Electric Power Systems Research

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1 Electic Powe Systems Reseach 89 (2012) Contents lists available at SciVese ScienceDiect Electic Powe Systems Reseach jou n al h om epage: Diect field oiented contol of linea induction motos Macello Pucci ISSIA-CNR, Section of Palemo, Via Dante, , Palemo, Italy a t i c l e i n f o Aticle histoy: Received 24 Febuay 2011 Received in evised fom 28 Novembe 2011 Accepted 22 Januay 2012 Keywods: Electical dives Linea induction moto (LIM) Field oiented contol (FOC) a b s t a c t This pape deals with diect field oiented contol of linea induction moto dives. Afte elaboating the inducto and induced pat space-vecto equations of the LIM in seveal efeence fames, some induced pat flux models taking into consideation the end effects ae pesented. In paticula, the so called voltage model based on the inducto equations in the inducto efeence fame and the cuent model based on induced pat equations in both the inducto and induced pat flux linkage efeence fames ae deduced and compaed to the otational induction machine countepat. Aftewads, afte a pope tuning of such models based on both FEA (finite element analysis) and expeimental measuements, some simulation and expeimental tests have been pefomed. Simulations show that the poposed flux models taking into consideation the LIM end effects pemit the dive to achieve bette dynamic pefomance. Moeove, some compaative expeimental esults, adopting both the cuent and the voltage flux models, have been pefomed on a suitably devised test set-up Elsevie B.V. All ights eseved. 1. Intoduction Linea induction motos (LIM) have been lagely studied fo seveal yeas [1 6]. Among the main easons of thei inteest thee is the oppotunity to develop a diect linea motion without the need of any gea-box fo the motion tansfomation (fom otating to linea). This advantage pesents, as countepat, the disadvantage of an incease of complexity of the machine model, pesenting the so-called end effects and bode effects. The pope dynamic modelling of LIMs, taking into consideation the end effects, has been faced up to by people fom academia and industy [7 12]. These effects, which ae due to asymmeties in the inducto stuctue both in the longitudinal and in the tansvesal diections, futhe cause an incease of the difficulty in contolling with good pefomance the main electo-mechanical chaacteistics of the machine, in paticula the thust, the magnetic flux and consequently the linea speed and position. Some of these mathematical models ae based on the electical paametes of the machine including the so-called end facto [8,9], while othes ae based on the stuctual chaacteistics of the machine (length, numbe of slots and coils etc.) with a pope definition of the ai-gap function [10,12]. Stating fom these kinds of models, pope high pefomance contol technique have been developed, in paticula field oiented contol (FOC) [13 16] o diect thust contol (DTC) [17]. These contol techniques have been patially adapted fom the otating induction Coesponding autho. Tel.: ; fax: addess: pucci@pa.issia.cn.it machine (RIM) countepat [18 23], so to take into consideation the end effects of the LIMs. In paticula, [13] poposed a FOC stategy, with a closed-loop contol of both the magnetic flux and the machine speed, whee the induced pat flux linkage angle is computed as the sum of the measued linea speed and the slip speed, obtained fom the estimated induced pat flux amplitude and the quadatue component of the stato cuent in the field oiented efeence fame. Some expeimental esults ae shown. [16] basically poposes the same solution, but only some simulation esults ae pesented. The FOC scheme poposed in [14] is based on the same kind of computation of the induced pat flux linkage angle but, as a diffeence, the PI contolles ae integated with suitable decoupling cicuits (feed-fowad contol actions), which implicitly take into consideation the end effects. Even in this case expeimental esults ae pesented. A diffeent appoach has been followed by [17], which stating fom the classic diect toque contol (DTC) based on switching table [24], develops an analogous methodology suitable fo LIMs pemitting to take into consideation even the attactive foce between the inducto and the aluminium tack. This pape, that is the famewok of FOC contol, stating fom the steady-state (static) equivalent cicuit of the LIM pesented in [8], poposes the elated space-vecto based equivalent cicuit witten in the inducto efeence fame. This has been obtained embedding in the well-known space-vecto electical diagam of the RIM, the modified electical paametes depending on the end-effect facto Q, thus obtaining the space-vecto equivalent cicuit of the LIM. Scientific liteatue usually pesents space-vecto equations of LIMs, witten in the synchonous (induced pat flux oiented) efeence fame [9,13,14]. This is done with the scope of diectly finding the expession of the slip pulsation needed to pefom a /$ see font matte 2012 Elsevie B.V. All ights eseved. doi: /j.eps

2 12 M. Pucci / Electic Powe Systems Reseach 89 (2012) Nomenclatue List of symbols u s = u sd + ju sq inducto voltages space-vecto in the inducto efeence fame; i s = i sd + j i sq inducto cuents space-vecto in the inducto efeence fame; i = i d + j i q inducto cuents space-vecto in the inducto efeence fame; s = sd + j sq inducto flux space-vecto in the inducto efeence fame; = d + j q induced pat flux space-vecto in the inducto efeence fame; u s = u sd j u sq voltages space-vecto in the induced pat flux linkage efeence fame; is = i sd j i sq cuents space-vecto in the induced pat flux linkage efeence fame; phase angle of the induced pat flux space-vecto; L s, L, L m inducto, induced pat and thee-phase magnetizing inductances; L s, L inducto and induced pat leakage inductances; R s, R inducto and induced pat esistances; p numbe of pole pais; ω angula oto speed (in electical angles pe second); v linea speed; m length of the inducto; p pola pitch; coect field oientation in pesence of end-effects. Raely, both voltage and cuent flux model of LIMs have been deived in a systematic way. In this pape, fistly the so-called voltage model of the LIM has been deived, pemitting to estimate the induced pat flux linkage space-vecto (both amplitude and phase) on the basis of the inducto equations in the inducto efeence fame. Aftewads, the so-called cuent model of the LIM has been deived, pemitting to estimate the induced pat flux linkage space-vecto on the basis of the induced pat equations in the inducto efeence fame. Finally, afte a coodinate tansfomation, the cuent model in the induced pat flux linkage efeence fame has been deduced. Afte a pope tuning of such models on the basis of both FEA (finite element analysis) and expeimental measuements, both the poposed models have been implemented expeimentally on a suitably developed test setup. Expeimental esults have been obtained making the LIM dive wok in both speed and position closed-loop contol. In synthesis, the main oiginal contibutions of this manuscipt ae: (1) The genealization of the electic scheme of the LIM embedding the end effects in a space-vecto fom. (2) The definition of the voltage and cuent flux models of the LIM, taking into consideation the end effects, in seveal efeence fames. (3) The pope chaacteization of the flux models by means of expeimental measuements and FEA (finite element analysis). (4) The veification, in numeical simulation, of the inceased dynamic pefomance of the dive, achievable thanks to the adoption of any flux model taking into consideation the LIM end effects in place of the coesponding ones fo the RIM. (5) The expeimental implementation, on a suitable devised test set-up, of the poposed flux models in a diect field oiented contol scheme and elated compaison of the esults. 2. Space-vecto equivalent cicuit of the LIM including end effects In a LIM, diffeently fom a RIM (otating induction machine), the seconday (induced pat), consists of a sheet of aluminium with a back coe of ion. Duing the motion of the inducto, a continuous vaiation of the aluminium sheet happens, while the inducto pesents a limited length. This causes a vaiation of the induced cuents in the sheet and coesponding magnetic flux density in the ai-gap, in poximity of the entance (font of the motion) and exit (back of the motion) of the inducto. This flux modification is diffeent in the entance espect to the exit. When the moving inducto faces a new pat of aluminium sheet, new induced cuents ae geneated stating fom a null value. This gowth, which is quite fast because of the absence of peviously induced cuents, ty to oppose to the inducing flux vaiation. The effect is a deep eduction of the esulting flux in poximity of the entance. At the same time, at the exit the induced cuent oppose to a sudden flux eduction fom the induce, ceating an oveall flux incease. The highe the speed of the inducto, the highe the end effect phenomenon. This last has been taken into consideation in liteatue by a so-called end effect facto Q [8,9], defined as: Q = m R (L m + L )v It can be obseved that, the highe the machine speed, the highe the ai-gap thickness (highe leakage inductance) and the lowe the inducto length, the lowe the facto Q. It means that the end effects incease with the machine speed, with the ai-gap thickness and educes with the inducto length. Coespondingly, the thee-phase magnetizing inductance vaies with Q in the following way: L m = L m (1 f (Q )) (2) with: f (Q) = 1 e Q Q telling that the inductance vitually educes with the end effects. A computation of the oveall losses of the machine shows that an additional esistance appeas in the tansvesal banch taking into consideation the eddy cuent joule losses. This esistance is equal to: R = R f (Q ) (4) Coespondingly, the space-vecto equivalent cicuit of the LIM can be deduced ad it is shown in Fig. 1. It could be obseved that the main diffeences with the equivalent cicuit of the RIM ae in the magnetizing inductance and in the eddy cuent esistance, both pesent in the tansvesal banch. 3. Induced pat flux-linkage estimation Stating fom the space-vecto equivalent cicuit in the inducto efeence fame in Fig. 1, afte applying the Kichhoff law, the following set of space-vecto equations of the LIM can be witten, valid in the inducto efeence fame: u s = R s i s + R f (Q ) [i s + i ] + d s 0 = R i + R f (Q ) [i s + i ] + d jω with: { (a) (b) s = [L s + L m (1 f (Q))] i s + L m (1 f (Q )) i (a) = L m (1 f (Q)) i s + [L + L m (1 f (Q ))] i (b) (1) (3) (5) (6)

3 M. Pucci / Electic Powe Systems Reseach 89 (2012) Fig. 1. Space-vecto equivalent cicuits of the LIM. whee u s, i s, i, s and ae espectively the inducto voltage and cuent, the induced pat cuent, and the inducto and induced pat flux linkage space-vectos witten in the inducto efeence fame, R s and R ae the inducto and induced pat phase esistances, L s and L ae the inducto and induced pat leakage inductances, ω is the electical otating speed of the inducto, Q the above defined end effect facto and j is the imaginay unit. Fo all the symbols, please efe to the list at the beginning of the manuscipt. Eqs. (5)a and b ae espectively the inducto and induced pat voltage equations, while Eqs. (6)a and b ae espectively the inducto and induced pat flux equations. Witing Eq. (6)b as a function of i and substituting it in Eq. (5)a, the space vecto equations of the voltage model can be deduced: d = u s R s i s L s d i s R f (Q) L m (1 f (Q)) (7) Eq. (7) has been deduced unde the simplifying assumption that the leakage facto of the induced pat is null, L = 0, and the ate of change of the linea speed is almost null, i.e. dv/ = 0, othewise some additive tems depending on the linea acceleation of the moto would appea. If Eq. (7) is decomposed in its eal and imaginay pats, the two following scala equations could be obtained: d d d q = u sd R s i sd L s d i sd = u sq R s i sq L s d i sq R f (Q ) L m (1 f (Q)) d R f (Q ) L m (1 f (Q)) q Eqs. (8) define the so called voltage model of the LIM, including the end effects. It should be noted that this model pesents some diffeences with espect to its countepat fo the RIM. In paticula, an additional tem depending on the end effect facto and on the induced pat flux linkage components appea. This tem vaies with the linea speed of the machine and is null when the linea speed of the LIM is null, since the end effects ae not visible in this case. In pactice, fom a dynamical point of view, the voltage model behaves as a fist ode system with a pole in (R f(q))/(l m (1 f(q))) moving with the speed. Moeove, while in the RIM model the flux estimation is based on an open-loop integation, with consequent numeical poblems due to the integato initial conditions and to the dift effects, in the LIM model the integation of the flux is closed-loop because of the pesence of this additional tem. As a matte of fact, the open-loop integation of the voltage model in a RIM causes a dift of the flux due to the offsets in the measuements and initial conditions of the integation [26]. This featue makes, fom the numeical integation point of view, the LIM model advantageous with espect to the RIM one. Fig. 2 shows the block diagam of the voltage model of the LIM, descibing Eqs. (8). (a) (b) (8) Witing Eq. (6)b as a function of i and substituting it in Eq. (5)b, the space vecto equations of the cuent model can be deduced: ( ) d = R i s + j p v R (1 + f (Q)) p L m (1 f (Q)) (9) whee p is the pola step of the inducto and p the numbe of pole pais. Even in this case, Eq. (9) has been deduced unde the simplifying assumption that the leakage facto of the induced pat is null, L = 0, and the ate of change of the linea speed is almost null, i.e. dv/ = 0. The tem (p/ p )v = ω epesents the equivalent otational speed of the induced pat in electical angles. If Eq. (9) is decomposed in its eal and imaginay pats, the two following scala equations could be obtained: d d d q = R i sd p p v q R (1 + f (Q )) L m (1 f (Q)) = R i sq + p p v d R (1 + f (Q)) L m (1 f (Q)) d q (a) (b) (10) Eqs. (10) epesent the so-called cuent model of the LIM, including the end effects. A compaison between these last equations with the countepats of the RIM shows that the main diffeence lies in the flux feedback tem. This tem in the LIM depends on the linea speed of the machine by the end effect facto Q. As in the voltage model, even the cuent model behaves as a fist ode system with a pole in (R (1 + f(q))/l m (1 f(q))) moving with the speed. On this basis, an equivalent induced pat time constant of the machine taking into consideation the end effects, depending on the linea machine speed itself, can be defined as: = L m (1 f (Q)) R (1 + f (Q )) (11) Fig. 3 shows the block diagam of the cuent model of the LIM in the inducto efeence fame, descibing Eq. (9). If Eq. (9) is tansfomed fom the inducto to the induced pat flux linkage efeence fame by the vecto otation = e j, whee is the induced pat flux angle, the following space-vecto equation can be deduced: d = R i s j ( ) ω m p v R (1 + f (Q )) p L m (1 f (Q )) (12) whee the tem in apex means that the vaiable is expessed in the induced pat flux efeence fame and ω m is the induced pat flux vecto otational speed. If Eq. (12) is decomposed in its eal

4 14 M. Pucci / Electic Powe Systems Reseach 89 (2012) Fig. 2. Block diagam of the LIM voltage model. and imaginay pats, the two following scala equations could be obtained: d = R isx R (1 + f (Q )) L m (1 f (Q )) ω m = p v + R isy p (a) (b) (13) Eqs (13) define the cuent model in the induced pat flux linkage efeence fame. In paticula Eq. (13)a confims that the induced pat flux amplitude depends only on the diect component of the inducto cuent and its dynamics depends on the above defined equivalent induced pat time constant, which is a vaying quantity with the linea speed. Eq. (13)b confims that the slip speed, fom which the steady-state value of thust diectly depends, has a diect popotionality with the quadatue component of the inducto cuent. This is the classic equation adopted fo guaanteeing the coect field oientation in LIM dives [9,12,13]. Eqs. (13) basically define the coect conditions of field oientation fo a LIM. Fig. 4 shows the block diagam of the cuent model of the LIM in the induced pat flux efeence fame, descibing Eqs. (13). 4. Field oiented contol of the LIM A diect induced-pat-flux oiented contol of the LIM has been adopted. Field oiented contol (FOC) is the most pefoming and accepted at an industial level contol stategy suitable fo RIM dives. It has been hee diectly adopted fo LIM dives, with the adoption, as fa as the block flux model is concened, of both the flux models descibed in Section 3. The adoption of any of the flux models taking into consideation the LIM end effects, in place of any classic flux model suitable fo RIMs, pemits to pefom a moe accuate estimation of both the amplitude and phase of the induced pat space-vecto. With paticula efeence to phase estimation of the induced pat flux space-vecto, taking into consideation the end effects of the LIM is paticulaly impotant, since it guaantees the conditions of coect flux oientation and theefoe maintains the full thust capability of the dive. The block diagam of the adopted contol scheme is dawn in Fig. 5. In the adopted FOC scheme, cuent contol is pefomed in the induced pat flux oiented efeence fame. On the diect axis (x), a flux contol loop commands the diect cuent loop and a voltage contol loop commands the flux loop to pemit the dive to wok automatically in the field weakening egion by maintaining constant the poduct of the induced pat flux amplitude and the absolute value of the oto speed. On the quadatue axis (y), the position loop contols the speed loop, which then contols the Fig. 3. Block diagam of the LIM cuent model in the inducto efeence fame.

5 M. Pucci / Electic Powe Systems Reseach 89 (2012) Fig. 4. Block diagam of the LIM cuent model in the induced pat flux efeence fame. Fig. 5. Block diagam of the diect field oiented contol scheme. quadatue cuent loop. If the position contol loop is disabled, the LIM dive can wok in speed contol mode. Both these woking mode of the LIM dive have been implemented and tested, as shown in the following. It should be noted that the angula position, needed fo the coect field oientation, is povided by the block flux model. This block can implement any of the thee models descibed in Section 3, the choice depending mainly on the speed woking ange in which the dive is supposed to wok and coespondingly on the load foce. As a matte of fact, the voltage model pesents a highe sensitivity to the inducto esistance vaiation, paticulaly fo low values of the linea speed of the machine, while the cuent model pesents a highe sensitivity to the induced pat time constant vaiation, paticulaly fo high values of the load foce [26]. Fig. 6, efeing to the classic cuent model of the RIM, shows the set of cuves epesenting the atio between the eal and the estimated oto flux vecto amplitude / ˆ vesus the slip pulsation ω sl fo seveal values of the atio between the eal induced pat esistance and that given to the flux model R /ˆR. It can be clealy obseved that the cuent model is paticulaly sensitive to induced pat esistance vaiations and that this sensitivity inceases with load (slip pulsation). Fig. 7, efeing to the classic voltage model of the RIM, shows the set of cuves epesenting the atio between the eal and the estimated induced pat flux vecto amplitude / ˆ vesus the induced pat speed ω fo seveal values of the atio between the eal stato esistance and that given to the flux model R s /ˆR s. It can be clealy obseved that the voltage model is paticulaly sensitive to stato esistance vaiations and that this sensitivity inceases at educed otational speeds. In this case, both the voltage model and the cuent model in the induced pat flux efeence fame have been adopted and compaed in tems of pefomance and applicability in speed and position contol loops. Fig. 6. Set of cuves giving / ˆ vesus ω sl fo seveal values of the atio R /ˆR.

6 16 M. Pucci / Electic Powe Systems Reseach 89 (2012) B, Tesla Length, cm Fig. 7. Set of cuves giving / ˆ vesus ω fo seveal values of the atio R s/ˆr s. All contolles used in the contol loops ae PI (popotional integal) type, but the position contolle, that is of the P (Popotional) type, because of the pesence of a pole in the oigin of the openloop tansfe function. The sampling fequency of all the contol loops has been set to f s = 10 khz. An asynchonous space-vecto pulsewih modulation (SV-PWM) with f PWM = 5 khz has been used to command the invete. 5. Finite element analysis veification A finite element analysis (FEA) of the moto unde test has been pefomed to popely chaacteize the machine in tems of its electical paametes. As a matte of fact, diffeently fom a RIM whee the no-load and locked oto tests can easily be made and ae sufficient to compute all the electical paametes of the equivalent cicuit, in a LIM with a shot tack it is not possible to pefom the no-load test, since the machine can hadly get its ated speed and then block within such a shot path. As a consequence, the no-load test is usually pefomed keeping the inducto away fom the tack Fig. 9. Flux density magnitude at the ai-gap. and making the magnetic flux density lines close in ai, instead of the back ion; it implies that the tansvesal electical paametes of the LIM ae measued in woking conditions which ae diffeent fom those of the nomal opeation, with consequent measuements eos and potential eduction of the dive pefomance. The FEA pemits the no-load test to be popely simulated keeping the aluminium path away fom the tack, while maintaining the back ion, thus guaanteeing measuement conditions equal (o vey simila) to those of the nomal opeation of the dive. As a consequence, the vitually measued paametes of the LIM ae moe accuate and both the flux model and the contolles adopted in the FOC scheme ae popely tuned, with esulting bette dynamic pefomance of the dive in tems of thust esponse. The FEA has been done with the softwae FEMM 4.2., which is a feewae tool pemitting to pefom a steady-state analysis of the machine unde ideal sinusoidal supply and with the inducto not in movement. Giving the softwae the complex values of the theephase inducto cuents, the FEA softwae computes the elated inducto voltages, active, eactive and appaent powes as well as the powe facto. At the same time, the FEA softwae computes the magnetic flux density contou lines. Fig. 8. Longitudinal coss-section and flux density lines of the LIM unde locked inducto conditions.

7 M. Pucci / Electic Powe Systems Reseach 89 (2012) Table 1 Rated data of the adopted LIM. Rated thust Fn [N] 200 Rated voltage Un [V] 380 Rated fequency fn [Hz] 50 Pole pais 3 Stato esistance Rs [] 11 Stato inductance Ls [H] Roto esistance R [] 20.3 Global magnetising inductance Lm [H] Rated speed [m/s] 6.85 Mass M [Kg] 20 Fig. 8 shows the flux density lines obtained in a locked induction conditions and Fig. 9 shows the coesponding pofile of the ai-gap flux density amplitude. This last figue shows the pesence of significant high ode space hamonics, as expected in this kind of moto. 6. Test set-up A test setup has been suitably built to test the diect field oiented contol with the above descibed flux models consideing the end effects of the LIM. The machine unde test is a LIM model Baldo LMAC1607C23D99, whose ated data and electical paametes ae shown in Tab. I. The tansvesal paametes of the equivalent cicuit of the LIM (L s, R s ) have been computed by the no-load test pefomed with the FEA. The longitudinal paametes of the equivalent cicuit of the LIM (L s + L, R s + R ) have been computed by the expeimental locked-inducto test, futhe confimed by the coesponding FEA. Futhe simple algebaic manipulations of the measued data pemit to etieve, s,, L m, L s, L and R. The LIM has been equipped with a linea encode Numeik Jena LIA seies, used to close the speed/position contol loops of the dive. The LIM pesents an induced pat tack on length 1.6 m. The entie contol scheme has been integated with a method fo compensating both the invete dead-time and the invete on-state voltage dops [25]. The employed test set up consists of: A thee-phase linea induction moto with paametes shown in Table 1; A fequency convete which consists of a thee-phase diode ectifie and a 7.5 kva, thee-phase VSI; A dspace cad (DS1103) with a PowePC 604e at 400 MHz and a floating-point DSP TMS320F240. Fig. 10 shows a photogaph of the test setup. Fig. 10. Photogaph of the expeimental test setup. Fig. 11. Refeence and measued speed with the flux models with and without endeffects (simulation). 7. Results 7.1. Simulation esults The behaviou of FOC dive with the LIM has been fistly veified in numeical simulation. With this egad, the Matlab Simulink softwae has been adopted. To suitably check the poposed flux models taking into consideation the LIM end effects, as machine unde test used in the simulations, the dynamic model of the LIM poposed in [10,11] has been implemented. This last model is based on the stuctual chaacteistics of the LIM (length of the inducto, numbe of slots, thickness of the induced pat tack etc.) diffeently fom any of the above flux models, which ae based on the electical paametes of the machine. In this way, the machine unde test does not implement the same equations of the flux model; consequently the eal capability of the poposed flux models to accuately estimate the amplitude and phase of the induced pat flux linkage vecto can be popely tested. The LIM dive has been fistly magnetized. Then, the following set of speed step efeences have been given m/s, including a speed evesal at ated speed. The flux model based on the cuent model of the LIM in the induced pat flux efeence fame (Fig. 4) has been adopted fo this test. It has been pefomed twice, espectively with the flux model taking into consideation the LIM end effects and with same flux model without consideing the end effects [18,22], to highlight the inceased dynamic pefomance of the LIM achievable with the poposed flux model. Fig. 11 shows the efeence and the measued speeds obtained duing such a test. It can be obseved that the poposed flux model taking into consideation the end effects pemits the dive to achieve a highe dynamic pefomance, with the speed evesal accomplished in about 0.4 s less than the case with the classic flux model fo the RIM. In paticula, it is clealy obsevable that the distance between the two speed cuves inceases at highe speed (in absolute value), confiming the fact that a pope consideation of the end effects influences the pefomance of the dive at high speed. Fig. 12 shows the thust of the LIM duing the same test. It can be noted a highe value of the thust duing the speed evesal, obtained with the flux model consideing the end effects. This highe value of the thust, due to the bette field oientation condition, is cause of the bette dynamic pefomance of the speed contol loop. Fig. 13, showing the eal and estimated induced pat flux angle as obtained at speed steady-state, highlights that the flux model taking into consideation the end-effect pemits a

8 18 M. Pucci / Electic Powe Systems Reseach 89 (2012) Fig. 12. Thust with and without end-effects (simulation). Fig. 15. Refeence and measued speed of the LIM with the cuent model (expeiment). highe the speed of the LIM, the highe the value of the R, the lowe the value of L m and the lowe the values of T Expeimental esults Fig. 13. Estimated and eal induced pat flux angle with and without end-effects (simulation). significantly moe accuate estimation of the induced pat flux angle (angle eo of 0.45 ad against 0.8 ad), with esulting bette field oientation. Finally, Fig. 14 shows the modified electical paametes of the cicuital scheme of the LIM. As expected, the The LIM dive has been tested expeimentally on the test setup descibed in Section 6. To compae the achievable esults, both the cuent model of the LIM in the induced pat flux efeence fame and the voltage model have been implemented expeimentally and compaed in speed contol mode. As fa as position contol mode is concened, only the test with the cuent flux model has been pefomed, since the voltage model hadly pemits the dive to wok in position contol mode, because of its sensitivity to inducto esistance vaiations at vey low and zeo speeds. With egad to the speed contol mode, a speed squae efeence with steps fom 0.8 to 0.8 m/s has been given the dive. The fequency of the squae speed efeence is detemined by the length of the tack: wheneve the inducto eaches the end of the tack, the sign of the speed efeence is changed. Figs. 15 and 17 Fig. 14. Modified electical cicuit paametes with end-effects. Fig. 16. Refeence and measued speed of the LIM with the voltage model (expeiment).

9 M. Pucci / Electic Powe Systems Reseach 89 (2012) Fig. 17. Refeence and measued i sx, i sy cuent components with the cuent model (expeiement). show the efeence and the measued linea speed duing this test, obtained espectively with the cuent and the voltage flux models. Figs. 16 and 18 show the coesponding i sx, i sy cuent components. It should be noted that the measued speed popely tacks its efeence with high dynamic pefomance, as expected. No appeciable diffeence in the speed behaviou is obsevable between the two flux models. As fa as the cuent wavefoms ae concened, both flux models pemit a pope field oientation obsevable fom the coect decoupling the two cuent components. In paticula, i sx is coectly contolled to a constant efeence value and i sy pesents steps vaiations at each speed evesal being contolled to a constant non-null value at speed steady-state because of the Fig. 18. Refeence and measued i sx, i sy cuent components with the voltage model (expeiement).

10 20 M. Pucci / Electic Powe Systems Reseach 89 (2012) fiction load. In this case, it can easily be obseved that the cuent components obtained with the cuent flux model pesent a lowe ipple with espect to the coesponding obtained with the voltage model. The diffeence in the behaviou between the cuent and voltage flux models can be explained with the following consideations. The LIM dive in the pefomed tests woks at low speed and with low load foce (only the fiction foce acts). In these woking conditions, the cuent model, which is sensitive only to the vaiations of the induced pat time constant and only in loaded conditions, does not pesent sensitivity poblems. On the contay, the voltage model, which is sensitive to the vaiations of the inducto esistance and in paticula at low speed, pesents sensitivity poblems. Moeove, the voltage model pesents the additional poblems of open-loop flux integation caused by the DC off-sets of sensos and initial conditions; it has been solved hee adopting the classic solution of substituting the pue integato with a 1st ode low-pass filte with low cut-off fequency. The DC dift poblem is not suffeed by the cuent model in the synchonous efeence fame, since in this case the flux is closed-loop integated. The paticula woking conditions ae those making the cuent model pefeable to the voltage one as fa as the sensitivity to paametes vaiation is concened. With egad to this specific issue, the consideations valid fo the RIM can be extended to the LIM without lack of accuacy, since the tests ae made at low speed whee end effects ae not significant. A wong estimation of the induced pat flux, both in magnitude and in angle, causes a wong level of magnetization of the machine, which wongly woks eithe in deep satuation o altenatively in the linea egion of the magnetic chaacteistic of the machine. Coespondingly, the wong knowledge of the induced pat flux angle causes a wong field oientation, with consequent lowe dynamic pefomance of the dive and lowe load capability of it. As a emedy, methodologies fo the on-line estimation of the electical paametes to be used fo the coesponding on-line flux model adaptation, should be used [27,28]. To confim the goodness of the poposed flux modelling appoach taking into consideation the LIM end effects, a speed evesal test simila that that shown in Fig. 11 has been pefomed expeimentally. Even in this case, the test has been made twice, espectively with the cuent model taking into consideation the LIM end effects and with the same kind of model, without the end effects taken into consideation (the classic cuent model fo the RIM). Fig. 19 shows the efeence and measued speeds, obtained with both flux models duing a speed evesal fom 0.8 to 0.8 m/s. This is the maximum speed amplitude at which the evesal can be expeimentally accomplished, due to the limited length of the tack of the test set-up. Even fo this evesal, it can be obseved that the flux model including the end effects pemits the evesal to be accomplished in s less, which is almost 6% less of the entie speed evesal time. With egad to the position contol mode, only the test with the cuent model has been pefomed, since the voltage model hadly woks in position contol mode. The following position step efeences have been given the dive: m. Figs. 20 and 21 show the efeence and measued position as well as the measued speed. Fig. 22 shows the i sx, i sy cuent components. It can be obseved that the measued position popely tacks its efeence vaiations, with high dynamic pefomance. The measued speed pesents a wavefom coheent with the position efeence vaiations. Coespondingly, the inducto cuent cuves highlight a pope field oientation obsevable fom the coect decoupling the two cuent components. In paticula, i sx is coectly contolled to a constant efeence value and i sy pesents steps vaiations at each non-null speed command. Fig. 19. Refeence and measued speed with the flux models with and without endeffects (expeiment)il tu. Fig. 20. Refeence and measued position with the cuent model (expeiment). Fig. 21. Measued speed with the cuent model (expeiment).

11 M. Pucci / Electic Powe Systems Reseach 89 (2012) Fig. 22. Refeence and measued i sx, i sy cuent components with the cuent model (expeiement). 8. Conclusions This pape poposes and discusses some issues about diect field oiented contol (FOC) of linea induction moto (LIM) dives. Afte elaboating the inducto and induced pat space-vecto equations of the LIM in seveal efeence fames, some induced pat flux models taking into consideation the end effects ae pesented. In paticula, the so called voltage model based on the inducto equations in the inducto efeence fame and the cuent model based on induced pat equations in both the inducto and induced pat flux linkage efeence fames ae deduced and compaed to the otational induction machine countepat. Aftewads, afte a pope tuning of such models based on both FEA (finite element analysis), some simulation and expeiments have been pefomed. Simulation and expeimental esults show the impovements in tems of dynamic pefomance achievable with the poposed flux models taking into consideations the LIM end effects. Moeove, expeimental esults show a bette behaviou of the cuent model with espect to the voltage model, in paticula in the low speed egion. Refeences [1] E.R. Laithwaite:, Linea electic machines a pesonal view, Poceedings IEEE 63 (2) (1975), febbaio. [2] S. Yamamua, Theoy of the Linea Induction Moto, John Wiley & Sons, [3] I. Boldea, S. Nasa, Linea Electic Motos, Pentice-Hall Inc., [4] I. Boldea, S. Nasa, Linea Motion Electomagnetic Devices, Taylo & Fancis, New Yok, [5] I. Boldea, S.A. Nasa, Linea Electic Actuatos and Geneatos, Cambidge Univesity Pess, [6] M. Poloujadoff, The Theoy of Linea Induction Machiney, Claendon pess, Oxfod, [7] R.M. Pai, I. Boldea, S.A. Nasa, A complete equivalent cicuit of a linea induction moto with sheet seconday, IEEE. Tans. (1988) , MAG-24, n 1. [8] J. Duncan, C. Eng, Linea induction moto-equivalent-cicuit model, Poceedings IEE 130 (1983) 51 57, Pt. B, n. 1. [9] E.F. da Silva, E. Bueno dos Santos, P.C.M. Machado, M.A.A. de Oliveia, Dynamic model fo linea induction motos, in: ICITO32, Intenational Confeence on Industy Technology, Maibo, Slovenia, Decembe, 2003, pp [10] G. Gentile, N. Rotondale, M. Scaano:, Analisi del funzionamento tansitoio del motoe lineae, L Enegia elettica (5) (1987), Novembe. [11] G. Gentile, N. Rotondale, M. Scaano:, The linea induction moto in tansient opeation, L Enegia Elettica 7 8 (1988). [12] D. Hall, J. Kapinski, M. Kefta, O. Chistianson, Tansient electomechanical modeling fo shot seconday linea induction machines, IEEE Tansactions on Enegy Convesion 23 (3) (2008), Septembe. [13] J.H. Sung, K. Nam, A new appoach to vecto contol fo a linea induction moto consideing end effects, Industy Applications Confeence, 1999, Confeence Recod of the 1999 IEEE Thity-Fouth IAS Annual Meeting 4 (1999) [14] G. Kang, K. Nam, Field-oiented contol scheme fo linea induction moto with the end effect, IEE Poceedings: Electic Powe Applications 152 (6) (2005), Novembe. [15] V. Isastia, G. Gentile, S. Meo, A. Ometto, N. Rotondale, M. Scaano, A voltage feeding algoithm fo vectoial contol of linea asynchonous machines taking into account end effect, Powe Electonic Dives and Enegy Systems fo Industial Gowth 2 (1998) [16] E.F. da Silva, E. Bueno dos Santos, J.W.L. News Neys, Field oiented contol of linea induction moto taking into account end-effects, in: 8th IEEE Intenational Wokshop on Advanced Motion Contol, AMC 2004, Mach, 2004, pp [17] I. Takahashi, Y. Ide, Decoupling Contol of thust and attactive foce of a LIM using a space vecto contol invete, IEEE Tansactions on Industy Applications 29 (1993) [18] W. Leonhad, Contol of Electical Dives, Spinge-Velag, [19] W. Leonhad, Field oientation with contolled AC dives the natual solution to a tavelling wave poblem, Automatica 36 (1 2) (1995) 5 9. [20] W. Leonhad, Contolled AC dives, a successful tansition fom ideas to industial pactice, Contol Engineeing Pactice 4 (7) (1996) [21] W. Leonhad, 30 Yeas space vectos, 20 yeas field oientation, 10 yeas digital signals pocessing with contolled AC-dives, a eview (pat 1), EPE Jounal 1 (1) (1991), July. [22] W. Leonhad, 30 yeas space vectos, 20 yeas field oientation, 10 yeas digital signals pocessing with contolled AC-dives, a eview (pat 2), EPE Jounal 1 (2) (1991), Octobe. [23] P. Vas, Sensoless Vecto and Diect Toque Contol, Oxfod Science Publications, [24] I. Takahashi, T. Noguchi, A new quick esponse and high-efficiency contol stategy of an induction moto, IEEE Tansactions on Industy Applications 22 (5) (1986) [25] Holtz, Sensoless contol of induction moto dives, Poceedings of the IEEE 90 (8) (2002) , August.

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