APPROXIMATION OF STRONG ELECTRIC FIELD

Transcription

1 APPROXIMATION OF STRONG ELECTRIC FIELD PROF. RNDR. ING. MILOSLAV KOŠEK, CSC. ING. JIŘÍ PRIMAS ING. MICHAL MALÍK PROF. ING. ALEŠ RICHTER, CSC. Abstct: Since stong electic field is used now in mny es, simle methods fo exct clcultion of electic field should be found. Fo the tus, whee the stong electic field is oduced by thin stight conducto ove conducting body, sevel oximting fomule (tht use conducting lne insted of the body) wee deived. sing the fomule, the field stength hs been clculted in detil nd tyicl esults e esented. The comison of ll the methods shows tht in the e ne the conducto (ne zone) ny of the fomule cn be used. In the vicinity of the conducto diffeence exists. Key wods: Electosttic field of thin wie, mio method, field of finite length conducto, conducting lne, electosttic obe INTRODCTION At the esent time stong electic field is used in mny lictions in science, design, industy etc - fom dust setos o electosttic industil inting systems, though nno-fibes oduction in textile science [1, 8], to highly sohisticted devices in ticle hysics. Theefoe, the tsk of the electosttic field modeling is vey imotnt. Fom the ll ossible souces it is cle, tht this t of hysics is unjustly neglected. The ossible eson my be the difficulties both in theoeticl nd exeimentl esech. It is shme, becuse it could led to mny cticl lictions of this effect. Thee is n ctive esech t Technicl nivesity in Libeec in the field of the symmeticl ccitos - e.g. thin wie ove metl lte electode. To find fomul descibing the foce on the symmeticl ccito [, 3, 4] connected to high voltge (tens of kilovolts) we need to obtin descition of the electosttic field. In this ticul configution the exct 3D descition of this electosttic field is not ossible without using the finite element method (FEM), but fo ou esech the oximte descition would be sufficient the geomety is simle enough nd the liction of bsic equtions of electosttic theoy leds to ccetble esults. This e dels with sevel methods of stong electosttic field oximtion. 1 THEORY The oblem cn be fomulted like this: Ove conductive othogonl lne of negligible thickness (length, width b) nd in distnce d thee is wie of the sme length nd dius R susended in llel wy to the lne. The wie is connected to DC voltge, the lne is gounded. The wie dius R is vey smll comed to the distnce d. Ou tsk is to find fomul descibing the stength of the electic field on the xis eendicul to the lne nd ssing though the middle of the wie. Fom the theoeticl oint of view the oblem cn be descibed s electosttic field of chged conductos. It is ossible to solve this oblem using otentil φ, which cn be in tun used to find the electic field stength: E = gdϕ. (1) Potentil hs to fulfill the following conditions: 1. The Llce eqution is fulfilled ound the conducto ϕ =. (). The otentil in the infinity equls zeo. ϕ ( ). (3) lim =

2 3. The otentil hs unknown but constnt vlue φ on the sufce of the conducto. 4. The noml comonent of the electic field stength on the sufce of the conducto is equl to sufce chge density, which cn be witten in integl fom thus: = ϕ Q, (4) E ds n gd n ds = ( S ) ( S ) whee Q is chge on the conducto nd the S symbol shows us, tht we e integting ove the sufce of the conducto. So the following ules must ly to electic field stength: 1. Inside the conducto it equls zeo. On the sufce of the conducto its tngent comonent is equl to zeo (othewise thee would be stong cuent flow inside nd on the sufce of the conducto) 3. On the sufce thee is only noml comonent of the field stength see fomul (4) Fomulted this wy the oblem cnnot be solved nlyticlly. Tyiclly this oblem could be oximtely solved using finite element method (FEM). In simlified fom the FEM is lied on the Llce eqution (). The boundy conditions e the secified vlues of otentil φ on conductos. But thee is oblem with the zeo vlue of the otentil in infinity. The lst condition the totl chge of the system of conductos is equl to zeo - cn be used to veify the esults. But becuse we need the fomul s stting oint fo futhe deivtions, it is bette to solve the oblem nlyticlly, though only with oximte esults tht is, by using simlified but sufficient models. The oblem cn be nlyticlly solved only if we ssume tht the lne could be substituted by infinite conductive sufce. Anothe simlifiction uses the fct, tht the wie dius is vey smll, so we cn elce it with chged line segment. Line chge density is consideed to be constnt. So fo the nlyticl solution we cn use these fou gdully moe ccute models: 1. Infinite single wie. Infinite wie nd conductive lne 3. Finite length single wie 4. Finite length wie nd conductive lne Othogonl system of coodintes is oientted so, tht its zeo is in the middle of the wie xis, X-xis is llel to the wie xis nd it is oientted fom left to ight nd Z-xis is eendicul to the chged lne. If we e to use cylindicl symmety we will be using the dil distnce insted of coodinte z. Following fomule e vlid only fo the field on the lne cossing the wie xis nd eendicul to the lne tht is in the XZ lne. Then the system of coodintes is shown on Fig. 1. It is ossible to deive the nlyticl fomule fo the 3D field, but they e much moe comlicted. Fig.1: Geomety of the exeimentl ngement. sing the Guss lw fom electosttics [5], [6] we cn deive the following fomul fo otentil of single line with the line chge density η: η ϕ ( ) = ln + A, (5) πε whee ε is emittivity of vcuum nd is the distnce fom the line, whee the field is clculted. Constnt A could be witten like this: η A = ln, (6) n πε whee the new constnt n must fulfill the following condition: n >. When we ly (6) on (5) we get hysiclly moe illusttive genel fomul fo otentil: η η η n ϕ ( ) = ln + ln n = ln. (7) πε πε πε Constnt n eesents the distnce of the oint, whee the otentil is equl to zeo. Moe ecisely, it is the dius of coxil gounded conductive cylindicl sufce. Alying stndd ocedue bsed on the fomul fo the otentil of the oint chge nd sueosition incile [7] we get fomul (7). If we elce the chged line with the infinite chged wie of cicul coss-section nd dius R, due to cylindicl symmety the electosttic field will emin the sme. On the wie sufce thee will be constnt otentil φ(r). Between this wie nd the cylindicl conductive sufce in the distnce n thee will be the voltge, which cn be, using fomul (7), defined s: η n = ϕ ( R) = ln. (8) πε R As the line chge density η is difficult to define, we cn elce it, using (8), with voltge between electodes o conductos: whee dimensionless constnt K is πε η =, (9) K d K = ln. (1) R It gives the eltion between the geometicl oeties defining the electicl field. The oughest

3 oximtion of electic field esides in ou elcing the oute cylindicl conductive sufce with zeo otentil with conductive lne in the distnce d. Tht is n = d. Afte using (9) on (7) we get cticl oximtive fomul fo otentil: d d. (11) ϕ( ) = ln = ln d K ln R Fo the oximtive electic field stength of single infinite wie in the distnce fom its xis we cn use the Guss lw to get simil fomul: 1 E ( ) =. (1) K When we ly the mio method, which elces the infinite conductive lne, fte twice lying (1) we get following fomul fo stength: 1 1 E ( ) =. (13) K d Axil comonent of the electic field stength (in diection of X-xis) is due to the symmety (cused by the infinite wie length nd infinite conductive sufce) in both cses equl to zeo. This lso wnts the fulfillment of the boundy condition on the conductive sufce, which equies the tngent comonent of the electic field to be equl to zeo. Fo the single wie of finite length (ssuming the line chge density on the wie is constnt) we cn deive the following fomul fo the dil comonent of the electic field stength: 1 E = (, x) K whee the constnt + x + x ( ) ( ) + x + x +, (14) = (15) eesents hlf of the wie length nd constnt K ws defined bove (1). The cente of the system of coodintes (zeo vlue) is in the middle of the wie xis, the x coodinte eesents the osition on the wie xis nd eesents the distnce fom the wie xis. So the field stength now deends on both coodintes. If we conside wie of finite length susended in llel wy bove conductive lne, the fomul (14) fo dil comonent of the electic field stength chnges to: 1 E = (, x) K + x + x + x ( + x) + ( x) + ( + x) + ( d ) ( x) + ( d ) x. (16) In the cse of finite length wie thee lso exists n xil comonent of the electic field stength in the diection of wie xis (o the X-xis) in the exmined lne XZ, see Fig. 1. Fo the single wie we cn deive the following fomul fo the xil comonent of stength: E = (, x) K 1 1 ( ) ( ) + x + x +. (17) The x coodinte eesents the osition on wie xis; the cente of the system of coodintes (zeo vlue) is in the middle of the wie. It is cle, tht the tngent comonent of the stength is equl to zeo only in the oint x =, tht is in the cente of the wie xis ojection. Fo the finite length wie bove the conductive lne the following fomul fo the xil comonent of the stength cn be deived: E = (, x) K 1 1 ( + x) + ( x) + ( + x) + ( d ) ( x) + ( d ) (18) Also in this cse the tngent comonent of the stength is equl to zeo only in the oint x =, tht is in the cente of the wie xis ojection. These simle models of electic field wee deived to comute the wek dynmic foce on symmeticl ccito with i dielectic. Positive ions e cceleted by stong electic filed in the vicinity of the wie. Becuse of the consevtion of momentum lw the cceleted ticles exet foce on the wie. As the field is due to the conducting lne stongly symmeticl, the individul foces e not nullified. (Fee electons move in the oosite diection.) In the esent time we e eing the theoy of the foce oiginting on the symmeticl ccito nd the electicl field descition will be its min t. EXPERIMENTS Mesuing the electic field stength is one of the most comlicted mesuements, so it is elly difficult to veify ou theoeticl esults. Thee e sevel tyes of mesuing obes, but they e ll t lest x mm in size, so they do not mesue the oint vlues, but only vege vlue ove cetin e. Tht is why fo ou geomety those obes e simly too big nd useless. Becuse of the size of the obes we would hve to use the exeimentl setu t lest ten times bigge nd

4 mybe highe voltge too. Even in this cse we would hve to conside the size of the obe, eliminte othe sitic effects nd mybe conside the conductive obe chnging the she of the field. We did not succeed to get suitble obe, but we ln diect exeimentl veifiction fo the ne futue. As ws mentioned bove ou esech is concened with foces on symmeticl ccitos. To find fomul descibing eltion between the foce oiginting theeon nd the electicl stength ound the electodes, we fist needed method of descibing the electicl field. The bove mentioned fomule wee used fo tht uose. We used them to theoeticlly comute the foce on the symmeticl ccito (wie dimete.1 mm, lge electode: 1 x 5 x 1 mm, wie distnce 3 mm, see Fig. ) nd we comed this vlue with the one we exeimentlly mesued. Mesuing of this wek foce is vey comlicted. Howeve to decide between the individul models - not only fo dynmic descition of the foce, but lso fo the electic field models - we need to ech high ccucy of mesuement. Fig.3: Effect of conducting lne on the field of infinite length wie. The conductive lne emoves the xil symmety of the field nd mkes the field stongly symmetic. It is confimed in Fig. 4 tht esents the dil comonent of the field stength on oosite wie sides. The cuve Down shows the dil comonent between conducto nd lne. The cuve is fo the dil field stength comonent on the oosite side of conducto, fom conducto to emty sce. In the oosite side the field stength deceses oximtely ccoding to the lw 1/. Fig.: Model of Asymmeticl ccito. 3 RESLTS The clcultions wee mde on the model fo conducto with dimete of.1 mm, totl length of 1 mm nd distnce fom conducting lne of 3 mm. The lied voltge ws kv. The field ws clculted only in the e contining the wie xis nd noml to the conducting lne, since the key infomtion is contined in this e. Stndd ghs e used fo field comonent descition in the diection noml to the conducting lne nd llel to it. Logithmic scles e often used, since the distnce nd field stength vies by sevel odes. Only ghs of hysicl o technicl imotnce, o ghs with tyicl o illusttive contents e esented hee, no systemtic esenttion of lot of clcultions is chosen. The effect of conducting lne in the cse of infinite length wie is in Fig. 3. The dil comonent chnges only ne the lne, but the chnge is vey fst. Ne the wie, the dil comonent of field stength deceses ccoding to the lw 1/, whee is the distnce fom wie xis. Fig.4: Comison of dil fields on oosite conducto sides. Effect of conducting lne is elly vey stong in the zone ne the conducto. The diffeence is lmost 3 odes t the conducto sufce. This is vey imotnt to exlin the wokings of some secil devices (e.g. symmeticl ccito). In the cse of infinite wie only single coodinte, the distnce fom wie xis, is used in the bsic investigted e, see Fig. 1. The descition of the field stength of finite length wie equies two coodintes, the distnce fom wie xis in veticl diection nd the distnce x fom wie cente in hoizontl diection. A oul och is to use sufce ghs. nfotuntely, they exhibit only qulittive infomtion. Theefoe we efe ghs with sevel metic cuves. Fo the field stength function in dil diection the eltive distnce of veticl line fom wie cente is used. At the wie

5 cente the eltive osition is nd t the wie end this metes chieves vlue 1. Tyicl esults fo the cse of finite length conducto e in Fig. 5. The diffeence is only t the wie edge nd in the zone ne the conductive lne. Fig.7: Effect of single wie length t the edge. Fig.5: Rdil comonent of single finite length conducto. Comison of the field of single infinite nd finite length conductos is in Fig. 6. The fields e comed fields t the conducto edge. The extended dil distnce is used in ode to show diffeences. The comison of field in the ne zone nd conducto cente is in Fig. 8. In this cse the finite length hs cticlly no effect. Fig.8: Effect of single wie length t the cente. Fig.6: Comison of dil field of finite nd infinite length single conducto It is evident fom Fig. 6 tht the sloe of dil field stength fo finite length conducto is diffeent t diffeent distnces. The distnce fom conducto cente cn be divided into thee zones: 1. Ne zone, in which the sloe is the sme nd constnt.. Middle zone, whee the sloe of field stength of finite length wie vies. 3. F zone with constnt sloe but of vlue diffeent fom ech othe. Accoding to Fig. 6 the middle zone stts oximtely t osition = 3 mm nd stos t = mm. Comison of fields in the exeimentl equiment in Fig. 7 shows tht in the exeiment thee is the ne zone only. Theefoe the definition of middle zone is ccetble. The single conductos e comed in Fig. 7. The effect of conducting lne of the finite length wie is vey simil to tht on infinite wie shown in Fig. 3. Theefoe we omit the detils hee. Comison of ll the oximtions in the line going though conducto cente is in Fig. 9. The only diffeent one mong them is the oximtion, eesenting the conducting lne of the finite wie length. The sme cse fo the wie end is in Fig. 1. The effect of wie length is in the entie ne zone nd the effect of conducting lne is only ne its vicinity.

6 Fig.9: Comison of ll oximting methods fo conducto cente. Fig. 11: Field stength of single wie of finite length. Fig.1: Comison of ll oximting methods fo conducto cente. In evious ghs (Fig. 3 to 1) the deendence of electic field stength on dius ws shown nd the distnce x fom conducto cente ws the mete. In ode to get full infomtion in next ghs the field stength on lines llel to conducto xis will be esented nd the distnce fom conducto xis will be the mete. Sevel cuves exhibiting the field stength of finite length single wie e in Fig. 11. Vey close to the conducto xis the field dil comonent is unifom long lmost ll the conducto length nd then deceses idly. At lge distnces the field comonent is not unifom. In Fig. 1 thee is the comison of single infinite nd finite length conducto t eltively lge distnce fom the conducto xis nd ne to the conducting lne. At lge distnce fom conducto xis the effect of finite length is vey lge. Fig. 1: Comison of the field of single wie of finite nd infinite length. The dil field comonent in the esence of conducting lne is in Fig. 13. In comison with Fig. 11 the field is moe unifom nd its decese is moe id. Fig. 13: Field stength of single wie of finite length llel to the conducting lne. The effect of conducting lne fo finite length conducto is shown in Fig. 14. The field stength deceses, but the field is moe unifom.

7 comonent of the field stength long the dius is in Fig. 17 fo the cse of single conducto. Its vlue is vey low ne the conducto cente nd chieves high vlue t its end. Fig. 14. The effect of conducting lne. Comison of ll the oximtions vey ne to the conducto is in Fig.15. Along the wie the diffeence is vey smll. Fig. 16: Axil field stength comonent of single conducto long the dius. The effect of conducting lne on the xil comonent is in Fig, 17. The chnge is visible only vey ne to the lne. Fig. 15: Comison of ll the oximtions ne the conducto Comison of ll the oximtions eltively ne the conducting lne is in Fig.16. In this cse the diffeences e clely evident. Fig. 17: Axil field stength comonent of conducto nd lne long the dius. The behvio of xil comonent on lines llel to conducto xis is in next two Figues. In Fig. 18 thee is the xil comonent of single conducto. The comonent is lge only t wie ends. Fig. 16: Comison of ll the oximtions f fom the conducto The finite length wie oduces lso the undesible field with xil comonent. Axil (o tngentil)

8 Fig. 18: Axil field stength comonent of single conducto on lines llel to conducto xis. The effect of conducting lne on the xil comonent is in Fig. 19. The only smll chnge is fo the cuve vey close to conducting lne. Fig. 19: Axil field stength comonent of conducto nd lne on lines llel to conducto xis. 4 DISCSSION Duing the study bsic simle nlyticl fomule wee deived. All of them e only oximte fomule nd wee obtined by the use of sevel simlifictions. The most imotnt neglecting is in the considetion of the infinite conducting lne insted of el box of smll finite dimensions. The second imotnt neglecting is the constnt line density of chge on the finite length conducto. At the conducto end the chge density is in elity not unifom. The thid neglecting is the elcement of the cicul coss section conducto by the chged line. The finite dimete of conducto nd esence of conducting lne leds to diffeent sufce chge density on the side ne to lne nd the oosite one. We must sy tht the models tht llow nlyticl solution e the simlest fist ode models. Moe ccute solution cn be obtined by two wys. The simle wy is the use of Finite Element Method. Howeve, thee e two disdvntges t lest s it hs been ledy mentioned in the t devoted to the theoy. The fist comliction is in the difficult fomultion of boundy conditions. The second difficulty is in the nlysis of the esults. To get ghs simil to the ones esented bove needs detiled knowledge of the FEM system. The second imovement is to chnge the sufce chge density on both the conducto sufce nd the finite dimension lne to get moe exct solution. nfotuntely, due to finite dimensions of lne nd the non-unifom chge density, the solution must be mde only numeiclly, using the numeicl integtion. Numeicl integtion cn be ogmmed eltively esily (e.g. in MATLAB) nd the comuting is vey fst on moden comutes. So this is not oblem. But the oblem lies in the fct, tht thee is no stightfowd wy how to find the coect chge density. Iesective of oximte solution, the chieved esults e inteesting nd imotnt both fom the hysicl nd cticl oint of view. Fo xis the esults esented in the fom of ghs hel the bette undestnding the function of el devices. The bove mentioned fomule deived fo ll tyes of models eveled tht esecilly fo the e close to the wie the esults e cticlly the sme. Theefoe the simlest model of single wie is fully sufficient nd thus the fomul (7) o (8) cn be used. The model cn be used with esonble ccucy in othe cses too. The esults obtined fom the models in egion close to the conducting lne e not in efect geement with elity, if we focus to the xil comonent of electic field. This comonent must be zeo on the conducing lne. Fig. 19 shows tht thee is smll but nonzeo xil comonent t distnce of 3 mm, e.g. t the conducting lne. It is due the model simlicity. Futhe imovements equie moe coect chge distibution, which is vey difficult to mke. The numeicl integtion is necessy in this cse. Vey imotnt fct fo the xis is the stong symmety of dil comonent on oosite sides of the wie, s it follows fom Fig. 4. The field stength on the e between wie nd lne is highe by sevel odes. The ngement elizes the symmeticl ccito. If we conside the foce due to the cceletion of ions, the foce cting on the wie is much highe in the e between wie nd lne. The weight of the ccito deceses. The foce ws mesuble in ou exeiments. Ghs on the Fig. 6 e imotnt fom the hysicl oint of view. It ws ledy mentioned tht thee zones of the field of finite length conducto exist. In ne zone vey close to the conducto the field deceses by the fist negtive owe of distnce, 1/, which is the field of infinite length wie. Relly, in the ne zone the conducto cn be teted s n infinite one. In the f zone the dil comonent of the field deceses by the second negtive owe of distnce, 1/. It coesonds to the field of oint chge. At lge distnce fom the finite length wie its dimensions e negligible nd it cts s oint chge. In the middle zone no oximtion is vlid. The boundy of zones deends on the wie length. Fig. 6 gives distnces fo the used ngement. In ou exeiment only ne zone is to be consideed. It hs been ledy mentioned tht in secil cses (comuting the intensity ne the ends of the wie) we hve to tun to moe comlicted models. But s thee is

9 lwys some sitic effect esent in the liction of electosttic field, it is hdly eve decidble whethe the devitions e due to the sitic effect o the model. Thee e t lest two esons, why the exeimentl confimtion of the theoy is difficult. Mny distubing effects mke the mesuement ccucy low. The size of the electosttic field obe is eltively big; theefoe, the mesuement on ctul ngement is not ossible. At lest 1 times lge exeimentl setu should be mde in ode to efom elible mesuement in incile. Even fte the dimension incese of the exeimentl model the integting effect of the obe should be consideed. The exeimentl ngement is now in the design stge but thee is comliction to get oe obe fo the eltively ccute electosttic field mesuement. 5 CONLSIONS Sevel vey simle nlyticl models wee eed fo the modeling of exeimentl device. Simle fomule deived fom the models llowed illusttive nd detiled descition of the field. The qulittive geement with elity is efect nd in imotnt es lso good quntittive geement exists esecilly in the e vey close to the conducto. Since exeimentl veifiction is difficult, futhe efinement of the models is not necessy in esent time. Anlyticl fomule wee used in the deivtion of the simle oximted fomul fo the foce cting on the wie in symmetic ccito. The fomul ws veified by eliminy exeiments. We wee unble to diectly veify the fomule exeimentlly, but we hve found n geement between the theoeticlly comuted foce nd exeimentlly mesued foce on the symmeticl ccito. Thus we hve eson to believe tht the bove deived fomule e coect nd they cn be used in fomule fo the exeimentl device. Ou next wok is heding to comlete the simle but sufficient model of dynmic foce on the symmeticl ccito nd to find moe ecise exeimentl methods of mesuing the foce. This in tun cn be used to veify the dynmic foce model nd lso electic field models. [5] Hňk, L.: Theoy of electomgnetic field, ublished in Ph, SNTL, 1975 (in Czech) [6] Inn, S, Inn A, S.: Engineeing electomgnetics, ublished in Menlo Pk, Clifoni, Addison-Wesley, Inc. [7] Dufek, M, Mikulec, M.: Poblems of theoeticl electicl engineeing, ublished in Ph, SNTL, 197 (in Czech) [8] Lukáš, D, Sk, A, Pokoný, P.: J. Al. Phys., 13 (8), Pof. RND. Ing. Miloslv Košek, CSc. Pof. Ing. Aleš Richte, CSc. Ing. Jiří Pims Ing. Michl Mlík Technicl nivesity of Libeec Studentská Libeec Czech Reublic miloslv.kosek@tul.cz les.ichte@tul.cz jii.ims@tul.cz michl.mlik@tul.cz 6 ACKNOWLEDGEMENTS This wok ws suoted by the fund of Ministy of Industy No. FT-TA3/17 nd the fund of Czech Science Foundtion No. GAČR 1/8/H81. 7 REFERENCES [1] Fong, H, Reneke, D, H.: Electosinning nd the fomtion of nnofibes, In: Slem DR, edito, Stuctue fomtion in olymeic fibes, ublished in Munich; Cincinnti, Ohio: Hnse Publishing; [] Bown, T, T.: A Method of nd n tus o mchine fo oducing foce o motion, Bitish Ptent no. 3311, ublished in 197 [3] Bhde, T, B.: Foce on symmetic ccito, ublished in Adelhi, Amy Resech Lbotoy, 3 [4] Cnning, F, X.: Asymmeticl ccitos fo oulsion, ublished in West Vigini, NASA, 4