposed fault-tolerant SPMSM drive is built upon a three-phase two-level VSI, in which the DC-source u in is directly connected to the neutral point 'N', as shown in Fig. 3. The fast-acting fuses (F j and F j , j = A, B, C) are used to isolate the faulty phase when short-circuit failures appear. However, neither redundant components (split capacitors and auxiliary legs) nor TRIACs are employed. u bus is the DC-bus voltage; C is the DC-bus capacitor; S j and S j (j = A, B, C) are the PWM signals of switching devices; i A , i B and i C are the phase-currents; i N is the neutral current; i bus is the DC-bus current.

A. Average Model

In order to study the characteristics of the proposed drive, the average model under the healthy operation will be established in this subsection.

In one PWM period (T s ), the average value of a variable x(t) is defined as

x(t) = 1 T s t t-Ts x(τ )dτ (1)
Therefore, the average voltage equations of the drive system can be formulated by

  u AO u BO u CO   =   α A α B α C   u bus =   u AN u BN u CN   +   u in u in u in   (2)
where u AO , u BO and u CO are the terminal voltages of the inverter; u AN , u BN and u CN are the motor's phase-voltages; α A , α B and α C are respectively the duty-cycles of S A , S B and S C . The phase-voltages of the SPMSM can be formulated by

               u AN = Ri A + L AA di A dt + M AB di B dt + M AC di C dt + e A u BN = Ri B + L BB di B dt + M BA di A dt + M BC di C dt + e B u CN = Ri C + L CC di C dt + M CA di A dt + M CB di B dt + e C (3) with      e A = -ω e ψ f sin(θ e ) e B = -ω e ψ f sin(θ e -2π/3) e C = -ω e ψ f sin(θ e + 2π/3) (4)                            L AA = L σ + L Σ + L ∆ cos 2θ e L BB = L σ + L Σ + L ∆ cos 2 θ e -2π/3 L CC = L σ + L Σ + L ∆ cos 2 θ e + 2π/3 M AB = M BA = -L Σ /2 + L ∆ cos 2 θ e + 2π/3 M BC = M CB = -L Σ /2 + L ∆ cos 2θ e M AC = M CA = -L Σ /2 + L ∆ cos 2 θ e -2π/3 L Σ = (L d + L q -2L σ ) /3 L ∆ = (L d -L q ) /3 L σ = L 0 (5)
where R is the phase resistance; L AA , L BB and L CC are the self-inductances; M AB , M BA , M AC , M CA , M BC and M CB are the mutual inductances; e A , e B and e C are the back-EMFs; L d , L q and L 0 are respectively the d-axis, q-axis and zero-sequence inductances; L σ , L Σ and L ∆ are respectively the leakage inductance, average inductance and inductance fluctuation (for a SPMSM, L ∆ = 0); ψ f is the permanent magnet flux linkage; θ e is the electrical position; ω e is the electrical speed.

The average current equations can be expressed by

i bus = C du bus dt = -   α A α B α C   T   i A i B i C   (6)
i A + i B + i C + i N = 0 (7)
By using the Park transformation, (2) can be transformed from A-B-C to d-q-0 coordinate systems, as

  u dO u qO u 0O   =   α d α q α 0   u bus =   u d u q u 0   +   0 0 u in   (8)
where u dO , u qO and u 0O represent the inverter's terminal voltages in d-q-0 coordinate system; α d , α q and α 0 are the values of α A , α B and α C converted into d-q-0 coordinate system; u d , u q and u 0 are the motor's voltages in d-q-0 coordinate system. In d-q-0 coordinate system, the voltage equations of the SPMSM can be reformulated by

  u d u q u 0   =   R 0 0 0 R 0 0 0 R     i d i q i 0   +   L d 0 0 0 L q 0 0 0 L 0   d dt   i d i q i 0   + ω e   -L q i q L d i d + ψ f 0   (9)
where i d , i q and i 0 are the currents in d-q-0 coordinate system.

By substituting the zero-sequence term of ( 9) into (8), we obtain

α 0 u bus = u 0 + u in = Ri 0 + L 0 di 0 dt + u in (10)
According to the Park transformation, i 0 and α 0 can be expressed by

i 0 = (i A + i B + i C ) 3 = - i N 3 (11)
α 0 = α A + α B + α C 3 (12
) By substituting ( 11) into [START_REF] De Araujo Ribeiro | Fault-tolerant voltage-fed PWM inverter AC motor drive systems[END_REF], we obtain

L E + - u in S 2 C u bus i N + - N Motor Inverter (1-¡ 0 ) DC/DC converter R s S 1 (¡ 0 ) + - u L E = L 0 3
u in -α 0 u bus = -u 0 = R 3 i N + L 0 3 di N dt u L E (13)
According to [START_REF] Zhao | An approach to modeling and field-oriented control of a three phase induction machine with structural imbalance[END_REF], the motor can be seen as an equivalent inductor (L E ) whose resistance and inductance are respectively R/3 and L 0 /3 (by looking from the input u in to the output u bus ). For convenience, the voltage of L E is defined as u L E . Because the duty-cycles (α A , α B and α C ) are strictly bounded between 0 and 1, α 0 is limited between 0 and 1. Therefore, it is interesting to find that the zero-sequence equation of the drive corresponds to a DC/DC boost converter, where L 0 /3 functions as the filter inductor and (1-α 0 ) represents the dutycycle of the boost's chopper. By assuming the positive circuit (without any zero-sequence component) of the drive as an equivalent load (R s ), the proposed drive is equivalent to a DC/DC boost converter followed by a conventional SPMSM drive system (the neutral point is disconnected), as shown in Fig. 4. It needs to note that all the zero-sequence components are embodied in the equivalent boost converter. S 1 and S 2 are two virtual switches for visualizing the zero-sequence dutycycle α 0 . In steady-states, the DC-bus voltage approximates to

u bus ≈ u in α 0 (14)
In addition, (13) also indicates that the DC-bus voltage is only related to these zero-sequence components (u 0 , i 0 , L 0 and α 0 ). Meanwhile, according to [START_REF] Welchko | Fault tolerant three-phase AC motor drive topologies: a comparison of features, cost, and limitations[END_REF], the motor performance is determined by the components located in d-q plane. This means that the motor and the equivalent DC/DC boost converter are uncoupled in d-q-0 coordinate system. Consequently, it is possible to separate the controls of the motor and the equivalent boost converter.

B. Power Balance Analysis

Because the power of the proposed drive is provided by the DC-source, it is obvious that i N or i 0 is nonzero.

The input power (p in ) of the drive can be calculated by

p in = u in i N = -3u in i 0 ( 15
)
The torque of the SPMSM is formulated by

T e = 3 2 pψ f i q ( 16
)
where p is the number of pole-pairs. Thus, the output power (p out ) of the motor can be formulated approximately as

p out = 2πT e n 60 = πpψ f i q n 20 (17)
where n (rpm) is the rotor speed.

According to energy conservation law, we obtain

t2 t1 p in dt E in = t2 t1 p out η dt E out + C 2 u 2 bus2 - C 2 u 2 bus1 E C (18)
where t 1 and t 2 represent two operating moments; η is the system efficiency; E in is the input energy; E out is the output energy (including system losses); u bus1 and u bus2 are respectively the DC-bus voltages at t 1 and t 2 ; E C means the energy absorbed by the DC-bus capacitor from t 1 to t 2 .

In steady-states, u bus is constant thereby E C is null. In this case, the power balance equation can be derived as

p in = p out η (19)
In order to quantify the proportion of the zero-sequence current, a new variable γ is defined. By combining ( 15), ( 17) and ( 19), γ can be formulated by

γ = i 0 i q = πpψ f n 60u in η (20)
The parameters of the SPMSM drive used in this paper are shown in Table . I. Because the rated DC-bus voltage is 30 V, a 15 V DC-source is sufficient due to the equivalent DC/DC boost function. Under this situation, the trend of γ with respect to n and η is illustrated in Fig. 5. The value range of n is considered as (0, 4000) rpm and the value range of η is chosen as (0.6, 1). It can be seen that γ increases with the rise of speed and the fall of efficiency. To be specific, γ varies between 0.31 and 0.52 under the nominal speed. Within the nominal operating area, γ is less than 0.52. By using the inverse Park transformation, the representations of phase-currents under the healthy operation can be expressed by

     i A = i d cos(θ e ) -i q sin(θ e ) -γi q i B = i d cos(θ e -2π/3) -i q sin(θ e -2π/3) -γi q i C = i d cos(θ e + 2π/3) -i q sin(θ e + 2π/3) -γi q (21)
Apparently, it can be seen that a DC component (-γi q ) is superposed to the fundamental current of each phase.

III. FAULT-TOLERANT SCHEME

For the state of art presented in Fig. 2, TRIACs and split capacitors (or an auxiliary leg) are necessary to provide a path to the zero-sequence current when an OPF occurs. Thanks to the original connection of motor's neutral point, the proposed drive does not need any additional component to construct the path.

When the drive system encounters an OPF, it has only two available phases left. In order to preserve motor's performance, the current references of remaining phases must be reformulated. For convenience, an OPF occurring in phase A is taken into account throughout.

When the OPF occurs, i A = 0. By considering this and using the Clark transformation, we obtain

i α = -i 0 (22)
According to the rotation transformation, i d and i q can be reformulated by

i d = i α cosθ e + i β sinθ e i q = -i α sinθ e + i β cosθ e (23)
where i α and i β are the currents in α-β plane. By substituting ( 22) into (23), i d and i q become

i d = -i 0 cosθ e + i β sinθ e i q = i 0 sinθ e + i β cosθ e (24)
By eliminating i β , i 0 is expressed as

i 0 = i q sinθ e -i d cosθ e (25)
In order to preserve the torque, i q must remain unchanged before and after the OPF. Defining the d-q axes currents respectively as i dh and i qh before the OPF, i 0 under the postfault operation should follow

i 0 = i qh sinθ e -i dh cosθ e (26)
Most of fault-tolerant strategies in the 3L-NCM and (3+1)L-NAL are on the basis of ( 26). In this case, i 0 is a periodic function. Its period is consistent with the fundamental period (T f = 2π/ω e ) and its mean is 0.

In regard to the proposed drive, if i 0 is chosen as [START_REF] Tousizadeh | Performance Comparison of Fault-Tolerant Three-Phase Induction Motor Drives Considering Current and Voltage Limits[END_REF], the input power (p in ) will be periodic. In one T f , the mean of p in can be calculated by

< p in > T f = 1 T f T f 0 p in dt = ω e 2π
2π ωe 0 -3u in i qh sin(ω e t) -i dh cos(ω e t) dt = 0

(27)
This indicates that the average input power will be null. In this case, it is impossible to provide a constant output power to the motor. Therefore, the typical representation ( 26) is infeasible in the proposed drive.

With the considerations of a non-constant d-axis current and a non-null average input power, we propose a novel d-axis

current reference (i * d ), as i * d = -2i 0h cosθ e ( 28
)
where i 0h is the zero-sequence current before the OPF. By substituting i * q = i qh and ( 28) into ( 25), the new i * 0 can be calculated by

i * 0 = i qh sinθ e + i 0h 1 + cos(2θ e ) (29)
With i * 0 , the average input power is

< p in > T f = 1 T f T f 0 p in dt = ω e 2π 2π ωe 0 -3u in i qh sin(ω e t) + i 0h [1 + cos(2ω e t)] dt = -3u in i 0h = u in i N h ( 30
)
where i N h is the neutral current before the OPF. Obviously, [START_REF] Zhang | Neutral Point Supply Scheme for PMSM Drive to Boost DC Voltage[END_REF] demonstrates that the same average input power can be held in one T f . I.e. the proposed i * d and i * 0 make it possible to provide the required power to the motor for maintaining invariable torque and speed (output power). However, it should be noted that, the instantaneous input power is non-constant because of the non-constant i * 0 . Therefore, according to [START_REF] Liu | A strategy for improving reliability of field-oriented controlled induction motor drives[END_REF], the DC-bus capacitor has to absorb or release some instantaneous energy in one T f .

Based on the new i * d and i * 0 , the theoretical phase-currents under the post-fault operation can be expressed by

               i A = 0 i B = 2γi
qh cosθ e cos(θ e -2π/3) -i qh sin(θ e -2π/3) + i qh sinθ e -γi qh 1 + cos(2θ e ) i C = 2γi qh cosθ e cos(θ e + 2π/3) -i qh sin(θ e + 2π/3) + i qh sinθ e -γi qh 1 + cos(2θ e ) (31) Theoretical current waveforms are visualized in Fig. 6 where γ = 0.25 is considered as an example. It can be seen that i dh is 0 for SPMSMs; i d is a cosine function of θ e ; i q remains i qh for maintaining the desired torque; The shape of

+ 2π 3 ) + i 0h 1 + cos[2(θe + 2π 3 )] 0 0 0 0 0 -4π -3π -2π -π π 2π 3π 4π 2i qh -2i qh 2i qh i qh 2i qh -2i qh 4i qh -4i qh 2i qh -2i qh i d i dh i q i qh = i 0 i 0h -0.25i qh = i A i B i C
(θ e ) Fig. 6. Theoretical current waveforms of the proposed drive under the postfault operation (an OPF occurs in phase A).

i 0 is non-sinusoidal and the mean of i 0 is equal to i 0h ; i A is 0 because of the fault; i B and i C are no longer the standard sinusoidal waves with a 60 • phase-shift.

Regarding the faulty cases appearing in phase B and C, the post-fault i * d , i * q and i * 0 can be derived in the same way. For clarity, all the representations are summarized in Table . II.

IV. OVERALL CONTROL STRATEGY

In order to control the motor as well as the DC-bus voltage, an overall control strategy is proposed for the drive system, as shown in Fig. 7. It is divided into the healthy and the fault-tolerant modes. The speed controller, Park transformation and PWM block are consistent both in the healthy and faulttolerant modes.

A. Healthy Mode

In the healthy mode, the conventional vector control (VC) framework is still employed. It consists of one speed-loop PI controller and two current-loop PI controllers followed by decoupling compensations. n * is the target speed. The speed-loop PI outputs the q-axis current reference (i * q ). d-axis current reference (i * d ) is normally set as 0 for SPMSMs. daxis current-loop PI outputs d-axis voltage reference (u * d ) and q-axis current-loop PI outputs q-axis voltage reference (u * q ).

In addition to speed and d-q axes currents, the zero-sequence current and DC-bus voltage also need to be regulated. As shown in Fig. 7, an additional PI controller is used to control the zero-sequence current. Meanwhile, another PI controller is employed to control the DC-bus voltage. The detailed design of the two PI controllers are presented in the following.

Considering that L 0 is rarely used in most of conventional drives, Fig. 8 depicts the measuring method of L 0 . As shown in Fig. 8(a), three phases of the motor are connected in parallel. A voltage pulse (1 V) is applied between the neutral point and the terminals. By implementing a voltage-step test, the step response of neutral currents is obtained, as shown in Fig. 8(b). Accordingly, L 0 can be calculated as

L 0 = 3 • R 3 ∆t 0.632 = 3 • 1(V) 5.9(A) • 1.7(ms) = 0.8(mH) (32)
where i N 0.632 means the value of i N reaching 0.632 times the stable value 5.9 (A); ∆t 0.632 is the time spent when i N increases from 0 to i N 0.632 . According to the zero-sequence model presented in [START_REF] Welchko | Fault tolerant three-phase AC motor drive topologies: a comparison of features, cost, and limitations[END_REF], its transfer function (a first-order inertia element) is expressed by

H0(s) = i 0 (s) u 0 (s) = 1 R + L 0 s (33)
The transfer function of a PI controller is defined as

P I(s) = K p • (T i • s + 1) T i • s = K p + K i • 1 s (34)
where K p is the proportional gain; T i is the integral time;

K i = K p T i is the integral gain.
By choosing T i = L 0 R = 0.0016, the closed-loop transfer function of the zero-sequence current-loop can be expressed by

H1(s) = 1 1 + R • T i K p • s (35)
Apparently, H1 is also a first-order inertia element. By designing the time constant (T 0 ) of H1 as 0.01 s, K p can be calculated by For the DC-bus voltage-loop, its PI parameters can be designed according to the small-signal model of the equivalent boost converter, as

K p = R • T i T 0 = 0.08 (
i N ubus =     0 -α 0s L 0 /3 α 0s C -1 R s C     ĩ N ũ bus +     u * bus L 0 /3 -i N s C     (1 -α0 )
(37) where i N s , α 0s are the equilibrium states of the equivalent boost converter.

Under rated condition, the equivalent load R s of the equivalent boost converter can be calculated approximately by

R s ≈ u * 2 bus P n = 30 2 52.5 = 17 Ω (38)
where u * bus is the DC-bus voltage reference. Due to the equivalent boost function, a 15 V DC-source is sufficient for the drive [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF]- [START_REF] Zhang | Neutral Point Supply Scheme for PMSM Drive to Boost DC Voltage[END_REF]. Thus, i N s , α 0s can be calculated by

       α 0s = u in u * bus = 0.5 i N s = u * 2 bus u in R s = 3.5 (A) (39)
Consequently, the transfer function of the DC-bus voltageloop is derived as

H2(s) = ũ bus (s) ĩ N (s) = -i N s C s + α 0s (L 0 /3)C u * bus u * bus L 0 /3 s + u * bus R s (L 0 /3)C + α 0s (L 0 /3)C i N s
(40) For this plant, the proportional gain and integral gain of its PI controller are tuned as 19.9 and 0.003. After the tuning, the open-loop bode diagram is shown in Fig. 9. The gain margin (G.M.) is 80.5 dB and the phase margin (P.M.) is 60 degree. The corresponding step response is shown in Fig. 10. It can be seen that the setting time is less than 0.1 s.

Therefore, the DC-bus voltage-loop PI outputs the neutral current reference i * N . i * 0 can be obtained according to the proportional relation in [START_REF] Errabelli | Fault-Tolerant Voltage Source Inverter for Permanent Magnet Drives[END_REF]

B. Fault-Tolerant Mode

The d-0 axes current references in the fault-tolerant mode are non-constant. It is well known that deadbeat controller has the advantages of high bandwidth, fast response and optimum tracking ability [31] [32]. Therefore, a deadbeat current controller is developed to track the changeful trajectories.

By applying Euler forward approximation to the SPMSM model, the d-q-0 currents at the (k + 1) th moment can be expressed by

  i d (k + 1) iq(k + 1) i0(k + 1)   =        1 - RTs L d ωe(k)LqTs L d 0 - ωe(k)L d Ts Lq 1 - RTs Lq 0 0 0 1 - RTs L0        A   i d (k) iq(k) i0(k)   +       Ts L d 0 0 0 Ts Lq 0 0 0 Ts L0       B   u d (k) uq(k) u0(k)   +    0 - ωe(k)ψ f Ts Lq 0    C (41)
where A, B and C are the matrices of the discrete equation.

In order to eliminate current errors at the (k +1) th sampling interval, the d-q-0 currents at the

(k + 1) th moment should follow      i d (k + 1) = i * d i q (k + 1) = i * q i 0 (k + 1) = i * 0 (42)
By replacing (42) into (41), the required d-q-0 voltages at the k th moment are

  u d (k) u q (k) u 0 (k)   = B -1     i * d i * q i * 0   -A   i d (k) i q (k) i 0 (k)   -C   (43)
In addition, as analyzed in Section III, the DC-bus capacitor has to absorb and release some instantaneous energy under the post-fault operation. That is to say the DC-bus voltage will oscillate with the frequency of the neutral current. Therefore, low pass filter (LPF) is employed to obtain the mean of u bus . The cut-off frequency of the LPF is lower than the fundamental frequency. The DC-bus voltage PI controller is still used to obtain i * N h and i * 0h . i * qh is still obtained by the speedloop PI. Then, the post-fault current references are calculated according to these formulas presented in Table . II. With the proposed deadbeat current controller (43), u * d , u * q and u * 0 are obtained. Finally, by using the inverse Park transformation and CBPWM, the control signals of the remaining two phases are generated.

V. EXPERIMENTAL RESULTS

Experimental validations are carried out on the SPMSM whose parameters are consistent with Table. I, to verify the effectiveness of the proposed scheme and its overall control strategy. Thanks to the equivalent boost function, a 15 V DCsource is supplied to the proposed drive. The algorithms are implemented on a dSPACE MicroLabBox platform. The testbench is shown in Fig. 11. The OPF is simulated by a circuit breaker connected in series with phase A.

A. Healthy Operation

Fig. 12 depicts the experimental results of the proposed drive operating under the healthy condition with speed and load step tests. The experimental settings are as follows: n * is 1000 rpm before 1 s. It is increased to 2000 rpm at 1 s while it is reset to 1000 rpm after 5 s. A 100 mN•m load is imposed at 10 s and removed at 15 s. In Fig. 12, it can be seen that the rotor speed can track the reference well during the acceleration and deceleration processes. Facing with load disturbances, it shows about 50 rpm fluctuations but turns to stable state rapidly. In the whole experiment, the ripple of electromagnetic torque is around 11 mN•m. The DC-bus voltage reaches 30 V even if a 15 V DC-source is supplied. This demonstrates the existence of an equivalent boost function in the drive system. u bus shows around 0.5 V voltage dip and rise during the speed step tests, but it recovers to 30 V quickly. In addition, it can be seen that i N increases from 0 to 3.8 A when the system has an acceleration request. Then, it decreases to -0.6 A when the motor decelerates sharply. This is because the motor operates as a generator in this case. Under the condition of 1000 rpm and 100 mN•m, the mean of i N is around 1.8 A and the the ripple of u bus is around 0.4 V. Consequently, this experiment verifies the effectiveness of the proposed overall control strategy, in which the motor and the DC-bus voltage can be regulated simultaneously. In order to clarify the advantages and disadvantages of the proposed drive, a contrast experiment is carried out on the same motor but fed by the conventional topology (the neutral point is unused but the parameters of VC are consistent). For the conventional drive, a 30 V DC-source is required. Relevant experimental results are shown in Fig. 13. By comparison, it can be seen that the motor performance (including the speed control and torque performance) is similar between the two drives. Therefore, it can be concluded that the motor performance can be maintained when its neutral point is connected to a DC-source.

Fig. 14(a) and Fig. 14(b) illustrate the detailed phasecurrents of the two drives under the healthy operation (1000 rpm and 100 mN•m). First, it can be seen that the mean of phase-currents in the proposed drive is around -0.6 A. This means that a 0.6 A zero-sequence current flows into the motor. In terms of the conventional drive, the mean of its phase-currents is 0 A. Second, it is visible that the current ripple is larger in the proposed drive. In order to quantify the total harmonic distortion (THD), fast fourier transform (FFT) analyses are shown respectively in Fig. 14(c) and Fig. 14(d). According to the FFT results, it can be concluded that the current THD in the proposed drive increases by 2.12% compared to that in the conventional drive. This is a drawback to the proposed drive in order to win over a free boost function.

B. Post-Fault Operation

Fig. 15 shows the fault-tolerant test of the proposed drive operating under 2000 rpm without load condition. First, under the healthy operation, n is stable at 2000 rpm. The ripple of T e is about 10 mN•m. u bus is boosted at 30 V. i dh is 0 A and i qh is 0.58 A. i 0h and i N h are respectively -0.15 A and 0.45 A. Thus, it can be calculated that the zero-sequence current proportion γ is around 0.26. Second, when the OPF occurs, i A becomes 0 A instantly. Without the fault-tolerant control strategy, T e and i q show big fluctuations with 50 mN•m and 1.6 A, respectively. The ripple of u bus increases to 2 V as well. The ripples of i d , i 0 and i N are respectively 1.4, 0.4 and 1.2 A. Third, once the fault-tolerant mode switches on, the performance of T e and i q is preserved rapidly. The ripple of T e decreases to 12 mN•m, which is in the range of the values obtained under the healthy operation. For greater detail, enlarged views are displayed to the right of each graph. It can be seen that i d is a cosine curve, which is in line with the theoretical waveform presented in Fig. 6. Similarly, i 0 is also similar to the theoretical curve. The period of i d and i 0 is the same as that of i B and i C , which is the fundamental period (T f = 7.5 ms). As analyzed in Section III, for maintaining the constant output power, the DC-bus capacitor has to absorb and release some transient energy. Therefore, it can be seen that u bus is synchronized with i N and the ripple of u bus increases to 3 V. However, it needs to be emphasized that the mean of u bus is still around 30 V.

For verifying the tracking performance of the proposed deadbeat current controller, Fig. 16 illustrates the detailed comparison between the theoretical and actual phase-current waveforms. It can be seen that the actual i B and i C show good agreement with the theoretical results. As analyzed before, they are no longer the standard sinusoidal waves with a 60 • phase-shift.

In order to check the fault-tolerant performance when the motor operates under load condition, Fig. 17 depicts relevant experimental results. First, under the healthy operation, n is controlled at 2000 rpm steadily. T e is about 60 mN•m, which is almost half of the nominal torque. The torque ripple is around 9 mN•m. u bus is still boosted at 30 V stably. i dh is 0 A and i qh is about 1.79 A. i 0h is -0.46 A and i N h is 1.37 A. Second, when the OPF occurs, the ripple of T e increases to 150 mN•m dramatically. Meanwhile, the ripple of i q arises to 4.2 A. The ripples of i d , i 0 and i N are respectively 4, 1.0 and 2.5 A. Third, thanks to the fault-tolerant controller, T e returns to 60 mN•m quickly with oscillations of around 13 mN•m. In the enlarged views, i d is still a cosinoidal waveform as analyzed. Moreover, the period of i d and i 0 is still consistent with the fundamental period. With the increase of motor's power, the peak value of i N increases to 5.6 A while it is about 1.8 A under the no-load condition. This means the transient energy absorbed or released by the DC-bus capacitor also increases. As a result, it can be seen that the ripple of u bus increases to 6 V. However, the mean of u bus is still about 30 V due to the DC-bus voltage controller. The detailed comparison between the theoretical and actual phase-current is illustrated in Fig. 18. It can be seen that the actual i B and i C are still consistent with the theoretical waveforms.

When the motor operates at the fault-tolerant mode, it still has dynamic requirements, such as speed regulation and load change. Thus, in order to further validate the practicability, dynamic tests under the post-fault operation are implemented, as shown in Fig. 19. The setup of this experiment is as follows: n * is 1000 rpm before 2 s. It is set to 1500 rpm between 2 and 5 s while it is reset to 1000 rpm after 5 s. A 50 mN•m load is imposed at 10 s and removed at 15 s. According to the experimental results, it can be seen that i A is always 0 A because of the OPF. The motor speed can still track the referenced signal well. The torque performance is still maintained during the acceleration, deceleration, loading and deloading processes. When the motor accelerates instantly, u bus declines to 22 V firstly. Thanks to the control for the mean of u bus , i * 0h decreases from -0.07 to -0.37 A (i.e. i * N h increases from 0.21 to 1.11 A) rapidly in order to provide more power to the drive system. Therefore, the mean of u bus can return to 30 V. Similarly, i * 0h can also adjust cleverly during the load change processes. It changes from -0.07 to -0.28 A then returns to -0.07 A again. This demonstrates that it is feasible to obtain suitable i * 0h by using a controller to regulate the mean of u bus .

Consequently, according to the theoretical analyses and experimental validations, the characteristics of the proposed fault-tolerant three-phase SPMSM drive compared to the existing topologies are summarized in Table. III.

VI. CONCLUSION

In this paper, an innovative fault-tolerant three-phase SPMSM drive and its overall control strategy were proposed. First, the proposed drive is able to reach a higher DCbus voltage thanks to the equivalent DC/DC boost converter function. However, the rise (2.12%) of current THD even the issue of iron core saturation caused by the zero-sequence current should be noticed in practical applications. In order to control the DC-bus voltage, a new control-loop is added into the conventional VC framework by considering the zerosequence components. Second, the proposed drive has the selfhealing ability towards OPFs neither using split capacitors nor adding auxiliary inverter legs. Moreover, it needs no TRIAC to reconfigure the circuit structure. In order to maintain the motor performance under the post-fault operation, a cosinoidal d-axis current reference and a nonsinusoidal zero-sequence current reference were proposed. For tracking the non-constant references, a deadbeat current controller was developed. The convincing experimental results verified the effectiveness of the innovative drive and its control strategy. In the healthy operation, the DC-bus voltage can be boosted to twice the DC-source voltage. In the fault-tolerant operation, the torque performance can recover rapidly once the fault-tolerant control strategy works.

Consequently, the proposed drive has the advantages of low cost, volume, weight and structure simplicity compared to these existing fault-tolerant three-phase topologies. The study of this paper enriches the categories of fault-tolerant threephase drives. From the view of application, the proposed solution seems to have significant value in EVs because a free DC/DC boost converter function and an economical faulttolerant solution can be provided simultaneously by one threephase two-level VSI.

VII. APPENDIX

The Clark transformation is

T c = 2 3         1 1 2 1 2 0 √ 3 2 - √ 3 2 1 2 1 2 1 2         (A.1)
The rotation transformation from α-β-0 to d-q-0 coordinates is

Fig. 2 .

 2 Fig. 2. Four typical fault-tolerant topologies for three-phase AC motors. (a) 3 legs with phase to capacitor midpoint (3L-PCM); (b) (3+1) legs with phase to an auxiliary leg ((3+1)L-PAL); (c) 3 legs with neutral point to capacitor midpoint (3L-NCM); (d) (3+1) legs with neutral point to an auxiliary leg ((3+1)L-NAL).

Fig. 3 .

 3 Fig. 3. Proposed fault-tolerant three-phase SPMSM drive.

Fig. 4 .

 4 Fig. 4. Equivalent circuit of the proposed drive under the healthy operation.

Fig. 5 .

 5 Fig. 5. Trend of γ with respect to n and η.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Overall control strategy of the proposed drive.

Fig. 9 .

 9 Fig. 9. Open-loop bode diagram of the DC-bus voltage-loop after the tuning.

Fig. 10 .

 10 Fig. 10. Step response of the DC-bus voltage-loop after the tuning.

Fig. 11 .

 11 Fig. 11. Test-bench.

Fig. 12 .

 12 Fig. 12. Experimental results of the proposed drive operating under the healthy condition with a 15 V DC-source.

30

 30

Fig. 13 .

 13 Fig.[START_REF] Zhao | An approach to modeling and field-oriented control of a three phase induction machine with structural imbalance[END_REF]. Experimental results of the conventional drive (neutral point is unused) operating under the healthy condition with a 30 V DC-source.

Fig. 14 .

 14 Fig. 14. Phase-current comparisons between the proposed and the conventional drives under the healthy operation (1000 rpm and 100 mN•m). (a) i A and i B of the proposed drive. (b) i A and i B of the conventional drive. (c) FFT analysis of the phase-currents in the proposed drive. (d) FFT analysis of the phase-currents in the conventional drive.

Fig. 15 .

 15 Fig. 15. Experimental results of the fault-tolerant test under 2000 rpm without load condition.

  i A ,i B ,i C [1 A/div] theoretical waveforms actual waveforms i A ,i B ,i C [1 A/div] [12 ms/div]

Fig. 16 .

 16 Fig. 16. Detailed comparison between the theoretical and actual phasecurrents under the fault-tolerant operation (2000 rpm without load).

Fig. 17 .Fig. 18 .

 1718 Fig. 17. Experimental results of the fault-tolerant test under 2000 rpm with load condition.

Fig. 19 .

 19 Fig. 19. Experimental results of the proposed drive under the post-fault operation with dynamic tests.

  e ) sin(θ e ) 0 -sin(θ e ) cos(θ e ) (θ e ) -sin(θ e ) 1 cos(θ e -2π 3 ) -sin(θ e -