THE POG TECHNIQUE FOR MODELLING MULTI-PHASE PERMANENT MAGNET SYNCHRONOUS MOTORS

Transcription

1 THE POG TECHNIQUE FOR MODELLING MULTI-PHASE PERMANENT MAGNET SYNCHRONOUS MOTORS Roberto Zana, Federca Gro DII, Unverty of Modena and Reggo Emla Va Vgnolee 95, Modena, Italy (Roberto Zana) Abtract In the paper the Power-Orented Graph (POG) technque ued for modellng n-phae permanent magnet ynchronou motor The POG technque a graphcal modellng technque whch ue only two bac block (the elaboraton and connecton block) for modellng phycal ytem It man charactertc are the followng: t keep a drect correpondence between par of ytem varable and real power flow; the POG block repreent real part of the ytem; t utable for repreentng phycal ytem both n calar and vectoral fahon; the POG cheme can be ealy tranformed, both graphcally and mathematcally; the POG cheme are mple, modular, eay to ue and utable for educaton The POG model of the condered electrcal motor how very well, from a power pont of vew, t nternal tructure: the electrc part of the motor nteract wth the mechancal part by mean of a connecton block whch nether tore nor dpate energy The dynamc model of the motor a general a poble and t conder an arbtrary odd number of phae and an arbtrary number of harmonc of the rotor flux waveform Generalzed orthonormal tranformaton allow to wrte the dynamc equaton of the ytem n a very compact way The model fnally mplemented wth Matlab/Smulnk oftware The Smulnk tructure of the motor clearly reflect t POG repreentaton Smulaton reult are then preented to valdate the machne model Keyword: Modellng, Smulaton, mult-phae ynchronou motor, arbtrary rotor flux, Graphcal modellng technque, Power-Orented Graph Preentng Author Bography Roberto Zana He graduated wth honor n Electrcal Engneerng n 986 Snce 998 he ha been workng a Aocate Profeor of Automatc Control at Unverty of Modena and Reggo Emla In he became Profeor n Automatc Control He held the poton of Vtng Scentt at the IRIMS of Mocow n 99, at the MIT of Boton n 99 and at the Unvert Catholque de Louvan n 995 H reearch nteret nclude: mathematcal modellng, mulaton, automotve, control of varable-tructure ytem, ntegral control, robotc, etc Paper preented at EUROSIM 7, 6-th EUROSIM Congre on Modellng and Smulaton, September 9-, 7, Ljubljana, Slovena

2 Introducton A well known graphcal technque for modellng phycal ytem the Bond Graph technque, ee [], [] and the reference theren Th modellng technque ue power nteracton between ytem a the bac concept for modellng It ha alo a formal graphcal language to repreent the bac component that may appear n a broad range of phycal ytem However th technque ha few drawback: the graphcal chematc repreentaton need more than ymbol to repreent phycal ytem and t not ealy readable; the power varable mut be clafed n effort and flow varable and fnally the mplementaton of the bond graph on a general purpoe computer mulator may requre a non trval tranlaton (caualty problem) A for Bond Graph, the bac dea of the Power- Orented Graph (POG) modellng technque to ue the power nteracton between ubytem a bac concept for modellng Pleae refer to [], [] and [5] for further detal Th approach allow the modellng of a wde varety of ytem nvolvng dfferent energetc doman Dfferently from the bond graph technque, the POG modellng technque ue only three bac ymbol, t doe not need to clafy the power varable and t olve drectly the caualty problem By th way, the POG cheme are ealy readable, cloe to the computer mplementaton and allow relable mulaton ung every computer mulator A lt of reference of example of applcaton of the POG technque can be found n [6] The paper organzed n the followng way Secton tate the bac properte of the POG modellng technque Secton how the detal of POG modellng of n-phae permanent magnet ynchronou motor Fnally, n Secton ome mulaton are reported Notaton In the paper the followng notaton wll be ued: - Row matrce: [ R ] = :n [ R ] = [ R R R n ] :n - Column and dagonal matrce: R R R n, [ R ] = :n R R R n - Full matrce: R R R m j R R R m [ R,j ] = :n :m R n R n R nm - The ymbol b n=a:d c n = c a + c a+d + c a+d + c a+d + wll be ued for repreentng the um of a ucceon of number c n where the ndex n range from a to b wth ncrement d that, ung the Matlab ymbology, n = [a : d : b] - The ymbol x wll be ued for denotng the nteger part of x rounded toward below - The ymbol I m wll be ued for denotng the dentty matrx of order m - Functon mod (θ, π) the remander of varable θ after dvon by coeffcent π The bae of Power-Orented Graph The Power-Orented Graph are gnal flow graph combned wth a partcular modular tructure eentally baed on the two block hown n Fg The bac charactertc of th modular tructure the drect correpondence between par of ytem varable and real power flow: the product of the two varable nvolved n each dahed lne of the graph ha the phycal meanng of power flowng through the ecton The x y G() x y x K x y K T y Fg Bac block: elaboraton block (eb) and connecton block (cb) two bac block hown n Fg are named elaboraton block (eb) and connecton block (cb) The crcle preent n the eb a ummaton element and the black pot repreent a mnu gn that multple the enterng varable There no retrcton on x and y other than the fact that the nner product x, y = x T y mut have the phycal meanng of a power The eb and the cb are utable for repreentng both calar and vectoral ytem In the vectoral cae, G() and K are matrce: G() alway quare, K can alo be rectangular Whle the elaboraton block can tore and dpate energy (e prng, mae and damper), the connecton block can only tranform the energy, that, tranform the ytem varable from one type of energy-feld to another (e any type of gear reducton) In the lnear vectoral cae when G() = [M + R] -, (M ymmetrc and potve defnte) the energy E tored n the eb and the power P d dpatng n the eb can be expreed a: E = yt M y, P d = y T R y There a drect correpondence between POG repreentaton and the correpondng tate pace decrp-

3 ton For example, the ytem { L ẋ = Ax + Bu y = B T x L = L T > () can be repreented by the POG cheme hown n Fg PSfrag replacement u B y B T L - x Fg POG block cheme of a generc dynamc ytem When an egenvalue of matrx L tend to zero (or to nfnty), ytem () degenerate toward a lower dmenon dynamc ytem In th cae, the dynamc model of the reduced ytem can be drectly obtaned from () by ung a mple congruent tranformaton x = Tz (T contant): { TT LTż = T T ATz+T T Bu y = B T Tz A { Lż = Az+Bu y = B T z 5 q 5 d θ φ(θ) Fg Structure of a fve-phae motor n the cae of ngle polar expanon (p = ) m : number of motor phae; p : number of polar expanon; θ r : rotor angular poton; ω r : rotor angular peed; θ : electrc angle: θ = p θ r ; N c : number of col for each phae; R : -th phae retance (p = ); L : -th phae elf nducton coeffcent (p = ); M j : mutual nducton coeffcent of -th phae coupled wth j-th phae (p = ); φ(θ) : rotor permanent magnet flux; φ c (θ) : total rotor flux chaned wth tator phae ; φ c (θ) : total rotor flux chaned wth tator phae -th; ϕ r : maxmum value of functon φ(θ); ϕ c : maxmum value of functon φ c (θ); J r : rotor nerta momentum; b r : rotor lnear frcton coeffcent; τ r : electromotve torque actng on the rotor; τ e : external load torque actng on the rotor; γ : bac angular dplacement; Fluxe φ(θ) and φ c (θ) atfy relaton: φ c (θ) = p N c φ(θ) = p N c ϕ r φ(θ) = ϕc φ(θ) where L = T T LT, A = T T AT and B = T T B If matrx T tme-varyng, an addtonal term T T LṪz appear n the tranformed ytem When matrx T rectangular, the ytem tranformed and reduced at the ame tme Electrcal motor modellng In th paper we wll refer only to permanent magnet ynchronou electrcal motor wth an odd number of phae In Fg t hown the electromechancal tructure of a fve-phae motor n the cae of ngle polar expanon (p = ) The condered mult-phae electrcal motor characterzed by the followng parameter: where ϕ c = p N c ϕ r and φ(θ) the rotor flux functon normalzed repectvely to t maxmum value ϕ r Let γ = π m denote the bac angular phae dplacement for electrcal motor wth m phae Referrng to the condered mult-phae electrcal motor, the followng hypothee are aumed: H) Functon φ(θ) perodc wth perod π; H) Functon φ(θ) an even functon of θ; H) Functon φ(θ + π ) an odd functon of θ; H) For θ = the rotor flux φ c (θ) chaned wth phae maxmum; H5) The electrcal motor homogeneou n t electrcal charactertc The electrcal couplng of a generc couple of phae and j of the motor hown Fg The dfferental equaton decrbng the dynamc behavour of the -th

4 V Φ R E p Φ c(θ) ω r J r b r L - I p Φ T c (θ) τr τ e Electrcal part Energy converon Mechancal part Fg 5 POG cheme of the mult-phae motor dynamc model I p R p R j V V φ(θ) V j V p L p M j p L j Fg Electrc couplng of a generc couple of phae and j of the motor phae of the motor the followng: d φ c = V V p R I, {,, m} () dt where V V the voltage appled to the -th phae and φ c the total magnetc flux chaned wth the -th phae: m φ c = p L I + p M j I j + φ c (θ) () I = I I m j=, j V repreent the reference common voltage for all the phae If the m phae are tar-connected t can be hown that V the voltage at the tar centre and V = m m = V However, when the ytem tarconnected the nput voltage V are defned wth a degree of freedom, o wthout lo of generalty n () one can et V = If the m phae are ndependent, V the ground voltage and n th cae too one can et V = Let ntroduce the vector: I V V, V = V m and let ndcate wth Φ c (θ) the vector of chaned rotor fluxe Due to the magnetc ymmetry of the electrcal I j motor (ee H5) we have that: φ φ c (θ) c (θ) φ φ c (θ) Φ c (θ) = = c (θ γ) φ c (θ γ) () φ cm (θ) φ c (θ (m )γ) Let Φ c denote the vector of chaned total magnetc fluxe defned n (): Φ c (I, θ) = L I + Φ c (θ) (5) where L > the followng potve-defnte ymmetrc matrx: L M M M m M L M M m L = p M M L M m M m M m M m L m The ytem of m dfferental equaton () can be rewrtten n matrx form a: d Φ c (I, θ) dt where R the dagonal matrx: = V R I (6) R = p [ R ] :m From Eq () and (5) we have that: d(l I) dt = V RI E {}}{ p Φ c(θ) } {{ } K τ (θ) ω r (7)

5 V t T T b b T T ω ω V Φ J ω ω L ω R ω E ω K(θ) ω r J r b r ω L - I t T b b T ω ω I ω K T (θ) τr τ e Σ t Σ b Σ b Σ ω ˆ Electrcal part Energy Converon Mechancal part Fg 6 POG cheme of a mult-phae electrcal motor n the tranformed pace Σ ω where K τ (θ) = p Φc(θ) the vector that multpled by ω r gve the counter-electromotve force E = K τ (θ)ω actng on the electrcal part of the motor The dfferental equaton decrbng the mechancal part of the motor : d(j r ω r ) dt = τ r τ e b r ω r (8) Snce the energy E(I, ω r, θ r ) tored n the electromechancal ytem : E(I, ω r, θ r ) = IT LI + Φc(pθ r) I T dφ c (pθ r ) + J rω r the electromotve torque can be calculated a follow: τ r = E(I, ω r, θ r ) r = p ΦT c (θ) I = K T τ (θ) I }{{ } K T τ (θ) Relaton (7) and (8) can be graphcally repreented by the POG cheme hown n Fg 5 The elaboraton block preent between the power ecton and repreent the Electrcal part of the ytem, whle the block between ecton and repreent the Mechancal part of the ytem The connecton block preent between ecton and repreent the converon of energy and power (wthout accumulaton nor dpaton) between the electrcal and mechancal doman Functon φ c (θ) even and perodc of perod π o t can be developed n Fourer ere of cone wth only odd harmonc: φ c (θ) = ϕ c φ(θ) = ϕc n=: a n co(nθ) (9) From () and (9) t follow that vector Φ c (θ) can be rewrtten n a compact form a: h [ ] Φ c (θ) = ϕ c a n co[n(θ h γ)] () n=: :m From (7) and () one obtan vector K τ (θ): h [ K τ (θ) = p ϕ c n=: :m ] n a n n[n(θ h γ)] () Let u now conder the two followng orthonormal tranformaton: t T b = h [ co(hkγ), n(hkγ) k h [ ] ], m b T ω = :m k [ co(kθ) n(kθ) n(kθ) co(kθ) ::m ] ::m :m,, Matrx t T b mlar to the generalzed Concorda tranformaton but wth a permutaton on column Th matrx tranform the ytem varable from the orgnal reference frame Σ t to an ntermedate reference frame Σ b Matrx b T ω repreent a multple rotaton n the tate pace a functon of the electrcal motor angle θ Th matrx tranform the ytem from the ntermedate reference frame Σ b to fnal rotatng reference frame Σ ω The product of thee two matrce t T ω = t T b b T ω tll an orthonormal tranformaton, from Σ t to Σ ω, wth the followng tructure: k [ co(k (θ h γ)) t T T ω = ω n(k (θ h γ)) T t = ::m m [ ] h :m :m ] h Snce the mult-phae motor homogeneou n t elec-

6 trcal charactertc, ee hypothe H5, we can et M j = M co(( j)γ) L = + M R = R, j {,,, m} where and M are proper parameter characterzng the elf and mutual nducton coeffcent of the motor phae Applyng tranformaton t T ω to matrce L, R and K τ (θ) one obtan the followng tranformed matrce ω L = ω T t L t T ω, ω R and ω K τ (θ): + m M + m M ω L = p, The POG cheme decrbng the condered multphae electrcal motor n the tranformed pace Σ ω hown n Fg 6 Note that the POG cheme eay to read, t clearly how the ecton where power flow (the thn dahed lne) and t ha a drect correpondence wth the tate pace decrpton () Moreover, th POG cheme can be drectly ntroduced (more or le a t ) and mulated n Smulnk Fourer ere of the rotor flux The Fourer ere of ome perodc rotor fluxe of practcal nteret are lted below - Trapezodal waveform, ee Fg 8: φ(θ) ω R = ω T t R t T ω = R = p R I m, PSfrag replacement - m ω K τ (θ) = ω T t K τ (θ) = p ϕ c k [(n+k) a n+k +(n k) a n k ] n(nθ) n=:m [(n+k) a n+k (n k) a n k ] co(nθ) () n=:m ::m na n n (nθ) n=m:m Note that vector ω K τ (θ) compoed only by harmonc n(nθ) and co(nθ) where n an nteger number multple of m Moreover, vector ω K τ (θ) can be ealy computed knowng the coeffcent a n of the Fourer ere of the rotor flux, ee Eq (9) PSfrag replacement In the tranformed pace Σ ω, the dynamc equaton of the mult-phae electrcal motor (both mechancal and α electrcal part) can be repreented n compact co πθ form + a: π α π α [ ω ] L [ ] ω [ İ ω R+J ω ω L ω ][ K ω ] [ τ I ω ] Fg 9 Conuodal nterpolatated waveform φ(θ) V = J r ω r ω + K T τ b r ω r τ e ( ) () α πθ where ω I = ω T t I, ω V = ω T t V and matrx J ω : φ(θ) = π co + π α, θ [, α] α θ, θ [ ] α, π k [ ] k ω J ω = k ω It Fourer ere : ::m π co(nα) φ(θ) = n (π n α ) co(nθ) where ω = θ If the m phae of the rotor are tarconnected then m = I = and o ω I m = In th cae the m-th equaton of ytem () a tatc relaton that can be elmnated lettng the ytem reduce to order m Fg 8 Trapezodal waveform φ(θ) wth α π φ(θ) = π n=: ( n n π ) n (nα) α n co(nθ) () The Fourer ere of the Square and Trangular waveform can be ealy obtaned from () by ettng, repectvely, α = and α = π - Conuodal nterpolatated waveform The gnal φ(θ) hown n Fg9 defned a follow: n=: φ(θ) The normalzed functon : co φ(θ) φ(θ) = ( α π + π (5) α) θ

7 b n (r) = r ( + j) r j= ( r α r π n r+ ( ) ) (αn) = r ( r + ( ) + ) = r j= (αn) + ( + j) n(αn)+ r j=+ ( + j) co(αn) Fg 7 General formula for computng coeffcent b n (r) of Fourer ere of gnal φ(r, θ) φ(r, θ) φ(q, θ) q r frag replacement - PSfrag replacement Fg Perodc gnal φ(q, θ) for q =,, 6 Fg Perodc gnal φ(r, θ) for r =, 5, 7 - Odd polynomal nterpolaton Let u conder the famly of perodc gnal φ(r, θ) hown n Fg { φ(r, θ), θ [, α] φ(r, θ) =, θ [ ] α, π where φ(r, θ) a polynomal (r odd) wth the followng tructure: φ(r, θ) = d (r)θ + d (r)θ + + d r (r)θ r The polynomal coeffcent d (r) can be computed, for {,,, r}, conderng the followng contnuty condton: φ(r, α) = φ (r, α) = φ ( r+ ) (r, α) = The Fourer ere of gnal φ(r, θ) φ(r, θ) = b n (r) n(nθ) (6) n=: where coeffcent b n (r) can be calculated ung the general formula reported n Fg 7 - Even polynomal nterpolaton Let u conder the famly of perodc gnal φ(q, θ) hown n Fg: φ(q, θ), θ [, α] φ(q, θ) = π θ, θ [ ] α, π where φ(q, θ) a polynomal (q even) wth the followng tructure: φ(q, θ) = c (q) + c (q)θ + c (q)θ + + c q (q)θ q The polynomal coeffcent c (q) can be computed, for {,,, q}, conderng the followng contnuty condton: φ(q, α) = π α φ (q, α) = φ (j) (q, α) = for j {,, q } The Fourer ere of gnal φ(q, θ) : φ(q, θ) = n=: a n (q) co(nθ) (7) where the ere coeffcent a n (q) can be obtaned from the coeffcent b n (r) reported n Fg 7 a follow: a n (q) = b n(r) n r=q The normalzed gnal φ(q, θ) = φ(q, θ) (8) c (q) can be obtaned from gnal φ(q, θ) dvdng by coeffcent c (q): c (q) = π α q = q+ ( + ) q! q

8 frag replacement Note that the dervatve φ(,θ) of gnal φ(q, θ) when q = ha a trapezodal hape and o for th type of rotor flux the counter-electromotve force E = K τ (θ)ω have a trapezodal hape proportonal to the motor velocty ω, ee eq (7) Smulaton The POG cheme hown n Fg 6 ha been mplemented n Smulnk Smulaton decrbed n th Sec- Electromotve torque τ 5 r generated by current ω I ton have been obtaned ung the followng PSfrag replacement electrcal 5 and mechancal parameter: m = 5, p =, R = Ω, L = H, M = 8 H, N c =, N a =, ϕ r = W, J m = 6 kg m, b m = 8 Nm /rad 5 Parameter N a the number of harmonc condered n Fourer ere expanon of the rotor flux functon Tme [] The mulaton have been obtaned wth tar connected phae and ung for the rotor flux the conuodal nterpolated waveform φ(θ) Fg Angular velocty ω r of the motor and electromotve torque τ r generated by the motor current ω I wth α = π 5, ee (5) Symmetrc nput voltage have been condered, t to ay: V = V M co(π t + γ), {,,, m} wth V M = 8 V At tme t = the load torque τ e wtche from to Nm Fg, and how the mulaton reult obtaned for the followng varable: phae current I and ω I n the Σ t and Σ ω reference frame; motor angular velocty ω r ; electromotve torque τ r ; ymmetrc m-phae nput voltage V and counter-electromotve voltage ω E n the tranformed Σ ω reference frame Current I [A] Current ω I [A] Phae current I n the reference PSfrag frame Σ replacement t Phae current ω I n the tranformed frame Σ ω Tme [] Fg Phae current I and ω I n the orgnal Σ t and n the tranformed Σ ω reference frame repectvely Let u now conder the control law V = V M S, for {,,, m}, where: θ [π + α, π α] S = θ [α, π α] otherwe (9) θ = mod (θ + ( )γ, π) Th control law can be appled to the POG cheme hown n Fg 5 and 6 by connectng at the power ec- Velocty ωr [rad/] Torque τr [N m] Voltage V [V] Voltage ω E [V] Angular velocty ω r of the motor Symmetrc m-phae nput voltage V Back electromotve voltage ω E generated by velocty ω r Tme [] Fg Symmetrc m-phae nput voltage V and counter-electromotve voltage ω E n the tranformed Σ ω reference frame ton of thee cheme the connecton block hown n Fg 5 The mulaton reult hown n Fg 6 and 7 refer to the ame ytem parameter ued n the prevou mulaton except for: number of polar expanon p = ; the load torque appled at tme t = 5; for the rotor flux t ha been choen the normalzed even polynomal nterpolaton functon φ(q, θ) wth q = and α = π 5, ee (8) In th cae the hape of the counterelectromotve voltage E trapezodal a t evdent n the upper part of Fg 7 Th type of control generate hgh torque τ r when the angular velocty ω r mall, ee the upper part of Fg 6 and the lower part of Fg 7 The behavour of the controlled ytem table wth compared to the varaton of the external torque τ e, n fact the motor torque τ r ncreae to face the ncreaed external torque τ e, ee Fg 7

9 V M I M [ S ] :m [ S ] :m V I 5 Concluon In th paper a n-phae permanent magnet ynchronou motor ha been modelled ung the Power-Orented Graph (POG) technque Th approach exhbt ome advantage n comparon wth other graphcal technque and allow to realze a very compact model cheme Moreover t can be ealy tranlated nto a Smulnk model Some mulaton are carred out n order to how the effectvene of the realzed model n the cae of a fve-phae motor wth tar connecton confguraton A bref lt of Fourer ere of the rotor flux gnal ha been preented and then ued n the Smulnk model frag replacement Fg 5 POG connecton block that can be ued to apply the control law (9) to the POG cheme hown n Fg 5 and 6 Velocty ωr [rad/] Current I [A] Angular velocty ω r of the motor Phae current I n the reference frame Σ t Tme [] Fg 6 Angular velocty ω r and phae current I of the motor Voltage E [V] Counter-electromotve voltage E Motor torque τ r 6 Reference [] Paynter, HM, Analy and Degn of Engneerng Sytem, MIT-pre, Camb, MA, 96 [] D C Karnopp, DL Margol, R C Roemberg, Sytem dynamc - Modelng and Smulaton of Mechatronc Sytem, Wley Intercence, ISBN -7--8, rd ed [] R Zana, Power Orented Modellng of Dynamcal Sytem for Smulaton, IMACS Symp on Modellng and Control of Technologcal Sytem, Llle, France, May 99 [] R Zana, K Salbury, Dynamc Modelng, Smulaton and Parameter Identfcaton for the WAM Arm, AI Memo No 87, MIT, Cambrdge, USA, Augut 99 [5] Zana R, Dynamc of a n-lnk Manpulator by Ung Power-Orented Graph, SYROCO 9, Capr, Italy, 99 [6] R Morell, R Zana, Modelng of Automotve Control Sytem Ung Power Orented Graph, nd Annual Conference of the IEEE Indutral Electronc Socety, IECON 6, Parg, 7- Novembre, 6 [7] E Semal, X Ketelyn, A Boucayrol, Rght Harmonc Spectrum for the Back-Electromotve Force of a n-phae Synchronou Motor, Indutry Applcaton Conference,, 9th IAS Annual Meetng, ISBN: [8] E Semal, X Ketelyn, A Boucayrol, Sentvty of a 5-phae Bruhle DC machne to the 7th harmonc of the back electromotve force, 5th Annual IEEE Power Electronc Specalt Conference, frag replacement Torque τr [Nm] Tme [] Fg 7 Counter-electromotve voltage E and motor torque τ r