Transient and Static Modeling of Tubular Linear Induction Motors

Transcription

1 Proceengs of the 5th WSEAS nt. Conf. on Power Systems an Electromagnetc Comatblty, Corfu, Greece, August -5, 5 (55-557) ransent an Statc oelng of ubular near nucton otors EZA HAGHAA(,) ABBAS SHOAE() () () Deartment of Electrcal Engneerng mam Hussen Unversty Babae Hghway, ehran AN Deartment of Electrcal Engneerng ran Unversty of Scence echnology Narmak, ehran Abstract: - hs aer s concerne wth the transent moelng of tubular lnear nucton motors wth blocke rotor n the assgnable resectve ostons of rotor an stator. he motor s a travelng-wave generator- rven tye. hs work s base on calculaton of self nuctances of a lnk an a col, mutual nuctance an mutual nuctance graent of coaxal cols wth current flament metho, assumton of the rotor consttute of some rngs, an current calculaton of stator cols an rotor rngs wth resect to tme. We have wrtten some comuter coes for calculatng nuctances an analyzng the erformance. Fnally the nstantaneously force ale on rotor s calculate. Keywors: - Ar Core ubular near nucton otor, Colgun, nucton auncher ntroucton ubular near nucton otors (s) can have unque alcatons as electromagnetc launchers wth very hgh see an acceleraton abltes, an controllablty []. As all motors, a conssts of two man arts: statonary (stator) an movng (armature here) members. he motor sulyng s by stator whch conssts of several coaxal cols an number of them s etermne wth resect to number of hases, oles, an sectons. Usually t s necessary that we esgn the cols n such a way that coul be changeable nvually []. When analyzng the motor by current flament metho, the rotor whch s usually a conuctve an thn crcular layer, eene to requre accuracy, length an thckness of rotor s ve to several current sectons, calle as rngs. nks of the stator cols an rngs of the rotor are assume as crcular an arallel lnks wth unform current enstes, calle as current flaments. utual nuctance equaton between two current flaments s []: µ ab m K m m Where ( m ) E ( ) (١) a raus of the frst flament b raus of the secon flament axal stance between two flaments 4ab m ( a + b) + K (m) an E (m) are frst an secon tyes of elltcal ntegral. nuctance Calculatons. Self nuctance of a nk For calculaton of a lnk self nuctance, we assume the lnk s consttute of n flaments wth the same current, an use the metho of current flament mutual nuctance. For carryng out ths calculaton, we assume a lnk s consttute of three flaments (Fg. ), an gve the self- nuctance of a lnk as below: oltages of all three flaments are equal; the voltage of the frst flament s equal to: v + + t t Wth regar to the assumton: t ()

2 Proceengs of the 5th WSEAS nt. Conf. on Power Systems an Electromagnetc Comatblty, Corfu, Greece, August -5, 5 (55-557) hen v ( Fg. : a nk Consttute of hree Current Flaments + + ) t eq t So eq j j An wrtng the voltages of flaments an, eq j eq j j j hen the equvalent nuctance woul be eq ( eq + eq + eq ) j j () (4) (5a) (5b) (5c) (6) Assumng a lnk consstng of n flaments, one can n n eq j (7) n j Calculatng mutual nuctance of a flament wth tself, usng current flament metho, one obtans an nefnte value [4]. So for calculatng self nuctance of each lnk, we calculate the mutual nuctance of each flament only wth ajacent able : esults of the Coe, ef [5] an [6] Col Dout Dn nuctance (mh) nks No. (mm) (mm) (mm) [5] [6] coe flaments. n ths case, self nuctance of a lnk woul be calculate from the below equaton: n n j j (8) n n ( ) j. Self nuctance of a Col Assumng a col consste of three lnks (Fg. ), the voltage an current relaton of col s v v + + v v + + t t t t t t t t t j eq j t t Assumng a col wth n lnks, one obtans n eq j n j (9) () Obvously, nuctance calculaton of a col requres calculatons of self nuctance of lnks as well as mutual nuctance between each two lnks. Comrsng the results of the wrtten comuter coe n our work, results of the reference [5] an results of the reference [6], for self nuctance calculaton of a col wth table characterstc, our coe was verfe (table ). Wth regar to the coe format, col arrangement was assume to be 4 layers an 6 lnks n each layer ( 4 6 4), col arrangement was assume to be 4 layers an 44 lnks n each layer ( ), an actual number of lnks was taken nto conseraton by alyng a sutable coeffcent. Fg. : a Col wth hree Seres nks Fg. : an Ar-Core Cylnrcal Col Dmensons for Calculaton of ts nuctance [6]

3 Proceengs of the 5th WSEAS nt. Conf. on Power Systems an Electromagnetc Comatblty, Corfu, Greece, August -5, 5 (55-557) he reference [6] calculates the self nuctance of a tubular ar-core col(fg. ) usng number of lnks, N,mean ameter,,length,,thckness, C, as Where s n meters an Φ.45 + α + l α ΦN ( µ H ) ().π ρ + ( ) αρ[( α + ) /( α + )] c an ρ (). utual nuctance of wo Cols We begn nvestgaton of mutual nuctance of two cols by a col of two lnks an other col of three lnks (Fg. 4). he voltage of the two lnks col s : v v + v t t t t t t t t t t ( ) t ( ) t t t () Consequently, mutual nuctance of a col of n lnks an a col of n lnks s obtane by the equaton of n n+ n j j n+ (4) Currents of Cols an ngs For an ntal esgn of the requre comuter coe, a motor wth 6 cols n stator an rngs n rotor was assume (table an Fg. 5). utual nuctance matrx of cols an rngs s : (5) Where nuctance matrx of rve cols nuctance matrx of rotor rngs mutual nuctance matrx of cols an rngs able : he Smulate otor Characterstcs Stator Outut raus of cols (mm) 45.5 aal thckness of cols (mm) 4.9 ongtunal length of cols (mm) 7. ayers number of cols 8 nks number of each layer 4 Dstance between each two cols (mm).9 Dameter of cuer wre.7 Number of cols 6 otor Outut raus (mm) 9.5 aal thckness (mm) ongtunal length (mm) 45 materal Alumnum Suly oltage (S) (v) Frequency (Hz) 5 Fg. 4 : a Col of wo nks an Another Col of hree nks Fg. 5 :a otor wth 6 Stator Cols an otor ngs

4 An voltage relaton of cols an rngs s (6) + N N N N N r r v Where v voltage vector of rve cols current vector of rve cols current vector of rotor rngs r resstance matrx of rve cols r resstance matrx of rotor rngs n summary (7) + Cols currents are relate to hases currents as: (8) h S ] [ Where h s the vector of hase currents an s calle transformaton matrx. Generally, vector of cols an rngs currents,, s relate to vector of hases an rngs currents as : (9) ]' ][ [ S O One can wrte from equatons of (7) an (9) () () ultlyng left ses of equaton () by [] t t t () () herefore state equatons governng on the motor are : { } (4) Where []' s voltage vector of hases an rngs : (5) 8 7 S By solvng equaton (4), one can calculate hases an rngs currents. hs soluton requres erformng [], [] an then [] an [] matrces. hese equatons have been solve by ung-kutta metho an the results for hases an rngs currents are reresente n Fg. 6. As you seen. Phases S, currents have transent states, an rngs currents are strbute unequally as a result of en effects. Fg. 6: Phases an ngs Currents 4 Force Calculatons Calculaton of force ale to the rotor s base on mutual nuctance changng wth resect to longtunal slacement between stator cols an rotor rngs. For more calculaton see, one can frst calculate sum of mutual nuctances between Proceengs of the 5th WSEAS nt. Conf. on Power Systems an Electromagnetc Comatblty, Corfu, Greece, August -5, 5 (55-557)

5 Proceengs of the 5th WSEAS nt. Conf. on Power Systems an Electromagnetc Comatblty, Corfu, Greece, August -5, 5 (55-557) flaments of two cols, an then obtan ervaton of the sum wth resect to longtunal slacement. 4. utual nuctance Graent ongtunal force between two current flaments s [7] : FZ P (6) Z A coe was wrtten for calculaton of mutual nuctance an graent of mutual nuctance between a col an a rng for a range of longtunal ostons whch results of t were obtane for the col an rng wth reference [5] secfcatons (Fg. 7)an t s seen a goo agreement between ths fgures an fgures of reference[5] (Fg 8). Smulaton results also emonstrate that assumng a constant thckness an length for col, the arrangement of lnks oes not affect the results for cols self an mutual nuctances. able shows the mutual nuctances for several arrangements of table cols. N refers to layers number an N refers to lnks number n each layer. he mutual nuctance calculate by [5] for axal stance of zero s 685 uh. Fg. 7: utual nuctance an utual nuctance Graent Between Col an ng (coe results) able : utual nuctances of several arrangements of table cols N N utual nuctance Col 5 Col uh Col 4 6 Col uh Col 4 Col uh 4. nstantaneous Force Wth regar to Fg. 5, nstantaneous force ale on rotor wth rngs s f f + + f 7 + f f + f + f + + f 7 + f8 + f, ,8 6,8 7 8, Z Z Z 7 8, (7) Z Z Z Fg. 8: utual nuctance an utual nuctance Graent Between Col an ng[5] An n matrx form f [ ] An n summary 6 z z 7 67 z z,8 6,8 7 8 (8)

6 Proceengs of the 5th WSEAS nt. Conf. on Power Systems an Electromagnetc Comatblty, Corfu, Greece, August -5, 5 (55-557) f [ ] [ P ] Z (9) So, t s necessary to obtan /z. nstantaneous longtunal force obtane of a wrtten coe for a motor wth table characterstcs s seen n Fg. 9. After a transent ero, the force reaches to ts steay state value wth a snusoal form an a constant mean value. As t s seen, the force at ntal nstants s negatve. Also, a coe was wrtten for mean value calculaton of force ale on the rotor at steay state, an was run for the motor wth table characterstc an 4 rotor rngs at several longtunal ostons of rotor, an Fg. was obtane. We see that the maxmum mean force ale on the rotor s obtane when rotor s comletely nse the stator, an the force becomes nearly zero when the rotor les entrely outse the stator. Smulaton results showe that: ) assumng mnmum 4 rngs for rotor s necessary for obtanng suffcent accuracy, ) ncreasng rotor thckness an changng rotor materal of alumnum to cuer result n ncrease force, ) n ths level of voltage, the temerature effect s gnorable. Fg. 9: nstantaneous ongtunal Force Ale on otor Fg. : ean alue of Force at Gven Postons 5 Concluson Assumng a constant thckness an length for col, the arrangement of lnks for rvng the number of col lnks oes not affect the results for cols self an mutual nuctances. he coulng between stator an rotor s very low, so that three hases currents when rotor s comletely nse the stator s aroxmately as equal as the currents when rotor s comletely outse the stator. ransent state of force s very short, an n the steay state, force has an alternatng form. he maxmum mean value of force on rotor s obtane when rotor s comletely nse the stator, an the force s nearly zero when the rotor s comletely outse the rotor. Conserng mnmum 4 rngs for rotor s necessary for suffcent calculatons accuracy. ncreasng rotor thckness as well as changng rotor materal from alumnum to cuer ncreases the force. At low voltage levels, the effect of temerature s nsgnfcant. eferences: []. Haghmaram, A. Shoulae, nucton Col Electromagnetc aunchers, Proceeng of the n nternatonal an 5 th Natonal Conference of ran Socety of Ar an Sace, sfahan, ran,, Persan. []. Haghmaram, A. Shoulae, terature evew of heory an echnology of Ar-Core ubular near nucton otors, Proc. 9 th nternatonal Unverstes Power Engneerng Conference, UK, 57-5,4. [] B. Azzerbon, E. Carell,. aug, A. elln," Some emarks on the Current Flament oelng of Electromagnetc aunchers, EEE rans. ag., vol. 9, Jan. 99, [4] A. Hemmat, Dynamc moelng an best esgn of fast tubular lnear nucton motors, PhD hess, ran Unversty of Scence echnology, ran,, Persan. [5] A.. Shayeste Far, Stuy an Smulaton of Pulse ubular nucton otor,.s hess, ran Unversty of Scence echnology, 996, Persan. [6] K. horborg, Power electroncs, 9, secton. [7].. tajan, Stuy of Electromagnetc Force Ale to Conuctors an Current Carryng Elements,.S. hess, ran Unversty of Scence echnology, 998, Persan.