Calculation of Stray Capacitances of MCG Coil include One Turn, Single-Layer and Conductor Wire Filaments in Rectangular form

Transcription

1 Cacuation of Stay Capacitances of MCG Coi incude One Tun, Singe-Laye and Conducto Wie Fiaments in Rectangua fom M. E. Moseh* and M. R. Besmi** Abstact: This pape pesents a new method caed vespiay egua hexagona (VRH) mode in ode to cacuate paasit capacitance between conducto wie fiaments of one tun of coi (OTC) and between conducto wie fiaments and ine and aso tota capacitance of one tun of the heix magneto fux cumuative geneato (MCG) coi incude singe-aye conducto wie fiaments in fom of ectangua co-section. In this pape, wie fiaments of the coi ae sepaated into many vey sma simia eementay ces. In this stuctue, an equiatea ozenge-shape bas ce (ELBC) with two tapezium-shape egions is consideed between two adjacent conducto wie fiaments in one tun of the geneato coi. This method appies to cacuate stay capacitance of one tun of the coi with muti conducto wie fiaments (CWFs). Keywods: Magneto Fux Cumuative Geneato, Singe-aye, Stay Capacitance, Vespiay Mode. 1 Intoduction 1 In the design of a heix magneto fux cumuative geneato (MCG) (see Fig. 1.), in ode to we and coect pefomance of the geneato, we need to obtain a geometa mode with suffient accuacy and appopiate chaacteists of geometa components. An appopiate geometa mode wi incease the accuacy of geneato paametes cacuation and hence aises effectivene and effiency of the geneato. In the past studies the MCG coi was simuated ony by esistance and inductance [1]-[]. In ou studies we have neve seen the effects of stay capacitance in the MCG modeing. In this pape, fo the fist time, the effects of stay capacitance in the MCG ae discued and anayzed. Howeve in the past, high-fequency behavio magnet components ae discued in vaious papes and iteatues. But the pobems eated ony to the stay sef-capacitances and H.F. esistance of a singe aye and muti ayes inductos, cois, cuits of eecton and eect in vey sma dimensions [3]-[6]. Aso behavio of a tansfome and detemination and cacuation of the stay capacitances in the high- Ianian Jouna of Eecta & Eecton Engineeing, 011. Pape fist eceived 19 Aug. 010 and in evised fom 15 Nov * The Autho is with the Depatment of Eecta Eng., Shahed Univesity, Tehan, Ian. E-mai: omidezati000@yahoo.com. ** The Autho is with the Depatment of Eecta Eng., Shahed Univesity, Tehan, Ian. E-mai: besmi@shahed.ac.i. fequencies and high-votages ae discued and anayzed in [7]-[11]. Effects of stay and eddy cuents in tansfome ae descibed in [1]. The aspects eated to the paasit ac winding esistances and coe oes fo singe and mutipe-aye inductos, RF cuits and EMI fites, eectomagnet fied theoy ae discued and anayzed in [13]-[18]. The most ecent impotant eseaches ae pesented about high fequency magnet components in [19]. But in the iteatues eview we have not seen any pape about stay capacitance modeing fo the MCG. The aim of this pape is to pesent a mode to cacuate stay capacitance of MCG coi fo one tun with singe aye and ectangua co section. Paasit capacitance of the coi has signifant effect on geneato function. The pupose of this pape is to pesent a new method fo pedting and cacuating stay capacitance of OTC with singe-aye, k-conducto wie fiaments (k-cwfs) (k=numbe of the conducto wie fiaments) in fom (aangement) of ectangua co-section (Fig. ). The poposed method in this pape is a function of vaious paametes, tun geometa coodinates and the numbe of CWFs in OTC. By using the pesented method, we can cacuate stay capacitance of OTC. Futhe in MCG, effects of eecta fied wi be too much, because of two easons; the votage incease in each tun duing exposion poge and decease of the numbe of coi tuns. Ianian Jouna of Eecta & Eecton Engineeing, Vo. 7, No. 1, Mach

2 It shoud be noted that eecta fied poduced by opeation of the MCG poce and the ines of eecta fied poduced by it, between tuns of the coi and aso between tun and cyinda ine paced at the coi cente, make tun-tun ( tt ) C and tun-ine ( C t ) capacitances. Anaysis pesented in this pape appies fo cois incude singe-aye with fiamentay conductos in fom of ectangua co-section. Fequency incease in the coi of MCG makes skin effect and poximity effects whh finay causes incease of esistance of the coi tuns. Aso, because of capacitance in the system, esutant impedance of the geneato wi incease. Theefoe, to cacuate tota capacitance of one tun of coi (OTC), we need to cacuate: Stay capacitance between conducto wie fiaments in one tun, wie fiament - wie fiament capasitance ( C ). Stay capacitance between wie fiaments of one tun coi and ine, wie fiament ine capasitance ( C s ). Agoithm and Method of Capacitance Agoithm used in this pape is as foows: (a) Consideing an ovea mode fo equivaent cuit of OTC. (b) Cacuation of fiament-fiament capacitance ( C ) and fiament-ine ( C s ) capacitance. (c) Cacuation of fiament-ine capacitance ( C s ). (d) Cacuation of tota paasit capacitance C(k) fo OTC. (e) Simuation of tota paasit capacitance C(k) fo OTC. (f) Tota stay capacitance of OTC of the MCG can be modeed by a umped capacitance connected between the teminas of OTC, as shown in Fig. 3, [5]. Fig. 1 Heix geneato (MCG) with exposive detonato. (a)befoe exposion, (b) duing exposion. Fig. View of OTC of the MCG incudes singe-aye k- CWFs in fom of ectangua co-section. (a) (b) Fig. 3 Equivaent cuit of OTC of the MCG. a) Equivaent cuit of OTC of the MCG b) Simpified equivaent RLC cuit of the geneato OTC.1 Aumptions fo OTC Incuding k-cwfs In ode to daw detais of Fig. 3, we suppose foowing aumptions. A) We negect the capacitance between nonadjacent wie fiament tuns of OTC. B) Ai gap between a CWFs in a tun of the geneato coi and centa cyinda ine ae equa and consideed to be Z. Because ai gap between CWFs and centa cyinda ine is much bigge than the diamete of the co-section of one tun of the coi, so it coud be negected by a sight eo. See Fig. 5. C) We negect capacitance eakage effects between edges of two CWFs and aso between edges of CWF and centa cyinda ine of the MCG. Figue 4 shows capacitances poduced between CWFs of ony one tun with ine, whh incudes the foowing: 1- Fiament-fiament capacitance ( C ) between non-adjacent wie fiaments in a tun of the coi. - Fiament-fiament capacitance ( C ) between adjacent wie fiaments in a tun of the coi. 3- Fiament-ine capacitance ( C s ). Figue 5 shows the co-section of OTC with one aye of CWFs whh has been wounded unifomy. It is ceay seen that, in ode to cacuate the capacitance between two CWFs of OTC, fist we conside a egua hexagon with pefecty equa anges, π /3 adian and pefecty equa sides that each side be equa to y in ound co-section of each CWF so that the cente of this hexagon oveaps with the cente of the ound co-section of each CWF. Acs opposite to sides of this hexagon ae pefecty equa because of geomet equaity of sides. Now, conside an equiatea ozenge-shaped bas ce (ELBC) composed of two tapezoid shape egions (egions 1 and of Fig. 7). Extension of the sides of this equiatea ozenge wi pa fom two adjacent apex of hexagon and wi be tangent to extena sufaces of adjacent CWFs. 0 Ianian Jouna of Eecta & Eecton Engineeing, Vo. 7, No. 1, Mach 011

3 Fig. 6 An abcd equiatea ozenge-shape bas ce fo cacuating fiament-fiament capacitance. a) Vespiay egua hexagon (VHR) egion b) abcd bas ce. Fig. 4 View of tota stay capacitance of one tun incudes sevea wie fiaments of the MCG coi in non ea scae. Fig. 5 Rectangua co-section of one tun of the MCG coi incude singe-aye CWFs in non ea scae. Hee, the hexagon egion whh is simia to a egua hexagon netwok and aso simia to vespiay is caed vespiay egua hexagona (VHR) egion. Figues 6 and 7 show ozenge bas ce abcd eated to fiament-fiament capacitance ( C ). By ooking at Figs. 5 and 6, we can find geomet symmety of CWFs of OTC. By consideing this geomet symmety of CWFs, faction of eect fied ines whh exit fom a CWF, is competey suounded by two adjacent CWFs aound this CWF. By consideing the geomet symmety of the coi, eect fied ines shoud be divided equay among two adjacent CWFs. If we conside two cose adjacent CWFs in one tun, then dc eementay capacitance between two eementay sufaces of these two CWFs with aea of ds whh ae opposite to each othe wi be: ds (1) dc = g( β) ε = ε * ε () 0 1 whee, ε is eative pemittivity and ε 0 = 8.85*10 is vacuum pemittivity and g(β) is ength of eect fied ines between two eementay conducto sufaces whh ae opposite to each othe. In this pape, "g" is a function of two CWFs eementay suface positions of OTC with ange "g" and is not constant. Thus, position of each eementay suface fo each cua CWF in a tun of the geneato coi coud be stated by ange "g" coodinates (Fig. 7). 3 Fiament-Fiament Capacitance Between Two CWFs in OTC 3.1 Stuctue of ELBC of Vespiay Mode Figue 7 shows an equiatea ozenge bas ce 'abcd' whh foms fiament-fiament capacitance ( C ). It was ceay seen that the geomet stuctue of the bas ce is the same fo two adjacent CWFs of the same aye and two adjacent CWFs of diffeent ayes. Thus, intena egion of the co-section of one tun of coi coud be divided into simia bas ces and ony adjacent ces to centa cyinda ine wi be diffeent fom fiament- fiament ce. (a) (b) Fig. 7 View of path g(β) of eecta fied ines E unde ange β between two adjacent CWFs in OTC of the geneato. a) View of g(β) path of eecta fied ines unde β ange b) Two tapezium-shape egions 1 and in ELBC Moseh & Besmi: Cacuation of Stay Capacitances of MCG Coi 1

4 By an appoximation and simpe peiminay aumption, we can suppose a bas ces to be simia and identa. These bas ces incude a egion of the peiphey of the CWFs, whh coespond to ange of π /3 adian. (See Fig. 6). Theefoe, in ode to obtain fiament-fiament capacitance of one tun of coi, we shoud integate (1) unde ange of π /3 adian. This expeion can be accepted fo CWFs whh ae suounded by two othe adjacent CWFs, because of geomet symmety as shown in Figs. 5, 6 and 7. By appoximation, we can negect the edge capacitance and finging effects. Theefoe, we can use the same ange of π /3 adian fo CWFs whh ae not pefecty suounded. Fo equiatea ozenge-shape bas ce (ELBC) shown in Fig. 6, thee diffeent egions ae coed by eect fied ines that they ae: 1- Insuating coating egion of the fist CWF. - Insuating coating egion of the second CWF. 3- Ai gap egion between two above CWFs in OTC whh ae adjacent. Theefoe, eementay capacitance of dc between adjacent CWFs in OTC of the geneato is equivaent to seies combination of thee eementay capacitos in thee above egions. In othe wods, fist capacito is eated to insuating coating of the fist CWF, second capacito is eated to ai gap between two adjacent CWFs and thid capacito is eated to insuating coating of the second CWF. Sufaces of CWFs ae consideed co-potentia by a eativey good appoximation. As you can see, the diection of eect fied ines between two adjacent CWFs in OTC is in adia fom. By a good appoximation, we can conside the shotest poibe path whh is paae to the ine connecting two centes of two adjacent cua CWFs, as the path of the eect fied ines unde consideation. This appoximation appies fo sma amounts of β ange whh has a main oe in fiament-fiament capacitance ( C ). Fo big amounts of β, eo of this appoximation wi incease, so that big amounts of β ead to incease the capacitance vaue in compaison with the actua vaue. Anyway, the fiament-fiament capacitance of sufaces wi decease when β amounts incease. Aso by inceasing of β amounts, eo amount poduced in capacitance wi be negigibe. 3. Capacitance of Insuating Coating ( C ) Between Two Adjacent CWFs in OTC of the Geneato In this section, a method is pesented to cacuate capacitance of insuating coating in one tun of the MCG coi. Fig. 8 shows a cyinda eementay suface paced between the suface of CWFs and the suface of insuating coating. Eementay capacitance eated to this cyinda insuating wi be: Fig. 8 View of cyinda eementay co-section paced in insuating coating in non-ea scae. *d β*dh dc = ε ds = ε * ε * g( β) 0 (3) d Integation of this equation fo the ange of adius () fom the adius of CWF without insuating coating ( w ) to oute adius of CWF with insuating coating ( w ) and fo h (tun ength of CWF) fom zeo to tun ength of CWF ( L ts ), shows the capacitance of insuating coating imited to eementay ange of dβ as: ts w dc = ε * ε *d * dh 0 β (4) 0 d w So we have: dc = ε * ε * ts dβ 0 n w (5) Futhemoe since two insuating coating of two CWFs in one tun of the coi ae combined in seies fom, so insuating coating eementay capacitance of a bas ce of two adjacent CWFs with β ange wi be: Ianian Jouna of Eecta & Eecton Engineeing, Vo. 7, No. 1, Mach 011

5 1 1 1 = + = dc dc dc dc Theefoe eementay capacitance of insuating coating of a bas ce is dc = ε * ε * ts dβ (7) 0 *n w As a esut, by integation of (7) on β, capacitance of insuating coating of two adjacent CWFs in OTC of the geneato is obtained by: C ( β) = ε * ε * ts 0 β (8) *n w 3.3 Ai Gap Capacitance ( C ag ) Between Two Adjacent CWFs in OTC of the Geneato Consideing Figs. 6 and 7, ange of β is: g( β) cos( β) = 1 D (6) (9) Theefoe the ength of the paths of the eect fied whh is a function of β ange wi be: g( β) = D *(1 cos( β)) (10) The aea of eementay suface of the CWF whh has insuating coating and is in fom of an eementay ing of ength ( ts ) (tun ength of the CWF) wi be: 1 (11) ds = * *D dβ ts Capacitance between two adjacent CWFs eation to ai gap in one tun wi be: ds dc = ε * ag 0 g( β) (1) Tota capacitance of dcag eated to ai gap between two adjacent CWFs wi be: ts *D w ts dc ag ( β) = ε * d β = ε * dβ 0 *g( β) 0 *(1 cos( β)) (13) Theefoe, we can obtain ai gap capacitance ( C ag ) between two adjacent CWFs in OTC of the geneato by integation of (14) in β= π/6. Thus we wi have: π 1 β= 1 (14) 6 C ag ( β) = * ε * 0 ts dβ β 0 = 1 cos( β) 4 Cacuation of Fiament-Fiament Capacitance C Between Two Adjacent CWFs in OTC of the MCG Since capacitances of two egions 1 and between two adjacent CWFs wi be combined in paae fom, so we have: dc ( β) = dc ( β) + dc ( β) = *dc ( β) (15) 1 1 whee, dc ( β ) is tota capacitance of egion 1 and 1 dc ( β ) is tota capacitance of egion between two CWFs that by consideing the geometa symmety of two egions 1 and in Fig. 7, the capacitances of these two egions ae equa. Since eementay capacitance of insuating coating and eementay capacitance of ai gap between two adjacent CWFs of each two tapezoid shaped egions 1 o ae combined in seies fom, so by consideing (15), tota capacitance between two adjacent CWFs wi be: dc *dc ( β) ag dc ( β) = *dc ( β) = (16) 1 dc + dc ( β) ag Theefoe by consideing the effect of insuating coating, tota fiament-fiament capacitances ( C ) of two adjacent CWFs in OTC of the MCG is: ε * C ( ) 0 ts β = dβ (17) D 1 cos( β) + *n ε 1 D By integation of (17) and by consideing ange of ange (β) fom zeo to π /6 and by epacing paametes, vaue of capacitance ( C ) between two adjacent CWFs wi be: π Ac tan tan * K 4 (18) C = ε * ε * * 0 ts K3 D whee K1 =, K = ε *( n(k1) ) and D w ( ) K3 = ε *n(k1) + n(k1). 5 Fiament-Line Capacitance Between one CWF in OTC and Line To cacuate fiament-ine capacitance ( C s ), fist we use (5) to cacuate the capacitance of insuating coating whh is paced between CWF and ine. Hee we emind that we have ony one insuating coating fo cacuating fiament-ine capacitance. So we have: ε * ε * dc = dc = 0 ts dα (19) s n w Besides, path ength of eect fied ines in ai gap between CWF and imaginay ψ ine whh on aveage paes fom the midde of coi tun, is equa to Z. Aso, Moseh & Besmi: Cacuation of Stay Capacitances of MCG Coi 3

6 the bas ce of fiament-ine capacitance is bigge and wide than the bas ce of fiament-fiament capacitance and a pat of the peimete of the CWF whh by an appoximation and simpified aumptions, confoms to π/3 adian, wi incude the bas ce of fiament-ine (Fig. 7). Consideing Figs. 6 and 7, the vaue of ange α is: g( α ) (0) cos( α ) = 1 D w Thus, path ength of eect fied that is function of ange α wi be: D *(1 cos( α)) (1) g( α) = The aea of eementay suface of the CWF whh has insuating coating and is in fom of an eementay ing of ength ts (tun ength of the CWF) wi be: 1 () ds = * *D dα ts w Aso, capacitance between CWF and centa cyinda ine eated to ai gap is: ds (3) dc ag s = ε * 0 g( α ) + Z whee dc is tota capacitance eated to egion 3 ag s of ai gap between CWF and centa cyinda ine. So, by consideing above expanations and equations (1), (), and (3), the ai gap capacitance of fiamentine of egion 3 is: ε * *D (4) 0 dc ( α) ts w d ag = α s (g( α) + Z) Thus to epace (1) in (4), ai gap capacitance between wie fiament and ine in egain (3) wi be: ε * *D 0 dc ( α) = ts w dα (5) ag s D *(1 cos( α)) + Z Consideing Figs. 6 and 7 and consideing that α is between zeo to π /3 and aso by integation of (5), vaue of the ai gap capacitance between ine and CWF of egion (3) that pace in one tun wi be: (D + Z) Ac tan 3+ Z (6) C ( α) = ε * *D * ag s 0 ts w D + Z Z* 1+ Z Since capacitance between CWF and centa cyinda ine of two egions 3 and 4 ae combined in paae fom, we have: C ( α) = dc ( α) + dc ( α) = *dc ( α) (7) s s3 s4 s3 whee dc ( ) s3 α is tota capacitance of egion (3) and dc ( α ) is tota capacitance of egion (4) between CWF s4 and centa cyinda ine and consideing the geometa symmety of two egions 3 and 4 in Fig. 7, capacitances of these two egions ae equa. Consideing the seies combination of ai gap eementay capacitance and insuating coating of tapezoid shaped egion 3 o 4, then the tota equivaent capacitance between CWF in OTC and centa cyinda ine with consideing (7) is: dc * dc ag s s dc ( α) = *dc ( α) = (8) s s3 dc + dc s ag s Theefoe we have: *D dc ( α) = * ε ε * ts w dα (9) s 0 K4 + (K5*[ ε ε ]) 0ts whee K4 = ( ε *D * )*n(k1) and 0 w ts K5 = Z + D (1 cos( α)). So C ( ) s α wi be: K9 1 Ac tan * (30) K6 + K7 + K8 3 C ( α) = K10* s K6 + K7 + K8 whee K6 = ε *Z(D + Z), K7 = D * ε (D + Z)n(K1), w w K8 = D *n (K1), K9 = ε (D + Z) + (D *n(k1)) and w w K10 = * ε * ε * * D. 0 ts w 6 Tota Stay Capacitance of OTC of the MCG In ode to obtain tota stay capacitance of one singe-aye tun of the MCG coi in fom of ectangua co-section, as it is shown in Fig. 9, fist by using fiament-fiament capacitance ( C ) and fiament-ine capacitance ( C s ) coesponding to Fig.10, we cacuate tota stay capacitance C(k), singe-aye adjacent k- CWFs of the MCG coi. A netwok composed of umped capacitos accoding to Fig. 10 can descibe tota stay capacitance (fiament-fiament pus fiament-ine). So equivaent cuit can be modeed by a tota stay capacito netwok C(k) [5]. Fig. 11 shows a netwok of umped capacitos fo one tun that has singe-aye k conducto wie fiaments (k-cwfs) in fom of ectangua co-section and is wounded aound the centa cyinda conducto ine, that fo one tun, capacito netwok of Fig. 11 shoud be soved. Centa cyinda conducto ine can be consideed as a node that fiament-ine capacitance ( ) s C is connected to this node. 4 Ianian Jouna of Eecta & Eecton Engineeing, Vo. 7, No. 1, Mach 011

7 C C(k = ) = C(1, ) = C + s (31) tt Fo moe numbe of wie fiaments, fist we conside thee adjacent wie fiaments of the MCG coi. Equivaent capacitance of this capacito netwok with thee wie fiaments (k=3) can be obtained by spitting capacitance ( C ) into two haves and appying the s() Fig. 9 One tun of the MCG coi incudes singe-aye k- CWFs in fom of ectangua co-section with cyinda ine paced in cente of the geneato coi. Fo OTC of the geneato that is composed of many CWFs in itsef, fist we conside two adjacent wie fiaments among wie fiaments in one tun of the geneato coi fo these two fiaments (k=), capacitance of netwok consists of capacitance ( C ) between two wie fiaments 1 and that is paae with the seies combination of the fiament-ine capacitances ( C ) s(1) and ( C ), whh C = C = C, whee ( C ) s() s(1) s() s s(1) and ( C ) ae capacitance between wie fiaments 1 s() and with centa cyinda ine of the geneato espectivey. So, equivaent capacitance of these two adjacent wie fiaments is: Fig. 10 Geometa space of one tun incude singe-aye k- CWFs and cyinda ine paced at cente of the coi tun and fiament-fiament paasit capacitance (C ) and fiamentine paasit capacitance (C s ) in non-ea scae. Fig. 11 Lump capacitos of one tun incude singe-aye k- CWFs of the MCG coi in non-ea scae. Δ / Υ tansfomations (Fig. 1). So, we wi have: C C (3) C(k = 3) = C(1,,3) = + s In ode to obtain tota capacitance of fou wie fiaments in one tun, we can conside one moe wie fiament at the side of thee wie fiaments. So, tota capacitance is equa to capacitance of pevious aangement (i.e. two wie fiaments), whh is in seies fom with fiament-fiament capacitance and paae to the seies combination of fiament-ine capacitance. Theefoe, fo k=4, we wi have: C *C() C C(k = 4) = C(1,,3, 4) = + s (33) C + *C() In ode to obtain the tota capacitance of k-cwfs, we can add one wie fiament to pevious wie fiaments in each time. Theefoe, tota stay capacitance of one tun of the MCG coi with any numbe of wie fiaments can be cacuated by mentioned method. So fo OTC of the MCG composed of singe-aye k-cwfs, we have: C *C(k = ) C C(k) = + s (34) C + *C(k = ) whee, C(k-) is stay capacitance of (k-) wie fiaments of OTC of the MCG composed of k-cwfs and k is the tota numbe of wie fiaments existing in OTC of the geneato. Two ayes k-cwfs tun is affected by highe stay capacitance than singe-aye tun. Aso it is affected by highe esistance at high fequency opeation. Thus, using two-ayes k-cwfs tun is not pope fo the MCG inductos designed fo high-fequency opeation. On the othe hand, the ovea stay capacitance of cois with thee ayes deceases, but the soution of the umped capacito netwok becomes moe compated. Aso the poximity effect inceases as the numbe of ayes incease. Theefoe by consideing the above causes, singe-aye cois shoud be used at high fequencies at the MCG system. 7 Resuts of Simuations Consideing Figs and simuation eated to the tota capacitance C(k) of OTC, if tun ength of CWFs ( ts ) and co-section diamete of CWFs in coi tun (D w ) inceases and the ai gap between CWFs and Moseh & Besmi: Cacuation of Stay Capacitances of MCG Coi 5

8 consideing above esuts, to decease the tota stay capacitance of singe-aye k-cwfs, we shoud incease the numbe of CWFs in tun and ai gap between CWFs and centa cyinda ine (z), on the othe hand, we shoud decease co-section diamete of CWFs in coi tun (D w ) and tun ength of CWFs( ts ). Fig. 1 Capacitance between fou adjacent CWFs in one tun wounded aound the centa cyinda ine. Fig. 13 Tota capacitance of one singe-aye tun of the MCG ( ) ts fo diffeent tun engths of CWFs. 8 Veifation of the Resuts In this pape by consideing the imitations and pobems of the MCG exposion, in ode to examine and study the esuts accuacy, the poposed method to cacuate stay capacitance is compaed with the esuts of efeence [5]. In this case study we aume that the MCG has one tun with paametes as foow: D = 49.5*10 mm, D = 45*10 mm, w t =.5*10 mm, ε = 3.5, ε 8.85*10 1 π =, β =, 0 6 = 4.495*10 mm, Z 50 mm ts =. By using equation (17) and with consideing to above mention chaacteists capacitance C wi be: 8.85*10 1 * 4.495*10 π/6 C = 0 dβ (3.5) *10 = 1 cos( β) + *n 45* pf. Aso by using the esuts of [5] with same above paametes we wi have C =3.934 pf. So we can see that the cacuated fiament-fiament capacitance C by equation (17) is amost same as fiament-fiament capacitance C in efeence [5]. Fig. 14 Tota capacitance of one singe-aye tun of the MCG fo diffeent co-section diametes of CWFs. Fig. 15 Tota capacitance of one singe-aye tun of the MCG fo diffeent ai gaps between tun and ine. centa cyinda ine (z) deceases, then the tota stay capacitance vaue of singe-aye k-cwfs in fom of ectangua co-section in OTC wi incease. On the othe hand, we can see decease of tota stay capacitance vaue of singe-aye k-cwfs by inceasing the numbe of singe-aye CWFs in one tun. Theefoe 9 Concusions In this pape, we pesented a new method that caed VRH mode, in ode to cacuate stay capacitance between singe-aye CWFs in OTC in fom of ectangua co-section in the MCG system. The tota stay capacitance of OTC of the MCG that incude singe-aye k-cwfs was obtained fom two simpified equivaent cuits with distibuted paametes. Fina mode incudes the esistance seied with equivaent inductance of each CWF and paaeed with fiament-fiament capacitances between adjacent CWFs. The pesented method appies we and is tue to cacuate the stay capacitance of singe-aye CWFs tun in fom of ectangua co-section. It was seen that the numbe of CWFs existing in one tun has effect on tota stay capacitance of the MCG. The pesented mode is simpe and the numbe of CWFs, diamete of CWFs existing in one tun of the MCG coi, kind of the insuating coating, thkne of insuating coating of CWFs and tun ength of CWFs wi be effective on tota stay capacitance of the geneato. 6 Ianian Jouna of Eecta & Eecton Engineeing, Vo. 7, No. 1, Mach 011

9 Aso, in ode to decease tota stay capacitance of singe-aye k-cwfs in one tun of the coi, we shoud decease the diamete of CWFs cua co-section ( D ) and CWFs ength ( ) and aso incease ai gap w ts between CWFs and centa cyinda ine (z) and numbe of CWFs in one tun. On the othe hand, by consideing the simuations esuts, we saw that by nceasing the CWFs numbe in one tun of the coi, the capacitance vaue of one tun of the MCG coi wi decease. Finay the veifation of simuation fo fiament-fiament capacitance is obtained with compae to esuts of efeence [5]. Refeences [1] Neube A. A., Exposivey Diven Pused Powe Hea Magnet Fux Compeion Geneatos, 1nd ed., Vo. 1, New Yok: Spinge-Veag, 005. [] Atgibes L. L., Bown M. D. J. and Novac B. M., Magnetocumuative geneatos, 1nd ed., Vo. 1, New Yok, USA: Spinge-Veag, 000. [3] Gandi G. and Kazimieczuk M. K., Stay capacitances of singe-aye ai-coe inductos fo high-fequency appations, IEEE Tans. Industy appations, Vo. 35, No. 5, pp , Sep./Oct [4] Yu Q. and Homes T. W., A Study on stay capacitance modeing of inductos by using the Finite Eement method, IEEE Tans. Eectomagnet Compatibiity, Vo. 43, No. 1, pp , Feb [5] Maaini A. and Kazimieczuk M. K., Sef- Eectons, Vo. 1, No. 4, pp , Juy capacitance of inductos, IEEEE Tans. Powe [6] Ajayi O., H. F. Resistance of ound-wie singe- aye inductance cois, Poc. IEE, Vo. 10, No. 9, pp , Sep [7] Duedoth W. T., Equivaent capacitances of tansfome winding, Wiee Eng., Vo. 3, pp , Jun [8] Lu H., Expeimenta detemination of stay capacitances in high fequency tansfomes, IEEE Tans. Powe Eectons,, Vo. 18, No. 5, pp , Sep [9] Daeando L., Cavacante F. and Koa J. W., Cacuation of the Stay Capacitance of High- Votage Tansfomes, Int. Rep. 03/04, ETH Zuh, Zuh, Switzeand, Ma [10] Daeando L., Siveia F. and Koa J. W., Sef-Capacitance of High Votage Tansfomes, IEEE Tans. Powe Eectons, Vo., No. 5, pp , Sep [11] Gone N. R., Tansfome fo Eecton Cuits, Second Edition, McGaw-Hi: New- Yok, [1] Dowe P. L., Effects of eddy cuents in tansfome windings, Poc. IEE, Vo. 113, No. 8, pp , Aug [13] Chen R., Van Wyk J. D., Wang S. and Odendaa W. G., Appation of stuctua winding capacitance canceationn fo integated EMI fites by embedding conductive ayes, in Poc. Ind. App. Soc.., pp , 004. [14] Yu Q., RF Equivaent cuit modeing of feite- coe inductos and chaacteization of coe mateias, IEEE Tans. Eectomagnetc Compatibiity, Vo. 44, No. 1, pp. 58-6, Feb. 00. [15] Maaini A., Kazimieczuk M. K. and Gandi G., Lumped paamete modes fo singe and mutipe-aye inductos, Poc. Powe Eectons, Speciaist Conf., Baveno, June [16] Azaky A. and Hawey R., Fundamentas of Eectomagnet Fied Theoy, Haape Co Ltd.: London, [17] Batoi M., Reatti A. and Kazimieczuk M. K., Modeing of ion powde inductos at highh fequencies, IEEE Industy Appations Conf. Denve, Co, pp , [18] Sneing E. C., Soft Feites, Popeties and Appations, London, U.K.: ILIFFE Books, [19] Kazimieczuk M. K., High Fequency Magnet Components, U.K.: John Wiey & Sons, Ltd, 009. Majid Ezati Moseh was bon in Tehan, Ian, in He is a Membe of IAEEE in 005 in Ian. He eceived the B.S. degee in eecta engineeingg fom Eecton Techna Univesity of Shiaz, Ian, in 005 and the M.S.. degee in eecta engineeing fom Shahed Univesity in 009. His eseach inteests ae in high-fequency machines, magneto cumuative geneatos, magnet. He has two inventions and sevea county awad of Ian. He has pubished sevea books and papes in eectons engineeing. Mohammad Reza Besmi was bon in Tehan, Ian, in He eceived the B.S. degee in eecta engineeingg fom Amikabi Univesity, Ian, in 1989, the M.Tech degee in eecta engineeing fom Univesity of Indiann Institute of Technoogy, Dehi in June 199 and the Ph.D. degee in eecta engineeing fom Newcaste upon Tyne Univesity in Septembe Fom 1997 to 1999, he was with the Nioo Reseach Institute in Ian. He is cuenty Aistant Pofeo of Eecta Engineeing Depatment and head of powe goup at Shahed Univesity. His cuent eseach wok is in specia eecta machine design. Moseh & Besmi: Cacuation of Stay Capacitances of MCG Coi 7