Modelling and simulation of a signal injection self-sensored drive

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1 Modelling and imulation o a ignal injection el-enoed dive Alen Poljugan Faculty o Electical Engineeing and Computing, Depatment o electic machine, dive and automation, Univeity o Zageb Unka, Zageb, Coatia, ax. 8697, tel , [email protected] Abtact- Thi pape peent a new induction moto model peciically developed o imulating el-enoed vaiable peed dive which employ ignal injection baed enole contol technique. The model i to be ued to invetigate impoved ilteing algoithm and cloed loop peed contolle. The model employ a imple tate pace equivalent cicuit baed model o the induction machine, which i enhanced to include oto lotting and main lux atuation eect. The impovement i obtained by including a vaiation o machine inductance vaiation with the oto poition and the lux poition. The new model i veiied againt expeimental eult. Keywod- Electical machine, Induction moto, Adjutable peed dive, Sel-ening contol, Simulation I. INTODUCTION In ignal injection el-enoed dive, poition and peed etimation i obtained by tacking the machine epone to high equency (HF) voltage upeimpoed onto the nomal undamental voltage. A poition dependent ignal caued by the lotting o the oto can be extacted om the HF cuent waveom. Howeve atuation eect in the machine and non-linea invete eect uch a dead time can add igniicant ditotion to the etimated poition ignal [,,] which can igniicantly impai the epone o the dive. To enhance etimation accuacy, dieent ditubance compenation method can be ued, uch a hamonic and ideband ilte [], compenation table [-], adaptive ditubance identiie [] etc. The main dawback o thee technique i thei inluence on the dive contol peomance, mainly obeved a a eduction o the contol loop bandwidth. In the commiioning poce o el-enoed dive, the contolle paamete need to be tuned, and thi i uually achieved by tail and eo. In thi wok the aim i to ceate a machine model which can be ued to invetigate the inluence o the peed etimation algoithm (including ilte etc) on the oveall dive contol peomance with the ultimate objective being to achieve a high bandwidth peed contolle. It hould be noted that the aim i not to ceate a phyically accuate machine model. Thi can be achieved uing inite element (FE) modelling package, but thei complexity when added to eal time contol algoithm mean that they equie vey long imulation time [9]. The impliied equivalent cicuit model i computationally at but doe not contain any phyical detail o the machine. The Dynamic Meh eluctance Model (DMM) povide a cetain compomie by combining detailed machine analyi and elatively at computational time [7,8]. It i howeve, till too low to be ued o HF injection ytem. [9,,,] combine a FE analyi with a phae vaiable model. A machine inductance vaiation with oto poition i obtained om FE analyi and i then incopoated a look up table. [9,] epot a at computation time and accuate modelling o the machine atuation, oto lotting eect and cogging toque. A pace-vecto tate model o an induction machine including the oto lotting eect i decibed in []. Thi model conide a vaiation o the tato and oto leakage inductance due to oto and tato lotting and develop a pace-vecto tate omulation ueul o application to obeve. The oto lotting hamonic (SH) equency i accuately modelled but thee i an amplitude mimatch when compaed to expeiment, and atuation eect ae not included. In the winding unction appoach (WFA) a machine model i deived by mean o winding unction whee no ymmety i aumed [,]. All machine inductance ae calculated diectly om the geomety and the winding layout o the machine. The WFA how good modelling accuacy. It computational time i ate than FEM ( hou v hou []) howeve it i till compaatively low when complex ealtime contol i equied. The induction machine model deived hee i an extenion o the imple equivalent cicuit model, modiied to include oto lotting and main lux atuation eect. The impovement i obtained by including a vaiation o machine inductance with the oto poition and the lux poition. The pape i tuctued uch that ection II intoduce the equation o the machine model while ection III decibe the key point o the modelling and implementation. In the ection IV the tuning o the model paamete i peented along with a compaion o the imulation with expeimental eult. II. THE INDUCTION MACHINE MODE The machine model popoed in thi pape i deived om a geneal phae model o the induction machine i.e. in the a-b-c ame o eeence [6]. In the ollowing ection, the moto phae voltage and cuent will be eeed to a winding vaiable. Fom the equivalent cicuit a geneal om o the equation in the a-b-c ame can be witten in matix om a ollow:

2 U U abc abc = = i i i i abc abc + abc abc d dt + d dt ' T Ψ Ψ = i i whee ubcipt denote tato and ubcipt oto vaiable. The eitance matice and a well a the inductance matice, and in () can be witten a ollow: a = l + = ba ca b m l + = ba ca m, c l l ab + cb ab cb m + coθ ' = m co θ π co θ + π m = l l a ac bc ac bc whee and epeent the tato and oto eitance epectively, l and l ae the tato and oto leakage inductance, m and m ae the magnetizing inductance, ab,, cb ae the tato mutual inductance and ab,, cb ae the oto mutual inductance. The oto electical angle i denoted by Θ. The model i implemented in the Matlab/Simulink envionment. In thi baic model the oto lotting eect i not epeented and theeoe it cannot be ued in the imulation o a ignal injection dive. In the next ection thi model will be extended to incopoate oto lotting and atuation eect. III. MODEING OF THE MACHINE NONINEA EFFECTS A. oto lotting and atuation eect incopoated in the tato leakage inductance In an induction moto, a non-contant ai gap pemeance eult om the peence o the tato and/o oto lot, aymmety in the tato and/o oto, eccenticity o the oto and/o tato and due to magnetic atuation. A geneal expeion decibing the total aigap pemeance wave i quite complex and unneceay o thi wok, a only peed-elated tem ae equied. Fo thi eaon, it can be aumed that the aigap o the machine i bounded by a mooth tato and lotted oto, and the vaiation o the aigap pemeance i ceated by the oto lot opening only. The oto lot will + ' ' m + abc abc b m, co θ + π coθ co θ π abc abc, c co θ π co θ + π coθ () () poduce aigap pemeance wave with a patial ditibution dependent on the numbe ued []. A typical pemeance wave deciption will conit o a contant component (dc tem) and a upeimpoed ipple (ac tem). The undamental component o the oto lot pemeance inteact with the magnetizing component o the aigap MMF and thi modulation poce geneate two hamonic component in the aigap lux denity. The SH ae impeed on the tato cuent a a conequence o the modulation o the machine undamental magnetic ield and the aigap pemeance wave poduced by the oto lot. The magnitude o thee component vaie little with applied load, except in machine with the cloed oto lot. The equencie o thee hamonic component will be given by [,6]: SH N =υ ± k pp whee h i the oto lot hamonic equency, υ i the SH ode, N i the total numbe o oto lot, pp i the numbe o pole pai, i the mechanical equency o the oto otation and e i the upply equency. A can be een om equation (), the SH equencie ae peed dependent and can be ued to tack the oto peed [6]. In the ollowing dicuion thee two component will be denoted a +SH and -SH, epectively. To model the oto lotting eect on the phae cuent, only the tato leakage inductance in the equivalent induction moto model i conideed to change in a peiodic manne with oto poition. Fom the theoetical point o view, the oto lotting eect will alo aect othe inductance in the machine [9,,,] but hee, o implicity only the contibution o the tato leakage inductance wa conideed. The poile o the tato leakage inductance, peented in Fig., i choen ollowing [9, ] and alo by uing a DMM imulation. The peented poile can be decibed with the ollowing mathematical expeion: o l_dc l/l_dc () N N in[ Θ + φ ] in[( Θ pp Θ ) + φ ] pp pp () la = l _ DC Fig.. Stato leakage inductance a a unction o the oto poition. (top) and the equency pectum o the inductance poile (bottom)

3 ab/m o m/ Fig.. Stato mutual inductance a a unction o the tato lux angle (top) and the equency pectum o the inductance poile (bottom) whee la i the modiied total tato leakage inductance o the phae a, l_dc i the dc component o the leakage inductance, Θ i the tato lux poition, k,k and Φ,Φ ae the amplitude and phae diplacement o the ac inductance component. The new poile conit o a dc component upeimpoed with two ac component, a hown in Fig.. The component 8Θ (N /pp=6/) epeent an aigap pemeance wave caued by a lotted oto, while the component 8Θ -Θ epeent a atuation ideband modulated by the SH []. B. Satuation eect incopoated in the tato mutual inductance Peviou tudie [7,8] how that atuation o the tato and oto teeth i impotant a a much highe lux denity exit in thee location. Since the lux denity vecto i otating, the atuation egion will otate too, modiying the tooth pemeability in a cyclic manne eulting in a vaiation o the tato winding inductance. Fo implicity, it will be aumed that only the tato mutual inductance will be modulated by the cyclic vaiation o tooth atuation. The autho in [7] popoe a vaiation decibed with a imple mathematical equation. Although they peent good eult, thee i only one tuning paamete available to adjut the magnitude o the atuation hamonic. Fo the imulation o a ignal injection dive, a lage degee o ine tuning o the atuation hamonic component i deiable. Following analyi o the poile obtained om FEM analyi [9,,,], the ollowing expeion wa choen to incopoate the atuation eect in the tato mutual inductance: ab m = ba = [ in( pp Θ + ρ) in( pp Θ in( pp Θ + ρ ) in( pp Θ + ρ ) + 6 in( pp Θ + ρ )] 8 + ρ ) + whee k, k, k 6, k 8 and k ae amplitude and ρ,.., ρ ae phae diplacement o the pemeance wave intoduced by tooth atuation. The mutual inductance poile ac = ca and bc = cb ae a imila hape. Fo a ou-pole machine the inductance poile conit o component at,, 6, 8 and Θ a hown in Fig.. The paamete o the tato leakage () and mutual inductance poile ae adjuted accoding to expeimental meauement, a peented in the next ection. IV. SIMUATION ESUTS AND EXPEIMENTA VEIFICATION A ou-pole, kw delta connected machine with 6 oto lot i modelled. The machine ha non-kewed emi-cloed lot, and i pat o an expeimental ytem. The baic model i extended with the tato leakage and mutual inductance vaiation decibed in ection III. The inductance vaiation ae incopoated uing () and (). The imulation and expeimental ytem alo incopoated an indiect oto ield oientated contol (IFO) a well a the αβ ignal injection poition etimation cheme decibed in [-]. A. Model paamete tuning The paamete o the oto lotting and atuation model ae adjuted a ollow. Fit the paamete o the oto lotting model wee tuned. The model i it imulated without the ignal injection algoithm. The paamete k o the oto lotting model i adjuted in uch a way a to match the level o the +SH in the imulation to the one meaued om the expeimental line cuent. The -SH i peent only in the phae cuent. When opeation with ignal injection i conideed, expeimental invetigation howed that the epone o the machine model to the HF voltage doe not match the expeimental meauement. Fo the ame injection voltage ignal (778Hz and 7V peak ), it can be noticed that the cuent epone at the HF equency i appoximately ou time malle than the expeiment (.A compaed to.9a). Nevethele the elative amplitude o the SH component modulated by injected ignal i the ame (appoximately.%). The elative amplitude i deined a the pecentage o the excitation cuent component at the injection equency. Although the poblem with the mall abolute magnitude may lead to ignal poceing iue, the initial eult uing the poition etimation algoithm indicated that the abolute amplitude mimatch wa not impotant when conideing the ultimate application o the model. Once the amplitude o the lotting hamonic ha been adjuted, the atuation model can be tuned to match undamental equency elated atuation component (i.e. th, 7 th, etc.). Howeve, when thi method i ued, the elative amplitude o the atuation component aound the injection equency will not match thoe obeved in the expeimental ytem. Thi again i due to the act that the numbe o paamete adjuted in the model i kept low o implicity, and theeoe the model doe not neceaily epeent the tue behaviou o the eal machine. Non-linea equency dependent vaiation neglected hee include kin eect and eddy cuent to add thee would ove-complicate the model o the pupoe intended. Theeoe a thi model i ued peciically o invetigation o ignal injection technique, the atuation model hould be tuned o the epone at the injection equency.

4 i a i ß i bandpa extaction o the poition dependant cuent epone to injection voltage Ditubance compenation and oto poition extaction Fig.. Poition etimation cheme in the ignal injection dive i pa i pß theta_ imulated undamental atuation component (the th and 7 th ) how a tendency to inceae with load wheea the oppoite occu in the expeimental ytem a een in Table II. Once again thi can be attibuted to the act that coe atuation ha not been modelled. In the poition etimation poce the meaued line cuent i poceed uing a demodulation cheme decibed in [-] which ultimately give the cuent poition ignal i pα and i pβ, a hown Fig.. Fom the cuent poition ignal the oto lot poition i obtained uing an actan unction o a phae locked loop. In the ideal cae the cuent poition ignal pectum will conit only o the oto lotting component 8. Howeve, atuation component modulated by the injected ignal will appea in the cuent poition ignal and they epeent a igniicant ditubance to the etimation poce []. A the main aim o thi eeach i to invetigate ditubance elimination technique and thei inluence on the dive contol peomance, the egion aound the injection ignal equency i vey impotant and the atuation model i adjuted accoding to the pectum o the cuent poition ignal. By adjuting paamete k, k, k 6, k 8 k and k all the atuation hamonic component in the cuent poition ignal can be matched to expeimental meauement. An adjutment o the atuation hamonic component in the cuent poition ignal will alo caue a change in the hamonic pectum aound undamental lotting hamonic +SH. B. Opeation aound Fundamental Fequency In the Fig.. and Fig.. eult at pm and no load ae peented. Fom the eult it can be een that paamete o the model have been adjuted uch that the magnitude o the imulated +SH component matche the expeimental eult, alo hown in Table I. Thee i light mimatch in the amplitude o the +/-SH atuation ideband becaue the atuation model ha been tuned to match the hamonic content aound the injection equency, mentioned ealie. The bigget mimatch i in the magnitude o the th and 7 th hamonic component. With egad to the machine model application tooth atuation i the mot impotant atuation phenomena and coe atuation in thi cae i neglected. By intoducing a coe atuation component to the model thi mimatch can be educed, at the expene o a moe complicated model [7,9]. In Fig. 6. and Fig. 7. eult at pm and % o the nominal load ae peented. Fom the expeimental eult it can be een that loading will caue a elative amplitude deceae in all o the hamonic component conideed, a hown in Table II. Although the abolute amplitude o each hamonic i inceaing, thei amplitude elative to the undamental cuent component i deceaing. In the imulation, the magnitude o the +SH component and +HS atuation ideband how a tendency to deceae with load, but by a malle amount compaed to expeiment. Thi pobably eult om the impliication made in the modelling o the oto lotting and atuation eect. The C. Opeation at ignal injection equency The imulation model wa alo invetigated o opeation with HF voltage injection. In the dive a mall magnitude (7V peak ) voltage ignal wa injected in the αβ ame at the equency o 778 Hz. The equency pectum o the imulated cuent poition ignal wa compaed with the expeimental eult, noting that the hamonic magnitude i expeed a a pecentage o the cuent component at the injection equency. In Fig. 8. and Fig. 9. eult at pm and no load ae peented. The SH component at the 8 and thee main atuation component e, e and 8 e match thoe obtained om the expeiment (Table III). Fo the component at e and 8 - e an amplitude mimatch can be noticed. By adjuting the paamete (k and k ) the amplitude o thee hamonic can be matched to expeimental eult but thi will then ceate ome additional hamonic component aound the undamental lotting hamonic +SH, namely the -SH- e, +SH- e and mot impotantly the +/-SH- e component. A can be noticed om Table I thi component i aleady lage than that een in the expeimental ytem. I o example the paamete k i tuned to match the 8 - e component, the -SH- e component will inceae by up to % (moe then hal o the +SH), a een in Fig.. A imila eect i peent when the e component i tuned. Futhe invetigation will hopeully pove that opeation o the cuent contolle i not igniicantly aected by thi phenomenon, and the paamete can then be tuned to match the e and 8 - e component. In Fig.. and Fig.. eult at pm and % load ae peented. Fom the expeimental eult, the atuation component e, e and 8 e deceae igniicantly with load (Table IV). In the imulation eult thi deceae can alo be een but it i vey mall. It i obviou that in the high equency egion and unde the load condition the atuation model i not modelling accuately the atuation phenomenon. But om the FEM analyi o [,,,] it i clea that the magnitude o the inductance component a well a it phae diplacement will vay with the load. Futhe invetigation o the inductance vaiation poile can be ued o additional ine tuning, eithe though a FE imulation o expeimental teting, i equied. Alo it can be een that unde load two new atuation component appea at and 6 e, a hown in Fig. 9. In the imulation the magnitude o thee component ae negligible. A modiied atuation model wa invetigated which included the new hamonic component in the mutual inductance poile. The eult howed that the and 6 e component can be modelled but again the tuning o thee component caued additional component aound undamental SH. A mentioned beoe, thi only meit uthe invetigation i a moe detailed model i equied.

5 e Hz ;.9 A +SH e Hz ;.9 A +SH e 7e -SH - e +SH + e e 7e -SH - e +SH + e Fig.. Expeimental eult - equency pectum o the line cuent at pm and no load e Hz ;.9 A e 7e -SH - e +SH +SH + e Fig.. Simulation eult - equency pectum o the line cuent at pm and no load TABE I HAMONIC SPECTUM OF THE INE CUENT FO NO OAD CASE (%) th 7 th +SH -SH-e +SH+e expeiment imulation Fig. 6. Expeimental eult - equency pectum o the line cuent at pm and % o nominal load e Hz ;.9 A e 7e -SH - e +SH +SH + e Fig. 7. Simulation eult - equency pectum o the line cuent at pm and % o nominal load TABE II HAMONIC SPECTUM OF THE INE CUENT FO % OAD (%) th 7 th +SH -SH-e +SH+e expeiment imulation e e 8 e e 8 - e Fig. 8. Expeimental eult - equency pectum o the cuent poition ignal at pm, no load and h injection e e 8 e e 8 - e Fig. 9. Simulation eult - equency pectum o the cuent poition ignal at pm, no load and h injection TABE III HAMONIC SPECTUM OF THE INE CUENT FO NO OAD CASE AND HF INJECTION (%) 8 e e 8 e e 8 - e 8 expeiment imulation e e 8 e e e 6 e 8 - e Fig.. Expeimental eult - equency pectum o the cuent poition ignal at pm, % o nominal load and h injection e e 8 e e 8 - e Fig.. Simulation eult - equency pectum o the cuent poition ignal at pm, % o nominal load and h injection TABE IV HAMONIC SPECTUM OF THE INE CUENT FO NO OAD CASE AND HF INJECTION (%) 8 8 e e 8 e e 8 - e 8 expeiment imulation

6 Expeimental eult hown in Fig. 7 and 9, alo include hamonic component at the 7, 6 and 67 Hz. The equency o thee component i not dependent on the upply equency but the magnitude do inceae with load. The autho upect that thee component oiginate om aymmety in the expeimental ytem. The eult om nomal opeation alo howed mall even hamonic component in the line cuent which alo ugget an aymmety poblem, Fig. and. A mentioned beoe, the main aim o thi poject wa to ind an induction machine model which gave a good compomie between the imulation time and modelling accuacy. The imulation model peented hee i at 7 computation time o a imulation o a DO machine. Fo a DMM imulation (peviouly epoted to be at ), a computation time o 96 i needed. V. CONCUSION Thi pape ha peented a new induction moto model o ue with developing ignal injection baed el-ening induction moto contol ytem. A imple equivalent cicuit model i impoved with the oto lotting and atuation eect. The oto lotting eect i incopoated a a oto poition dependant tato leakage inductance vaiation. The magnitude o the oto lotting hamonic can be adjuted to match thoe meaued expeimentally by mean o tuning two paamete. The atuation eect i incopoated a a lux poition dependant component in the tato mutual inductance. The inductance poile conit o ive hamonic component and the magnitude and phae diplacement o each component can be adjuted. Conideing the machine model application, the atuation model wa tuned to match the expeimental eult aound the injection equency. The eult compaion how that a elative magnitude o the oto lotting hamonic match thoe obtained om the expeiment o all loading condition. The atuation model o the no load condition give good eult with a light magnitude mimatch o two hamonic component ( e and 8 - e ). Thee component can be additionally tuned to match the expeiment but thi eult in detuning o ome hamonic component aound the undamental SH. A uthe invetigation will how how thee component aect the cuent contolle. The popoed poile o the machine inductance ae not conideed to be load dependant. Theeoe unde load condition the atuation model how poo eult. I a moe detailed model i equied, vaiation o the inductance poile with a load can be included (at exta computational cot). EFEENCES [] J.I. Ha, S.K. Sul, Senole Field-Oientation Contol o an Induction Machine by High-Fequency Signal Injection, IEEE Tanaction on Induty Application, Vol., No., Jan/Feb 999 [] N. Teke, G. M. Ahe, M. Sumne, and K. J. Badley, Suppeion o Satuation Saliency Eect o the Senole Poition Contol o Induction Moto Dive unde oaded Condition, IEEE Tanaction on Indutial Electonic, vol. 7, no., pp., Oct.. [] N. Teke, G.M. Ahe, K.J. Badley and M. Sumne, Analyi and Suppeion o Invete Clamping Saliency in Senole Poition Contolled Induction Machine Dive, IEEE-IAS, Poc. On CD OM [] Q. Gao, G. Ahe and M. 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