J. Electrical Systems 1-3 (2005): Regular paper

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1 K. Saii D. Rahem S. Saii A Miaoui Regula pape Coupled Analytical-Finite Element Methods fo Linea Electomagnetic Actuato Analysis JES Jounal of Electical Systems In this pape, a linea electomagnetic actuato with moving pats is analyzed. The movement is consideed though the modification of bounday conditions only using coupled analytical and finite element analysis. In ode to evaluate the dynamic pefomance of the device, the coupling between electic, magnetic and mechanical phenomena is established. The displacement of the moving pats and the inducto cuent ae detemined when the device is supplied by capacito dischage voltage. Keywods Actuatos, analytical analysis, coupling model, finite element method, movement, tansient feeding. 1. INTRODUCTION In electomagnetic devices, the electomagnetic field can be obtained by solving patial diffeential equations. The solution of these equations can be done though numeical methods such as finite element one [1]-[2]. Nevetheless in the case of dynamic studies of these devices, this method fail due to the flexion and the defomation of the solution domain subdivision when the mobile pats ae moved. To solve this poblem, geneally a new mesh at each displacement step is equied, and in this case, this method becomes cumbesome and vey expensive. Seveal fomulations ae developed in ode to take account of the movement in mobile systems such as the electic machines, the actuatos, the induction heating systems, etc... Geneally, the existing fomulations, allowing the movement simulation, make use of special elements o meshing modifications and lead to costly models [1]-[2]. An othe appoach is used fo movement consideation in linea electomagnetic systems. This appoach is based on only one finite element meshing fo all the displacement steps as descibed in [3]. The majo defect of this technique is the fact to simulate the movement to discontinuous steps. This led to numeical noises in calculation if finite element mesh is not well efined on the level of the movement zone. Coesponding autho : ksaii@yahoo.f Laboatoie de Modélisation des Systèmes Enegétiques Dépatement d'electotechnique, Univesité de Biska BP 145, Biska 7, Algéie Copyight JES 25 on-line : jounal.esgoups.og/jes

2 To ovecome this poblem, a coupled model based an analytical and finite element solution is poposed in this pape. In this model, the movement is taken account though the modification of bounday conditions only. The analytical solution is detemined in a simple shape egion consideed between a moving pat and a fixed one, and called «MZ : Movement Zone». The est of the domain is finite element meshed. The analytical solution is coupled to the numeical one though the continuity condition fo the field tangential component (Ht). The movement is then consideed by only the modification of the points coodinates of the inteface between both analytical and numeical sub-domains. In this way, the matix due to the finite element discetization is calculated once and used fo evey elative position of the moving pat. The fomulation is elaboated in the case of an axisymmetical stuctue and its validity is achieved when applying it to study an electomagnetic actuato. In the othe hand, the main design factos of the actuato : the displacement of the moving pats and the electical cuent in the coil, ae detemined by the coupling between electic, magnetic and mechanical phenomena [4]-[7]. The poblem is investigated by the paameteization coupling model. Measuements ae caied out and compaed to computed data when the device is supplied by capacito dischage voltage. 2. FIELD EQUATIONS AND FORMULATION Let conside an axisymmetical electomagnetic system (Fig. 1). The load is a body which moves unde the effect of the electomagnetic foces. z (Symmetic axis) A ϕ = A 2 FEDA A ϕ = A 1 Moving object (load) Coil b MZ a Fig. 1. Electomagnetic system objects. Note that : FEDA is the Finite Element Discetized Aea and MZ is the Movement Zone. In this case, the magnetic vecto potential equations ae as follows [3]: 25

3 K. Saii et al: Coupled Analytical-Finite Element Methods fo Linea... (( ). ( ϕ) ) ( ).(( ) ( ϕ) ) ν A + z ν A z ( )( ) = μ ( J σ V ϕ J = σ A t + (vx u l(a)) = σ DA t c ϕ ϕ ϕ (2) A ϕ and J ae the components following the angula diection ( ϕ ). V : electical potential, v : velocity, σ : electic conductivity, ν : magnetic eluctivity, μ : vacuum pemeability. In MZ zone (Fig. 1), equation (1) becomes : ( ) ( ) ( ϕ) ( ) ( ) ( ϕ) 1 A + z. 1 A z = The vaiables sepaation technique is applied to solve (3). When consideing the following axisymmetical conditions : (1) (3) Aϕ( + az, ) = Aϕ( az, ) = A1 Aϕ(, b 2 ) = Aϕ(, b 2) = A2 = the solution of this equation using Bessels functions is given as follows : k = { k (( ) 1 ) (( ) ) Aϕ (, z) = 1.4A 2k+ 1π ( ) } (( k )( b) ) z ( J1( )) J1 ( a).cos π. λ + λ (4) (5) λ= + π. J 1 is Bessel s function of fist ode and fist kind. In ode to couple the analytical solution to the finite element analysis, one has to put the continuity condition in tem of tangential component ( H t ) of the magnetic field : with j ( 2k 1)( b) Ht = ( ν ). A n ϕ (6) To detemine H t, we calculate the deivation espect to the nomal n : 26

4 ( ϕ) ϕ ( ϕ ) ( ϕ ) A n= A A n A z + + nz k = A 1 { ( 4. ( 1) ) ( 2k 1) + π cos ( ( 2k+ 1)( π b) ) z k =. (. λ. J( λ )) ( J1 ( + a) λ ) }. n k + A 1 { ( 4. ( 1) ) ( b) sin ( ( 2k+ 1)( π b) ) z k =. ( J1( λ ) ) ( J1( ( + a) λ) ) }. n z ( ) 2 2 m m Whee J ( λ ) = ( 1) ( λ 2 ) ( m! ) is the Bessel s function of n ae the nomal vecto n components in the ( z, ) plane. m= zeo ode. n and z In the finite element discetized aea (FEDA), the Galekin fomulation of equation (1) is given by the following elation: Ω ( A )( ) ( A z )( z ) ( ddz ) ν αi + αi ( ) ( ) + σ DA Dt α ddz H αdγ Ω i t i Γ ( 1 )( ) = σαi V ϕ ddz Ω whee α is the pojection function, Ω is the FEDA and Γ is the inteface between the FEDA and the MZ. So, the integal tem consideed on Γ can be expessed using the field H fomulae (6). t When the load moves espect to the inducto, only the MZ fomulae (6) has to be changed though the modification of the coodinates and z. It can be noticed that the MZ fomulae can be associated to standad softwae since it can be put as Newmann bounday condition. (7) (8) ELECTROMACHANICAL COUPLING MODEL The dynamical behavio of linea electomagnetic actuato can be basically descibed by the following electomechanical equations system [6]-[7] : 27

5 K. Saii et al: Coupled Analytical-Finite Element Methods fo Linea... ( ) V () t = Ri + N φ( z, i) t φ ( z, i) = il( z, i) 2 2 Fmag ( z, i) = M( d z dt ) + α ( dz dt) ± Fg v = dz dt whee V ( t ) is the exciting voltage applied to the coil, R is the coil esistance, i is the coil cuent, N is the numbe of tuns, ( zi, ) coil, z is the displacement, t is the time, Lzi (, ) is the inductance, F ( z, i ) (9) φ is the flux though the is the global magnetic foce, M is the mobile pat mass, α is the fiction coefficient, F g is the foce of gavity and v is the mobile pat velocity. The unknown vaiables of the electomechanical poblem ae the cuent (i) and the mechanical displacement ( z ). The method consists of simultaneously solving the equations system (1); that equies the knowledge of magnetic foce ( F mag ) and flux ( φ ) which ae functions of the displacement and the cuent. These vaiables ( F mag, φ ) ae paameteized using intepolation functions and finite element solution fo the electomagnetic equation. This solution is caied out fo seies of discete values of the excitation J and the displacement z in the anges of thei eal vaiations. The displacement of the moving pat is consideed by only the modification of the points coodinates of the inteface between both analytical and numeical sub-domains. In this way, the matix due to the finite element discetization is calculated once and used fo evey elative position of the moving pat. mag APPLICATION AND RESULTS Figue 2 descibes the test poblem. This is an axisymmetical actuato used to poduce stike foces. It is composed by too coils and a cylindical steel amatue moving following -z- axis when a voltage stem fom a capacito dischage is applied to coil 1 o coil 2. The chaacteistics of the system ae : M = 5.52 kg, ν= (the 6 elative eluctivity of the amatue), σ= S / m (amatue), R 1 = 3.21 Ω, R 2 = 1.22 Ω and λ= Ns. / m. Figue 3, 4, 5 and 6 show espectively the supplying voltage, the electical cuent in the coil, the mechanical displacement of the amatue and the velocity as functions of time. Note that, the coil 1 is excited fom t = s to t =.3 s, afte that, the coil 2 will be excited fom t =.72 s to t =.125 s. 28

6 In figue 4, the coupling model is compaed to the expeimental data. One obseves that in this figue, the cuent in coil 1 eaches it s maximum in advance of 62.4 ms then that of the mechanical displacement (Fig. 5). The electomagnetic system is moe apid then the mechanical one. Figue 6 shows the velocity as a function of time. One obseves that, the moving pat (amatue) aived at its initial position with a velocity of v 2.3 m/ s, that coesponds to kinetics enegy of W 14.6 J. On the othe hand, the electical conductivity (σ) of the moving pat is weak, which pemits to neglect the eddy cuents in the conducto and allowing the use of the paameteization method. The computed esults based on coupled analytical and finite element analysis fo linea electomagnetic actuato having moving pats ae in good ageement with the expeimental ones (Fig. 4). The diffeence between measued and calculated values is essentially due to the appoximation of the actuato physical popeties. Supplying cicuit z C1 Contol cicuit C2 V1 V2 A m a t u e z Coil 1 (R1) Coil 2 (R2) Fig. 2. The axisymmetical epesentation of the electomagnetic actuato. Note that : V 1 and V 2 ae the supplying voltages. z is the diection of the mechanical displacement of the moving pat (amatue). Supplying Voltage (V) Time (seconds) Fig. 3. Supplying voltage as a function of time. 29

7 K. Saii et al: Coupled Analytical-Finite Element Methods fo Linea... Measued Supplying Cuent (A) Calculated Time (seconds) Fig. 4. Supplying cuent as a function of time. 6 Displacement (mm) Time (seconds) Fig. 5. Mechanical displacement as a function of time. 2 1 Velocity (m/s) -1 CONCLUSION Time (seconds) Fig. 6. Velocity as a function of time. The poposed method is elaboated fo the modeling of the electomagnetic systems having vaiable configuations in time. The movement simulation is caied out though the modification of the inteface continuity conditions only. Such conditions ae obtained fom analytical solution, leading to an accuate and economic modeling. 3

8 Acknowledgment The authos would like to expess thanks to the ministy of the highe education and scientific eseach fo its financial assistance. REFERENECES [1] A. Razek, J. L. Coulomb, M. Féliachi and J. C. Sabonnadièe, Conception of An Ai-Gap Element fo The Dynamic Analysis of The Electomagnetic Machines., IEEE Tansaction on Magnetics, vol. 18, No. 5, pp , [2] M. Janieux, D. Genie, G. Reyne and G. Meunie, F.E.M. Computation of Eddy Cuent and Foces in Moving Systems, Application to Linea Induction Launches, IEEE Tansaction on Magnetics, vol. 29, No. 2, pp , Mach [3] K. Saii, M. Féliachi and Z. Ren, Electoimagnetic Actuato Behavio Analysis Using Finite Element and Paametization Methods., IEEE Tansaction on Magnetics, vol. 31, No. 6, pp , Novembe [4] P. S. Sangha and D. Rodge, Design and Analysis of Voltage Fed Axisymmetic Actuatos., IEEE Tansaction on Magnetics, vol. 3, No. 5, pp , Septembe [5] Li Eping and P. M. McEwan, Analysis of Cicuit Beake Solenoid Actuato Systems Using The Decoupled CAD-FE-Integal Technique., IEEE Tansaction on Magnetics, vol. 28, No. 2, pp , Mach [6] S. J. Salon, Coupling of Tansient Fields, Cicuit and Motion Using Finite Element Analysis., Poceedings of IMACS 91, Dublin, July [7] B. Aldefeld, A Numeical Solution of Tansient Nonlinea Eddy Cuent Poblems Including Moving Ion Pats., IEEE Tansaction on Magnetics, vol. 14, No. 5, pp , Septembe