Leakage flux analysis in toroidal core with circular cross section based on analytic calculation

Transcription

1 Leaage fux anaysis in tooida coe with cicua coss section based on anaytic cacuation Schoo of Automation and ectica engineeing Lanhou Jiao Tong Univesity 88 West Anning Rd. Lanhou Gansu China Abstact: It is impotant to undestand the eaage fux distibutions of tooida coe unde the condition of the coi is wound patiay on the suface of the coe, and the magnetic pemeabiity is not vey age. In this pape, fomuas of the eectica fied intensity and magnetic fied intensity in tooida coe with cicua coss section ae deived, as the eaage fuxes to be consideed. The fomua of eddy-cuent powe osses ae aso deived based on the eectica fied intensity fomua. The cacuated esuts of the eddy-cuent powe osses ae in good ageement with the fomua which is deived fom the sef impedance of coi. ffects of fequency, magnetic pemeabiity and distibution of windings on eaage fuxes ae anayed. The esuts show the fomuas accuatey pedicted the case when magnetic pemeabiity of the coe is vey age o the coi is wound densey and unifomy on the suface of the coe, the eaage fuxes ae vey sma and can be negected. Magnetic fuxes deceases with fequency inceasing, whie the popotion of eaage fuxes incease with fequency inceasing. It has been shown that eaage fuxes imit the uppe fequency of opeation. Keywods: tooida coe;eectomagnetic fied distibutions; eaage fuxes;eddy-cuent powe osses Intoduction The tooida coe has been widey used in eectonic cicuits [-]. Powe osses and eaage fuxes in the coe become moe ponounced with enegised fequency inceasing. Theefoe the nowedge of powe osses and eaage fuxes at high fequencies becomes moe and moe impotant in the design of magnetic components, such as tansfomes and inductos. ectomagnetic fied distibution and powe osses in tooida coe have attacted consideabe attentions [-8]. Ying Baiqing investigated magnetic fied distibutions and eddy-cuent powe osses in tooida coe with an abitay coss section by using FM ( finite eement method) []. K V Namjoshi et a pefomed eseaches on magnetic fied distibutions and eddy-cuent powe osses in tooida coe with ectangua coss section, and then they gave anaytica soutions [-6]. ectomagnetic distibutions and ion oss in tooida coe with cicua coss section wee aso studied by Saotome et a, and the anaytica soutions wee given [7-8]. Wheeas a the pevious studies have been mainy focused on the case of magnetic pemeabiity of the coe is vey age ( when δ =S m, M f =, > )o the coi is wound densey and unifomy on the suface of the coe, negecting the eaage fuxes. w h Coss section of coi (a) Coi on tooida coe a a a = Length of the coe A A b Coss section though A A (b) Coss-section of tooida coe and coi and coodinates system Fig. Tooida coe unde study In this pape, the diagam of the tooida coe -ISSN: -66X Voume, 6

2 unde study is shown in Fig., whee coi is wound patiay on the suface of the coe and magnetic pemeabiity of the coe is not vey age - (when δ=s m, f = M, < ). In this case, eaage fuxes can t be negected. Theefoe, the aim of this study is to pesent an anaytica method fo the eectica fied and magnetic fied distibutions in tooida coe with cicua coss section, as the eaage fuxes to be consideed. Based on the fomua of eectica fied intensity, a cosed-fom expession is aso deived fo the eddy-cuent powe osses. Deduction of fomua fo the cacuation of eectica fied in tooida coe. A fiamentay tun on an infinite coe The physica aangement unde study is shown in Fig.. The coe, taen to be infinitey ong, is teated as a homogeneous isotopic inea medium of conductivity σ and magnetic pemeabiity. The coe adius is b. A fiamentay tun of adius a encices the coe at =, and caies a sinusoida cuent i, i.e. i j t Ie ω =, whee ω is the angua fequency. Cyinde coodinates,, ae utiied hee. The foowing identities appy on the gound of symmety: =, =, = =, =, = ai b coe a i = enegising tun Fig. Fiamentay tun on infinite coes Fomua of eectica fied intensity in the coe is deived in Ref. [9], whee it is shown as I( Γ) = j ω I aβ K( βa) I( Γb) () I ( Γ b){ I( βbk ) ( βb) + I( βbk ) ( βb)} ΓI( Γ bk ) ( βb) + βk( βbi ) ( Γb) whee I, K, I, K the modified Besse functions. Γ = β + m m = jω σ magnetic pemeabiity of medium outside -7 the coe( =π ). indicate Fouie intega tansfom function of β the tansfom paamete. Soution of is obtained by Fouie tansfom invesion. j ω I I( ΓI ) ( βb) = ( ) π ak βa I( Γb) K( βb) I( βb) () β + β K( βb) I( βb) jβ e dβ K( β b) I( Γb) β + Γ K( β b) I( Γb) with the auxiiay functions defined by: g( β ) + f ( β ) () Φ( β ) = g( β ) + f ( Γ) I ( xb) f ( x) = x () I( xb) K( xb) g ( x) = x () K( xb) Witing the fomua of qn. as jω I = ( ) π ak βa (6) I( ΓI ) ( βb) jβ Φ( β)e dβ I( Γb) Fo witing convenience, the fomua of qn.6 was witten as jβ a (,, )e = β dβ (7). A fiamentay tun on tooida coe In many magnetic components, the coe wi fom a cosed magnetic cicuit, usuay ectangua o tooida, as shown in Fig.. The issue of shape is sidestepped by imagining the coe to have been cut open and staightened out to its ength (noting that in pactice / b > fo tooida coe [], thus negecting the diffeence between the inne diamete and oute diamete). A etun fux path of eo euctance is then povide by pacing the staightened-out coe between two infinite pates of pefect magnetic mateia ( =, δ = ) [9], as shown in Fig.(a). Let the enegiing tun be ocated -ISSN: -66X Voume, 6

3 hafway aong the staightened coe (by choosing to open the actua coe at the ocation most emote fom the tun invoved), the fied is then symmetica about the pane =. (a) quivaent ayout (b) Images Fig. Simuation of tooida coe and image tuns Since the pates ae infinitey pemeabe and non-conducting, the ines of fux ente o eave the pates at ight anges. Futhemoe, a ine of fux eaving at the ight-hand pate may be egitimatey consideed to immediatey e-ente fom the coesponding position on the eft-hand pate. Accoding to mio image method, the equied bounday condition wi be equivaenty satisfied, whee π β = () On the basis of qn.6 jω I = ak( βa) = () I( ΓI ) ( β b) jβ Φ( β )e I( Γb) whee Γ = β + m () Thus, the cosed-coe case fomay convets the integas of the infinite-coe case to staightfowad summations. Noting that singe-sided summation is justified, then we have = + () whee j ω I I( m) = () mi( mb) jω I = ak( a) β = () I( ΓI ) ( β b) Φ( β)cos( β) I( Γb) = eative pemeabiity of the coe. A coi on tooida coe Coi is composed of many tuns. We speciaie on the usua case of ectange coss-section of coi as iustated in Fig.. The numbe of tuns is N. Then, eative to, if the bounday pates ae emoved and an infinite numbe of images intoduced aong an infinite coe in the manne shown in Fig.(b) (the soid cice indicate actua enegiing tun at =, the dotted cices indicate image tuns). Thus, on the basis of qn.7 witing the coesponding eectica fied intensity fomua fo the cosed coe is simpy = j j a (,, ) β β e e d = β β (8) Noting that π π e j β = δ ( β ) = = qn.8 immediatey educes to π jβ a (,, )e = β (9) = Fig. Simuation of tooida coes and image cois in a simia manne as shown in Section., the fiamentay tuns ae epaced by cois, as shown in Fig.. Negecting gap among tuns, then the numbe of tuns in pe unit aea of coss section of the coi is N. The fomua of eectica fied intensity can be hw obtained by integating ove the coss-section of the coi pe unit aea. It is shown beow: -ISSN: -66X Voume, 6

4 jω I N = hw w a I( m) w ak( a) a + β mi( mb) = I( ΓI ) ( β b) Φ( β)cos β( τ) dτda (6) I( Γb) The intena integas ae eadiy evauated to give the fomua: = + (7) whee j ω NI I( m) = (8) mi( mb) jω NI = P( a, a) R( w) β β β hw = (9) I( ΓI ) ( βb) Φ( β)cos( β) I( Γb) whee P ( βx, βy) = [ p( βx) p( βy) ] β πα p ( α) = [ K( α) L ( α) + L ( α) K ( α) ] L ( x), L ( x) the modified Stuve functions ( βx R β x) = sin β In qn.7 is decomposed into and. It is noted that is uneated to, i.e. it is a constant vaue aong the axia diection. But is a function of. Magnetic fied intensity in the coe On the basis of Maxwe s equation = and qn.7, we have jω NI = P( βa, βa) R( βw) hw = () I( ΓI ) ( βb) Φ( β) β sin( β) I( Γb) = + () whee NIφ I( m) = () I( mb) NI = P( βa, βa) R( βw) hw = () ΓI( ΓI ) ( βb) Φ( β) β cos( β) I( Γb) is the component of the magnetic fied intensity in the adius diection and is the component of the magnetic fied intensity in the axia diection. Simia to, is aso decomposed into and. is uneated to, indicating it is a constant vaue aong the axia diection. But is a function of. ddy-cuent powe osses in the coe Fomua about eectica fied intensity and aveage powe P in isotopic inea medium is witten as foows: P = σ dv () V Substituting qn.7 into qn., eddy-cuent powe osses in the tooida coe can be obtained as J b σ dd ω I N σ P = π π( ) b = mi ( mb) m m [ mi ( mb) I ( mb) mi ( mb) I ( mb) ] I( βb) + P( βa, βa) R( βω ) Φ( β) h ω = I( τb) b [ τi( τbi ) ( τb) τi( τbi ) ( τb) ] τ τ () Veification The ea pat of impedance give the eddy-cuent osses in the coe in the fom of a seies esistance []. Theefoe, in ode to vaidate accuacy of qn. anothe fomua of eddy-cuent powe osses is given heein PZ = I Re( Zmm ) (6) whee Z mm is sef- impedance which is given in Ref.[9]: Z mm jω πb I( mb) = N mbi( mb) jω π N + [ P ( βa, βa)] hw = I ( β b) Q w F ( β ) ( β) K( βb) whee f ( β ) + f ( Γ) F( β ) = g( β ) + f ( Γ) Q ( β w) = ( cos β w) β Whee function f( x ) and gx ( ) ae defined by qn. and qn.. -ISSN: -66X Voume, 6

5 Tab. Dimensions and popeties of coe and coi Coe adius b =.8mm Numbe of tuns N = Coi inside adius Coi outside adius Coi width a = mm a = mm w = mm Magnetic path ength = mm Reative pemeabiity = 7 Coe conductivity σ = S m Fo the compaison, cacuations wee caied out on a Micometas T-6 cm O.D. powdeed coe with a distibuted gap. Fu detais of the coe and coi ae given in Tab., which ae the same as those used in Ref.[]. In the foowing, we use qn. and qn.6 to cacuate eddy-cuent powe osses at diffeent fequency, espectivey. The esuts ae shown in Tab.. It is found that the cacuated esuts agee we with each othe, vaidating the accuacy of qn.. Thus the accuacy of qn.7, and have aso been veified. Tab. Compaison of esuts fom qn. and qn. 6 f () P J (W) P Z (W) Resuts and discussion 6. ectomagnetic fied distibutions Fo magnetic component in section, eectica fied intensity and magnetic fied intensity ae cacuated fo vaious position of the coe,a /m,a /m.... -axis,m. Fig.6 Ampitude of.. -axis,m.. at. -axis,m -axis,m.. f = K Fig.7 Ampitude of at f = K at f = K. The cacuated esuts ae shown in Figs.-7. Fom the cuve shown in Fig. and Fig.6, it is.. found that with the inceasing of, both and decease ponouncedy. This esut is due to the fact that thee ae eativey age eaage fuxes in the coe. With the inceasing of, and both incease too. This esut is due to sin effect. is noma component of magnetic fied intensity. Magnetic fuxes which aise fom ae eaage φ,v /m axis,m. Fig. Ampitude of at. -axis,m.. f = K fuxes. Fig.6 and Fig.7 indicate that smae than is much. When is eo, is eo too. Then it has a substantia incease and eaches the maxima vaue at about = mm. Theeafte it has a substantia decease. At = mm its vaue appoximatey decease to eo. It has been shown that noma eaage fux effects ae concentated mainy -ISSN: -66X Voume, 6

6 aound the windings. 6. Leaage fux anaysis On the basis of qn., fomua of magnetic fuxes in homogeneous inea isotopic medium can be obtained as foows Φ= ds = πd b πni b I π ( mb) NIφ = + m I ( mb) hw R ( β w) bi ( β b) Φ( β ) β cos( β ) P( βa, βa) (7) = 6.. ffect of fequency on eaage fuxes Fo magnetic component shown in Fig., except magnetic pemeabiity, the est paametes ae the same as those given in Tab.. Magnetic fuxes ae eaage fuxes incease with the inceasing of fequency, whie magnetic fuxes decease. At highe fequencies this phenomenon becomes moe obvious. 7 Fo exampe, as f = the vaue of magnetic fuxes is ony % compaing to the fequency is f =, the popotion of eaage fuxes eaches %. The deteioation in pefomance, accuatey pedicted by the fomua, expain why the manufactues deem the coe unsuitabe fo opeation above K 6.. ffect of magnetic pemeabiity on eaage fuxes Fo magnetic component shown in Fig., except cacuated at = fo diffeent fequency. The magnetic pemeabiity, the est paametes ae the cacuated esuts ae shown in Tab.. same as those given in Tab.. Magnetic pemeabiity Φ is defined as magnetic fuxes at = is vaied. Magnetic fuxes ae cacuated at f = M and Φis magnetic fuxes at eaage fuxes at = mm = mm. Thus can be defined as Φ = Φ Φ. The popotion of eaage fuxes can be defined as Φ Φ. The cacuated esuts of the eaage fuxes and the popotion of eaage fuxes ae shown in Tab.. Tab. Magnetic fuxes fo diffeent f () Φ(Web) (mm) f () Φ Φ Φ fequency at = Tab. eaage fuxes and the popotion of eaage fuxes at =. 6.8% % 9.6%.6%.% Tab. and Tab. indicate that the popotion of fo diffeent magnetic pemeabiity. The cacuated esuts ae shown in Tab.. Definition of Φ Φ Φ Φ Φ is same as above. The cacuated esuts of eaage fuxes and the popotion of eaage fuxes ae shown in Tab.6. Tab. Magnetic fuxes fo diffeent magnetic (mm) Φ(Web) pemeabiity at.67 f = M Tab.6 eaage fuxes and the popotion of eaage fuxes at f = M Φ Φ Φ.6%.6% 7.9%.% Tab. and Tab.6 indicate when magnetic pemeabiity is sma magnetic fuxes ae sma, the popotion of eaage fuxes ae age. Fo -ISSN: -66X 6 Voume, 6

7 exampe, at = the popotion of eaage fuxes eaches.6%, moe than haf of the magnetic fuxes. With the inceasing of magnetic pemeabiity magnetic fuxes incease, whie the popotion of eaage fuxes decease. At = eaage fuxes ae sma, the popotion of eaage fuxes educe to.%, indicating when magnetic pemeabiity of the coe is vey age, eaage fuxes ae vey sma and can be negected, as efeed to befoe. On this occasion, eectica fied intensity and magnetic fied intensity ae cacuated at = mm. The cacuated esuts ae shown in Tab.7 and Tab.8. Tab.7 Ampitude of eectica fied intensity at f = M, =, = mm (mm) 8 (V/m) (V/m) Tab.8 Ampitude of magnetic fied intensity at f = M, =, = mm (mm) Fom Tab.7 and Tab.8 it is found that <<... It has been shown that in this case, and can be negected. Thus magnetic fied ony has axia diection component. Both and ae constant vaue aong the axia diection. 6.. ffect of windings distibution on eaage fuxes Fo magnetic component shown in Fig., paametes of the coe ae the same as those given in Tab., whie paametes of the coi ae vaied. ee Magnetic fuxes ae cacuated ony fo a paticua case when the coi is wound densey and unifomy on the suface of the coe. Paametes of the coi ae given as foows: NIφ = A, w = = mm, a b=.8mm, h =mm,thus a =.8mm. Magnetic fuxes ae cacuated at cacuated esuts ae shown in Tab.9. (mm) Φ(Web) f = M. The Tab.9 Magnetic fuxes when the coi is wound densey and unifomy on the suface of the coe at f = M, = Fom Tab.9 it is found that when the coi is wound densey and unifomy on the suface of the coe, even if magnetic pemeabiity is sma(in this numeica exampe = 7)and fequency is high, eaage fuxes ae vey sma, theefoe they can be negected. On this occasion, eectica fied intensity and magnetic fied intensity ae cacuated at = mm. Cacuated esuts ae shown in Tab. and Tab.. Tab. Ampitude of eectica fied intensity when the coi is wound densey and unifomy on the suface of the coe at (mm) 8 f = M, = 7, = mm (V/m) (V/m) Tab. Ampitude of magnetic fied intensity (mm) 8 when the coi is wound densey and unifomy on the suface of the coe at f = M, = 7, = mm Fom Tab. and it is found that ISSN: -66X 7 Voume, 6

8 Simia to the case when magnetic pemeabiity is vey age,, and can aso be negected. Thus magnetic fied ony has axia diection component. Both and ae constant vaue aong the axia diection. 7 Concusion In this pape, tooida coe with cicua coss section is consideed. An Anaytic method is used fo eectomagnetic fied distibutions and eddy-cuent powe osses fo tooida coe, in which the fied vaiabes ae expessed as a singe seies in tems of Besse functions and tigonometic functions. It is not ony appicabe to the case when coi is wound patiay on the suface of the coe, with taing into account eaage fuxes, but aso appicabe to the case when magnetic pemeabiity of the coe is vey age o the coi is wound densey and unifomy on the suface of the coe,with negecting eaage fuxes. Resuts pesented in the study show a cassica sepaation of fomuas of eectica fied intensity and magnetic fied intensity into two components espectivey, one component (, ) is constant aong the axia diection, wheeas the othe component (, ) is vaiabe. The popotion of the vaiabe component deceases with inceasing of magnetic pemeabiity. When magnetic pemeabiity of the coe is vey age (in the numeica exampe of this pape > ) o the coi is wound densey and unifomy on the suface of the coe, the vaiabe component is so sma that it can be negected. Refeences [] Beove C. andboo of moden eectonics and eectica engineeing[m]. New Yo : Wiey,986,9-6,8. [] Wang Zhao an, Zhang Mingxun. Appication handboo of Powe eectonic devices [M]. Beijing: China Machine Pess,,7-7. [] an Shuai, Zhang Li, Tan Xingguo. Mateia seection based on oss chaacteiation fo high-powe high-fequency tansfome coes [J]. igh Votage ngineeing,,8(6):86-9. [] Ying Baiqing. FM soutions of eddy-cuent powe oss in tooida coes with ahitay coss section [J]. Micoeectonics & Compute,,(): -. [] K V Namjoshi, J Dougas aves, P P Biinge. ddy cuent powe oss in tooida coes with ectangua coss section [J]. I Tansactions on Magnetics, 988, (): [6] Ma Xiui, Zhao Yanhen, Dai Dong. Anaytica soution of eddy-cuent powe oss in tooida coes with ectangua coss section fo highfequency eectonic cicuits [J]. Poceeding of the CS,, (6): -8. [7] Saotome, Saai Y. Ion oss anaysis of Mn-Zn feite coes[j]. I Tansactions on Magnetics,997, (): [8] Daming Zhang, Che Fo Foo. Theoetica anaysis of the eectica and magnetic fied distibutions in a tooida coe with cicua coss section[j]. I Tansactions on Magnetics, 999, (): 9-9. [9] Wicox D J, Conon M, uey W G. Cacuation of sef and mutua impedances fo cois on feomagnetic coes[j]. I Poceedings, 988, (7): [] Wiiam G uey, David J Wicox. Cacuation of eaage inductance in tansfome windings[j]. I Tansactions on Powe ectonics, 99, 9(): -6. [] W M Gead uey, David J Wicox. Cacuation of shot cicuit impedance and eaage impedance in tansfome windings [A].Powe ectonics Speciaists Confeence, PSC 9 Recod nd Annua I[C] ISSN: -66X 8 Voume, 6