CALCULATION OF BRAKING FORCE IN EDDY CURRENT BRAKES

Transcription

1 CALCULATION OF BRAKING FORCE IN EDDY CURRENT BRAKES By P. HANYECZ Department f Theretical Electricity. Technical University Budapest Received March Presented by Prf. Dr. I. V,\GO Intrductin The eddy current braking f linear mtrs is usually required fr btaining mving frce - velcity characteristics ptimal frm the pint f view f a particular drive. This necessitates a braking device with prescribed characteristics which can be attained with the aid f braking ples f definite dimensins and f ple flux-density f apprpriate value. This paper presents a prcedure fr the calculatin f braking frce at ples f given dimensins and at given flux-density. In the curse f the slutin, Ritz prcess based n variatinal principles is presented fr the calculatin f statinary cnductive fields excited by mtinal inductin, and a technique is frmulated fr treating singular excitatins which imprves the cnvergence f the numerical prcedure. An alternative methd based n an infinite number f images fr the slutin f the prblem is als presented thus permitting the examinatin f the Ritz prcess. Finally, the results f the measurements carried ut t check the calculatins are presented. 1. Mdelling f the prblem, derivatin f the describing equatins The scheme f the studied arrangement is shwn in Fig. 1. A plate f width 2d, thickness s and cnductivity (J mves between the ples excited by direct current at a unifrm velcity v. The task is t determine the braking frce acting n the plate. T this end, the eddy currents resulting frm mtinal inductin have t be calculated. The fllwing simplifying presumptins are made fr the slutin: The flux-density B f the magnetic field generated by the exciting cils is taken t be perpendicular t the plate and f cnstant magnitude B under the ples, and zer utside the ples. The reactin f the eddy currents n the ple's flux-density is neglected, thus the electric field is slely induced by the fluxdensity B. In the directin perpendicular t the surface f the plate unifrm

2 250 HANYECZ P. Cl N x Fig. 1. Scheme f eddy current braking f linear mtr current distributin is assumed. The cnductivity (j is cnsidered t be cnstant and the lngitudinal dimensin f the plate t be infinite. As a cnsequence f the simplifying presumptins, the electric field f eddy currents can be discussed in the crdinate system f the ples as a statinary cnductive field generated by the impressed field intensity Ei = = V x B cnstant in time. In this crdinate system the current distributin and the vectr Ei f the impressed field intensity seem t be cnstant. The prcess is linear, thus the electric field belnging t ne pair f ples will nly be discussed in the fllwing. The Maxwell equatins describing the phenmenn are div J =0 (1.1 ) curl E=O (1.2) ( 1.3) J. E and Ei dente the vectrs f the field generated by ne pair f ples. The crdinate system is chsen as shwn in Fig. 1. In this system, v = va x, B = = - Ba=, thus Ei = B va,. under the ples and zer elsewhere. Since unifrm current distributin has been presumed in directin z, the prblem can be discussed as a tw-dimensinal ne in the plane xy. The fllwing equatin is derived fr the scalar ptential intrduced as E = - grad qj: LlqJ=div Ei (l.4) where Ll is the planar Laplace peratr.

3 CALC(;LATlOS OF BRAKISG FORCE 251 In the fllwing, relative crdinates and dimensinless, relative quantities are intrduced t simplify the numerical calculatins and t btain results independent f cncrete dimensins. The relative crdinates are x c=- - d v '1 = - d (1.5) (1.6) In the system f relative crdinates the riginal regin is mdified as shwn in Fig. 2, and it des nt vary n the variatin f gemetrical dimensins which nly affects the relative ple-dimensins a' and b'. The definitins f relative quantities are: et,',, I -0', \ : 0' 1-' Fi~I' :C, The studied regin in the system f relative crdinates The relative ptential and field intensity: ( 1.7) e= -grad (P' ( 1.8) E. The relative value f the impressed field intensity is e i = E',. thus where j dentes the relative current density. The relative current flwing thrugh a curve l' f the plane ~11 I is: (1.9) i = S j(a~ x dl') ( 1.10) ['

4 252 HANYECZ, p, as is the unit vectr nrmal t the plane (1]. The relative pwer lss due t the tw pairs f ples is: 1 :xc p=4s S Ij((,lj)-j((+2c',rf)1 2 d(d1] (1.11) - '" C' is here the relative value f the dimensin c. In frmula (1.8) as well as in the fllwing the differential peratrs relate t relative crdinates. The relatinships between the riginal and relative quantities are: (1.12) (1.13) (1.14) If the image f the curve l' is I in the plane xy, the current flwing thrugh a crss sectin f the plane determined by the curve I is: ( 1.15) (1.4) yields the equatin ( 1.16) fr the relative ptential functin. The fllwing bundary cnditins shuld be prescribed n the bundary f the regin fr the determinatin f the ptential functin: cq/ - = 0 at il± 1 C'I ( 1.17) lim e = 0 fr - 1 ::;; '1 ::;; 1 ( 1.18) In accrdance with the definitin f e i and the relatinship Ei = = Bva p the right-hand side f (1.16) is zer except n the lines -a'::;; ::;; ( ::;; ai, 'I = ± b', and it is singular alng these lines. Let us exclude the singular lcatins frm the regin. Thus, the Laplace equatin Llq/ =0 ( 1.19)

5 CALCULATION OF BRAKING FORCE 253 is btained fr the ptential functin. Hwever, the nrmal cmpnent f current density and the ptential functin have t be cntinuus alng the lines excluded frm cnsideratins. This means the fllwing cnditins fr cp' and e: lim cp'(~, 1,)= lim cp'(~, 1]); -a'::; ~ ::;a' (1.20),/--+±b' 0,/--+±b'+O lim e,/(~,17)- lim et/(~,1])= 1; -a'::;~::;a' (1.21),/--+b' - 0,/--+b' + 0 lim '/--+ b' et/(~,1])- lim et/(~,1,)=1;,/--+-b'+o - a' ::; ~ ::; a' (1.22) S, the nrmal cmpnent f field intensity has a discntinuity f unit value alng the singular lines. It is knwn frm the thery f electrstatic field that discntinuity f the nrmal cmpnent f the displacement vectr r, in hmgeneus media, f the nrmal cmpnent f the field intensity appears alng surfaces with surface charge density, In accrdance with the analgy between the statinary cnductive field and the static electric field, the ptential functin cp' can be cnsidered t be excited by surface charges f density alng the line - a' ::; ~::; a', 17 = b' and 1'=80.1 [~J m V= -8 '/[~J m alng the line - a' ::; ~ ::; a', '7 = - b' and f infinite length in the directin perpendicular t the plane ~17, with the bundary cnditins (1.17) and (1.18), prvided the permittivity f the medium in the analgus static field is taken t be 8 0, Cnsequently, the ptential field sught is excited by a singular arrangement f charges. In the fllwing, the relative ptential, field intensity and pwer lss will be determined with the aid f the analgus static electric field.

6 254 HANYECZ. P. 2. Slutin by variatinal methd The Ritz prcess based n variatinal principles serves fr the numerical slutin f bundary value prblems f certain types [2, 3]. Let us presume that the peratr equatin described by -LJu= f (2.1 ) in a regin V bunded by the surface S = S! + S 2 and by the hmgeneus mixed bundary cnditins (2.2) (2.3) are t be slved. In (2.1) u and f are functins defined in the regin V and ~u n dentes the derivative f u in the nrmal directin. Accrding t Ritz prcess, the apprximate slutin is sught as an expansin (2.4) The functins CfJ!, CfJ2'..., CfJn are linearly independent, and they are elements f a functin-set entire in the energy space f the peratr [3]. The cefficients a k are btained as the slutin f a set f linear equatins: n L [CfJbCfJj]ak (j,cfj). j=i,2,...,n. (2.5) k=! In this expressin [CfJk> CfJ J dentes the energy prduct f the functins CfJk and CfJj which is defined in ur case as [2,3]: [CfJhCfJJ = fgrad CfJkgrad CfJjdV. (2.6) v The right-hand side f (2.5) is the scalar prduct defined in the regin V f the functins f "md CfJj: (j, CfJ) = f fcfjjdv (2.7) v

7 CALCULATION OF BRAKING FORCE 255 It can be shwn that the sequence f the apprximate slutins thus btained tends t the slutin f the peratr equatin in the energy nrm which means that lim S grad 2 (un-uo)=o (2.8) n-+ X) V where U is the exact slutin. The cnvergence f the variatinal methd depends upn the nature f the exciting functin. The cnvergence is expected t be slwer if the exciting functin is discntinuus r, r as in ur case, is singular. T imprve the cnvergence f the numerical slutin the singular cmpnent is extracted frm the ptential functin sught. T this end, the relative ptential functin cp' is sught as the sum f three functins: (2.9) Here, us(c;, 1]) is the ptential excited by the surface charges in free space, t/i(c;, 1]) is an arbitrary functin with at least tw cntinuus derivatives cmplying with the symmetry cnditins n cp' and having a derivative in directin 1/ alng the lines 1] = ± 1 which is f equal abslute value and ppsite sign as the same f the functin Us' li(c;,i/) is the unknwn functin whse determinatin is pssible with the aid f the equatins relating t cp'. The functin Us can be btained by elementary methds: 1 (- f3 A f Arctan. -,' rctan - 2IT Arctan -:- 'Y. + A rctan -'Y.) " (2.1 0) Here, 'Y. = c; - a', fj = c; + a', t' = 1/-b', 6 = '/ + b'. The derivative f the functin Us in directin 1] n the lines 1/ = ± 1 is 1 ( 'Y. fj g( c;) = - Arctan - Arctan - - 2IT e e 'Y. A fj) - Arctan - + rctan - /( /( (2.11)

8 256 HANrECZ, p, where 8 = 1 - b', K = 1 + b'. Thus a functin cmplying with the cnditins is e.g. (2.12) In accrdance with the cnditins n q/ and the prperties f Us and 1jJ, the Laplace-Pissn equatin (2.13) is btained fr the unknwn functin 1/, and the derivative f 1/ in directin tl shuld be zer alng the lines rt = ± 1. The Ritz prcess is applied t determine the functin 1/. Since the exciting functin in (2.13) has an infinite number f derivatives and 1/ is t satisfy hmgeneus bundary cnditins. the cnvergence f the numerical prcedure is fast. Due t the symmetry f the regin shwn in Fig. 2, the ptential functin q/ is even in ~ and dd in '1 if the ptential n the line rt = 0 is chsen zer. Since Us and IjJ als have this prperty. v must have it as well. Accrding t [2J, the apprximate slutin in the regin infinite in directin ~ is sught in the frm ;:. ~. (2k + 1 )1[11 cs (21 Arctan ~) L II m = L 1... a kl SIll k=oi=o 2 /1+,"2 'V '> (2.14) The elements f the matrix in the set (2.5) are in accrdance with (2.6): ' (2k +,1 )ml ' (2i + 1 h!ll Akl. ij Sin 2 Sin 2 d/i ' 1 f x 'J" [2Isin(21 Arctan~) + ~cs(21 Arctan ~)J~:ysin(2j Arctan~) + ';cs(2jarctan m d-: + (\+;-)' - 1 (2k+l)(2i+l)rr: r (2k+l)rrl) (2i+l)rrl) + 4 cs 2 cs 2 d/i. xc, f cs (21 Arctan :;') cs (2j Arctan ~) de 1 +~2 - (2.15)

9 CALCULATION OF BRAKING FORCE 257 The elements n the right-hand side are accrding t (2.7): I. (2k+ 1)m1 b kl = - I] SIn 2 dl1. f :c. f d2g(~) cs (21 Arctan ~) d v d ( v 2 /1 r-----;:- +,~2 ( - - V s (2.16) It has been utilized that it is sufficient t integrate ver a quarter f the regin due t symmetry. In cnnectin with the numerical prcedure, it is nted that the terms in the series (2.14) belng t a strngly minimal functin-set [2]. This feature ensures that the prcess is insensitive t the accumulating numerical errrs and the matrix f the set f equatins is easily inverted. A further advantageus prperty is that the matrix has nn-zer elements in square blcks f rder (n+ 1)(n+ 1) alng the main diagnal nly, and particularly in the main diagnal and in the tw rws under and ver the latter. In knwledge f the relative ptential functin q/, the relative field intensity and relative current density can be calculated and hence (1.11) and (1.14) yield the pwer lss and braking frce F = P. Finally, it is nted that if v the pwer lss is calculated frm Us nly as in (3.3), the lss in a plate f infinite dimensins in bth directins is btained. 3. Slutin with the aid f images The simple gemetry permits the calculatin f the analgus electric field by an infinite number f images. Namely, the hmgeneus Neumann bundary cnditin alng the lines I] = ± 1 can be satisfied by an infinite number f image charges (Fig. 3). It is sufficient t give the expressins f the relative field intensity which is written as an infinite series: (3.1 )

10 258 HANYECZ, p, b' ~;+y -'i IQ' ---v -2 -_ +v --+\-' -I. -- -\! Fig, 3. Satisfactin f the bundary cnditin with the aid f an infinite number f images I ( p Cl. P CI.) eh = Arc tan - - Arc tan - - Arc tan -;;- + Arc tan s: + "2IT Y you 1"" [ papa + - L (-It Arctan" -Arctan-,,- -Arctan-_- +Arctan-_-- 2IT k=1 Ik1 Ik1 k! k! p Cl. P Cl. ] - Arc tan. + Arc tan, + Arc tan _ - Arc tan -_- Yk2 Yk2 0k2 0k2 (3.2) The meaning f a, p, y and b is the same as befre and b kl = t/ + 2k + b', c5 k2 = =ry+2k-b', Ykl =ry-2k-b', Yk2=ry-2k+b'. In the knwledge f e the pwer lss can be determined as explained previusly, but it is mre effective t derive a simpler expressin with the aid f Green's therem: b' p=4 J ry [e~(a', t/)-e~(a' +2e', ry)] dry b' -4 J r7[e~( -a', ry)-e~( -a' + 2e', ry)] d1/+ +4b' lim ry~b'+o a' J [ery(~,1/)-ery((+2e',ry)]d( -a' (3.3) It is evident that single integrals rather than duble nes are t be evaluated here.

11 ~~~_~~ CALCULATION OF BRAKING FORCE The results f the calculatin Accrding t the abve discussin, the relative lss has been determined at different ple dimensins and ple distances. The dependence f the parameters has been examined by keeping the prduct a'b' and the distance c' -a' cnstant, and the rati a'ib' has been varied. The results thus btained with the aid f images have bee.l pltted in Figs 4.1,2 and 3. The cnstant value f the prduct a'b' means a cnstant ple surface and, in case f well utilized ~l 21 ~~0b'=07 1 ~ ~ 00=0.6 I ~ 00=05 00= b'=03-00=02 '--.1 ~b'= G L. 160/0 Fig Relative pwer lss vs. the rati a'/h' at cnstant ple surface and c' -a'=o ~r I I j ~ ~Ob=07 ~ -~~cib=o.6 OS = b=oi = 0 D= D=02 l ---- ci O= U 1.6 a'/b Fig Relative pwer lss vs. the rati a'lh' at cnstant ple surface and c' - a' = 0.2 irn, a cnstant ple-flux. The dimensin 2(c' - a') is the distance between the inner edges f the ple pairs. The curves thus indicate the dependence f the relative pwer lss n the rati f the ple-width and ple length if the distance between the ples and the ple flux are kept cnstant. A dtted line in Fig. 3 indicates the lss at c' - a' ~ (f). This shws that the interactin between the ple pairs is negligible if the relative distance between the inner edges f the ple pairs is greater than 1. Cmparisn f Figs 4.1, 2 and

12 260 HANYECZ, p, ~r 2J,1 _~~~:~:~~ _-~~t:i=05 '11 ~-~~:~~g~ ~~0'ti:02 0~1 ~_-----~ -~-_-_-_O_'t:i_'=_O_'~ I. 1.50'/0 Fig. 4.3, Relative pwer lss vs, the rati a'/b' at cnstant ple surface and c' - a' = 0.5 and c' - a' -+ J: Fig, 5, Current distributin in the plate 3 reveals that the lss and thus the braking frce increase significantly, if the ple pairs are drawn nearer t each ther. Therefre, if the aim is t attain a braking frce as great as pssible, the tw ples shuld be set as near t each ther as pssible. The minimal distance is determined by the space requirement f the exciting cil and by the leakage between the edges f the tw ple pairs which decreases the ple-flux. In Fig. 5 the current distributin in the plate has been pltted in a particular case. The relative dimensins are a' = 0.358, b' = 0.713, e' = The relative current between tw adjacent lines f current is

13 CALClLATlOS OF BRA/(fSCi FORCE Testing measurements T test the mdel and the numerical prcess, measurements have been carried ut n an experimental linear mtr at the Department f Electrical Machines f the Technical University Budapest. The essential parameters were: a=22.5 mm, n=47mm. c=40mm. d=71.5mm, s=1 mm. 6 0 =3mm, v= UQm. T determine the air gap flux density Ba the flux f the ples has been measured. Ba was nt calculated frm the riginal irn dimensins, since the leakage at the edges causes a part f the flux t run utside the ple edges. The leakage has been examined n the basis f [4J with the presumptin f infinite permeability and prismatic ples by Schwarz-Christffel transfrmatin. Accrding t this, in case f 6 0 /2a < 5 and 6 0 /2b < 5 a mre exact value f Ba is btained if the transversal dimensin f the ples is mdified by utside and by n the side between the tw ples, and the lngitudinal dimensin by 1.3 (50 n each side, and the whle flux is related t this surface. The mdified dimensins are: a=25.6mm. b=51 mm, c=41 mm. At these dimensins. the relative pwer lss was 1.36 by variatinal methd with 14 terms in the apprximatin and 1.34 by the methd f images with 10 pairs f image charges taken int accunt n bth sides. These were fund t be between 1.28 and Measurements have been carried ut as fllws. The plate has been accelerated by a cnstant frce. initially withut braking. then at different values f air gap flux density. With the aid f the punch tape attached t the plate and the pht dide munted n the ple the displacement time functin was btained in a large number f pints. A five-degree plynmial was fit n these pints by least square methd. Hence. the velcity time and acceleratin time functins were determined. The frictinal frce at different velcities was calculated frm the acceleratin withut braking and thus the braking frce was btained frm the acceleratin f braked mtins. The ttal mass f the Table I Cmpuced and measured L'alues f hraking frce Br, F rric F br mca~. Fhrcmp, [Vsm 2 ] [m s] [N] [N] [N] Peridica Plytechnica El

14 262 HAl'YECZ. P. mving part was 14.7 kg. It was assumed that the acceleratin has n effect n the dependence f braking frce n velcity. The measured and calculated values f braking frce are shwn in Table 1. Their cmparisn shws the maximal difference t be lwer than 5~/~. 6. Cnclusins A methd has been presented in the paper which permits the calculatin f braking frce in an eddy current brake f a linear mtr. The results f the testing measurements indicate that the mdel and the numerical prcess describe the phenmena with acceptable accuracy. As regards the Ritz prcess presented. it is nted that althugh its cmputatinal requirements are excessive, it is still very effective and can be applied t bundary value prblems f far greater cmplexity than the ne slved in this paper. The mst imprtant result f the methd described fr the treatment f singular excitatins is the faster cnvergence f the numerical prcess in energy nrm, and experience shws the field characteristics cmputed frm the apprximate slutin t apprximate the values btained by image charges, which can be regarded as exact, even in the vicinity f the singular lcatins. This is prved by the fact that in case f a sufficient number f terms in the apprximatin the pwer lss can be cmputed frm (3.3) instead f (1.11) which is f great numerical significance. Namely, the field intensity is btained as a finite series and its square appears in the frmula f lss, thus a twvariable numerical integratin is necessary in (1.11), while nly ne-variable appear in (3.3). If the Ritz prcess had been applied directly t singular excitatins, frmula (3.3) culd nt have been emplyed. Finally, it is nted that cmputatins have been carried ut n the desk calculatr EMG 666. The cmputatinal time necessary t btain the relative pwer lss in a particular arrangement was abut 4 minutes with the aid f image charges and minutes by variatinal methd applying frmula (3.3). Summary Ritz numerical prcess fr slving variatinal prblems is applied t the calculatin f the electric field in an eddy-current brake f a linear mtr. An alternative methd based n images fr the slutin is als discussed. The eitect f mtr dimensins n braking frce is examined. The results f the testing measurements carried ut t check the calculatin are als presented. -

15 CA LCL'LAT10N OF BRAKISG FORCE 263 References I. FODOR, GY.,-V.l.GO. I.: Electricity. VJ. 12. Static and statinary field. (In Hungarian). Tank6nyvkiad6. Budapest, MIKHLIN, S. G.: ~umerical realizatin f variatinal methds. (In Russian). Izd. "Nauka", Mscw, MIKHLIN, S. G.: Variatinal methds in mathematical physics. (In Russian). Izd. "Nauka", Mscw, SIMO:-1YI. K.-FDOR. GY.-VAGO. I.: Cllectin f examples n theretical electricity. (In Hungarian). Tanknyvkiad6, Budapest ANTAL. 1. (editr): Manual f physics fr technical experts. VJ. I. (In Hungarian). Miiszaki K6nyvkiad6, Budapest Pill HANYECZ H-1521 Budapest 4*