Iteratioal Scietific Collouium Modellig for Electromagetic Processig Haover, March 4-6, 3 Electromechaical Oscillatios Ifluece to Iductace of Arc Furace Secod Circuit V.S. Cheredicheko, A.I. Aliferov, R.A. Bikeev Abstract Mathematical model for electromechaical oscillatio ivestigatio of arc furace cable chais is elaborated. Cable chai i each phase is represeted as oe coductor. Mass ad stiffess are localized i lower poit of cable chai loop. Electrodyamical iteractios are determied by Biot-Savart's low. Arc furace is represeted three-phases usymmetrical circuit. Electric arc is described differetial coductivity euatio. Mutual iductace betwee phases is calculated by euatio, which allow to varies ucertaities i dispositio of coductors. This model calculates phase iductace chages as a fuctio of time ad electromechaical oscillatios of cable chais. Itroductio The iductace of power supplyig electric cables for arc electric furaces (AEF) is calculated ow by geeralizatio method. []. The essece of this method is to collect the partial solutios of Maxwell s euatio for differet power supplyig schemes ad cables geometry of simple type ito oe table ad combie this solutios for complex geometry. But if we have the wires beig situated arbitrary relative each others, this method leads to great error. Numerical method beig suggested i [3, 4] is also based o aalytical expressios ad tables with the same shortcomigs. By this reasos, it is preferable to calculate iductace of the wires directly from Maxwell s euatios usig umerical methods.. Mathematical model The iductio impedace of cables chais is ofte prevails the 5% of total iductace of secodary power supplyig cables. It is proofed ow [], that decreasig of cable s iductio impedace leads to better electric characteristics of AEF, but icrease the mechaical oscillatios. The cables chais decliatio agle α from its euilibrium positios ca achieves 3 4 degrees uder the actio of electrodyamics forces. The iductio impedace of AEF s cables has the dyamic compoet ω i (τ) durig its displacemet. It is ecessary therefore to brig it out the ifluece of mechaical oscillatios oto iductace of AEF s cables. I order to clarify this pheomeo there was developed the mathematical model of electromechaical oscillatios of cables chais. The chai of each phase represets itself the sigle wire with its mass ad stiffess. Cartesia coordiates is liked to the poit of middle phase chai fasteig. All wires are decomposed oto rectiliear pieces. The iductio i each space poit W with coordiates (x W, y W, z W ) is carried out accordig to Biot-Savart- aplace law [] ad the the vector of magetic iductio B with its compoets B x, By, Bz is defied: 85
I AB I AB I AB B { B,B,B x y z} k, x k, y kz, (.) where k x, k y, k z are the dimesioless coefficiets which values deped o relative displacemet of cable s rectiliear piece ad the poit where the vector of magetic iductio should be calculated; I AB is the istataeous curret s value beig ruig o a cable piece aalyzed, А; -7 is the magetic costat, H/m. Itegral euatio for electrodyamics force F e [6], which is actig o a acute cable piece of legth l, from a side of aother cable piece has a follows solutio: { F,F,F } F e x y z I ABICDKx, I ABICDK y, I ABICDKz, (.) where K x, K y, K z are the dimesioless coefficiets which values deped o relative displacemet ad geometry of actig currets. l α X А C Fig.. Scheme of cable chai deviatio from static euilibrium. Z y O Y d α m l + D l dt dα dt It is well kow from practice, that cable s chais have the maximal curvature i a plae YOZ (see fig.), while the others deformatios are egligible. That is why the cable s euilibrium positio is defied by the agle α of cable s plae deviatio from its static euilibrium. The calculatio of euilibrium positio is provided uder coditio of all forces to be focused to the lowest poit of cable s loop. The horizotal displacemet of the lowest poit y is defied by: y l si(α), (.3) where l is the distace from the actual poit to rotatio axes beig spaed betwee poits А ad О. The model euatio of electromechaical oscillatios is as follows: m& y + Dy& + Ky + mg si( α) Fe ( I, y), (.4) which is trasformed after takig ito accout the euatio (.3) ito: + K l si ( α) + mg si( α) F ( I,α), (.5) where m is the itrisic mass of the cable; D is the dampig coefficiet; K is the stiffess of the cable; F э is the geeralized electromagetic force. Self iductace ad the mutual iductaces are defied by follows euatios []: + i M ij, M M kl, j k l j i where i is the self iductace of i-th sectio of the cable; M ij is the mutual iductace betwee i-th ad j-th cable s sectios; M kl is the mutual iductace betwee k-th sectio of oe flexible cable ad l-th sectio of aother oe; is the total umber of sectios the cable is disitegrated to. The total iductace of each phases the will be writte as: e 86
A B C Ai Bi Ci + M + M + M A B C ( τ), M A( τ) M Aij ( τ) ± M BA( τ) ± M CA( τ) j j i ( τ), M B( τ) MBij ( τ) ± M AB( τ) ± M CB( τ) j j i ( τ), M C ( τ) MCij ( τ) ± M BC ( τ) ± M AC ( τ) j j i, (.6) where the A, B, C are the total iductaces of flexible cables of each phases; M A (τ), M B (τ), M C (τ) are the dyamics iductaces of each phases. Self iductace of rectiliear wire is defied by follows formula: N G, N l[ l( l) ] 7, G l l( g) 7, (.7) where l is the legth of the wire, м; g are the averaged distace from the ceter of wire s cross sectio to its border, м. The mutual iductace of two rectiliear wires l ad l are as follows []: dl ( dl M cos ϕ ), (.8) R ll where dl ad dl are the differetials of the wires l ad l ; R is the distace betwee the wires; ϕ is the agle betwee the wires l ad l. The solutio of euatio (.8) was take from [] ad represet itself the follows expressio: Fp x p l a Ap si R p ( ϕ ) M cos( ϕ ) ( ) p + Fp, (.9) p ( y x cos( ϕ ) + R ) + y l x y cos( ϕ ) p arctg tg a x + p y x p y cos p ( + R ) p ( ) ϕ + a p + Ap. (.) Direct usig of solutios (.9) ad (.) is impossible for some wires displacemet because of idefiite values of A p for the followig cases: ) the wires are parallel ad its currets are cocurret i its directio or cotra curret (a, ϕ ; ϕ π); ) the wires are crossed i oe poit or they have a mutual lie (a <ϕ<π); 3) the whole displacemet of the wire are coicides ad their curret are cotra curret (a ϕ π). For the cases metioed, the ucertaities of expressios A p from formulas (.9), (.) we have solved as follows: ) the parallel wires a, ϕ : a lim ϕ p p a ( ) p + arctg tg ( ) ( ) si ϕ a the parallel wires a, ϕ π: p+ R p, (.) ( ) + + + ( ) ( ) + lim p a xp y Rp ϕ p a arctg tg, (.) ϕ π p si ϕ a p x + + p y Rp a 87
x p + y + R p For the case ϕ π the limit is defied oly if >. This expressio is valid for all a cases of parallel wires displacemet. ) the wire is crossed i a poit a <ϕ<π: a lim a si < ϕ< π ( ϕ ) arctg tg a 3) the wires are displaced o the same lie a ϕ π: a lim arctg tg a si( ϕ ) a ϕ π, (.3), (.4) The expressios (.9), (.) together with (.)-(.4) are defied the mutual iductace of rectiliear wires which are displaced arbitrary i a space. The parameter а i (.9)-(.4) is the distace betwee the parallel plaes to which the both wires belog []. The euivalet electrical scheme of AEF is preseted i fig., where ei are the istataeous voltage values of each phase; Ri active resistaces of each phases; Xi are the impeduces of secodary wires without the cables; i are the self iductaces of the cables; Mij(τ) are the mutual iductaces betwee the cables; Rdi are the arc s resistaces. Fig.. Three phase euivalet electrical circuit of AEF. The currets of each phase i euivalet asymmetrical electrical circuit was defied from the follows system of differetial euatios: di di di3 di di ir + + M( τ ) + M3( τ ) + ir d ir M( τ ) dt dt dt dt dt di3 M 3 ( τ ) ir d e () t e () t, dt di di di3 di3 di di ir + M M3 ir d ir 3 3 3 M M3 dt + dt + dt + dt dt dt (.5) ir 3 d 3 e() t e3() t, i+ i + i 3. The euatios (.), (.), (.5)-(.7), (.9)-(.5) represet itself the mathematical model of AEF, which permits to calculate the iductio impedace Х АВС f(τ) of power supplyig cables durig the electromechaical oscillatios of cables chais. 88
. Summary Fig.3 presets the time depedece currets i three phases of AEF. Fig.4 presets the decliatio agels, the dyamics of mutual iductaces betwee the flexible cables ad the total impedace of each phase. The umerous calculatios of euatios (.5) revealed the defiite liks betwee the amplitudes of curret perturbatio, mechaical oscillatios of cables chais ad dyamics impedace chagig of AEF s secodary wires i a wide rage of amplitudes ad freuecies. The developed mathematical model have bee used for aalyzig the electrodyamical processes of idustrial AEFs.. 5. 5 ia, A. 5. 5 3 4 5 6 7 а). 5. 5 ib, A. 5. 5 3 4 5 6 7 b). 5. 5 ic, A. 5. 5 3 4 5 6 7 c) Fig. 3. Time depedecies of the currets: а) curret i a phase А; b) curret i a phase B; c) curret i a phase C. 89
ϕ y, 4 ϕ A Aa, Ab, Ac, ϕ C ϕ B 4 3 4 5 6 7 а) m xy, H.43 m ab m bc.85.7 m ca.7 3 4 5 6 7 b) X Y, mohm xa 4.5 4.4 X B X A xb xc 4.3 3.9 X C 3.8 3 4 5 6 7 c) Fig.4. The ifluece of flexible cables agle decliatio o iductio impedace of secodary wires: а) the decliatio agels of flexible cables of all three phases over the time; b) the dyamical mutual iductace betwee the phases for each pair of flexible cables; c) the iductio impedace of the short wire. Refereces [] Short wires ad the electrical parameters of AEF. Had book / Edited by B. Datsis. М.: Metallurgiya, 987 [] Tseytli.A. The iductace of the wires ad loops. Goseergoizdat, 95. [3] Haiso A.V., Treyvas V.G, Pirogov N.A. The mathematical modelig of secodary wires of AEF // Elektrotehicheskya Promyshleost, 979, p.-. 9
[4] Pirogov N.A. The iductace computatio of secodary wires of AEF // Trudy VNIIETO.- M. Eergiya, v., 979, p.-. [5] Cheredicheko V.S., Elmaova.P. The electrodyamics of tree phase AEF // The electrotechology processes ad istallatios / Novosibirsk, 995, p.3-9. [6] Holyavskiy G.B. The calculatio of electrodyamical forces i electrical devices. Eergiya, 97. [7] Kalatarov P.., Tseytli.A. Calculatio of iductace. M.: Eergiya, 97. Authors Prof. Cheredicheko, Vladimir S. Novosibirsk State Techical Uiversity Karl Marx av. 639 Novosibirsk, Russia E-mail: elterm@tatra.power.stu.ru Prof. Aliferov, Alexadr I. Novosibirsk State Techical Uiversity Karl Marx av. 639 Novosibirsk, Russia E-mail: elterm@tatra.power.stu.ru Eg. Bikeev, Roma A. Novosibirsk State Techical Uiversity Karl Marx av. 639 Novosibirsk, Russia E-mail: elterm@tatra.power.stu.ru 9