Gerald Gerlach; Wolfram Dötzel. Solutions for exercises for Introduction to Microsystem Technology: A Guide for Students. Solutions for exercises

Transcription

1 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Solutions fo eeises Vesion fom 48

2 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Solutions to te eeises of pte 4: iofition Eeise 4 oing to ISO 86, te funtion of tolene ftos i is use to etemine te vlues of te si tolenes of qulities IT 5 IT 8 ges of si tolene fo te imensions of up to 5 mm Eqution i 45 D D is use to lulte tolene fto i Hee, D D D is te geometi men of te oesponing te imension nge n D, D e te nge limits o si tolene ge IT 5 IT 7, it pplies tt T IT5 7 i T IT6 i T IT7 6 i fo e suessive qulit, te si tolene ineses fto 6, eg T IT 6i o te imensions 6 mm, te following tle povies si tolenes fo si tolene ges IT 5 IT 7 s wl s tei popotion in peent of te te imension Te peentge in kets efes to te geometi men vlue D of te oesponing te imension nge It illusttes tt te popotion of te tolene pil ineses wit eesing te imension o te imensions lowe tn mm, tis tenen eomes stonge see tle n pitue Tle Bsi tolene ges IT 5 IT 7 n tei popotion in peent of te te imension N mm D mm i µm T IT5 µm T IT5 /N % T IT6 µm T IT6 /N % T IT7 µm T IT6 /N % Pitue Bsi tolene ges IT 5 IT 7 n tei popotion in peent of te te imension Tee is lge emn fo new evopments in te e of imensioning n tolenes s wl s test n mesuement tenolog fo mioomponents: Design of funtionoiente toleting sis espeill fo te imensions smlle tn mm, ontol of eisting s wl s intoution of new pmetes fo esiing mioomponent teistis eg imensions, fom evitions, wviness, ougness, tee-imensionl sufe pmetes, stutue-oiente pmetes, new mesuing sttegies n tenologies mesuing funtion sufes wit esolution in te nnomete nge, sot mesuing times, nume n istiution of mesuing points, ss fo tolenes, mesuing n ontol poeues Vesion fom 48

3 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Solutions to te eeises of pte 6: untion n om Elements in iosstem Tenolog Eeise 6 sing Eq 67, it pplies fo te eigenfequen tt N fe π m π N π o te given vlues of te etion n eigenfequen of te movle ement, it esults tt N 65 µm π f e Eeise 6 ppling Eq 655, it pplies fo te et flow tt moves ue to et onution fom te T mio plte to te fme tt Φ t, wit te teml esise fo et onution Rt eing s Rt λ Te two silion spings wit teml esises R t, R t t s pll iuit Totl teml Rt Rt esise tus eomes R ttot Due to R t R t R t,, te esult is Rt Rt t µ m R ttot R 78 K/W 6µ m 56 W/Km 6K Te esulting et flow is Φ t 7 mw 78K/W Eeise 6 Bulk miomining wit te osillto moving pepeniull to te wfe sufe: E E f e π m π 4l mlm ρ π 4l m lm ρ Wit f e S, it esults tt f e fe, ie tive tikness vition uses n ientil tive vition of te osillto s eigenfequen S is te solute sensitivit of te initil vlue f e in tion to pmete Sufe miomining wit te osillto moving ltell in tion to te wfe sufe: Tikness oes not ffet te osillto s eigenfequen s it is not inlue in te eqution fo f e : E E f e π m π 4l m lm ρ π 4l m lm ρ o te tive vition of te osillto s eigenfequen, it ten pplies tt fe sing te msk, it is possile to ve peis set te mesue fo f e Vesion fom 48

4 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Eeise 64 Te mping of movle miomenil ements n e ffete gs pessue, gs tpe, gp ise n te stutue of te osillting pltes pefotion, gooves If te plte moves pll to te fie sufe, te pessue in te i gp will not nge Te mping effet esults fom te gvit foes in te flui slie-film mping If te plte moves vetill to te fie sufe, pessue will t on te gs Te gs ptill espes toug te gp etween movle n fie plte n poues fition losses squeeze-film mping Espeill wen te gps e now n plte fequen is ig, pt of te gs nnot espe fom te gp n will e ompesse Tese two psil penomen poue foe on te plte wi n e epesente s omple epession Te omponent ts s mping k S, te imgin omponent s n itionl sping s, wi nges te sstem s stiffness Bot effets e te to pessue n fequen Te fequen of te osillting mss wen n imgin foe is ientil is lle ut-off fequen o fequenies ow ut-off te mping omponent is lge k eff k k s ω, fo woking fequenies ow ut-off, te stiffness omponent is ominting eff s ω Tpil plte imensions in te millimete nge n tpil gp ises of 6 µm poue ut-off fequenies ow khz o vetil plte movements, te effet of gses in now gps is esie s moleul o visous flow; tee is lso tnsitionl nge In oe to esie te tee nges, we use te KNDSEN nume, ie te tio etween gp size n men fee pt of te moleules see lso Setion 6 luii Elements Eeise 65 ppling te sping eqution, it esults fo te stiffness ting on te poe ll tt T m T, s, m Te spings e nge in seies: ot spings e loe te sme foe, te sping pts up T m T m Tis esults in T, s s s Wit, m T m s fo te glss fie n T m l s Sot, n l S S s S T ml fo te silion sping fom Sot,, it finll esults tt T m Sot, T, T m T m l l Sot, l Sot, Sot, Te following tle summizes te vlues esulting fo te sping stiffness of te glss fie n fo te effetive sping stiffness T, t te poe ll, se on te fie lengts 4 πe n imetes given in te tsk Te sping stiffness of te glss fie is Te 64 l sping stiffness ting on te poe ll tus eomes 4 πe ot T, see pitue 4 64 l πe l ot Vesion fom 48 4

5 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents 9 sing E 75 P, opts te following vlues: Tle Sping stiffness of te glss fie in N/m l 8 µm 4 µm 5 µm mm mm mm t ot 5-6 Nm, te following vlues esult fo T, : Tle Sping stiffness ting on te poe T, in N/m l 8 µm 4 µm 5 µm mm mm mm 4 68 Pitue Sping stiffness ting on te poe If te ttile foe fo s m s to lie in te nge of Tm 5 µn, using T m T,, te sping stiffness t te poe ll s to e in te nge of sm T, 5 5 N/m Teefoe, te fie lengts n imetes oesponing to te se es soul e tken into onsietion Te stti sgging of te senso ement ue to te silion plte s own weigt is G z stt Sz µ m Te own weigt of te glss fie n e neglete in tis lultion, it mounts to less tn % e Te eigenfequen of te senso ement in z-ietion is Sz f ez 44 Hz π m Eeise 66 Te ensit of te wte is ssume to e kg/m Te uli imete oespons to te imete of linil pill In tis se, te WEBER nume is 6 6 ρ u DH We 9 σ 7 P Vesion fom 48 5

6 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Te smll WEBER nume illusttes tt - on te mio-sle - te sufe tension is peominnt un n teefoe e use s n tuting piniple Eeise 67 Hee, te teisti geomet is te imete Teefoe, te KNDSEN nume is 8 λ 6 Kn D H Te Knusen nume lies etween n Teefoe, we will use te ontinuum mo wit te NVIER-STOKES eqution n gliing onition t te wll Eeise 68 Duing te opetion of te senso, te foe onvetion of te teml esise eeses esulting in eese of te teml time onst τ t t R t Eeise 69 Te tempetue t te lowe sie of te silion sustte in te nge of te ete stutue is illustte in te teml equivlent iuit fo te sttion se in igue 67 t te outgoing sie of teml esise R Si point ϑ lowe sie in te pitue o te sttion se, we ppl te simplifitions povie in Emple 66 Te totl effetive teml esise ten oespons to te seies onnetion of R O R Si, ie R t 56 7 KW - 6 KW - o supplie eting effet of P Het Φ 6 W, te esulting tempetue iffeene in tion to te ete tempetue is T P Het R t 5 K Pitue Teml equivlent iuit of te i flow senso Vesion fom 48 6

7 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Solutions to te eeises of pte 7: Sensos n tutos Eeise 76 Sufe n ne-sufe miomining often use pitive om stutues om sstem inteigitl finge stutue, inteigitl pito, om ive onsists of jent fie n movle finges, wi e esigne s epetitive stutue see pitue 4 Te lill epete setion is te ement l fie movle Pitue 4 om l om ls n e use fo sensoi n tuting funtion stutue eg fo foe genetion Te pitive sensitivities o effets, espetiv, lws eist in ll tee sptil imensions n ement l n ontin up to tee septe etill ontollle etoe es Te sping suspension efine in te esign will etemine te opetionl ietion In te opetionl ietion, te sping suspension s te lgest fleiilit In te ote sptil ietions, te suspension soul e s stiff s possile, in oe to eue te osssensitivities Opeting ietions n e oizontl in tion to te oienttion of te ige, oizontll tnsvesl to te oienttion of te ige o vetil Hoizontll move om stutues - oing to Tle 7 - sow ve goo lineit in te ovelpping nge of te movement s oppose to tis, tnsvesll moving om ls e lws non-line, oing to Tle 7 o oizontll move om ls oing to Tle 7, - e vitions, oesponing etoe es e etill onnete, wi mens tt finll tee will emin onl two etoes wit n tive pit, ie tee is pll iuit wit mn smll plte pitos Tis mens tt fo smll stutul imensions it is possile to genete omptiv lge pit nges / n teefoe omptiv lge foes fo W tuting pplitions n te oesponing tnsfomtion uents fo sensoi pplitions I t pol pol Te nge in pit t t n is onl te to te e, ut not to te etoe ise Te effet, ie te etostti iving powe ee te om ives is minl etemine te nume n of te om ms s wl s om ept n pito gp tt n e ieve given te tenologill fesile tolenes Vesion fom 48 7

8 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Vesion fom 48 8 om ls n e use to genete omptiv lge movements in te wfe plne s oppose to plte pitos wit ise vition, it is possile to ieve line etomenil tnsfe funtion Tnsvesll move om ls oing to Tle 7 ise vition onstitute iffeentil pitos om movement in te given ietion poues fo e om finge n its fie ounte-etoe iffeent, non-line pit funtions esun poues n opposite nge of ot pities, wile ote viles eg umiit, tempetue ϑ effet ot pities in te sme ietion It pplies fo te two pities of n ement l tt, Te iffeentil signl - is mesue s follows:, wit seies epnsion esults in K K o smll, te seies n e inteupte fte te line tem: Te iffeentil piniple tus uses te sensitivit to oule in tion to mesun In ition, it uses lineiztion of te output signl s wl s suppesses ote viles Te effet eomes ovious if te output signls of ot pities e epne in TYLOR seies s funtions of ot viles, : ±, ±, ±,, ±,,, It esults fo te iffeentil signl tt, -,,, Te line tem of vile n te pu sque tems o not ve n effet on iffeentil signl Wt emins is te effet of te mie sque tem igue 88 in te ook sows tt te output teisti of miomenil tuto fo iffeentil pito opetion is lineize te polistion voltge igue 89 illusttes te ouling of te sensitivit fo miomenil senso fo iffeentil pito opetion Te iffeene etween pitive tnsfomtion uents ue to pies, n te susequent uent-voltge-tnsfomtion genetes te output signl of te senso

9 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Diffeentil pitos e lso use fo sensos in lose-loop opetions in oe to mesue pit n - t te sme time - etill genete ompenstion foes Eeise 77 Te limit etion G s to poue isplement of te mss µm, ie etion foe s to ompenste te sping s estoing foe t isplement of µm: wit m n tot It esults tt tot S E E G m ρ l ρ l ρ Sping stiffness tot tee esults fom te pll onnetion of two S-spings, wose sping stiffness S n e lulte oing to Eq 69 om te given numeil vlues, it esults tt: mss m 9 - kg, sping stiffness tot 5 - N/m, G 544 m/s 554g t G 544 m/s, te ontt foe is K Te mounting position of te sensos etemines wete n ow te own weigt of te mss G mg 9-9 N s to e inlue into te lultion of G : Gm 64 m/s, Gmin 44 m/s lose K In oe to poue ontt foe K µn, we nee n etion, m wit lose tot 5-9 N ppling te given numeil vlues, it esults tt 4 m/s 6 g nt e oing to Eq 757 Te voltge s to e lge enoug to poue isplement of µm Tt equies foe of lose tot 5-9 N Wit nume of om ms n 8, etoe tot ise µm, stutue ept t µm, it esults tt 84 V n t f If ontt foe K µn is to e poue uing te sf-test, te equie voltge ineses to sf test 85 V Eeise 78 o etemining te tion of isplement ngle of D tosionl mio see pitue 5 n voltge t te etoe sufes, we n use te toque equiliium me : / Te sping stiffness of te tosionl sping is I t G / L, wit I t eing te tosionl esise n G te se moulus o tosionl sping wit egul oss-setion w, it esults tt I t w k Depening on te tio w/, fto k n ssume vlues of k 4 o tpeoil sping oss-setions fite wet emil eting, te tem fo te tosionl esise woul e mu moe omple Te iffeene in I t is omptiv smll if ou nge wit w ut keep tikness onst, fo inse Teefoe it is usull suffiient to onsie te egul oss-setion In te following, we will lulte te eti toque fo te se epesente in te figue Tt mens tt onl te etoe une te igt sie of te mio plte is ontolle voltge Vesion fom 48 9

10 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Vesion fom 48 Pitue 5 D tosionl mio It pplies fo te etostti foe ting on te movle mio plte tt W Te pit funtion fo te igt sie of te plte pito is, ie Note: oe stives to inese te pit, ie to eue plte ise s te ietion of te etostti foe ting on te etoes oinies wit te ietion of te ise eution, it pplies tt - n / -/ If te mio plte is isple ue to te ting foe >, te tul ise etween movle mio plte n se etoe eeses epening on ise to te ottionl is epening on wete we tke te fie se etoe o te movle mio etoe s efeene se, we n lso wite sin inste of o te ngles we look t, it is suffiientl peise to ppl sin, teefoe te istintion is not itil We ive t te eti toque ting on te igt sie of te mio plte if we sum up ll ptil toques tt t on te now stips wit e n tt e ve ise to te se etoe of o e ptil toque, it pplies tt Te summtion is ie out witin te limits of Wit it esults tt Sustituting z, it esults tt z/ n susequentl z z z z z z z z ln ln ln

11 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Vesion fom 48 Wit, it esults tt ln Note: o stti isplement, it will ppl tt < / ue to te pull-in effet Teefoe, will lie in te nge of o esonnt opetion, te pull-in effet oes not pl n oll In tt se, we ve to inlue te tion es Q stt It finll esults tt G k w L ln- 4 ln- Stting te tion of ngle n voltge f is not tivil, s tee e sevel ste sttes fo given voltge Reves it is possile toug to unmiguousl etemine te voltge equie fo given isplement ngle : Te tio voltge-ngle is L G I L G k w ln- ln- 4 ln- 4 t o sque mio plte mio lengt, it esults tt L G I L G k w ln- ln- ln- t Emple: φ teisti, nltill lulte wit t see pitue 6 4 mm, mm, sping lengt L 9 mm, sping tikness µm, sping wit w 4 µm, etoe se ise 8 µm, ontolling voltge 55 V Pitue 6 φ teisti

12 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents untion f is not efine fo Te effetive toque t n e eive fom te effet on te non-isple mio plte,, fo wi it pplies tt W Tis foe is pplie t te ente of gvit of te igtn mio plte, ie t ise / to te ottionl is: 4 o sque mio plte, it esults tt Eeise 79 Te pesente fme of te ppo opening is omplemente wit u-spe P ligt onuto tt oinies wit te fme Its opening fes te tosionl mio Te lse em is ouple to te u-spe ligt-onuting P fme un le to poto senso tt is situte t te font of te P fme Wen fling ojet inteupts te ligt utin poue te lse t n point, te ligt signl t te poto senso will e inteupte too In oe fo te lse em to it te fling ojet t lest one, te minimum smpling l 5m te s to e f smpling wit t 6 ms t v 5km/ It esults tt f smpling 4Hz In oe to inese etetion poilit, we use f e Hz s te eigenfequen of te esonne opete miomenil tosionl mio Te ise etween mio n ppo opening s to e 5m 5m 5m L 56 m o αopt α me 5 Te ege lengt of te sque mio s to e lge tn te imete of te lse em, eg io 4 4 mm sete inimum tikness D of te mio wfe esults fom summing up etoe se ise n mio tikness: D Eletoe se ise s to e suffiientl lge fo te full isple mio plte not to it te ounte-etoe: > α me Wit mm n α me 5, it esults tt > 44 µm Sete: 45 µm io tikness esults fom solution e e sing te eqution fo te eigenfequen of te tosionl mio G It fe π J π l J we ive t lengt l of tosionl sping G I t l 4 π J fe Inseting into tis eqution te espetive tions fo I t n J n using I t w k, m J 4 ρ 4 4, w/, k w/ 9 n 4 >> it esults tt G w l 77 4 ρ f e Vesion fom 48

13 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Hee, is te sping stiffness fo tosion, J te mss moment of ineti, te ege lengt of te mio, te tikness of mio n tosionl sping, w te wit of te tosionl sping, G te se moulus, m te mio mss, ρ te ensit n I t te tosionl esise ppling te vlues Gw/ 55 GP, mm, ρ kg/m n f e Hz, fo te fee esign pmetes sping lengt l n sping wit w, it esults tt tio l / mm 57 6 w/ µm, ie l w, Tle 4 iomio esign pmetes w/µm l/mm Te vlues pefee ue to tenologil esons e igligte in ge We set w 4 µm, wi esults in µm, n fo tikness D of te mio wfe see tsk, te esult is D 47 µm Toque equiliium me esults in voltge, tt is equie to sttill isple te sque mio α stt α me α w k G α α l ln- α wit Note: See lso solution 78 In Eeise 79, isplement ngle n sping lengt ve iffeent enomintions: isplement ngle α inste of, sping lengt l inste of L Due to α es Qα stt n Q, te ove eqution s to solve fo n ngle of α stt α me 5 Te vlues of α 5, w 4 µm, µm, k 9, G 55 GP, mm, 45 µm, L 8 mm, s/vm esult in stt 5 8 V Reging te etoe se ise, we ve to tke into ount te iffeene etween stti n nmi opetion of te miomenil tosionl mio see lso Eeise 78: o stti isplement, α < / ue to te pull-in effet Teefoe, it pplies tt Duing esonnt opetion of te mio, te pull-in effet oes not pl n ole In tt se, usull ssumes ve smll vlues o te etoe se ise, we ve to tke into ount te tion es Q stt toug Wit te lulte o sete vlues fo, l,, w,, te equiements eging eigenfequen n isplement ngle e met f Te spee of te fling ojet etemines oing to solution te equie minimum smpling fequen n tus te neess eigenfequen of te mio g If we wnt to etet te fligt ietion in ition to te pssing of fling ojet, we oul nge two of te ppo openings epesente in igue 77 one fte te ote Vesion fom 48

14 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Eeise 7 Retining foe esults fom te etostti foe genete te ive, minus te estoing foe of te spings: sing te given vlues fo te nume of om ms n 4, effetive lengt of te pito pltes L µm, wfe tikness s wl s plte wit 5 µm, plte ise in ste position 9 µm, sift pt to en stop u 7 µm, opeting voltge B 5 V, sping onst 8 N/m n s/vm, te following equtions n numeil vlues esult: Eletostti foe plte pito B Te effetive pito e is n L 4 7 m n te plte ise t en position is 6 u m Teefoe, it esults tt 48 mn Te estoing foe of ot spings is u,5 mn Te etining foe is mn Eeise 7 Te mss is suspene ove ntileve spings Tee ntileves wit stiffness e nge pll, n tis ngement ompises fou pis onnete in seies Totl stiffness teefoe is s follows: / 4 ges ; 4 6 / 4 In e se, one ntileve sping is ent in n s-spe Sping stiffness of ntileve sping wit eigt n wit, eing ent wit in plne: E l Pitue 7 In-plne efletion of ntileve sing te eqution of te eigenfequen of te stutue ges f π m m n ppling tnsfomtions, te equie ntileve lengt eomes l E π f m m Vesion fom 48 4

15 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Numeil vlue: l 594 µm ppling W n q, it esults fo te etostti foe tt W q q q E finge of te om ive t te seismi mss s two etoe gps Te totl foe esults teefoe fom multipling n Two om ives wit finges e t in one ietion Te menil sping foe is eive te stti isplement: Sping 6 q Sping Te equiliium of foes n e use to n 6 q 4 n q n t 5 Volt, te stti isplement is 7 nm If mping is wek, te osilltion mplitue ineses in se of esonne te esonne qulit fto iomenil stutues e minl mpe te suouning gs Te pessue-te nmi visosit n e use to influene te mping of te stutue Reuing te pessue in te senso ousing n e use to inese te menil qulit m Ωv, wit v q& qˆ ω osω t ω m m Ωqˆ ω osωt q ω m Ω qˆ osωt wit ω ω ω esults q ω Ω qˆ q nt ω Tking into ount esonne qulit fto of, t /s te mplitue of te etetion movement of te seon osillto mounts to ppoimt µm e Te two pities e mesue iffeentill see lso Solution 76 Te pit is lulte s follows:, wit q q q q q q q q K K o smll q, it esults tt Vesion fom 48 5

16 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents q S q o n finges, te pit nge is 57 p/µm Tking into ount te esults in Eeise, it eomes 6 f/ /s f iet voltge poliztion voltge pol ove te etetion etoes n e use to ffet te menil stiffness of te seon osillto o smll isplements, te following etostti foe pplies: pol q q It opposes te sping foe: t ot q Te stiffness is efine s eivtion of te foe fte te isplement: tot pol q q q n, fo q, it esults tt: pol lim tot q q Te eqution sows tt te esulting stiffness n e eue using poliztion voltge Te eigenfequen of te seon osilltion n e imensione sligtl ige tn te pim fequen sing poliztion voltge, tis fequen n susequentl e juste Eeise 7 Te toque esults fom summing up ll ptil toques k, tt ve een poue te etonmi foe ting on te iniviul onutos k k n o te ptil toques, it pplies tt k k k I Bk lk k Hee, k sttes te ise of onuto to te ottionl is of te movle ement B k is te vlue of te uent ensit n l k te lengt of te onuto, te to te -oointe of k see figue Te uent ensit is ssume to e onst long onuto n ove te onuto s wit o smmeti uent ensit istiution n smmeti oienttion of te movle ement in te i gp, te totl toque is n k k ges, wee n is te nume of iuit loops in te pln oil sing te oizontl pojetion of te mio plte n pln oil, we n eive te following vlues: onuto wit 5 µm, s: wit of tosion ns 5 µm Nume of iuit loops: n mm/ Vesion fom 48 6

17 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents onuto eigt µm ltot ρ n Dm Lm OH s esise of pln oil: R ρ ppling te equtions fo te etonmill genete toques to te uent sitution, we ive t Dm ges n I B Lm n I B Lm Dm Inseting te given vlues L m 4 mm, D m 4 mm, it esults tt fo equie toque ges 56-6 Nm te pout s to e ni 4 o m 5 µm wit of tosion n n n mm, te nume of iuit loops eomes n 4, n onsequentl I m uent ensit /mm sing R D L Pitue 8 Pln oil ltot ρ n m m ρ, te OHi esise of te pln oil t ρ l 8 Ωmm /m eomes R 4 Ω OHi powe issiption tus eomes P I R 4 mw Te epete tempetue inese is lulte using I R T P Rt wit DL 9 mm n α K W/m K n eomes αk T K I R I ρ n Dm Lm mm No om T P Rt, it follows ue to tt αk αk n I ρ n Dm Lm T αk mm sing te known numeil vlues, it esults tt T / K 5 I / n n ue to ni 4, it follows tt T / K 5 6 K see ove, wi mens inepenentl of te sete pmetes I, n If te ise etween iuit loops of te pln oils s e 5 µm e n te nume of iuit loops emins onst t n 4, te onuto will eome smlle tn 5 µm Tis mens tt R n onsequentl P n T will inese n ltentive woul e to etin wit 5 µm n to eese te nume of iuit loops to ppoimt n 6 Tis w, te uent woul ve to e inese to I m in oe to genete te sme toque Te OHi esise woul eese to 6 Ω Te OHi powe issiption woul mount to P I R 448 mw, ie T woul lso inese Vesion fom 48 7

18 Gel Gel; Wolfm Dötz Solutions fo eeises fo Intoution to iosstem Tenolog: Guie fo Stuents Eeise 7 s n emple fo ot tuto piniples, we n use mio-mio wit ise vition We will pesent te oesponing vntges n isvntges: Tle 5 vntges n isvntges of etostti n etonmi tution piniples io-mio, etostti ive see Tle 7, igue 68 io-mio, etonmi ive see igues 7, 7 Pos: Pos: - Smll imensions, esulting in lge eigenfequen - Dmping n e ssume to e onst no gp - pitive loliztion is possile - oe in te omogenous mgneti fi is popotionl - Voltge-ontolle to te uent ttting n epulsive foes n e ons: genete - Onl ttting foes possile - Te genete toque epens onl on te ngle; it n - Dmping in te i gp stongl non-line e onsiee to e onst fo smll isplements - Eleti iving toque epens on te sque of te voltge Tis funtion n its invese nnot e ontinuousl - Te mgneti iving toque n e ffete fom te outsie nging te mgneti flu ensit iffeentite n ve points of isontinuit ons: - ontollilit epens on mio position - Lge imensions ue to pln oil on movle e - ontollilit lost in te nge of mimum isplement - Het issiption fom pln oil Reging ontol tenolog, te vntges of te etonmi iving piniple pevil non-line ontol of te etostti tuto is possile if ll non-line tions e etl known mping, tosionl sping, teisti voltge-ngle Vesion fom 48 8