Calculation of electromotive force induced by the slot harmonics and parameters of the linear generator

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Calculation of lctromotiv forc inducd by th lot harmonic and paramtr of th linar gnrator (*)Hui-juan IU (**)Yi-huang ZHANG (*)School of Elctrical Enginring, Bijing Jiaotong Univrity, Bijing,China

1 Calculation of lctromotiv forc inducd by th lot harmonic and paramtr of th linar gnrator (*)Hui-juan IU (**)Yi-huang ZHANG (*)School of Elctrical Enginring, Bijing Jiaotong Univrity, Bijing,China , (**)School of Elctrical Enginring, Bijing Jiaotong Univrity, Bijing,China , Ky word: linar gnrator, lctromotiv forc (EMF), lot harmonic, ritanc, inductanc Abtract: Th linar gnrator ud in magntic lvitation vhicl i invtigatd in thi papr. Firtly, th principl of th linar gnrator ar introducd, thn th formula of lctromotiv forc (EMF) inducd by th lot harmonic in th airgap and th formula of armatur ritanc and inductanc ar dducd, and th dign principl in which th pha of EMF ar matchd ar prntd. Finally, th EMF, armatur ritanc and inductanc ar calculatd uing tho quation.. Introduction inar ynchronou motor ha bn paid grat attntion a on of nw actuator in th magntic lvitation vhicl bcau of it high fficincy, nrgy aving and o on. Whn th vlocity of th long tator linar ynchronou motor ud in magntic lvitation vhicl xcd om valu, th linar gnrator can provid th xcitation currnt of th linar ynchronou motor. Th linar gnrator armatur winding ar placd in th lot of th pol-ho of th linar ynchronou motor, a hown in Fig.. Whn th magnt pol of th linar ynchronou motor ar moving, th lot harmonic from th fild winding of th linar ynchronou motor induc th altrnating lctromotiv forc into th linar gnrator armatur winding. Th ingl-pha bridg rctifir to upply th xcitation currnt can convrt th altrnating currnt into dirct. Thi papr propod th baic knowldg of linar gnrator ud in magntic lvitation vhicl. Fig. Th principl of linar gnrator On th firt tag, th formula of EMF in th armatur winding of linar gnrator ar dducd by thortical invtigation. Thn th dign principl in which th pha of EMF ar matchd ar obtaind. Finally, th formula of armatur ritanc and inductanc ar prntd for th armatur winding 37

2 of linar gnrator.. Th lctromotiv forc (EMF) Th typ of tator winding of th linar ynchronou motor i thr-pha ingl- layr winding. If th thr-pha ymmtric currnt in th winding of th linar ynchronou motor ar i i A i C B = = = I coωt I co( ωt ) I co( ωt 4 ) Thn th rulting th -harmonic componnt of magntomotiv forc (MMF) in th airgap () f i π f ( x, t) = f A + f B + f C F co( ωt + x) () τ whr f f f ar th -harmonic componnt of MMF of th A B C pha rpctivly. A B C Th -harmonic amplitud of MMF qual to th fundamntal MMF amplitud divid th numbr of harmonic (i.. F = ). Notic that, =5 or 7 and = or 3. F / Fig. how th location of conductor,,3,4 in th fild of 5 th and 7 th pac harmonic whnψ =. Fig. Th location of conductor,,3,4 in th fild of 5 th and 7 th pac harmonic whnψ = At thi tim, th EMF of pha A i maximum, aψ =, and th currnt of pha A i maximum. Th pak of th fundamntal MMF wav i locatd at th axi of pha A, and alo i locatd at th q axi. Th pak of 5 th and 7 th pac harmonic MMF i locatd at th q axi. St th lot harmonic of =5 or 7 a an xampl. Aum that th angl(ψ ) btwn. I and 38

3 . E of th linar ynchronou motor not qual to zro, i..ψ. So th angl btwn th fundamntal MMF and q axi iψ lctrical dgr, and th angl btwn th total MMF of =5 or 7 and q axi iψ lctrical dgr. A a rult, th EMF inducd by th =5 or 7 lot harmonic in th conductor,,3,4 of th linar gnrator will b ( ) ( ) 3( ) 4( ) E E E E co( ω t + ψ θ co( ω t + ψ θ co( ω t + ψ θ co( ω t + ψ θ ( ) ( ) ( ) ( ) ) α ( ) ) α ( ) ) 3α ( ) ) (3) Th conductor ar connctd in ( 3) ( 4) or in ( ) (3 4),th total EMF will b 4 nc E k y k q co( ω t + ϕ ) (4) 3 9 whr ϕ = ψ + θ( ) α ( ). Whn =5, quation (4) i poitiv. Whn =7, quation (4) i ngativ. Th EMF inducd by th = or 3 lot harmonic can b obtaind in th am way. 3. Condition of th pha of EMF ar matchd Aum that th pha of th 5 th harmonic EMF i matchd with th pha of th 7 th harmonic EMF, that i th um ofϕ 5 andϕ 7 mut qual to 8 dgr or th odd multipl of 8 dgr, i.. ϕ 5 + ϕ 7 = ( k + ) 8 (5) whr k=,,,3,. If θ rprnt th lctrical dgr from th q axi to th conductor, α rprnt th fundamntal lot pitch of th linar gnrator, thn θ (5) = 5θ, θ(7) = 7θ, α ( 5) 5α, α (7) = 7α =.Th quation (5) can b xprd ψ + θ + 3α = kπ () 3 Thi formula dcrib th condition of th pha of EMF ar matchd. Th condition of th pha of th th harmonic EMF and th pha of th 3 th harmonic EMF ar matchd can b obtaind in th am way. 39

4 4. Paramtr of th linar gnrator Th armatur dc ritanc pr pol r i r Z l z = ρt (7) NSa whrρ t i th conductor ritivity (ohm. m), Z i turn pr pha, l Z i th avrag half lngth of on coil (m), N i th paralll trand, S a i th conductor cro-ction ara (m ). Conidr th kin ffct in th conductor with altrnating currnt, th ac ritanc qual to K F tim dc ritanc, and th cofficint K F i K µ πfmnb 3ρb 4 F = + ( ) a (8) whr a b ar thickn and width of th conductor rpctivly (m), b i th width of th lot bottom (m), m i numbr of th ri conductor in th lot, n i th numbr of conductor in width, f i th frquncy of th currnt (Hz), μ =4π -7 H/m. If th conductor in lot hav n c turn, and th currnt flow in th conductor i i c, thn th armatur inductanc pr pol a i a Ψ ncφ = = = nc tlf µ (9) i i k δ c c c whr t i th width btwn th lot and th tth (m), l f i th ffctiv lngth of rotor cor (m), k c i a cofficint, δi th air gap lngth (m). Conidr th aturation of magnt circuit, th armatur ractanc pr pol qual to th fk tim th armatur inductanc, and th aturation cofficint K can b calculatd uing th finit lmnt mthod. Th armatur lakag inductanc pr pol i = Q () S = µ Z λ lf Q whr λ i th lot lakag magntic conductanc cofficint, Q i th pr pol lot numbr. Conidr th kin ffct, th ffctiv armatur lakag inductanc pr pol qual to K x tim, and th cofficint K x i 8 4 m 4 = ( ξ ) + ( ξ ) m 35 m 3 K x () whr bπfµ =. ξ αa, α = b ρt 4

5 Th armatur nd turn lakag inductanc pr pol i µ Z λ lf Q = () whrλ i th nd lakag magntic conductanc cofficint. If dfin l E πd ν = ym, whr y m i avrag pan pac of th coil in lot numbr, D i th p avrag diamtr of th winding nd, ν i nd cofficint, p i th pol numbr, thn th abov quation will b.5 E 7 = µ Z le =.5Z πp l p Uing th abov quation (4), (7), (8), (9), (), () and (3), th paramtr of th linar gnrator can b calculatd. (3) 5 An xampl of calculation Th data of th linar ynchronou motor ar hown in tabl. Th tabl how th pr pol rulting lctromotiv forc(mf), armatur ritanc and inductanc of th linar gnrator uing th abov quation, whn th opration tat of th linar ynchronou motor i f= 5Hz,I=A. Tabl th data of th linar ynchronou motor pol pitch (mm) 58 Slot width of tator (mm) pr pol pr pha lot numbr Tth width of tator (mm) pr pha turn Effctiv lngth of rotor cor (mm) fundamntal winding factor Rotor cor width (mm) 7 Pol numbr 3 Slot width of rotor (mm) Air gap (mm) Tth width of rotor (mm) Tabl th calculation rult ( 3) ( 4) ( ) (3 4) =5 or 7 = or 3 =5 or 7 = or 3 EMF (rm) ψ = 4.45V.9V.33V.8383V ψ = V.953V.833V.389V ac ritanc (Ω) ractanc (Ω) lakag ractanc (Ω) nd turn lakag ractanc (Ω)

6 . Concluion Purpo of thi papr i to propo th baic knowldg of th linar gnrator ud in magntic lvitation vhicl. Thortical invtigation about th formula of EMF in th armatur winding of linar gnrator ar dducd. Th dign principl in which th pha of EMF ar matchd ar obtaind. Th formula of armatur ritanc and inductanc ar prntd for th armatur winding of linar gnrator. Th calculation rult how that () th5 th and 7 th pac harmonic mf i largr than th th and 3 th pac harmonic mf whn th armatur winding connctd in ( 3) ( 4) () th diffrnc btwn th5 th and 7 th pac harmonic mf and th th and 3 th pac harmonic mf i mall whn th armatur winding connctd in ( ) (3 4), and th air fild i balanc, thi will ffct th lvitation forc of th magntic lvitation vhicl. Rfrnc. Huaqingiu, Zhiyi, EngngRn. Magntic lvitation vhicl in Grmany [M].Chndu, Elctronic tchnical univrity pr,995.. YunqiuTang, NaiShi. Elctrical machinry [M].Bijing, China machin pr,. 3. ShikunChn. Dign of Elctrical machinry [M]. Bijing, China machin pr, Nongniani, BaolingWang, RuhuangZhou. Dign of Elctrical machinry [M]. Bijing, Tinghua univrity pr,99. 4

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd