Understanding small, permanent magnet brushed DC motors.

Transcription

1 Undernding ll, pernen gne ruhed DC oor. Gideon Gouw Mrch 2008 L Upded 21 April 2015

2 Vrile B = Fixed gneic field (Tel) i () = rure curren, funcion of ie (A) i ( ) = edy e rure curren (A) J L = Ineri of he lod D L = Vicou dping coefficien of he lod J = Ineri of he rure nd roor J L = Ineri of he lod. J M = Tol ineri preened o he oor D = Vicou dping coefficien of he rure nd roor D L = Vicou dping coefficien of he lod. D M = Vicou dping coefficien of he rure plu lod. = ck ef conn = Moor conn = orque conn (N./A) l = lengh of he conducor () L = rure inducnce (Henrie) N 1 = Ger on oor ide N 2 = Ger on lod ide P E = Elecricl power (W) P M = Mechnicl power (W) = Arure reince (oh) = Speed regulion conn = Serie reince (oh) T() = Torque (newon.eer) T ll = Sll orque of oor (newon.eer) v () = pplied rure volge (vol) v () = induced ck ef (vol) θ () = ngulr diplceen of he oor (rdin) ω () = ngulr velociy of oor (rdin/econd) ω 0 = no-lod peed of oor (rdin/econd) Areviion ef: elecro oive force LT: Lplce rnfor TF: Trnfer funcion 2

3 1. Wh doe he ile en? In he who who in he elecric oor zoo, he group of oor h cn e clified ll, pernen gne ruhed DC oor i ju one of ny grouping. The docuen ile hu iplie h we re looking only hi one group, which i: Sll - of liied power oupu, ypiclly lo ll in ize. Pernen gne The gneic field i pernen, ypiclly fored y pernen gne. Bruhed he couor rrngeen ue ruhe for elecricl conc nd reverl of he rure curren. DC - ue direc curren for operion. Undernding - lef o he reder... Thi ype of oor i generlly inexpenive, ey o ue nd i ypiclly ued in echronic pplicion. 2. Principle of Operion nd Conrucion. A DC oor i good exple of n elecroechnicl device, coniing of oh n elecricl nd echnicl uye. I funcion idirecionl rnducer, convering elecricl energy ino echnicl energy (oor pplicion), or cn lo conver echnicl energy ino elecricl (generor pplicion). Moor exploi he phenoenon decried y Mxwell equion; curren flowing wihin conducor will elih gneic field round i. If hi curren-crrying conducor i now plced in pernen gneic field, he inercion eween he wo gneic field will produce force on he conducor. In he oor, he producion of hi force i ued o cree orque h urn he roor. The conrucion of hi ype of oor cn e divided ino he following pr: The or, or ionry pr h coni of he pernen gne ched o he oor houing. The roor (hf) h lo crrie he rure or coil winding. 3

4 The couor nd ruhe h provide elecricl conc o coil while he oor i roing nd provide ehod of wiching he direcion of curren in he rure. Bering h he hf run on. 3. Derivion of he rnfer funcion. The curren i () flowing in he rure will produce n inercion wih he gneic field, o h he rure will experience force f(), f() = B.l.i () Thi force will cue he rure o ove (roe), which will led o ck ef eing induced due o he gneic field: d() v() () (1) d The volge-curren relionhip in he rure i hen given y: di() v() i () L v() (2) d Thi cn e wrien in LT for: V = I + L I + V (3) The orque he oor develop will e proporionl o he rure curren, o h: or T = I (4) T I (5) where i he orque conn. Suiue (2) nd (5) ino (4): T L V (6) 4

5 or wriing he e equion in er he ngulr velociy ω(): T L V (6) Conider he echnicl uye of he oor, where he orque produced y he rure curren now urn he roor-rure coinion. Thi coinion h n ineri J nd liner dping (vicou) D. Oher poile dping effec re negleced. T () J 2 d () D 2 d d() d or in LT for: T=J 2 θ + D θ = (J 2 + D ) θ (7) Siilrly, in er of he ngulr velociy Ω: T=J Ω + D Ω = (J + D ) Ω (7) Suiuing (7) ino (6) o produce: 2 J D L V (8) The rnfer funcion cn hen e derived in er of ngulr diplceen or ngulr velociy : or θ ( ) V ( ) = /(J L ) +( J 3 +D L J L ) +( 2 D + J L ) (9) 5

6 Ω V ( ) = /(J L ) +( J 2 +D L J L ) + ( D + J L ) (9) If i i ued h he rure inducnce J i very ll copred o he rure reince, i cn e hown (fro eqn (8)) h he rnfer funcion reduce o: or θ ( ) V ( ) = /(J ) [ + 1 J ( D + )] Ω V ( ) = /(J ) [ 1 + J ( D + )] (10) (10) 4. elionhip eween nd. The ck ef conn of oor,, (lo clled he peed conn or he volge conn) w defined : v wih uni V..rd -1. Siilrly, he orque conn of he oor h een defined reling he oor orque o he rure curren, wih: T () nd uni N.A -1. i () The vlue of oh hee conn re deerined y he geoericl nd phyicl properie of he oor, wih fcor uch he phyicl dienion, nuer of urn on he coil winding nd he gneic flux deniy ll conriuing. The relionhip eween hee conn cn e e een y conidering he oor in generor pplicion. Alo neglec non-idel 6

7 echnicl nd elecricl loe ocied wih he oor/generor operion. If hi pproxiion i de, he echnicl power on he inpu of he generor hould equl he elecricl power he generor oupu. The echnicl power i given y: P = (Torque x ngulr diplceen)/ie = T ω. The elecricl power genered i defined : P e = oupu volge x oupu curren = v i. So h: T v i nd T i v Thi iplie h = when oh re eured in conien uni. Thi i rue ecue he e fcor (geoericl nd phyicl) h govern he vlue of one conn lo deerine he vlue of he oher. Thee conn decrie he fundenl coupling eween echnicl nd elecricl power nd hould e no differen for he direcion of energy converion. We cn hen iply define new conn,, he oor conn : = = nd he rnfer funcion previou derived cn e odified o reflec hi definiion. 7

8 5. The orque-peed curve. The echnicl conn of he oor cn e oined fro dynoeer e euring he roionl peed nd orque of he oor conn pplied volge while chnging he lod. Suiuing (2) nd (5) ino (4) yield: T T L V Wih he pproxiion L 0, hi will reduce o: T Ω V If we rnfor o he ie-doin nd rerrnge nd conidering edy e condiion: T v (12) Thi will e in he for of righ line y = x + c when we plo T v ω. In prcie, we will produce fily of line reling T o ω for differen vlue of he pplied volge. Two poin re of ignificnce on hi grph: (1) The ll orque (T ll ) preen he poin where he lod ecoe o gre h he oor ll (ω = 0). Thi preen he xiu orque h he oor cn deliver well he xiu curren h he oor drw. (2) The no-lod peed of he oor i he peed wih no pplied lod (xiu peed poile for hi pplied volge). The only orque i h needed o urn he rure nd roor nd he curren drwn i iniu (xiu ck ef). I cn e ued h he orque of he oor hi poin i ~ 0. The y-inercep of he grph fro (12) occur when he oor ll, o h: 8

9 T ll v If we cn hen eure he ll orque, we cn clcule. The lope of hi line will e: T, The invere of he lope i oeie clled he peed regulion conn, defined : B The ll orque cn hen lo e wrien : T ll no lod (check ou) The iniu orque poin (T ~ 0) on he x-xi will occur : no lod v A iniu, only he wo condiion of T ll nd ω no-lod need o e eured o genere hi line. In prcice, i i iporn o keep in ind h ll occur whenever he oor i eping o ove gin force h i igger hn he orque i cn genere. However, ll condiion exi every ie he oor r fro reing poiion or ny ie he oor revere direcion. Thi u e ken ino ccoun when deigning oor driver circui. 6. Block digr repreenion of he DC oor. The oor cn e repreened in lock digr for hown elow: 9

10 If he rnfer funcion of ech individul lock i deerined w decried in ecion 2, i will produce: Boh he elecricl nd echnicl uye re hu fir order nd cn e preened in er of heir ie conn where J D nd e L If we iplify nd ue h τ e << τ, he rnfer funcion cn e reduced o: Ω V 1 1 0

11 where i he iplified gin D nd τ i he iplified ie conn J D 7. Trnfer funcion eiion y explici eureen. In order o oin he oor rnfer funcion, we u e le o eure he following preer: = Arure reince L = Arure inducnce = Torque conn = Bck ef conn J = Ineri of he oor D = Dping of he oor. 7.1 Meuring. The rure reince cn iply e eured y euring he reince cro he oor erinl. However, eure for everl poiion of roion nd verge in order o ke ino ccoun he effec vrying conc of he ruhe on he couor. The reince cn lo e oined y clping he oor (prevening roion) nd euring he edy e curren o pecific pplied volge. Thi enure h no ck ef i induced. Agin, eure everl poiion of roion. Typicl vlue of hould e in he Ω rnge. 7.2 Meuring L. The inducnce of he rure cn e eured fro eure of he ie conn of he L circui h for he elecricl uye. Thi gin enil clping he oor nd hen ujecing i o ep in he inpu volge. An exernl reior,, hould e conneced in erie wih he rure o h he volge drop cro cn e eured in order o clcule he curren. 11

12 The ie conn of hi ep repone i hen given y: e L nd he vlue of L cn e clculed if i known. Typicl vlue for L hould e in he 10-3 H rnge. 7.3 Meureen of (nd ). The curren-volge relionhip in he rure edy e condiion i given y: v ( ) i ( ) ( ) If he curren nd he roionl peed cn hen e eured, cn e clculed : v i The ngulr velociy cn e eured wih choeer or y n opicl enor. A hould equl, we cn ue hi ge h we hve = =. 7.4 Meuring D. Fro he curren orque relionhip T = i () nd he equion for he echnicl uye T d() J D d () we cn wrie n expreion for he edy e condiion (dω/d = 0) : 1 2

13 i = D ω By euring he edy e roionl velociy nd he curren nd fro knowledge of we cn hen clcule he dping coefficien of he oor. 7.5 Meuring J. The oor ineri cn e oined fro eure of he echnicl ie conn of he ye. Thi cn e done y wiching off he oor y producing n open circui in he elecricl inpu. Fro he decy in ngulr velociy wih ie, he echnicl ie conn cn e eured. Thi i given y: J D nd J cn e clculed if D i known. 7.6 Soe furher poin. In ecion 6 he iplified (ignoring inducnce) gin of he oor w hown o e: D v If we hen e he oor repone o n inpu volge ep nd eure he edy e roionl velociy, we cn clcule hi fcor. Noe h he vlue of i independen of he ineri of he oor nd i only deerined y he dping of he oor (nd he oor conn well he rure reince). If we know he oher fcor, we cn hen clcule D fro he edy e repone. I cn e hown h hi equion for he dping reduce o he e h ued in Secion 7.4 D J ***** 1 3