K Boughrara

R Ibtiouen

T Lubin

Analytical Prediction of Magnetic Field in Parallel Double Excitation and Spoke-Type Permanent-Magnet Machines Accounting for Tooth-Tips and Shape of Polar Pieces

Keywords: Exact analytical calculation, finite element method, magnetic field distribution, parallel double excitation, Spoke-Type permanent magnet machines

This paper presents an analytical method based on subdomain method for the computation of open circuit, armature reaction and on-load magnetic field distribution in integer slot winding parallel double excitation and Spoke-Type tangential permanent-magnet machines. The proposed model takes into account for stator and rotor slots tooth tips and shape of polar piece. A 2D exact analytical solution of magnetic field distribution is established. It involves solution of Laplace's and Poisson's equations in semi-closed stator and rotor slots, airgap, buried permanent magnets into rotor semi-closed slots, and non magnetic region under magnets. Obtained exact analytical results of open circuit, armature reaction and on-load magnetic field distribution are verified with those issued from finite element method (FEM).

I. INTRODUCTION

ouble excitation machines are synchronous machines that have two coexisting excitation field sources: permanent magnets and wound field excitation. The goal behind the use of double excitation principle is to combine advantages of permanent-magnet excited machines and wound field synchronous machines. The combination of permanent magnets and wound field excitation constitutes an additional degree of freedom which can be used to optimize the energy consumption of the electric propulsion system. Among various double excitation machines, parallel double excitation machines have some advantages than series double excitation machines [START_REF] Amara | Hybrid Excitation Synchronous Machines: Energy Efficient Solution for Vehicles Propulsion[END_REF]- [START_REF] Bali | Analytical Modeling of Open Circuit Magnetic Field in Wound Field and Series Double Excitation Synchronous Machines[END_REF]. The flux created by excitation coils and permanent magnets have different trajectories and the flux created by excitation coils doesn't pass through permanent magnets, hence the demagnetization risk is avoided. Depending on DC excitation current direction, excitation coils can either be used to enhance or decrease excitation flux passing through armature windings. There are no authors who applied analytical model for predicting magnetic field in this type of hybrid excitation machines with slotted rotor and stator and buried tangential permanent magnet taking into account rotor and stator tooth tips and real structure of polar piece. Analytical methods are useful tools for a first evaluation of electrical motors performances and for design optimization. However, finite element method is used at the final stage of the design. In reference [START_REF] Bali | Analytical Modeling of Open Circuit Magnetic Field in Wound Field and Series Double Excitation Synchronous Machines[END_REF], exact analytical calculation of magnetic field distribution is developed for a slotted stator series double excitation machine, where coils current excitation are situated in the slotted rotor with radial surface mounted permanent magnets. There are other structures of double excitation permanent magnet machines with complicated geometry [START_REF] Owen | Hybrid-Excited Flux-Switching Permanent-Magnet Machines With Iron Flux Bridges[END_REF]- [START_REF] Sulaiman | Design Optimization of 12Slot-10Pole Hybrid Excitation Flux Switching Synchronous Machine with 0.4kg Permanent Magnet for Hybrid Electric Vehicles[END_REF], where the study is done by finite element method only.

Permanent magnet brushless dc (BLDC) motors have been widely used due to their many advantages such as high torque and high efficiency. In particular, the spoke-type BLDC motor, which can concentrate the flux from low cost ferrite permanent magnets, has a high torque density per unit volume resulting from the additional reluctance torque. The prediction of open circuit, armature reaction and on load magnetic field distribution of this type of motors is done generally by the finite element method (FEM) [START_REF] Hwang | A Study on Optimal Pole Design of Spoke-Type IPMSM With Concentrated Winding for Reducing the Torque Ripple by Experiment Design Method[END_REF]- [START_REF] Hao Qian | Multi-Domain Analysis of Tangential-set Permanent Magnet Synchronous Generator[END_REF] and there are very few authors which use analytical methods [START_REF] Lin | Analytical Prediction of Cogging Torque in Spoke Type Permanent Magnet Motors[END_REF]- [START_REF] Li | Analytical Calculation of Inductance Parameter of Tangentially Magnetized PM Motor Based on Basic Air-gap Magnetic Field[END_REF]. This is due to the structure of those machines, where buried tangential PM region has high height and low width with presence of non magnetic region under magnet. In reference [START_REF] Lin | Analytical Prediction of Cogging Torque in Spoke Type Permanent Magnet Motors[END_REF], Lin et all. used conformal transformation of rotor and stator slots to determine a relative permeance function to take into account the rotor and stator slotting effects. In reference [START_REF] Li | Analytical Calculation of Inductance Parameter of Tangentially Magnetized PM Motor Based on Basic Air-gap Magnetic Field[END_REF], authors used also a Schwarz-Christoffel transformation to determine d-axial and q-axial airgap magnetic field. Conformal transformation leads to an approximation of magnetic field distribution in a simplified structure of slotted stator spoketype permanent magnet motors. Investigations in applying analytical and numerical conformal mapping have been done in [START_REF] Boughrara | Modélisation Analytique et Numérique des Performances Electromagnétiques des Moteurs Synchrones à Aimants Permanents[END_REF] and show the inaccuracy of conformal mapping method in magnetic field modeling of this type of machines. This is not due to the stator slotting effect but to the presence of a deep and small thickness of permanent magnet region. To the author's knowledge, there is no study on the slotted or slotless Spoke-Type PM machines that solves analytically Poisson's and Laplace's equations. However, many authors have used exact analytical method based on subdomain model to take into account stator slotting effect and tooth tips in open circuit and armature reaction magnetic field with circular and tubular radial surface mounted and inset permanent magnet motors [START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF]- [START_REF] Wang | Analysis and Design Optimization of an Improved Axially Magnetized Tubular Permanent-Magnet Machine[END_REF]. Inset PM subregion in PM machines is not deep and the width is high. So, it can be modeled accurately D with conformal mapping [START_REF] Boughrara | Magnetic Field Analysis of Inset and Surface Mounted Permanent Magnet Synchronous Motors using Schwarz-Christoffel Transformation[END_REF]. In references [START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF]- [START_REF] Wu | An Improved Subdomain Model for Predicting Magnetic Field of Surface-Mounted Permanent Magnet Machines Accounting for Tooth-Tips[END_REF], authors have developed an exact analytical solution based on subdomain model for predicting open circuit and armature reaction in slotted stator radial surface mounted permanent magnet motor taking into account tooth tips. In [START_REF] Jian | Analytical Calculation of Magnetic Field in Surface-Inset Permanent Magnet Motors[END_REF] and [START_REF] Jian | Analytical Method for Magnetic Field Calculation in a Low-Speed Permanent-Magnet Harmonic Machine[END_REF], scalar potential magnetic field is used to predict open circuit magnetic field in slotted stator surface and inset radial permanent magnet motors. Authors in [START_REF] Amara | Armature Reaction Magnetic Field of Tubular Linear Surface-Inset Permanent-Magnet Machines[END_REF] and [START_REF] Amara | Analytical Modeling of Magnetic Field I Surface Mounted Permanent-Magnet Tubular Linear Machines[END_REF] have developed an exact analytical solution based on subdomain model for predicting open circuit and armature reaction in tubular surface mounted and inset radial permanent magnet machines. In [START_REF] Gysen | General Formulation of the Electromagnetic Field Distribution in Machines and Devices Using Fourier Analysis[END_REF]- [START_REF] Wang | Analysis and Design Optimization of an Improved Axially Magnetized Tubular Permanent-Magnet Machine[END_REF], the authors have developed respectively an exact analytical solution also based on subdomain model for flux switching permanent magnet machines and a tubular slotless stator with axially magnetized permanent magnet in rotor.

In this paper, an exact analytical prediction based on subdomain model of open circuit, armature reaction and onload magnetic field distribution in integer slot winding parallel double excitation and spoke-type tangential permanent magnet machines is presented. It involves the solution of Poisson's and Laplace's equations in semi-closed stator slots, buried permanent magnets placed in semi-closed slots, rotor double excitation semi-closed slots, airgap and non magnetic region under permanent magnets. To handle permanent magnets, rotor and stator slots current at the same time, The magnetic vector potential formulation is used. All results from the developed analytical model are then compared to those found by the finite element method (FEM).

II. MAGNETIC FIELD SOLUTION IN PARALLEL DOUBLE EXCITATION PM MACHINES

Figs. 1 and2 show the machine model where region I represents the air gap, region II the magnets, regions III and V the stator semi-closed slots, region IV a non magnetic material under magnets, region VI the rotor slots at the top of permanent magnets, regions VII and VIII the rotor excitation semi-closed slots. The model is formulated in vector potential and two-dimensional polar coordinates with the following assumptions.

The stator and rotor cores are assumed to be infinitely permeable.

Eddy current effects are neglected. The axial length of the machine is infinite i.e. end effects are neglected.

The current density has only one component along the zaxis.

The stator and rotor slots have radial sides. The partial differential equations for magnetic fields in a continuous and isotopic region in term of vector potential A which has one component in z direction and independent of z can be expressed by

A M µ ∇ = -∇ × , in region II (2) 2 0 A J µ ∇ = - , in region III (3) 2 0 f A J µ ∇ = - , in region VII (4)
where M is the magnetization of permanent magnets, J the stator slots current density, J f the excitation rotor slots current density and µ 0 the permeability of vacuum.

A. General Solution of Poisson's Equation in Stator Semiclosed Slot Subdomain (Region III)

In each slot subdomain (i) of region III (Fig. 3), we have to solve Poisson's equation

i i i i AIII AIII AIII J r r r r µ θ ∂ ∂ ∂ + + = - ∂ ∂ ∂ (8)
where J i is the current density in the slot i.

As shown in Fig. 3, the ith slot subdomain (region III) where i vary from 1 to Q s (Q s is the number of stator slots) is associated with boundary conditions at the bottom and at each sides of the slot as where α i is the angular position of the ith slot and c the slot opening in radian.

The boundary condition [START_REF] Hwang | A Study on Optimal Pole Design of Spoke-Type IPMSM With Concentrated Winding for Reducing the Torque Ripple by Experiment Design Method[END_REF], leads to the eigenvalues and eigenfunctions of the partial differential equation ( 8) as shown in details in [START_REF] Lubin | Exact Analytical Method for Magnetic Field Computation in the Air Gap of Cylindrical Electrical Machines Considering Slotting Effects[END_REF]. The eigenfunctions are called spatial frequencies of the considered region [START_REF] Gysen | Modeling of Flux Switching Permanent Magnet Machines With Fourier Analysis[END_REF]. The boundary condition [START_REF] Son | Maximum Torque Control for Optimal Design to Reduce Cogging Torque in Spoke Type Interior Permanent Magnet Synchronous Motor[END_REF], leads to the general solution of equation ( 8) with only two integration constants C i,0 and C i,m as shown in [START_REF] Hao Qian | Multi-Domain Analysis of Tangential-set Permanent Magnet Synchronous Generator[END_REF].

From above boundary conditions ( 9) and ( 10), the solution of (8) using the method of separation of variables is

( ) ( ) 2 2 ,0 0 4 0 1 1 , ln 2 4 i i i i AIII r C J r r J r θ µ µ = + - ( 11
) , 1 4 4 cos 2 m m c c i m i m r r m c C r r c π π π θ α - ∞ =             + - -+                       ∑
where m is a positive integer.

B. General Solution of Laplace's Equation in Stator Semi-Closed Slot Subdomain (Region V)

In each stator semi-closed slot subdomain (i) of region V (Figs. 2, 3), we have to solve Laplace's equation

i i i AV AV AV r r r r θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (12)
As shown in Fig. 3, the ith slot subdomain (region V) where i vary from 1 to Q s (Q s number of stator slots) is associated with the following boundary conditions

2 0 i i d AV θ α θ = - ∂ = ∂ and 2 0 i i d AV θ α θ = + ∂ = ∂ (13)
where d is the semi-closed slot opening in radian. From above boundary conditions [START_REF] Li | Analytical Calculation of Inductance Parameter of Tangentially Magnetized PM Motor Based on Basic Air-gap Magnetic Field[END_REF], the solution of (12) using the method of separation of variables is

( ) ( ) ,0 ,0 , 11 12
ln i i i AV r A A r θ = + (14)
, , 1 [START_REF] Hao Qian | Multi-Domain Analysis of Tangential-set Permanent Magnet Synchronous Generator[END_REF] 12 cos 2

k k d d i k i k i k k d A r A r d π π π θ α ∞ - =       + + -+             ∑
where k is a positive integer.

C. General Solution of Poisson's Equation in Permanent Magnet Subdomain (Region II)

In each permanent magnet subdomain (j) of region II (Figs. 2 and4), we have to solve Poisson's equation [START_REF] Shanming | State of the Art of Hybrid Excitation Permanent Magnet Synchronous Machines[END_REF]. The magnetization of parallel double excitation motor is purely tangential. Equation ( 2) is then reduced to

2 2 0 2 2 2 1 1 j j j AII AII AII M r r r r r θ µ θ ∂ ∂ ∂ + + = - ∂ ∂ ∂ (15)
where ( )

0 1 j rem j B M M θ µ = = -
For a 2p poles machine, j vary from 1 to 2p and B rem is the remanence of magnetization.

As shown in Fig. 4, the jth magnet subdomain (region II) is associated with the following boundary conditions where g j is the angular position of the jth magnet and a the magnet opening in radian.

From above boundary conditions ( 16), the general solution of (15) using the method of separation of variables is given by ( ) ( )

,0 ,0 0 , 5 6 ln

j j j j AII r A A r M r θ µ = + - (17)
, , 1 [START_REF] Bali | Analytical Modeling of Open Circuit Magnetic Field in Wound Field and Series Double Excitation Synchronous Machines[END_REF] 6 cos 2

m m a a j m j m j m m a A r A r g a π π π θ ∞ - =       + + -+             ∑ D.

General Solution of Laplace's Equation in Rotor Semi-Closed Slot Subdomain (Region VI)

In each rotor semi-closed slot subdomain (j) of region VI (Figs. 2,4), we have to solve Laplace's equation

2 2 2 2 2 1 1 0 j j j AVI AVI AVI r r r r θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (18)
As shown in Fig. 4, the jth slot subdomain (region VI) where j vary from 1 to 2p is associated with the following boundary conditions where b is the semi-closed slot opening in radian. From above boundary conditions [START_REF] Jian | Analytical Method for Magnetic Field Calculation in a Low-Speed Permanent-Magnet Harmonic Machine[END_REF], the solution of (18) using the method of separation of variables is

( ) ( )
,0 ,0 , 13 14 ln

j j j AVI r A A r θ = + (20)
, , 1 [START_REF] Li | Analytical Calculation of Inductance Parameter of Tangentially Magnetized PM Motor Based on Basic Air-gap Magnetic Field[END_REF] 14 cos 2

k k b b j k j k j k k b A r A r g b π π π θ ∞ - =       + + -+             ∑ E.

General Solution of Laplace's Equation in Airgap Subdomain (Region I)

The Laplace equation (1) in the airgap subdomain (region I) which is an annular domain delimited by the radii R m and R s (Fig. 2) is given by

2 2 2 2 2 1 1 0 AI AI AI r r r r θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (21)
Taking into account the periodicity boundary condition between θ=0 and θ=2π/p for the studied machine with integer slot winding, the solution of equation ( 21) is

( ) ( ) ( ) 1 , 1 2 sin np np n n n AI r A r A r np θ θ ∞ - = = + ∑ (22) ( ) ( ) 3 4 cos np np n n A r A r npθ - + +
where n is a positive integer.

F. General Solution of Laplace's Equation in the Non Magnetic Subdomain (Region IV)

The Laplace's equation (1) in the non magnetic subdomain region IV which is an annular domain delimited by the radii R i and R r and where the relative recoil permeability is equal to 1 is given by

2 2 2 2 2 1 1 0 AIV AIV AIV r r r r θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (23)
Taking into account the periodicity boundary condition between θ=0 and θ=2π/p, the general solution of ( 23) is

( ) ( ) ( ) 1 , 7 8 sin np np n n n AIV r A r A r np θ θ ∞ - = = + ∑ (24)
( ) ( )

9 10 cos np np n n A r A r npθ - + +
The magnetic vector potential must be finite in region IV. In this case, constants A8 n and A10 n are equals to zero and equation ( 24) is reduced to

( ) ( ) 1 , 7 sin np n n AIV r A r np θ θ ∞ = = ∑ (25)
( )

9 cos np n A r npθ + G.

General Solution of Poisson's Equation in Rotor Excitation Coil Slot Subdomain (Region VII)

In each rotor slot subdomain (ir) of region VII, we have to solve Poisson's equation ( 26)

2 2 0 2 2 2 1 1 ir ir ir f ir AVII AVII AVII J r r r r µ θ ∂ ∂ ∂ + + = - ∂ ∂ ∂ (26)
where J fir is the current density in rotor slot ir.

As shown in Fig. 5, the irth slot subdomain (region VII) where ir vary from 1 to N r (N r is total number of rotor excitation slots) is associated with the following boundary conditions where β ir is the angular position of the irth slot and c r the rotor slot opening in radian.

From the above boundary conditions ( 27) and ( 28), the solution of ( 26) using the method of separation of variables is

( ) ( ) 2 2 ,0 0 1 0 1 1 , 1 ln 2 4 ir ir f f ir ir AVII r C J r r J r θ µ µ = + - , 1 1 1 1 . r r m m c c ir m m r r C r r π π - ∞ =         + -               ∑ (29) cos 2 r ir r m c c π θ β     -+        

H. General Solution of Laplace's Equation in Rotor Excitation Coil Slot Subdomain (Region VIII)

In each semi-closed slot subdomain (ir) of region VIII (Fig. 5), we have to solve Laplace's equation

2 2 2 2 2 1 1 0 ir ir ir AVIII AVIII AVIII r r r r θ ∂ ∂ ∂ + + = ∂ ∂ ∂ (30)
As shown in Fig. 5, the irth slot subdomain (region VIII) where ir vary from 1 to N r is associated with the following boundary conditions where dr is the semi-closed slot opening in radian. From previous boundary conditions (31), the solution of partial differential equation (30) using the method of separation of variables is

( ) ( ) ,0 ,0 , 15 16 ln ir ir ir AVIII r A A r θ = + (32)
, , 1 [START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF] 16 .

k k dr dr ir k ir k k A r A r π π ∞ - =   + +     ∑ cos 2 ir k dr dr π θ β     -+        

III. BOUNDARY AND INTERFACE CONDITIONS

To determine Fourier series unknown constants A1 n , A2 n , A3 n , A4 n , A5 j,0 , A6 j,0 , A5 j,m , A6 j,m , A7 n , A9 n , A11 i,0 , A12 i,0 , A11 i,k , A12 i,k , A13 j,0 , A14 j,0 , A13 j,k , A14 j,k, C i,0 , C i,m , C1 ir,0 , C1 ir,m , A15 ir,0 , A16 ir,0 , A15 ir,k and A16 ir,k , boundary and interface conditions should be introduced.

The interface conditions between regions IV and II at R r are

( ) ( ) , , j r r AII R AIV R θ θ = (33) where 2 2 j j a a g g θ -≤ ≤ + . ( ) ( ) , , θ θ = (37) where 2 2 j j b b g g θ -≤ ≤ + .
The interface condition between regions I and VIII at R m is The interface conditions between regions I and V at R s are

( ) ( ) , , m ir m AI R AVIII R θ θ = (38)
( ) ( ) , , s i s AI R AV R θ θ = ( 40
)
where

2 2 i i d d α θ α -≤ ≤ + . ( ) ( ) , , s s i HI R HV R θ θ θ θ = (41) where 2 2 i i d d α θ α -≤ ≤ + . ( ) , 0 s HI R θ θ = elsewhere.
The interface conditions between regions III and V at r 3 are

( ) ( ) 3 3
, ,

i i AIII r AV r θ θ = (42) where 2 2 i i d d α θ α -≤ ≤ + . ( ) ( ) 3 3
, , Interface conditions (33) to (45) concern regions with different subdomain frequencies which need Fourier series expansions to satisfy equalities of vector potential and magnetic field at each interface radius.

Interface condition (33) represents also the continuity of radial flux density at the permanent magnet opening at R r . According to Fourier series expansion, we obtain two equations as ( ) ,0 ,0 0 5 6 ln

j j r j r A A R M R µ + - (46)
( )

2 2 1 , j j a g r a g AIV R d a θ θ + - = ∫ , , 5 6 m m a a j m r j m r A R A R π π     -        + (47) ( ) 2 2 2 , cos 2 j j a g r j a g m a AIV R g d a a π θ θ θ + -     = -+         ∫ Interface condition (34) gives ( ) 1 0 7 np n r np A R µ -   -     (48) ( ) ( ) 2 2 1 2 1 , sin j j a g p r j j a g HII R np d θ θ θ θ π + = - = ∑ ∫ ( ) 1 0 9 np n r np A R µ -   -     (49) ( ) ( ) 2 2 1 2 1 , cos j j a g p r j j a g HII R np d θ θ θ θ π + = - = ∑ ∫
Fourier series expansion of interface condition (35) between region II and VI at radius r 2 gives ( ) ,0 ,0 2 13 14 ln

j j A A r + (50) ( ) 2 2 2 1 , j j b g j b g AII r d b θ θ + - = ∫ , 2
, 2 [START_REF] Li | Analytical Calculation of Inductance Parameter of Tangentially Magnetized PM Motor Based on Basic Air-gap Magnetic Field[END_REF] 14

k k b b j k j k A r A r π π - + (51) ( ) 2 2 2 2 , cos 2 j j b g j j b g k b AII r g d b b π θ θ θ + -     = -+         ∫ From interface condition (36), we get ,0 2 0 6 1 j r A r µ µ - = ( ) 2 2 2 1 , j j b g j b g HVI r d a θ θ θ + - ∫ (52) 1 1 , 2 , 2 0 1 5 6 m m a a j m j m r m A r A r a π π π µ µ - - -   -     (53) ( ) 2 2 2 2 , cos 2 j j b g j j b g m a HVI r g d a a θ π θ θ θ + -     = -+         ∫
At radius R m , Fourier series expansions of the three interface conditions (37) to (39) between regions I, VI and VIII result in 6 equations. Interface condition (37) gives

( ) ( ) 2 ,0 ,0 2 1 13
14 ln ,

j j b g j j m m b g A A R AI R d b θ θ + - + = ∫ (54)
, , [START_REF] Li | Analytical Calculation of Inductance Parameter of Tangentially Magnetized PM Motor Based on Basic Air-gap Magnetic Field[END_REF] 14

k k b b j k m j k m A R A R π π - + (55) ( ) 2 2 2 , cos 2 j j b g m j b g k b AI R g d b b π θ θ θ + -     = -+         ∫
Fourier series expansion of interface condition (38) gives

β β π θ θ β + -     = -+         ∫
Fourier series expansion of interface condition (39) gives ( ) ( )

1 1 0 1 2 np np n m n m np A R A R µ - -- - + (58) ( ) ( ) 2 2 1 2 1 , sin j j b g p j m j b g HVI R np d θ θ θ θ π + = - = ∑ ∫ ( ) ( )
1 1 0 3 4 np np n m n m np A R A R µ - -- - + (59) ( ) ( ) 2 2 1 2 1 , cos j j b g p j m j b g HVI R np d θ θ θ θ π + = - = ∑ ∫ ( ) ( )
, , [START_REF] Hao Qian | Multi-Domain Analysis of Tangential-set Permanent Magnet Synchronous Generator[END_REF] 12

k k d d i k s i k s A R A R π π - + (61) ( ) 2 2 2 , cos 2 i i d s i d k d AI R d d d α α π θ θ α θ + -     = -+         ∫
Interface condition (41) gives ( )

1 1 0 1 2 np np n s n s np A R A R µ - -- - + (62) ( ) ( ) 2 1 2 1 , sin i s i d Q i s i d HV R np d α θ α θ θ θ π + = - = ∑ ∫
( )

1 1 0 3 4 np np n s n s np A R A R µ - -- - + (63) ( ) ( ) 2 1 2 1 , cos i s i d Q i s i d HV R np d α θ α θ θ θ π + = - = ∑ ∫
Fourier series expansion of interface condition (42) gives

( ) ( ) 2 ,0 ,0 3 3 2 1 11
12 ln ,

i i d i i i d A A r AIII r d d α α θ θ + - + = ∫ (64)
, 3 , 3 [START_REF] Hao Qian | Multi-Domain Analysis of Tangential-set Permanent Magnet Synchronous Generator[END_REF] 12

k k d d i k i k A r A r π π - + (65)
( )

2 3 2 2 , cos 2 i i d i i d k d AIII r d d d α α π θ θ α θ + -     = -+         ∫
Fourier series expansion of interface condition (43) gives

2 0 4 0 3 0 3 1 1 1 2 2 i i J r J r r µ µ µ   - +     (66) ( ) 2 3 2 1 , i i d i d HV r d c α θ α θ θ + - = ∫ , 3 3
0 3 4 4 m m c c i m C r r m cr r r π π π µ -           - +                   (67) ( ) 2
β β π θ θ β θ + -     = -+         ∫
Fourier series expansion of interface condition (45) gives

2 0 1 0 0 0 0 1 1 1 2 2 fir fir J r J r r µ µ µ   - +       (70)
        - + =               (71) ( ) 2
    -+        
∫ Some developments of equations ( 46) to (71) are given in appendix.

From equations ( 46)-( 71) we can calculate the 26 coefficients A1 n , A2 n , A3 n , A4 n , A5 j,0 , A6 j,0 , A5 j,m , A6 j,m , A7 n , A9 n , A11 i,0 , A12 i,0 , A11 i,k , A12 i,k , A13 j,0 , A14 j,0 , A13 j,k , A14 j,k, C i,0 , C i,m , C1 ir,0 , C1 ir,m , A15 ir,0 , A16 ir,0 , A15 ir,k , A16 ir,k with a given number of harmonics for n, m and k.

Spoke-Type PM machine shown in Fig. 6 is a particular case of parallel double excitation PM machine. The analytical developments are the same where subregions VII and VIII are omitted. Fig. 6. Studied Spoke-Type PM machine.

IV. RESULTS AND VALIDATION

The developed exact analytical method based on subdomain model for parallel double excitation and Spoke-Type permanent magnet machines taking into account rotor and stator slot tooth tips and real structure of polar piece is used to determine open circuit, armature reaction and on load magnetic field distribution in the middle of the airgap. The main dimensions and parameters of the studied machine are given in Table I. Then, analytical results are verified by finite element method (FEM) [START_REF] Meeker | Finite Element Method Magnetics[END_REF]. For the FE simulations, we have used 92882 nodes and 184906 elements.

In order to calculate armature reaction, on load and rotor double excitation magnetic field distribution, stator current density in slots is defined by a matrix connection between phase current and stator slots as 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1

0 1 0 0 1 0 0 1 0 0 1 0 C - -     = - -     - -   (72)
The studied machine is fed by 120° rectangular currents and has an integer number of slot per pole and per phase (Table I).

Stator current densities in slots are defined as

[ ] 0 T c i m m N J C I I S = - ( 73
)
where S is the area of slot, N c the number of conductors in slot and I m the stator phase peak current.

Rotor double excitation current density for one pole pair is defined as

[ ] 1 1 1 1 f f f nr f N I J S = -- (74)
where N f is the number of conductors in slot, I f the DC excitation current and S f the surface of rotor slot.

Analytical calculation of magnetic field can be done for one pole pair to minimize the number of harmonics for n, k and m and to reduce the computation time. It should be noted that the number of harmonics are defined carefully and including proper scaling of machine model dimensions to get good results and avoid solution to diverge [START_REF] Gysen | General Formulation of the Electromagnetic Field Distribution in Machines and Devices Using Fourier Analysis[END_REF]. Finite element method which uses a meshed geometry has a limited accuracy related to the density of the mesh. Analytical subdomain model based on Fourier theory exhibits a similar problem. The inaccuracies of the proposed method are related to the limited amount of harmonics included in the solution. When the number of harmonic terms increase, the system of equations written in matrix form can be ill-conditioned and the solution becomes inaccurate. This problem has been reduced in our calculations by including proper scaling of machine model dimensions in radial direction for all regions 10 xc scal real r r = (75) for a given real radii of the studied machine (Table I), new scaled radii are given by equation (75).

Limiting the number of harmonics for limiting computational time will lead to inaccurate field solutions, especially for PM machines where airgap length is small, PM subregion with high height and small thickness and high number of subregions with different spatial frequencies. For the studied parallel double excitation PM machine, the number of subregions is equal to 50. To obtain an accurate magnetic field solution, xc = 1.45 with the number of harmonics 250 for n and 80 for m and k. An optimum number of harmonics and scaling factor with limiting computational time can be found with using for example, different harmonics for all of the 8 regions. Of course, the number of harmonics in the airgap and PM should be high but the number of harmonics in the non magnetic region VI can be small. Figs. 7 to 14 show a comparison between analytical and FEM open circuit, armature reaction and on load magnetic field distribution results in the middle of the airgap of parallel double excitation PM machine accounting for tooth tips and real structure of polar piece. The results are in excellent agreement. This type of PM machines has a small magnet arc to pole pitch ratio (0.2) and a very high height (29.8 mm) in comparison with radial surface mounted and inset PM machines which present a small PM height and high width. Analytical method based on subdomain model give a very good accuracy even when PM subregion is deep. In Figs. 7 to 12, radial and tangential flux density in the middle of the airgap is given for 3 conditions: Permanent magnets alone; stator currents alone and rotor excitation currents alone. The radial and tangential flux density distribution for on-load condition is shown in Figs. 13 and14. The machine is fed by 120° rectangular stator current and we consider the permanent magnets and rotor DC current acting together. Figs. 15 and16 show the flux lines obtained by FEM due to rotor excitation current and stator currents. We can observe that all analytical results are in very good agreement with those obtained by FEM. In Figs. 17 to 20, we show radial and tangential flux densities for on load condition in the cases where the hybrid PM machine has stator and rotor slots fed by current and PM does not exist and only stator slots are fed and PM are considered. Those two cases represent the contribution of permanent magnet and rotor excitation current to magnetic field on load. In control process of magnetic field in the air gap, current excitation can be set to zero to decrease magnetic field. It is important to note here that the results are obtained for a recoil relative permeability of permanent magnet equal to unity, but the method take into account the effect of non unity permeability. Figs. [START_REF] Amara | Analytical Modeling of Magnetic Field I Surface Mounted Permanent-Magnet Tubular Linear Machines[END_REF] to 26 show open circuit, armature reaction and on load radial and tangential flux density in the middle of airgap in integer slot winding Spoke-Type PM machine accounting for tooth tips and real structure of polar piece. All analytical results are in very good agreement with those obtained by FEM, which confirm the accuracy of analytical subdomain model in predictions of this type of permanent magnet machines even when a PM subregion is very deep with small thickness and with taking into account tooth tips and real structure of polar piece. Fig. 27 shows magnetic field on load obtained by FEM in Spoke-Type PM machine. In this paper, we have proposed an improved analytical subdomain model for predicting open circuit, armature reaction and on-load magnetic field distribution in integer stator slot distributed winding parallel double excitation and Spoke-Type permanent magnet machines. The proposed model takes into account the rotor and stator slots tooth tips and the shape of polar pieces. The whole domain is divided into eight subregions for parallel double excitation motor and six subregions for Spoke-Type permanent magnet motor, stator semi-closed slots, rotor semi-closed slots, airgap, buried tangential permanent magnet and non magnetic region under magnet. Poisson's and Laplace's equations are solved analytically using the method of separation of variables. Analytical results are in excellent agreement with the ones obtained by FEM. From these results, we have shown the accuracy of analytical subdomain model to predict magnetic field accounting for rotor and stator tooth tips and shape of polar piece, even when PM subregion is deep with small thickness.

APPENDIX

Fourier series coefficients of general solution in different regions of parallel double excitation and Spoke-Type permanent magnet machines are determined by resolution of a system of equations as seen above. Some of those equations are detailed as follows. From equation (46), we get ( ) ,0 ,0 0 5 6 ln

j j r j r A A R M R µ + - (A-1) ( ) ( ) 2 1 2 1 7 sin j j a g np n r n a g A R np d a θ θ + ∞ = - = ∑ ∫ ( ) ( ) 2 1 2 1 9 cos j j a g np n r n a g A R np d a θ θ + ∞ = - + ∑ ∫ Development of equation (47) gives: , , 5 6 m m a a j m r j m r A R A R π π - + (A-2) ( ) 1 2 7 . np n r n A R a ∞ = = ∑ ( ) 2 2 sin cos 2 j j a g j a g m a np g d a π θ θ θ + -     -+         ∫ ( ) 1 2 9 . np n r n A R a ∞ = + ∑ ( ) 2 2 cos cos 2 j j a g j a g m a np g d a π θ θ θ + -     -+         ∫ where ( ) 2 2 sin cos 2 j j a g j a g m a np g d a π θ θ θ + -     -+         ∫ ( ) ( ) ( ) 2 2 2 2 2 2 1 1 cos cos 2 m j npa npa npg m p n a π   -+ -     = - ( ) ( ) ( ) 2 2 2 2 2 2 1 1 sin sin 2 m j npa npa npg m p n a π   ---     + -
    - + +         = - sin 2 2 j npa a npg   - +     - for 2 2 2 2 2 0 m p n a π - = and ( ) 2 2 cos cos 2 j j a g j a g m a np g d a π θ θ θ + -     -+         ∫ ( ) ( ) ( ) 2 2 2 2 2 2 1 1 sin cos 2 m j npa npa npg m p n a π   + -     = - - ( ) ( ) ( ) 2 2 2 2 2 2 1 1 sin cos 2 m j npa npa npg m p n a π   -+ -     - - for 2 2 2 2 2 0 m p n a π - ≠ and cos sin 2 2 2 4 j j npa npa a npg npg np     - + - +         = + 3 sin 2 4 j npa npg np   +     + for 2 2 2 2 2 0 m p n a π - = .
From equation (48), we have:

( )

1 0 7 np n r np A R µ -   -     (A-3) 2 1 1 , , 1 1 0 1 5 6 . m m p a a j m r j m r j m r m m A R A R a a π π π π πµ µ ∞ - - - = =   = -     ∑∑ ( ) 2 2 sin cos 2 j j a g j a g m a np g d a π θ θ θ + -     -+         ∫ ( ) 2 2 ,0 1 0 2 6 1 sin j j a g p j j a r r g A np d R θ θ πµ µ + = -   -    ∑ ∫ Equation (49) gives ( ) 1 0 9 np n r np A R µ -   -     (A-4)
    -+         ∫ ( ) 2 2 ,0 1 0 2 6 1 cos j j a g p j j a r r g A np d R θ θ πµ µ + = -   -    ∑ ∫ Equation (50) gives ( )
,0 ,0 2 13 14 ln

j j A A r + (A-5) ( ) ,0 ,0 2 0 2 5 6 ln j j j A A r M r µ = + - , 2 , 2 1 1 5 6 . m m a a j m j m m A r A r b π π ∞ - =   + +     ∑ 2 2 cos 2 j j b g j b g m a g d a π θ θ + -     -+         ∫ where 2 2 cos 2 j j b g j b g m a g d a π θ θ + -     -+         ∫ 2 sin cos 2 2 mp m a a m π π π             =
From equation (51), we have , 2

, 2 [START_REF] Li | Analytical Calculation of Inductance Parameter of Tangentially Magnetized PM Motor Based on Basic Air-gap Magnetic Field[END_REF] 14

k k b b j k j k A r A r π π - + (A-6) , 2 , 2 1 2 5 6 . m m a a j m j m m A r A r b π π ∞ - =   = +     ∑ 2 2 cos cos 2 2 j j b g j j b g m a k b g g d a b π π θ θ θ + -         -+ -+                 ∫ ( ) ( )
,0 ,0 2 0 2 2 5 6 ln .

π π θ θ θ + -         -+ -+                 ∫ ( ) ( ) ( ) 2 2 2 2 2 1 1 sin cos 2 2 k mb m mb a a m b k a π π π     + -         = - - + ( ) ( ) ( ) 2 2 2 2 2 1 1 sin cos 2 2 k m mb mb a a m b k a π π π     -+ -         - - + for 2 2 2 2 0 m b k a - + ≠ and 1 2 cos 4 2 bk k a bk k b π π π π   - +   =         1 3 sin sin 4 2 2 bk ka bk ka b b k b b π π π π π   - + +     + - +             for 2 2 2 2 0 m b k a - + = .
Development of equation ( 52) is reduced to ,0 ,0 2 0 0 2 6 14 1

j j r A A b r a r µ µ µ - = - (A-7)
From equation (53), we have

1 1 , 2 , 2 0 1 5 6 m m a a j m j m r m A r A r a π π π µ µ - - -   -     (A-8) 2 ,0 0 2 2 14 2 cos 2 j j b g j j b g A m a g d a r a π θ θ µ + -       = - -+             ∫ 1 1 , 2 , 2 1 0 2 13 14 . k k b b j k j k k k A r A r a b π π π µ ∞ -- - =   - - +     ∑ 2 2 cos cos 2 2 j j b g j j b g k b m a g g d b a π π θ θ θ + -         -+ -+                 ∫
From equation (54), we have ( )

,0 ,0 13 14 ln

j j m A A R + (A-9) ( ) ( ) 2 1 2 1 1 2 sin j j b g np np n m n m n b g A R A R np d b θ θ + ∞ - = - = + ∑ ∫ ( ) ( ) 2 1 2 1 3 4 cos j j b g np np n m n m n b g A R A R np d b θ θ + ∞ - = - + + ∑ ∫ Equation (55) development gives , , 13
k k b b j k m j k m A R A R π π - + (A-10) ( ) 1 2 1 2 . np np n m n m n A R A R b ∞ - = = + ∑ ( ) 2 2 sin cos 2 j j b g j b g k b np g d b π θ θ θ + -     -+         ∫ ( ) 1 2 3 4 . np np n m n m n A R A R b ∞ - = + + ∑ ( ) 2 2 cos cos 2 j j b g j b g k b np g d b π θ θ θ + -     -+         ∫ From equation (56), we have ( ) ,0 ,0 15 16 ln ir ir m A A R + (A-11) ( ) ( ) 2 1 2 1 1 2 . sin ir ir dr np np n m n m n dr A R A R np d b β β θ θ + ∞ - = - = + ∑ ∫ ( ) ( ) 14

∑ ∫

From equation (57), we have , , [START_REF] Lubin | 2-D Exact Analytical Model for Surface-Mounted Permanent-Magnet Motors With Semi-Closed Slots[END_REF] 16

k k dr dr ir k m ir k m A R A R π π - + (A-12) ( ) 1 2 1 2 . np np n m n m n A R A R dr ∞ - = = + ∑ ( ) 2
+ -     -+         ∫ ( ) 1 2 3 4 . np np n m n m n A R A R dr ∞ - = + + ∑ ( ) 2
+ -     -+         ∫ Development of equation (58) gives ( ) 1 1 0 1 2 np np n m n m np A R A R µ - -- - + (A-13) 2 1 1 , , 1 1 0 1 13 14 . k k p b b j k m j k m j k k A R A R b π π π πµ ∞ -- - = =   = -     ∑∑ ( ) 2 2 sin cos 2 j j b g j b g k b np g d b π θ θ θ + -     -+         ∫ ( ) 2
k A R A R dr π π π ∞ -- - = =   + -     ∑∑ ( ) 2
+ -     -+         ∫ ( ) 2 ,0 1 0 2 16 sin ir r ir dr N ir ir dr m A np d R β β θ θ πµ + = -   -    ∑ ∫
From equation (59), we have ( )

1 1 0 3 4 np np n m n m np A R A R µ - -- - + (A-14) 2 1 1 , , 1 1 0 1 13 14 . k k p b b j k m j k m j k k A R A R b π π π πµ ∞ -- - = =   = -     ∑∑ ( ) 2 2 cos cos 2 j j b g j b g k b np g d b π θ θ θ + -     -+         ∫ ( ) 2 2 ,0 1 0 2 14 cos j j b g p j j b m g A np d R θ θ π µ + = - -∑ ∫ 1 1 , , 1 1 15 16 . r k k N dr dr ir k m ir k m ir k k A R A R dr π π π ∞ -- - = =   + -     ∑∑ ( ) 2
+ -     -+         ∫ ( ) 2 ,0 1 0 2 16 cos ir r ir dr N ir ir dr m A np d R β β θ θ πµ + = -   -    ∑ ∫ Equation (60) development is ( ) ,0 ,0 11 12 ln i i s A A R + (A-15) ( ) ( ) 2 1 2 1 1 2 sin i i d np np n s n s n d A R A R np d d α α θ θ + ∞ - = - = + ∑ ∫ ( ) ( ) 2 1 2 1 3 4 cos i i d np np n s n s n d A R A R np d d α α θ θ + ∞ - = - + + ∑ ∫ From equation (61), we have , , 11
k k d d i k s i k s A R A R π π - + (A-16) ( ) 1 2 1 2 . np np n s n s n A R A R d ∞ - = = + ∑ ( ) 2 2 cos sin 2 i i d i d k d np d d α α π θ α θ θ + -     -+         ∫ ( ) 1 2 3 4 . np np n s n s n A R A R d ∞ - = + + ∑ ( ) 2 2 cos cos 2 i i d i d k d np d d α α π θ α θ θ + -     -+         ∫ 12
From equation (62), we have ( )

1 1 0 1 2 np np n s n s np A R A R µ - -- - + (A-17) 1 1 , , 1 1 0 1 11 12 . s k k Q d d i k s i k s i k k A R A R d π π π πµ ∞ -- - = =   = - - +     ∑∑ ( ) 2 2 cos sin 2 i i d i d k d np d d α α π θ α θ θ + -     -+         ∫ ( ) 2 ,0 1 0 2 12 sin i s i d Q i i d s A np d R α α θ θ π µ + = - -∑ ∫
From equation (63), we have ( )

1 1 0 3 4 np np n s n s np A R A R µ - -- - + (A-18) 1 1 , , 1 1 0 1 11 12 . s k k Q d d i k s i k s i k k A R A R d π π π πµ ∞ -- - = =   = - - +     ∑∑ ( ) 2

  in region I, IV, V, VI and VIII (1) 2 0

Fig. 1 .

 1 Fig. 1. Studied parallel double excitation PM machine.

Fig. 2 .Fig. 5 .µ

 25 Fig. 2. Simplified model.

  .The interface conditions between regions VII and VIII at r 0 are

  At radius R s , Fourier series expansions of interface condition (40) gives

Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .Fig. 17

 7891011121314151617 Fig. 7. Radial flux density due to permanent magnet alone in parallel double excitation PM machine

Fig. 18 .Fig. 19 .Fig. 20 .Fig. 21 .Fig. 22 .Fig. 23 .Fig. 24 .Fig. 25 .Fig. 26 .Fig. 27 .

 18192021222324252627 Fig. 17. Radial flux density due to rotor excitation and stator currents (on load) in parallel double excitation PM machine

Kamel Boughrara was born in Algiers, Algeria, in 1969. He received the engineer diploma from the University of Sciences and Technology Houari Boumediene and the magister degree from Ecole Nationale Polytechnique, Algiers, Algeria, in 1994 and 1997 respectively, and the Doctorat d'Etat degree from Ecole Nationale Polytechnique, Algeria, in 2008. He is currently a Lecturer at the Université de Khemis Miliana, Algeria, at the Laboratoire de l'Energie et des systèmes intelligents. His interests include modeling and control of electrical machines.

Rachid Ibtiouen was born in