Complex Dynamic Models of Multi-phase Permanent Magnet Synchronous Motors

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1 Preprint of the 8th IFAC World Congre Milano (Italy) Augut 8 - September, Complex Dynamic Model of Multi-phae Permanent Magnet Synchronou Motor R. Zanai, F. Groi, M. Fei Information Engineering Department, Univerity of Modena and Reggio Emilia, Via Vignolee 9, 4 Modena, Italy, {roberto.zanai, federica.groi, marco.fei}@unimore.it. Abtract: In the paper two power-invariant real and complex tate pace tranformation for modeling multi-phae electrical machine in a compact and general form are propoed. In particular the paper deal with the modeling of multi-phae permanent magnet ynchronou machine with an arbitrary number of phae and an arbitrary hape of the rotor flux. The dynamic model of the motor i obtained uing a Lagrangian approach and in the frame of the Power-Oriented Graph technique. The obtained model are equivalent from a mathematical point of view and can be directly implemented in Simulin. The complex tranformed model i quite compact and ue a reduced order tate vector. Some imulation reult end the paper. Keyword: Dynamic Modeling, Multi-phae ynchronou motor, Power-Oriented Graph, State pace tranformation.. INTRODUCTION In the literature many coordinate tranformation have been propoed in order to write the dynamic equation of ynchronou electric motor in a proper form. The tranformation typically ued are the Par, the Clare and Fortecue tranformation, ee Clare (9), Par (99) and Fortecue (98), repectively. Uually thee tranformation are exploited for three-phae machine, ee Paap (), but they can be extended to the cae of multi-phae machine, where the number of phae i greater than three. The extended Par and Clare tranformation are baed on a real matrix (ee Para and Toliyat () and Ketelyn et al. ()), the Fortecue tranformation i baed on a complex matrix which contain complex vector and their complex conjugate (ee White and Woodon (99)) while in the pace vector approach (ee Grandi et al. (6)) only one half of the Fortecue matrix i conidered (i.e. without the complex conjugate part). Unfortunately, all thee tranformation are not power invariant, i.e. after the tranformation the power flow cannot be expreed a the imple product of the tranformed coniugate power variable, but alway a correction coefficient mut be ued. In thi paper two power invariant tate-pace tranformation (one real and one complex) for modeling multi-phae ynchronou machine are propoed. They are uitable for machine with an arbitrary odd number of tator phae and for an arbitrary hape of the rotor flux. Uing the propoed tranformation one obtain the dynamic of the ytem written inaveryimple andcompact form, without the need of conidering a et of eparate fictitiou -phae machine a it i uually done in the literature (ee, for example, S et al. (4)). In thi paper the dynamic model of multi-phae ynchronou motor, uually obtained by mean of claical mathematical method i obtained uingalagrangian approach (ee Zanai and Groi (8)) in the frame of the Power- Oriented Graph (POG) technique. Thi graphical modeling technique how the power flow within the ytem, allow to write the tate pace equation of the ytem in a very compact form with a vectorial notation and provide a dynamic model that can be directly implemented in Simulin. The paper i organized a follow. Sec. introduce the main feature of POG technique. Sec. how the detail of the POG dynamic model of the multi-phae ynchronou motor, introduce the two propoed tate pace tranformation and how the obtained tranformed ytem. Some imulation reult are preented in Sec. 4 and concluion are given in Sec... POWER-ORIENTED GRAPHS BASIC PRINCIPLES The Power-Oriented Graph technique, ee Zanai (99) and Zanai (), i an energy-baed technique uitable for modeling phyical ytem. The POG are bloc diagram combined with a particular modular tructure eentially baed on the ue of the two bloc hown in Fig..a and Fig..b: the elaboration bloc (e.b.) tore and/or diipate energy (i.e. pring, mae, damper, capacitie, inductance, reitance, etc.), the connection bloc (c.b.) reditribute the power within the ytem x y G() x y a) e. b. b) c. b. Figure. POG baic bloc: a) elaboration bloc; b) connection bloc. x y K x K y Copyright by the International Federation of Automatic Control (IFAC) 8

2 Preprint of the 8th IFAC World Congre Milano (Italy) Augut 8 - September, u B y B L - x A { Lẋ = Ax + Bu y = B x (x = Tz) { Lż = Az+Bu y = B z () () R I I I I m L V V M M V V m τ m J m b m τ e Figure. POG cheme of a generic dynamic ytem in the complex domain. without toring nor diipating energy (i.e. any type of gear reduction, tranformer, etc.). The c.b. tranform the power variable with the contraint x y = x y. The circle in the e.b. i a ummation element and the blac pot repreent a minu ign that multiplie the entering variable. The POG eep a direct correpondence between the dahed ection of the graph and real power ection of the modeled ytem: the calar product x y of the two power vector x and y involved in each dahed line of a POG, ee Fig., ha the phyical meaning of the power flowing through that particularection. Another important apect of the POG technique i the direct correpondence between the POG repreentation and the correponding tate pace decription. For example, the POG cheme hown in Fig. can be repreented by the tate pace equation given in () where the energy matrix L i ymmetric and poitive definite: L = L >. For uch a ytem, the tored energy E and the diipating power P d can alway be expreed a follow: E = x Lx, P d = x Ax. The dynamic model () can be tranformed and reduced to ytem () uinga congruent tranformationx =Tz (matrix T can alo be rectangular and time-varying) where L = T LT, A = T AT + T LṪ and B = T B.. Notation The full and diagonal matrice will be denoted a follow: R R R m R i j [ R R R m R i,j =.. :n :m...., i [ R Ri =... :n.. R n R n R nm R n The ymbol i i Ri and [ R i will denote the column :n :n and row matrice. The ymbol b n=a:d c n = c a + c a+d + c a+d +... will be ued to repreent the um of a ucceion of number c n where the index n range from a to b with increment d, that i, uing the Matlab ymbology, n = [a : d : b. Uing the previou notation one can repreent, for example, the following matrix: [ R R 6 R 9 i j [ R i,j = ::6 :: R R 6 R 9 R R 6 R 9 Given a complex matrix A, the conjugate matrix will be denoted by A, the tranpoe matrix by A T and the conjugate tranpoe matrix by A. The following relation hold: A = (A ) T = (A T ). The ymbol will be ued to repreent a zero bloc matrix of proper dimenion. V Figure. Baic tructure of a multi-phae ynchronou motor. m number of motor phae p number of polar expanion θ, θ m electric and rotor angular poition: θ = p θ m ω, electric and rotor angular velocitie: ω = p R i-th tator phae reitance L i-th tator phae elf induction coefficient M maximum value of mutual inductance between tator phae φ c(θ) total rotor flux chained with tator phae ϕ c maximum value of function φ c(θ) J m rotor moment of inertia b m rotor linear friction coefficient τ m electromotive torque acting on the rotor τ e external load torque acting on the rotor γ baic angular diplacement (γ = π/m ) Table. Parameter of the multi-phae ynchronou motor.. ELECTRICAL MOTORS MODELLING The baic tructure of a permanent magnet ynchronou motor with an odd number m of tar-connected phae i hown in Fig.. The parameter of the conidered ytem are hown in Tab.. Let t I and t V denote the following current and voltage tator vector: t I =[I I I m T, t V =[V V V m T. () Uing a Lagrangian approach, ee Zanai and Groi (8) and Zanai and Groi (9), the dynamic equation S t of the conidered electric motor expreed with repect to the external fixed frame Σ t are: [ t [ [ L t I t R t [ = K τ (θ) t [ I t V J m t + (4) K T τ (θ) b m τ e Thee equation, in the literature, are uually rewritten and tranformed uing the Par and Fortecue tranformation, ee White andwoodon (99). ThePar tranformation, alo nown a dq tranformation, i baed on the following matrix: t TP = h [ co((θ hγ )) in((θ hγ )) m = :m co(θ) in(θ) co(θ) in(θ) co(θ γ ) in(θ γ ) co(θ γ ) in(θ γ ) co(θ γ ) in(θ γ ) co(θ 6γ ) in(θ 6γ ) co(θ γ ) in(θ γ ) co(θ 9γ ) in(θ 9γ ) co(θ 4γ ) in(θ 4γ ) co(θ γ ) in(θ γ ) and the Fortecue tranformation, alo nown a ymmetrical tranformation, i baed on the following matrix: m = 84

3 Preprint of the 8th IFAC World Congre Milano (Italy) Augut 8 - September, t TF = h [ e j h γ = e jγ e jγ e jγ e j4γ m e jγ e j4γ e j6γ e j8γ :m :m e jγ e j6γ e j9γ e jγ e j4γ e j8γ e jγ e j6γ In Grandi et al. (6) the complex conjugate term of Fortecue tranformation are neglected in order to reduce the number of the dynamic equation. The main drawbac of thee tranformation i that they are not power invariant. In the following two power-invariant tranformation are propoed.. Sytem S ω in the rotating frame Σ ω Uing the POG approach and applying to ytem S t in (4) the following tranformation matrix: t T ω = h [ co((hγ θ)), in((hγ θ)) m :m one obtain a tranformed and reduced ytem S ω decribed by real variable: [ ω [ L ω [ ω İ R = + ω L ω J ω [ K τ (θ) ω [ I ω V J m ω K T + τ (θ) b m τ e (6) where ω I = t T T ω t I, ω L = t T T ω t L t T ω, ω R = t T T ω t R t T ω = t R, ω J = t T T ω t Ṫ ω, ω K τ = t T T ω t K τ and ω V = t T T ω t V. Matrix t T ω (θ) i a function of the electrical angle θ and tranform the ytem variable from the original reference frame Σ t to a tranformed rotating frame Σ ω. For m = the tranformation matrix t T ω ha the following form: t T ω = co( θ) in( θ) co( θ) in( θ) co(γ θ) in(γ θ) co(γ θ) in(γ θ) co(γ θ) in(γ θ) co(6γ θ) in(6γ θ) co(γ θ) in(γ θ) co(9γ θ) in(9γ θ) co(4γ θ) in(4γ θ) co(γ θ) in(γ θ) m= (). The POG bloc cheme of a multi-phae ynchronou motor (6) in the tranformed rotating frame Σ ω i hown in Fig. 4. The POG cheme clearly put in evidence five different power ection. The connection bloc between ection - repreent the tate pace tranformation t T ω between the fixed reference frame Σ t and the rotating reference frame Σ ω. The elaboration bloc preent between the power ection - repreent the Electrical part of the ytem, while the bloc preent between ection 4 - repreent the Mechanical part of the ytem. The connection bloc preent between ection - 4 repreent the energy and power converion (without accumulation nor diipation) between the electrical and mechanical part of the motor. Let p(t) = t V T t I and (t) = ω V T ω I denote the intantaneou power in ection and of the POG cheme in Fig. 4. It can be eaily proved that the tranformation t T ω i power-invariant, indeed the power p(t) in ection i equal to the power (t) in ection : p(t) = t V T t I = ω V T t T T ω t T ω ω I = ω V T ω I = (t). The quare matrice ω L, ω J and ω R are defined a: [ [ ω L = L, L ω J = i [, ω R = R m = :m t V t I t T T ω V ω t T ω Σ t Σ ω ω L - ω I ω R + ω L ω J Electrical part (real variable) ω E ω K τ J m b m ω K T τ τ e τ m Energy Mechanical part Converion 4 Figure 4. POG bloc cheme of a multi-phae ynchronou motor in the rotating frame Σ ω. where ω= θ and parameter L are: { L L = + m M for = L for {,,...,m } Vector ω I and ω V in (6) are defined a: [ ω I = I = Id, I ω V = q ω V (7) [ = Vd (8) V q where I d, I q, V d and V q are, repectively, the direct and quadrature component of the current and voltage vector ω I and ω V. Note that uing tranformation ω V = t T T ω t V the original tate pace Σ t i tranformed into (m )/ two-dimenional orthogonal ubpace named Σ ω with { : : m }. It can be hown, ee Zanai and Groi (8), that when the rotor flux function φ(θ) ha the following tructure: φ(θ) = m i=: a i co(iθ) (9) the torque vector ω K τ (θ) i contant: [ K τ =pϕ c [ = a K τ = Kd. () K q Thi expreion clearly how that the motor torque τ m = ω K T τ ω I depend only on the quadrature component ω I q of the current vector ω I. Note that () hold for a generic number of phae m, while uually in the literature only a few particular cae for the torque vector ω K τ can be found, ee for example Para and Toliyat () and?. The dynamic model of a tar connected multi-phae ynchronou motor in the rotating frame Σ ω for m = i hown in Fig... Sytem S ω in the reduced complex rotating frame Σ ω Acomplex andreducedmodelofthemulti-phae ynchronou motor expreed in the rotating frame Σ ω can be obtained uingthefollowing reducedandcomplex peudotranformation matrix t m m T ωn C : t T ωn = t T ω N, () where t T ω and N are defined a: t T ω = m h [ e j(θ hγ), N = I m. () :m 8

4 Preprint of the 8th IFAC World Congre Milano (Italy) Augut 8 - September, L I R p L d L I q pω L m L R pϕ c I L d = R p L I q p L R pϕ c J m pϕ c a pϕ c a b m a a Figure. Dynamic model of a tar connected -phae ynchronou motor in the rotating frame Σ ω. L ω R İ L + jp L jpϕ c ω İ = a R + j p L jpϕ c a J m jpϕ c a jpϕ c a b m ω I ω I I d I q I d I q ω V + ω V τ e V d V q + V d V q τ e Figure 6. Dynamic model of a tar connected -phae ynchronou motor in the reduced complex rotating frame Σ ω. For a -phae motor the matrice t T ω and N are: t T ω = e jθ e j(θ γ) e j(θ γ) e j(θ γ) e j(θ 4γ) e jθ e j(θ γ) e j(θ γ) e j(θ γ) e j(θ 4γ) [, N =. Applying the peudo-tranformation () to ytem (4), one obtain the following tranformed ytem S ω expreed in the complex reduced rotating frame Σ ω : [ ω [ [ L ω ω İ R = + ω L ω J ω [ K ω τn I ω V +[ J m ω K τn b m τ e () The tranformed vector ω I, ω V and ω K τn are obtained uing matrix t T ωn : ω I = t T ωn t I = I, ω V = t T ωn t V = ωv ω K τn = t T ωn t K τ = ωkτ [ = K d + jk q while the tranformed matrice ω L = t T ω t L t T ω, ω J = t T ω tṫ ω and ω R = t T ω t R t T ω : [ L = L, [ J = j p, ω R = R I m. are obtained uing the tranformation matrix t T ω. The component ω I and ω V of vector ω I and ω V are complex number which atify the following relation: ω I = I d + ji q, ω V = V d + jv q where I d, I q, V d and V q are the direct and quadrature component of the current and voltage vector defined in (8). When the rotor flux function φ(θ) ha the tructure defined in (9), the torque vector ω K τn ha a complex and contant form: ω K τn (θ) = [ K τn = jpϕ c a. The original m -dimenion model in the fixed frame Σ t ha been tranformed and reduced to a (m )/-dimenion complex model in the rotating frame Σ ω. The dynamic modelofatarconnected multi-phae ynchronoumotor in the reduced complex rotating frame Σ ω for m = i hown in Fig. 6. Note that the electrical part of ytem S ω ha dimenion (m )/ while the electrical part of ytem S ω ha dimenion m. The POG cheme correponding to the dynamic ytem S ω given in () i hown infig. 7. ThefunctionRe( ) preent in the connection bloc between the power ection - and - 4 convert the tranformed complex vector ω I in the real vector t I : t I = Re ( t T ωn ω I ) = Re ( t T ω N ω I ) and in the real motor torque τ m : ( ) τ m = Re ω K τn ω I = Re ( t K τ t T ω N ω ) I. The mechanical part of the cheme i unchanged while theinternalvariable oftheelectricalpart (between power ection and ) are complex vector of reduced dimenion. Note that thi tranformation i power-invariant. Indeed it can be eaily proved that the intantaneou power p(t) = t V T t I in ection of Fig. 7 i equal to the real part of the complex intantaneou power (t) = ω V ω I in ection : p(t) = t V T t I = t V T Re( t T ωn t T ωn ) t I = Re( t V T t T ωn t T ωn t I )=Re( ω V ω I )=Re((t)). Suppoing the mechanical dynamic lower than the electrical one, i.e. auming a contant velocity, the eigenvalue of the following reduced ytem ω L ω İ = [ ω R + ω L ω J ω I + ω K τn (θ) + ω V are the eigenvalue of matrix A = ω L [ ω R + ω L ω J, i.e. the root of the following polynomial (): () = det [ ω L ( ω R + ω L ω J ) [ = det L R j p L = ::m (L R j p L ) 86

5 Preprint of the 8th IFAC World Congre Milano (Italy) Augut 8 - September, t V t I Re( ) t T ω V ωn Re( ) t T ωn Complex- Σ Real t Σ ω converion ω L - ω R + ω L ω J ω E ω I Electrical part (complex variable) ω K τn (θ) Re( ) ω K τn(θ) Re( ) τm Energy converion Complex- Real converion 4 J m b m τ e Mechanical part (real variable) Figure 7. POG cheme of a multi-phae electrical motor in the reduced complex rotating frame Σ ω. ω I [A Current in the complex reference frame T a T a Ih [A Time [ Current in the original reference frame Figure 8. Simulin bloccheme of themulti-phae electric motor and the correponding uer interface. Note that the characteritic polynomial () ha complex coefficient and therefore it (m )/ root are not necearily complex conjugate. In thi cae the root of polynomial () can be eaily found becaue ω L, ω R and ω J are bloc diagonal matrice. The (m )/ root λ of () are: λ = R + j p for = : : m (4) L and the correponding ettling time T a are: T a = for = : : m () Re(λ ) For the tranformed ytem S ω obtained uing the tate pace tranformation () the following property hold: when ω n i contant, the m eigenvalue of the conidered multi-phae ynchronou motor coincide with the (m )/ eigenvalue of ytem S ω, i.e. the complex root (4) of polynomial (), and their (m )/ complex conjugate value. 4. SIMULATION The dynamic model of the multi-phae electric motor hown in Fig. 4 and in Fig. 7 have been implemented in Simulin. The Simulin bloc cheme of the dynamic model of Fig. 7 and the correponding uer interface are howninfig. 8. Averyimportantadvantage ofexploiting the POG technique with vectorial notation i that the Simulin cheme i very compact and there i no need to Time [ Figure 9. Current in the Σ ω and Σ t reference frame. conider eparate fictitiou machine. With thi approach a unique general model i ued whatever the number of phae m i. Than to the graphical interface hown in Fig. 8, the uer can modify the number of phae and the parameter of the model without changing the tructure of the model. The imulation reult hown in Fig. 9 and Fig. have been obtained uing the following electrical and mechanical parameter: m =, p =, R =.Ω, L =. H, M =. H, ϕ r =. Wb, J m =. g m, b m =. Nm /rad, V max = V, a =.9, a =. and the external torque τ e = Nm. The ytem ha been fed with thefollowing input, ee the POGcheme of Fig. 7: ω V =( ω R + ω J ω L ) ω I d + ω K τ d where ω I d and d are the deired current vector and motor velocity, repectively. The deired current vector ω I d that provide the deired torque τ d minimizing the current diipation i the current vector parallel to the torque vector ω K τn : ω I d = ω K τn ω K τn τ d = [ jτd K, K = K q ω K τn The deired torque i τ d = Nm for t [,4[ and τ d = Nm for t 4 therefore at t = 4 a tep i applied to voltage vector ω V to ee the open-loop tep repone of the ytem. The time behavior of tator current t I and ω Ī in frame Σ t and Σ ω are hown in Fig. 9. The tranient of the component ω I of vector ω I i deeply related to the pole (4) of the electrical part 87

6 Preprint of the 8th IFAC World Congre Milano (Italy) Augut 8 - September, ωm [rad/ τm [Nm Time [ Motor velocity T am Motor torque τ m Time [ Figure. Motor velocity and motor torque τ m. I [A Amplitude [A Time [ t Phae current pectrum -rd ω [rad/ Zoom T a Phae current in the original reference frame Figure. Phae current waveform and harmonic pectrum in teady-tate condition. of the ytem: λ = 8.6 ± j and λ = ± j when = rad/. The correponding ettling time T a obtained uing () are: T a = Re(λ )., T a =.. Re(λ ) The time behavior of motor velocity and motor torque τ m are hown in Fig.. The torque tranient i deeply related to the electrical dominant pole of the ytem located in λ, while the motor velocity tranient i related to the mechanical pole located in λ m = b m /J m =.67. The mechanical ettling time i: T am = 4. Re(λ m ) For t > T a the torque variation i mainly due to the increaing of motor velocity which caue the variation of bac-emf. Since the component ω I and ω I of vector ω I are both different from zero, then the harmonic pectrum of the phae current waveform contain the - t and the -rd harmonic, a it i hown in Fig.. The ame imulation reult have been obtained alo uing the Simulin implementation of the POG cheme of Fig. 4 which decribe the multi-phae electric motor with repect to the rotating frame Σ ω : the order of the error between the two imulation i 4.. CONCLUSION In thi paper the modeling of multi-phae ynchronou motor with an odd number of phae ha been invetigated with a pecific focu on the coordinate tranformation which allow to write the ytem dynamic in a compact and reduced form. Two tate pace tranformation (one real and one complex) have been preented. The propoed tranformation have ome advantage with repect to thoe ued in the literature: they are powerinvariant and they ue a very compact vectorial notation. The POGcheme uedtorepreenttheconideredmodel ha the important advantage that they can be directly implemented in Simulin and therefore there i no need to conider eparate fictitiou machine. The preented imulation reult have hown the effectivene and the preciion of the propoed tranformation. REFERENCES D. C. White and H. H. Woodon, Electromechanical Energy Converion. New Yor: Wiley, 99 Para, L. and Toliyat, H.A., Five-Phae Permanent- Magnet Motor Drive, IEEE Tranaction on Indutry Application,, Vol. 4, No., pp. -7. X. Ketelyn, E. S , JP. Hautier, Vectorial Multimachine Modeling for a Five-Phae Machine, in Proc. Int. Conf. Electrical Machine (ICEM), Bruge, Belgium,, CD-ROM, Paper 94. E. S , X. Ketelyn, A. Boucayrol, Right Harmonic Spectrum for the Bac-Electromotive Force of an-phae Synchronou Motor, Indutry Application Conference, 4, 9th IAS Annual Meeting, ISBN: Paap, G.C., Symmetrical Component in the Time Domain and Their Application to Power Networ Calculation, IEEETranactiononPower Sytem, Volume, Iue, pp.-8,. C. L. Fortecue, Method of Symmetrical Coordinate Appliedtothe Solution ofpolyphae Networ. Tran. AIEE, pt. II, vol. 7, pp.7-4, 98 Grandi G.; Serra G.; Tani A., General Analyi of Multi- Phae Sytem Baed on Space Vector Approach, Power Electronic and Motion Control Conference, EPE-PEMC 6, pp E.Clare, Circuit Analyi of AC Power Sytem. New Yor: Wiley, 9, vol.i R. H. Par, Two-reaction theory of ynchronou machine. Tran. AIEE, vol. 48, pp. 76, 99. R. Zanai, F. Groi Optimal Rotor Flux Shape for Multi-phae Permanent Magnet Synchronou Motor, International Power Electronic and Motion Control Conference, 8, Poznan, Poland. R. Zanai, Power Oriented Modelling of Dynamical Sytem for Simulation, IMACS Symp. on Modelling and Control of Technological Sytem, Lille, France, 99. R. Zanai, The Power-Oriented Graph Technique: ytem modeling and baic propertie, Vehicular Power and Propulion Conference, Lille, France, Sept. R. Zanai, F. Groi. Vectorial Control of Multi-phae Synchronou Motor uing POG Approach. IECON 9, th Annual Conference of the IEEE Indutrial Electronic Society, 9 Porto, Portugal. 88