Thierry Lubin

Smail Mezani

Abderrezak Rezzoug

Analytic Calculation of Eddy Currents in the Slots of Electrical Machines: Application to Cage Rotor Induction Motors

Keywords: Eddy currents, analytical solution, diffusion equation, rotor bars, induction motor

This paper describes a two-dimensional analytic model for predicting the induced currents within conducting material placed in the slots of electrical machines. Slot-embedded conductors are subjected to a rotating magnetic field including space harmonics. The analytical method is based on the resolution of time-harmonic homogeneous Helmholtz's equation in polar coordinates by the separation of variables method. The global solution is obtained using boundary and continuity conditions between the different subdomains (rotor slots and air-gap). The proposed model is applied for magnetic field and torque calculations in a rotor cage induction motor under steady state condition. The analytical results are verified with those obtained from finite element analyses.

I. INTRODUCTION

CCURATE modeling of eddy current effects is important to design electrical machines and especially to calculate the torque-speed curve of induction motors. Eddy-currents can be evaluated by numerical or analytical methods. With numerical methods [START_REF] Salon | Finite Element Analysis of Electrical Machines[END_REF]- [START_REF] Mezani | Saturation and space harmonics in the complex finite element computation of induction machine[END_REF], complex geometric shapes (as rotor bars in induction motor) and non-linear effects (saturation of the magnetic parts) can be taken into account. However, these methods are computer time consuming and are not suitable for the first step of design stage of electrical machines. To be fast and efficient, analytical methods suppose simple geometries (conducting plates or cylinders) and linear magnetic behavior [START_REF] Tegopoulos | Eddy Currents in Linear Conducting Media[END_REF][START_REF] Stoll | The Analysis of Eddy Currents[END_REF]. Due to their low computational time, they are often preferred for the first evaluations of electrical motors performances and for design optimization.

Two-dimensional (2D) analytical models for induced currents prediction in induction motors (IM) have been proposed in the literature. They deal with solid-rotor topology (cageless rotor) which present cylindrical symmetry [START_REF] Chalmers | General theory of solid-rotor induction machines[END_REF]- [START_REF] Shah | Rapid analytical optimization of eddycurrent shield thickness for associated loss minimization in electrical machines[END_REF]. The induced currents distribution in the rotor (conducting cylinder) is obtained by solving the magnetic diffusion equation in polar coordinates. Analytical models are also proposed for the evaluation of slot impedance of a rectangular conductor placed in a semi-closed slot [START_REF] Swann | Effective resistance and reactance of a rectangular conductor placed in a semi-closed slot[END_REF][START_REF] Binns | The Analytical and numerical solution of electric and magnetic fields[END_REF]. However, the models are developed for a single rotor bar (with imposed net current), so the interaction between neighbouring bars is ignored. These models are well suited for the determination of equivalent circuit parameters of IM.

Exact analytical models for the magnetic field prediction in the slots of electrical machines have been proposed recently [START_REF] Ackermann | Analytical modeling of the cogging torque in permanent magnet motors[END_REF]- [START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF]. However, they deal with permanent magnet machines and are based on the magnetostatic approach which neglects the eddy-current reaction field. To the author's knowledge, there has not been anything reported in the literature on 2D exact analytic models for the prediction of current density induced in the rotor bars of squirrel cage induction motor.

In this paper, we present an original analytical model for eddy-currents computation within conducting material placed in rotor slots of electrical machines and subjected to a rotating magnetic field. Laplace's and Helmholtz's equations are solved respectively in the air-gap and in the rotor slots regions. The global solution is obtained using boundary and interface conditions between the different regions. The proposed analytical method is used to investigate the eddy current distribution in the rotor bars of an induction motor.

II. PROBLEM DESCRIPTION AND ASSUMPTIONS

A. Problem Description

The cross-section of the studied machine is shown in Fig. 1. The rotor contains Q slots (Q = 6 in Fig. 1) with conducting material (rotor bars) of electrical conductivity σ and magnetic permeability µ 0 . We consider that the rotor bars are permanently shorted to each other through end rings. The geometrical parameters are the inner radius of the rotor slots R 1 , the radius of the rotor surface R 2 and the stator bore radius R 3 . The slot opening angle is β. In a frame linked to the rotor, the angular position of the i-th rotor bar is defined as

2 2 i i Q β π θ = -+ with Q i ≤ ≤ 1 (1)
In order to simplify the theoretical analysis, we assume that the source term is represented by a current sheet distributed over the stator inner radius (r = R 3 ). It is well-known that the stator ampere-conductor can be approximated by an equivalent current sheet disposed over the slot openings. If the stator currents are sinusoidal in time, the expression of the current sheet (A/meter) for the m-th space harmonic component is ( )

( , ) cos m s m s s J t J mp t θ θ ω = - (2)
where J m is the peak value of the m-th harmonic component, p is the number of pole pairs, θ s is the angular displacement on the stator bore, ω s is the angular frequency of the stator supply, t is time.

For a balanced three-phase stator windings distribution with an integer number of slot per poles and per phase, the m-th space harmonics component is given by

1 6 m l = ± where 0,1, 2,3.... l = (3)
The peak value of the m-th harmonic component of the current sheet is 3 [START_REF] Vassent | Simulation of induction machines using complex magnetodynamic finite element method coupled with the circuit equations[END_REF] 2

ph m wm N I J K R π = (4)
where N ph is the number of series turns per phase and I is the stator RMS current. The winding factor K wm associated with the m-th harmonic is given by

( ) ( ) ( ) ( ) ( ) sin 6 sin 2 sin 2 sin 2 2 sin 6 wm m m K m m m q m q π ε π λ ε π = (5)
where q is the number of slots per pole per phase, λ is the coil pitch angle, ε is the width of the slot opening in radians (λ and ε are electrical angles).

Each harmonic component of the current sheet produces a rotating magnetic field which rotates forward (m=1,7,13…) or backward (m = -5, -11,…) with respect to the stator with an angular mechanical speed given by

s sm mp ω Ω = (6)

B. Assumptions

The model presented here is based on the following main assumptions

• The machine is infinitely long (end effects are neglected).

• Stator and rotor iron cores are infinitely permeable (saturation effect is neglected) and the stator core is slotless.

• The rotor iron circuit is perfectly laminated so that eddy currents are negligible.

• The rotor slots have radial sides as shown in Fig. 1.

• The current density has only one component along z-axis.

• The stator windings are fed with sinusoidal current of angular frequency ω s .

• The angular velocity of the rotor noted Ω is constant.

• The electrical conductivity of the rotor bars is constant.

As shown in Fig. 1, the whole domain is divided into 1+Q subdomains: the airgap subdomain (regions I) which has an annular shape and Q rotor bars subdomains (regions i). The ith rotor bar subdomain is shown in Fig. 2. To describe the problem, a magnetic vector potential formulation is used in 2D polar coordinates. According to the adopted assumptions, the magnetic vector potential has only one component along the zdirection and only depends on the r and θ coordinates. The magnetic vector potential in the different subdomains is noted as ( , , )

C. Current Sheet Expression in the Rotor Reference Frame

If the rotor moves with a constant angular velocity Ω, the current sheet expression (2) can be transformed from the stator reference frame (angle θ s ) to the rotor reference frame (angle θ) such that

s t θ θ Ω = + (7) which leads to ( ) ( , ) cos m m rm J t J mp t θ θ ω = - (8)
where ω rm is the angular frequency of the rotor currents induced by the m-th harmonic rotating field. Its expression is given by ( ) ( )

1 1 rm s m s ω ω = - - with 1 1 s s Ω Ω = - (9)
s is the rotor slip for the fundamental component of the rotating field (m = 1) and Ω s1 is given by [START_REF] Tegopoulos | Eddy Currents in Linear Conducting Media[END_REF].

In the next subsections, the magnetic vector potentials in the air-gap and rotor slots sub-domains will be computed in the rotor reference frame for each space harmonic component. According to the form of the source term (8), the magnetic vector potentials in the whole space vary sinusoidally with time at an angular frequency ω rm . Therefore, it is convenient to use the complex notation

{ } ( , , ) ( , ) rm j t I I A r t A r e ω θ θ = ℜ (10) { } ( , , ) ( , ) rm j t i i A r t A r e ω θ θ = ℜ (11)
where ℜ denotes the real part of the complex number and 1 j = -. Using complex notation, the m-th harmonic component of the current sheet [START_REF] Chalmers | General theory of solid-rotor induction machines[END_REF] becomes

{ } ( , ) ( ) rm j t m m J t J e ω θ θ = ℜ with ( ) jmp m m J J e θ θ - = (12)

III. ANALYTICAL SOLUTIONS

A. Solution of Helmholtz's Equation in the i-th Rotor Bar

The i-th rotor bar sub-domain and the associated boundary conditions are shown in Fig. 2. From the quasi-static Maxwell's equations, we have to solve the complex Helmholtz equation ( 13) in a domain of inner radius R 1 and outer radius R 2 delimited by the angles θ i and θ i +β

2 2 0 2 2 2 1 1 i i i rm i A A A j A r r r r ω σµ θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (13)
The tangential component of the magnetic field at the sides and at the bottom of the rotor bars is zero (the iron is infinitely permeable). In terms of magnetic vector potential, the boundary conditions are

0 i i A θ θ θ = ∂ = ∂ and 0 i i A θ θ β θ = + ∂ = ∂ ( 14
) 1 0 i r R A r = ∂ = ∂ (15)
The continuity condition between the i-th rotor bar and the air-gap leads to

2 2 ( , ) ( , ) i I
A R A R θ θ = (16)
The general solution of ( 13) can be found by using the method of the separation of variables. The magnetic vector potential is expressed as the product of two functions, each of them having only one variable ( , ) ( ) ( )

i i i A r r θ ρ Θ θ = (17)
Fig. 2. i-th rotor bar domain with its boundary conditions.

Substituting ( 17) into (13), we obtain two ordinary differential equations where λ is the separation constant [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF] " 0

i i Θ λΘ + = (18)
( )

2 2 2 " ' 0 i i i r r r ρ ρ α λ ρ + + - = (19)
where 2

0 rm j α ω σµ = - , (19)
is a Bessel differential equation. Using [START_REF] Zhu | An accurate subdomain model for magnetic field computation in slotted surface-mounted permanent magnet machines[END_REF], the boundary conditions ( 14) become '( ) 0

i i Θ θ = and '( ) 0 i i Θ θ β + = (20)
We are confronted to the problem of finding the values of the parameter λ for which non-trivial solutions of [START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF] satisfying the boundary conditions [START_REF] Meeker | Finite Element Method Magnetics[END_REF] exist. This is a standard Sturm-Liouville problem [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]. The eigenvalues of the problem ( 18), ( 20) are

0 0 = λ (21) 2 k kπ λ β   =     with ,..... 3 , 2 , 1 = k (22)
The eigenfunctions corresponding to 0 λ and k λ are given by 0 ( )

1 i Θ θ = (23)
( ) cos ( )

ik i kπ Θ θ θ θ β   = -     ( 24
)
For λ 0 and λ k , the general solutions of ( 19) are ( ) ( ) Writing the general solution as a linear combination of the previous solutions, one gets

0 0 0 0 0 ( ) i i i r A J r B Y r ρ α α = + (25) ( ) ( )
( ) i i ik k k k k r A J r B Y r π β π β ρ α α = + (
( ) ( ) ( ) ( ) ( ) 0 0 0 0 1 ( , ) cos ( ) i i i i i k k k k i k A r A J r B Y r k A J r B Y r π β π β θ α α π α α θ θ β ∞ = = +   + + -     ∑ (27)
Considering the boundary condition [START_REF] Ackermann | Analytical modeling of the cogging torque in permanent magnet motors[END_REF] and the continuity condition [START_REF] Gysen | General formulation of the electromagnetic field distribution in machines and devices using Fourier analysis[END_REF], the general solution of the magnetic vector potential in the i-th rotor bar domain can be rewritten as ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 0 0 0 2 1 0 2 2 2 1 ( , ) cos ( ) i i k k k i k i k k k k J r F Y r A r C J R F Y R J r F Y r k C J R F Y R π β π β π β π β α α θ α α α α π θ θ β α α ∞ =   - =     -   -   + -   -   ∑ (
1 1 1 1 1 J R F Y R α α = (29) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 k k k k k k J R J R R F k Y R Y R R π β π β π β π β π α α α β π α α α β + + - + = - + (30)
The integration constants 0 i C and i k C are determined using a Fourier series expansion of the air-gap magnetic vector potential ( , )

I A r θ over the slot interval [θ i , θ i +β ] at r=R 2 . 0 2 1 ( , ) i i i I C A R d θ β θ θ θ β + = ∫ (31) 2 2
( , ) cos ( )

i i i k I i k C A R d θ β θ π θ θ θ θ β β +   = -     ∫ (32)
The expressions for the integration constants 0 i C and i k C are developed in the appendix.

B. Solution of Laplace's Equation in the Air-Gap

The air-gap domain and the associated boundary conditions are shown in Fig. 3. We have to solve the Laplace equation in an annulus of inner radius R 2 and outer radius R 3

I I I A A A r r r r θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (33)
Considering the current sheet ( ) m J θ (12) and an infinite permeability of the stator back iron, the boundary condition at

3 R r = is 3 0 ( ) I m r R A J r µ θ = ∂ = ∂ ( 34
)
where µ 0 is the vacuum permeability.

Due to the presence of the rotor slots, the boundary condition at 2 r R = is more complex. It can be written as follows [START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF] 2 ( )

I r R A f r θ = ∂ = ∂ (35)
0 i i i r R A r f elsewhere θ θ θ β θ =  ∂ ∀ ∈ +   ∂ =     (36)
where ( , ) i A r θ is the magnetic vector potential in the ith rotor bar given by (28).

Taking into account the boundary conditions (34) and (35), the general solution of the magnetic vector potential in the airgap can be written as ( )

( ) 0 3 3 2 2 2 3 3 2 1 3 3 2 2 2 3 3 2 1 ( , ) ( , ) ( , ) cos ( , ) ( , ) ( , ) ( , ) sin ( , ) ( , )
I I I I n n n n n n n I I n n n n n n n A r A r R R r R R A B n n R R n R R r R R r R R C D n n R R n R R θ Ρ Ρ θ Ε Ε Ρ Ρ θ Ε Ε ∞ = ∞ = =   + +       + +     ∑ ∑ ( 37
)
where n is a positive integer, the functions ( , )

n n n R r r R R r Ρ     = +         (38) 3 2 2 3 3 2 ( , ) n n n R R R R R R Ε     = -         (39)
The complex integration constants I

Q I i n i r R A A n d r π θ θ π = = ∂ = ∂ ∑ ∫ (44) 2 2 1 0 2 sin( ) 2 Q I i n i r R A C n d r π θ θ π = = ∂ = ∂ ∑ ∫ (45)
The expressions for the integration constants I

I I R R A A R d R d r r π π θ θ ∂ ∂ - = ∂ ∂ ∫ ∫ (46)
The treatment of (46) yields to the following relations between the coefficient 0 i C defined in (31) 0 1 0

Q i i C = = ∑ (47)
The radial and tangential flux density distribution in the airgap can be deduced from the magnetic vector potential (37) by

1 I Ir A B r θ ∂ = ∂ I I A B r θ ∂ = -∂ (48)

IV. EDDY-CURRENTS AND TORQUE EXPRESSIONS

A. Current in the i-th Rotor Bar

The current density induced in the i-th rotor bar by the m-th space harmonic of the rotating magnetic field is given by ( , ) ( , )

i rm i J r j A r θ ω σ θ = - (49)
where ( , ) i A r θ is given by (28).

The current flowing through the i-th rotor bar is determined by

2 1 ( , ) i i R i i R I J r rdrd θ β θ θ θ + = ∫ ∫ (50)
Considering ( 28) and ( 49), (50) yields to

( ) ( ) ( ) ( ) ( ) 2 1 0 0 1 0 0 2 1 0 2 R i rm i R jC I r J r F Y r dr J R F Y R σω β α α α α - = - - ∫ (51)
The development of (51) leads to

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 1 2 1 1 0 2 0 2 1 1 1 1 0 2 i rm i J R Y R J R Y R jC R I J R Y R J R Y R α α α α σω β α α α α α   - =     -   (52)
where 2 0 rm j α ω σµ = -. With this relation, it is worth noting that the condition (47) imposes that the sum of the currents in the rotor bars is zero.

B. Electromagnetic Torque

The electromagnetic torque is obtained using the Maxwell stress tensor. A circle of radius R e in the air-gap is taken as the integration path. The electromagnetic torque due to the m-th space harmonics of the rotating field is expressed as follows

  = ℜ     ∫ ( 53
)
where L is the axial length of the motor. Substituting (48) into (53), the analytical expression for the electromagnetic torque becomes

2 * * 0 1 ( . . ) e e n n n n n LR T W X Y Z π µ ∞ = = + ∑ (54) with 3 3 2 2 2 3 3 2 3 3 2 2 2 3 3 2 3 3 2 2 2 3 3 2 3 3 2 2 3 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( ,
R R R R R R W A B R R R R R R R R R R R R X C D R R R R R R R R R R R R Y C D R R R R R R R R R R Z A B R R R R Ρ Ρ Ε Ε Ε Ε Ε Ε Ρ Ρ Ε Ε Ε Ε = - - = - - = + = - - 2 3 2 ( , ) ( , ) n e e n R R R R Ε Ε (55)
and * n X is the complex conjugate of n X .

V. RESULTS AND DISCUSIONS

In order to validate the proposed model, the analytical results have been compared with those obtained from 2D timeharmonic FEM simulations [START_REF] Meeker | Finite Element Method Magnetics[END_REF]. The finite-element solutions were obtained by imposing the natural Neumann condition along the iron boundaries, as in the analytical study. The mesh in the different subdomains has been refined until convergent results are obtained. The analytical and finite-element simulations are carried out for the whole motor cross section. Obviously, if periodicity of the magnetic field distribution exists, it can be taken into account.

The geometrical parameters given in Table 1 are used in the simulation studies. These parameters have been chosen arbitrary. The objective of this subsection is not to study a real induction motor but to show the effectiveness of the proposed analytical method. The analytical solutions in the airgap and in the rotor bars regions have been computed with a finite number of harmonic terms N and K as indicated in Table I.

For the equivalent current sheet, we consider a full pitch stator winding (λ = π) with a number of slots per pole per phase q = 1. Moreover, we suppose that the stator slot opening ε is small. So, the value of the winding factor ( 5) is equal to 1 for each harmonic component (the peak values J m of the current sheet harmonics are all the same as it is indicated in Table I). As the peak value of the stator current sheet is imposed whatever the slip, the results presented here correspond to an induction motor fed from a constant current source.

A. Results for the fundamental component of the stator current sheet

We first consider the fundamental component of the current sheet ( )

1 1 1 ( , ) cos r J t J p t θ θ ω = - with 1 r s s ω ω = ( 56
)
Figure 4 shows the flux lines in the machine under no-load condition (s ≈ 0). The corresponding radial and tangential components of the flux density distribution in the middle of the air-gap are shown in Fig. 5. The effects of the rotor slotting can be clearly observed in these figures. A good accordance can be seen between the finite element and the analytical computations. The flux lines and the eddy-current distribution within the rotor bars for locked rotor condition (s = 1) are shown in Fig. 6. Due to the induced current reaction, one can observe that the magnetic field is rejected at the surface of the rotor. Fig. 7 presents the corresponding radial and tangential flux density distribution in the middle of the airgap for s = 1. Compared to the no-load results of Fig. 5, one can see clearly the influence of the eddy-current reaction field on both the radial and the tangential flux densities (as expected, the radial component of the flux density decreases for locked rotor condition).

The current density distribution from the bottom to the top of a rotor bar (slot Q = 18; θ = 0) is given in Fig. 8. Due to the well-known skin effect, the current density tends to flow at the outer surface of the rotor bar. The real part of the total current in each rotor bar computed with (52) is shown in Fig. 9.

Comparison between numerical and analytical prediction of the torque-slip curve for the fundamental component of the current sheet is given in Fig. 10 for slip values ranging from -0.2 to +0.2.

It can be seen from Fig. 7 to Fig. 10 that the results obtained with the proposed analytical method are consistent with those obtained by FEM.

B. Results for the 5 th harmonics of the stator current sheet

To show the effectiveness of the proposed model for the modeling of space harmonics effects, we consider now the 5 th harmonic of the current sheet ( )

Figure 11 shows the flux lines (with 10 pole-pairs) and the eddy-current distribution in the rotor bars for locked rotor condition (s = 1). The corresponding radial and tangential components of the flux density distribution are shown in Fig. 12. The current density distribution along a rotor bar (Q = 18) and the total current in each rotor are given respectively in Fig. 13 and Fig. 14.

As it is known, the fifth harmonic field rotates in a direction opposite to the one of the rotor with an angular speed The torque due to the 5 th harmonic is shown in Fig. 15. As expected, the torque is null for s = 1.2. The operating region for which s < 1.2 corresponds to a generator mode (negative torque).

The results presented in Fig. 12 to Fig. 15 show that the analytical model is able to predict the effect of space harmonics with a very good accuracy.

C. Torque with 5 th and 7 th space Harmonic Effects

We now use the analytical model to compute the torque for the fundamental, the 5 th and the 7 th space harmonic components of the current sheet. For a given slip value, the number of computations needed is equal to the number of considered space harmonics. Indeed, the angular frequency is different for each harmonic [START_REF] Perry | Eddy current induction in a solid conducting cylinder with a transverse magnetic field[END_REF].

The torque produced by the fundamental component and the 5 th and 7 th harmonic are shown in Fig. 16. The total torque is the sum of all the components. We can observe the wellknown parasitic effects of the 5 th and 7 th space harmonics on the torque-slip characteristic of an induction machine.

In order to have a good precision in the analytical torque prediction (harmonic 1 of Fig. 16), it is sufficient, for the studied machine, to take the fundamental component in the airgap region (N = 1) and the two first harmonic terms in the rotor bars region (K = 2). For a given slip value, the computation time is about 70 ms with the analytical model whereas the linear FEM takes about 2 s for a mesh of 16200 elements. The analytical computations being much faster, the presented model can advantageously be used in a preliminary design of induction motors. The analytical model leads to lower computational time than FEM. Therefore, it can be used to investigate rapidly (under linear condition) the influence of the design parameters such as rotor slots dimensions (height and slot opening angle), number of rotor slots, electrical conductivity of the rotor bars and stator winding configuration. π

θ θ β β n k n n n i n k f i i k - - + - - = (A.5) ( ) 2 2 2 2 2 ) cos( ) ( cos ) 1 ( ) , , ( β π θ θ β β n k n n n i n k g i i k - - + - = (A.6) -for β π n k = ( ) 1 ( , , ) cos( ) sin ( 2 ) sin( ) 2 2 i i i f k n i n n n k β θ θ β θ π   = + + -     (A.7) ( ) 1 ( , , ) sin( ) cos ( 2 ) cos( ) 2 2 i i i g k n i n n n k β θ θ β θ π   = - + -     (A.8)
The development of (A.3) and (A.4) gives the following functions ( )

) sin( ) sin( 1 ) , ( i i n n n n i n r θ β θ - + = (A.9) ( ) ) cos( ) cos( 1 ) , ( i i n n n n i n s θ β θ + + - = (A.10)
The solution for the seven unknown integration constants can be determined by solving a linear system of equations which can be written in matrix form as

[ ]( ) ( ) M X S = (A.11) ( ) ( ) 0 0 t i i I I I I I k n n n n X C C A B C D A = (A.12)
The matrix [M] and the source vector (S) are developed in (A. [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF] where N and K represent respectively the number of harmonics used in the air-gap and in the rotor bar regions. Q is the number of rotor bars. I Q is the identity matrix of dimension Q. A numerical solution of (A. [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]) can be obtained by using mathematical software (Matlab).

• Expression of the integration constants 0 i C and i k C for the i-th rotor bar sub-domain

The treatment of (31) yields to the following linear relation 0 0

2 3 3 2 2 3 3 2 1 2 3 3 2 2 3 3 2 1 ( , ) 2 ( , ) ( , ) ( , ) ( , ) 2 ( , ) ( , ) ( , ) i I
I I n n n n n n I I n n n n n n C A R R R R A B r n i n R R n R R R R R R C D s n i n R R n R R Ρ β Ε β Ε Ρ β Ε β Ε ∞ = ∞ = =   + +       + +     ∑ ∑ (A.13)
which can be rewritten in vector and matrix form as indicated in the first line of the matrix [M] in (A. [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]). The treatment of (32) yields to the following linear relation

2 3 3 2 2 3 3 2 1 2 3 3 2 2 3 3 2 1 ( , ) 2 4 ( , , ) ( , ) ( , ) ( , ) 2 4
( , , ) ( , ) ( , )

i I I n k n n n n n I I n n n n n n R R R R C A B f k n i n R R n R R R R R R C D g k n i n R R n R R Ρ β Ε β Ε Ρ β Ε β Ε ∞ = ∞ =   = +       + +     ∑ ∑ (A.14)
which can be rewritten in vector and matrix form as indicated in the second line of the matrix [M] in (A. [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]. It is worth noting that the vector ( ) where p and m are respectively the number of pole pairs and the m-th space harmonics of the current sheet.

(A.15) and (A. [START_REF] Gysen | General formulation of the electromagnetic field distribution in machines and devices using Fourier analysis[END_REF]) can be rewritten in vector and matrix form as indicated in the fourth and the sixth lines of the matrix [M] and vector source (S) in (A. [START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF].

The development of (44) and (45) gives ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 1 1 2 0 0 2 1 0 2 1 ' ' 2 2 2 2 1 1 1 ( , ) 1
( , , )

Q I i n i Q k k k i k k k k i k J R F Y R A C r n i J R F Y R J R F Y R C f k n i J R F Y R
Q I i n i Q k k k i k k k k i k J R F Y R C C s n i J R F Y R J R F Y R C g k n i J R F Y R
Q Q N Q N Q N Q N KQ KQ N KQ N KQ N KQ N N N Q N KQ N N N Q N KQ N Q i Q Q i k KQ t C I M M M M C I M M M M M M I I M M I I × × × × × × × × × × × ×   -     -       -     × -     -     -         ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
                               =                                   (A.19)

Fig. 1 .

 1 Fig. 1. Geometry of the studied model (with Q = 6 rotor bars).

  i-th rotor bar domain

  J π β is the Bessel function of the first kind and order kπ/β and k Y π β is the Bessel function of the second kind.

C

  are new integration constants and

Fig. 3 .

 3 Fig. 3. Air-gap domain with its boundary conditions.with

  θ over the air-gap interval[0, 2π]

D

  are developed in the appendix. The Laplace equation (33) with the Neumann boundary conditions (34) and (35) must verify[START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]

βK 5 Fig. 4 .Fig. 5 .

 545 Fig. 4. Flux lines for no-load condition (s ≈ 0) and m = 1 (fundamental component of the current sheet) obtained with FEM.

Fig. 6 .

 6 Fig. 6. Flux lines and eddy-current distribution (A/mm 2 ) for s = 1 (locked rotor condition) and m = 1 (fundamental component of the current sheet) obtained with FEM.

Fig. 8 .

 8 Fig. 8. Current density distribution along the 18 th rotor bar (θ = 0) for locked rotor condition (s = 1) and m = 1.

Fig. 9 .

 9 Fig. 9. Current distribution in the rotor bars (real part) for locked rotor condition (s = 1) and m = 1.

Fig. 11 .

 11 Fig. 11. Flux lines and eddy-current distribution (A/mm 2 ) for s = 1 (locked rotor condition) and m = -5 (5 th harmonic component of the current sheet) obtained with FEM.

Fig. 13 .

 13 Fig. 13. Current density distribution along the 18 th rotor bar (θ = 0) for locked rotor condition (s = 1) and m = -5.

Fig. 14 .

 14 Fig. 14. Current distribution in the rotor bars (real part) for locked rotor condition (s = 1) and m = -5.

Fig. 15 .

 15 Fig. 15. Torque-slip characteristic for the 5 th harmonic component of the stator current sheet (m = -5).

Fig. 16 .

 16 Fig. 16. Torque-slip curve with 5 th and 7 th harmonic components effects (analytical model).

  .[START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]) is of dimension K•Q and takes into account the Q rotor bars.• Expressions of the integration constants

  [START_REF] Zhu | An accurate subdomain model for magnetic field computation in slotted surface-mounted permanent magnet machines[END_REF]) and (A.[START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF]) can be rewritten in vector and matrix form as indicated in the third and the fifth lines of the matrix [M] in (A.[START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]).The seventh line in (A.[START_REF] Farlow | Partial Differential Equations for Scientists and Engineers[END_REF]) corresponds to the relation (47).

  )

		I	n	e	I	n	e
	n	n			n
		e	n		e	n
		I	n	e	I	n	e
	n	n			n
		e	n		e	n
		I	n	e	I	n	e
	n	n			n
		e	n		e	n
		I	n	e	I
	n	n			n
		e	n

TABLE I PARAMETERS

 I

		OF THE MODEL
	Symbol	Quantity	value
	R1	Inner radius of the rotor slots	4 cm
	R2	Radius of the rotor surface	6 cm
	R3	Stator bore radius	6.3 cm
	L	Axial length	20 cm
	Q	Number of rotor bars	18


> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1

Keywords: Induction heater, analytical solution, eddy current, induced heating power, linear motion, superconducting, magneto-thermal

This paper presents a new structure of an induction heater for aluminium billets. The device consists in using a strong DC magnetic field created by a superconducting inductor in which the conducting billet is actuated by a linear motion at a constant velocity. The analysis, based on the analytical model for the magnetic field in the billet, has been developed, and then the eddy currents and induced heating power have been evaluated. The analytical results are compared to those issued from a finite element software (COMSOL). The magneto thermal analysis shows that the temperature profile in the billet is homogenous.

I. INTRODUCTION

NDUCTION heating is widely used in industrial processes to preheat aluminium billets before press-forming operations [START_REF] Rudnev | Handbook of Induction Heating[END_REF]. The analysis of induction heating requires the development of simulation and modeling tools. It depends on several physical phenomena: electromagnetic, thermal, and mechanical ones.

In conventional induction heaters, the billet is exposed to an AC magnetic field produced by a water-cooled copper winding. The thermal power is due to the eddy currents induced in the billet. It is well known that the efficiency of the process is related to the power transferred from the inductor to the billet. The efficiency depends on the resistivity and the magnetic permeability of the material to be heated. For magnetic steel billets, the efficiency is in the range of 80 to 90% below the Curie temperature and remains relatively important (approximately 70%) above this temperature when steel becomes nonmagnetic [START_REF] Rudnev | Handbook of Induction Heating[END_REF]. In the same way, the efficiency of an induction heater for nonmagnetic metal billets (aluminium and Copper) drastically drops to about 50%. In fact, it is difficult to heat nonmagnetic conducting materials Manuscript received February 4, 2011. H. Bensaidane and Y. Ouazir are with the University of science and technology (USTHB), BP N°32 El Alia Bab Ezzouar, Algiers, Algeria (phone: +213 661 46 55 15; fax: +213 21 24 71 87; e-mail: hakimbensaidane@ yahoo.fr and youazir@yahoo.fr).

T. Lubin, S. Mezani and A. Rezzoug are with GREEN-UHP, Faculté des Sciences et Technologies, BP 239, 54506, Vandouvre-lès-Nancy, France (email: Thierry.Lubin@green.uhp-nancy.fr).

with conventional heating devices.

In order to improve the efficiency, Runde and Magnusson [START_REF] Runde | Induction heating of aluminium billets using superconducting coils[END_REF]- [START_REF] Magnusson | Efficiency analysis of a hightemperature superconducting induction heater[END_REF] have proposed a new type of induction heater by replacing the conventional copper inductor by a hightemperature superconducting (HTS) coil excited with 50/60Hz AC current. However, because of the AC losses in the HTS coil, the efficiency obtained with a small-scale demonstrator does not exceed 60% [START_REF] Magnusson | Efficiency analysis of a hightemperature superconducting induction heater[END_REF]. More recently, a new DC induction heating concept with a superconducting coil has been proposed. In this approach, the billet is rotated inside a transverse DC magnetic field generated by superconducting coils [START_REF] Magnusson | A 200 kW MgB2 induction heater project[END_REF]- [START_REF] Runde | Commercial induction heaters with high temperature superconductor coils[END_REF]. Velocity induced currents react to the driving torque and generates a thermal power within the billet [START_REF] Fabbri | Experimental and numerical analysis of DC induction heating of aluminium billets[END_REF]- [START_REF] Lubin | Induction heating of aluminium billet subjected to a strong rotating magnetic field produced by superconducting windings[END_REF].

In this paper, we develop an electromagnetic and thermal analysis of an original induction heater for aluminium billets using superconducting coils. In this topology, the billet is subjected to a linear motion inside a strong DC magnetic field produced by a superconducting inductor. The topology described here was in its principle, patented independently by Magnusson [START_REF] Magnusson | An apparatus and a method for induction heating of pieces of electrically conducting and non-magnetic material[END_REF] and Müller et al. [START_REF] Müller | Energy efficient metal heating installation[END_REF]. The description of the studied induction heater is given in section II. An analytical electromagnetic model is then developed in section III and IV to find the eddy current density and induced power density in the billet. Finally, the temperature profile inside the billet is obtained with two dimensional finite elements software (COMSOL).

II. DESCRIPTION OF THE INDUCTION HEATER

Fig. 1 shows the cross section of the studied induction heater. It is composed of an inductor made from superconducting coils which create a strong DC magnetic field. The aluminium billet to be heated is placed inside the warm hole of the cryostat and moves linearly at a low speed. The linear displacement of the billet in the DC magnetic field generates induced currents which produce a Joule heating in it. As will be shown in the paper, the use of a low linear speed and a strong DC magnetic field allow obtaining a large skin depth and therefore a homogeneous temperature distribution inside the billet, which is necessary before the press-forming step. The physical parameters of the studied device are given in Table I. The geometrical parameters have been chosen arbitrarily in order to test the analytical model developed in the paper.

III. EQUIVALENT CURRENT SHEET

To reduce the study to a 2D axisymmetric problem, we consider that the induction heater (the inductor and the billet) has an infinite axial length. The billet is subjected to a linear movement along the z axis with a constant velocity V. In the analysis, the current in the superconducting coil is replaced by a current sheet of radius R 2 +e/2 with an infinite axial length as shown in Fig. 2 [12]- [START_REF] He | Analysis of induction type coilgun performance based on cylindrical current sheet model[END_REF]. This current sheet is chosen to carry the same current as the original coil.

The superconducting winding is excited with a DC current density J. The electrical current in a coil is given by

I a eJ = (1)
The corresponding value of the equivalent current sheet in A/m is then

s J e J = (2)
where J and J S have only one component in the θ direction. The current sheet distribution versus the coordinate z is shown in Fig. 3. Its Fourier series expansion is ( ) cos( )

s sn n J z J nkz = ∑ with 2 k π τ = ( 3
)
where J sn is the peak value of the nth harmonic component of the current sheet given by 4 sin sn e J n a J n π π τ Due to the cryostat and thermal insulation thicknesses, the radius of the billet is relatively small compared to the inner radius of the superconducting coil. Hence, the higher harmonic terms of the flux density strongly decrease with the air gap thickness. In this study, only the fundamental term (n=1) of the current sheet distribution is considered.

  =     (4)
( ) ( )

1 4 sin cos s e J a J z kz π π τ   =     (5)

IV. MATHEMATICAL FORMULATION

A general multiregion problem is analyzed. The whole space is divided into three regions as shown in Fig. 4.

-Region 1 corresponds to the conducting billet (workpiece).

-Region 2 corresponds to the space between the billet and the current sheet.

-Region 3 corresponds to the space outside the current sheet.

A. Diffusion equation

By considering the movement of the billet, the electromagnetic behaviour of the system is described by Maxwell's equations expressed as . 0 . 0

B H J E t B E ∂ ∇ × = ∇ × = -∂ ∇ = ∇ = (6)
with the material properties ( )

0 = × = = J V B with V V u z B H σ µ (7)
The magnetic vector potential A is defined by

B A = ∇ × (8)
The combination of ( 6), ( 7) and ( 8) gives the magnetic diffusion equation which takes into account the movement of the conducting billet ( )

0
in region 1 0 in regions 2 and 3

A V A A σµ  ∇ × ∇ × = × ∇×   ∇ × ∇ × =  (9)
Due to the axial symmetry, the studied problem is invariant according to the θ-coordinate. So, the electric and magnetic quantities will be independent of θ. Under these conditions, the magnetic vector potential, in each region, presents only one component in the azimuthal direction, and depends only on r and z variables. Then, (9) becomes in region 1

2 2 1 1 1 1 1 0 2 2 2 1 1 A A A A A V r r z r r z σµ ∂ ∂ ∂ ∂ + - + = ∂ ∂ ∂ ∂ (10)

B. Magnetic vector potential expressions

According to the form of the current sheet excitation for the fundamental term of the Fourier series [START_REF] Magnusson | A 200 kW MgB2 induction heater project[END_REF], the solution of the magnetic vector potential in region i (i=1,2,3) can be written as ( ) , ( ) cos( ( ))

i i i A r z A r kz r ϕ = + (11)
Since the magnetic vector potential is varying sinusoidally in space, we can use the complex notation

( ) ( ) { } , i i jkz A r z A r e = ℜ ( 12
)
Where ℜ denotes the real part of a complex number. Using the complex representation, (10) becomes a modified Bessel differential equation

2 1 1 2 2 2 1 2 ( 1) 0 d A d A r r r A dr dr λ + - + = (13) with 2 2 0 k jk V λ σµ = + and 1 j = - (14)
In the conducting billet (Region 1), the general solution of ( 13), knowing that 1 A must be finite when r → 0, has the following form [START_REF] Abramowitz | Handbook of Mathematical Function[END_REF]

1 1 1 ( ) ( ) A r a I r λ = (15)
where I 1 is the modified Bessel function of the first kind and order 1 and a 1 is a complex coefficient which will be determinate using the interface conditions. For region 2 where the electrical conductivity σ = 0, the general solution is

2 2 1 2 1 ( ) ( ) ( ) A r a I kr b K kr = + (16)
where K 1 is the modified Bessel function of the second kind and order 1.

For region 3, knowing that 3 A must be finite when ∞ → r , the solution is

3 3 1 ( ) ( ) A r b K kr = (17)
The complex coefficients a 1 , a 2 , b 2 and b 3 in ( 15), ( 16) and (17) are determined using the interface conditions between the different regions

at 1 r R = : 1 2 1 2 A A d A d A dr dr  =   =   (18) at 2 2 e r R = + (current sheet interface): 2 3 2 3 0 1 s A A d A d A J dr dr µ  =   = +   (19)
We obtain a system of four linear equations from which we extract the expressions of the coefficients a 1 , a 2 , b 2 and b 3 . The magnetic vector potential in region 1 is then expressed as

0 1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) s J I kR K kR A r I r kC I R D I R µ λ λ λ   = +     (20) with 0 2 1 2 0 2 1 2 2 2 2 2 e e K k R I k R e C I k R e K k R           + +                         = + +               +             (21)
and

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 kI R K kR K kR I R D kI kR I R I kR I R λ λ λ λ λ λ + = - (22)
where I 0 and K 0 are respectively the modified Bessel function of the first kind and the second kind, of order 0.

C. Eddy current expression

The induced current density in the aluminium billet (region 1) can be obtained from the following relation ( )

1 1 1 ( , ) ( , ) A r z J r z V A V z σ σ ∂ = × ∇ × = - ∂ (23)
which becomes in the complex domain

1 1 ( ) ( ) J r j Vk A r σ = - (24)
Finally, the expression of the induced current density can be written as

{ } 1 1 ( , ) ( ) jkz J r z J r e = ℜ (25)

D. Induced heating power expression

The induced power density in the billet is expressed as:

2 1 ( , ) ( , ) J r z p r z σ = ( 26
)
So the induced power in the billet of length L is

1 2 / 2 2 1 0 0 /2 ( , ) R L L J r z P rdrd dz π θ σ - = ∫ ∫ ∫ (27)
Substituting (20) and ( 24) into (25) and then (25) into (27), we obtain the analytical expression of the induced heating power in the conducting billet

2 2 * 1 1 2 . . P V k a a E F πσ = (28)
where

0 1 1 1 1 1 1 1 1 1 1 ( ) ( ) 1 ( ) ( ) s J I kR K kR a kC I R D I R µ λ λ   = +     (29)
and

( ) ( ) ( ) ( ) * * * 1 0 1 1 1 0 1 1 1 * R E I R I R I R I R λ λ λ λ λ λ λ λ   = -   - (30) 1 1 sin( ) cos(2 ( )) 1 2 kL R F L k ϕ   = +     (31)
where 1 1 ( ) R ϕ is the argument of 1 1 ( ) A R given by (20). The analytical expression of the induced heating power in the billet (28) depends on the geometrical and physical parameters of the induction heater. This expression will be used in the next section to study the effect of the geometrical parameters on the performance of the induction heater.

V. RESULTS AND DISCUSSION

A. Eddy current density

Figs. 5 and6 show the flux lines and the magnetic flux density distribution obtained by finite element method (COMSOL) when the conducting billet is moving respectively at a linear velocity of V=0.24 m/s and V=0.62 m/s. The superconducting coil is excited with a current density J=200 A/mm 2 . One can observe that the flux lines are driven in the direction of the motion and are rejected from the billet when the speed increases.

The critical current in the superconducting coil is closely dependent on the maximum value of the flux density. Hence, it is important in the design of the HTS induction heater to carefully consider this maximum value. In the case of the simulations of Figs. 5 and6, a maximum value of around 5T appears in the inner surface of the coils. Notice that this value is acceptable for DI-BSCCO HTS tapes [START_REF]Superconductivity[END_REF] for which the critical current density is higher than 200 A/mm² for operating temperatures of 15-20 K. Fig. 7 and8 show the spatial distribution of the eddy currents in the billet for V=0.24 m/s and V=0.62 m/s.

The eddy current density versus the radial distance r at z = 0 for a linear speed of V = 0.24 m/s and V = 0.62 m/s are given respectively in Figs. 9 and 10. We can notice that the results obtained with the analytical method (current sheet approximation with the first harmonic) are in good agreement with those obtained by FEM (the real coil geometry is considered in the FE computations).

B. Induced heating power

The power density variation with respect to radius is plotted in Fig. 13 for two values of the velocity. As for the current density (Figs. 9, 10), the penetration of the power density is higher in the case of low velocity values.

The induced heating power in the conducting billet obtained for various values of the linear velocity V and for the coil excited by a DC current density J = 200 A/mm 2 is given in Fig. 14. These results have been obtained using the analytical expression (28) and compared with those obtained by FEM (FEM with real coil distribution). We can observe that the analytical model can predict with a very good precision the induced power in the billet. Furthermore, the analytical computations are much faster than the numerical ones. Therefore it could be used as an efficient tool to study the influence of the geometrical parameters on the induced heating power.

C. Design analysis of the induction heater 1) Effect of the distance between two coils (parameter b)

Using the analytical expression (28), Figs. 15 and16 show the effect of the distance between two superconducting coils (inductor) on the induced power for two values of the linear velocity V. We can observe that the induced power presents a maximum value for b = 7.5 cm, then decreases while increasing the distance b.

2) Effect of the distance between the inductor and the conducting billet

Figs. 17 and 18 show the effect of the ratio (R 2 +e/2)/R 1 on the induced power for various values of the linear velocity V. A certain distance is necessary between the inductor and the billet. Indeed, the superconducting coils must be placed inside the cryostat and a thermal insulation must be placed between the cryostat and the billet. As expected, one can observe on Figs. 17 and 18 that the induced power decreases rapidly when the distance between the inductor and the conducting billet increases.

VI. TEMPERATURE FIELD ANALYSIS

Taking into account the movement of the conducting billet, the equation describing the heat transfer phenomenon in the billet to be heated is given by [START_REF] Lubin | Induction heating of aluminium billet subjected to a strong rotating magnetic field produced by superconducting windings[END_REF], [START_REF] Ho | A novel crossed traveling wave induction heating system and finite element analysis of eddy current and temperature distributions[END_REF] .

( , )

∂   + ∇ = + ∆   ∂   t T c V T p r z T t ρ λ (32)
where T, λ t , ρ and c respectively represent the temperature, the thermal conductivity of the billet, the specific mass and the specific heat. The term T . V ∇ in (32) is the convective term due to the motion of the billet and ( ) z r p , is the heat source produced by eddy current in the billet. In 2D cylindrical coordinates, the boundary value problem describing the temperature evolution is defined only in region 1 (adiabatic condition is considered on the surface of the billet)

2 2 2 2 1 ( , ) t T T T T T c V p r z t z r r r z ρ λ   ∂ ∂ ∂ ∂ ∂   + = + + +       ∂ ∂ ∂ ∂ ∂     (33) 1 0 ( , , 0) 30 r R T r T r z t C = ∂ = ∂ = = ° (34)
The problem is solved with 2D finite element software (COMSOL). The induced power density is evaluated using the electromagnetic problem with a constant linear velocity V of the billet. The source term for the heat transfer problem is the one given by (26).

Fig. 19 gives the transient temperature profile at the centre of the conducting billet. The linear velocity of the billet is constant and equal to V = 0.62 m/s and the current density in the superconducting coil is fixed at J = 200 A/mm 2 . The corresponding induced heating power in the billet is about 200 kW. We observe that the temperature inside the billet increases almost linearly because the boundary condition on the surface of the billet is adiabatic. The rise time to heat the conducting billet from 30 °C to 500 °C is about 145 s.

Fig 20 shows the temperature distribution inside the conducting billet along the radial distance r at t = 150s. The maximum temperature is T=543°C on the surface of the conducting billet and the minimum one is T=517°C at the centre of the billet. This result shows that the temperature is sufficiently homogeneous in the billet (within 5%), which is suitable for press forming operation.

It is important to keep in mind that the analysis developed in the paper is a simplification of a 3-D problem to a 2-D one. Indeed, we have supposed an infinitely long cylinder whereas the length of the cylinder is finite. The end effect is ignored so the power density in the end regions of the cylinder will be smaller than in the centre of the billet. This can be shown only with a 3D analysis.

VII. CONCLUSION

In this paper, we have presented a new structure of induction heaters for aluminium billets. The device consists in using a strong DC magnetic field created by a superconducting inductor in which the conducting billet is actuated by a linear movement at a constant velocity.

An analytical model for the magnetic field distribution in the billet has been developed, and then the eddy currents and the induced heating power have been evaluated. The results obtained using this model are validated through finite element computations. The analytical model has been used to study the effect of the geometrical parameters on the performance of the induction heater. It has been shown that an optimum distance exists between the superconducting coils to obtain the maximum induced power in the billet. By solving the thermal problem, we have also shown that the heat power is distributed in the heart of the billet, so the temperature is sufficiently homogeneous in the billet and satisfies the requirement for press forming operation.

The considered model being a 2D one, it will probably be necessary to develop a full 3D model in order to consider the end effects. Furthermore, the choice of a constant velocity leads to a very long furnace which may be a limitation for this heating system. The use of an oscillating system would probably be more suitable. Finally, it will be important to consider the end effects as well as the reaction field since they affect the performances of the superconducting coils (losses and critical current density).

Fig. 2 .Fig. 3 .

 23 Fig.2. Simplified model used for the analysis

+

  the spatial period of the current sheet.

Fig. 4 .

 4 Fig.4. Multiregion problem used for the analysis

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig.5. Flux lines distribution and magnetic flux density distribution (T) for V = 0.24 m/s and J = 200 A/mm 2

Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .

 89101112 Fig.8. Eddy current distribution in the billet for V = 0.62 m/s and J=200 A/mm 2 .

Fig. 13 .Fig. 14 .

 1314 Fig.13. Induced power density versus the radius r at z = 0 a) V = 0.24 m/s b) V = 0.62 m/s

Fig. 15 .Fig. 16 .

 1516 Fig.15. Induced heating power versus the distance between two superconducting coils for V = 0.24 m/s and J = 200 A/mm 2

Fig. 18 .

 18 Fig.17. Induced power versus the distance between the inductor and the billet for V = 0.24 m/s and J = 200 A/mm 2

Fig. 19 :Fig. 20 :

 1920 Fig.19: Transient temperature profile at the center of the billet for V=0.62 m/s and J=200 A/mm 2

TABLE I PHYSICAL

 I AND GEOMETRICAL PARAMETERS OF THE STUDIED INDUCTION

		HEATER
	Symbol	Quantity	Value
	R1	Billet radius	8 cm
	R2	Superconducting coil inner radius	13 cm
	e	Superconducting coil thickness	4 cm
	a	Width of the coil	15 cm
	b	Distance between two coils	10 cm
	L	Billet axial length	1 m
	V	Linear velocity of the billet	variable
	σ	Electrical conductivity of the billet (Al)	37 10 6 Ω -1 m -1
	µ0	Magnetic permeability	4π 10 -7 H/m
	λt	Thermal conductivity of the billet	206 Wm -1 K -1
	ρ	Specific mass of aluminium	2707 Kg m -3
	c	Specific heat of aluminium	936 J Kg -1 K -1
	J	Current density in the superconducting
		coils

Hakim Bensaidane was born in Tizi-ouzou, Algeria, in 1977. He received the engineer diploma and the magister degree from the university Mouloud Mammeri, Tizi-ouzou, Algeria in 2003 and 2006 respectively.

He is currently Ph.D. student at the University of Sciences and Technology Houari Boumediene, Algiers, Algeria, at the laboratoire des systems électriques et industriels de l'USTHB d'Alger where his research interests include modeling and electromagnetic computation of electromechanical devices and power electric system.

Youcef Ouazir was born in Setif, Algeria, in 1970. He received the engineer diploma,the magister and Ph.D degree from the Ecole Nationale Polytechnique d'Algier, Algeria in 1994, 1997 and 2006 respectively.

He is currently a lecturer at the University of Sciences and Technology Houari Boumediene (USTHB), Algiers, and a member of the laboratoire des systems électriques et industriels. His research interests include the modeling and electromagnetic computation of electromechanical devices and power electric system.

Thierry Lubin was born in


Elsa Tolone
email: tolone@univ-paris-est.fr

Stavroula Voyatzi
email: voyatzi@univ-mlv.fr

Extending the adverbial coverage of a NLP oriented resource for French

 1990). This work relies on the exploitation of fine-grained linguistic information provided in existing resources. Various features are encoded in both LG tables and they haven"t been exploited yet. They describe the relations of deleting, permuting, intensifying and paraphrasing that associate, on the one hand, the simple and compound adverbs and, on the other hand, different types of compound adverbs. The resulting syntactic resource is manually evaluated and freely available under the LGPL-LR license.

Introduction

Recognising adverbs such as extrêmement "extremely" and à long terme "in the long run" in texts is likely to be useful for information retrieval and extraction because of the information that some of these adverbials convey.

Adverbs, or more generally circumstantial complements, have often been overlooked in the compilation of lexical resources (Nølke, 1990: 3). Several reasons explain this lack of interest. Firstly, adverbials are usually felt as less useful than nouns for information retrieval and extraction. Secondly, compound adverbs in particular are difficult to distinguish from prepositional phrases assuming other syntactic functions, such as arguments or noun modifiers: the distinction is hardly correlated to any material markers in texts and lies in complex linguistic notions [START_REF] Villavicencio | Learning to distinguish PP arguments from adjuncts[END_REF][START_REF] Merlo | Generalised PP-attachment Disambiguisation using Corpus-based Linguistic Dia-gnostics[END_REF].

The availability of large-coverage lexicons providing lexical, syntactic and semantic information is essential in order to gain insight on the recognition of adverbs, including the dual problems of variability and ambiguity. In addition, it is likely to help solving prepositional phrase attachment during shallow or deep parsing [START_REF] Agirre | Improving parsing and PP attachment performance with sense information[END_REF].

In this paper, we present work on extending the adverbial entries of LGLex, a NLP oriented syntactic resource for French. These adverbs were extracted from the Lexicon-Grammar tables (hereafter LG tables) of both simple adverbs ending in -ment "-ly" (Molinier and Levrier, 2000) and compound adverbs [START_REF] Gross | Lexicon-Grammar. The representation of compound words[END_REF]1990).

The paper is organized as follows. In section 2, we provide an overview of the three resources used in our work. In section 3, we describe how we enhanced LGLex thanks to various features that are encoded in LG tables. In section 4, we present results and discuss the manual evaluation process. Finally, we conclude in section 5 by pointing out several possible extensions and issues for future research.

Resources

Lexicon-Grammar tables (hereafter LG tables) are currently one of the major sources of lexical syntactic information for the French language; several LG tables exist for other languages (see section 2.1). Their development was initiated as early as the 1970s by Maurice Gross, at the LADL [START_REF] Gross | Méthodes en syntaxe : Régimes des constructions complétives[END_REF]1994), and then at the LIGM, University Paris-Est [START_REF] Boons | La structure des phrases simples en français: Constructions intransitives[END_REF][START_REF] Guillet | La structure des phrases simples en français : les constructions transitives locatives[END_REF].

Lexical information is represented in the form of tables. Each table puts together elements of a given lexical-grammatical category (for a given language) that share a certain number of defining features, which usually concern subcategorization information. These elements form a class. These tables are represented as matrices (see section 2.1.2): each row corresponds to a lexical item of the corresponding class; each column lists all features that may be valid or not for the different members of the class; at the intersection of a row and a column, the symbol + (resp. -) indicates that the feature corresponding to the column is valid (resp. not valid) for the lexical entry corresponding to the row.

The resources described in this paper correspond to the LG tables of both simple and compound adverbs, in which previously implicit features have been made explicit 1 for more convenient use in NLP. All tables are fully available 2 under a free license (LGPL-LR).

The LG tables of adverbs

LG tables of adverbs are available in several languages, notably in English [START_REF] Gross | Lexicon-Grammar. The representation of compound words[END_REF], German (Seelbach, 1990), Spanish [START_REF] Blanco | Quelques remarques sur un dictionnaire électronique d"adverbes composés en espagnol[END_REF][START_REF] Blanco | Quelques remarques sur un dictionnaire électronique d"adverbes composés en espagnol[END_REF]), Italian (De Gioia, 2001), Portuguese [START_REF] Baptista | Some Families of Compound Temporal Adverbs in Portuguese[END_REF], Korean [START_REF] Jung | Grammaire des adverbes de durée et de date en coréen[END_REF] and Modern Greek [START_REF] Voyatzi | Description morphosyntaxique et sémantique des adverbes figés en vue d'un système d'analyse automatique des textes grecs[END_REF].

In French, there are two resources of adverbs that follow different principles both in classification and in representation within the Lexicon-Grammar framework. That is, firstly, tables of simple adverbs ending in -ment "-ly" [START_REF] Moliner | Etude syntaxique et sémantique des adverbes de manière en -ment[END_REF][START_REF] Molinier | Grammaire des adverbes : description des formes en -ment[END_REF], which are mainly derived from adjectives and, secondly, tables of compound adverbs [START_REF] Gross | Lexicon-Grammar. The representation of compound words[END_REF]1990). In this section, we describe briefly the different classes, morphosyntactic structures and features provided in the tables of both types of adverbs.

The LG tables of simple adverbs

According to Molinier and Levrier (2000), adverbs ending in -ment "-ly" form a large class of French adverbs. Moreover, as opposed to other types of adverbs, they form a quite open class.

These adverbs constitute a morphologically homogeneous class, since most of them are created according to the pattern adjective + -ment '-ly'. In total, there are 3,203 simple adverbial entries which are represented in sixteen LG tables.

The first partition has been established between sentential adverbs and adverbs integrated into the sentence; that means attached to the predicate or any other component of the sentence.

1 In order to make previous implicit features explicit, a table of classes has been created [START_REF] Tolone | Les tables du Lexique-Grammaire au format TAL[END_REF][START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du français[END_REF]. Its role is to assign features when their value is constant over a class, e.g. class definition features. Each row stands for a class and each column stands for a feature. Each cell corresponds to the validity of a feature in a class. In particular, the table of French adverb classes is composed of 32 different classes and 163 features. 2 http://infolingu.univ-mlv.fr/english > Language Resources > Lexicon-Grammar > Download.

Three major classes of sentential adverbs are worth mentioning: conjuncts, style disjuncts (or utterance-level adverbs) and attitude disjuncts (or statement adverbs). The latter are divided into four classes: evaluative adverbs, e.g. curieusement "curiously", adverbs of habit, e.g. habituellement "usually", modal adverbs, e.g. certainement "certainly", and subject oriented attitude adverbs, e.g. bêtement "foolishly".

On the other hand, there are six major classes of adverbs integrated into the sentence: (i) adverbs of subject oriented manner, e.g.

gentiment d'un point de vue linguistique "linguistically = from a linguistic point of view"; (vi) focus adverbs, e.g. essentiellement "essentially". Features included in the LG tables can be organized into four main groups: distributional features (e.g. the possibility for an adverb to occur at the beginning of a negative clause), local syntactic features (e.g. the possibility for an adverb to have a function as an indefinite determiner), semantic features (e.g. knowing what type of interrogative adverb corresponds to an adverb allows it to be categorized semantically), and paraphrasing features (e.g. Adv, Pindicatif = C'est Adj que Psubjonctif =: Bizarrement, Marie n'est pas venue à la soirée = C'est bizarre que Marie ne soit pas venue à la soirée " Oddly, Mary did not come to the party = It is odd that Mary did not come to the party").

The LG tables of compound adverbs

The scope of the LG tables of compound adverbs is delimited by the intersection of two criteria: (i) multiword expressions and (ii) adverbial func-tion. As Laporte and Voyatzi (2008) state "a phrase composed of several words is considered to be a multiword expression if some or all of its components are tied together, that is, if their combination does not obey productive rules of syntactic and semantic compositionality". This criterion ensures a complementarity between lexicon and grammar. In other words, it tends to ensure that any combination of linguistic elements which is correct in the language, but is not represented in common syntactic-semantic grammars, should be stored in lexicons.

In terms of the adverbial function, LG tables of compound adverbs deal only with circumstantial complements, namely, complements that modify the predicate or any other element of the sentence in which they occur. Sentential adverbs modify or enhance the entire sentence. They are identified through criteria [START_REF] Gross | Lexicon-Grammar. The representation of compound words[END_REF]1990) involving the fact that they are optional, they combine freely with a wide range of predicates and some of them typically answer questions such as comment? "how?", où? "where?", quand? "when?", etc. The compound adverbs described in LG tables [START_REF] Gross | Lexicon-Grammar. The representation of compound words[END_REF]1990) take several morphosyntactic forms: unsuffixed adverbs, e.g. demain3 "tomorrow", suffixed adverbs, e.g. nuitamment4 "by night", prepositional phrases, e.g. à la dernière minute "at the last minute", noun phrases, e.g. le cas échéant "if necessary" or adverbial clauses, e.g. jusqu'à ce que mort s'ensuive "until death comes".

These compound adverbs are classified according to their internal morphosyntactic structure which is described at the elementary level of sequences of parts of speech. The classification is based mainly on the number, type and position of the fixed and variable lexical components of adverbs. These classificatory morphosyntactic structures have a three-fold aim: first, they are intended as an aid to organize the heterogeneous compound adverbs in an electronic lexicon; second, they are intended as an aid to identify the compound adverbs in a parser; finally, they have impact on the syntactic-semantic subcategorization of compound adverbs. For instance, most of the adverbial clauses represented in table PF are interpolated clauses, e.g. si ma mémoire est bonne "if my memory serves me well".

Table 1 illustrates the sixteen formal classes, together with their defining internal morphosyntactic structure, an illustrative example 5 and the number of entries listed: Table 1. Morphosyntactic structures of French compound adverbs

Compound adverbs are represented in sixteen LG tables 6 , one for each of the defining morphosyntactic structures 7 . Unlike simple adverbs, compound adverbs are represented in the tables within a structure of elementary sentence which is composed of a verbal predicate (intransitive in most cases) and its arguments. This representation takes into account the combination of the 5 It is possible that one or more of the components defined in a morphosyntactic structure are absent. For instance, in the adverb à cent pour cent "one hundred percent", which is assigned the structure PCPC, Dét1 and Dét2 are empty. 6 The LG tables of adverbs described in this paper have been updated by the members of LIGM, so that previously implicit linguistic information becomes explicit and convenient for NLP applications. [START_REF] Tolone | Les tables du Lexique-Grammaire au format TAL[END_REF][START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du français[END_REF] adverb with a structure of elementary sentence, and, thus provides precise information about various types of constraints occurring between compound adverbs and the predicate they modify. An example of time constraint is given below:

Les tablettes (remplaceront + *ont remplacé + *remplacent) les PC dans un avenir proche "Ipads (will replace + *replaced + *replace) desktop PC in a near future"

The LG tables of French compound adverbs contain 7,283 entries. -a semantic feature holding binary values, for instance, Conjonction points out whether the compound adverb has a connector function in discourse, i.e. it links the clause in which it occurs with the previous clause as, for example, dans le cas contraire "otherwise";

-an item of information provided as an aid to help human readers find examples of sentences containing the compound adverb: features with identifiers Ppv and Prédicat type give an example of a verbal predicate that combines commonly with the adverb.

Unlike simple adverbs, the sixteen classes of compound adverbs, represented in sixteen LG tables, are both syntactically and semantically heterogeneous. For instance, the table PAC encodes adverbs that are defined by the morphosyntactic structure Prép Det Adj C, but belong to different syntactic and semantic classes of [START_REF] Molinier | Grammaire des adverbes : description des formes en -ment[END_REF]: conjuncts, e.g. dans un premier temps "initially", disjuncts, e.g. à Posss0 humble avis "in Poss0 humble opinion", adverbs of time, e.g. depuis cent sept ans "one hundred and seven years since", adverbs of verbal manner, e.g. n'importe comment "no matter how", etc.

Despite their differences, both types of adverbs are often related by productive and regular relations such as, for example, paraphrasing relations allowing the creation of pairs of synonyms, as shown in Table 3:

The syntactic lexicon LGLex

The current version of the French LG tables has to consider the use of these lexical data in NLP tools [START_REF] Tolone | Les tables du Lexique-Grammaire au format TAL[END_REF]. Therefore, the tables have been converted into an interchange format, based on the same linguistic concepts as those handled in the tables. This conversion is based on LGExtract: a generic tool for generating a syntactic lexicon for NLP from the LG tables [START_REF] Constant | A generic tool to generate a lexicon for NLP from Lexicon-Grammar tables[END_REF]. It relies, first, on a global table of classes in which we added the missing features and, second, on a single extraction script including all operations related to each feature to be performed for all tables. Thanks to LGExtract, a French lexicon for NLP has been generated from all LG tables and for most lexical-grammatical categories: verbs, predicative nouns, idioms and adverbs. This syntactic lexicon is named LGLex [START_REF] Constant | A generic tool to generate a lexicon for NLP from Lexicon-Grammar tables[END_REF][START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du français[END_REF] and it is freely available 8 under the LGPL-LR license in both plain text format and XML.

LGLex is currently composed of 13,867 verbal entries (from 67 tables), 12,696 nominal entries (from 78 tables), 39,628 idioms (from 69 tables) and 10,487 adverbial entries (from 32 tables) of which 3,203 are simple adverbs (from 16 tables) and 7,284 are compound adverbs (from 16 tables).

Extending the LGLex

Each entry of the lexicon includes three sections:

(i) section Lexical information identifies the lexical entry, for instance, the adverb jusqu'à la fin des (=de les) temps "until the end of time" which is encoded in table PCDC, and also gives the category of each lexical component. We added the information of paraphrases, other structures and other entries with intensification (see section 3.2);

(ii) section Arguments gives information about the arguments of the predicate: for instance, the subject argument N0, assigned to the predicate that can be modified by the adverb jusqu'à la fin des temps "until the end of time" is a non human noun phrase, represented by N0 = N-hum;

(iii) section Constructions enumerates the identifiers of all constructions of the lexical entry (e.g. N0 V Adv W or Adv parlant, P) and of all internal morphosyntactic structures, that is Adv for all simple adverbs or Prép1 Det1 C1 Prép2 Det2 C2 for compound adverbs like jusqu'à la fin des temps "until the end of time", but also Prép1 Det1 C1 for its variant without prepositional noun phrase modifier, e.g. jusqu'à la fin "until the end".

So, we first extended LGLex with respect to adverbial entries by using various types of features that are encoded in the tables of both simple and compound adverbs. We added 11,328 entries (+108%), so the lexicon is now composed of 21,815 adverbial entries in total.

8 http://infolingu.univ-mlv.fr/english > Language Resources > Lexicon-Grammar > Download.

Using the paraphrasing features

The viewpoint adverb linguistiquement "linguistically" accepts the following paraphrasing constructions [START_REF] Sekine | Automatic Paraphrase Discovery based on Context and Keywords between NE Pairs[END_REF] that are encoded in the table ADVMP by means of binary features:

linguistiquement "linguistically" = (du+d'un+au) point de vue linguistique "from a linguistic point of view" = du point de vue de la linguistique "from the point of view of linguistics" = au niveau linguistique "at the linguistic level" = (au+sur le) plan linguistique "on the linguistic level" = en linguistique "in linguistics" = linguistiquement parlant "linguistically speaking". For this work, we first dealt with paraphrasing constructions that are described directly in the LG tables through explicit features, as it is shown in Table 4:

Paraphrasing features encoded in the LG tables

Lexical values of the paraphrasing features in the script

Adj-ment = avec Adj-n "avec @Adj-n@" Adj-ment = (du+d'un) point de vue Adj "d'un point de vue @Adj@", "du point de vue @Adj@" Adj-ment = du point de vue de Ddef Ndomaine "du point de vue de Ddef @Ndomaine@" Adj-ment = au niveau Adj "au niveau @Adj@" Adj-ment = au point de vue Adj "au point de vue @Adj@" Adj-ment = au plan Adj "au plan @Adj@" Adj-ment = sur le plan Adj "sur le plan @Adj@" Adj-ment = de source Adj "de source @Adj@" Adj-ment = en Adj-n "en @Adj-n@" Adj-ment = en Ndomaine "en @Ndomaine@" Adj-ment = en termes Adj "en termes @Adj@" Adj-ment = en toute Adj-n "en toute @Adj-n@"

Table 4. Paraphrasing features encoded directly in LG tables

The notation @...@ specifies the lexical value of a lexical feature which is encoded systematically in a LG table. For instance, when @Adj@ refers to table ADVMP, it can take the following values: linguistique "linguistic", politique "political", informatique "computational", etc . Thanks to these features, we added to LGLex 2,084 adverbial entries (+20%).

However, a certain number of paraphrases are part of construction features, and thus need to be extracted from them. Construction features are encoded in LG tables by means of binary features, as it is shown in Table 5:

Construction features encoded in the LG tables

Lexical values of the paraphrasing features in the script Adv parlant, P "@<ENT>Adv@ parlant" N0 V W C-a-ment "@C-a-ment@" N0 V W avec Adj-n "avec @Adj-n@" N0 V W de (E+une) (façon + manière) C-a "de façon @C-a@", "de manière @C-a@", "d'une façon @C-a@", "d'une manière @Ca@" N0 V W de (E+une) (façon + manière) Adj "de façon @Adj@", "de manière @Adj@", "d'une façon @Adj@", "d'une manière @Adj@" N0hum V W avec Adj-n "avec @Adj-n@" N0hum V W de (E+une) (façon + manière) Adj "de façon @Adj@", "de manière @Adj@", "d'une façon @Adj@", "d'une manière @Adj@" à Adv parler, P "à @<ENT>Adv@ parler"

Table 5. Paraphrasing features embedded in construction features

Using this type of features, we enhanced LGLex with 7,125 adverbial entries (+68%). Using both types of paraphrasing features, the number of adverbial entries in LGLex raised from 10,487 to 19,695 (+88%).

Other structures

Within the other structures, we distinguish three types of features: deletion, permutation and transformation. They are all described in the following sections.

Using the deletion features

The adverb jusqu'à la fin des (=de les) temps "until the end of time" is defined by the morpho-syntactic structure Prép1 Det1 C1 Prép2 Det2 C2. It accepts also the substructure jusqu'à la fin "until the end", which is obtained after deletion of the prepositional noun phrase modifier des (=de les) temps "of time", and without loss of information. The new adverb has the structure Prép1 Det1 C1. "@<ENT>Det1@ @<ENT>C1@ @<ENT>Prép2@ @<ENT>C2@" Prép Det C "@<ENT>Prép@ @<ENT>Det@ @<ENT>C@" Prép1 Det1 C1 "@<ENT>Prép1@ @<ENT>Det1@ @<ENT>C1@" Table 6. Deletion features encoded in LG tables

The notation @<ENT>...@ specifies the lexical value of an entry described in a LG table. In the case of compound adverbs, an entry is composed by two or more lexical components. Thanks to the deletion features, we added to LGLex 1,519 adverbial entries (+14%).

Using the permutation features

The adverb dans un avenir proche "in a near future", which is defined by the morphosyntactic structure Prép Det C Modif pré-adj Adj, can also take the form dans un proche avenir due to the permutation of the adjectival phrase proche "near". The new adverb has the structure Prép Det Modif pré-adj Adj C, as it is shown in the Table 7: Permutation features encoded in the LG tables Lexical value of the permutation features in the script Prép Det Modif pré-adj Adj C "@<ENT>Prép@ @<ENT>Det@ @<ENT>Modif@ @<ENT>pré-adj@ @<ENT>Adj@ @<ENT>C@" Prép Modif préadj Adj C "@<ENT>Prép@ @<ENT>Modif@ @<ENT>pré-adj@ @<ENT>Adj@ @<ENT>C@"

Table 7. Permutation features encoded in LG tables

Let us consider now the compound adverb dans les délais les plus brefs "as soon as possible", which is defined by the morphosyntactic structure Prép Det C Modif pré-adj Adj. Although it accepts the permutation of the adjectival phrase les plus brefs "the shortest", it produces an agrammatical entry *dans les les plus brefs délais. This happens only when Modif préadj corresponds to a complex unit, that is, in our example, the superlative determiner les plus "the most". To remedy this situation, we added to table PCA a specific feature enabling both the deletion and permutation of certain components without loss of information, i.e. the feature Prép Modif pré-adj Adj C.

This kind of problem is rather widespread since multiword units raise specific difficulties as regards their representation 9 in an electronic resource. Semantically, by definition, compound adverbs cannot be decomposed into simple units. In other words, the resulting overall meaning of the compound adverb cannot usually be deduced from the sum of the meaning of its component elements. Consequently, the various lexical components of compound adverbs are sometimes represented in the tables in an ambiguous or even arbitrary way. This is due, of course, to their irregular syntax and internal word combination constraints.

By means of the permutation features, we enhanced LGLex with 103 adverbial entries (+1%).

Using transformational features

Finally, other structures can result from general transformations applied to the free prepositional noun phrase modifier de N "of N", which is part of semi-fixed adverbial expressions. For instance, the adverb pour le bénéfice "for the benefit", defined by the morphosyntactic structure Prép1 Det1 C1 de N2, can take the following two forms: pour le bénéfice général "for the general benefit" and pour son benefice "for his benefit". Table 8 displays the transforma-9 According to Gross (1986: 1), the unit of representation in a linear lexicon is roughly the word, defined as a sequence of letters separated from neighboring sequences by boundary blanks or other kind of separators, e.g. apostrophe. As a consequence, multiword units cannot be put directly into a lexicon the way simple words are. An identification procedure is needed for their occurrences in texts, and this procedure will make use of the various simple lexical components included in the multiword unit (and described in the LG tables at the elementary level of sequences of parts-ofspeech). Hence, the formal linguistic properties of multiword units will determine both the procedure of identification in texts and the type of storage they require. tional features that are encoded in the corresponding LG tables:

Transformation features encoded in the LG tables Lexical value of the transformational features in the script Prép1 Det1 C1 de N2 = Prép1 Det1 C1 général "@Prép1@ @Det1@ @C1@ général" Prép1 Det1 C1 de N2 = Prép1 Poss2 C1 "@Prép1@ Poss2 @C1@" Table 8. Transformational features encoded in

LG tables

Thanks to these features, we added to LGLex 288 adverbial entries (+3%).

Taking into account the three different types of features included in other structures, the number of adverbial entries in LGLex increased from 10,487 to 12,397 (+18%).

Using the intensifying features

The focus adverb particulièrement "particularly" can be modified by two specific intensifiers conveying a greater emphasis to its meaning, and thus produce the two following entries: tout particulièrement "quite particularly" and plus particulièrement "more particularly". Intensifying features are represented in LG tables as shown in Table 9:

Intensifying features encoded in the LG tables Lexical value of the intensifying features in the script bien Adv "bien @<ENT>Adv@" fort Adv "fort @<ENT>Adv@" plus Adv "plus @<ENT>Adv@" tout Adv "tout @<ENT>Adv@" très Adv "très @<ENT>Adv@" (plus+moins) Adv "plus @<ENT>Adv@", "moins @<ENT>Adv@"

Table 9. Intensifying features encoded in LG tables

Thanks to these features, the number of adverbial entries in LGLex augmented from 10,487 to 10,697 (+2%).

Evaluation of the extended LGLex

The The results are quite satisfactory as we obtain more than double, precisely 108% new entries in the lexicon, only by exploiting precise linguistic information of high coverage, which is freely available in existing resources.

We manually evaluated the new entries of the lexicon in order to detect errors.

Indeed, a manual validation was necessary when the generated entries corresponded to one single word. For example, the adverb pourboire compris "tip included", which is defined by the morphosyntactic structure Prép Det C Modif pré-adj Adj, accepted the deletion of the participial phrase modifier compris "included", but produced a non-adverbial entry *pourboire "tip".

Then, we mention the problem of duplicates that may be produced once the generation of new adverbial entries completed. They concern a few pairs of tables, notably, the pairs PCDC and PCDN, PCA and PAC or PDETC. For example, ces temps derniers "recently", defined by the morphosyntactic structure Prép Det C Modif pré-adj Adj and encoded in table PCA, can also take the form ces derniers temps due to the permutation of the adjective derniers "recent". This latter is already encoded in table PAC.

We can also evoke the case of the deleting relation that associates the adverbs of table PCDC with those of table PCDN. For example, en l'état actuel des choses "in the current state of things" accepts also the substructure en l'état actuel 10 "in the current state", which is obtained after deletion of the prepositional noun phrase modifier des choses "of things", and without loss of information. The generated substructure is already an entry of table PCDN. In a similar manner, en l'état actuel des connaissances "in the current state of knowledge" produces the same substructure. Moreover, the adverbs dans l'état actuel des choses "in the current state of things" and dans l'état actuel des connaissances "in the current state of knowledge", which are both encoded in 10 Normally, in table PCDN, the fixed part of the semi-fixed adverbial expression comprises also the second preposition de "of" which is constant for all entries of the table, and for that reason it is not encoded explicitly. table PCDC, accept the substructure dans l'état actuel "in the current state".

In fact, each substructure provides information about the corresponding entry, and the generated substructures can be filtered automatically to easily delete duplicates.

Last, errors in the new entries are sometimes due to the way initial adverbial entries are encoded in tables. Considering the adverb à cette heure-ci "at the present time" represented in table PCA: the noun component heure "time" is encoded together with the hyphen, and thus form an amalgam that is automatically reproduced in the substructure à cette heure-"at this time".

Conclusion and future work

At a time when the lack of large scale lexical syntactic resources for French impedes on NLP research, we showed the interest of using finegrained linguistic information, which is provided in existing resources, in order to enrich or diversify their content. This work led to an increase of 108% of the adverbial entries in LGLex.

These encouraging results confirm it is worthwhile exploiting features such as paraphrases. Therefore, we plan to complete the LG tables in that direction, starting, for example, with the table of verbal manner adverbs: Adj-ment = en tout Nabstrait =: amicalement = en toute amitié "friendly" "in all friendship" Adj-ment = par Nmoyen_communication =: téléphoniquement = par téléphone "by telephone" "by telephone" Furthermore, we plan to convert the new adverbial entries into the Lefff format [START_REF] Sagot | The Lefff, a freely available and large-coverage morphological and syntactic lexicon for French[END_REF], in order to integrate them into a parser, following similar work by Tolone and Sagot (2011) and [START_REF] Tolone | Analyse syntaxique à l'aide des tables du Lexique-Grammaire du français[END_REF].

Besides, the new adverbial entries will be added to the French morphological lexicon DELA, which will enable us to evaluate their accuracy by means of both a corpus" annotation practice and a detailed comparison with related work by Laporte et al. (2008).

Finally, we can also consider enhancing the French Wordnet with respect to adverbial entries [START_REF] Sagot | Extending the adverbial coverage of a French wordnet[END_REF].

  Table 2 displays a sample of the table PCA which is defined by the morphosyntactic structure Preposition, Determin-

er, Constrained noun, Pre-adjectival Modifier, Adjective:

Table 2

 2 Prép Det Modif pré-adj Adj C describes the possible (without loss of information) of the adjectival phrase represented in this table as Modif pré-adj Adj; moreover, Neg obl encodes the constraint that the adverbial occurs obligatorily in a negative clause;

	Adj;
	-a syntactic feature holding binary values, for
	example:

. Compound adverbs of table PCA

In this table, each row corresponds to a lexical item with adverbial function, and each column corresponds to:

-one of the components in the morphosyntactic structure of the items, i.e. features with identifiers Prép, Det, C, Modif pré-adj, and

Table 3 .

 3 Paraphrasing relations between simple and compound adverbs

  Table 6 displays the deletion features present in LG tables of adverbs:

	Deletion features encoded	Lexical value of the dele-
	in the LG tables	tion features in script
	ConjC Det1 C1
	Prép2 C2 = Det1 C1
	Prép2 C2

  Table 10 shows the number of the initial adverbial entries in LGLex and the detail of the 11,328 new entries:

	Initial entries	10,487
	Paraphrases	9,208
	Other structures	1,910
	Intensifying features	210
	Final entries	21,815

Table 10 .

 10 Number of entries in LGLex

According toGross (1990: 153), demain "tomorrow" is a compound form from an etymological point of view: from the Latin expression de mane which means literally "in the morning". But, it is now regarded as a simple form since it is represented by a single word. These forms are derived from noun or prepositional noun phrases previously analyzed that were tied at various times.

 4 According toGross (1986: 2), nuitamment "by night" can also be considered as an idiomatic compound, though not constituted of words but of a word and a suffix. Lack of compositionality stems from the observation that quotidiennement "daily', mensuellement "monthly', etc. which are derived adverbs of the same formal type have a regular formation, in the sense that their interpretation is homogeneous. Thus, nuitamment "by night" is an isolated case, as opposed to an open series of identical forms with a different interpretation.