STUDY OF THE THEE-PHASE INDUCTION MACHINE UNDE DYNAMIC BAKING ALECSANDU SIMION, LEONAD LIVADAU, ADU-VOINEA COCIU, ADIAN MUNTEANU Key wod: Induction machine, Mathematical model in total fluxe, Tanient dynamic baking duty. The thee-phae induction machine that opeate unde the pecific condition of dynamic baking can be tanlated a an induction machine unde vaiable tate, tanient and unbalanced. Thi duty can be analyed by uing fit ode diffeential equation that contain nothing but the total fluxe of the phae winding and the otation angle of the oto. Conequently, the cuent ae excluded and thee i a educed numbe of vaiable. One of the mot impotant concluion of the tudy efe to the dynamic baking time, which i ignificantly influenced both by the initial moment when the d.c. ouce i connected to the tato winding and by the ouce polaity.. INTODUCTION It i a fact that the thee-phae induction machine have ome emakable qualitie a egad baking, which eventually lead to the top of the diven mechanim at a pecie pot when it come to poitioning ytem. The inceaingly utilization of induction moto i facilitated, among othe thing, by the poibilitie of an efficient baking with educed invetment. One of the method ued fo the induction machine baking i the o-called dynamic baking, o d.c. baking. [,, 3]. The next comment efe to thi baking pocedue, Fig.. If the machine ha cage oto (Fig. a) then the dynamic baking involve diconnection of the tato winding fom the main (K i opening the main cicuit) and almot intant connection of the K and K3 contacto. The tato winding i now connected to a d.c. ouce and the induction machine become a ynchonou geneato with eveed tuctue. The d.c. excitation i placed on tato, and the mechanical enegy fom the haft i tanfomed in electic enegy, which ubequently poduce themal effect upon cage winding. The toque developed now inide the machine ha an oppoite diection to the otation of the oto and Gh. Aachi Technical Univeity of Iay, Bd. D. Mangeon 5 53, 75, Iay, omania, E-mail: aimion@ee.tuiai.o. ev. oum. Sci. Techn. Électotechn. et Éneg., 58, 3, p. 73 83, Bucaet, 3
74 Alecandu Simion et al. conequently thee i a deceleation o, in othe wod, a baking. Fig. b how diffeent connection type of the tato winding coeponding to the initial connection: Y (b, b) and (b3, b4). The tudy peented in thi pape ue the b connection: two tato phae ae eie connected and upplied fom the d.c. ouce and the thid phae i hot-cicuited. L L L3 K d K T b ) +U U b ) +U U K4 a MA 3~ a) b M e c n K3 M M ML +U U b 3 ) b 4 ) b) +U U Fig. Dynamic baking of thee-phae induction machine: a) Schematic diagam, b) Connection type of the tato winding to d.c. ouce.. EQUATIONS OF THE THEE-PHASE INDUCTION MACHINE BY USING THE MATHEMATICAL MODEL IN TOTAL FLUXES The tudy by imulation unde tanient dynamic baking can be pefomed by uing togethe the 6 equation of the electic cicuit 3 fo tato (with the ubcipt a, b, c) and 3 fo oto (with the ubcipt a, b, c, accoding to eq. 6. fom [4]) and the movement equation (eq. 6.7 fom [4]): ( ) ( ) ( ) ( ) ( ) +νt ψ a = ua ν ψ b c + ν 3 σ a b c co 3 c b in ψ ψ ψ θ + ψ ψ θ ( +νt) ψ b = ub ν( ψ c a) + ( ν 3) σ ( a b c) co 3( a c) in ψ + ψ ψ θ + ψ ψ θ ( +νt ) ψ c = uc ν ( ψ a b ) + ( ν 3) σ ( a b c ) co 3( b a ) in ψ ψ + ψ θ + ψ ψ θ ( +νt) ψ a = ν ( ψ b c ) + ( ν 3) σ ( a b c ) co 3( b c ) in ψ ψ ψ θ + ψ ψ θ ( +νt ) ψ b = ν ( ψ c a ) + ( ν 3) σ ( a b c ) co 3( c a ) in ψ + ψ ψ θ + ψ ψ θ ( +νt) ψ c = ν ( ψ a b ) + ( ν 3) σ ( a b c ) co 3( a b ) in ψ ψ + ψ θ + ψ ψ θ + k J ω = p J p Λ inθ ψ ψ ψ ψ ψ ψ ψ + ( z ) ( ) ( ) 3{ [ a( a b c) b( b c a) ( ) 3co [ ( ) ( ) ( ) } ψ ψ ψ + θ ψ ψ ψ ψ ψ ψ ψ M c c a b a b c b c a c a b t () () (3) (4) (5) (6) (7).
3 Study of the thee-phae induction machine unde dynamic baking 75 θ, =ω = pω (8) ( ddt), whee the electomagnetic toque [4] i given by eq. (6.4), and whee the paticula value of the tato voltage coeponding to the d.c. upply ae: u a = + U; uc = U ; ub = ; ua = ub = uc =. (9) The tudy of the thee-phae machine with cage oto which opeate unde balanced o unbalanced upply ytem (including baking) can be pefomed with one of the poibilitie decibed in [4 7] by mean of the ymbolic method (Laplace). Aftewad, on the bai of a tuctual diagam, the analyi i pefomed in Libay: Simulink/Continuou. It i alo poible to ue othe diffeent oftwae envionment uch a Ppice [8]. All quantitie ue SI unit. The expeion of the hybid paamete, ν ij ae identical to thoe fom [4, 6, 7]. Both the pemeance, Λ 3 (H ) and the hybid paamete, ν ij ( - ), can be expeed in coelation with the meauable paamete of the machine: tato and oto phae eitance (, ), leakage and linkage inductance (L h =L h, L σ, L σ ), that i: LhLσ + 3LhLσ + L L σ σ LhLσ + 3LhLσ + L L σ σ ν t = ; ν t = ; 3LhLσ + 3LhLσ + Lσ Lσ Lσ 3LhLσ + 3LhLσ + Lσ Lσ Lσ LhL LhL σ σ ν = ; ν = ; 3LhLσ + 3LhLσ + Lσ Lσ Lσ 3LhLσ + 3LhLσ + Lσ Lσ Lσ () 3L 3L σ σ ν σ = ; ν σ = ; 3Lσ + 3Lσ + Lσ Lσ Lh Lσ 3Lσ + 3Lσ + Lσ Lσ Lh Lσ ν =ν ν ; ν =ν ν ; λ = 3 L + L + L L L. ( )( ) t t 3 σ σ σ σ h Heeinafte we deduce the equation that ae neceay to daw up the chematic diagam of a thee phae induction machine (TPIM) with impoed paamete (SI unit); = ; = ; L h =.9; L σ = L σ =.; J =.5; p = ; k z =.; ω = 34.. The hybid paamete have conequently the following value: ν t = 35.5 = ν t ; ν = 3.6 = ν ; ν = 3.4 = ν ; ν σ = 96.43 = ν σ ; λ 3 = 3.6. Thee value ae ubequently intoduced in (,,..., 8) to give the final equation ytem [6, 7]. Then, the applied voltage and the value of the load toque coeponding to a cetain equence of opeating dutie ae filled in [6, 7]. On the bai of thi lat equation ytem, the chematic diagam of the TPIM can be etablihed [7]. 3. SIMULATION ESULTS A) In thi ection we peent the eult obtained fo the vaiable egime V, denoted with the aconym P-SS-FD-D, pefomed in Matlab envionment. It contain the following dutie: (P) no-load tat-up of the cage moto unde ymmetical condition, t =. The load toque i epeented by the vicou fiction toque, which depend
76 Alecandu Simion et al. 4 popotionally to oto peed. It value aie to appox. 3 Nm, which coepond to ated peed. Thi toque act up to t =.5. (SS) toque enfocement with a value of Nm (.5 M en ). It i about an oveload duty within the ange (.5 t fd ), whee t fd.4 coepond to the moment of baking. (FD) dynamic baking (within t fd and t d, whee t d coepond to the diconnection of the d.c. ouce). The d.c. voltage applied to a and c phae ha the value U a = +6 V and U c = 6 V while the b phae i hot-cicuited. Simultaneouly, the load toque i diminihed time and it value attain 5 Nm (which i a eaonable value fo the pactical ituation when thee i a mechanical tep-down gea). The baking poce end when the peed become null (t d.53 ) but thi moment time i not clealy detemined ince it depend, among othe, on the poition of oto coeponding to the moment of the d.c. ouce connection. (D) diconnection of the d.c. ouce and hot-cicuit of the a and c phae (opening of K and K3 and cloing of K4) at the moment time t d.53 when the oto peed i null and the winding diipate the emanent magnetic enegy. The magnetic fluxe deceae in time and become null at t =.75. Fig. peent the time vaiation of the applied voltage on a phae. One can ee the peence of the hamonic voltage (pulation of 34. ad/) up to the moment of baking. At t fd =.4 the d.c. voltage ae applied to a and c phae (U a = +6 V, U c = 6 V). The oto deceleate to zeo value (t d.53 ) and aftewad the machine i diconnected fom the d.c. ouce and the tato phae ae hot-cicuited. Within the ange t d.53 to t f =.75 the machine i enegy-depleted and the magnetic enegy i themally diipated. 5 u a[v] 4 θ [ad] 8 6-5..4.6 4..4.6 Fig. Stato phae voltage, u a = f(t). Fig. 3 otation angle, θ = f(t). If the machine dive a poitioning ytem then the vaiation of the oto angle θ ove the full cycle i neceay (Fig. 3). Ove the initial aea, the value
5 Study of the thee-phae induction machine unde dynamic baking 77 inceae cuai paabolic. Then thee i a linea inceae coeponding to no-load opeation. The lope iing become malle fo oveload opeation and then thee i a quick deceae of it value duing baking. Finally, the oto top coepond to the pecibed poition. Fig. 4 how the vaiation of angula peed ove the full cycle, P-SS-FD-D. One can notice that the time of dynamic baking i of t fd =.3 and i imila in value to no-load tat-up time. Thi length depend obviouly on d.c. upply voltage. The vaiation of the electomagnetic toque (Fig. 5), put in view a ignificant toque value (appox. Nm) that aie when the d.c. ouce i connected and detemine the apid deceleation. Ω [ad/] 5 5..4.6 Fig. 4 oto angula peed, Ω = f (t). M e [Nm] 5 5-5 -..4.6 Fig. 5 Electomagnetic toque, M e = f (t). Fo cetain dive ytem, when the electic machine i placed at a long ditance fom the upply ouce and ha often tat-up dutie fo example, the input powe i a key paamete mainly fo the dimenioning of the powe tanfome. Fig. 6 how the vaiation of the input powe aco a phae and Fig. 7 how the vaiation of the total input powe, P t. One can clealy ee that the input active powe, P t, coeponding to the fit. i 3 time geate than ated output powe (7 kw up againt P N = 9 kw). The tanient poce can be alo analyzed by mean of dynamic mechanical chaacteitic, Ω = f (M e ) Fig. 8. The point opeation tat fom O point (oigin) and tack ucceively OMSNFQO (accoding to aow). It ha to be noticed that the connection of the d.c. ouce (t fd =.4 ) move the point opeation fom N to F whee the toque ha a much geate negative value then the pull-out toque and the baking poce i tonge. Then, the baking toque deceae cloe to zeo value (Q point) and keep a value of appox ( 5) Nm. Fig. 9 how the vaiation of the
78 Alecandu Simion et al. 6 output mechanical powe, P mec = f (t), which a expected become negative duing baking and i diipated inide the machine electic cicuit by themal effect. P a [kw] ( ) 3 5 Pt [kw] ( ) 4 3 -..4.6 Fig. 6 Active powe on a phae, P a = f (t)...4.6 Fig. 7 Total input powe, P t = f (t). 5 Ω [ad/] S.5 P mec [kw] ( ) F Q N.5 5 O M - M e [Nm] Fig. 8 Dynamic mechanical chaacteitic, Ω = f (M e ). -.5 - -.5..4.6 Fig. 9 Total output mechanical powe, P mec = f (t). The total oto flux of the a phae (Fig. ) ha a hamonic vaiation duing oveload duty with amplitude of. Wb and a fequency which i impoed by the lip. Duing the baking poce, the flux vaiation become apeiodic and it value deceae to zeo in a damped way. It i about the enegy diipation tage. Inteeting infomation concening the phenomena that take place inide the machine i offeed by the hodogaph of the eultant oto flux (Fig. ). The time evolution involve tack of diffeent cycle, denoted fom to 3: O-S, no-load tat-up; S-F, oveload opeation; F-D, dynamic baking till d.c. ouce decoupling; D-O, fee top and enegy diipation.
7 Study of the thee-phae induction machine unde dynamic baking 79.5 -.5 Ψ a [Wb] -..4.6 Fig. Total flux of a oto phae, Ψ a = f (t). Wb 4 9 O 3 - D 3 F 8 6 7 5 S Wb - - - Fig. eultant oto flux hodogaph. Finally, the cuent vaiation a deivative quantitie of voltage and total fluxe: i jk = ( u jk d Ψjk dt) jk ae peented in Fig. coeponding to i a, and Fig. 3 coeponding to i a. The tato cuent keep the fequency of the main fo no-load and unde load duty and have an amplitude that depend on the load facto. Duing baking poce, they become moe continuou and have ignificant amplitude. The oto cuent have much malle fequency value with vey low amplitude duing no-load opeation but with high value at oveload. Duing baking duty they become moe hamonic with a vaiable fequency that depend on oto peed. Thei amplitude i cloe in value with the tato cuent one. 8 6 4 - -4-6 i a [A]..4.6 Fig. Stato cuent along a phae, i a = f (t). 6 4 - -4-6 -8 i a [A]..4.6 Fig. 3 oto cuent along a phae, i a = f (t). B) The next eult coepond to a econd vaiable duty V, denoted with the aconym P-SS-FD*-D, which i athe imila with the above peented one. The diffeence conit in the invee polaity of the d.c. ouce, that i U a = 6V and U c = + 6V (Fig. 4).
8 Alecandu Simion et al. 8 A fit conequence i the ignificant inceae of the baking toque and conequently the deceae of the baking time. The otation angle of the oto, Fig. 5, i alo diminihed fom ad to ad. Thi phenomenon i viible on the time vaiation chaacteitic of the oto peed, Fig. 6. Now the baking poce top at t d =.475, which mean a length of appox. t' df =.75, that i only 6% fom the peviou baking time that coepond to V. At the ame time, the a and c phae ae diconnected fom the d.c. ouce and then hot-cicuited. Then, up to the t f =.75 when the imulation end, the machine diipate in a themal way the magnetic enegy. The vaiation of the electomagnetic toque i imila to the cae V in the ange t = to t =.4 (Fig. 7), but it ha a udden vaiation at the baking moment when it eache appox. Nm, that i twice the value of the peviou cae. Thi toque keep high negative value, which detemine a udden deceleation of the oto up to it top. 5 u a [V] 8 6 4 θ [ad] -5..4.6 Fig. 4 Stato voltage u a = f(t) V...4.6 Fig. 5 otation angle of the oto, θ = f(t) V. 5 5 Ω [ad/] - M e [Nm]..4.6 Fig. 6 oto angula peed, Ω = f(t) V. -..4.6 Fig. 7 Electomagnetic toque, M e = f(t) V. The vaiation of the mechanical quantitie with time can be analyzed by mean of the dynamic chaacteitic Ω = f(m e ) (Fig. 8). The opeation point
9 Study of the thee-phae induction machine unde dynamic baking 8 tat fom the oigin O, tack the OMM'SNFO cycle (accoding to the aow) and end again in oigin. It ha to be noticed that the connection of the d.c. ouce to the c and a phae, at the moment t fd =.4 and with invee polaity detemine a jump of the opeation point fom N to F whee thee i a negative toque of much highe value (appox. Nm). The baking effect i conequently much poweful. The value of thi toque deceae omehow in time but ha till mean value of Nm. Simila to V cae, ueful infomation egading the phenomena that take place inide the machine ae offeed by the hodogaph of the eultant oto flux (Fig. 9). The time evolution tat fom oigin, O, and tack the following cycle accoding to the numbeed aow ( to ): O-S (no-load tat-up vey cloe to ynchonou peed); S-F (oveload duty); F-D (dynamic baking up to diconnection of the d.c. ouce); D-O (fee deceleation and enegy diipation). 5 5 F Ω [ad/] S M' M O M e [Nm] - - Fig. 8 Dynamic mechanical chaacteitic, Ω = f (M e ), V. N Wb 5 8 O Wb 3 9 - F D 6 7 S 4 - - - Fig. 9 eultant oto flux hodogaph, V. One can ee that in contadiction to V cae, now the hodogaph i placed on a moe ditanced fom oigin aea duing the baking poce. Thi mean that the baking efficiency depend eentially by the value of the oto flux and it i popotional with them. Moe peciely, the vaiation of the eultant oto flux duing baking i fundamentally influenced by it initial poition and by the flux ceated by the two tato phae, which ae d.c. enegized. C) The dependency of the baking with the moment when the tato phae ae d.c. enegized can be tudied in a imila way with the peented analye. A a matte of fact, a diffeent moment fo d.c. connection implie a diffeent poition of the oto flux. The upepoition of the d.c. tato flux with the oto flux give now a diffeent evolution of the hodogaph, which detemine ditinct value of the baking toque and baking time.
8 Alecandu Simion et al. 4. CONCLUSIONS The tudy of the thee-phae induction machine unde dynamic baking duty, which i an unbalanced tanient duty, can be pefomed with high accuacy accoding to the popoed appoach. Unlike the uual appoach that can be found in cientific liteatue whee the tudy of the opeation of the induction machine unde diffeent dutie tat with two-phae model [, 3, 8 ], which i a model with etictive application, the popoed model allow an accuate analyi of diffeent pecial dutie, including fo example the d.c. baking. It i ued a ytem of fit ode diffeential equation that contain a quantitie nothing but the total fluxe of the winding and the otation angle of the oto. The phae cuent ae excluded and conequently the numbe of vaiable i educed. Thi new and oiginal appoach allow the obtaining of accuate infomation concening the intantaneou phenomena that take place inide the electic machine. On the bai of thi infomation one can adopt pope tategie concening the mot favoable moment time of connecting the d.c. ouce in ode to get a contollable baking time, uually the minimum time, which i impotant mainly fo tacking and poitioning ytem. A evaluation tool, the hodogaph of the eultant oto flux i ued ince it ha eal didactic advantage in the ight intepetation of the phyical phenomena that define the behaviou of the machine unde the mentioned duty and the time tacking of the opeation point on the dynamic mechanical chaacteitic. eceived on Januay 8, 3 EFEENCES. H. entzch, Handbuch fu Elektomotoen, 3, ubeab. Auflage, Bown Bovei and Cie Aktiengeellchaft, Mannheim, W. Giadet Velag, Een, 98.. I. Boldea and L. Tutelea, Electic machine Steady tate, Tanient and Deign with MATLAB, CC Pe,. 3. Kovak K. Pal, Analiza egimuilo tanzitoii ale maşinilo electice (tanlation fom Hungaian), Edit. Tehnică, Bucueşti, 98. 4. Al. Simion, Maşini electice III, Maşina ainconă, Edit. PIM Iaşi,. 5. Al. Simion, Study of the Induction Machine Unymmetical Condition Uing In Total Fluxe Equation, Advance in Electical and Compute Engineeing,, pp. 34 4,. 6. Al. Simion, L. Livadau and A. Munteanu, Mathematical Model of the Thee-Phae Induction Machine fo the Study of Steady-State and Tanient Balanced and Unbalanced egime, in: Induction Moto Modelling and Contol, InTech, ijeka,, p. 3 44; http://www.intechopen.com/book/induction-moto-modelling-and-contol/mathematicalmodel-of-the-thee-phae-induction-machine-fo-the-tudy-of-teady-tate-and-tanient-. 7. Al. Simion, L. Livadau,.-V. Cociu and A. Munteanu, Analyi of Thee-Phae Induction Machine Opeating unde Tanient Dynamic Baking by Uing the Mathematical Model in Total Fluxe, Simp. Nat. de Electot. Teoetică (SNET ), Conf. Poceeding, Dec.4,, Politehnica Univeity of Buchaet, pp. 38 43; http://net.elth.pub.o/net/volume/snetpoceeding.pdf.
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