Transformer leakage flux modeling

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1 Trnsforer lekge flux odeling Thor Henriksen SINTEF Energy Reserch Se Selndsvei N-7465 Trondhei, Norwy Abstrct The conventionl trnsforer odel seprtes the gnetizing ipednce fro the lekge inductnces. One of the lekge inductnces becoes negtive for soe three-winding trnsforers nd y result in n unstble tie doin circuit odel. This pper shows how to void this proble by odifying the trnsforer odel. Vrious lterntives re presented nd the selection ong the depends on the geoetricl dt vilble for the trnsforer. It is for trnsforers with concentric cylindricl windings better to connect the gnetizing brnch to the terinl of the inner winding thn to the internl node in the conventionl odel. The loction of the gnetizing brnch y hve significnt influence when the trnsforer is in hevy sturtion. Keywords: Trnsforer odeling, negtive lekge inductnce, unstble circuit, physiclly bsed odel, EMTP. I. INTRODUCTION Fig. shows the Sturble Trnsforer Coponent in EMTP []. This odel corresponds to the conventionl trnsforer odel nd its in chrcteristic copred to other odels is tht it seprtes the gnetizing ipednce fro the short-circuit (lekge) inductnces. The lekge inductnces re norlly deterined fro esureents. Only one such esureent cn be perfored for two-winding trnsforer nd it is coon prctice to ssue the two lekge inductnces to the equl (in p.u.). Three independent esureents cn be de for three-winding trnsforer. The three lekge inductnces re thus deterined uniquely. For trnsforers with four or ore windings the nuber of independent esureents is higher thn the nuber of inductnces nd the odel (Fig. ) cn in generl not reproduce correctly ll the esured vlues. The trnsforer odel in Fig. includes winding resistnces nd idel trnsforers. The winding resistnces re neglected in this pper, but they cn esily be included s prt of the network connected to the trnsforer s terinls. The idel trnsforers re oitted s well. This iplies tht ll ipednces re referred to the se voltge level. It is well known tht one of the lekge inductnces in Fig. norlly becoes negtive for three-winding trnsforers. Ref. [] shows tht this negtive vlue y result in n unstble circuit tht gives copletely unrelistic tie doin responses. One exple of such cse is found R or g ψ i g Fig.. Sturble Trnsforer Coponent. when n idel voltge source is connected to the winding with the negtive inductnce (e.g. ) nd the other windings re open. Assuing constnt gnetizing inductnce gives n eigenvlue equl to ( )/ ( ) λ = -R ( ) This eigenvlue becoes positive nd the nuericl vlue is typiclly in the rnge 0 8 s -. This iplies tht prt of trnsient response shows n exponentil increse with tie constnt in the rnge 0-8 s. Ref. [] shows soe exples of unstble responses obtined with resistive lod connected to the winding with the negtive inductnce. Ref. [] shows tht the proble due to the unstble circuit cn be voided by representing the gnetizing losses by resistnces connected to the trnsforer s terinls or by introducing series inductnce in the gnetizing brnch. Those odifictions were however not derived fro physicl conditions of the trnsforer. This is the in purpose of this pper. II. PASSIVE NETWORK, INDUCTANCES WITH MUTUA COUPING A pssive network is network tht does not generte ny power. It does therefore not cuse ny unstble condition. The trnsforer is pssive coponent nd ust therefore correspond to pssive network. One wy of chieving this is to restrict the odel to non-negtive RC coponents only. This pproch is however not good solution since it will not gree with the esured short-circuit inductnces when they result in negtive vlue for one of the lekge inductnces in Fig.. A negtive coponent y on the other hnd be prt of pssive network. A positive resistnce y for Idel R R N N

2 instnce be odeled s series connection of negtive resistnce nd positive one. Fig. shows n equivlent for two inductnces with utul coupling. The inductnces correspond to pssive network when nd b re positive nd the coupling fctor is less or equl to one, i.e.: M ( ) b It is worth noting tht b M y becoe negtive s long s condition () is fulfilled. (Nuericl exple: = b nd M =. b (coupling fctor 0.9) gives b M = - 0. b ). III. TRANSFORMER MODE DERIVED FROM A PHYSICA BASIS Fig. shows cross section of the trnsforer tht is ssued to hve three concentric cylindricl windings. The core is ssued cylindricl s well. All windings re ssued to hve one turn only. (The ctul nuber cn be ccounted for by idel trnsforers). A odel for the trnsforer will now be derived ssuing tht the gnetic flux density outside the core is verticl (i.e. end effects re neglected). The flux linkges of the windings cn be expressed s contribution fro the core (Ψ ) plus liner contribution due to the field outside the core. M b Inductnces with utul coupling - M M b - M Equivlent without utul coupling Fig.. Inductnces with utul coupling. Z Ψ Ψ Ψ = Ψ Ψ Ψ i i i ( ) The eleents in the inductnce trix () cn be deterined colun by colun by pplying current in one winding nd deterining the flux linkges with Ψ = 0. This lst condition is chieved by introducing fictitious winding (gnetizing winding) with zero thickness t the surfce of the core nd pplying current equl to the one in the first winding but with opposite sign. The lekge inductnce forul in the ppendix cn be used to deterine the digonl eleents in the inductnce trix. The forul ssues hoogenous flux density between the two windings involved (i.e. in the lekge chnnel) nd it is possible to deterine the off-digonl eleents in the inductnce trix utilizing this ssuption. The trix becoes: d / 6 = k where d / 6 k = µ 0 π D h D is the en dieter of the windings. d / 6 (4) ( 5 ) The flux in the core ust be produced by gnetizing current i. This current cn be considered to be supplied by the fictitious gnetizing winding. The totl current in this winding is thus i i i i. However, this current ust be zero, which iplies tht i is the su of the currents in the other windings. The reltion between Ψ nd i is nonliner nd is norlly represented by prllel connection of resistnce nd non-liner inductnce. Fig. 4 shows n equivlent circuit tht cn be derived fro () nd (4) nd the gnetizing ipednce. The decoupling technique in Fig. hs been pplied s well. The dotted lines show how the equivlent cn be extended to tke fourth winding into ccount. It should be noted tht the following iniu vlues pplies (see Fig. ): d d d in = d / ( 6 ) ( d d )/ in = ( 7 ) h ( d d )/ in = ( 8 ) core r Fig.. Trnsforer with three cylindricl windings. Fig. 4 is bsed on geoetricl odel but severl siplifictions hve been introduced nd discrepncies ust be expected between the odel nd rel trnsforer. It is however iportnt to note tht Fig. 4 represents pssive network. This cn be verified fro (4) by considering the coupling fctor for ny pir of windings.

3 4 -kd 4 /6 nd = k ( ) -kd /6 A k A -kd /6 k -kd /6 Fig. 4. Trnsforer equivlent bsed on () nd (4). IV. ADAPTING TRANSFORMER MODE TO MEASURED DATA The gnetizing current hs norlly no influence on the short-circuit ipednces. Ignoring this current in Figs. nd 4 shows tht the two odels re equivlent when: ( d / 6) k = k ( 9 ) k 4 = k d / 6 ( 0 ) ( d / 6) = k ( ) Fig. is equivlent to Fig. 4 under these conditions if the inductnce k is ignored nd the gnetizing brnch in Fig. 4 is connected to node A insted of node A. nd in Fig. 4 cn be djusted in such wy tht the two gnetizing brnches give the se ipednce t stedy stte noinl voltge. The discrepncy between the two brnches is then insignificnt when the trnsforer is not sturted. Soe discrepncy occurs during sturtion but it is ost probbly insignificnt copred to the error tht is introduced by representing the gnetizing losses by liner resistnce. The in difference between the two odels is the influence on the gnetizing current fro the voltge drop cross the inductnce k. This influence y be significnt when the trnsforer is sturted. The inductnce in Fig. 4 corresponding to cn be decoposed into two coponents: = k d / 6 ( ) Eq. (7) shows tht is t lest. becoes t lest 6 if one ssues tht d is pproxitely equl to d. This iplies tht it is better to connect the gnetizing brnch to terinl thn to the node A. Ref. [] shows tht the trnsforer odel in Fig. becoes pssive one when the gnetizing brnch is connected to one of the terinls of the odel or contins sufficiently gret series inductnce. The trnsforer odel in Fig. 4 is bsed on physicl considertions nd it is therefore ssued to be better bsis for odeling the trnsforer thn Fig.. The odel should however gree with the short-circuit esureents. This iplies tht the lekge inductnces, nd re to be deterined fro esureents nd not fro geoetricl dt. Those dt could be used to divide into the two prts nd. The geoetricl odel gives ( 6 ) / d / = ( 4 ) This rtio is n estite nd it should be used only if the reltive vlues of, nd fro the esureents show resonble greeent with vlues clculted fro the geoetricl dt using (9) (). If no resonble greeent is obtined, it is recoended to connect the gnetizing brnch to the terinl of the inner winding. Fig. 5 shows the vrious odels tht hve been described so fr (Note tht the vlues of nd re not the se for ll odels)., nd re deterined fro esureents nd the nubering of the windings is ssued done in such wy tht nd re greter thn (or equl). The choice ong the odels depends on the vlue of nd the dditionl geoetricl dt tht re vilble. Tble shows the proposed selection. Tble Selection of trnsforer odel Cse Description Select trnsforer odel I > 0 A II < 0 Winding is not the center B or C winding III < 0 Position of the windings B or C unknown IV < 0 Position of the windings D (or E) known nd winding is the center winding V < 0 Distnces known nd reltive vlues of, nd re in resonble greeent with the geoetricl dt E

4 A Conventionl odel B Mgnetizing brnch with series inductnce C Mgnetizing losses represented by resistnces connected to the terinls. D Mgnetizing brnch connected to terinl of inner winding E Model fro geoetricl dt Fig. 5. Trnsforer odels. Coents to the tble: Cse I: does not gree with Fig. 4. One possible reson is tht the windings re not rrnged s three concentric windings. The conventionl odel corresponds in this cse to pssive network since ll inductnces re non-negtive. Cse II: Fig. 4 shows tht the negtive inductnce corresponds to the center winding, but the esured vlues give nother result. Tht ens tht there is fundentl disgreeent between the esured vlues nd the odel bsed on geoetricl dt. Fig. 5 B or Fig. 5 C should then be pplied since they correspond to pssive networks tht re derived without tking geoetricl dt into ccount. The series inductnce in Fig. 5 B should be selected in such wy tht it is greter thn the bsolute vlue of the prllel connection of nd the inor one of nd []. Fig. 5 B devites in generl less fro Fig. 5 A in the frequency doin thn Fig. 5 C. The difference is however not very significnt nd Fig. 5 C is often selected becuse it is esier to ipleent in existing progrs tht includes the trnsforer odel in Fig. 5 A. Cse III: The negtive inductnce y or y not correspond to the center winding. A cutious pproch is therefore to use the se odel s in cse II. It ight however be worth to consider to pply the odel in Fig. 5 D if it is resonble to ssue tht corresponds to the center winding nd. Cse IV: Fig. 5 D is the nturl choice in this cse. However, Fig. 5 E could be used if is used to estite. Eqs. (0) nd () give: d / d = ( 5 ) The rtio d / d cn be estited by ssuing the se noinl current density in the windings nd king soe correction for the incresed ount of insultion when the noinl voltge increses. V. MAGNETIZING IMPEDANCE The gnetizing ipednce is very high copred to the short-circuit ipednces when the trnsforer is not sturted. There is in this cse no significnt devition ong the vrious lterntives in Fig. 5 s long s they correspond to pssive network. The sitution y be quite different when sturtion occurs. The gnetizing ipednce in hevy sturtion is of the se order of gnitude s the short-circuit ipednces nd the lterntives in Fig. 5 re not equivlent ny ore. Ref. [] (section 6.6.) shows (in Tble 6.) coprison between esureents nd coputtions with the gnetizing brnch connected s in Fig. 5 A or connected to the terinl of the inner winding. The trnsforer ws in hevy sturtion nd ws energized fro one winding with the other windings open. The voltge ws esured t ll terinls nd ll windings were energized (one by one). Connecting the gnetizing brnch to the terinl of the inner winding in the coputtion odel gve differences less thn 5%. Differences in the rnge 60% (one cse even 8%) were obtined when connecting the brnch s in Fig. 5 A. Ref. [] concludes tht the Sturble Trnsforer Coponent (in EMTP) could becoe ore useful if the code were chnged so tht the gnetizing brnch could be connected to ny terinl. An EMTP-user cn however overcoe this obstcle by ignoring the gnetizing current in the Sturble Trnsforer Coponent nd connecting resistnce nd non-liner inductnce to the ctul terinl. The gnetizing ipednce is norlly deterined by esureents, t lest prtly. The ipednce in 4

5 Fig. 5 corresponding to the esureent y include lekge inductnce in series with the gnetizing brnch. It is iportnt to tke this inductnce into ccount when deterining the gnetizing ipednce to be used in the odel. The gnetizing brnch includes prllel resistnce nd Fig. 4 shows tht this resistnce should not be in prllel with k tht is prt of the gnetizing inductnce. There is however no need distinguish between the two lterntives except when one of the results in trnsforer odel tht does not correspond to pssive network. VI. TWO-WINDING TRANSFORMERS Section V showed tht the loction of the gnetizing ipednce in the odel ight hve gret influence even when there is no stbility proble. It y therefore be of soe interest to nlyze two-winding trnsforer. Fig. 6 shows the odel obtined fro Fig. 4 for two-winding trnsforer. The odel norlly pplied is shown in Fig. 6 b. The negtive inductnce between nodes nd A in Fig. 6 cn be clculted if the geoetricl dt re known. The inductnce between nodes A nd should then be selected to give lekge inductnce tht corresponds to the esured vlue. If it is not possible to deterine the vlue of the negtive inductnce between nodes nd A, then the best estite is zero, i.e. to connect the gnetizing brnch to the node corresponding to the inner winding (Fig. 6 c). This choice is uch better thn pplying the conventionl odel (Fig. 6 b). The conventionl odel should therefore be used only when it is ipossible to identify the inner winding. phse. Fig. 7 corresponds to Fig. in [4] nd shows cross section of core nd winding ssebly s well s the derived gnetic circuit. The odeling of the lekge inductnces is not discussed in [4] nd it is therefore nlyzed in this pper. The current sources I H nd I X in Fig. 7 correspond to the current in phse of the high voltge nd the low voltge winding respectively. The low voltge winding is the inner winding. It is found fro the figure tht HX equls the lekge inductnce between the two windings. The inductnce XC represents the lekge flux between the inner winding nd the core. Copring Figs. 6c nd 7 it is found tht the lekge inductnces HX nd XC re connected in the se wy s (corresponding to HX ) nd k (corresponding to XC ). The se odel is used for the lekge flux of ll phses in Fig. 7. The odel in [4] corresponds therefore to Fig. 6 c nd not to the conventionl odel (Fig. 6 b). Modeling the linkge fluxes s in Fig. 6 y give ore ccurte odel provided tht the relevnt geoetricl dt re known. VIII. CONCUSIONS One of the lekge inductnces in the conventionl trnsforer odel becoes norlly negtive for VII. SOPHISTICATED FIVE-EGGED TRANSFORMER MODE Ref. [4] presents topologiclly correct odel for five leg three-phse trnsforer with two windings per -kd /6 -kd /6 k A = k k A B C Model fro geoetricl dt Conventionl odel Mgnetizing brnch connected to node Fig. 6. Two-winding trnsforer odels. Fig. 7. Model presented in [4] (Fig. ). Top: Cross section of core nd winding ssebly. ekge flux tubes re lbeled. Center: Corresponding luped-preter gnetic circuit. Botto: Dulity derived equivlent circuit. 5

6 three-winding trnsforers nd could cuse n unstble tie doin circuit odel. This pper shows how to void this proble. Alterntive odifictions of the originl odel re proposed nd the choice should be de depending on geoetricl dt vilble for the ctul trnsforer. The in devition ong the vrious lterntives is the loction of the gnetizing brnch. This loction does not hve ny significnt influence except when the trnsforer is sturted. The influence y be very significnt in hevy sturtion nd it is in generl better to connect the gnetizing brnch to the terinl of the inner winding thn to the internl node in the conventionl odel. The lekge inductnces in the two-winding version of the conventionl trnsforer odel re norlly ssued to be equl. A ore ccurte odel is obtined for trnsforers with concentric windings if the gnetizing brnch is connected to the terinl of the inner winding. IX. REFERENCES [] Cndin / Aericn EMTP User Group, Alterntive Trnsients Progr (ATP) Rule Book, [] T. Henriksen; How to Avoid Unstble Tie Doin Responses Cused by Trnsforer Models, pper no. 000 TR 48 subitted (My 000) to IEEE for publiction in IEEE Trnsctions on Power Delivery. [] H. W. Doel; Electrognetic Trnsients Progr (EMTP). Theory Book, Bonneville Power Adinistrtion. Portlnd, Oregon 987 [4] B. A. Mork; Five-egged Wound-core Trnsforer Model: Derivtion, Preters, Ipleenttion nd Evlution, IEEE Trnsctions on Power Delivery, Vol. 4, No. 4, pp 59-56, October 999 APPENDIX. EAKAGE INDUCTANCE FORMUA Ref. [A] presents in (6) forul for the shortcircuit rectnce between two concentric windings. The forul is bsed on the flux density vrition shown in Fig. A. The lekge inductnce corresponding to the windings with thickness d nd d in Fig. A cn be found fro (6) [A]: (Ech winding is here ssued to hve one turn). Φ s δ δ δ d d d d 4 Fig. A. ekge flux density vrition, fro Fig. 6 [A]. µ 0 π D = d δ d ( A ) where µ 0 is perebility of vcuu, D is the en dieter nd is the high (h) of the windings plus contribution ( δ d ) d. Ignoring this lst contribution nd substituting the distnce δ by (see Fig. A) gives: µ 0 π D d d = ( A ) h 6 6 This forul ignores the vrition in the circuference s the distnce fro the core increses. One version of the forul ws presented by Kpp in 898 [A]. Experience hs shown tht (A) is resonbly correct. Ref. [A] presents se coprisons between the forul nd esureents nd the devition is typiclly in the rnge 0%. REFERENCES [A] K. Schlosser; Eine uf physiklischer Grundlge erittelte Erstzschltung für Trnsfortoren it ehreren Wicklungen, Brown Boveri- Nchrichten, vol. 45, pp. 07-, Mrch 96 (in Gern) [A] G. Kpp; Ein Beitrg zur Vorusberechnung der Streuung in Trnsfortoren, ETZ bd. 9 (898), H5, pp (in Gern) [A] K. Schlosser; Anwendung der Erstzschltung eienes Trnsfortors it ehreren Wicklungen, Brown Boveri-Nchrichten, vol. 45, pp- 8-, June 96 (in Gern) 6