Sensorless Control of AC Machines

Transcription

1 esorless Corol of AC achies Werer Zimmerma Fachhochschule fuer Techik Esslige Fladersraße, D Esslige, Germay Eail: Absrac Iverer fed AC machies are used i may applicaios, where high dyamic performace is eeded, bu posiio corol is o required. Coveioal field orieed corol fulfills he dyamic requiremes, bu usually eeds a cosly posiio or speed sesor. imple volage-frequecy-corol avoids hese sesors, bu will oly achieve a moderae dyamic performace. For such applicaios a self corolled scheme based o he measured flux vecor may be used. I his paper a flux phase corol loop ad a mehod of backig he flux model wih a observer o improve he behaviour a low saor frequecies are described. The flux observer may be easily corolled by sigals from he he iverer's curre corol sysem, which by usig he 'free wheelig sae' efficiely reduces he hermal sress of he iverer. suied for cosiderig rasie effecs i saor curre corol loops. a) u e j ϕ b) i e j ϕ i d R u R e j ϕ σl u L e j ϕ R R ~ L R u i = d (- σ e j ϕ ) d d L. Iroducio oder microelecroics have chaged he world of elecrical drives, where AC machies seem o have compleely coquered he field of speed variable drives, which formerly was a domai of DC machies. Wih moder power rasisors, high performace iverers, pioeered i such applicaios by Boehriger ad ohers i he lae 97s [], ca be realized a reasoable cos. Today's moder microcorollers, DPs (digial sigal processors) ad AICs (applicaio specific iegraed circuis) allow he implemeaio of he ecessary complicaed corol srucures based o he work of Blaschke, Leohard e.al. [2]. Wih Depebrock's priciple of 'direc self corol' (DC) wo remaiig problems of he field orieed corol could be overcome [3]. DC reduces he sesiviy of he corol algorihm o variaios of he machie parameers by iroducig closed loop flux corol ad akes io accou he limiaios of he iverer, i.e. he fiie swichig saes, swichig frequecy ad limied oupu volage. Especially his laer feaure makes DC well suied for use i high power racio drives. As a posiive side effec, DC does o iherely eed a posiio sesor. I his paper a varia of DC for iverers wih uderlyig curre corol ad squirrel cage AC machies wihou speed or posiio sesor is described, which is very well suied for applicaios, where high dyamic performace is demaded, bu posiio corol is o eeded ad speed accuracy requiremes are o exreme. 2. Dyamic behaviour of asychroous machies Fig. shows simplified equivale circuis of he asychroous machie. Fig.a describes he behaviour of he saor circui, c) u L u jq u R u i i q jb i i d ~ i = d d iq Figure : Equivale circui of he asychroous ma chie a) saor b) roor c) space vec or diagram (abbreviaios see appedix A) The easies way o corol he machie is o apply a saor volage U ad frequecy ω =2πf =dϕ/d proporioal o he desired speed (Fig.3). This esures a early cosa, speed idepede mageizaio. A low speeds he saor volages mus be slighly icreased o compesae for he resisive volage drops. A high speeds, he saor volages are held cosa o he iverers maximum oupu value, operaig he machie i he field weakeig rage. Due o he slip, here will be a sligh, load depede deviaio of he he machie's speed from he se saor frequecy. This ca also be compesaed for by measurig he ampliude of he saor curres. This corol scheme works wihou a speed sesor, bu does oly have a moderae dyamic performace. Durig fas load ad speed ϕ d a

2 chages here may be heavy rasie oscillaios of he machie's orque, as he phase relaios of he curres ad volages are o ake io accou. U much smaller ha he roor ime cosa T R = L R /R R (Fig.b), he idea of field orieed corol is o use i q for high dyamic orque corol ad o hold i d ad hus d cosa - a leas as log as here is o eed for field weakeig - o avoid slow rasies i he machie's mageic field. U N base speed rage field weakeig rage 3. Torque, flux ad curre corol f,n Figure 2: Volage - frequecy corol f ~ As ca be see from he space vecor diagram i Fig.c, oly ha compoe i q of he saor curre i = i d +j i q, which is i phase wih he volage u i iduced by he flux likage = d (per defiiio is q =) direcly produces mechaical orque i. The orhogoal compoe i d corols he mageic field ad hus he ampliude of he flux likage d, show i he roor equivale circui i Fig.b. The agle ϕ describes he agular posiio of d i a saor fixed referece frame a,b (Fig.c), hus ϕ() = ω (τ) dτ, () where ω is he agular frequecy of d wih respec o he saor (please oe: uder dyamic operaig codiios he agular frequecy ω of d ad ha of u may be differe). As he saor rasie ime cosa T σ = σl /R (Fig.a) is Accordig o Blaschke's idea he se values of i q ad i d come from a speed coroller ad a flux coroller, respecively, ad are rasformed io phase curre se values i ν * (ν=,2,3) wih saor frequecy by a vecor modulaor (Fig.3). The uderlyig curre corol loop cosiss of simple comparaors. The oupu sigals of he comparaors are sampled i regular clock iervals ad ur he iverer swiches o ad off. To reduce he average swichig frequecy ad hus he hermal sress of he iverer wihou affecig is dyamic behaviour, he machie widigs are shor circuied (free wheelig sae), wheever he magiude of he curre deviaio is below a cer ai limi i all hree phases. 3. Curre corol wih free wheelig ad iverer sress Wih his ype of curre corol, he maximum swichig frequecy of he iverer is f,max =/2T, where T is he clock period of he iverer, e.g. 25µs. Due o he hermal capaciies, however, he hermal sress of he iverer's semicoducors does o direcly deped o he maximum swichig frequecy, bu is more or less proporioal o he average swichig frequecy. curre corol d() flux corol speed corol i* d i* q vecor modulaor i ν & D Q D Q clock T free wheelig iverer u ν i ν A () si ϕ cos ϕ ϕ () Figure 3: Curre, orque (speed) ad flux corol A... asychroous machie *... se values... measured or observed values

3 i ν i R,av / (U T/L σ ) 2, R L,5 W I u ν ν, Figure 7: Flux ideificaio measurig ν (ν=,2,3) As durig seady sae operaio he iverer oupu curres ad he fudameal cycle of he oupu volages are siusoidal, he average may be calculaed over a period of he saor frequecy. Fig.4 shows he average swichig frequecy f,av versus he ampliude of he (fudameal cycle) of he oupu volage û s. This diagram clearly shows, ha he average swichig frequecy i a sysem wihou free wheelig (W I) is much higher ha ha wih free wheelig (W II). The posiive effec of usig free wheelig saes o he swichig frequecy is achieved wihou icreasig he ripple of he oupu curres (Fig.5). However, a very low saor frequecies he hermal ime cosas of he iverer's semicoducors are ologer grea compared o he period of he saor curres ad volages. I his operaig rage he hermal sress of he iverer is beer described by he peak swichig frequecy f,peak, which is he average over a small umber of swichig cycles. As ca be see from Fig.6, a low oupu volages he curre corol wih free wheelig (W II) also leads o lower peak swichig frequecies. Wih AC machies low saor frequecies correspod wih low volages. Thus, he curre corol sysem wih free wheelig esures a lower sress for he iverer semicoducors i he whole operaig rage. A more deailled,5 descripio of his corol scheme is give i [4,5]. 4. Ideifyig he flux vecor The key i field orieed corol lies i ideifyig he ampliude d ad he agle ϕ of he flux. As ca be see i Fig. a, i is possible o compue W II u^ / u^,max,2,4,6,8 Figure 5: Average curre ripple for curre corol wih ou (W I) ad wih (W II) free wheelig ν () = [u ν (τ)-r i ν (τ)] dτ - σl i ν () (2) by measurig ad iegraig he volages ad curres u ν ad i ν i he hree phases ν=,2,3 (Fig.7). From hese hree sigals a quadraure volage pair wih a = = d cosϕ() f,av / f,max,5 f,peak / f,max,4 W I,8 W I,3,6,2 W II,4 W II, u^ / u^,max,2,4,6,8,2 u^ / u^,max,2,4,6,8 Figure 4: Average iverer swichig frequecy for curre corol wihou (W I) ad wih (WII) free wheelig Figure 6: Peak iverer swichig frequecy for curre corol wihou (W I) ad wih (WII) free wheelig

4 a b si ϕ cos ϕ Figure 9: Flux ideificaio by compuig d d d Wih a hird vecor modulaor d d is compued (Fig.9). uch vecor modulaors may be realized combiig aalog ad digial echiques [6] ad are ow available as iegraed circuis [7]. By his phase corol loop, he flux agle ϕ ad hus he saor frequecy ω are self corolled. The resulig saor frequecy ω may be measured by compuig he derivae dϕ/d or by low pass filerig he oupu of he comparaor, as show i Fig. 8. ad b = ( 2-3 )/ 3 = d siϕ() (3) is derived by simple addiios (Fig.8). I a secod vecor modulaor hese sigals are ras formed via 5. peed calculaio Wih asychroous machies speed is q = - a siϕ + b cosϕ ~ (ω - s ω ) (6a) = d [cosϕ() siϕ ()-siϕ() cosϕ ()] = d si(ϕ-ϕ ) (4) ad fed o a comparaor ad iegraor (Fig.8). If q is posiive, ϕ is icreased, if q is egaiv, ϕ is decreased. Due o he closed loop cofiguraio he comparaor ad iegraor will eforce q ad hus ϕ ϕ. (5) where s ω = R R L L R i q d is he so called slip frequecy. For machies above kw oupu power, he slip frequecy ypically is less ha 5% of he saor agular frequecy ω N a omial speed. Therefore he slip may be egleced a higher speeds. To improve he accuracy of he speed regulaio a lower speeds, may be correced accordig o eq. (6). As a low saor frequecies d is kep cosa, he corecio may be liearized as show i Fig.8. (6b) a phase corol loop 2 3 b q 3 * speed calculaio i q cos ϕ si ϕ ~ R L L R R d ω ϕ ϕ Figure 8: Flux ideificaio by measurig ϕ, ω ad

5 () speed corol 6. Problems i ideifyig he flux vecor Ideifyig he flux vecor by iegraig he saor volages accordig o eq.(2) works very well a medium ad higher saor frequecies ad eve wih field weakeig [8]. As log as he volage drops across he saor resisors ad he leakage iducaces are small compared wih he volage iduced by he flux, deviaios of he machie's parameers from he parameers used i he flux calculaio circui have oly a mior effec. However, a high saor frequecies ad exreme field weakeig he volage across he leakage iducace is i he same order of magiude as he volage iduced by d. A very low saor frequecies he volage across he saor resisors becomes eve greaer ha he iduced volage. I hese operaig rages he ifluece of mismaches bewee he acual parameers of he machie ad of hose parameers, used i he flux measurig circui, mus be cosidered. 6. Ifluece of he leakage iducace a very high frequecies If here is a deviaio i σl, he d-compoes used i he corol algorihm are o loger i phase wih he acual roor flux of he machie. Neverheless he orque sill is i ~ d i q - q i d = d i q (7) ad is direcly corolled by i q, because he phase corol loop esures, ha q =. Bu corol acios of he speed corol loop wih i q will ow also cause rasies of he flux d [9]. If he leakage iducace σl was oally egleced i he flux model of eq.(2), he d-axis would be i phase wih he saor flux isead of he roor flux. To avoid ay ifluece of flux rasies o he machie's orque ad he speed corol loop, he simple decouplig measure show i Fig. may be used. d() Figure : Decouplig he speed corol loop from flux rasiios i d 6.2 Ifluece of he saor resisors a very low frequecies The flux iegraio of eq.(2) lies wihi he closed loops for d ad ϕ, so ha drif effecs, caused by operaioal amplifier offses ec., are ucriical. However a very low speeds, he volage drop across R becomes domiaig ad ay mismach bewee he real value ad he value used i he flux calculaio, which will especially be due o hermal effecs, leads o phase ad ampliude deviaios i he measured flux values. A firs measure o reduce his effec is o corol he low frequecy gai of he iegraors by a frequecy depede feedback k(ω ). Fig.a shows a possible realizaio. The iegraig capacior C is bypassed by resisor R p, whe swich is closed, hus leadig o a low pass filer isead of a iegraor. To achieve a smooh rasiio o higher saor frequecies, where R P is o eeded, swich is modulaed wih a duy cycle, which is depede o he saor frequecy. I combiaio wih he iverer corol scheme described i secio 3, here is o eed of addiioally producig such a sigal. The sigal 'free wheelig' of he iverer's curre corol logic ca be direcly used, because 'free wheelig' will be very freque a low saor volages, i.e. low saor frequecies, ad very seldom a high volages, i.e. high saor fre quecies. A secod measure is o use he flux model show i Fig.b isead of ha i Fig.7. This model does oly give he ampliude d, o he agular posiio ϕ (Fig.b). I has a limied dyamic accuracy ad should o be used a higher saor frequecies, especially i he field weakeig rage, because i uses he ime cosa T R =L R /R R, which is also exremely emperaure depede. Bu i is very well suied as a observer for backig he values, derived by he iegraors, a low saor frequecies. To simplify he corol circui, he se value for i d isead of he acual value is fed io he back up model. Agai a smooh rasiio o higher frequecies is achieved by makig he correcio facor v depede o ω. A exremely low saor frequecies, whe ω is much smaller ha he omial slip frequecy (s ω ) N, he phase corol loop for ϕ may o loger esure siusoidal saor curres. For ha operaig rage, he corol srucure is revised (Fig.2). q is corolled o zero by i q * wih a secod flux coroller ad he speed corol acs direcly o he ipu of he comparaor of he phase corol loop. A more advaced soluio for his operaig rage, which also avoids a speed sesor, was preseed i [], bu eeds cosiderably more sigal processig power, especially wih iverers wih high swichig frequecy. The same applies o mehods ryig o deec roor speed uilizig slo harmoics [].

6 ω free wheelig a) k( ω ) R P R C + ω b) * i d L T R do v( ω ) see Fig.2 * flux corol * i d d from Fig.8 Figure : Improvig he flux ideificaio a low saor frequecies a) saor frequecy depede 'iegraio' circui b) flux observer based o saor curre isead of saor volage 7. Coclusio ad resuls By usig he ampliude ad he agular posiio of he machie's flux a self corolled operaio of he asychroous machie is achieved, which does o eed a speed or posiio sesor. The ifluece of he machie's elecrical parameers o he corol behaviour a medium ad high saor frequecies is small. To improve he behaviour a low saor frequecies, he flux ideificaio circui, which iegraes he saor volages, is modified o make is low frequecy gai depede o he saor frequecy. Addiioally he flux coroller is backed by a secod flux model, which is based o he saor curres. Of course oe of hese measures esures seady sae operaio of he drive wih full orque a saor frequecies ear zero, bu i allows acceleraig or deacceleraig a drive from ad o sadsill (Fig.3) ad also guaraees very good performace, whe saor frequecy crosses hrough zero durig speed reversal (Fig.4). modified phase corol loop of Fig.7 q() flux corol speed corol 2 i* q vecor modulaor ad curre corol of Fig. 2 q ϕ 2 () Figure 2: Revised corol srucure for exremely low saor frequecies swich posiio : ormal srucure swich posiio 2: revised srucure

7 N d N d q q 4A id i q 3A i q i d -4 /mi 3 /mi 2ms -4 /mi 2ms Figure 3: peed profile icludig field weakeig ad sadsill Figure 4: peed reversal a low speeds A. Appedix Fig.3 ad 4 refer o a AC machie wih he followig omial daa: saor volage saor curre saor frequecy U N = 23 V I N = 22 A f N = 5 Hz speed N = 445 mi - orque in = 73 Nm mome of ieria J N =,5 kg m 2 saor resisace R =,35 Ω leakage iducace roor resisace roor iducace σl = 2 mh R R =,46 Ω L R = 58 mh Refereces [] Boehriger, A.; ue, G.; Ruppma, C.; Würsli, R.: Ewicklug eies drehzahlgeseuere Asychromaschieariebs für Werkzeugmaschie. w Zeischrif für idusrielle Ferigug,(979) H.8, p [2] Leohard, W.: Corol of elecrical drives. priger Verlag, 985 [3] Depebrock,.: Direke elbsregelug (DR) für hochdyamische Drehfeldariebe mi romricherspeisug. ezarchiv, (985) H.7. p.2-28 [4] Roh-ielow,J.; Zimmerma,W.; Boehriger, A.; chwarz,b.: Zwei zeidiskree euerverfahre für Pulsumricher im Vergleich. ezarchiv (989) H.2, p [5] Roh-ielow, J.: Beiräge zu Pulsumrichersyseme großer Leisug mi ahezu siusförmige Ausgagssröme hoher Frequez. Disseraio Uiversiä ugar (99)

8 [6] Deusche Paeoffelegugsschrife DE , 983; DE , 984; DE , 99; DE 44 64, 994 [7] Fle, F.P.: Vecor corol usig a sigle vecor roaio semicoducor for iducio ad permae mage moors. Techical publicaio, Aalog Devices, 992 [8] Joee, R.; chierlig, H.: Corol of he iducio machie i he field weakeig rage. IFAC Corol i Power Elecroics ad Elecrical Drives, coferece paper, Lausae, wizerlad, 983, p [9] Bader, Uwe: Hochdyamische Drehmomeregelug eier Asychromaschie im säderflußbezogee Koordiaesysem. ezarchiv (989) H., p.-6 [] Depebrock,.; aud, Volker: Deermiaio of he saor flux space vecor of sauraed AC machies. ezarchiv (99) H., p [] Haemmerli, B..: Roor speed deecor for iducio machies uilizig slo harmoics. ACEI Coferece, Turie, p