Acquisition of Rotor Anisotropy Signals in Sensorless Position Control Systems

Transcription

1 IEEE Transacions on Indusry Applicaions, Vol. 4, 24 aaa Acquisiion of Roor Anisoropy Signals in Sensorless Posiion Conrol Sysems Joachim Holz, Fellow, IEEE, and Hangwen Pan Elecrical Machines and Drives Group Universiy of Wupperal 4297 Wupperal Germany Panasonic Elecronic Componens 2337 Lüneburg Germany Absrac Sensorless posiion conrol of inducion moors relies on he deecion of roor anisoropies. The repeiive ransien exciaion hrough he swiched volages of he PWM inverer provides he es signals o idenify he spaial orienaion of he anisoropies ha indicae he roor posiion. The posiion informaion is conained in he inverer oupu volages. These are measured agains he neural poin of he sar conneced saor winding. The signals are heavily corruped by disurbances. These originae from he propagaion of ravelling waves on a long moor cable and from he influence of high-frequency common mode currens. The paper describes how a clean posiion signal is exraced in he presence of such noise. Keywords: Inducion moor, sensorless posiion conrol, roor anisoropy, common mode currens, long moor cable, conrol a zero saor frequency I. INTRODUCTION Sensorless vecor conrol of inducion moors a zero saor frequency consiues a problem of specific naure. As he elecromagneic field does no roae in his operaing poin, raher mainaining a fixed posiion in space, he roor induced volages in he saor windings assume zero values. This makes i impossible o esimae he spaial orienaion of he roor field based on measured elecrical quaniies volages and currens of he saor windings. While he fundamenal field model of he inducion moor hus fails o provide full conrollabiliy a zero saor frequency, soluions o his problem can be found by making use of he anisoropic properies of he machine roor. Squirrel cage roors generally exhibi wo kinds of anisoropies []. A firs anisoropic propery is caused by magneic sauraion. The spaial variaions of he main field change he magneic permeabiliy of he roor and he saor iron. The inducance values of he hree saor phase windings herefore vary as a funcion of he displacemen angle beween he field axis and he respecive phase axis. This effec permis esimaing he spaial orienaion of he main field. The field angle hus obained is he key quaniy for implemening a field oriened conrol sysem. A second anisoropic propery of he roor is caused by he exisence of N r discree conducors along he circumference of he cage roor. Such srucure creaes wo effecs: Firs, he inducors, having low magneic permeabiliy while being embedded in he roor iron of high magneic permeabiliy, creae a periodic modulaion along he roor circumference of he spaial magneic characerisics. Secondly, he discree conducors hemselves, being shor circuied, exer an influence on he leakage inducances of he respecive phase windings. The leakage inducance of one phase winding varies coninuously, assuming a minimum and a maximum value as he roor angle displaces by one roor bar pich. Since he number of roor bars is generally no a muliple of hree, he phase angles of he saor leakage inducance signals are displaced wih respec o each oher by 2p/3. Measuring he oal leakage inducances of he hree saor phases herefore permis deermining he absolue roor posiion wihin one roor bar pich [2]. I is advanageous o measure he spaial orienaions of he roor anisoropies while subjecing he machine o a ransien condiion. This creaes magneic field componens ha propagae in space a velociies differen from he velociy of he fundamenal field, hus making he resul independen of he prevailing load condiion. Transien condiions are esablished by injecing frequency componens oher han he fundamenal frequency, and generally higher, ino he saor windings. The resuling ransiens can be exploied in wo differen ways: One class of mehods evaluaes he response of he machine o an injeced periodic high-frequency signal, generally referred o as carrier signal [3, 4, 5]. An alernaive approach uilizes he response o he sepped saor volage waveforms ha occur a pulsewidh modulaed (PWM) conrol [2]. The laer mehod provides a high spaial resoluion of he acquired angular posiion signal, and exhibis high dynamic bandwidh and a high signal-o-noise raio. I is herefore given preference in he following. The paper discusses firs how he roor bar posiion angle is deermined from he measured volages a he inverer oupu erminals. I shows furher ha ravelling wave phenomena and high-frequency common mode currens end o disurb he measured signals, especially when a longer cable exiss beween he inverer and he drive moor. Mehods o eliminae he disurbances are presened, permiing o exrac clean

2 6 mh l s a 4 l s a aaa 2 l s b l s c 6 l s b mh 4 6 l s c mh 2 4 Ud u ib i b ia l sa u ia N u ic u b l sb i c u a l sc u c p 2p J/p 2 Fig. 2. Equivalen circui of he inducion moor when he inverer swiching sae u is applied Fig.. Variaions of he phase values l sa, l sb, l sc of he oal leakage inducance caused by an anisoropic roor; J/p = 2p corresponds o he angular displacemen of he roor by one pole pair. N r = 44, 2p = 4 and accurae roor anisoropy signals from a PWM conrolled moor drive. These signals define he field angle, or he roor bar posiion angle, even a zero saor frequency or zero speed. Using hese signals for sensorless posiion conrol of an inducion moor is he subjec of a previous publicaion [6]. II. ACQUISITION OF THE ROTOR ANISOTROPY SIGNAL The anisoropic properies of a squirrel cage roor are refleced in variaions of he oal leakage inducances l s of he saor phase windings. Fig. shows how he respecive phase componens change when he cage roor displaces by a spaial angle corresponding o one pole pair, i.e. i displaces over a full elecrical revoluion. Wih he machine operaed a no load and full magneic exciaion, he sauraion anisoropy accouns for he wo respecive minimum and maximum values per elecrical revoluion. Superimposed o his is he effec of he roor cage anisoropy, characerized by N r /p oscillaions per mechanical revoluion, where N r is he number of roor bars and p is he number of pole pairs. I is seen in addiion ha he ampliude of he cage anisoropy componen depends on he sauraion level. Noe ha he wo anisoropies roae a differen angular velociies when he machine is loaded. A signal reflecing he anisoropic characerisics of a squirrel cage roor can be acquired following a ransien exciaion of he machine. Such exciaion is perpeually generaed by he swiching of he inverer. The respecive inverers saes are characerized by he eigh swiching sae vecors shown in he righ of Fig. 3. Le a paricular inverer swiching sae vecor be u (+ ). The noaion u (+ ) indicaes ha he phase poenials u a = U d /2 and u b = u c = U d /2, where U d is he dc link volage. This siuaion is represened by he equivalen circui Fig. 2. The following equaions hold dis u = lσ + u dτ i () where i s and u i are he space vecors of he saor curren and he roor induced volage, respecively, and lσ a l σ = lσb lσc is he leakage inducance ensor, he hree-phase componens of which are he respecive phase values l sa, l sb, l sc of he oal leakage inducance. These inducances would equal each oher wih an isoropic roor, l sa = l sb = l sc. In such case, he zero sequence componen, expressed by he sum of he hree phase volages, amouns o zero, ua + ub + uc =, (3) since he induced volages u ia, u ib and u ic, being almos sinusoidal, do no exhibi a zero sequence componen, and and consequenly (2) uia + uib + uic =, (4) ia + ib + ic = (5) dia dib dic + + =. (6) dτ dτ dτ An anisoropic roor inroduces an unbalance o he phase values of he oal leakage inducances as demonsraed in he simulaed curves Fig.. I is assumed ha he unbalances vary sinusoidally when he anisoropy displaces in space as he roor and is magneic field roae. The ransien volage componens l sa di a /d, l sb di b /d, l sc di c /d of he firs erm in () hen vary as he oal leakage inducance values vary. As a consequence, he derivaive di s /d in () assumes a differen direcion han he driving volage vecor u u i. To illusrae he effec, i is assumed ha he anisoropic, e.g. sauraed roor in Fig. 3 does no move: w s = and hence u i = in (). The leakage inducance ensor l s makes he derivaive vecor di s /d poin in a differen direcion han he exciing volage vecor u. The physical explanaion is ha he low leakage inducance in he d-axis creaes a greaer curren derivaive componen han he high leakage inducance in he q-axis. One way of idenifying he spaial orienaion of he anisoropy is o deermine he ransien componen di r /d = u u i

3 jq J d u 3 jim u 2 u Re 5V () u s 5 u 4 u, u 7 u 5 u 6 di r d 2d 2s Fig. 4. The anisoropy signal u s () a an angular displacemen of he roor by one pole pair, compued a no-load and Hz saor frequency Fig. 3. Anisoropic, sauraed roor causing he ransien curren derivaive di r /d o deviae from he direcion of he exciing swiching sae vecor u of he saor curren derivaive di s /d. Is locus moves on a circle as shown in Fig. 3, having an angular displacemen 2J where J is he spaial displacemen angle of he anisoropy. A simpler mehod consiss in analyzing only he scalar componen di r /d in he direcion of he exciing volage vecor, which is u in his example. This componen induces he ransien volage l sa di a /d in saor phase a. Also he componens l sb di b /d and l sc di c /d of he oher wo phases are affeced by he respecive leakage inducance values l sb and l sc, which are as well anisoropy dependen. These dependences have an effec on he poenial of he neural poin N in Fig. 2, which makes he zero sequence volage assume nonzero values. Equaion (3) convers o ua + ub + uc = u σ (7) which expression defines an anisoropy relaed signal. This signal is acquired by measuring and adding he phase volages u a, u b, and u c. To invesigae he properies of he anisoropy signal u s (7), equaions () hrough (4), (6) and (7) are solved, which yields ( ) u u l l l l l () σa σb + σc 2 σb σc σ = d lσalσb + lσblσc + lσalσc where he erm + uσi, (8) lσalσbuic + lσalσcuib + lσblσcuia uσi = 3 (9) lσalσb + lσblσc + lσalσc represens he conribuion of he roor induced volages u ia, u ib and u ic. These volages are small a low speed and hence can be negleced. The influence of u si a higher speed is discussed in [6]. The superscrip in he noaion u () s in (8) indicaes ha he anisoropy signal refers o a ransien exciaion by he swiching sae vecor u. I is obvious from he inspecion of Fig. 3 ha ransien exciaions by differen swiching sae vecors produce differen values of he anisoropy signals u (2) s, u (3) s,..., however, u (4) s, = u () s, u (5) s, = u (2) s, and u (6) s, = u (3) s. The anisoropy signal u () s is displayed in Fig. 4. The signal exhibis very favorable characerisics: The ampliude of he roor cage relaed anisoropy componen, which is he high-frequency oscillaion, is much more uniform han he corresponding componens of he phase leakage inducances, Fig.. Oher han he phase leakage inducance curves l sa, l sb, and l sc in Fig., he anisoropy signal u () s does no exhibi a dc offse. The formal reason for hese favorable properies is in he nonlinear mapping of he phase leakage inducances on he anisoropy signal expressed by (8). The summing of he phase volages in (7) furher eliminaes all nonsignifican large fundamenal componens, and he dc offse as well. The small changes in he curve l sa herefore ransform o a balanced ac signal having a remarkable ampliude of more han 5 V. This ensures a very high signal-o-noise raio. ~ ~ ~ u al u bl u cl PE l D m D6 2C d a u am 2C d Fig. 5. Drive sysem opology b c u ag u bg l s u cg C g g III. SIGNAL DISTURBANCES 3. Parasiic effecs The discussion in he preceding paragraph are based on idealized condiions; he resuls were obained by simulaions. The condiions in a real inverer drive sysem are quie differen. The common mode volages generaed by he swiching of he inverer produce oscillaing currens of high frequency ha close

4 V 2 ms (a) simulaed waveform V envelope: u al + U d /2 u bl + U d /2 u cl + U d /2 envelope: u cl U d /2 u al U d /2 u bl U d /2 2 ms (a) simulaed waveform V V 2 ms (b) measured waveform 2 ms (b) measured waveform Fig. 6. Common mode volage wih he inverer in one of he zero swiching saes Fig. 7. Common mode volage wih he inverer operaed in he pulsewidh modulaion mode hrough he sray capaciances of he moor windings versus ground, from where hey reener he drive sysem hrough he feeding uiliy ransformer, he line-side diode bridge converer and he dc-link circui [7]. In addiion, ravelling waves propagae along he hreephase cable ha connecs he inverer o he drive moor [8]. They are parially refleced from he moor erminals back o he inverer; anoher porion peneraes ino he saor windings, hus creaing oscillaions of he common mode volages. Wih hese disurbances exising, i is almos impossible o measure he anisoropy signal (7) wih he required accuracy. Seps mus be aken, herefore, o reduce he influence of he parasiic oscillaions. 3.2 Common mode currens An analysis of he ransien processes caused by he common mode volage of he inverer is based on he equivalen circui Fig. 5. The moor is represened here by he leakage inducances of is hree phases. The disribued capaciances of he inverer, he moor cable and he moor windings agains ground are approximaed by he lumped capacior C g. The balanced fundamenal hree-phase volage sysem ha feeds he drive moor represens one componen of he pulsewidh modulaed waveforms. The respecive poenials of he hree inverer erminals referred o he cener poin m of he dc link circui are resriced o eiher +U d /2 or U d /2. The sum of he hree phase poenials is always nonzero, alhough ime-variable. Is common mode componen agains ground is uabc,g = ( uag + ubg + u 3 cg ) () which can be also wrien as uabc,g = ( uam + ubm + ucm)+ u 3 mg () where u am, u bm, and u cm are he volages beween he re- specive moor erminals and he dc link mid poin, and is he volage of he mid poin agains ground. Table shows he values of he common mode volage u abc, g as a funcion of he eigh possible swiching sae vecors, where u = u ( ) and u 7 = u 7 (+ + +). The Table indicaes ha u abc,g does no only depend on he respecive swiching sae vecors, bu also on. The volage iself is variable which will be analyzed in he following Zero swiching sae vecors. The line-side recifier in Fig. 5 operaes predominanly in he disconinuous conducion mode. Owing o he dc link capacior C d, he dc link volage U d is nearly consan and mosly higher han he phase-o-phase volages of he mains. The line currens hen end o decrease o zero afer each shor inerval in which he capacior ges charged. I is assumed firs ha he inverer mainains one of is zero saes u or u 7. Considering he paricular charging inerval in which he diodes D and D6 conduc, he following equaion holds u U l di mg = d + + u u u u d bl = abl + bl = 2 τ 2 2 cl (2) where l is he line inducance per phase and i is he charging curren in he loop D C d D6. Hence, during he charging inerval equals u c /2. The volage of he capacior C g is hen u cl /2 U d /2 when he inverer is in swiching sae u ; i equals u cl /2 +U d /2 when he inverer is in swiching sae u 7. When he diodes reurn o a blocking sae afer he shor charging inerval, he swiching sae vecor u u, u 3, u 5 u 2, u 4, u 6 u 7 u abc, g U d /2 U d /6 + U d /6 + U d /2 Table. Common mode volage u abc, g a differen swiching saes

5 5 A G'/dx i L'dx R'dx C'dx dx 5 A 5 µs 5 µs a) wihou common mode b) wih common filer mode filer Fig. 8. Measured waveforms of he common mode curren following an inverer commuaion inverer ges disconneced from he feeding line, and he volage of he capacior C g remains consan. Also remains consan, keeping he value i had when he diodes disconneced. Fig 6 shows he respecive simulaed and measured waveforms Acive swiching sae vecors. Addiional common mode volages are generaed when he inverer operaes in he pulsewidh modulaed mode. Table shows ha he common mode volage u abc, g, in addiion o, seps up by U d /3 incremens during one PWM half-cycle, e.g. u u u 2 u 7, sepping down by U d /3 decremens in he subsequen half-cycle u 7 u 2 u u. I is assumed firs ha he line-side bridge recifier is in a sae of conducion. The sray capaciances of he sysem, represened by he lumped capacior C g in Fig. 5, are hen periodically charged or discharged, following he changes of he common mode volage a each commuaion of he inverer. The charging loop closes hrough ground, he uiliy ransformer, and he diode bridge. Owing o he inducances of his circui, he charging curren is superimposed by damped high-frequency oscillaions. The siuaion is differen when he diodes of he line-side bridge recifier are in he blocking phase of he disconinuous 2 2 moor cable saor winding Fig. 9. Propagaion of ravelling waves on he moor cable and in he saor windings l f Fig.. Equivalen circui per uni lengh dx of one saor phase winding conducion mode. Considering he same phase inerval of he line volages as in Secion 3.2., i is he diodes D and D6 ha are exposed o he leas blocking volage from he line according o uabl Ud = ud + ud6 (3) where u ab L < U d, and hence u D2, u D2 <. Taking he pulsewidh modulaed operaion of he inverer ino accoun, posiive sep changes of he common mode volage u abc, g occur in he course of one of he swiching paerns ha are lised as a downward sequence in Table. Fig. 5 shows ha such sep changes will forward-bias diode D. Following each sep, a common curren flows hrough D, firs charging is reverse blocking capaciance and hen ransferring D o he conducion mode. Alhough originally nonconducing, he diode D connecs he dc link o he mains wih he effec ha he volage ges limied o u a U d /2. Conversely, negaive sep changes of he common mode volage u abc, g occur in he course of any upward sequence of he swiching paerns as per Table. The resuling common mode curren has hen iniially he opposie polariy, hus forcing D6 ino conducion which limis he volage o u b + U d /2. Similar condiions exis when he phase angle of he line volage advances and he leas blocking volage occurs across oher diode pairs. As a consequence, he waveform of exhibis he characerisic paern shown in Fig. 7. The envelops of he high-frequency oscillaions are formed by he respecive uiliy phase volages. Whenever he common mode volage changes in a sep fashion, he common mode currens, afer an iniial ransien, develop a fas ringing componen as he disribued capaciances of he machine winding, represened by C g, and he sray inducances resonae. The respecive power diodes of he bridge recifier, having once iniiaed conducion, canno reac fas enough when he ringing currens reverse; hey end o mainain heir inernal carrier disribuion and hence remain heir original sae of conducion. An oscillogram of he common mode curren is shown in Fig. 8(a). Is frequency is abou 2 MHz. This high value indicaes ha he leakage inducances of he moor and he mains are no fully effecive; hey are parially bypassed as he seep curren gradiens find a parallel pah hrough he winding ca-

6 6 V u an 3 paciances. 3.3 Travelling wave propagaion The inverer is generally mouned in a conrol cubicle, while he drive moor, forming par he plan, is siuaed in a differen locaion. The connecing hree-phase cable can be of considerable lengh. I can be modelled as a disribued parameer sysem [7]. The cable is excied by he volage seps of he inverer. These propagae along he cable lengh as individual forward ravelling waves per phase, uf = uinv, if = uf Zc (4) where he subscrip f refers o a forward ravelling wave and Z c is he naural impedance of he cable. i f is he curren associaed o he forward wave. The waves ravel a a finie velociy of abou 6 m/ms. Their propagaion is schemaically illusraed in Fig. 9, assuming ha a deenergized moor cable is excied by he swiching sae vecor u (+ ). A ime insan, he forward waves have reached he marked posiions in Fig. 9. Since he saor winding impedance Z s is higher han he naural impedance Z c of he moor cable, he reflecion facor a he moor erminals 2C d m 2C d C p 2 3 µs 2 µs a) wihou filer b) wih an RC filer a he moor erminals Fig.. Measured phase o neural volages a he inverer oupu; moor cable lengh 5 m C p2 inverer u a,b,c N common mode inducor L c long cable g Fig. 2. Filer arrangemens and definiion of he inverer oupu volages ha deermine he anisoropy signal; he long cable is no shielded. 5 5 Zs Zc G s = <. (5) Zs + Zc where Z s is he erminal impedance of he saor winding. The ransien condiions produce ravelling waves in he moor windings as well. Therefore, as he forward waves reach he moor erminals, hey bifurcae o iniiae wo new ravelling waves per phase: Assuming in a firs sep ha Γ s is frequency independen, he moor cable develops he reverse ravelling waves ur = G suf, ir = ur Zc, (6) while a se of forward waves sars peneraing ino he saor windings. The respecive volages and currens a he moor erminals are defined by PE u = u + u, i = u Z (7) fs f r fs fs s The propagaion of forward waves in he moor winding is marked in Fig. 9 as corresponding o ime insan 2. The phase windings of he moor behave as a disribued parameer sysem, similar o ha of an elecric line. There is a difference, hough, since addiional capaciances exis beween he individual urns of a muli-urn winding. The equivalen circui per uni lengh dx, Fig., models he winding capaciances by he addiional elemen G'/dx [8]. An inciden wave appearing a he moor erminals is herefore spli ino wo porions: An undercriical and an overcriical componen [9]. I is apparen from Fig. ha frequency componens above a criical frequency are predominanly shor-circuied by he wo disribued capaciances G'/dx and C'dx. The do no reach furher ino he winding han he firs few urns. Hence he reflecion facor for hese componens is Γ so, which indicaes ha higher frequency waves are fully refleced a he moor erminals. They ravel back o he inverer. Only he lower frequency componens penerae ino he saor windings as described by (7). The inermediae frequency componens in he neighborhood of he criical frequency ω c =, (8) ls L' G' where l s is he conducor lengh of one saor moor phase winding, are parially refleced o reurn on he moor cable, and parially ransmied o N penerae ino he saor windings. The reflecion facor Γ R s in (6) is herefore frequency dependen. This influences upon he shape of he rav- f C f elling waves ha propagae in he saor windings. Their sep fron convers ino a wedge-like C g shape as shown in he righ of Fig. 9. The fron lengh [9] of he modified wave is lf =πls G'. (9) L'

7 5 V u s (a) 2 V 5 V u ab.5 s 6 u s 5 V 5 V (b) 5 ms Fig. 3. (a) sampled anisoropy signal u s, showing a 3-Hz disorion. (b) enlarged secion of he same signal; he low frequency componen is he anisoropy signal; moor cable lengh 5 m The reverse wave on he moor cable is again refleced a he inverer erminals, where G inv <. The reflecions beween he moor and inverer erminals coninue while he wave energy reduces, being parly absorbed in he saor windings and parial consumed by losses. Similar processes ake place in he saor windings. The modified forward waves are refleced a he sarpoin N, which is shor-circuied and hence G N =. The reverse waves are hen again refleced a he moor erminals, which process coninues unil a seady-sae is reached. Since he disribued capaciances and inducances of he moor windings are much higher han hose of he moor cable, he propagaion velociy and hence he frequency of he periodic reflecions in he moor is lower han ha of he cable. Two differen frequencies are herefore superimposed on he saor phase volage, Fig. (a). They are caused by ravelling delay effecs. The moor cable generaes he high-frequency oscillaions, while he saor windings produce he low frequency componen. IV. MEASUREMENT OF ROTOR ANISOTROPIES To avoid running a separae measuremen cable from he moor erminals o he inverer conrol uni, he anisoropy samples are sampled a he inverer erminals iself, Fig. 2. The anisoropy signal (7) hen becomes uσ = uan + ubn + ucn. (2) A connecion o he moor sarpoin N is sill needed for signal acquisiion. The approach requires he following addiional measures: Reducing he influence of high-frequency common mode currens. Reducing he influence of signal disorions caused by signal reflecions on he moor cable. Inroducing a delay inerval beween a ransien exciaion by he inverer and he signal sampling insan o accoun for he signal delay on he moor cable and in he saor windings. 2 µs 2 µs a) wihou filer b) wih filer Fig. 4. Effec of a hree-phase RC filer a he moor erminals on he phase-o-phase volage a he inverer erminals 4. Common mode currens The anisoropy signal is measured following he ransien exciaions of he sysem during changes of he inverer swiching sae. According o Table, such changes are accompanied by he changes uabc,g =± Ud + u 3 mg (2) of he common mode volage u abc,g, where ±U d /3 are single sep funcions while is he high-frequency volage shown in Fig. 7. While he diode bridge recifier in Fig. 5 operaes in he disconinuous conducion mode, he changes of he common mode volage (2) force one paricular recifier diode during each 6 -degree inerval of he line volages ino conducion. The effec was explained in Secion I leads o a 3- Hz ampliude modulaion of he volage componen in (2) as seen in Fig. 7. The resuling common mode curren superimposes a clear 3-Hz componen on he anisoropy signal as shown in Fig. 3. The circui arrangemen in Fig. 2 eliminaes his disurbance. The wo capaciors C p and C p2 deviae he oscillaing common mode curren from he line-side recifier. The curren is hen impeded from forcing he pair of near-conducing recifier diodes (D and D6 in Secion 3.2.2) ino alernaing conducion. The volage remains almos consan as a consequence. In addiion, a se of coupled inducors L c beween he inverer oupu and he moor cable consiues a high impedance for he common mode currens. These reduce in ampliude as shown in Fig. 8(b). 4.2 Reflecions of ravelling waves A beer impedance maching a he moor side of he cable can be achieved by a hree-phase RC filer as shown in Fig. 2. Such filer adjuss he reflecion facor Γ s in (5) close o zero for higher frequencies if R f = Z is chosen. The value of he filer capacior C f, on he one hand, deermines he frequency range for which impedance maching is achieved, and he filer losses and he oher. A good compromise is C f = (3

8 u σ 2 V u σ 2 V u σ ' 2 V 2 V T d occur a a very shor duraion when he saor frequency, and hence he fundamenal saor volages, are low. The on-imes of he acive vecors may be even lower han he minimum commuaion inerval of he inverer, which is in he microsecond order. I appears impossible, even wih he improvemen achieved by addiional filers in he power circui, o ake accurae measuremen samples afer such shor ime delay. The soluion consiss in aking he samples while he lowfrequency oscillaions sill persis, bu having assumed a defined form wih he high-frequency oscillaions already disappeared. The approach is illusraed wih reference o he oscillograms of he anisoropy signal u s, Fig. 5. The RC filer a he moor erminals reduces much of he high-frequency disurbance caused by he moor cable. The signal is sampled a s + T d, a which ime insan he high-frequency oscillaions have died ou. T d = 3.4 ms was chosen in our seup, which means ha he sample is aken a he firs overshoo of he low-frequency oscillaion. This sample is ermed u s '. A correcion is hen made o esimae is seady-sae value u s. The following modificaion is inroduced for his purpose uσ' uσ = + k (22) s where k s is he overshoo percenage. Noe ha he ampliude of u s is small as compared wih he ampliudes of he highfrequency disurbances. The correced anisoropy signals are displayed in Fig. 6. The low-frequency componen is caused by magneic sauraion. Is separaion from he high-frequency signals is described in [6]. The laer signals are induced by he roor cage anisoropy. The waveform u sa is composed of eiher u () s a ransien exciaion by u as per (8), or u (4) s a ransien exciaion by u 4 = u, likewise is u sb composed of eiher u (3) s or u s (6), and u sc of eiher u (5) s or u s (2). Accordingly, u sa relaes o a ransien exciaion in phase axis a, u sb in axis b, and u sb in axis c. The highlighed secion in Fig. 6 indicaes ha he roor anisoropy signals u sa, u sb, u sc form a balanced hree-phase sysem ha defines a unique phase angle, which is he roor bar posiion angle. The signals are sampled a he high repei µs s µs a) wihou common mode filers b) wih common mode filers; inse: enlarged porion Fig. 5. Oscillograms of he anisoropy signal u s... 4)l c C c ', where l c is he cable lengh and C c ' is he cable capaciance per uni lengh [7]. The oscillaions and overvolages a he inverer erminals are significanly reduced as he refleced waves from he moor side are minimized. A comparison is shown in Fig Delayed sampling The propagaion of he swiching ransiens in he saor windings occurs a relaively low velociy. Travelling wave reflecions hen produce he low-frequency oscillaions in he phase-o-neural volages ha are seen in Fig.. Alhough i appears desirable searching for means o eliminae he oscillaions, i mus be realized ha he wave propagaion process and he expansion of he ransien leakage fields are physically inseparable. The fac requires he anisoropy signal be sampled before he ransien leakage fields of he machine decay []. However, he high-frequency componens of he phaseo-neural volages inroduce an error when he samples are aken oo early. An even more sringen reason for aking he measuremen samples wih minimum delay afer he ransien exciaion is given by he PWM algorihm. The acive swiching sae vecors, used for ransien exciaion prior o signal acquisiion, u σ a u σ b u σ c 5V 5 5V 5 5V s Fig. 6. Phase componens of he anisoropy signal u s. The shaded area highlighs one cycle of he roor bar anisoropy signal; his signal forms a balanced hree-phase sysem

9 ion rae of 2 khz which ensures a high dynamic bandwidh of he roor posiion measuremen. They are coninuous and smooh which guaranees a high spaial resoluion. 4.4 Moor Daa The inducion moor used for he experimens has he following daa: P R = 4.5 kw, U R = 36 V, 5 Hz, I R = 9.26 A, cos j =.86, 73 rpm, 2p = 4, n rp = 22, R s =.26 W, R r =.639 W, L s = 4.48 mh, L r = 6.29 mh, L m = 4 mh, no skew. SUMMARY The squirrel cage roor of an inducion moor is an anisoropic srucure. Is spaial orienaion influences on he respecive phase values of he oal leakage inducances of he saor winding. The swiched waveforms of he pulsewidh saor volages subjec he machine o repeiive ransien exciaions. These serve o exrac a signal ha represens he absolue angular posiion of he roor bars. The mehod requires measuring he phase-o-neural volages a he inverer erminals. Disurbances originaing from high-frequency common mode currens and ravelling wave phenomena on long moor cables are eliminaed by appropriae filer and signal sampling echniques. REFERENCES. J. Holz, Sensorless Conrol of Inducion Moors, Proceedings of he IEEE Vol. 9, No. 8, Aug. 22, pp J. Holz, Sensorless Posiion Conrol of Inducion Moors an Emerging Technology, IEEE Transacions on Indusrial Elecronics, Vol. 45, No. 6, Nov/Dec. 998, pp P. L. Jansen and R. D. Lorenz, Transducerless Posiion and Velociy Esimaion in Inducion and Salien AC Machines, IEEE Transacions on Indusry Applicaions, Vol. 3, No. 2, Mar/Apr 995, pp M. W. Degner and R. D. Lorenz, Using Muliple Saliencies for he Esimaion of Flux, Posiion and Velociy in AC Machines, IEEE Transacions on Indusry Applicaions, Vol. 34, No. 5, Sep/Oc. 998, pp N. Teske, G. M. Asher, K. J. Bradley, and M. Sumner, Analysis and Suppression of Inverer Clamping Saliency in Sensorless Posiion Conrolled of Inducion Moor Drives, IEEE Indusry Applicaions Sociey Annual Meeing, Chicago, Sep. 3 - Oc. 4, 2, on CD-ROM. 6. J. Holz and H. Pan, Eliminaion of Sauraion Effecs in Sensorless Posiion Conrolled Inducion Moors, IEEE Indusry Applicaions Sociey Annual Meeing, Pisburgh, Oc. 3-8, A. von Jouanne, P. Enjei and W. Gray, Applicaion Issues for PWM Adjusable Speed AC Moor Drives, IEEE Indusry Applicaions Magazine, Sep/Oc. 986, pp V. Chura, Travelling Waves in he Windings of Roaing Elecrical Machines, (in German), ez-archiv, Vol., 989, pp R. Rüdenberg, Elecric Travelling Waves, (in German), Springer, Berlin J. Holz, On he Spaial Propagaion of Transien Magneic Fields in AC Machines, IEEE Transacions on Indusry Applicaions, Vol. 32, No. 4, July/Aug. 996, pp