Analytical Electromagnetic Analysis of Multi-Phases Cage Rotor Induction Motors in Healthy, Broken Bars and Open Phases Conditions

Transcription

1 Progress In Electromagnetics Research B, Vol. 7, , 216 al Electromagnetic Analysis of Multi-Phases Cage Rotor Induction Motors in Healthy, Broken Bars and Open Phases Conditions Lazhar Rouache *, Kamel Boughrara, and Rachid Itiouen Astract This paper presents an analytical calculation of magnetic field and electromagnetic performances of 3-, 5-, 7-, 9-, 11-phases cage rotor induction machines in healthy, roken ars and open phase s conditions. This model is formulated to consider all types of multi-phase/multipoles windings and used for the identification of electrical equivalent circuit (EEC) parameters It s ased on the sudomain model and the resolution of Poisson s, Laplace s, and Helmholtz s equations in each sudomain issued from Maxwell equations using the method of separation of variales and Fourier series when the machines are fed with sinusoidal current and voltage. The developed analytical model permits the calculation of magnetic field distriution, eddy current, circuit model parameters, and unalanced magnetic radial force due to roken ars, electromagnetic torque and asored stator current. A comparative analysis etween the studied five multi-phases machines is done with considering identical power rate. The analytical results are validated y those issued from the finite-element method (FEM). 1. INTRODUCTION Polyphase cage rotor induction motors have many advantages, thanks to their easy and roust construction. This type of machine is eing extensively used in several applications [1 3]. In the literature many multi-phases machine topics have een addressed, such as multi-phase machine advantages, properties, modeling, and performance with different control techniques [2 5]. One further advantage is that multi-phase machines are said to e fault-tolerant, offering greater security in missioncritical applications [5]. The designs and developments of multi-phase induction machines depend on the existing computeraided analytical design tool. It s possile to use numerical methods (i.e., finite elements or the finite differences) [3, 6, 7] to analysis the electromagnetic performances of multi-phase induction machines, and their react against fault in rotor ars or in ones of their phases. However these approaches are time consuming and not suitale for the optimization prolems. Some of recent 2-D static analytical predictions of magnetic field distriution are proposed in [8 12]. The model presented in [8] which takes into account rotor and stator semi-slots gives good results in the calculation of magnetic field, eddy-currents, EEC parameters, and steady-state performances of three phase s induction motors in healthy and roken ar conditions. In this paper, we generalize this model for n-phases cage rotor induction motors without rotor and stator semi-slots in healthy case, roken ars and open phase s conditions, and use it for analysis of 3-, 5-, 7-, 9-, 11-phases identical power rated multi-phase machines. To facilitate the evaluation of various stator designs, the same phase current as well as flux per pole is used in all machines. Received 25 July 216, Accepted 11 Octoer 216, Scheduled 3 Octoer 216 * Corresponding author: Lazhar Rouache (rouache.lazhar@gmail.com). The authors are with the Ecole Nationale Polytechnique (LRE-ENP), Algiers, 1, Av. Pasteur, El Harrach, BP 182, 162, Algeria.

2 114 Rouache, Boughrara, and Itiouen The contriutions of this paper are divided among the various sections. Section 2 includes the machine design and specifications, and the differential equations to e solved in each part of the machine. Section 3 presents the analytical solutions of Laplace s, Poisson s, and Helmholtz s equations. Section 4 shows the identification of the EEC of the multi-phases machine parameters and the calculation of radial force in the air gap created in roken ars condition. It is noticed that the radial force is null for open phases condition. Symmetry of magnetic flux lines is also realized in this case. Results and validation of the analytic model with FEM [14], and comparative study etween the n-phase machines designed is given in Sections 5 and 6. Finally, the conclusion in Section PROBLEM DEFINITIONS In this paper, a sudomain analytical model ased on Maxwell equations is applied to analyze the electromagnetic performances of 3-, 5-, 7-, 9-, and 11-phases machines in healthy and fault conditions and fed with current and voltage. The stator winding of the studied machines is distriuted in slots with integer numer of slot per pole and per phase. The five machines have the same output power, to get a comparative study and in the same time validate the analytical multi-phase model Machines Design In this section, a low-voltage three-phase machine is used as reference for comparison with other multiphase machines. Fig. 1 shows the machine model where Region I represents the air-gap, Region II the rotor ars, Regions III the stator slots. Using principles of the design in [13], the other multi-phase machines are designed to accommodate the constant volume rotors and otain the same output power and phase current level. Hence, over the parameters given in the following equations, all the other parameters are the same for all machines, the five designed machines and their corresponding parameters are shown in the Tale 1. Figure 1. Studied multi-phases cage rotor induction machine. 1 The numer of stator slots Q s which is selected for having an integer numer of slots per pole and per phase, and get the values close among all machines. 2 Opening angle slot (in degrees) is calculated from c(n) = 36 (1) Q s (n) 3 Numer of turns per phase N ph assumed to e equal to N ph (n) = 3N ph(3) (2) n 4 Numer of conductors per stator slot N c can e expressed as shown in N c (n) = 2nN ph(n) (3) Q s

3 Progress In Electromagnetics Research B, Vol. 7, The stator slot area is given y 6 External radius of stator slot S(n) = N c(n) S(3) (4) N c (3) R 4 (n) = 2S(n)c(n)+R 2 3 (5) Tale 1. Design details of the five studied machines. Parameters Symol Unit 3-ph 5-ph 7-ph 9-ph 11-p Numer of pole pairs p Numer of stator slots per pole per phase q Numer of stator slots Q s Numer of turns per phase N ph Numer of conductors per stator slot N c Stator slot opening width c deg Radius of external stator surface R ext mm External radius of stator slot R 4 mm Internal radius of stator R 3 mm Numer of rotor slot Q r External radius of rotor slot R 2 mm Internal radius of rotor slot R 1 mm Rotor slot opening width deg Stack length L u mm Conductivity of rotor ars σ MS/m Peak phase current I m A Peak phase voltage V m V Magnetic Field Equations The analytical model is formulated in complex vector potential and two-dimensional polar coordinates with the following assumptions. The stator and rotor core are assumed to e infinitely permeale with zero electrical conductivities; All multi-phases machines are supplied with a alanced n-phases sinusoidal currents or a alanced n-phases sinusoidal voltage; The time variation of the magnetic field is assumed to e sinusoidal; The resistance of the end-rings connecting the rotor ars are not considered; The current density in the stator slots has only one component along the z-axis; The stator and rotor slots are open and have radial sides without semi-closed slots, as shown in Fig. 1. The general partial differential equation issued from Maxwell s equations in a continuous region in term of vector potential A, which has one component in z-axis and independent of z, can e expressed y [7] as ( Az (r, θ, t) Δ A z (r, θ, t) =μ μ r J s (r, θ, t) μ μ r σ +Ω A ) z(r, θ, t) (6) t θ In static analysis of induction motors, the first harmonic of the magnetic potential is considered dominant, and the relative movement etween the stator and the rotor is represented y relative time

4 116 Rouache, Boughrara, and Itiouen pulsation. Therefore, we assume that the pulsation of the potential vector is ω in the air-gap/stator regions and ω rm in the rotor regions y mean of speed addiction ω rm = sω = ω pω where s is the slip, p the numer of pole pairs, and Ω the constant angular velocity of rotor. From the preceding considerations, Eq. (6) ecomes Δ A z (r, θ, t) =μ μ r J s (r, θ, t) μ μ r σ A z(r, θ, t) t (7) and A z (r, θ, t) =Re{A z (r, θ) e jωt } (8) J s (r, θ, t) =Re{J s (r, θ) e jωt } (9) in the air-gap and stator (i.e., Regions I, III) A z (r, θ, t) =Re{A z (r, θ) e jωrmt } (1) in the rotor (i.e., Regions II). From Eqs. (7) to (1), the magnetic vector potential A(r, θ) in each region is given y ΔA z (r, θ) = In Regions I (11) ΔA z (r, θ) =jμ μ r σω rm A z (r, θ) In Regions II (12) ΔA z (r, θ) = μ J s (r, θ) In Regions III (13) where J s (r, θ) =J i is the stator slots current density, which is constant in each slot, σ the electrical conductivity of the rotor ars, μ r = 1 the relative permeaility of the rotor ars, j = 1thecomplex numer, and μ the permeaility of vacuum. The field vectors B and H, in the various regions, are coupled y B (r, θ) =μh (r, θ) (14) The radial and tangential components of the magnetic flux density are deduced from A(r, θ) [1]. 3. MAGNETIC FIELD SOLUTION Using the Fourier series analysis and the method of separation variales, the solution of Eqs. (11) (13) leads to determining the magnetic field distriution in all regions Solution of Laplace s Equation in the Air-Gap In Region I, which is located etween the radii R 2 and R 3 in Fig. 1, the solution of complex Laplace s Equation (11) is ( A zi (r, θ) =A 1 + A 2 ln(r)+ A 1k r k + A 2k r k) ( sin(kθ)+ A 3k r k + A 4k r k) cos(kθ) (15) k=1 where k is the spatial harmonic orders, and A 1 A 4k are complex integration constants in Region I. This general solution accounts for the periodic oundary condition etween and 2π considering all types of multi-phase/multi-poles windings Solution of Helmholtz s Equation in the Rotor Bars The jth rotor ar (i.e., Region II) is a slot with the classical oundary conditions shown in Fig. 2. The solution of the complex Helmholtz s Equation (12) is given y [8] ( ( mπ A ziij (r, θ) = B jm g m (r)cos θ g j + )) (16) 2 m= k=1

5 Progress In Electromagnetics Research B, Vol. 7, () Figure 2. The oundary conditions of rotor and () stator slots regions. where B jm are complex integration constants, and m is the spatial harmonic orders. For a Q r rotor ars, j varies from 1 to Q r and g j is the angular position of the jth rotor ar and the ar opening in radians. The expressions of g m (r) depend on the expression of α. For α =( jω rm σμ ) 1 2 For α =(jω rm σμ ) 1 2 (αr), Y mπ g m (r) =J mπ J (αr) mπ+ (αr 1 )αr 1 mπj mπ Y mπ+ K mπ+ (αr 1 )αr 1 mπy mπ (αr 1) (αr 1) Y mπ I g m (r) =I mπ (αr) mπ+ (αr 1 )αr 1 + mπi mπ (αr 1) (αr 1 )αr 1 + mπk mπ (αr), I mπ (αr) andk mπ where J mπ first and second kinds of the order mπ. (αr 1) K mπ (αr) (αr) (αr) are the Bessel and modified Bessel functions of the 3.3. Solution of Poisson s Equation in the Stator Slots The magnetic field in the stator slots is calculated analytically y solving Poisson s Equation (8). The solution of Eq. (13) in the ith slot suject to the oundary conditions shown in Fig. 2() is given y A ziiii (r, θ) =C i μ J i R4 2 ln(r) 1 ( lπ ( 2 μ J i r 2 + C il h l (r)cos θ β i + c ) ) c 2 (17) h l (r) =r lπ c + R 2 lπ c 4 r lπ c where C i C il are complex integration constants and l is the spatial harmonic orders Interfaces Conditions To determine Fourier series unknown constants, A 1 A 4n, B j B QrM and C i C QsL of magnetic field solutions in all regions, interfaces conditions should e introduced. The interface conditions etween Regions I and II at R 2 are Az I (R 2,θ)=Az II (R 2,θ) (18) H I (R 2,θ)=H II (R 2,θ) (19) The interface conditions etween Regions I and III at R 3 are Az II (R 3,θ)=Az III (R 3,θ) (2) H II (R 3,θ)=H III (R 3,θ) (21) l=1

6 118 Rouache, Boughrara, and Itiouen The preceding four interface conditions permit to determine a linear system of equations which depend on the harmonic orders (k, m, andl) choice in each region. Some details of the development of interface conditions can e found in [8] and [1] Broken Bars Condition The roken rotor ars are represented y Region II d having an electrical conductivity equal to zero. The resistance of rotor ars is infinity in this case and the eddy current is null. The solution of Equation (12) ecomes ( ( mπ A ziijd (r, θ) =B j + B jm h m (r)cos θ g i + )) 2 (22) h m (r) =r mπ m=1 mπ 2 1 r mπ + R 3.6. Rotor Bar Current and Winding Definition The eddy-current density induced in the jth rotor ar is given y J j (r, θ) = jω rm σ j A ziij (r, θ) (23) The current in the jth rotor ar is I j = R 2 g j + 2 R 1 g j 2 J j (r, θ)drdθ (24) It should e noticed that in the roken ars, the electrical conductivity is null, and the current density is zero. The stator current density in the slots for multi-phase single layer winding is defined y a connection matrix etween phase currents and stator slots as (for one pole) C = i {}}{ 1 2. n+1 2 n+3 2. n 1:q {}}{ 1..1 q+1:2q {}}{ 2q+1:3q {}}{ (n 2)q+1:(n 1)q {}}{ (n 1)q+1:nq {}}{ where n is the numer of phases and q the numer of slots per pole and per phase. It is important to notice that the proposed analytical model can e easily extended to doule layer winding. For n-phase induction motors supplied y alanced sinusoidal current, stator current vector is defined as [ ] I s = I m 1 e j 2π n... e j 2(n 1)π n Notice that in the open phase condition the current of phases is zero, and the current vector will e. I s = I m 1 e j 2π n... ph Open {}}{... e 2(n 1)π j n (25) The current-density in stator slots is J i = N c S [I s] [C] (26) where S is the surface of the stator slots and N c the numer of conductors in the slot.

7 Progress In Electromagnetics Research B, Vol. 7, CIRCUIT MODEL PARAMETERS IDENTIFICATION, STATOR CURRENT, TORQUE, AND RADIAL FORCES The determination of the fluxes permits the calculation of lumped parameters of the equivalent circuit of induction motor. Neglecting the iron losses and assuming the first harmonic hypothesis, the EEC showninfig.3caneconsidered. Figure 3. EEC of one phase for the cage rotor multi-phases induction motors. In this circuit, the flux leakage inductance is lumped in the secondary ranch. The electrical components of the circuit are defined y: 1) R s is the stator resistance which can e determined y classical analytical formula; 2) L m is the stator inductance (i.e., the magnetizing inductance); 3) X r (s) is the leakage inductance expressed in the rotor frame which depends on the slip; 4) R r (s) is the equivalent resistance expressed in the rotor frame which depends on the slip. The magnetizing inductance L mi can e determined y performing a simulation of no-load operation (i.e., s = ). With an input n-phase currents, the resolution of the field equations allow determining the magnetic flux vector Ψ = [ψ 1...ψ n ]. So we have. L mi = ψ i /I si The complex phases flux vector is given y Ψ=N c [φ 1...φ Qs ] [C] T (27) where the flux linkage is calculated from the method ased on Stokes theorem using the vector potential in the stator slots φ i = L u S R 4 β i + c 2 R 3 β i c 2 A ziiii (r, θ)rdrdθ (28) where A ziiii (r, θ) is the vector potential given y (14). The calculation of the rotor parameters needs the simulation of the electromagnetic ehavior for a given value of the slip s. For load operation, an input phase current is imposed. The resolution of the field equation allows determining the magnetic fluxes. Then, the secondary current I r is determined for each phase y I ri (s) =I si ψ i /L mi (29) The secondary impedance Z ri (s) can e calculated for each phase y Z ri (s) =jωψ i /I ri (s) R ri (s) =s Re{Z ri (s)} (3) X ri (s) =Im{Z ri (s)} This approach is adopted for the calculation of the EEC parameters y using oth FEM and analytical model presented in this paper. When the machine is fed with a constant n-phases voltage V s = V m [1 e j 2π 2(n 1)π j n... e n ] the stator current for each slip can e determined for y the EEC as I si (s) =V si /(R si +(Z ri (s)//l mi ω)) (31)

8 12 Rouache, Boughrara, and Itiouen In healthy condition the EEC parameters are the same for all phases, although in the case of roken ars or open phase s conditions, the EEC parameters are different etween the phases, which induce different phases current. The torque-slip characteristic can e otained analytically and y FEM. Of course, for a given stator current at a given slip, the torque is calculated y the Maxwell stress tensor method in Eq. (33). The electromagnetic torque can also e determined using the EEC as T em = P tr(s) ω (32) p where P tr(s) is the active power transformed to rotor. According to the Maxwell stress tensor method, the electromagnetic torque T em is computed using the analytical expression T e m = L ur 2 g μ 2π 1 2 Re{B Ir(R g,θ) B Iθ (R g,θ)}dθ (33) where R g is the radius of a circle placed at the middle of the air-gap, L u the axial length of the motor, and B Ir (R g,θ)andbiθ (R g,θ) are, respectively, the radial and tangential components of the flux density in the middle of air gap. The radial and tangential-components of magnetic pressure in function of the spatial angle of the machine θ in the middle of air gap for giving slip s are defined as [11]. P r (s, θ) = 1 ( BIr (R g,θ) 2 B Iθ (R g,θ) 2) (34) 2μ P θ (s, θ) = 1 μ B Ir (R g,θ) B Iθ (R g,θ) (35) The x- and y-components of the radial force for giving slip are calculated as The radial force magnitude is F x (s) =R g L u F y (s) =R g L u 2π 2π F r (s) = [P r cos(θ) P θ sin(θ)]dθ (36) [P r sin(θ)+p θ cos(θ)]dθ (37) F 2 x + F 2 y (38) Notice that in the healthy ars conditions, the radial force is null ecause of the symmetry of magnetic pressure created in each pole, ut when ones of rotor ars roken, this pressure will e non-symmetric and a radial force will appear. 5. RESULTS AND VALIDATION In this section, the machine is taken as reference to validate the analytical model. All results are verified y FEM [14] No-Load Condition with Current Supply Under no-load condition (i.e., s =.1), Fig. 4 shows the flux lines in the machine otained with FEM. The radial and tangential components of the flux density distriution in the middle of the air-gap are shown in Fig. 5. A very good agreement can e seen etween the numerical and analytical results.

9 Progress In Electromagnetics Research B, Vol. 7, Figure 4. Flux lines for machine at no-load condition (i.e., s =.1) otained with FEM m B r (T) B θ (T) Angle (mech. degrees) Angle (mech. degrees) () Figure 5. Radial and () tangential components of the flux density in the middle of the air-gap for machine and no-load condition (i.e., s =.1) Locked-Rotor Condition with Current Supply The flux lines, the real part of eddy-current density distriution in the middle of the rotor ar No. 8 and the real part of the current in each rotor ars are shown in Fig. 6. The radial and tangential flux density waveform in the middle of the air-gap is shown in Fig. 7. Compared with the no-load results, one can oserve the influence of the eddy-currents on oth the radial and the tangential flux densities. It can e seen that the radial component of the flux density decreases Circuit Model Parameters, Stator Current, and Torque The values of the self-inductances calculated y FEM at s =.1 for the machine is L m ω =7.48 while the analytical model gives L m ω =7.49. Fig. 8 shows the evolution of parameters R r and X r versus the slip. There is a good agreement etween analytic and FEM results. The comparison of FEM, circuit model, and analytical calculation of the current-slip and torqueslip characteristics under constant voltage are shown in Fig. 9. It can e seen that the results otained analytically are in good agreement with FEM.

10 122 Rouache, Boughrara, and Itiouen Current density (A/mm 2 ) Radius (m) (c) Bar current (A) () Bar numer (d) Figure 6. Flux lines for machine, () flux lines for machine, (c) real part of eddycurrent density distriution (A/mm 2 ), and (d) the current in the rotor ars for and machines at locked rotor condition (i.e., s = 1) B r (T) B θ (T) Angle (mech. degrees) Angle (mech. degrees) () Figure 7. Radial and () tangential components of the flux density in the middle of the air-gap for machine and locked rotor condition (i.e., s = 1).

11 Progress In Electromagnetics Research B, Vol. 7, R' r (Ω).25 X' r (Ω) Figure 8. Comparison etween analytical and finite element circuit model rotor parameters versus slip for and machines. I 1 (A) T em (Nm) Circuit model () Figure 9. Current-slip and () torque-slip characteristics for and machines under constant alanced voltage Magnetic Field and Eddy-Current On-Load at s =.2 Under constant voltage, on-load flux lines and current in each rotor ars at s =.2 are shown in Fig. 1. The corresponding radial and tangential flux density waveforms in the middle of the air-gap are shown in Fig. 11. A very good agreement can e seen etween the numerical and analytical results Broken Bars Condition In this section, the ars Nos. 9 and 1 are supposed roken as a machine fed with a alanced constant voltage, the stator currents are not alanced (Fig. 13). The flux lines, eddy-current density distriution and the current in the rotor ars are shown in Fig. 12 in the case of ars eing roken (i.e., σ 9 = σ 1 = ). Fig. 13 shows the stator current, electromagnetic torque, and created unalanced radial force versus slip. A comparison etween one roken ar and two roken ars is shown in Fig. 14. One can see that the influence of roken ars has een affected the torque and create unalanced force.

12 124 Rouache, Boughrara, and Itiouen () Bar current (A) Bar numer (c) Figure 1. Flux lines for machine, () flux lines for machine, and (c) the current in the rotor ars for and motors on-load at s =.2. B r (T) Angle (mech. degrees) B θ (T) Angle (mech. degrees) () Figure 11. Radial and () tangential components of the flux density in the middle of the air-gap for machine on-load at s =.2.

13 Progress In Electromagnetics Research B, Vol. 7, Bar current (A) Bar numer () Figure 12. Flux lines, and () the current in the rotor ars for machine on-load at s =.2 and ars 9 and 1 roken. Current(A) Phase1 4 Phase2 Phase3 2 Phase4 Phase T em (Nm) Circuit model () 1 8 F radial (N) Figure 13. Phases currents, () electromagnetic Torque, and (c) radial force versus slip for machine and two roken ars condition. (c)

14 126 Rouache, Boughrara, and Itiouen 12 9 One roken ar Two roken ars F radial (N) 6 T em (Nm) Healthy ar One roken ar Two roken ars () Figure 14. Comparison etween radial force, () Torque versus slip characteristic for machine in healthy, one roken ar, and two roken ars conditions Open Phases Condition In this section, the phases Nos. 1 and 2 are supposed open; therefore, their currents are null (Fig. 16). The flux lines, eddy-current density distriution and the current in the rotor ars are shown in Fig. 15 in the case of 2 phases open (i.e., I 1 = I 2 = ). Fig. 16 shows the stator current, the electromagnetic torque versus slip. A comparison etween one phase and two phases open are shown in Fig. 16(c). One can see that the influence of open phases which has affected the torque. 2 Bar current (A) Bar numer () Figure 15. Flux lines, and () the current in the rotor ars for machine on-load at s =.2 and phases 1 and 2 open. 6. INFLUENCE OF NUMBER OF PHASE IN BROKEN BARS AND OPEN PHASES CONDITIONS In this section, a comparative study using the analytic model among all polyphase induction machines in several states is presented. This study allows showing how the numer of phases influences in the fault case of polyphase machines in the rotor or in the stator. In healthy case (Fig. 17) all machines have close characteristics for nominal running. The 11-phase machine has a more important maximum,

15 Progress In Electromagnetics Research B, Vol. 7, Current(A) 12 8 Phase1 Phase2 4 Phase3 Phase4 Phase T em (Nm) () 12 Tem(Nm) Healthy phases One open phase Two open phases Figure 16. Phases currents, and () (c) electromagnetic Torque versus slip for open phases condition in machine. (c) Tem(Nm) phase 3 9-phase 11-phase I 1 (A) phase 9-phase 11-phase () Figure 17. Electromagnetic Torque, () stator current versus slip in healthy case. and start-up torque, and the same oservation is noticed for stator currents. Figure 18 shows torque versus slip characteristics for roken ars conditions. The electromagnetic torque is not highly affected for nominal running, although the maximum and start-up torques decrease y 37.5% for machine and only 17.24% for 11-phase machine.

16 128 Rouache, Boughrara, and Itiouen T em (Nm) phase 3 9-phase 11-phase Tem(Nm) phase 3 9-phase 11-phase () Figure 18. Electromagnetic Torque for one roken ar, () two roken ars. T em (Nm) phase 9-phase 11-phase T em (Nm) phase 9-phase 11-phase () Figure 19. Electromagnetic Torque for one open phase, () two open phases. F radial (N) phase 9-phase 11-phase F radial (N) phase 9-phase 11-phase () Figure 2. Radial force versus slip for one roken ar, and () two roken ars conditions.

17 Progress In Electromagnetics Research B, Vol. 7, For the open phase s condition shown in Fig. 19, the torque is affected for all values of slip. This effect is reduced for higher numer of phases machine. For 11-phase machine, the torque decreases y 6% for nominal running and y 17% for start-up torque. Fig. 2 shows the radial force created y the roken of one of the rotor ars. For a slip smaller than.2, the 11-phase machine is the least affected. 7. CONCLUSION In this paper, we propose an improved analytical method for predicting magnetic field distriution, electromagnetic performances and parameters of electrical equivalent circuit in healthy, roken ars, and open phase s conditions of cage rotor multi-phases induction motors. The presented analytical model ased on sudomain method is used to determine analytical expressions of eddy current, electromagnetic torque, stator current and unalanced magnetic force (UMF) in healthy, roken ars and open phase s conditions. Both cases are considered: the machine fed with sinusoidal current or voltage. Five identical powers rated machines are designed with analyzing the effect of a numer of open phases and roken ars on electromagnetic torque, stator current and UMF. We have seen that the augmentation of numer of phases reduces the radial force due to roken ars which is null in healthy case for any multi-phases machine. It is noticed that the start-up torque of the machine in roken ars or open phase s conditions is not highly affected for higher multi-phases machines which offer a greater security in several applications. REFERENCES 1. Levi, E., Multiphase electric machines for variale-speed applications, IEEE Trans. Ind. Elec., Vol. 55, No. 5, , May Pereira, L., A., Scharlau, C., C., Pereira, L., F., A., and Haffner, J., F., General model of a five-phase induction machine allowing for harmonics in the air gap field, IEEE Trans. Energy Conversion, Vol. 21, No. 4, , Decemer Adel-khalik, A. S., M. I. Masoud, A. Sheha, and A. M. Massoud, Effect of current harmonic injection on constant rotor volume multiphase induction machine stators: A comparative study, IEEE. Trans. Ind. Apps., Vol. 48, No. 6, , Novemer Decemer Matyas, A. R., K. A. Biro, and D. Fodorean, Multi-phase synchronous motor solution for steering applications, Progress In Electromagnetics Research, Vol. 131, 63 8, Septemer Apsley, J. and S. Williamson, Analysis of multiphase induction machines with winding faults, IEEE Trans. Ind. Apps., Vol. 42, No. 2, , March April Vaseghi, B., N. Takoraet, and F. Meiody-Taar, Transient finite element analysis of induction machines with stator winding turn fault, Progress In Electromagnetics Research, Vol. 95, 1 18, Mori, D. and T. Ishikawa, Force and viration analysis of induction motors, IEEE Trans. on Magnetics, Vol. 41, No. 5, , May Boughrara, K., N. Takoraet, R. Itiouen, O. Touhami, and F. Duas, al analysis of cage rotor induction motors in healthy, defective, and roken ars conditions, IEEE Trans. on Magnetics, Vol. 51, No. 2, , Feruary Luin, T., S. Mezani, and A. Rezzoug, calculation of eddy currents in the slots of electrical machines application to cage rotor induction motors, IEEE Trans. on Magnetics, Vol. 41, No. 11, , Novemer Boughrara, K., F. Duas, and R. Itiouen, 2-D analytical prediction of eddy currents, circuit model parameters, and steady-state performances in solid rotor induction motors, IEEE Trans. on Magnetics, Vol. 5, No. 12, , Decemer Kim, D. Y., G. H. Nam, and G. H. Jang, Reduction of magnetically induced viration of a spoketype IPM motor using magnetomechanical coupled analysis and optimization, IEEE Trans. on Magnetics, Vol. 49, No. 9, , Septemer 213.

18 13 Rouache, Boughrara, and Itiouen 12. Boutora, Y., N. Takoraet, and R. Itiouen, al model on real geometries of magnet ars of surface permanent magnet slotless machine, Progress In Electromagnetics Research B, Vol. 66, 31 47, Pyrhonen, J., T. Jokinen, and V. Hraovcova, Design of Rotating Electrical Machines, John Wiley & Sons, Ltd., Meeker, D. C., Finite Element Method Magnetics, Version 4.2, April 1, 29 Build,