Calculation of fields of magnetic deflection systems with FEM using a vector potential approach - Part II: time-dependent fields
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1 Avlle ole Pyss Poed (8) Poeedgs of e eve Ieol Cofeee o Cged Ple Os Clulo of felds of ge defleo syses w M usg veo oel o - P II: e-deede felds. lse*. l d P. H Vse leo Be GH Göswe ße Je Gey Reeved lseve Julyuse 8; oly: eeved Reeved de evsed ee; fo evsed July de ee; 8; eed de July ee 8 As A oedue s eseed fo lulg e-deede felds ge defleo syses (eselly sddle ols) oolly sye suoudg ossg of els w y eely d oduvy usg e M eod d veo oel o. e veo oel d e ue dsuo e exded s oue sees w ese o e ul oode. Cosequely e o e dled s see wo-desol ole. e kwd ule eod s used fo e ego. Coeos of e lol M equos e luled w oge fo e e deedee. e glol M equos ode w e oesodg equos e soy se. 8 lseve B.V. Oe ess ude CC BY-NC-N lese. PAC: ; 8.7.x; 47..g Keywods: leo os; efleo syse; ddy ue; elg e; e elee eod. Ioduo Mge defleo syses e used deves su s eleo e logy syses. e e-deede feld of ese defleos ees eddy ues odug els su s ole ees of leses. e selg e of ese defleos w s used y eddy ues s of ul ees se fluees ougu. e-deede ee-desol feld oes w e ge veo oel ve ee desed elsewee [4] d oelly vlle M sofwe e used o sule ee-desol eddy ue effes. Howeve e fully ee-desol o s vey e-osug d eques lge ou of oue esoues. I e sef se of defleo oled ge les suoudg w seldg fees e. e ee-desol o s ely ossle. I fs e of s sees [] veo oel o ws eseed o lule soy felds of ge defleo syses feoge suoudgs usg e e lee Meod (M). Aloug s * Coesodg uo. el.: l ddess: os.else@vse-se.o do:.6/.o.8.7.4
2 58. lse e l. / Pyss Poed (8) lse e l. / Pyss Poed (8) o s lso useful fo e ouo of soy felds e ul eso fo usg e veo oel ws e lo o e-deede felds. e ge sl oel o ([] []) sule fo lulg soy felds o loge e eloyed fo e-deede felds euse eddy ues e eed odug els d ul H s o loge eo eddy ue egos. eefoe e e-deede se e veo oel s o e used ougou. I s e we e foussg o e e-deede felds of sddle ols. Howeve e se of oodl ols e dled slly. ouely eleo os oly few feld os e of ees d eefoe e ee-desol ole e eled y few wo-desol oles. e o e ge veo oel d e ue dsuo e exded o oue sees w ese o e u gle of e yldl oode syse e oue o of e dled s see wo-desol M ole. eefoe oly few wo-desol M oles ve o e solved sed of u oe oled ee-desol M ole s log s e dsuo of feoge d/o odug els s oolly sye (see [] [] fo e ge sl oel se).. egy fuol fo e veo oel foulo e Mxwell equos fo e-deede ge feld e ul H B ul dv B dv () ogee w e el equo B H. () As usul e qus-soy oxo e dslee ue s egleed ( ). Moeove we ssue ee e o fee ges ( ). e ue desy e we s e () wee s e sefed ue desy e ols d e s e eddy ue desy dued oduos y gg ge felds. e le s gve y (4) e wee s e ele oduvy. e d of equo () s ssfed w e s B ul wee s e ge veo oel. Iseg (5) () gves: o (4) d (6) we ge: (5) (6)
3 . lse e l. / Pyss Poed (8). e (7) Usg () () (5) d (7) we ge e dffuso equo:. ul ul (8) quo (8) e deved s ule-lgge equo fo eegy fuol dv () og e Lgge desy. ul (). e ego oedue I fs se fo e ego we eloy e kwd ule eod ([5] [6]). Le e e sequee of e ego os d e e se se. e ge veo oel e e e we s: () olvg () fo d usg (8) gves e e-dseed dffuso equo:. ul ul () quo () s uely se-lke dffeel equo (.e. os oly sl devves w ese o e oodes ( )) fo e veo oel e e ovded e veo oel e e s ledy kow. e Lgge desy ow kes e fo: ul () I e followg we use yldl oodes ( ). I ese oodes () e we s: ) ( ) ( ) ( (4) e ow exd e ooes of e veo oel oue sees of os e ul oode.e. s os ) ( 5... (5). lse e l. / Pyss Poed (8)
4 4. lse e l. / Pyss Poed (8) wee Geek des... e ug oug e oodes ( ) ougou e e. I e se of del defleo syses oly odd oue os oue o e exso (5). I e followg we ofe ouselves o e se of ge sddle ols loug e ee of oodl ols e dled slly. As sow y Muo d Cu [] d Leov e l [] e ue desy of sddle ol e deved fo sgle fuo w s o-vsg oly sde e ol wdgs.e.. (6) I s ow sgfowd o follow e oedue ouled ou fs e of s sees [] o deve e eegy fuol of e o. Iegg ove d usg e ooolo elos of e gooe fuos we flly o e exesso dd gf f g (7) wee e ge of ego exeds ove e wole ego e --le osdeed. e ls u oe e e es e ewe y usg ego y s. e defo of e fuo g d e oeffes f e gve []. e oeffes f o e ue I sde e ol e we kess R d e segle of e sddle wdg. 4. seo of e - le d lol M equos e ule e fs ode M eod (OM) d sudvde e ego e --le o qudlels. qudlel s sudvded o lef ue gle d g lowe gle see g. []. e qudlels e ueed y = e = e wee e s e ue of es os deo d e s e ue of es os deo. e deoe y lowe e oxe exesso of e eegy fuol of e oue o eged ove e g lowe gle of e qudlel ) (. (o lef ue gle e eegy fuol s ue.) I s ow g sgfowd o follow e dseo oedue ouled [] o deve e oxe exesso fo e eegy fuol eged ove e g lowe gle of e qudlel 6. lse e l. / Pyss Poed (8) 57 64
5 . lse e l. / Pyss Poed (8) 5 lowe e es soy (8) (slly fo e lef ue gle) wee soy es e deoes e es ledy deeed fo e soy se [] ke e. e d e e vlues of e ooes of e veo oel of e o e o (ode) (= ) of e gle osdeed. e lulo of e lol M equos follows e oedue desed []. e lol M equos sg ow e vlue of lowe ges f e oel e oes of e gles ges e lowe lowe lowe lowe lowe lowe lowe lowe R G H () (slly fo e lef ue gles). e oeffes e gve y es soy es soy es soy es soy es soy es soy lowe lowe lowe lowe lowe lowe () (ll oe lowe lowe lowe lowe R H e uged oed w e soy se) d. lse e l. / Pyss Poed (8)
6 6. lse e l. / Pyss Poed (8) lowe lowe lowe lowe lowe lowe es soy es soy es soy es soy G es soy G es soy G () (slly fo e lef ue gles). oy es es we ve o se ee e es ledy luled [] fo e soy feld. e lol M equos ( ) e e ss fo e M sofwe. Ovously e lol M oeffes ve e syey (s e soy se): lowe lowe lowe lowe lowe lowe lowe lowe R R H H () s s o euse uses e x of e glol M syse of equos o e sye (see ex seo). Hee s we e e of e gle () d e e oode dffeees () d d e e oodes of e oes of e gle. deoes e oodes of e eod of e gle. 5. Glol M yse of quos e Mxwell equos e fulflled f e eegy fuol s ed. s leds o le syse of equos fo e ukows. ese equos e we s (4) wee deoes o e fe elee es see g. [] d e e e eegy fuols of e 6. lse e l. / Pyss Poed (8) 57 64
7 . lse e l. / Pyss Poed (8) lse e l. / Pyss Poed (8) 7 de gles 6 of. e oel ooes d ( = = e*e) e ged e soluo veo s follows:... e* e e* e e* e e* e e* e e* e A sgfowd lulo s desed [] yelds e se exessos fo e x of e le syse of equos M 6 6q d e g d sde R6 ( q = e*e deoe e o e fe elee es = oesod o e os d s es = deoe e ooe of e veo oel es o). (5) 6. Alo As vey sle lo g. sows sddle ol sde ell ylde. e sddle ol s dus of e ell ylde s dus of d s k. e elve eely of e ylde s s oduvy s 65. A/V/ (oe). A e = e ue e sddle ol s seously swed off. e feld o e xs e ee of e defleo syse follows selg uve. e selg uve ws luled y wo eods. g. sows e selg uve deeed w e M sofwe sed o e eod desed s e. o oso e selg uve ws lso luled usg fouls fo ef. [7] (see g. ). o s se defleo syse d ylde of fe leg e ssued. e /e deese of s e fs se s of que good geee w e 4 s e le se. Codug Cylde ddle Col g.. Mge sddle ol suouded y ell ylde. () () g.. elg uves fo e ge sddle ol suouded y ell ylde deed g.. () elg uve luled w e M sofwe odg o e eod desed s e. () elg uve luled w fouls fo Ref. [7]. o eleo e logy e e ul e feld s deesed o oe llo of s l vlue s of
8 64. lse e l. / Pyss Poed (8) lse e l. / Pyss Poed (8) ees euse e yl feld defleo s o e ode of gude of w desed uy of. s oly e eved w ele selg e f sevel eods e oed.e. lyg fees s well s feug e defleo ol w ddol ols fo (l) oeso of e oue feld [8]. I s ovous e M eod lso e led o s oe oled gee. e ouo e (Peu GH esol oue) fo lulg e e-deed feld of el les w e ses s oxely. s s oe wo odes of gude fse e se of oveol () M lulo. 7. Coluso e s fouls fo e deede felds oduve els ve ee develoed o e leeed M-og. e eduo fo e -se o -se gve d dvge oug e. (A fo of ws eved e ses of ees.) e eduo s ossle euse fo eddy ue lulos oly e fs o s of ees eleo ol ools (e.g. eleo e de we). Usg dffee ses of els offes e oouy o ly e M og o fl leses wee e ole ee s ufued fo sof o d seld of fees s used. e M sofwe ws used o desg ew olu fo eleo e logy []. Akowledgees e wok fo s e s of e MA+ Poe 7 65 CMO eg dled ude e g ue M 47B d s fuded y e BMB (Ge Msy fo duo d Rese). Refeees []. lse. l d P. H Poeedgs of CPO7 (fs e of s sees ulsed s volue). []. Muo d H. C. Cu Ok 6 (8) 7. [] B. Leov M. Le d K.. v de Ms J.V..eol. B7 (8) 846. [4] M. Klee d. Rege I sos o Mges 8 () 5. [5] R. Bek Kod-Zuse-Zeu fü Ifoosek Bel e C -4 (). [6] [7] H. Kde elsoee de Neek ge (5). [8] K. Kslk e l. x. e. Pys. 8 (8) 45. [] P. H e l. Resoluo ee fo Vle-ed Be es Moeleo geeg 84 (7) 774.
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