J F Brudny

SATURATED SINGLE PHASE MAGNETIC CIRCUIT UNDER SINUSOIDAL VOLTAGE: IRON LOSSES -DEFINITION OF A NEW PHYSICAL CONCEPT -EQUIVALENT SCHEMES

Keywords: Magnetic saturation, Power balance, Iron losses I)

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 marked "x" is characterized by its x ˆ peak value and its "X" rms value . The harmonic component of rank "k" of "x" is denoted k x , its rms value k X and its phase angle / / k x , defined with respect to a single reference phase angle which is that of "v", counted positively in the clockwise direction. When relevant, k

x will also be characterized by its k X time phasor. The considered MCs are assumed to be supplied by a voltage "v" of frequency f = 50 Hz (angular frequency ω, period T).

II) PRELIMINARY EXPERIMENTAL CONSIDERATIONS

This preliminary analysis considers a MC whose toroidal core is made from istrope steel sheets. This MC, presented in Fig. 1, carries a main winding of n turns. In order to guard against possible disturbances which may affect the supply voltage, a sinusoidal "v" voltage is generated by implementing a function generator associated with a power amplifier which is like having an infinite power supply grid.

II.1) Equivalent scheme, notations

The conventional configuration of the equivalent scheme of this CM is given in Fig. 2 where also appear the notations used. "r" and "l" represent the main winding respectively resistance and leakage inductance. These elements are assumed to have constant values. µ L corresponds to the magnetizing inductance and the EM block identifies with a il R resistive module. For given "V" and "f", il R and µ L are generally constant values. To overcome the uncertainties generated by "r" and "l", the experimental determination of the iron losses uses the "i" current and the internal "e" emf taken at the terminals of an auxiliary winding. The power measurement then leads, to the ratio close to the numbers of turns of the main and auxiliary windings, to the k P int quantities which are supposed to represent the iron losses. As is to be expected, even for sinusoidal "v", saturation will affect the waveforms of "i" and "e" which will include, in addition to the fundamental, harmonics. Implicitly, this means that the waveform of "b" will also be affected by harmonics even if their amplitudes are low. These harmonics are at the origin of additional losses which are difficult to apprehend because, a priori, there is no equivalent scheme which can translate their effects in terms of powers insofar as "v" is sinusoidal. Faced with this difficulty, the MC are generally designed by considering that the rated operating point is located at the level of the saturation bend of the b (H) characteristic. The problems raised are then simply ignored by introducing the notion of equivalent sine current to take account of the distortions on "i". This seems regrettable because better understanding the behavior of such MC subjected to high flux densities placing the rated operating point well beyond the saturation bend can, perhaps, lead to interesting properties.

II.2) Example of a measurement table

From an experimental point of view, to quantify the effects generated by natural harmonics, a wattmeter referenced YOKOGAWA WT230 or a pulse system from Brüel & Kjaer is used. These devices digitally process (FFT) the signals they pick up. The phase shifts are determined using the "cross product" function. The simultaneous use of these two devices during the measurements allows to ensure that the information obtained was reliable. Table 1 ( b ˆ(T) V(V) E(V) I(A) P ext P int P int1 P int3 P int5 P int7 Pi nt9 0,28 3,10 3,06 0,25 0,45 0,43 0,43 0,00 0,00 0,00 0,00 0,83 9,25 9,16 0,59 2,76 2,66 2,65 -0,01 0,00 0,00 0,00 1,10 12,32 12,21 0,89 4,44 4,22 4,23 -0,02 -0,00 0,00 0,00 1,21 13,59 13,48 1,10 5,31 4,96 5,00 -0,03 -0,00 0,00 0,00 1,35 15,30 15,00 1,56 6,73 6,03 6,14 -0,10 -0,01 -0,00 -0,00 1,48 16,63 16,46 3,03 9,99 7,30 8,22 -0,68 -0,18 -0,05 -0,01 1,58 17,79 17,55 6,11 18,60 8,13 13,00 -3,52 -1,06 -0,22 -0,02 1,62 18,28 18,01 7,01 22,07 8,31 14,86 -5,06 -1,11 -0,06 -0,08 deduced from the emf recorded at the terminals of the auxiliary winding not shown in Fig. 1. The iron losses deduced from EM are noted ilk P and il P .

II.3) Problems

For b ˆ low values (up to 0,80T), therefore in the absence of saturation, one can note that int P is practically equal to 1 int P . The quantities k P int for k> 1 exist but they are small and, having regard to the the resolution of the measuring device, zero values have been assigned to them. So it is possible to consider that: For b ˆ higher values, therefore for a MC affected by saturation, the quantities k P int for k> 1 become quite significant. Their contributions can therefore no longer be ignored. The major problem is that these quantities are negative. In addition,

1 int
P is greater than int P . For a saturated MC, the experimenters are then confronted with several problems. 1) How to justify the peculiarities which have just been mentioned concerning the active powers? Isn't there a problem with the experimental procedure used?

2) It is generally considered that Without presenting numerical data at this stage, it is possible to specify that

1 int
P is greater than 1 il P . Hence the studies presented in the literature intended to characterize in a more suitable way the average length of iron to be considered at the level of the Epstein frame [START_REF] Ph | A Méthod for Defining the Mean Path Length of the Epstein Frame[END_REF][START_REF] Pfützner | Concept for more correct iron loss measurements considering path length dynamics[END_REF], even to adapt this length according to b ˆ [14 -16]. In fact, the analysis carried out shows that saturation can, perhaps, intervene in a much more subtle way as for this correction.

0 int < k Q for k> 1.

III) PRESENTATION OF THE SEMI-ANALYTICAL METHOD

After having characterized the considered MC and presented the relationships used, the steps which define the developed semi-analytical method are exposed. Note that from this paragraph, the results presented mainly result from the use of the data obtained following the implementation of the proposed method. As a reminder, the abbreviation SAM (Semi-Analytical Method) is introduced at the level of the tables and figures concerned.

III.1) Characterization of the analyzed MC

• The MC used to validate the semi-analytical method and highlight the main features, is presented in Fig. 3 which specifies the different geometric dimensions, its L MC iron average length as well as its S MC cross-section. This MC, whose configuration of the equivalent scheme is always that presented in Fig. 2, is of industrial design and therefore overcomes the reservations that could possibly be expressed following the implementation of handcrafted design prototypes. Fig. 3 also presents the values of "r", "l" and x = lω.

• The nonlinear effects of the magnetic material will be taken into account by considering its characteristic of first magnetization b (H). It is the M400 50A which can be considered as isotropic. Fig. 4 presents its b(H) curve until the flux density at saturation with a zoom at the level of the inflection point.

In fact, it is not this characteristic which will be directly exploited but rather that which gives the variations of the relative magnetic permeability ) (b r µ [START_REF] Cossale | Modeling the Degradation of Relative Permeability in Soft Magnetic Materials[END_REF].

Z Z r β + α = µ
(1) where "Z" takes the whole values between 1 and 5. T2 specifies the numerical values which characterize these segments as well as the values of "b" which delimit the various zones. It should be noted that this characteristic has already been exploited with some success to take into account the effects of very high saturation in the context of applications which implemented virtual air gap [START_REF] Brudny | Series current limiter using a magnetic circuit comprising holes and windows[END_REF]19].

III.2) Epstein Model (EM)

• In order to be able to conclude as to the problems that can arise from measuring iron losses on a practical device, the theoretical analysis will consider an EM to characterize iron losses. As the peculiarities which will be highlighted are completely independent of the model precision, it is possible to conclude: -that implicitly, this credits without ambiguity and with a certain "almost universal" scope the behaviors which will be described,

4

-that to characterize the EM, it is possible to be satisfied with a fairly rudimentary model. Consequently, the EM 1 model for the fundamental, resulting from the experiments carried out on the Epstein frame, which generally considers 3 components: static, dynamic and by excess [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF][START_REF] Bertotti | Physical interpretation of induction and frequency depence of power losses in soft magnetic materials[END_REF], will be in this case defined by: -static losses of the form:

2 1 1 ) (b f K P stat stat = ,
-dynamic losses of expression:

2 1 1 ) ( b f K P dyn dyn = .
The expression which characterizes ilk P for k> 1 is quite complex, in particular concerning the term which takes into account the effects of minor loops [22 -26]. This analysis will be limited to considering at the EM k level only the dynamic effects:

2 ) ( k dyn dynk ilk b kf K P P = = .
The available data show that W P P P dyn stat il

5 1 1 1 = + =
for f = 50Hz and T b 1 ˆ1 = . Assuming further that, under these conditions:

 ≠ + = ) 1 ( 1 2 k k dynk dyn il P P P (2)

III.3) Relationships exploited

Although the device which will be modeled is elementary, the determination of the variations of the various variables in steady state is not obvious taking into account, on the one hand, "r" and "l" and, on the other hand, non-linearities which characterize the magnetic material. The procedure developed consists of working by successive approximations after an initialization phase. All developments have been made by adopting a single time origin which coincides with an instant "t" such that "v" has a maximum positive value. It comes:

t V v ω = cos 2
(3) The generated variables will be defined by their FSs but also by means of sequences of points on "T". The sequences consist of Y points regularly distributed over "T" therefore distant from each other by ∆t=T/Y. The notations used to characterize a variable "x" will be } {x for the sequences and FS(x) for the Fourier series. The value of "x" for any point "y" in the sequence will be defined by y

x .The transition from } {x to FS(x) and vice versa poses no particular difficulty.

The main hypothesis formulated assumes that, ∀t, "b" is distributed uniformly over S MC . On the other hand, the nonlinear effects which occur at the level of the joints will be ignored which, implicitly, boils down to considering a toroidal MC.

To distinguish the variables according to the iteration concerned, they will be assigned a higher index in parentheses which corresponds to this iteration.

ℜ

of the MC and by using the basic relations of electromagnetism, it is possible to characterize successively the temporal expressions of

) (j b , ) ( j r µ , ) ( j il P , ) ( j r µ , ) ( j a i , ) ( j r i , ) (j ℜ , ) ( j L µ and ) ( j th v . The variable ) ( j th
v corresponds to a theoretical voltage which is that it would be necessary to apply to the main winding in order to effectively obtain an internal emf which is identified with imposed ) (j e at the launching of the iteration "j". The different relationships that lead to these quantities are grouped in Annex A and referenced (AN with N = 1, 2, 3 .....). Are also specified at the level of this annex the notations which characterize the developments in Fourier series as well as the relations which define the THD and the rms value of a variable. III.3.2) Convergence of the method. In order to judge the convergence of the semi-analytical method, one uses the variable ) ( j th v defined by (A12). The approach adopted consists, at the end of each iteration, to compare ) ( j th v and "v" and,

when v v j th = ) (
to stop the calculation. In fact, under these conditions, it is possible to affirm that the "exact" response from the MC is obtained. Insofar as the correspondence ) ( j y b → ) ( j ry µ is unequivocal, this response is unique.

However, since the values of "k" and "Y" are finite, it is obvious that numerical errors will affect the variables. As a result, strict equality v v j th = ) ( will never be satisfied regardless of the number of iterations. It is therefore advisable to define a limit max j of "j" for which it is possible to consider that the MC response is sufficiently representative of that which one can qualify as exact. This max j value is difficult to predict because it depends on the elements of the equivalent scheme and the excitation (value of V). Indeed, when the saturation of the MC is more pronounced, the convergence of the results obtained towards the solution will be much slower and will require more iterations. To assess from which iteration it is possible to stop the calculation, 3 conditions on

) ( j th v
which must be satisfied simultaneously are imposed. 1) THD( ) ( j th v ) must be less than an imposed THD max value.

The other two conditions relate ) ( 1

j th v more particularly the relative values of ) ( 1 ˆj th v and / / ) ( 1 j th v . 2) Let v v v j th j th / ˆ) ( 1 ) ( 1 = ∆
this relative amplitude be, the difference in percent between

) ( 1
j th v ∆ and v ∆ ( 1 = ∆v
) is expressed by:

) 1 ( * 100 ˆ) ( 1 ) ( 1 j th j th v v ∆ - = δ (4)
3) Introduce the relative phase angle between )

( 1 j th v and "v" expressed in percent and defined by:

T v v v j th j th / /) / / (/ * 100 ) ( 1
) ( 1 - =  (5)
As / v / = 0, by adopting the degree as a unit, ( 5) is written:

360 / /) / ( * 100 ) ( 1
) ( 1 j th j th v v - =  (6)
In absolute values,

) ( 1 ˆj th v δ and ) ( 1
j th v 
must be respectively less than

max 1 ˆth v δ et max 1 th v  .
If the method converges, these three quantities must decrease when "j" increases. Note on the other hand that ) ( ) ( j th v FS also has the advantage, following the determinations of ) ( j extk P and

) ( j extk Q
, of verifying whether the Boucherot's theorem is satisfied or not at each iteration and, if so, thus validating all the intermediate calculations.

III.3.3) Exploitation of the results obtained following the iteration "j".

For each harmonic, the various active and reactive powers are calculated. Then, by summing the harmonic powers over "k", the overall powers are determined. The variables:

-

) ( j k e and ) ( j k i lead to ) ( int j k P and ) ( int j k Q defining ) ( int j P and ) ( int j Q , -) ( j k u and ) ( j k
i lead to (j) k p and (j) k q defining (j) p et (j) q , -) ( j thk v and )

( j k i lead to ) ( j extk P and

) ( j extk Q defining ) ( j ext P and ) ( j ext Q , -) ( j k e and ) ( j rk i lead to ) (j k P µ and ) (j k Q µ defining ) (j P µ and ) (j Q µ .
The validation of Boucherot's theorem consists in checking whether:

(j) k j k j extk p P P + = ) ( int ) ( and (j) k j k j extk q Q Q + = ) ( int ) ( .
The numerical values considered mainly exploit the coefficients of the FSs. However, in order to test the validity of certain results, more direct methods have also been implemented:

-Computation of the overall active powers by integrating the numerical sequences given by the products:

) ( ) ( j j i e , ) ( ) ( j j i u , …. -(j)
k p and (j) k q determined using "r" and "x" according to the relations:

2 (j) k (j) k ) r(I p = and 2 (j) k (j) k ) kx(I q = .
The determination of the behavior of this saturated MC essentially results from the analysis of the distribution of the various powers at the level of the configuration of the equivalent scheme presented in Fig. 2. The fundamental point concerns the powers ) (j k P µ and ) ( j k Q µ "consumed" by the magnetizing inductance knowing that, usually, it is considered that: 0

) ( = µ j k P ∀k.
It should be noted that the rms values and the overall active powers can, indifferently, be calculated directly from the numerical sequences (numerical integrations) or by using the coefficients of the FSs which have just been defined. The direct determination of the reactive powers is not possible so that, for these, the FSs are exploited by passing through the apparent harmonic powers or the phase angles between the variables concerned. As the problem of ambiguity regarding the signs which characterize the harmonic components of the powers arises, they are the phase angles defined by (A15) which will be used. However, the latter respond to the conventions initially defined (counted positively in the trigonometric direction).

Considering, for example, the time phasors

) (j k U and ) ( j k I , (j) k
p conventionally results from the relation:

) ( ) ( ) ( cos j k j k j k (j) k I U p ζ = where ) ( j k ζ is the phase angle of ) ( j k I with respect to ) (j k
U positively counted in the opposite direction of the trigonometric one. These considerations lead to:

/ / / / ) ( ) ( ) ( j k j k j k i u + - = ζ (7)

III.4) Principle of the semi-analytical method

This paragraph specifies how for each iteration "j" the choice of )

(j e to be imposed is made.

Iteration 0: initialization phase. Considering Fig. 2 it is assumed that: v e = ) 0 ( . This makes it possible to determine the ) 0 ( b variations therefore those of all the other variables of the MC (See Annex A). Finally ) 0 ( th v is compared to "v" in order to check if the criteria defined in III.3.2 are satisfied. As this will not be the case, iteration 1 is considered. Iteration 1. For this iteration, first assume that the supply current identifies with ) 0 ( i calculated for j = 0 and therefore the voltage being across "r-l" identifies with ) 0 ( u . The mesh law:

) 0 ( ) 1 ( u e v + =
, defines the variations on T of During the iteration "j" it is assumed that that the supply current identifies with

) 1 ( - j i : ) 1 ( ) *( - = j j i i
and therefore that the voltage being across "r-l" identifies with

) 1 ( - j u
. The mesh law:

) 1 ( ) ( - + = j j u e v defines the ) (j
e variations related to this iteration "j" which will be the starting point to calculate all the variables relating to this iteration "j" but also to check if the criteria on ) (J th v are satisfied. If it is not the case one passes to the iteration j + 1 and if it is the case one stops computation. Note. In this exploratory phase but also for pedagogical reasons, the various calculations just presented were carried out for each "j". However, practically, the validation and test procedures may be carried out, for example, only every 5 iterations.

IV) RESULTS DEDUCED FROM THE SEMI-ANALYTICAL METHOD

This procedure is implemented in Excel. Programming poses no particular difficulty. The calculation lines for j = 0 are taken as is for the other values of "j" by modifying the input variable "i" which imposes "e". From numerical data point of view, as:

2 3 ) ( 10 s W K dyn - =
, (A7) leads to il R =237,85*10 2 Ω. By adopting ∆t = 0,1ms, it comes : Y = 200. The harmonic components for k = 10 will be defined over a period of these signals using 20 points which is compatible with good accuracy.

The V value is 380V which leads to b ˆ around 1,5T. This rather pronounced saturation makes it possible to better highlight the peculiarities initially stated while avoiding too small differences which could be attributed to numerical errors. This relatively large flux density also makes it possible to appreciate the impact of saturation on the convergence of the method.

IV.1) Definition of j max

The values of

) ( ) ( j th v THD , ) ( 1 ˆj th v δ and ) ( 1
j th v 
for "j" varying from 0 to 15 are grouped in T3. Considering the absolute values and imposing a maximum value for these quantities of 0.1%, this means: -that harmonics alone (k> 1) intervene, at most, for 0,38V rms for ) ( j th v around 380V rms, -as a maximum, )

( 1 ˆj th v will differ by 0,53V compared to the peak 537,40V imposed on " v ˆ",

-at most, / / ) ( 1 j th v
will differ by 0,36° compared to / / v . T3 shows that the 3 constraints are satisfied for j = j max = 9 leading to harmonics which occur alone (k> 1) for 0,268V rms, an amplitude and a phase angle of

) 9 (
1 th v which will differ from 0,0053V and 0,004° respectively from those of "v". Let us specify that for j = 15 it is possible to consider that

) 15 ( th v
is identified perfectly with a sinusoid. Nevertheless, from an pedagogical point of view, it will be considered subsequently that the steady state corresponds to j = 9.

IV.2) Presentation of the results

This presentation concerns the cases j = 0, 2 and 9. The case j = 9 is representative of the MC response under sinusoidal voltage therefore in the presence only of natural harmonics. The case j = 2 can be interpreted as the MC response under disturbed supply and therefore that the natural harmonics interact with the supply harmonics. The case j = 0 is specific since no harmonic affects the MC in accordance with the procedure required when implementing the Epstein frame. Figs. 6a,b and c present the variations on "T" respectively of

) (j b , ) ( j i and ) (j u .
To present all the numerical values relating to these 3 iterations would be tedious. Hence the choice to show in the reduced tables grouped in T4 (reported in the annex B), only the values of the variables which highlight the MC particular behaviors. The amplitudes and phase angles of these variables are specified in the upper parts (lines 3 to 12) where the last columns (6, 12 and 18) specify the rms values of the electrical variables. The powers appear on lines 13 to 21. The overall powers, which result from the summations on "k" of the harmonic powers, are given in columns 6, 12 and 18. Note that the analysis of the complete tables shows that : -the different methods used to calculate the rms values and the powers lead to practically identical results, regardless of the iteration considered, -Boucherot's theorem is satisfied at all levels of the equivalent scheme and that ∀j. The THDs of the different variables for these 3 values of "j" are grouped in T5. It is possible to note, when j increases, that the THDs on ) (j e and ) (j b increase while the THDs on the other variables decrease.

-For j≠0 and for k> 1, the ) ( j ilk P , compared to ) ( 1 j il P , are very small having regard to the low values of ) ( ˆj k b whereas b ˆ is of the order of 1.5T. This remark can have important practical consequences because, at first sight, it is possible to consider that "b" presents a sinusoidal temporal repartition although this is not the case for "i".

-For k = 1, ) ( 1 ˆj a i defined from . This remark is also valid for k> 1. As an example, consider the case j = 9. It comes: , i (j) and u (j) for V=380V and j=0,

A i a 04 , 0 ˆ) 9 ( 1 = , ) ( ) ( A i j t(ms) (b) ) ( ) ( V u j t(ms) (c) ) (
2 et 9 j=0 j=2 j=9
T5: SAM -THD in percent of MC variables for j = 0, 2 and 9 j

) ( j e ) ( j b ) (j i ) (j u ) ( j r i ) (j th v 0 0 0 59,

P P >

. One find the peculiarities that characterized, from an experimental point of view, the toroidal MC operating (See T1) which, implicitly, validates the experimental procedure implemented as well as test results.

In addition, examination of Figs. 6, especially for cases j = 0 and 9, already raises questions as to • For j≠0 several quite disconcerting results appear. 5) The corollary of the above equality is that: Overall L µ plays its role of inductance but it is not the case harmonic by harmonic for which the elements which constitute this "inductance" need to be redefined knowing that for certain values of k, these elements present an active behavior ( ) ( j k P µ and

) ( ) ( int
) ( j k Q µ negative). 6
)

) ( ) ( int j k j k Q Q µ = ∀j leading to ) ( ) ( int j j Q Q µ =
. One can note that (validated for j = 15) which is not the case for j = 2. This indicates that the external powers are carried only by the fundamental although ) 9 ( i it is not sinusoidal which, implicitly, leads to admitting that ) 9 ( v identifies with "v". Relatively specific behaviors therefore appear for j≠0. Like these are dependent only on µ L validates the concept evoked in III.2 namely that the precision of the EM had no impact on the phenomena. The presented results highlight the effectiveness of the proposed semi-analytical method despite the fact that it must act simultaneously on the amplitudes and phases of several signals. Considering cases j = 0, 2 and 9, Fig. 7 presents the variations on T of ) ( j r µ and of ) ( j r η . Regardless of the differences which appear on certain extremums, which are perfectly justified by the fact that ) ( ˆj b decreases when "j" increases (See Fig. 6a), it should be noted that the extremums do not coincide from a temporal point of view like the the zoom of Fig. 7a underlines it. The analytical exploitation of these quantities uses their FSs which will include a mean term to which will be added harmonics of even ranks. By entering the quantity "h" (h = 1, 2, 3, 4, ....), it comes:

V) JUSTIFICATION OF THESE SPECIFIC BEHAVIORS V.1) Characterization of quantities

/) / 2 cos( ˆ) ( 2 ) ( 2 ) ( ) ( j h r h j h r j r j r t h µ - ω µ + > µ =< µ  (8) /) / 2 cos( ˆ) ( 2 ) ( 2 ) ( ) ( j h r h j h r j r j r t h η - ω η + > η =< η  (9)
The numerical values of the coefficients which characterize these FSs for j = 0 and 9 are grouped in T6 where the quantities which appear in the column relating to h = 0 correspond to the mean values. Although are quite significant. More particularly, concerning the phase angles, these take values of 0 or 180 ° for j = 0 while this is not the case for j = 9. This means that the distortions on "b" caused by the natural harmonics generate disorders at the level of ) ( j r µ and of ) ( j r η for j≠0 which should be quantified in terms of powers relative to ) (j L µ .

V.2) Characterization of the powers and currents relative to

) (j L µ
Given relationships (A1) to (A3), (A9) and (A13), by defining:

MC MC MC L n S K / ) ( 2 0 µ = (10)
and by introducing k', which plays the same role as "k", to characterize the ) (j e harmonics, it comes:

        π - - ω ω η =  ' ) ( ' ) ( ' ) ( ) ( ) 2 / / / ' cos( ' k j k j k MC j r j r e t k k e K i (11)
(9) and [START_REF] Janicke | Numerical modeling for anisotropic media including saturation effects[END_REF] show that ) ( j r i will be composed of two series of terms. The first, defining

) ( ' j r i , is a function of > η < ) ( j r
. The second, defining ) ( " j r i , is linked to the 2h rank harmonics of ) (j r η . It comes:

) ( ) ( ) ( " ' j r j r j r i i i + = (12
) By considering the components of rank "k" and the concept of time phasor, (12) leads to:

) ( ) ( ) ( " ' j rk j rk j rk I I I + = (13)
The same formulation will be used to characterize the powers:

     + = + = µ µ µ µ µ µ ) ( ) ( ) ( ) ( ) ( ) ( " ' " ' j k j k j k j k j k j k Q Q Q P P P ( 14
)
which, considering [START_REF] Mallard | Increasing the Energy Efficiency of Induction Machines by the Use of Grain-Oriented Magnetic Materials and Die Casting Copper Squirrel Cage in the Rotor[END_REF], exploit the phase angles ) • First series of terms. Since at given "V", > η < ) ( j r is a constant, one can deduced that, for k = k ':

°= ς 90 ' ) ( j rk . 0 ' ) ( = µ j k P ∀k.
Only the ) ( ' j k Q µ components exist. This allows the concept of inductance to be introduced. This, noted

) ( ' j L µ , considering (A11)
and [START_REF] Qu | Precise magnetic property measurement on electrical sheet steels under deep saturation[END_REF], is defined by: >

η < = µ ) ( ) ( / ' j r MC j K L ( 15
) Let ω = µ µ ) ( ) ( ' ' j j L X
the magnetizing reactance relative to the fundamental. In accordance with [START_REF] Janicke | Numerical modeling for anisotropic media including saturation effects[END_REF], that leads to:

) ( ) ( ) ( ' / ' j j k j rk jkX E I µ = ( 16
) for j=0 and 9

) ( 2 ) ( ) ( ' / ) ( ' j j k j k kX E Q µ µ = (17)
• Second series of terms. [START_REF] Janicke | Numerical modeling for anisotropic media including saturation effects[END_REF] shows that this second series results from the expression:

ω         - ω         η - ω η =   MC k j k j k j h r h j h r j r K e t k k e t h i / /) / ' sin( ' /) / 2 cos( " ' ) ( ' ) ( ' ) ( 2 ) ( 2 ) ( (18)
which can still be written:

{ } /] / / / ) ' 2 sin[( /] / / / ) ' 2 sin[( ' 2 1 " ) ( ' ) ( 2 ) ( ' ) ( 2 ' ) ( ' ) ( 2 ) ( j k j h r j k j h r h k j k j h r MC j r e t k h e t k h k e K i + η - ω - - - η - ω + η ω =  (19)
It appears that a harmonic of rank "k" of

) ( " j r i
results from several combinations of "h" and k'. The analytical expression of this harmonic is therefore composed of several terms. By limiting the values of k' to 9 and those of "h" to 4, T7 presents, according to the combinations of k' and "h", the ranks "k" resulting from (19). As an example, consider

) ( 3 " j r i . T7
shows that this component relating to k=3 results from taking into account 7 terms which are highlighted in gray in this table. Hence the " j k Q µ analytical determinations become tedious so that these powers will be processed numerically. ' k Q µ is given by ( 17). ( 16 • η η η η r components linked to the 2h terms. 2 distinct methods can be implemented.

) ( 3 " j r i expression:                                       + η - ω - η         + η - ω - - + η - ω - η         + η - ω - - + η - ω - η         + η - ω - - η - ω η ω ) ( 8
) ( 5 ) ( 8
) ( 9
) ( 6
) ( 9 ) ( 3
) ( 6 ) ( 3 ) ( 6 ) ( 7 ) ( 4
) ( 7 ) ( 1 ) ( 4 ) ( 1 ) ( 4 ) ( 5
) ( 2
) ( 5 ) ( 1 ) ( 2 ) ( 1 ) ( 2 )

V.3) Balance of powers relating to

-The first, qualified as external, supposes ) 9 ( i known and operates the EM. It consists first of all in determining: ) 9 ( r i : -The second method, called internal, consists in programming the relations deduced from (19). This leads to the parameters ( k e components lead to the determinations of the instantaneous partial powers tributary, for given "k", of each value taken by "h" (See [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF]). Then, by summations,

) 9 ( ) 9 ( ) 9 ( a r i i i - = then: ) 9 ( " r i : ) 9 ( ) 9 ( ) 9 ( ' " r r r i i i - = with
9 11 13 15 17 -7 -5 -3 -1
T7: SAM -Definitions of ranks "k" according to the k' and "h" values -for k = 9, there is only one significant term relative to h = 4, the second being, by deduction, defined for h = 5. The values for k = 9 obtained with the internal method are therefore not significant since the maximum value retained for "h" is 4. The developments carried out clearly show the advantage of implementing the 2 methods simultaneously to detect and, possibly, correct an inconsistency.

VI) DISTRIBUTION OF POWERS FOR j = 9: EQUIVALENT SCHEMES

The analyzed case leads, concerning the harmonic powers (k> 1), to inequalities:

) 9 ( ) 9 ( k ilk p P << and ) 9 ( ) 9 ( ' k k q Q << µ . It follows,
apart from the numerical uncertainties, considering for example the active powers, that:

) 9 ( ) 9 ( ilk k P p +
can practically be confused with ) 9 ( k p (See T4). To remove the ambiguity as to certain equalities which will be formulated, the semi-analytical method method was exploiting by significantly increasing the harmonic iron losses by reducing the ilk R value (ratio of 20) which occurs in (A6). The results of this analysis are not presented in this article.

VI.1) Considerations on powers

This sub-paragraph presents the different relationships which characterize the behavior of this saturated MC. One can distinguish : -the classical mathematical formulations which govern the MC behavior and, more particularly, the variables accessible to the measurement.

-the relations which characterize the behavior of the "magnetizing inductance" which are, inter alia, deduced from the results obtained by the semi-analytical method. Pragmatically, these different relationships are based on the following general formalism:

       = =   k indk ind k indk ind Q Q P P (21)        + = + =   > > ) 1 ( 1 ) 1 ( 1 k k indk ind ind k k indk ind ind Q Q Q P P P (22)
where the lower indication "ind" corresponds to "ext", "int", ... VI.1.1) Preliminary considerations. The application of Boucherot's theorem leads to the following relationships:

     + = + = ) 9 ( ) 9 ( int ) 9 ( ) 9 ( ) 9 ( int ) 9 (
k k extk k k extk q Q Q p P P (23)
As 0

     > - = > - = 1 1 ) 9 ( ) 9 ( int ) 9 ( ) 9 ( int k for q Q k for p P k k k k (24)
As the for k>1 are negative quantities as shown by T4. The following equalities may also be formulated:

     = + = µ µ ) 9 ( ) 9 ( int ) 9 ( ) 9 ( ) 9 ( int k k k ilk k Q Q P P P (25) (9) μk P and
" k P µ (W) 1 1,97 (2) -0,20 (2)
-0,00 (2) 0,00 (2) 1,77 (8) 3 -0,55 (2) -0,82 (2) -0,00 (2) 0,00 (1) -1,37 (7) 5 -0,00 (2) -0,36 (2) -0,03 (1) -0,00 (1) -0,39 (6) 7 -0,00 (2) -0,00 (1) -0,02 (1) 0,02 (1) -0,01 (5) 9 0,00 (1) -0,00 (1) -0,00 (1) -0,08 (1) -0,08 (4) T10: SAM -Internal Method : Partial powers in W defining

) 9 ( " k P µ H H r H H H r H I E Q I E P ( 36
)
The numerical values of

) ( 1 " H P µ and ) ( 1 " H Q µ lead to ) 9 ( ) ( 1 " H r I = 0,017A.
Fig. 9 presents the variations on "T" of and its components

) 9 ( 1 " r i , ) 9 ( ) ( 1 " F r i , ) 9 ( ) ( 1 " H r i (A) t(ms) ) 9 ( ) ( 1 " F r i ) 9 ( ) ( 1 " H r i ) 9 ( 1 " r i
This synthetic distribution, which considers the powers affected by their signs as they appear in T4, is the subject of a practical use in the following subparagraph. This theoretical analysis made it possible to formalize more rigorously the behavior of this saturated MC knowing that, for a MC considered to be unsaturated, it suffices to assign zero values to the ) 9 ( k I quantities for k> 1 and to consider that ∞ → µ ) 9 ( " X .

VI.3) Practical aspects: Experimental considerations

The analysis considers the steady state and the variables will be presented without referring to any iteration. VI.3.1) Exploitation of the balance of powers. At the level of paragraph VI.1, a certain number of relationships have been established. However, no relationship defines int Q . As an application, this quantity is defined using the synthetic diagram of Fig. 11.

-The diagram (Fig. 11a), by counting the powers in the positive direction of transfer, shows that:

) ( 1 1 1 int " H il P P P µ + =
. To define k P int , consider Fig. 11c and count the powers in the opposite direction to the positive direction of transfer. This leads to:

k k p P - = int .
Considering the first relation of ( 22) and (28), it comes: -For reactive powers, the implementation of a similar approach, defines:

) ( 1 ) ( 1 1 1 int " " ' H F Q Q Q Q µ µ µ + + = and: k k q Q - = int .
Taking into account the second relation of ( 22) and (33), leads to:

) ( 1 int " ' F Q Q Q µ µ + = (38)
Numerical applications validate (38). VI.3.2) Partial experimental validation. The presented equations allow, without introducing new data, to partially exploit the experimental numerical values presented in T1 which concerns the toroidal MC. First, identify "r". The fact of not having numerical information concerning the harmonics of "i", requires to exploit the first relation of [START_REF] Steentjes | Iron-Loss Model With Consideration of Minor Loops Applied to FE-Simulations of Electrical Machines[END_REF]

- = .
1) One can note that the difference between il P determinated from (39) and the latter is of the order of 18% while T4 defines a difference of 15.5%. Note that this comparison of the deviations makes sense since T1 and T4 are defined using the same magnetic material.

2) The ) ( int P D difference is very small whatever the b ˆ value, in particular, for 1,5T b = ˆ, it is close to 0%. It is possible to assume that the fact of satisfying simultaneously two relations resulting from presented developments, constitutes a relatively significant element to validate the proposed theory.

Note that for b ˆ high values, the harmonic iron losses result in values of 1 il P which will be lower than those of 1 i P . Therefore, the difference between 1 int P et 1 il P will be greater than that announced, which is 18%. Thus, confuse fundamental iron losses with 1 int P in saturated state leads to committing a significant error on the evaluation of these losses.

The developments carried out do not reveal a relation which allows simple access to the ilk P values. On the other hand, their estimation requires theoretically determining the k P µ quantities in accordance with the developments presented in V. similar formulation would lead to the definition of an exponent on b ˆ which would be much greater than 2 which it would be difficult to justify. This simple observation has the merit of giving additional credit to the remark previously formulated regarding the validity of the presented theory.

VII) ENERGY BALANCE

Examination of T4 for j = 0 and 9 shows that: -From a practical point of view, it is very simple to predict whether certain components of internal active powers should have negative values or not. For this, it suffices to refer to paragraph VI and more particularly to the basic relations ( 23) and [START_REF] Fiorillo | An improved approach to power losses in magnetic lamination under nonsinusoïdal induction Waveform[END_REF]. They show that if certain harmonics of "i" are present when they do not exist at the level of the power supply, certain components of the internal power will have negative values. In fact, to consider that, under sinusoidal voltage, the current absorbed by the CM is not sinusoidal, is traditional. If despite this there has been no development at the scientific level, it is quite simply because, on a transformer for example, the magnetizing current is weak compared to the rated current. Generally, when the current harmonics manifest themselves a little too significantly, the notion of equivalent sinusoidal current is used. This justification may not be the only one because it is very difficult, at first sight, to appreciate how the harmonic powers are distributed. Indeed, this distribution obeys a rather surprising protocol which is very difficult to imagine. It is the exploitation of the numerical values obtained by the proposed semi-analytical method which allows us to glimpse this fairly complex distribution knowing that, from an experimental point of view, it is only possible to have information at the level internal powers without having a clue how these quantities are distributed.

T b 56 , 1 ˆ) 0 ( 1 = , T b 51 , 1

CONCLUSION

The developed semi-analytical method, which is based on the configuration of the traditional equivalent scheme, validates certain rather disconcerting results obtained during experiments on saturated single-phase MCs subjected to sinusoidal supply voltages. The mathematical exploitation of the variables thus obtained leads to characterize the specific behaviors generated by saturation allowing, therefore, to formulate some practical results: -Exercise a critical mind on measurements of iron losses as soon as the supply current begins to present perceptible distortions even if the flux density seems to be almost sinusoidal. Analysis indeed shows that disturbances on "b" which can be considered insignificant can have relatively significant consequences.

-Use the measurements in a relevant way to characterize unambiguously the iron losses.

-Define the procedure intended to characterize what is qualified as "magnetizing inductance".

-Appreciate the contribution of saturation on the performance of these CMs. This article, although based on experimental results, was mainly intended to theoretically justify unconventional behaviors such as, for example, the use as a support of fundamental powers to convey harmonic powers. More imaginatively, it is possible to justify these results by the simple fact that the MC, in the presence of saturation, adopts its behavior so as to be able to satisfy the basic rules of electromagnetism and of Kirchhoff's laws while respecting the mathematical definition which characterizes the active powers The partial verification presented seems encouraging enough to validate these developments and, in particular, to foresee many other avenues of scientific investigation. 12,14 0,00 0,00 0,00 0,00 12,14 11,51 0,00 0,00 0,00 0,00 11,52 11,39 0,01 0,01 0,00 0,00 11,41 20

) (j k P µ
-0,00 0,00 0,00 0,00 0,00 -0,00 -1,77 0,45 1,08 0,20 0,05 -0,00 1,78 -1,37 -0,39 -0,01 -0,01 -0,00 21

FigFig. 2 :

 2 Fig. 1: Toroidal MC

Fig. ] 5

 5 presents the calculated points which made it possible, in turn, to model )

Fig. 4 :

 4 Fig. 4: Characteristic b(H) of the M400 50A

Fig. 3 :Fig. 5 : 5 ZIII. 3 . 1 )

 35531 Fig. 3: Considered MC -Numerical values T2: Charactérisation by segments of the M400 50A µt(b)

  is this quantity which is imposed to launch the calculations during iteration 1. Finally, one compares

  v". Failure to comply with the imposed criteria leads to iteration 2. .............................. Iteration j.

  that, by opting to present the numerical values by considering only 2 decimal places, there is no perceptible difference between )

Fig. 6 :

 6 Fig. 6: SAM -Variations on "T" of b (j)

.

  The difference on the current peak values seems far too great in comparison with the difference which characterizes the flux density peak values.• For j = 0 and for k> 1:

  µ positive for k> 1.

  µ are negative for k> 1.

  power identifies with what EM gives.

  priori, taking into account the significant distortions which affect ) is possible to admit, having regard to the numerical uncertainties near and the high values of

η

  As the operating anomalies come from ) (j L µ , they are, in accordance with (A11), only linked to) ( j r µ .It is therefore necessary to analyze the variations of this variable as well as those of its inverse

  Fig. 7: SAM -Variations on "T" of ) (j r µ and ) (j r η for V=380V and j=0, 2 et 9

  )

•

  Components linked to <η η η η r >. Considering the MC given in Fig. in T6, it comes:

  µ are obtained.

  sinusoidal "v"), one can deduce 2 typical relations of this operating:

Q

  µ resulting from the above considerations. Insofar as 0

  Fig. 9: SAM -) 9 ( 1 " r i and its components

P

  deduced from the measurements is very small up to 1,3T and that it then increases very quickly as a function of b ˆ. The plot of the curve giving this )

3 .

 3 However, at this stage, these cannot be undertaken due to a lack of information concerning certain variables and their variations with measured b ˆ values. Nevertheless, T4 shows that the harmonics of "b" for 1,5T b = ˆ are very weak as well as the ilk P quantities. It is therefore possible, as a first approximation, to confuse il P and 1 il P what, implicitly, leads to admitting sinusoidal variations of "b". Assuming that this condition is satisfied over the entire range of b ˆ variations presented at the level of T1, it is possible to estimate a variation. By considering a formulation of the form: by successive approximations makes it possible to define the constants κ et ϑ : κ=3,68 and ϑ=1,7. The last line of T11 presents the values of the ϑ value, seems rather coherent insofar as the literature considers 3 categories of iron losses: dynamic losses which involve 2 ) (b , static losses function of ξ ) (b (with the Steinmetz coefficient ξ varying from 1,6 to 2,2) and the excess losses due to

  -V=380V -Numerical values obtained by the semi-analytical method on a saturated MC for j = 0, 2 and 9ANNEX B

3 )

 3 Given the non-linear effects, is Boucherot's theorem still applicable? In view of the previous remark concerning

k P int for k> 1, how to characterize k Q int relative to µ L for k> 1? Are these quantities positive or negative? Although the values obtained are not shown in T1, experiments show that:

  [START_REF] Lopez | Grain oriented Steel Assembly Characterization for the Development of High Efficiency AC Rotating Electrical Machines[END_REF]

				) 0 2 ˆh r ( µ			6167,7 2859,4 3777,4 1761,2 241,7
	/	µ	( r	) 2h 0	/	(°)	0,00	180,0	180,0	180,0
	) 2 10 0 * ˆh r ( η	4	2,84	2,34	2,43	0,70	0,87
	/	η	( r	) 2h 0	/	(°)	180,0	0,00	180,0	0,00
				) 2 9 ( ˆh r µ			6608,8 2119,7 3532,0 1773,7 190,9
	/	µ	( r	) 2h 9	/	(°)	354,5	172,7	171,2	174,1
	) 9 2 10 * ˆh r ( η	4	1,99	0,77	1,20	0,17	0,32
	/	( r η	) 2h 9	/	(°)	174,3	351,8	2,72	344,9
	T6: SAM -Coefficients of	( FS µ	) (j r	)	and of	( FS η	) (j r	)

  which leads to:

													P ext	-	int P	=	r	) (I	2	. It can be deduced that: r = 0,285Ω. The first relation of (24) makes it
	possible to determine the k I values for k> 1 as well as the k p quantities. These currents k I for k>1 and " I " define 1 I in
	accordance with (A17). For	1,48T ˆ, it comes: b =	I 1 =	2,44A	,	,54A I 3 1 =	,	I 5 0 =	,79A	,	I 7 =	0,42A	,	I 9 =	0,19A	.
	By noting that (30) can still be written:		int P	1	=	il P	+	) 1 >  ( k k p	k	(39)
	one can deduce il P . If the analytical developments presented are correct, it should be noted, in accordance with (31), that
	equality	P il =	int P	must also be satisfied
	Are grouped in T11, according to the b ˆ data values given in T1, the measured values of int P and	int P	1	as well as the
	S	=	) 1 >  (k k p	k	and il P quantities deduced from the proposed theory. Fig. 12 shows the	1 P , il int P and int P variations with b ˆ. Are
	also calculated the relative deviations in absolute values expressed in % of	D	P ( int	1	)%	=	100	il P	-	P int	1	/	il P	and of
	D	P ( int	)%			100	il P	P int	/	il P

The internal method frees itself from EM and from ) 9 ( ' µ X but it is more difficult to implement insofar as it requires programming relationships deduced from (19). The advantage of having recourse simultaneously to the 2 proposed methods makes it possible, for example, to validate the EM both in terms of the fundamental and the harmonics. V. 3.1) Results obtained by the external method. Fig. 8 represents the variations on "T" of ) 9 ( r i and of its 2 components. T8 specifies the numerical values of the coefficients of the FSs of these currents. It is possible to notice, for k = 1, that , to values practically identical to those which appear in T8 for k ranging between 1 and 7 with a problem for k = 9. The powers calculated with this method also show good agreement with those given in T9 with always a problem for k = 9. Although the numerical values for k = 9 are small, the discrepancies noted are too substantial to attribute them to the numerical inaccuracies. To determine where the error came from, T10 presents, for each value of "k", the partial powers dependent on the ) 9 ( 2 ˆh r η quantities determined by integrating the instantaneous powers calculated from (19). The non-shaded numerical values given in parentheses in T10 specify the numbers of terms, deducted from T7, which make up each row (See. [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF]). The shaded numerical values in parentheses in the last column give the total number of terms involved in the definition of terms of rank "k". Taking into account the values which have been retained for "k" and "h", T10 shows that:

-the number of total terms which define ) 9 ( " k P µ at given "k" decreases when "k" increases, -at given "k" between 1 and 7, the number of significant terms (values marked in bold) is 2 and that these terms appear for values of "h" which increase when "k" increases, -these significant terms occur in pairs corresponding to h and h +

' k Q µ defined by [START_REF] Cossale | Modeling the Degradation of Relative Permeability in Soft Magnetic Materials[END_REF].

VI.1.2) Active powers. The first relations of ( 24) and (25) as well as (26) lead to define In accordance with [START_REF] Amar | A general formula for prediction of iron losses under non sinusoidal voltage waveform[END_REF], it comes:

Insofar as 0 " ) 9 ( = µ P (Cf. T4), one can deduce that:

" "

Let us consider the first relation of ( 25) defined for k = 1. Given (29), it is written: Taking into account the first relation of [START_REF] Amar | A general formula for prediction of iron losses under non sinusoidal voltage waveform[END_REF], by considering (30), ( 27) and ( 24), leads to:

equality which is verified if T4 is considered. Numerical applications define: 1,78W P" (9) μ1(H) = .

VI.1.3) Reactive powers. Examination of T4 and T9 shows that

By analogy with (28), let us introduce the variable:

As 612,53Ω X' (9) μ = , numerical applications lead to:

One does not find an equality as that given by (29). Let us introduce

defines as a component of fundamental magnetizing power linked to the effects of saturation and therefore to the ) 9 ( r η harmonics. This quantity results from equality:

" r i will also be characterized using 2 terms:

, just use the relationships:

VI.2) Equivalent schemes -Balance of powers

The saturated MC presents a fairly specific behavior which responds to the following concept: these are the fundamental power components

, the active and reactive powers required by natural harmonics (k> 1) so that they can manifest themselves at the MC level. This transfer of powers is accompanied by conversions, in terms of voltages and frequencies, carried out using a "converter" which is none other than the magnetic material itself. VI.2.1) Equivalent schemes. This rather unusual concept is reflected in the equivalent schemes in Fig. 10. The part consisting of the association of blocks "a" and "b" corresponds to the complete structure and shows that the exploitation of the variables relating to the fundamental (block a) requires knowledge of the powers consumed by the harmonics (block b).

These are identified for k> 1 to

according to ( 27) and (32) . The characterization of these quantities is based on equivalent schemes relating to harmonics (k> 1) presented in Fig. 10c. The fact of considering these powers as leaving the "converter" (gray area) indicates that these must be counted negatively at the level of

It should be noted that the box referenced: "additional part linked to saturation" in Fig. 10a only depends on the ' µ X characterize the reactance associated with µ L presented in Fig. 2. It appears that saturation significantly complicates the function that this reactance must perform.

VI.2.2) Balance of powers.

In order to concretize these different relationships in a more practical way, Fig. 11 presents the synthetic distribution of powers by considering the sets a (fundamental) and c (harmonics) of Fig. 10. Let us adopt as a convention a positive sense of power transfer when the latter flow from the supply (v) to the load (MC). Taking into account the concept previously stated, the magnetic core behaves like a load with respect to the supply so that the fundamental components of powers are counted positively. This same magnetic core behaves as a generator for harmonics (k> 1) insofar as the harmonic powers flow from the magnetic core to the loader made up of the power supply which identifies with a short circuit for these harmonics. This transfer of harmonic powers must be counted negatively. When one considers an electrical motor (or a generator) and that it is characterized by its power (λkW) then the Joule losses, for example, correspond to a percentage of λ (per unit system). These Joule losses will therefore be counted positively if λ is positive and negatively if it is negative. As the MC simultaneously performs the motor and generator functions and the convention for counting powers is unique, the fundamental Joule losses are counted positively and the harmonic Joule losses, although positive, are counted negatively. This is also true for the equivalents in terms of reactive powers ( ) 9 ( 1 q and = .

These differences have repercussions on the overall quantities by increasing. Considering for example "i" it comes: A î 13 , 4

) 0

) 9 ( = (See remark formulated in IV.2). One thus notes a certain advantage for the case j = 9 with incidences in terms of harmonics on "b" and potential generated negative effects (on the no-load voltage of a single-phase transformer for example) which will be relatively weak. If these natural harmonics impact, by the phase shifts they introduce, the powers carried by the fundamental and therefore the currents, they also contribute, through > η < j r , to this balance by acting on ) (j L µ defined by [START_REF] Cheng | Modeling of magnetic properties of g.o. electrical steel based on Epstein combination and loss data weighted processing[END_REF]. As:

L µ is lower for j = 0 than for j = 9 which leads to:

This analysis is only an observation which requires more specific developments to be validated. Nevertheless, this offers at this stage possible new perspectives in terms of scientific developments.

VIII) POSITIONING OF THE PROPOSED ANALYSIS

A priori, given the absence of scientific reference concerning the analysis carried out, it seems logical to ask the question of knowing if this abnormal experimental behavior does not correspond to a specific case. In fact, it is possible to affirm that it is a natural phenomenon which affects all the magnetic circuits used for the manufacture of electric machines when the saturation affects the magnetic material. This therefore concerns single-phase and three-phase transformers as well as rotating AC machines. The following elements provide certain justifications for these claims.

-The first argument is related to what was clarified in the introduction, namely the numerous experiments that were carried out in the laboratory by considering various magnetic materials (isotropic, anisotropic sheets, of different qualities, of different thicknesses, etc.) . The bibliography which concerns us and presented in this article shows that indeed the investigations in which we participated, were not limited to a single single-phase static structure. Each time we faced quite surprising results which, in some cases, did not manifest as dramatically as what appears in T1 for j = 9 (see the case j = 2 in T1). At the level of the presentation of articles and PhD dissertations, we avoided putting forward the results obtained which were similar to those which appear in T1 on the one hand, because we did not know how to justify them and, on the other hand, because that did not enter within the framework of the investigations undertaken which concerned the fundamental quantities. There is nevertheless at least 1 article which raises this problem, it is the reference [START_REF] Parent | Analysis of the magnetic flux distribution in a new shifted non-segmented grain oriented AC motor magnetic circuit[END_REF] which analyzes the impact of the PWM on iron losses. We imposed on a toroidal type device a given 50Hz sinusoidal fundamental voltage to which was superimposed a Hf sinusoidal voltage of imposed amplitude and of variable frequency. Under saturated conditions, experiments showed that, which was supposed to represent the fundamental iron losses, varied significantly when the switching frequency changed. I am practically persuaded to this day that these rather disconcerting variations of

P

could, at least partially, be explained using the theory presented.

-The second argument is based on a fairly simple observation, namely that the developments exposed differ from traditional considerations by the simple fact of taking µ r variable into account over a period of time instead of considering it constant. In order for these operating "anomalies" to manifest themselves in a sufficiently significant way, the operating point, in instantaneous values must go beyond the peak of the curve µ r (b) in Fig. 5. Indeed, these anomalies exist even when the operating point only moves on the "linear" part of µ r (b) (sections 1 and 2 in Fig. 5) but the effects are very weak and therefore ignored because, from a an experimental point of view, they are the order of measurement errors. It is therefore the harmonics of µ r which generate harmonic components at the level of magnetizing currents ((19), [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF]) which add to those produced by X' µ which are well 90 ° phase shifted with respect to the components of corresponding emf. At given "k", when these 2 types of components are added, then the phase shift of the resulting component of the magnetizing current takes, for example, a value of 90 °+ε. For ε> 0, the active power which was zero becomes negative. It should be emphasized that these current components dependent on the harmonics of µ r are, for k> 1 and compared to the components related to <µ r >, preponderant for k> 1 (Cf. T8).

For example, for j = 9, if (A3) shows that . This relatively simple explanation which shows that the "magnetizing inductance" consumes active power shows that the case considered is not a particular case. On

• The next step is to characterize ) ( j i which will be defined from its 2 components (See Fig. 2):

The expression which defines ) ( j dynk P as well as (A4), show by identification that the quantities ilk R are, ∀k, all identical. They will therefore be simply noted il R with:


Patrick Brest

Sadal Refae

Baharia Mograbi

Paul Hofman

Gerard Milano
email: gerard.milano@nice.unicancer.fr

Host Polymorphisms May Impact SARS-CoV-2 Infectivity

Keywords: pharmacogenetics, SNP, genetic variation, coronavirus, infection

Based on a broad public database compilation, we support the hypothesis that germinal polymorphisms may regulate the expression of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cellular target itself and proteases controlling the process of its shedding or, conversely, its internalization. Consequently, a genetic influence on individual susceptibility to COVID-19 infection is strongly suspected.

General Background

. Thus, in this context, here we develop two complementary themes: (i) a more thorough examination of the membrane shedding of angiotensinconverting enzyme 2 (ACE2), the SARS-CoV-2 cellular target, and its potential repercussion on virus propagation; and (ii) a description of the interindividual variability of the genes (SNPs) involved in ACE2 processing and their potential impact on the risk of contracting COVID-19.

ACE2 Expression and COVID-19

. The authors stressed that, counterintuitively, expression

of the SARS-CoV-2 target was inversely related to certain risk factors, showing higher levels in Asian females compared with Asian males and a significant decrease in patients with type 2 diabetes mellitus. Globally, at a population level, there was a negative correlation between ACE2 expression and COVID-19 severity. Recent data provide evidence that ACE2 is effectively shed from membranes, a process that is finetuned at different levels [START_REF] Heurich | TMPRSS2 and ADAM17 cleave ACE2 differentially and only proteolysis by TMPRSS2 augments entry driven by the severe acute respiratory syndrome coronavirus spike protein[END_REF] involving two cell membrane proteases: disintegrin and metalloproteinase domain-containing protein 17 (ADAM17) and transmembrane protease serine 2 (TMPRSS2) [START_REF] Heurich | TMPRSS2 and ADAM17 cleave ACE2 differentially and only proteolysis by TMPRSS2 augments entry driven by the severe acute respiratory syndrome coronavirus spike protein[END_REF]. More precisely, ADAM17 acts directly on ACE2 and leads to ACE2 shedding into the extracellular cellular space, while TMPRSS2 cleaves not only ACE2, but also the S protein of SARS-CoV-2, thus leading to membrane fusion and cellular uptake of the virus. Consequently, while ADAM17 and TMPRSS2 both act on ACE2, they may have opposite effects on net ACE2 shedding (Figure 1). When the respective proteolytic activities of ADAM17 and TMPRSS2 result in more ACE2 shedding than internalization, it follows that this situation may constitute a natural barrier to infection. This could be due to the interaction between soluble ACE2 with the virus at a distance from sensitive tissues. Subsequently, on this basis, one can hypothesize that, when the viral load is high, the shedding barrier effect is overwhelmed, thus facilitating subsequent infection.

ACE2, TMPRSS2, and ADAM17 Gene Polymorphisms

Cao et al. [START_REF] Cao | Comparative genetic analysis of the novel coronavirus (2019-nCoV/SARS-CoV-2) receptor ACE2 in different populations[END_REF] compiled a database analysis of all 1700 variants in the region of the ACE2 gene located on the X chromosome. They identified 15 unique expression quantitative trait loci (eQTLs; see Glossary) variants [14 SNPs and 1 insertion/deletion (INDEL)] with higher minor allele frequencies (MAF) in the Asian population than in a European population (MAF of 0.05 versus 0.35-0.48 for the top six most common variants). Interestingly, their data suggested that the 11 most common variants (MAF >0.05) were associated with increased expression of ACE2 in tissues, suggesting, according to the authors, a different sensitivity to SARS-CoV-2 infectivity. However, the functional basis of the influence of SNP on ACE2 expression remains to be established.

In this context, we performed a complementary in silico study including SNPs regulating gene expression not only for ACE2, but also for ADAM17 and TMPRSS2 (Table S1 in the supplemental information online).

Overall, and based on the ACE2 expression-associated MAF between ethnic populations, it appears that Asians express a higher level of ACE2 than whites, while Blacks show an intermediary level of ACE2 expression. This is consistent with the findings previously reported by Cao et al. [START_REF] Cao | Comparative genetic analysis of the novel coronavirus (2019-nCoV/SARS-CoV-2) receptor ACE2 in different populations[END_REF]. It is still debatable whether these differences should be taken into consideration in epidemiological studies on COVID-19 covering ethnic associations with disease occurrence [START_REF] Yi | COVID-19: what has been learned and to be learned about the novel coronavirus disease[END_REF]. Importantly, the diseases associated with a high level of SARS-Cov-2 infection (hypertension and diabetes) were found to be related to a lower expression of ACE2, in relation to the respective allelic distribution. This relationship concurs well with the study by Chen et al. pointing towards a negative correlation between ACE2 expression and COVID-19 severity [START_REF] Chen | Individual variation of the SARS-CoV-2 receptor ACE2 gene expression and regulation[END_REF].

It has been reported that subjects with rs383510/T and rs2070788/G genotypes of TMPSRSS2 located on chromosome 21q22.3 are more prone to develop a severe form of A (H1N1) influenza and acute respiratory distress syndrome [7]. Of note, males have been shown to be more likely to develop a severe form of H1N1 influenza and there is evidence that androgens are positive regulators of TMPRSS2 [START_REF] Lin | Prostate-localized and androgen-regulated expression of the membrane-bound serine protease TMPRSS2[END_REF]. Importantly, the alleles at risk (T for rs383510 and G for rs2070788) are linked to increased gene expression (Table S1 in the supplemental information online), logically supporting the hypothesis of a higher level of viral cell entry. It is tempting to extrapolate this SNP influence to SARS-CoV-2 infectivity.

The ADAM17 locus on chromosome 2p25.1 presents two clusters and three unique SNPs that induce strong differences in terms of allelic profiles between Asian and European populations that are associated with hypertension [START_REF] Li | Association between ADAM17 promoter polymorphisms and ischemic stroke in a Chinese population[END_REF] and/or sepsis [START_REF] Shao | Association study between promoter polymorphisms of ADAM17 and progression of sepsis[END_REF]. Of note, most of these SNPs are located in the promoter region of ADAM17 and are associated with either positive or negative eQTL, depending on the SNP and the tissue (Table S1 in the supplemental information online). Therefore, there is a strong possibility that genetic polymorphisms influencing ADAM17 expression also contribute to the modulation of ACE2 shedding intensity.

Practical Consequences

Taken together, the above-discussed data advocate in favor of a multifactorial genetic impact on the risk of SARS-Cov-2 infectivity and possible disease severity. A relatively simple and easy-to-perform test, such as quantitative PCR [START_REF] Katsanis | Molecular genetic testing and the future of clinical genomics[END_REF] or MASSarray [START_REF] Refae | locus (eQTL): a genomic locus that explains the variation in gene expression of nearby genes. Insertion/deletion (Indel): an insertion or deletion of bases in the genome. Linkage disequilibrium: nonrandom association of alleles at two or more loci in a general population. When alleles are in linkage disequilibrium, haplotypes do not occur at the expected frequencies. Minor allele frequency (MAF): frequency at which the second most common allele occurs in a given population[END_REF], would allow large-scale individual SNP profiling for ACE2, ADAM17, and TMPRSS2 to identify possible at-risk populations vulnerable to viral infection. On this basis, a 'multiSNPs risk score' could be established that would be applicable to large populations and, thus, it might then be possible to identify subjects carrying a combination of favorable alleles for ACE2, ADAM17, and TMPRSS2 conferring a lesser risk of contracting SARS-Cov-2 infection, and vice versa. Such an analytical strategy was recently developed based on patient genetic characteristics for immunogenetic profiling designed to personalized immunotherapy [START_REF] Refae | locus (eQTL): a genomic locus that explains the variation in gene expression of nearby genes. Insertion/deletion (Indel): an insertion or deletion of bases in the genome. Linkage disequilibrium: nonrandom association of alleles at two or more loci in a general population. When alleles are in linkage disequilibrium, haplotypes do not occur at the expected frequencies. Minor allele frequency (MAF): frequency at which the second most common allele occurs in a given population[END_REF]. A similar supervised genetic approach to COVID-19 risk assessment could complement the current unsupervised GWAS investigations, which require large population studies exploring the whole patient genome for DNA variations in an attempt to explain interindividual differences in COVID-19 severity [e.g., the Howard Hughes Medical Institute (HHMI) genetic project i and the COVID-19 Human Genetic Effort [START_REF] Casanova | A global effort to define the human genetics of protective immunity to SARS-CoV-2 infection[END_REF]). Ideally, in a final step, a multifactorial predictive index could be established incorporating SNP analysis and other, more established, risk factors.

In summary, until now, genetic influences on COVID-19 interindividual susceptibility have been largely underestimated; thus, we hope that the discussion might fill this gap and will pave the way for confirmatory investigations at experimental and clinical levels.

Figure 1 .

 1 Figure 1. Graphic Support for the Hypothesis of Angiotensin-Converting Enzyme 2 (ACE2)-Related Virus Neutralization at Distance due to the Presence of Enzymatic Membrane Mechanisms Regulating ACE2 Shedding and Virus Entry. Abbreviations: ADAM17, disintegrin and metalloproteinase domain-containing protein 17; SARS-CoV-2, severe acute respiratory syndrome coronavirus 2; TMPRSS2, transmembrane protease serine 2.

Acknowledgments

The authors would like to thank Jocelyn Gal, Emmanuel Chamorey, George Morgan, and Christiane Brahimi-Horn for helpful comments. Funding is acknowledged from the French Government (Agence Nationale de Recherche, ANR) through the 'Investments for the Future' LABEX SIGNALIFE (ANR-11-LABX-0028-01) and (AD-ME project R19162DD); CANC'AIR Genexposomic project, Canceropole PACA; DREAL PACA, ARS PACA, Région Sud, INSERM cancer; INCA Plan Cancer; and Children Medical Safety Research Institute (CMSRI, Vaccinophagy project R17033DJA).

Resources

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