A NEW APPROACH TO INDUCTION MOTOR DYNAMICS WITH PARAMETER PREDICTION

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1 A NEW APPROACH TO INDUCTION MOTOR DYNAMICS WITH PARAMETER PREDICTION Shayak Bhattachajee Depatment of Phyic, Indian Intitute of Technology Kanpu, NH-91, Kalyanpu, Kanpu , Utta Padeh, India. * * * * * KEYWORDS Induction moto Electomagnetic theoy Iteative tuctue Dynamic model ABSTRACT In thi wok the dynamic model of an induction moto i deived fom electomagnetic conideation. A moto model i contucted, and Maxwell equation olved fo it by an iteative method. The angula hamonic of the cuent and magnetic field ae witten a complex quantitie analogou to the pace phao epeentation. Laplace tanfomation of the cuent and field, followed by a ummation ove all ode of the iteation lead to the tanfe function of the moto. An appoximate cae i conideed fit and i ued to lead intuitively to the geneal cae. The tuctue of ou equation i identical to that of the exiting model. In the poce we povide theoetical etimate of the vaiou inductance and eitance ued in a taditional model. Thee etimate ae in good ageement with the eult of numeical evaluation. * * * * *

2 I. INTRODUCTION Induction moto ae the backbone of induty due to thei high pefomance and low maintenance equiement. The iue of contolling thee moto give ie to conideable inteet in thei dynamic behaviou. Below we peent two popula moto model. The d-q model [1]-[4] ue a two-phae moto in diect and quadatue axe. The oto α and β axe ae othogonal and α make an angle θ with the d axi of the tato. The teminal voltage of the tato and oto winding ae expeed a the um of voltage dop aco the eitance and the ate of change of the flux linkage. Thi latte tem i expeed in tem of the elf and mutual inductance and the oto and tato cuent. Unde the aumption of unifom ai-gap the elf inductance ae independent of the angula poition and the mutual inductance between any pai of othogonal winding i zeo. Thee aumption eult in a ytem of diffeential equation with time-vaying inductance. The tanfomation to obtain contant inductance i achieved by eplacing the actual oto with a fictitiou one on the d and q axe. The fictitiou oto cuent ae elated to the eal one though an idempotent matix, which i applied on the ytem with time vaying inductance, and finally efeed to the tato. Thi eult in the diffeential equation connecting the d and q component of oto and tato cuent to the applied voltage. Fo the abitay fame model, the cuent and voltage vecto (d and q component) ae multiplied by a matix and ubtituted into the tato fame model to obtain the dynamical equation. Thi model i deived fo a two phae moto, o it i neceay to tanfom fom the actual thee-phae to the hypothetical two phae machine. Thi i achieved by a matix multiplication, known a Pak tanfomation. The pace phao model due to Kovac and Racz [5] exploit the fact that inuoidal ditibution in pace can be epeented by complex vaiable. The pace vecto notation convet the voltage vecto [V d,v q ] T into the complex numbe V d +jv q (whee j i imaginay unit). The advantage i that the numbe of diffeential equation educe fom fou to two. Futhemoe, the hift fom one efeence fame to anothe i elatively eay. The equation ae typically cat in tem of the flux linkage, though they may alo be witten in tem of the cuent. The claic example of the latte fomulation i in Takahahi and Noguchi wok [6] on DTC the equation they have ued fo a 2-pole moto i v1 R1 + pl = 11 pm 0 (p j ɺ θ + ɺ m)m R (p j θ )L 2 m 22 i1 i2, (1) whee v 1 i the tato voltage vecto, R 1 the tato eitance, p the diffeential opeato, L 11 the tato elfinductance, M the mutual inductance, ɺ θ the oto angula velocity, R m 2 the oto eitance, L 22 the oto elfinductance and i 1 and i 2 the tato and oto cuent. Recently, wok by Holtz [7] ha given pecial impotance to the tanfe function obtained fom the dynamic a he ha ued them to plot complex ignal flow gaph. Thee gaph have been ued to undetand the natue of the oto-tato inteaction inide the moto. In thi wok we popoe a diffeent appoach to moto dynamic. We deive it tating fom Maxwell equation fo the moto. It i not poible to olve the equation diectly o we develop an iteative method, Sec. II-1. The application of thi method i hown in Sec. II-2. A limiting cae i conideed fit o a to implify the calculation involved. The anwe of thi eticted cenaio can be intepeted in a phyically meaningful way, which i extended to apply to the geneal cae, II-3. Ou equation ae found to be identical in tuctue to (1) above. Equating the coefficient of coeponding tem in the two fomalim, we obtain theoetical fomulae fo the vaiou eitance and inductance involved. In Sec. III we compae ou finding with the eult obtained fom finite element analyi and demontate good ageement between the two.

3 II. DERIVATION Since the entie deivation i athe long and involved, we plit it into vaiou pat, doing a paticula chain of calculation in each pat. 1. MODELING AND ITERATIVE DEVELOPMENT A chematic diagam of the induction moto i hown in Fig. 1. The co ection of the cylindical moto i viible thee. The oute pat i the tato which ha been hown to have a polaity of 2. The conducto ba in each labelled phae can be clealy een. The inne pat i the quiel cage oto whee the ba ae hoted by the end ing, viible in the figue. In thi pape we will not deal with wound oto. We define the vaiou geometical paamete in Table I. Paamete Height Radiu Mateial conductivity Thickne Angula velocity (anticlockwie) Pemeability of coe Pemeability aco ai gap Conducto epaation facto Roto h σ b ω µ c µ g c Stato h R σ b NULL µ c µ g c Table 1 : Moto paamete involved in modeling. The polaity i 2n. Some paamete will be explained duing the coue of the analyi. The angula velocity acibed to the tato in the above table how that we will pefom the entie analyi in the tato fame. A few geometical obevation lead to the mathematical model. The hape of the moto motivate the ue of cylindical co-odinate ρ,θ,z with z along the moto axi. Since mot moto ae heavily elongated in tuctue, we aume the magnetic field and cuent to be unifom along z. Futhemoe we aume that the cuent flow excluively along z. Thi amount to neglecting end effect and oto ba kew. Fo a 2n-pole moto we note that the pincipal patial hamonic of the field and cuent ae expeible in the fom a co(nθ)+b in(nθ) whee θ i the azimuthal angle. Analogou to the pace phao epeentation, we wite uch a quantity a a+jb (whee j 2 =-1). A egad the conducto ba, we ae not going to ue the cicula ba geomety hown in Fig. 1. Rathe, at fit we will model the tato and oto a continuou cylindical hell of thickne b and b. We will ubequently intoduce a coection which will incopoate the dicetized natue of the ba but will implicitly aume them to look like ecto of an annulu. An appoximate peciption fo genealizing to othe ba hape will be given late. Since the thicknee b and b ae in geneal much le than the adii and R, we wok in tem of the uface cuent (we may continue to ue the wod cuent fo it, fo bevity) flowing in the oto and tato. Thi i a convenience a the magnetic field fo a cylindical hell caying a inuoidal uface cuent can be calculated eaily. We have witten two pemeabilitie in the above table, o we explain the eaon fo it. Suppoe ome cuent i flowing though the tato. Then the magnetic field poduced at a point within the tato coe will cay a facto of µ c which i the natual pemeability of the coe. When the ame cuent ceate a field in the oto howeve, the pemeability i educed becaue of the ai gap, and thi educed pemeability will be denoted by µ g. We hall wok in tem of the vecto potential A uch that A = B whee B i the magnetic field. Until thi point, the uface

4 cuent, vecto potential and magnetic field ae all teated a vecto in eal pace. Howeve, we ealize that the only elevant component ae the z component of the cuent and potential and the ρ component of the magnetic field. Becaue of thi, we now teat them a patial cala and intead ue the vecto notation to denote the pace phao. A diect olution of Maxwell equation i hindeed by ou ignoance of the contitutive elation between electic field and cuent. In paticula, the σ of J=σE may be a complex quantity. Ou knowledge of the inductive chaacte of the tato and oto convince u that it i indeed o. We note that at the fequencie involved, complex σ ae not an inheent popety of any mateial (they may become an inheent popety at optical fequencie). Rathe, they ae a popety of the way in which the conducting mateial i aanged, i.e. the pecific cuent ditibution. They aie fom electomagnetic effect which occu within the concened cicuit, and can be conideed a an abtaction which allow u to obtain the cicuit behaviou without analying thee effect in detail. Hee we pefom the detailed analyi to bypa the complex σ. The analyi will conit of a tep-by-tep invetigation of the cuent and magnetic field in the moto. Ultimately, the dynamic will emege by combining all tep. In the zeoth tep (it will oon become clea what the tep exactly i) voltage i extenally applied only on the tato. Let it poduce a cuent a it would in a pue eito. We make thi aumption a ight now we ae concened only about the mateial popetie; any ingle tep doe not involve the complex σ which aie only afte combining all tep. Now thi cuent will ceate a magnetic field at the uface of both oto and tato. If the applied voltage i not contant in time (which it geneally in t) the magnetic field will alo change. By Faaday law, electic field will be induced in both the oto and tato. Thee electic field will give ie to cuent in both element, once moe though eal conductivitie. Now thee cuent will ceate magnetic field of thei own, and again thee field will give ie to moe cuent, o the ceation of field and cuent continue infinitely. Each tep mentioned at the tat of thi paagaph i of coue the ceation of a new ound of cuent. Now by the pinciple of upepoition, the eultant cuent and field will imply be the um of all the contibuting tem. Becaue of thi tuctue we will contuct eie development of the oto and tato cuent. The zeoth ode cuent (K 0 fo the oto and K 0 fo the tato) will be the eult of the applied voltage and will be popotional to the voltage though ome facto dependent on the geomety of the moto and the mateial ued in it. K 0 i obviouly zeo, and K 0 will be denoted by the ymbol V, a dimenional minome whoe phyical ignificance outweigh thi mino contadiction. The fit ode cuent K 1 and K 1 will follow fom the effect of the zeoth ode tem, the econd fom the fit and o on. Summation of the eie to infinity, if poible, will give an explicit fomula fo the dynamic of the cuent. The poblem ha thu been epaated into two component : (a) detemination of the geneal tem K k and K k and (b) pefoming the infinite ummation. The next ubection deal with the fit of thee objective. 2. DETERMINATION OF THE GENERAL TERM We note that the phyical pocee in going fom the et [ K k, K k ] T to the et [ K (k+1), K (k+1) ] T will be the ame fo all k, and o will the mathematical opeato epeenting thi tanition. Hence it uffice to deive the tuctue of thi opeato fo abitay k. Suppoe we ae given K k and K k ; we ty to etimate K (k+1) and K (k+1). The etimation i done a pe the phao diagam in Fig. 2 which how cuent, field and vecto potential phao in the d-q plane.

5 Hee all oto quantitie ae indicated in dak colo and all tato quantitie in light colo. Red, geen and blue tand fo cuent, vecto potential and magnetic field phao epectively. The left panel how the vaiou phao oiginating fom one oto cuent (dak ed phao maked 1). Unde the magnetotatic appoximation, valid o long a ω i much le than the velocity of light, the vecto potential due to thi cuent i calculated by 2 uing the equation A = µ J whee J i the cuent denity. Since the cuent ae confined to the tato and 0 oto uface, we get a Laplace equation. The bounday condition ae that the A i continuou aco a cuent uface while the tangential component of B expeience a dicontinuity popotional to the uface cuent. The fomulae fo the vecto potential inide and outide a ingle cylinde (oto) of polaity 2n caying a cuent vecto K ae given below. A A in out µ K c ρ = 2n n µ K g = 2n ρ n,. (2a) (2b) It i een fom (2) that the vecto potential phao ceated at the oto uface i paallel to the cuent phao. So i the potential ceated at the tato uface, but it magnitude i malle than the oto uface potential. The diffeential fom of Faaday law, E = A/ tgive the electic field. The field ae thu oppoite in diection to the potential. They ae maked with a 2 to indicate that they ae of the next ode to the phao maked 1. Now an additional component of oto electic field aie fom the fact that it i being dagged though a magnetic field. Altenatively, we mut apply J=σE in the oto fame, and fame tanfomation add on a v Btem to the tato fame field. The magnetic field ae obtained a the cul (in eal pace) of the vecto potential and evaluate to j n µ K c ρ B = in 2 B out = j µ g K 2 ρ 1 n+ 1,. (3a) (3b) The magnetic field i thu othogonal to the coeponding potential phao and the eulting oto electic field, alo maked with a 2, i given by ωb. The cuent denity J i obtained by multiplying E with the conductivity and the uface cuent i given by Jb. Thi complete the development of the oto cuent. The development of one

6 tato cuent phao i identical except fo the numeical value of the coefficient and i hown in the ight panel in Fig. 2. A pomied ealie we now efine the aumption that the oto and tato ae continuou cylindical hell. In eality they conit of ba of wie with pace between them. We aume that the hape of the ba i that of an annula ecto o that each ba ha a contant angula pan. When the effect of the pace i included, the cuent will no longe be of the fom Kinθ (o coθ) but will get multiplied by the Fouie expanion of the function which i unity in the angula inteval coeponding to the ba and zeo in the inte-ba pace. The fit tem of thi expanion i eay; it i imply the atio of the angula extenion of each ba to that of each ba-plu-pace unit. In othe wod it i like the duty cycle of the conducto in the cage cicumfeence. Thi ha been defined in Table 1 a the conducto epaation facto. Incopoation of thi facto will eult in the potential and field ceated by the oto getting multiplied by c, and thoe ceated by the tato by c. Putting all thi togethe yield the geneal tanfomation elation between ucceive ound of cuent phao, a follow. n 1 ( cµ cσb/ 2n)( p + jn ω ) (/R) ( cµ gσb/ 2n)( p + jnω) n+ 1( µ σ 2 )( ) ( µ σ g c 2 )( ) K (k+1) K = k K (k+1) (/R) c Rb / n p c Rb / n p Kk. (4) Letting T denote the tanfomation matix in the above equation, we at once have the elation between the k th and zeoth ode cuent phao. Since the zeoth phao ae [0,V] T we have Kk 0 = T k Kk V. (5) Thi complete ou objective of detemining the geneal tem of the iteative development, and the only emaining tak i that of 3. DOING THE INFINITE SUM The fom of (4) ugget the following vaiable definition. New vaiable Paamete combination Phyical ignificance τ cµ c σb/2n Roto time contant τ c µ c σ Rb /2n Stato time contant δ 1 (/R) n-1 (µ g /µ c ) Incopoate mutual and leakage δ 2 (/R) n+1 (µ g /µ c ) tem Table 2 : Convenient vaiable definition. The phyical ignificance of the paamete mentioned above i not yet appaent but will become clea a the analyi poceed. Since deivative ae involved and ummation ove epeated diffeentiation i ticky, we take the Laplace tanfom of (4), denoting the tanfom of f(t) by f(). The ummation ove all k then yield the tanfe function. The tanfomed (4) i ( + jn ) 1 ( + jn ) ( ) τ ( ) K(k+1) K τ ω δ τ ω k Kk = T = K(k+1) Kk δ2τ K k. (6) The limiting cae of δ 1,δ 2 <<1 i analyed fit. It aie fo moto with (a) wide ai gap and (b) high polaity. In thi cae we contuct two linea combination of the oto and tato cuent uch that T i eplaced by a diagonal matix. Thi call fo a detemination of the eigenvalue and eigenvecto of it tanpoe and the mallne of the δ i exploited in thi tep. The laget tem of the tato cuent will be independent of δ (eithe 1 o 2) while the laget tem of the oto cuent will be fit ode in δ 1 a it i diven by an inteaction fom the tato ide. Thee conideation detemine the peciion to which the calculation mut be caied out. The tanfe function fo

7 thee cuent combination ae detemined by ummation ove all coectional tem and ae uncambled to find the tato cuent tanfe function V K = 1 + τ, (7) and the oto cuent tanfe function δ ( τ + jnτ ω 1 ) V ( 1+ τ )( 1+ τ jnτ ω) K =. (8) Now thi pocedue i difficult to genealize to the cae whee thee ae no etiction on δ o we attempt an intepetation in tem of the electomagnetic inteaction going on inide the moto. The (1,1) th tem of T denote the poce by which a oto cuent phao poduce the next ode oto cuent, hence it ymbolie the oto inteaction with itelf. The (2,2) th element likewie indicate the tato elf-inteaction and the off-diagonal tem epeent the co-inteaction. In (7), the zeoth ode tato cuent phao i the applied V. The pimay facto govening the tato cuent i it inteaction with itelf the oto inteact with it though the δ 2 tem, which i vey mall. Hence the tanfe function i the eult of the tato inteacting with itelf an infinite numbe of time and poducing tem like V+(-τ )V+(-τ ) 2 V+... a geometic eie which quite obviouly convege to the value given in (7). Fo the oto, it might help to wite (8) in a eaanged way : K 1 V = δ1τ ( + ω) 1 τ ( ω) jn + jn 1+ τ. (9) Conideing the RHS of (9) we ecognize the tem in the fit culy backet a the eult of an infinite numbe of oto elf-inteaction. The econd culy backet which i eentially the zeoth ode oto cuent, ha two pat (box backet). The fit box backet i in fact the (1,2) th element of T which give the elation between a tato cuent and the next ode oto cuent. Hence the econd culy backet i like the oto cuent poduced by the tato cuent contained in the econd box backet. Now thi lat tem i nothing but the eultant tato cuent, (7). Thi motivate the following obevation, in which the etiction on the ize of δ i lifted. Let the eultant oto cuent K be known. Then it poduce a zeoth ode cuent phao δ 2 (-τ )K in the tato. The total zeoth ode tato cuent i obtained by adding V to thi contibution. Now by including the eultant oto cuent in the zeoth ode tato tem we have in effect taken into account all ode of the oto-tato inteaction. In othe wod we have effectively emoved the oto by incopoating it entie contibution into the initial tato tem. Hence we now need to conide only the tato inteaction with itelf, leading to the tanfe function V δ τ K 2 K =. (10) 1+ τ The identical pocedue fo the oto with known tato cuent lead to the tanfe function ( + jnω ) τ ( jnω ) δ τ K 1 K =. (11) 1+ The coeponding dynamical equation ae ( p jn ) ( p jn ) 1 + τ ω δ1τ ω K 0 = δ τ p 1+ τ p K V 2. (12)

8 Thi i identical in tuctue to (1). The coefficient can be matched when we ealize that n=1 fo a 2-pole moto. Moeove, ou equation have effectively nomalized the eitance to unity a we conide uface cuent on both the left and ight hand ide. Conveion of thee uface cuent to voltage and cuent though the neceay geometic facto will yield the eitance. It may appea that the coefficient on the off-diagonal tem ae unequal in (12) wheea they ae equal in (1). Howeve thi i an effect of not including the eitance. In paticula, the fit ow of the matix equation may be multiplied by any abitay contant a the oto voltage on the RHS i zeo. Thi contant can be choen o a to equate the coefficient of the off-diagonal element in the matix. We now make a bief note on the extenion of ou fomalim to ba of abitay hape. If the ba ae too deep, then of coue the entie model will have to be efomulated. If the depth i not a poblem but only the hape i not cloe to that of the annula ecto which we have aumed, then we ugget obtaining equivalent value of c and b (and c and b ) fom the aea of the ba. If ou oto (identical comment hold fo the tato o we conide 2πc only the oto) ha x ba, then the aea of each a pe ou model i a = b. Fo a eal moto x, and a ba will x ba be diectly meauable and we can take the effective poduct c b eff eff a x ba =. 2π It i quite eay to genealize the model to abitay fame. Let u view the moto fom a efeence fame otating anticlockwie at angula velocity β. Then the appaent angula velocity of the oto will become ω-β and the appaent angula velocity of the tato will be β. The motional emf effect, a dicued in the analyi of Fig. 1, will now contibute to the tato alo. The efomulated (6), whee the abitay fame voltage and cuent vecto ae denoted by a upecipt e, will become ( + jn( )) 1 ( + jn( )) ( jn ) ( jn ) e K τ ω β δ τ ω β e (k+1) Kk = e e δ2τ β τ β K(k+1) Kk. (13) The tep leading to (12) fom (6) can be etaced to obtain the abitay fame dynamic a ( p jn( )) 1 ( p jn( )) ( p jn ) 1 ( p jn ) 1 + τ ω β δ τ ω β e K 0 = e e δ2τ + β + τ + β K V. (14) We mut now wite the mechanical equation fo the moto i.e. the one decibing the time evolution of ω. The fit thing to take note of i that an integation ove θ i equied whee the quantity to be integated i the poduct of two angula ditibution the oto cuent and the magnetic field at the oto uface. Uing the tandad identitie fo integal of poduct of tigonometic function, we get that when two ditibution ae multiplied and integated, the poduct of the epective d component will have a nonzeo contibution to the eult a will the poduct of the epective q component. The co tem, featuing d of one phao and q of the othe, will howeve evaluate to zeo. Hence fo abitay ditibution X and Y, 2π 0 ( Xdconθ + Xqinnθ )( Yd conθ + Yqinnθ ) dθ = π ( XdYd + XqYq ). (15) The quantity of the RHS of (503) look vey much like the dot poduct of the phao X and Y, hence the integal of the poduct of two ditibution can be witten a the dot poduct of thei epeentative phao. The oto toque i thu popotional to the dot poduct of the oto cuent phao and the magnetic field phao at the uface of the oto. The oto cuent K i detemined fom (460) and the field will have two contibution one fom the oto itelf and the othe fom the tato. Equation (433) implie that the oto field phao i othogonal to K, hence it will not contibute to the dot poduct. The only contibution will be fom K, and thi will be obtained fom (437). Hence the key tem in the expeion fo the toque Γ will be

9 Γ K µ 1 c j δ K 2. (16) Fom thi point on, it i a matte of caeful book-keeping to take all the facto of, h etc. into account. The econd tem of the dot poduct in (504) give the magnetic field and the fit tem i the uface cuent. Multiplication of thi by dθ give the infiniteimal cuent the dθ i aleady accounted fo but we have collected a facto of. Then the infiniteimal foce on the conducto i obtained fomi l B(no pace phao hee) which thow in a facto of h. And toque involve Fo yet anothe facto of i accumulated. Finally we have to include the π fom the RHS of (503). Thi will give u the magnitude, o now let u obtain the diection. By default, K and K ae in the +z diection, and B in +ρ. Hence, if K and K ae both poitive, the foce will be zˆ ρˆ = + θˆ and the toque will be ρˆ θ ˆ = + zˆ. Combining all thi we have 2 πµ cδ1 h z K Γ = 2 ( K ) j. (17) Thi fomula give the output toque of the given moto. III. DISCUSSION AND CONCLUDING REMARKS Thi pape obtain the moto dynamic tating fom fit pinciple. In addition to the dynamic tuctue we have povided theoetical etimate of the vaiou eitance and inductance involved in the modeling. Pediction of thee paamete i difficult [9] and they ae geneally detemined fom expeiment on the moto, o fom numeical technique. Let u compae the eult deived hee with the pediction of the accuate Finite Element Analyi (FEA) method. The claic pape i by Williamon [10] whee they have, among othe thing, plotted the flux denity inide the oto. Thei flux denity plot (Fig. 6) i epoduced below a Fig. 3. Ou model of the 2-pole moto teat it a a cylinde with uface cuent Kinθ and it i well known that uch a ditibution poduce a unifom field inide. Thi i vey cloe to the field obtained in [10]. Yet anothe plot of the magnetic flux, thi time outide the moto, can be found in [11]. Like [10], FEA i ued hee. The moto i 4-pole and the magnetic field hould be like that of a cylinde with uface cuent Kin2θ which i the deciption of the moto a pe ou model. The imilaity between thei figue 2 (epoduced a Fig. 4) and the quadupole field, (3) with n=2 i eadily appaent. A combination of analytical and numeical method may be found in [12]. Thee the time contant can be obtained by dividing thei equation (14) by thei equation (8) L b = µ l λ (14) of [12] 0 b ba R ρ l b b b = (8) of [12] Ab to get a eult popotional to the conductivity, pemeability and the co ectional aea of the ba [ee comment afte (12)], and independent of the ba length. Thi i in diect ageement with ou pediction, which in addition pecify a numeical value of the indeteminate λ ba. Ou puely analytical appoach ha thu yielded eady-to-ue etimate of the vaiou moto paamete which ae in good ageement with highly ophiticated imulation.

10 Ou analyi can alo be ued fo a theoetical tudy of paamete vaiation duing opeation. The main caue of uch vaiation i the change in tempeatue of the moto duing unning. Knowledge of the tempeatue dependence of the vaiou conductivitie and pemeabilitie will allow the intantaneou inductance and eitance to be pedicted by meauing the moto tempeatue. A tempeatue feedback loop may then be intalled in a vaiable fequency dive [13] to take cae of thi vaiation. Ou wok will alo enable a tudy of econday effect uch a kin effect in the oto conducto o hyteei and atuation in the coe. Studie of thee phyical phenomena can be dawn on to caue uitable modification of the vaiou tem in (4). Let u elaboate a bit on kin effect. A the lip fequency inceae, the effect become moe and moe pominent. The eitance will tend to inceae a the effective co-ectional aea of the ba educe. Simultaneouly, the time contant, which i popotional to the aea, hould educe. Thi i exactly what i found in [14]. Thei Figue 3 clealy how the inceaing eitance and the educing time contant with inceae in lip. Ou wok alo help in undetanding the electomagnetic inteaction going on inide the moto. Such a tudy ha been undetaken by Holtz [15] who ha tated fom a taditionally deived dynamic model (pace phao) and intepeted it in tem of the magnetic enegy popagation inide the machine. Ou poce i in a ene the evee tating fom the magnetic behaviou we have obtained the dynamic model. Such a deivation poce ha alo enabled u to examine the limit δ 1,2 <<1 which i valid in moto of high polaity. In uch a cae, the dynamical equation ae natually uncoupled, which i a conideable implification ove the geneal cae. Though caied out fo the pincipal hamonic, ou method i applicable to all patial hamonic of a paticula moto. The value of the vaiou paamete will be diffeent fo each hamonic, a pe Table 2, but the dynamical tuctue emain unchanged. A detailed Fouie analyi of the applied voltage wavefom can be caied out, including many patial hamonic of the field and cuent. (12) o (14) can then be applied on each hamonic fo a paticulaly accuate tanient analyi. Thee elation can be ued in high pefoming vecto contolled pule width modulato (PWM) dive. Finally, we note that ou analyi i applicable not jut to induction moto (of all phae the phae did not ente the calculation anywhee) but to all kind of electical machine. The baic tenet of electomagnetim, on which ou entie tuctue et, ae the ame fo induction, ynchonou and dc moto, geneato and tanfome. Hence ou method i univeal in cope and can be pofitably employed fo an inightful tudy of all the electical wokhoe of moden induty. * * * * * ACKNOWLEDGEMENT

11 I am gateful to KVPY, Govenment of India, fo a geneou Fellowhip. REFERENCES [1] R. Kihnan, Electic Moto Dive Modeling, Analyi and Contol, PHI Leaning Pivate Limited, New Delhi (2010) [2] R. H. Pak, Two eaction theoy of ynchonou machine a genealied method of analyi, Tanaction of the Intitute of Electical Enginee (1929) [3] H. C. Stanley, An analyi of the induction machine, Tanaction of the Intitute of Electical Enginee (1938) [4] G. Kon, Equivalent Cicuit of Electic Machiney, Wiley (1951) [5] P. K. Kovac and I. Racz, Tanient behaviou in electic machiney, Velag de Ungaiche Akademie de Wienchaften (1959) [6] I. Takahahi and T. Noguchi, A new quick epone and high efficiency contol tategy of an induction moto, IEEE Tan. on Ind. Appl. 22 (5), (1986) [7] J. Holtz, The epeentation of AC Machine dynamic by complex ignal flow gaph, IEEE Tan. on Ind. Elec. 42 (3), (1995) [8] J. L. Kitley, Electic Machine, Maachuett Intitute of Technology : MIT OpenCoueWae (2005) [9] S. J. Chapman, Electic Machiney Fundamental, Fouth Ed. Mc Gaw Hill New Yok, USA (2005) [10] S. Williamon and M. C. Begg, Calculation of the ba eitance and leakage eactance of cage oto with cloed lot, IEE Poceeding 132 B (3), (1985) [11] M. R. Hachicha, N. B. Hadj, M. Ghaiani and R. Neji, Finite element method fo induction moto paamete identification, Poc Fit Intl. Conf. on Renewable Enegie and Vehicula Technology, [12] M. H. Gmiden and H. Tabeli, Calculation of two axi induction moto model uing finite element with coupled cicuit, 6 th Intl. Multi-confeence on ytem, ignal and device (2009) [13] D. Bae, D. Kim, H. Jung, S. Hahn and C. Koh, Detemination of induction moto paamete by uing neual netwok baed on FEM eult, IEEE Tan. on Magnetic 33 (2), (1997) [14] S. Bhattachajee, A modified cala contol tategy of an induction moto with application in taction, IAEME Intl. J. Elec. Engg. Tech. (IJEET) 3 (2), (2012) [15] J. Holtz, On the patial popagation of tanient magnetic field in AC machine, IEEE Tan. on Ind. Appl. 32 (4), (1996)