A Linear Synchronous-Motor Model for Evaluation of Sensorless Methods

Transcription

1 Linear SynchronousMotor Model for Evaluation of Sensorless Methods oberto Leidhold and Peter Mutschler Darmstadt University of Technoloy, Germany. bstract In this work, a model for the LonStator Linear SynchronousMotor is derived. The purpose of this model is the evaluation of sensorless control methods. The model is composed by a manetostatic and an electrical subsystem. The manetostatic subsystem is derived by usin the Manetic Equivalent ircuit (ME) and is expressed as a set of implicit nonlinear alebraic equations. The electrical subsystem is coupled with the former and is expressed by a set of ordinary differential equations. The complete model is solved with a numerical method for Differential lebraic Equations (DE). The ME is derived from the motor eometry, manetic characteristics of the involved materials, and the windin arranement. This allows doin analysis and simulation of a motor before it is build. In this work, an existin LonStator SynchronousMotor is modeled. First, the manetostatic subsystem is compared with Finite Element nalisys (FE), and second, the complete model is validated with experimental results. I. ITODUTIO n important number of sensorless methods have been proposed for rotative motors [][2], however few works can be found applyin it for linear motors e.. [3]. Linear motors present additional challenes to be solved for the implementation of sensorless methods, as like as asymmetries caused by end effects, or nonperiodicity of the flux linkae or of the saliencies. Moreover, lonstator motors are usually split in several sements to allow multiple movers and reducin the reactive power (e.. Transrapid), but makin the implementations of sensorless techniques more complex. It is also to be considered that there is a hiher diversity of linear motors than rotative ones, requirin a tareted analysis for each case. The first step for implementin a sensorless control method is to have an adequate model. When the motor is still under development and there isn t a prototype available to et the model parameters by experimental tests, the Finite Element nalysis (FE) is the usual tool. In [4] the use of FE is proposed for analyzin the ability to implement a sensorless method on a motor. The FE is however very timeconsumin makin it inadequate for a dynamic simulation. nother alternative is to use the Manetic Equivalent ircuit (ME) method [5], which can also be implemented startin from the motor eometry and manetic characteristics of the involved materials. In the present work, a ME based model is proposed to be used to evaluate sensorless methods on linear motors. The model is composed by a manetostatic and an electrical subsystem. The manetostatic subsystem is derived by usin the ME and is expressed as a set of implicit nonlinear alebraic equations. The electrical subsystem is coupled with the former and is expressed by a set of ordinary differential equations. The complete model is solved with a numerical method for Differential lebraic Equations (DE). Lon Stator Synchronous Motor, available in the authors institute, is modeled. In order to validate the proposed model, in a first instance the manetostatic subsystem is compared with FE, and in a second instance, the complete model is validated with experimental results. This paper is oranized as follows: In Section II the lonstator linear synchronousmotor is described. In Section III the ME based model is elaborated. In Section IV results of comparin the model with FE and experimental tests are presented. Finally, in Section V, some conclusions are drawn. II. MOTO GEOMETY The Lon Stator Synchronous Motor is arraned in multiple sements i.e. stators. Each sement is composed by two facin statorsides formin a slot between them where the mover translates. The windins of both sides are parallelconnected and electrically independent from the other sements. In Fi. the diaram of one sement of the motor is shown. In this fiure, the eometry of the stator and the mover as well as the windin distribution is shown. Each sement has 3 poles arraned in 39 slots, while the mover has 3 poles. III. MODEL. Manetostatic subsystem The model of the manetostatic subsystem is derived by usin the Manetic Equivalent ircuit (ME) method [5]. y modelin the manetic system only statically, effects like Eddy currents will not be considered. Even if some sensorless methods rely on hih frequency injection, it analysis can be well done nelectin Eddy current effects, e.. [4]. Other methods, however, are expressly based on the transient inductance [6], which would not be well modeled with only a manetostatic analysis. The ME method consists in dividin the manetic system in pieces in which the flux flows in (almost) one direction [5]. 26 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, includin reprintin/republishin this material for advertisin or promotional purposes, creatin new collective works for resale or redistribution to servers or lists, or reuse of any copyrihted component of this work in other works.

2 a a 2 a (n) a n a (n) Stator a a 2 a 3 a 4 a a 2 a (n) a n a (n) a a 2(m) a 2m a 2(m) a 2M Mover a 3(m) a 3m a 4(m) a 4m a 3(m) a 4(m) a 3M a 4M Stator x Fi. Diaram of the PM Linear Motor Geometry These pieces, also called flux tubes, are represented in the Equivalent ircuit by resistances, characterizin the manetic reluctance (or permeance). Windins are represented in the equivalent circuit by controlled ManetoMotiveForce (MMF) sources. While Permanent Manets (PM) are represented by constant flux sources. The motor of Fi. can be modeled with the equivalent circuit shown in Fi. 3. Fluxes are desinated by φ, manetomotiveforces by F, reluctances by (permeances by ) and the Manetic Potential at node {, } i j by a ij. The manetic potential at each node in Fi. 3 is related to the correspondin points shown in the diaram of Fi.. The equivalent circuit uses = 4 branches representin stator teeth, while the mover is divided into M = 222 branches representin virtual teeth of the actual mover and the air slice between both stator sides where the mover is absent. The lower half of the mover and the lower stator are evensymmetric to it upper counterparts. In the case of FE simulation, it is used to take advantae of this kind of symmetry by simulatin only one half of the system, and usin symmetry boundary conditions. Similarly, in the ME this boundary condition can be implemented by shortin toether all nodes layin on the axis of symmetry as shown in the circuit of Fi. 3. Leakae, yoke and movertooth reluctances ( SL, m, SY, Mam and Mbm ) are linear. Only the stator tooth reluctance ST is considered inherently nonlinear, as there is where the flux density is hiher. The airap reluctances Gm, n, which relate each node of layer with each node of layer 2, are parametric nonlinear, dependin on the mover position x. The airap permeance G between stator tooth and mover tooth can be calculated as, µ ls bg( x) G = () where bg ( x ) is the airap lenth covered simultaneously by both teeth, is the airap heiht, µ is the air permeability and l S is the stack heiht. The function bg ( x ) is implemented by spline interpolation from the points shown in Fi. 2, where b ST and b MT are the stator and mover tooth width respectively. In the same fiure, the resultin curve is shown with dashed line. For the remainin teeth, the function bg ( x ) is shifted by the distance between them..5b MT b G(x) b MT.5 b MT.5b MT Fi. 2 Function b ( ) teeth).5(b MTb ST) G b ST.5b MT b ST.5b MT x (airap lenth covered simultaneously by both In order to solve the ME, the correspondin node and potential equations are derived and expressed as an implicit function: = f( FS, a, a, a2, a3, a4, φ S, x) (2) Where matrix and vectors are desinated by bold symbols, a to a 4 are the potential vectors of each layer, φ S is the stator teeth flux vector, F S is the teeth MMF vector, and x the mover s position. The implicit function (2) has 3x2xM rows, 3x2xM unknown variables and known variables (e.. = 4; M = 222). The complete expression of (2) is presented in ppendix I. x

3 a ST SY SY SY SY φ S ST φ Sn ST φ S...n n... F S SL a a 2 a (n) SL F Sn a n SL a (n) a () SL a F S G Gmn GM a 2 a 2m a2m φ PM a 3 Ma φ PMm Ma...m ML m...m a 3m φ PMM a 3M Ma Mb Mb Mb xis of symmetry a 4 Fi. 3 Manetic Equivalent ircuit of the Linear Motor of Fi. a 4k = a 4i a 4m k=..m, i=..m a 4M In order to interface the manetostatic subsystem with the electric subsystem, the teeth MMF vector F S is related with the phase currents as follows, FS = w S i S (3) where i S = [ i i i] T is the phase current vector, and w S is the MMF transformation matrix, which can be derived from the windin distribution [5]. Similarly, the flux linkae can be related with the teeth flux vector φ S by, λs = ws φ S (4) where λ S = [ λ λ λ] T is the phase flux linkae vector, and w = w T. Substitutin (3) and (4) into (2), and solvin the invariant linear subsystem, it can be reduced to, = f2( is, a, a2, λ S,x) (5) Shrinkin to M3 rows, M3 unknown variables and 3 known variables.. Electric subsystem The linear motor is fed by a voltae source inverter. In Fi. 4, the electric circuit of such a system is shown. Even when the star point is not connected, the inclusion of a hih resistance is considered to et the star point voltae u in the simulation. It is included in this work, as several sensorless methods are based on measurin this voltae. From the circuit of Fi. 4 the complete set of differential equations can be derived, dλs dt = is u (6) with, = I3 3x3 (7) bein I 3 the identity matrix, and 3 a 3x3 matrix of ones. 3x u u u i i i Fi. 4 Electrical circuit of the linear drive dλ dt dλ dt dλ dt. omplete system The set of implicit nonlinear alebraic equations (2) cannot be directly solved in order to insert it in the differential equation (6) i.e. cannot be expressed as is = f( λ S ). onsequently, it must be solved numerically toether with the differential equation. Equations (6) and (5) form a set of differential alebraic equations (DE). In order to solve the DE, the alorithm presented in [7] is used. In order to start solvin the DE, consistent initial states must be provided. Therefore, the alebraic equation will be solved first (with the initial states of the ODE, λ S () ), usin a nonlinear alebraic solver. IV. ESULTS. Manetostatic ME simulations compared with FE The sensorless capability of a synchronous motor can be initially analyzed from the position dependence of its inductance or its induced Electro Motive Force (EMF), dependin on the considered sensorless method. The inductance matrix was derived as a function of the position and

4 Inductance Lαα and Lββ (H) x Mover position (mm) 5 mm offset x 3 e/ω (v s)..5.5 α β Mover position (mm) 5 mm offset. Inductance Lαα and Lββ (H) Mover position (mm) 5 mm offset Fi. 5 Inductance Lαα as a function of the position, a) by ME, b) by FE transformed into the in αβ coordinates (i.e. larke Transformation). It was obtained by usin the ME model on one hand, and by usin the FE on the other. The inductances L αα and L ββ obtained by the ME model are shown in Fi. 5.a, and by FE in Fi. 5.b. The inductance was derived with zero current. It is however also possible to analyze the inductance dependence on current by usin this model. esults were also obtained for the induced EMF. The normalized EMF e αβ ω is shown in Fi. 6.a and Fi. 6.b as obtained by usin the ME model and the FE model respectively (where ω is the electrical anular speed). It should be remarked that the processin time for the FE simulation took more than ten times the processin time for the ME simulation. The ME simulation was implemented in Matlab, in which the objective function is interpreted each iteration of the solver. This could be speedup even more by compilin the objective function.. Dynamic ME simulation compared with experimental results In order to analyze the complete model, the injection of a hihfrequency alternatin voltae is tested. This test allows e/ω (v s).5.5 α β Mover position (mm) 5 mm offset Fi. 6 ormalized EMF as a function of the position, a) by ME, b) by FE verifyin the capability to use a sensorless method with sinal injection as proposed in [8]. In this test, an alternatin voltae is applied in the daxis of an arbitrary reference frame γ, γ a cos( ω t) u = (8) where a is the amplitude of the injected voltae, and ω is the frequency. The resultin qaxis current, in the same reference frame, is bandpass filtered and demodulated as follows, γ idem = iq sin( ω t) (9) If the motor presents manetic saliencies, this sinal will be position dependent and can be used to detect the mover s position [8]. s can be appreciated in Fi., the mover has surface manets. onsequently, the manetic saliencies are mainly due to the saturation they produce in the stator. For test purposes, the sinal is injected at a constant reference frame anle γ, while the mover travels alon the stator span at a low speed ( v =.56 m/ s). The amplitude of the sinal is a = 8V and the frequency is khz. Simulation results of the demodulated sinal i dem are shown in Fi. 7

5 Demodulated sinal i dem () Demodulated sinal i dem () γ = γ = π/ Position x γ τ/π (m) Fi. 7 Demodulated sinal i dem as a function of the position obtained by simulation, a) γ=, b) γ=π /2 while in Fi.8 the experimental results for the same test are shown. It can be appreciated in both fiures that the sinal i dem presents an offset dependin on the referenceframe anle γ. This offset, which can also be seen as a stationary saliency, will produce a perturbation in the position detection. However, as the reference frame anle γ is a known value, the offset can be compensated. V. OLUSIOS dynamic model based on the Manetic Equivalent ircuit (ME) was derived for a LonStator Linear Synchronous motor. This allowed analyzin how sensorless methods would work on it. haracteristics like manetic saliencies, due to the saturation that the permanent manets produce in the stator, are well represented with this model. When the motor has wide yokes, this characteristic can be well modeled even includin nonlinearity only in the stator teeth. esults of the manetostatic subsystem were compared with results obtained by Finite Element nalysis (FE), showin ood areement between them. The complete ME based Demodulated sinal i dem () Demodulated sinal i dem () γ = γ = π/ Position x γ τ/π (m) Fi. 8 Demodulated sinal i dem as a function of the position obtained by experimentation, a) γ =, b) γ =π /2 model was also compared with experimental results, by injectin a hih frequency voltae as used for sensorless methods. When comparin the obtained demodulated sinal by simulation and by experimentation, sliht differences raised. There are however, some known differences between the experimental setup and the model (e.. the setup is arraned in a circular path, bein therefore one wedeshaped tooth per pole, while in the simulation all teeth are considered equal and rectanular). evertheless, with the proposed model it was possible to analyze the ability to apply a sensorless control method on the linear motor. PPEDIX I Equations of nodes a, a, a 2, a 3, a 4 and the equations of the teeth potentials, can be expressed in vector form, yieldin respectively: I φ a = () ()x S () φ S a = a 2a2 () φ G a a a a a = (2) ( ) PM Ma M 2 ML ML ML ML ML2 ML2 = ML( M 2) ML( M 2) ML( M ) ML( M ) ML( M ) ML( M )

6 ( ) ( ) φ G a a a G a I = (3) PM Ma Mb 3 M a4 M xm Mb 3 xm Mb Mxa4 = G a G (4) T a I()x a FS fst ( φs) = (5) where I X is the identity matrix of dimension X, XxY is a XxY matrix of ones, X is a column vector of dimension X, I()x = 2Y Y Y 2Y Y = Y 2Y Y Y Y L L L 2L L = L 2L L L L G G2 GM G2 G2M 2 = G G2 GM = T 2 2 M j = G j M G2 j = j= = dia 2 Mx ( ) M j = G j i= Gi Gi2 = i= 22 = dia 2 x ( ) i= GiM ([ 2 ]) ([ 2 ]) G = dia Ma Ma Ma MaM G = dia Mb Mb Mb MbM and f ST is the elementwise function that relates tooth flux with its manetic potential drop. Usually the manetization curve is provided as a vs. H table. From this table, a spline function can be implemented: H = ffe ( ) (6) From this function, the MMF can be related with the flux as follows: F = f φ ( l b ) h = f φ (7) ( ) ( ) Fe S S ST where l S is the stack lenth, b S is the tooth width, and h is the tooth heiht. Equations ()(5) form the implicit function f. The equations of the equivalent circuit are rearraned to allow transformations (3) and (4) to be applied. Therefore, φ S is solved from () yieldin, φ = S ( ) a 2 a 2 (8) ext, by substitutin (8) into () and (5), and applyin transformations (3) and (4), the equivalent circuit equations become, I a a a = (9) ()x (( ) 2 2 ) () ( ) 3 ( Ma ) Ma M Ma ( Ma Mb ) Mb Mxa4 M λ w a w a = (2) S S S 2 2 φ a G a G a = (2) PM φ G a G G a G = (22) PM xm Mb 3 xm Mb Mxa4 = G a G (23) T a I()x a ws is fst (( ) a 2 a2 ) = (24) y offline solvin the linear invariant subsystem, computational burden can be hihly reduced for solvin the DE. ode potentials a, a 3 and a 4 can be solved from (9), (22) and (23), requirin only constant matrices to be inverted (takin into account that, 2, 2 and 22 are position dependent). fter substitutin the solved expressions for a, a 3 and a 4 into the remainin equations (2), (2) and (24), the implicit function f 2 is obtained (eq. (5)). KOWLEDGMET This research results were attained with the assistance of the lexander von Humboldt Foundation, Germany. The authors would like to thank for the support. EFEEES [] K. ajashekara,. Kawamura and K. Matsuse, Sensorless ontrol of Motor Drives, IEEE Press, ew York, 996. [2] J. Holtz, Perspectives of Sensorless Drive Technoloy, in Proceedins of PIM Europe, ürnber, June 79, 25, pp [3] W. Limei and G. Qindin, Sensorless control of permanent manet linear synchronous motor based on nonlinear observer In Proceedins of the 8th IEEE International Workshop on dvanced Motion ontrol, 24. M'4, 2528 March 24. pp [4] U.H. ieder, M. Schroedl Simulation Method for nalyzin Saliencies with espect to Enhanced IFOMapability for Sensorless ontrol of PM Motors in the Low Speed ane includin Standstill EPE 25. Dresden. [5] V. Ostović, Dynamics of Saturated Electric Machines. SprinerVerla. ew York [6] J. Holtz and H Pan Elimination of saturation effects in sensorless positioncontrolled induction motors IEEE Transactions on Industry pplications, Vol. 4, o. 2, pp [7] Shampine, Lawrence; eichelt, Mark W.; Kierzenka, Jacek. "Solvin index DEs in MTL and Simulink". SIM ev. 4 (999), no. 3, pp [8] M. Linke,. Kennel, J. Holtz, "Sensorless position control of permanent manet synchronous machines without limitation at zero speed" IEEE 28th nnual onference of the Industrial Electronics Society IEO 2, Vol., 58 ov. 22 pp