EQUIVALENT CIRCUIT PARAMETERS AND OPERATING PERFORMANCES OF THE THREE-PHASE ASYNCHRONOUS MOTOR

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1 EQUIVALENT CIRCUIT PARAMETERS AND OPERATING PERFORMANCES OF THE THREE-PHASE ASYNCHRONOUS MOTOR MARIN MIHALACHE Key words: Asychroous otor, Equivalet circuit, Paraeters, Perforaces. Startig fro usual rated data (η, α η,, cosφ 1,, s, M ), a sile ad efficiet rocedure is roosed for idetifyig T-equivalet circuit araeters of three-hase asychroous otor. Subsequetly, by uerical solvig of equivalet circuit, the oeratig erforaces of a three-hase asychroous otor sulied fro voltage or curret sources, at costat or variable frequecy, ca be obtaied. A uerical exale (75kW, 1485 r asychroous otor) cofirs the validity of roosed ethod ad the relatively good agreeet betwee uerical ad aalytical characteristics. 1. INTRODUCTION Matheatical odel idetificatio (i.e. T equivalet circuit araeters) of three-hase asychroous otors, ost frequetly used otor i electrical drives, ca be a iortat objective with ractical beefices, i both aalysig ad sythesis robles of asychroous otors with iosed techical (η, α η,, cosφ 1,, s, M ),ad ecooical erforaces [1 3]. Aog aboveetioed erforaces, rated ower factor (cosφ 1, ), rated efficiecy (η ) ad.u. otial charge (α η, ) which corresods to axiu efficiecy are iortat requireets of the otor, ad costitute the ai data for the idetificatio algorith.. EQUIVALENT CIRCUIT PARAMETERS The iductio tye achies (trasforers, asychroous achies) are rereseted by the well kow T equivalet circuit atheatical odel, with r 1 ad x σ,1 the araeters of riary, r ad x σ, of the secodary ad r w ad x the araeters of the agetizatio circuit, all exressed i.u. Uiversity Politehica of Bucharest, Deartet of Electrical Egieerig, E-ail: aihro@yahoo.co Rev. Rou. Sci. Tech. Électrotech. et Éerg., 55, 1,. 3 41, Bucarest, 010

2 Equivalet circuit ad erforaces of three-hase asychroous otor RATED AND CATALOG DATA Asychroous otor rated o-load ad short-circuit data is ot stadardized as i trasforers. Therefore, atheatical odel idetificatio rocedure ust cosider oly the usual (oralized) data, as follows: rated ower factor (cosφ 1, ), rated efficiecy (η ) ad otial.u. charge (α η, ), rated sli (s ) ad.u. axiu torque of the otor, voltage sulied *, U =, U / η cos ϕ1,. Fro the silified vector diagra ad agular characteristic of asychroous achie [], e, =, U si ϕ, cos ϕ1,, it results the followig rated values: rotor (iduced) curret: i, = cos ϕ1, / cos ϕ,, ϕ, = 0.5 arcsi( e, /, U ) ; agetizatio curret: i, = si ϕ1, i, si ϕ, ; agetizig reactace: x 1/ i, ; (1) total leakage reactace: x σ 1/, U ; stator curret corresodig to axiu torque: i 1, M i1, k /, U. axiu torque N Fig. 1 Uiversal characteristics. I case of a asychroous otor with give ower factor (cosφ 1, ), the above values deed oly o axiu torque (,U u.), so, they are uiversal characteristics of the otor, voltage sulied, as i Fig. 1.

3 34 Mari Mihalache 3.. ELECTROMECHANICAL CHARACTERISTICS The electroechaical characteristics i 1 (α), cosφ 1 (α) ad η(α) at rated voltage ad frequecy, for differet sli values (s) or iteral agle (θ), ay be deteried agaist diesioless araeter α = P /P, s/s = 0 1; sile aalytical exressios for these characteristics have bee roosed i [4]: stator curret characteristic: i 1 ( α) i o + (1 io ) α, where i o i, ; () ower factor characteristic: cosϕ 1, ( α) α / α + ( i o + xσ α ), with a eak at α ϕ, = io / xσ, U io = f (,U ), a uiversal characteristic, Fig. 1; * * efficiecy characteristic: η( α) α /( α + o, ε + α k, ε ), with a axiu at α, o ε = o + εo * * η, = o, ε / k, ε * o Po, / P, * * * *,, k ε = k εo,, with ε o = i o j1,, =, P o, = Pfe, + P, v, ad k = Pk, / P,, P k, = Pj1, + Pj,. *.3. LOSSES DISTRIBUTION Startig fro the aalytical exressio of efficiecy, losses ca be searated: Joule losses i rotor widig: j, = s e, = s η cos ϕ1, /(1 s ) ; j, Joule losses i stator widig: j1, = ; (3) (1 + α η, ) (1 io ) (1 io ) α η, costat losses: o, = j1, io, (1 + α η, ) where = (1 η) cosϕ 1,. Ileetig the coditio of ositive losses, > 0, eas: a) fro J1, >0, the axiu rated sli value (s,ax ) ad the axiu value of costat losses ( o,ax ) result, both deedig o η ad α η, : s s, ax η, < = (1 η ) /(1 + η α ), o, o,ax η, η, < = α /(1 + α ). (4)

4 4 Equivalet circuit ad erforaces of three-hase asychroous otor 35 Fig. Losses distributio. b) fro o. > 0 results the axiu value of Joule losses i stator widig: α η, α η, (1 η ) cosϕ1, j1, < j1,ax = =, (5) 1+ α ) i (1 + α ) i ad the iiu value: α η, ( η, o η, o > ( α η, ) i j, = i o. i ( ) Therefore, for a asychroous otor with iosed data, the relatios (3) allows the searatio of losses i the achie (Fig. ), ad evetually a balaced choice, which could also cosider the referece losses ref /4. o.4. EQUIVALENT CIRCUIT PARAMETERS Subsequetly, the T-equivalet circuit araeters ca be obtaied: resistace of stator widig: r 1 = j 1, [.u.]; resistace of rotor widig: r = j, / i, = kr, s s ( k r, s 1); (6) total leakage reactace: xσ, 1 xσ, = xσ / 1/(4, U ), or k x, 1, = xσ, / xσ, 1; agetizig reactace: x u, / i,, u, = abs( u, ) 1 xσ,1 si ϕ1 ; equivalet costat losses (iro ad echaical losses): r w u, / 0, [.u.]. Note. I case of ecasulated iductio otors (i.e. IP44), echaical ad vetilatio losses ca be deteried fro their frae size (H,.). Based o relatio D e k H H [3], we obtai: j,

5 36 Mari Mihalache 5 4 P, v 1 1 k H H, v = (7) S S ad the equivalet arallel resistaces of equivalet circuit: Fe u, Fe, r, v u, /, v r /..5. APPLICATION A three-hase iductio otor with the followig rated data is cosidered: P = 75 kw, IP = 44, H = 80, D e = 45, =, s = 0.01, η = 0.94, (8) cosφ 1, = 0.88, M /M =.4; The distributio of losses versus α η, is show i Fig.. Balaced losses (close to ref ) are obtaied for α η, = The followig losses i [.u.] are couted for α η, = 1 (catalog data): j1, = , j, = , o, = 0. 05,, v = , ref = , ad corresodig araeters i [.u.], resectively: r = ; r = ; x α, 1 = ; x σ, = ( k x, 1, = ); x = ; (9) r w = 35.7 ( r Fe = 5.8 ad r, v = 110. ), u, = , i, = 0. 83, i, = MOTOR PERFORMANCES AT RATED FREQUENCY The electroechaical characteristics i 1 (α), cosφ 1 (α), η(α), ad echaical characteristic = f(α), are of ajor iterest i aticiatig the iductio otor erforace, whe fed at costat voltage ad frequecy. a) Electroechaical characteristics obtaied by solvig the T-equivalet circuit with costat araeters as show i (9) are rereseted i Fig. 3, where α = s/s, r /s = k r,s α, ad k r,s = r /s = The followig rated values were obtaied at i 1 = 1 [.u.]: s = , η = 0.941, α η, = 1, cosφ 1, = 0.876, (10) which are i a good agreeet with iosed values show i (8). Fig. 3. shows also the aalytical characteristics (). A good corresodece is agai observed, which cofirs the validity of roosed relatios.

6 6 Equivalet circuit ad erforaces of three-hase asychroous otor 37 Fig. 3 Electroechaical characteristics. b) Mechaical characteristics, or torque characteristics of the otor sulied by costat voltage ad frequecy, U = f(α), are rereseted i Fig. 4. Fig. 4 Mechaical characteristics. Relatively sigificat discreacies (10 15%) are observed betwee uerical (obtaied directly fro equivalet circuit) ad aalytical (Kloss, U,T ) results. For coariso, siilar otor characteristics whe fed at costat curret

7 38 Mari Mihalache 7 ad frequecy are rereseted i Fig. 4. I this case, a better agreeet of data is observed. c) Startig characteristics. At startig, the squirrel-cage otor is * ad startig curret characterized by startig torque i, which are deedet o each other. Ideed, torque ca be writte = r, i,, with r, = k R r rotor startig resistace ad k R, > 1 the icreasig factor that takes ito accout the ski effect. Assuig the aroxiatios r s ad i, i 1, = i, the icreasig factor ( k R, ) ca be iediately evaluated [4]: * ηcosϕ1 kr, = 1. (11) si si (1 s) A siilar relatio is obtaied for the startig decreasig factor k x of the leakage reactace ( x k x k 1 ), due to ski ad saturatio effects: σ, = x, σ, x, < x σ,, U k (1 ) x, = i + 1 r, xσ i where (1) r = r1 / r,, Alicatio. For the 75 kw asychroous otor (8), with i = 6. 7 ad * * =.1 or = η cosϕ 1 =1.74, it results: k R, = 3. 9 ad k x, = 0. 5, very iortat results for otor desig, esecially for sizig the squirrel-cage rotor. 4. ASYNCHRONOUS MOTOR PERFORMANCES AT VARIABLE FREQUENCY The frequecy characteristics are rereseted i Fig.5 for two sli values: s = s, the rated sli, ad s = s (f) = s, /f, which corresods to axiu torque. The otor is sulied at: u 1 = f for f 1, ad u 1 = 1 for f 1. A iortat decrease of the axiu torque (s ) is oticed i the low frequecy doai (f = f 1 /f 1, <1), i cotrast to usual estiatios (Kloss) [5, 8]: ( f ) ( s, f ), s / s ( f ) + s ( f ) / s

8 8 Equivalet circuit ad erforaces of three-hase asychroous otor 39 with ( f ), U u1 / f =, U = = cost. Fig. 5 Frequecy characteristics. Startig frequecy characteristics (s = 1). For asychroous otors sulied by variable frequecy it is iterestig to evaluate their startig erforaces (startig torque ad startig curret) at very low frequecies (f < ), whe ski effect is ot reset. Fig. 6 Startig frequecy characteristics (u 1 = f).

9 40 Mari Mihalache 9 Fig. 7 Startig frequecy characteristics (i 1 = 1). I case of u 1 = f it results P, 0.8 [.u.] at f P 0.1 (Fig. 6), which are very differet tha the estiated values ( P, =,U, ad f P = s, ), couted with Kloss forula. For coariso, i Fig. 7 there are also rereseted the startig (s = 1) characteristics of the otor sulied by a curret source (i 1,P = 1 [.u.]) ad variable frequecy. I the later, the results are i a good agreeet with the estiates values: P. =.I x, / 1.7 [.u.] at f P = s µ, = r /x, [.u.]. Also, the startig erforaces are uch better, which eas a greater torque at a lower frequecy, desite the iheret agetic saturatio at curret suly. 5. CONCLUSIONS The aer resets a sile ad reliable rocedure for deteriig the atheatical odel (T equivalet circuit araeters) of a three-hase asychroous otor o the basis of rated ad catalogue usual data η, α η,, cosφ 1,, s, ad,u. Cosiderig axiu torque (,U ) as a variable araeter, the uiversal characteristics of three-hase asychroous otor, i,, i,, x,, x σ, α α, = f(,u ), show i Fig. 1, are outlied. Based o roosed aalytical relatios for electroechaical characteristics (), there were obtaied a set of sile relatios for deteriig losses J1,, J,, o,, ad cosequetly for resistaces r 1, r, r w, with η, ad α η,, as a ai araeters.

10 10 Equivalet circuit ad erforaces of three-hase asychroous otor 41 A uerical exale of a three-hase asychroous otor with 75 kw, 1485 r, cofirs with a good accuracy the roosed aalytical relatios for electroechaical characteristics of the otor sulied by costat voltage ad frequecy (Fig. 3). There are, however, soe discreacies i torque evaluatio (Fig. 4), if coared to usual silified Kloss forula. Iterestig results are obtaied for variable frequecy whe iortat discreacies are agai idetified for axiu torque variatio at low frequecies (Fig. 5). The sae discreacies are idetified for startig torque (s = 1), at variable voltage ad frequecy (u 1 = f), Fig. 6. A better agreeet betwee roosed ethod ad Kloss forula are foud at costat curret. The results obtaied i this aer, esecially the araeters idetificatio ethod, ca be a useful tool ot oly i asychroous otors aalysis, but also i sythesis (desig) robles for asychroous otors with iosed techical ad ecooical erforaces (see the Poytig algorith []). Received o 10 Seteber 008 REFERENCES 1. M. Mihalache, Paraetrii scheei echivalete şi erforaţele de exloatare ale otorului asicro trifazat, S.N.E.T. 08, Bucureşti, M. Mihalache, Tratarea uitară a robleelor de aaliză şi siteză a aşiilor de curet alterativ, S.N.E.T. 07, Bucureşti, M. Mihalache, Maşia asicroă, roblee de aaliză şi siteză otială, Edit. Pritech, Bucureşti, M. Mihalache, Maşii electrice, Vol. ( Maşii electrice rotative ), Edit. U.P.B., Bucureşti, B.K. Bose, Power Electroics ad Variable Frequecy Drives, IEEE Press, New York, M. Mihalache, Cosideraţii rivid erforaţele eergetice ale otoarelor asicroe, Coferiţa Naţioală de Eergetică, Bucureşti, M. Mihalache, La sythèse du schéa equivalete du oteur asychroe à artir des ses doées oiales, AMSE Press, Vol. 1, No, Paris, A. Frasua, R. Magureau, Maşii şi acţioari electrice, Edit. Tehică, Bucureşti, I. Boldea, Gh. Ataasiu, Aaliza uitară a aşiilor electrice, Edit. Acadeiei, Bucureşti, C. Bala, Maşii electrice, Edit. Didactică şi Pedagogică, Bucureşti, C.V. Joes, The Uified Theory of Electrical Machies, Butterworth, Lodo, 1967.