Paamete dentification of DC Moto utho: Dipl.-ng. ngo öllmecke dvantage of the Paamete dentification Method Saving time and money in the teting poce: no anical coupling neceay Full infomation: Entie chaacteitic cuve and value of any moto www.imc-belin.com Belin: 9 ( 3-6 7 9- Hotline: 9 ( 3 6 7 9 6 Fankfut: 9 ( 6 7-9
DC-moto ae ued in a wide vaiety of application touching ou daily live, whee they eve to elieve u of much wok. Howeve, the lage numbe of DC-moto ued i attended by a lage amount of time and eouce devoted to inpecting them at the end of thei poduction cycle. The time needed fo thi poce hould be kept a bief a poible o that the inpection pocedue i not the lowet pat of the poduction poce. Due to the inceaing ma poduction of thee moto, pocedue fo inpecting them have been developed which ae able to dete-mine the tet object chaacteitic cuve within econd. Such pocedue ae known a Paamete dentification pocedue. They detemine the paamete without applying any extenal load, imply by meauing the cuent and voltage. The time and appaatu equied fo attaching a load and fo aligning the tet object with a loading anim can thu be totally omitted. ntoduction The dynamic behaviou of d.c. moto can be decibed uing two equation. The fit equation decibe the ctical behavio u i k ω di dt ( The econd equation decibe the anical behavio d ω k i k ω M dt ( whee the algebaic ymbol epeent the following: Symbol Unit Definition u ctic teminal voltage i ctic amatue cuent ω / otational feqency Ω Ohmic feule eito k geneato contant H inductivity kgm² moment of inetia k Nm liding fiction M Nm load Fig. : Electoanic equivalent cicuit. Enegy model To bette intepet Equation ( and (, we now poceed to model the enegy elationhip. Thu, Equation ( i multiplied with the cuent i and integated ove an indefinite inteval. u idt i dt k idt i di ω dt dt (3 The fit tem in Equation (3 decibe the enegy tanfeed ctically, the econd the Ohmic loe, the thid the anical enegy contained in the ytem and the lat the enegy toed a inductivity. Equation ( i multiplied by the otational fequency ω and integated ove an indefinite inteval. U(t d ω ω dt k i ω dt k ω dt M wdt dt ( The fit tem in Equation ( decibe the otation enegy toed in the anical ytem, the econd tem the anical enegy, the thid the peed-popotional enegy loe and the lat the anical enegy deliveed including wated enegy due to tatic fiction. E input E tanfe i(t U g(ω E ctical lo E magnetic E angula kinetic E fictional lo ctical pat anical pat Θ M i(i ω(t K M Note: The load alo eflect the moment of tatic fiction inheent in the ytem. Equation ( and ( can now be ummaized in a ingle equivalent ctoanic cicuit. E output Fig. : Diagam of enegy ditibution www.imc-belin.com Belin: 9 ( 3-6 7 9- Hotline: 9 ( 3 6 7 9 6 Fankfut: 9 ( 6 7-9
. Tanfe function of a DC moto Since only the teminal quantitie voltage and amatue cuent ae ued in identifying a DC moto' paamete, the coue of the pectum of the amatue cuent-to-teminal voltage atio i of paticula inteet. On the bai of thi tanfe function it i poible to make tatement about at which point in the fequency ange excitation i ueful, ince vant paamete change take effect in that fequency ange. imple example hould eve to affim thi: To identify the paamete of an ctic low-pa with a cutoff fequency of khz, it in't a pactical appoach to excite it with a Hz ocillating quantity, ince, conideing the meauement peciion, the input and output ignal ae appoximately the ame (tanfe facto appox., phae-hift between input and output ignal appox. degee. Only if the excitation appoache the cutoff fequency do the filte paamete become noticeable and i thee a chance of detemining them with eaonably high accuacy. Fom Equation (8, it follow that: ( km U( k We now intoduce the following identitie: Symbol Unit Definition k (9 k k k τ ctical time contant τ anical time contant k / un-down contant / gain U # ( U ( km k and ineting thee abbeviation eult in the following tanfe equation To detemine the tanfe function, Equation ( and ( ae tanfeed into the aplace domain. ( U ( # k k k τ τ τ ( U ( ( k Ω ( ( ( Ω( k ( k Ω( M (6 Fom Equation ( we thu find the otational fequency k ( M Ω( k By uing (7 in ( and making ome eaangement, we obtain the equation (7 voltage cuent otation peed ^3 PM.8.6.. 3. 3 3.8 9 3.6 3. 8 3. 3. 7.8.6 6. - τ.. -.8 -.6 -. 3 -. -3..8-3.6 -. - τ. -..7.8.9.3.3.3.33.3 ( k K k U( k km ( (8 k Equation (8 decibe the elationhip between the teminal quantitie U and and to the load M. Fig. 3 : Electical and anical time contant of the ctical DCmoto The contant τ i efeed to a the ctical time contant of the DC moto. The ctical time contant i a meaue of the cuent' eaction time upon change in the teminal voltage. www.imc-belin.com Belin: 9 ( 3-6 7 9- Hotline: 9 ( 3 6 7 9 6 Fankfut: 9 ( 6 7-9 3
The contant τ i efeed to a the anical time contant otation Speed Cuent output powe efficiency of the DC moto. The anical time contant i a meaue of the PM' eaction time upon change in the teminal voltage. Uing Equation (9, the DC-moto' fequency epone can be hown fo fixed paamete. The figue below how the fequency epone in tem of both magnitude and phae, a well a the chaacteitic cuve fo a moto with given paamete. Paamete Unit alue Ω.9 H. k.33 kgm² 7.e- k Nm e- PM 36 3 3 3 8 6 8 6 8 6 6 6 3 3...... Fig. 6: Chaacteitic cuve of a DC moto Nm W % 9 8 7 6 3 9 8 7 6 3 9 9 8 8 7 7 6 6 3 3 magnitude pole magnitude tau magnitude 6... n addition to the cuve of the epective fequency epone, the fequencie which ae popotional to the anical and ctic time contant, a well a the fequencie of the tanfe function' pole ae plotted... 3. 3.. The magnitude fequency epone of the DC moto coepond to a band-pa filte with a cente fequency lying between the moto' two time contant.. The phae fequency epone of the DC moto coepond to the... phae fequency epone of a band-pa. The zeo in the tanfe function' numeato caue the phae to appoach zeo at low. - - 3 6 fequencie. Hz Fig. : Magnitude fequency epone of a DC moto Gad 9 8 7 6 3 - - -3 - - -6-7 -8-9 - phae pole phae tau phae - - 3 Fig. : Phae fequency epone of a DC moto 6 Hz The DC moto in the example decibed ha two eal pole in the tanfe function. Thee pole appea in the fequency epone a imple but epaate pole. Conveely, thee ae alo moto with conjugated complex pole which can be epeented in the fequency epone a double-pole at the band-pa cente fequency. Taking a look at the tanfe function, it i now poible to detemine cetain egion in which the DC moto' paamete can be identified. indicated above, thi eult in a dependence on the numeato polynomial' zeo-coing in the lowe ange of the tanfe function; the zeo-coing in tun depend on the un-down contant. n addition, in thi ange the magnitude fequency i almot zeo, making etimation baically impoible and the un-down contant not to be detemined with adequate peciion. www.imc-belin.com Belin: 9 ( 3-6 7 9- Hotline: 9 ( 3 6 7 9 6 Fankfut: 9 ( 6 7-9
n the egion of the DC-moto' time contant, ufficiently lage amplitude eult fo the pupoe of etimating the ytem' pole. Then, the DC-moto' paamete can be calculated fom the pole. Fom Equation (, we ee that fo the zeo-coing:, k τ ± k τ k τ τ The ytem' pole become puely eal if the quae oot expeion i poitive, fom which follow: τ τ > ( ( τ k ( ll DC moto meeting the condition in Equation ( have puely eal pole. k k (7 f the numeato polynomial' pole have been detemined and the moto' un-down contant i additionally known, thee can be ued to find the moto' paamete. The lat actual pupoe of the etimate i to detemine the numeato polynomial' pole. Upon cloe inpection of the tanfe function and of the fequency epone, it i een that the ytem can be divided into a high-pa and a low-pa filte. k k ( # U ( ( T ( T T T (8 The fit tem in Equation (8 epeent a high-pa filte, the econd a low-pa..3 Etimating the tanfe function' pole Fo all moto fo which Equation ( applie, the following appoach to detemining the paamete i available: ( U ( # k ( T ( T thi equation can be multiplied out to yield ( U ( # k T T (3 ( By compaing coefficient with Equation (9, we obtain the following conditional equation fo the DC-moto' paamete. ( Fig. 7: epeentation of the DC-moto' individual fequency epone f the moto i now diven at fequencie f and f in ucceion, the facto in Equation (8 can be epaated and calculated individually. Figue 3 clealy how, when diven at ω, the low-pa etun a magnitude facto of appox. and a phae hift of zeo degee, o that the low-pa can be neglected at fit appoximation. Thu we obtain the following tanfe function: ( ω # U ( ω k jω jω T (9 By multiplying out Equation (9 and plitting it into it eal and imaginay pat, we find the ytem of equation (whee the indice and denote the eal and imaginay component of the epective quantity, e.g. U e{ U # (jω }. T T k (6 ω T U w T U * * ( www.imc-belin.com Belin: 9 ( 3-6 7 9- Hotline: 9 ( 3 6 7 9 6 Fankfut: 9 ( 6 7-9
The following identitie ae ued U U k * ω U U U k * w U ( and the equation fo detemining T and eult. * * U U T * * w ( U U * * U U ( Thu we obtain the fit value etimate fo detemining a pole of the tanfe function T and of the gain facto. Thee value etimate can now be ued in etimating the econd paamete. Fo thi pupoe, Equation (8 i eaanged a follow: k jw ( w ( j T U ( w ω (3 jw T uing the following identitie U U U k U T U U k ω ω ( ω ω T U k U T U U k ( ω ω ( ω ω T Uing Equation ( and (, the moto paamete can now be detemined by iteation. To do thi, the eult fom ( ae applied to ( until the iteation algoithm convege. Then the tet object' paamete can be found uing Equation ( though (7. dentifying a ample moto' paamete Fo the example of a eal moto decibed above, the following tanfe function eult: (. # 7 (6 U (.... 9 9 33. 7.. 7. 9. 7. which olve a We now olve fo T and, which, afte epaating the eal and imaginay component in Equation (, ae detemined by the conditional equation: ( U ( #. 666 (7 38. 666 79. and implie the following value fo chaacteitic quantitie: U U T w ( U U U U ( τ τ k. 63m 366. m. 666 www.imc-belin.com Belin: 9 ( 3-6 7 9- Hotline: 9 ( 3 6 7 9 6 Fankfut: 9 ( 6 7-9 6
Typical plot of cuent and voltage with the invee value of the ctical and anical time contant a the excitation fequencie ae diplayed in Figue 8. oltage Cuent 3 3 9 8 7 6 - - - - 3 - -3-3 - - -.8.9....3 The iteation ultimately etun the following paamete: T.3663 T.8.76.7679 Thi tanlate to the following paamete fo the tet object:.9. k.333 Thu the accuacy of the paamete detemined i within.%. Fig. 8: Plot of cuent and voltage fo detemining the paamete of a ample moto ^-3. 96. 87. 78.7 7. ^-3 3.6 3. 3. 3.7.9 ^-3 7.. 3..7 T T. 3 6 7 8 9 Fig. 9: Plot of identified paamete T, T,, Figue 9 how a plot of the paamete T, T, and. The convegence poduced by thi pocedue i clea to ee. www.imc-belin.com Belin: 9 ( 3-6 7 9- Hotline: 9 ( 3 6 7 9 6 Fankfut: 9 ( 6 7-9 7