U.P.B. Sci. Bull., Series C, Vol. 78, Iss., 16 ISSN 86-354 ELECTRODYNAMIC FORCES BETWEEN TWO DC BUSBARS DISTRIBUTION SYSTEMS CONDUCTORS Mri-Ctlin PETRESCU 1, Lucin PETRESCU This pper nlzes DC usr sstem used in power distriution unit. The spectrum of the mgnetic flu densit is investigted for different situtions (different distnce etween the rs) using comprison etween nlticl pproch nd numericl technique. Also, the forces etween the rs re investigted for oth norml (rted) nd short-circuit sitution. Investigtions re performed using nlticl nd numericl procedures. Kewords: Busr, sttic mgnetic field, FEM D, electrodnmic forces 1. Introduction Distriution sstems often contined usr conductors powered in high DC currents [1-]. These conductors re designed to offer protection to n user, ut in n out of ordinr sitution this might cuse dmges to the instlltions. In this kind of equipments, different prolems could pper: mgnetic field of medium intensit, high therml vrition, nd electrodnmicl forces [3]. DC usr sstems re used etensivel in the refining of chlorine, resulting products like custic sod, liquid chlorine, or chlorine dioide [3]. Another domin of intense usge of these distriution sstems is when high power DC driving utilit, like in pper mnufcture industr or trnsporttion, is needed. The im of this pper is to present n nlticl procedure vlidted numericl technique to evlute the mgnetic flu densit for DC usr distriution sstem. The electrodnmicl forces etween those conductors in rted nd high current regime re lso evluted [4]. The min concern ppers lthough when fult current (like short-circuit) is occurring. In one of these cses, the conductors m suffer mechnicl stresses which could generte considerle dmges of the entire distriution sstem [5-8]. The nlzed DC usr sstem is presented in Fig. 1. We hd considered two rectngulr copper conductors with cross section with the ssumption 1 Fcult of Electricl Engineering, Universit POLITEHNICA from Buchrest, e-mil: ctlin.petrescu@up.ro Fcult of Electricl Engineering, Universit POLITEHNICA from Buchrest, e-mil: lucin.petrescu@up.ro
4 Mri-Ctlin Petrescu, Lucin Petrescu tht the length l is much igger then the other dimensions (l>>, l>>), considering for the clculus s infinite long conductors. Fig. 1. DC Busr distriution sstems. Mgnetic Field Anlsis of DC Rectngulr Conductor First, we will consider rectngulr r with the cross section ( ). The pln-prllel electromgnetic prolem will generte mgnetic flu densit vector in the pln O of the cross section considered. Appling the Biot-Svrt- Lplce [8] formul, the B components in point P(,) in the usr proimit cn e found [9] - Fig. : d d B B μi = 8π μi = 8π ( ) ( ) ( ) ( ) d d d d (1) Fig.. Cross section of rectngulr r
Electrodnmic forces etween two DC usrs distriution sstems conductors 5 Using eqution (1), the module of the vector potentil t n point ner the rectngulr r is [9]: ( ) ( ) [ ] = d d I A ln 8π μ () Integrting nd simplifing, () ecomes: ( )( ) ( ) ( ) [ ] ( )( ) ( ) ( ) [ ] ( )( ) ( ) ( ) [ ] ( )( ) ( ) ( ) [ ] } ) ( ) ( ) ( ) ( ln ln ln ln { 8 = rcthn rcthn rcthn rcthn rcthn rcthn rcthn rcthn I A π μ (3) The components of the mgnetic flu densit will e determined s follow: = = A B A B (4) Accordingl, the totl vlue of the mgnetic flu densit is then: B B B = (5) For vlidting these epressions comprison etween the ove nlticl formuls integrted in common smolic softwre (Mple) [1] nd the numericl simultion free D softwre (FEMM) [11] ws performed.
6 Mri-Ctlin Petrescu, Lucin Petrescu We choose rectngulr r with the cross section of 36 mm ( = 3 mm nd = 3 mm). This kind of usr supports in DC over 869 A nd for the simultions we hd use rted current of 8 A. The simultions were mde ssuming = in oth situtions. The results re presented in Fig. 3. Fig. 3. Comprison etween the mgnetic flu densit vrition using the nlticl pproch (lue) nd the numericl simultion (red) The mgnetic flu densit spectrum for the rectngulr conductor is shown in Fig. 4. Fig. 4. Mgnetic flu densit spectrum for rectngulr conductor.
Electrodnmic forces etween two DC usrs distriution sstems conductors 7 3. Mgnetic Field of Sstem with Two DC Conductors Using the superposition nlsis of the sttic fields, the mgnetic flu densit etween two rectngulr conductors ws evluted. These two conductors re prt of usr distriution sstem powered with DC currents. Further we will consider lso the distnce d etween the centers of the conductors on the horizontl direction Fig. 5. Fig. 5. Busr distriution sstem composed of two rectngulr conductors. The stud ws crring out for three different vlues of the distnce etween the centers of the conductors: d (= 4) = 1 mm, d (=8) = 4 mm nd d (1) = 36 mm respectivel. In Fig. 6 one cn oserve the mgnetic flu densit spectrum etween the rs Fig. 6. Mgnetic flu densit spectrum for the distnce of 1 mm etween the rs
8 Mri-Ctlin Petrescu, Lucin Petrescu While Fig. 7 presents comprison etween the nlticl nd numericl computed vlues of the mgnetic field densit B over the distnce d = 1 mm etween the centers of the conductors. Fig. 7. Comprison etween the mgnetic flu densit vlues, computed numericl nd nlticl methods, over the distnce d = 1 mm etween the centers of the conductors In Figs. 8 nd 9 the comprtive vritions for the mgnetic flu densit etween the rs for other two different distnces: d = 4 mm nd d = 36 mm re presented.
Electrodnmic forces etween two DC usrs distriution sstems conductors 9 Fig. 8. Comprison etween the computed mgnetic flu densit for d = 4 mm Fig. 9. Comprison etween the computed mgnetic flu densit for d = 36 mm
3 Mri-Ctlin Petrescu, Lucin Petrescu The reltive error etween the mimum vlues of the mgnetic flu densit, evluted with (6), nd the verge reltive errors for these vlues on the distnce d, for ech cse studied computed with (7), were performed: nd Bm_ nlticl Bm_ numericl ε r ( Bm ) = 1[%] (6) B m_ nlticl n 1 Bnlticl Bnumericl ε r ( B) = 1 [%] (7) n B k = 1 nlticl where, n is the numer of the clcultion nd modeling points for the distnce etween the usrs. These reltive errors re presented in tle 1. Tle 1 Reltive errors etween nlticl nd numericl vlue for the mgnetic flu densit Studied Cse Distnce d [mm] ε r (B m ) [%] ε r (B) [%] 1 1 1.13 1.7 4.6.7 3 36 3.38 4.5 4. Electrodnmic Forces Determintion for the DC Busr Sstem The finl prt of the stud is focused on the nlticl nd numericl determintion of the electrodnmicl forces etween these two rs of the DC distriution sstem. The nlticl pproch uses some simplifing ssumptions: the height of the conductor is much igger then its width ( >> ), the distnce etween the conductors is much igger then their width (d >> ). Considering these, the force tht cts on the second conductor, due to the current from the first one, is to e computed with the following reltion [1]: μ l F d = I1I ϕ( d, ), (8) π d where d 4 4 ϕ ( d, ) = rcthn ln 1 (9) 4 d d d is lso known s the Dwight formul.
Electrodnmic forces etween two DC usrs distriution sstems conductors 31 The numericl determintion of the force ws mde using one conductor in FEMM nd selecting the Lorentz Force option (Fig. 1). Fig. 1. Numericl determintion of the electrodnmicl force in FEMM for the distnce of 1 mm etween the rs In oth nlticl nd numericl determintions, we considered the unit length l of the rs (per unit meter). As it ws epected, the verticl component of the force is much less thn the horizontl one. In tle, the forces clculted with formul (8), their numericl vlues determined with FEMM nd their reltive errors using (6) re presented. Tle Reltive errors etween nlticl nd numericl vlue for the electrodnmicl forces Studied Cse Anlticl vlue F n [N] Numericl vlue F num [N] ε r (F) [%] 1 4.469 4.618 3.1% 3.388 3.656 7.34% 3.695 3.86 1.69% It cn e oserved tht higher distnce etween the conductors genertes higher vlue of the reltive error etween the two determined forces. In Fig. 11 is presented vrition of the force etween the conductors s function of the two prmeters indicted in eqution (9): height of the conductors nd the distnce etween them.
3 Mri-Ctlin Petrescu, Lucin Petrescu Fig. 11. Vrition of the electrodnmicl forces etween the conductors s function of height nd the distnce etween them These forces re more importnt to e determinted in trnsient fult regime of the usr rther thn the rted one. As it cn e oserved in Tle, the vlues of these forces re ver week even for rted current of 8 A. On the other hnd, in short-circuit regime, these forces increse with the squre of the current vlue, so this m led to dmges for the usr sstem. In tle 3 re presented the vlues of the forces nlticll determined for the rted current of 8 A nd for presumed short-circuit current of 8 ka. Tle 3 Electrodnmicl forces in usr sstem working with rted currents, respectivel in short-circuit sitution Studied Cse Rted vlue for I=8A F n [N] Short-circuit vlue for I=8 ka F sc [N] 1 4.469 446.97 3.388 338.811 3.695 69.498 5. Conclusions The pper studied the mgnetic flu densit spectrum, which ws determinted nlticl equtions nd numericl techniques for single
Electrodnmic forces etween two DC usrs distriution sstems conductors 33 rectngulr conductor nd for DC usr sstem mde two rectngulr conductors. The results were ver like s it ws grphicll shown (Fig. 3, 7, 8 nd 9) for different situtions nd lso using reltive errors (Tle 1). The min ojective of the pper ws to determine the electrodnmicl forces etween the conductors in rted regime of high DC currents, using n nlticl formul nd numericl computtions. The results presented in Tle indicte tht, for smll distnce etween the conductors, the reltive error is ver feele. For distnce of 1 times the conductor s width, the error etween the forces numericll nd nlticll computed increses over 1%. The stud hs it relevnce for trnsient regime like short-circuit when the current increses over 1 times. R E F E R E N C E S [1]. Gou Zou, Zhengming Zho, Liqing Yun, Stud on DC usr structure considering str inductnce for the ck-to-ck IGBT-sed converter, Applied Power Electronics Conference nd Eposition (APEC), 13 Twent-Eighth Annul IEEE, 13, p. 113-118. []. Buschendorf, M.; Koe, M.; Alvrez, R.; Bernet, S., Comprehensive design of DC usrs for medium voltge pplictions, IEEE Energ Conversion Congress nd Eposition (ECCE), 13, p. 188-1885. [3] E. Czcu, Mrilen Stănculescu On the stilit issues of the min electromgnetic levittion techniques, The Sci. Bull. of the Elect. Eng. Fc., er 13, No. 1(1), pp. 9 13, 13. [4]. http://www.metle.cl/. [5]. Pop, I.; Doln, A.I., Numericl modeling of DC usr contcts, 13th Interntionl Conference on Optimiztion of Electricl nd Electronic Equipment (OPTIM 1), p. 188 193, 1. [6]. Yongsug Suh; Chngwoo Kim, A Stud on High-Current Rectifier Sstems With Mitigted Time-Vring Mgnetic Field Genertion t AC Input nd DC Output Busrs, IEEE Trnsctions on Power Electronics, vol. 7 (3), p. 11-119, 1. [7]. C. Lzroiu, C. Bulc, M.O. Popescu, N. Golovnov, Control nd protection of multiterminl low voltge dc sstem sed on VSC, U.P.B. Sci. Bull., Series C, Vol. 74, Iss. 1, 1, p. 173 18. [8]. E. Czcu, I. V. Nemoinu, M. C. Constntin Accurte Computtion of the Prospective Short Circuit Currents in Low Voltge Electric Instlltions, EEA,Vol. 59, Nr. 1, 11, pp. 41 48. [9]. C. Mocnu, Teori Cmpului Electromgnetic, Editur Didctic şi Pedgogică, Bucureşti, 1984.
34 Mri-Ctlin Petrescu, Lucin Petrescu [1]. http://www.mplesoft.com/. [11]. http://www.femm.info/wiki/homepge. [1]. G. Hortopn, Aprte Electrice de Comutţie vol. I. Principii, Editur tehnică, Bucureşti,.