Synthesis and modelling of an electrostatic induction motor Jean-Frédéric Charpentier, Yvan Lefèvre, Emmanuel Sarraute, Bernard Trannoy
To cite this version:
Jean-Frédéric Charpentier, Yvan Lefèvre, Emmanuel Sarraute, Bernard Trannoy. Synthesis and mod- elling of an electrostatic induction motor. IEEE Transactions on Magnetics, Institute of Electrical and Electronics Engineers, 1995, vol. 31 (n° 3), pp. 1404-1407. 10.1109/20.376290. hal-01792424
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To link to this article : DOI: 10.1109/20.376290 URL: http://dx.doi.org/10.1109/20.376290
To cite this version : Charpentier, Jean-Frédéric and Lefèvre, Yvan and Sarraute, Emmanuel and Trannoy, Bernard Synthesis and modelling of an electrostatic induction motor . (1995) IEEE Transactions on Magnetics, vol. 31 (n° 3). pp. 1404-1407. ISSN 0018-9464
Any correspondence concerning this service should be sent to the repository administrator: [email protected] SYNTHESIS AND MODELLING OF AN ELECTROSTATIC INDUCTION MOTOR
J.F. Charpentier, Y. Lefèvre,E. Sarrauteand B. Trannoy LEEI/ENSEElHT 2. Rue Charles Camichel 31071 Toulouse Cédcx FRANCE
A bs tract -- This paper deals wlth a new way of some previous works on the conception of micrometric synthesls and modelllng electrostatlc Induction actuators [6](7][8]. mlcromotors by means of duallty rules from the An E.I.M. requires a revolving electricfield on its stator. magnetic induction machine. An electromechanlcal Such as in Ùle M.I.M. this is achieved by applying a three model based on this method is given. Theo, a phasedvoltages source to electrodes equallyaround spaced the computational procedure based on a general Iumped stator. order 10 create an alternating electric field parameter model and an electrlc field calculatlon In code, bas been developed so as to simulate the êlistribution eachphase consists of pair of electrodes feeded dynamlc worklng of these actuators. A comparison with opposite sinusoïdal voltages (Fig. 1). Figure 2 shows 1s made between the computation results and the the final arrangement of the stator of a triphased, two poles model results. Satisfactory agreement between machine. theory and simulation is obtained in most respects. l.lNTRODUCTION -0,SV-sin(rot) Researchon a new typeof actuators have beenundertaken for a few years: the electrostatic microactuators. In these actuators the electromechanical conversion is based on the Fig 1. Creating an altemating electnc rieid electricfield rather than the magneticfield. Likethe classical magnetic machines, electrostatic machines can work on 0,5.V s.sin( wt-2013) -0,5.V s.sin( rot+211/3) differenl principles: electrostatic synchronous motor (E.S.M.), electroquasistatic induction motor (E.l.M.), variable capacitance motor (V.C.M.). For instance the )ç� V.C.M. have beenwidely studiedby many researchers: in the vu;,.,,, U.S.A., Japan and Europe. Sorne prototypes have been designed and fabricated using integrated-circuit �., foi-i � processing.[l][2][3] This paper is concerned with the E.I.M. Sorne authors have already studied this type of machine and developed �� theoretical models based on the theory of electromagnetic 0,5.Vs.sin(ox+2ill3) -0,5.Vs.sin(rot-2[1/3) waves. These models are mainlyadapted forE.I.M. wiÙI the !·1g. L Lreatmg a rotaung eleclnc 11c1<1 rotor made of smooth uniform conductor or a fluid and are not very easy to manipulate for the design. This type of In a M.I.M., due to Lenz's law combined with Ohm's E.I.M. uses charge relaxation to establish its rotor charge law, the revolving magnetic field induces voltages and distribution. It is therefore difficult to optimise its currentsin the rotor windings. According to Lenz'slaw these performances.[4](5] currents produce magnetic effects which counterbalance the The paper shows how to synthesize an another type of evolution of flux in the rotor coi!. In a wound rotor machine E.I.M. from considerations of duality with the familiar ilierotor coils are short-circuited. Relations between the flux magnetic induction machine (M.I.M.). In order to facilitate
a= -Msr.cos(0) b= -Msr.cos(0+4IT/3) (4) c= -Msr.cos(0+2I1/3)
Relations between the charge Qr and the voltage Vr where Msr is the amplitude of the coefficients of induction between the two electrodesforming a phaseare: between a phase on the stator and a phase on the rotor , and 0 theelectrical angular position of the rotor located fromthe Ir=dQr/dt andVr:= -Rr / Ir phasenumbered one of the stator. (2) As for the M.I.M. the coefficients of equation (3), These relations show that the induced voltages relating charges to voltages, are fonction of the angular counterbalance the evolution of charge on electrodes. The position of the rotor 0. In order to simplify these rel�tions current Ir between the two electrodes play the part of the • the three-phased variables are transformed to ?ew vanabl�s e.m.f. in M.I.M. and can be called current motive force. related to a reference framefixed to the revolvmg field. This reference frame is the analog of the d-q axis and the transformation is the classical Park's transformation. Equation (3) becomes: -Qr r Qs = Css.Vs - msr. Vr �t {Qr = Crr.Vr- msr. Vs (5) ,g.4. electrodes
To illustrateour approach we have chosen a three-phased where: rotor E.I.M. One arrangement of a two poles motor is Css=Cs-Ms/2,Crr = Cr-�/2 and msr=3..Msr/2 shown on figure 5. (6) Equation (5) relates the complex values of the ne� �leclrical 0,5.Ys.sin(rot-2TI/3) -0,5.Ys.sin( rot+2[l!3) variables to one another. Under steady cond1ttons and assuming balanced three-phase sinusoïdal fee�ing volta�es applied on the phases of the stator, the elec1r1cal pulsauon ros of the stator variablesand the electrical pulsation rorof the rotor variablesare related to the electrical angular speed of the rotor by theequation:
wr+w=ws (7) This equation is the non-zero torque condition. In these 0,5.Vs.sin(rot+2TI/3) -0,5.Ys.sin(rot-2TI/3) conditions all the new variables are constant in the new reference. The current of a phase of the rotor in the new referenceis: III. PARK'S EQUATIONS FOR THE SYNTHESIZED E.I.M dQr . da.r d Ir=--J-Qr = -J-. a.r Qr= J mr.. (C rr. Vr-rnsr. y s ) (8) In order to deduce a relatively simple model of this dt dt dt E.I.M. fewassumptions must be made: space harmonies and saliency effects are neglected. Thus for three-phased E.I.M. From equations (2) and (5) the expression of voltage of a the matrixcapacitance is: rotor phaseis deduced:
Me Ms Cs a b C j.ror.msr.Vs 2 2 Vr= (9) Ms Ms [j. ror.Crr+ 1 / Rr] Cs C a b 2 2 Qs2 Ms Ms Vs2v Cs b a Qs3 _ 2 2 • Vs3 We can comparethe equation (9) with theclassic expression Qrl - Mr Mr Vrl (3) a C b Cr forthe magnetic inductionmachine. Qr2 2 [Vr2"] Qr3 Mr Mr Vr3 b a C Cr 2 2 j.ror.msr.Is Mr Mr Ir=�-----= (10) n b a Cr 2 2 [j.oor.Lrr+ Rr] The torque is obtained by the derivative of the electric where reis the electrical torquegiven by (11); tfs,tfv, J and coenergy in the machine with respect to rotor position. lt's rm are respectively the viscous and static friction torque expression is: coefficients, the inertiaand the loadtorque ; [Vs] and [Vr] are the vector of the voltages on the stator and on the rotor. = .!. v T d[C(0)] [Crr], [Msr] are the matrix blocs components of the matrix r [ ] [V] (11) 2 d0 of coefficients of capacitance and induction [C(0)]. The components of [C(0)] are fonctionof therotor position. where [V] is the vector of voltages of the phases of the The [C(0)] matrix is determined by using an electricfield motor [C(9)] is the matrix of capacitances represented in computation code based on finite elements method. This equation(3). Thismatrix can be written by 4 blocs: code can take into account the movement of the rotor by means of the moving band technique [7][9] . Using this code and considering symmetry, ail the components of the [ Cs� • [ Msr �) [C(e)] = [ (12) matrix [C(0)] canbe determinedby only two simulations. [Msr @)] [crtj JJ The differential system (15) is solved by means of a Runge-Kutta method. For any given voltages supply Neglecting saliencyeffects, Css andCrr are not fonctions of waveforms,ail the electromechanicalvariables aredetermined rotor position. From equations (11) and (12) the expression as a fonction of lime. of torque is: With this procedure the working of a triphased, 6 potes E.1.M. has beensimulated. The arrangement of the structure 1 yd[Msr(0)] • of the machine is presented on figure 7. The diameter of the r= [Vs] [Vr]=-2.msr.Im(VrVs) (13) rotor is 120 µm. 2 de First the components of the matrix [C(0)] have been calculated. When only a unit volt is applied on a reference Assuming steady States operations and balanced sinusoïdal phase of the machine and zero on the other phases, the voltages supply, Vs can be supposed real and equation (13) calculation of chargeson each phasegives the coefficientof becomes: capacitance on the reference phase and the coefficients of 2 2 2 2 induction between the referencephase and the others phases. = 2.msr .Vs .wr/Rr 2.msr .Vs .ws/Rr r 2 2 2 = 2 2 2 (14) Figure 7 shows the map of the electric field calculated inside 1 / Rr + wr .Crr 1/ (Rr .g) + g.ws Crr the machine when the reference phase is a stator one's. The evolution of some coefficients versus the rotor position are Where g=wr/ws is the slip of the machine. The torque-slip shown on figure 8. The doted line represents the fondamental characteristic is qualitatively similar to M.I.M. one's (Fig. harmonie of one of the coefficients of induction between 6). stator and rotor. From these results, the parameters of the Park'smodcl are deduced: Css = 0.1226. 10-15 F Crr = 0.2238. 10-15 F (16) msr = 0.885. 10-16 F
0 ��gmax 1 g During the dynamic simulation the machine is fed by a three-phased balanced sinusoïdal voltages system with an amplitude of 200volts and a frequencyof 1 kHz. The values of theothers parameters are: IV. GENERAL LUMPED-PARAMETER ELECfROMECHANICALDYNAMIC Ifs= 0 ; tfv= 8. 10·16 13 = 20 2 To infer Park's mode! of the E.I.M., assumptions are rm = 8. 10· N.m J 40. J0· kg m (17) needed. In order to validate this mode! a more general model is elaborated. The general differential system of equations . We have simulated the starting of the machine from describing the dynamic working of the synthesised E.I.M. standstill to a 1900 rad/s steady state. The results are can bewritten in the form: presented on figure 9. The voltages of the phases on the rotor presen t, after a transient state, some perturbations around sinusoïdal curves forming a balanced three-phased d[Vr] d =(Crrr.[-[Rrnvr]- [Crr]ro(Vr] system. Likewise the torque and the speed show dt d8 perturbations around constant values. Perturbations are d[Vs] _ d[Msr] mainly due to harmonies of the coefficients of induction - Msr] ro[Vs]] [ dt d8 (15) between stator and rotor. The doted lines on figure 9 and dco figure 10 represent the results of simulation obtained with J dt= re-tfs.s1gne(ro)-tfv.w- rm thefondamental harmonie of these components. d8 -=ro During steady stateoperation the value of the slip can be dt deduced either from results on figure 9 or from results on figure 10. In both cases the slip calculated is about 8 per -Il cent. Relation (7). which is necessary to establish the 'IO pos1Ucn rad Xl01.o�torQueN.111 expression of Vr (9) and thetorque (14) in thePark's mode!, is thereforeverified. From the value of theslip and fromthe 20]······· .. / o.• values of the parameters (16), the value of torque obtained in doued line: from Park's mode! is 2.6 10-12 N.m which is roughly the simulation with � the fu damental �.000 o.oo, 0.010 0.01/·�.ooo 0.005 0.010 0.015 � value obtained by the general simulation procedure when t 111e uc ttme uc harmoruc of the fondamental harmonie of the coefficient of induction is 3000 speed radis coefficients of induction considered. bet ween stator The model defined in paragraph III seems to give a and roror relatively good representation of the electromechanical
behaviour of E.I.M.. This model is more adapted to the 1000 design of the machine and its supply than a more complicated mode! based with the electromagnetic waves 0.000 0.025 0.050 0.075 0.100 0.125 0.150 theoryused until now. )(10-1 UM HC
tg. an pos1t1on EFCAD VI. CONCLUSION At first some considerations of duality allows us to synthesizean electrostaticinduction motor andwe have also developed anelectromechanical mode!of thisactuators which is more simple than those developed in the previous works on this type of machine. Then we have elaborated a general procedurefor thesimulation of the dynamicworking of these actuators. The results obtained allows us to validate our mode! which can beuseful forthe design of these machines and theîr supply. Results obtained must be confirmed experimentally.
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