A FAST METHOD FOR CALCULATION OF TRANSFORMERS LEAKAGE REACTANCE USING ENERGY TECHNIQUE

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1 A FAST METHOD FOR CALCULATO OF TRASFORMERS LEAKAGE REACTACE USG EERGY TECHQUE A. aerian Jahrmi, Jawa Faiz an Hssein Mhseni Department f Electrical an Cmputer Engineering Center f Ecellence in Applie Electrmagnetics Faculty f Engineering, University f Tehran, P. O. B 7495/55 Tehran, ran, Abstract (Receive: June, 00 Accepte: February 6, 00) Energy technique prceure fr cmputing the leakage reactance in transfrmers is presente. This meth is very efficient cmpare with the use f flu element an image technique an is als remarkably accurate. Eamples f calculate leakage inuctances an the shrt circuit impeance are given fr illustratin. Fr valiatin, the results are cmpare with the results btaine using practical tests fr types f a small single-phase transfrmer, -phase istributin transfrmers an a high vltage test transfrmer were stuie. Key Wrs Leakage Reactance, Transfrmer, Electrmagnetic Energy, nuctance چكيده امپدانس اتصال كوتاه يك پارامتر مهم در طراحي و بهره برداري از ترانسفورماتور است. در اين مقاله با بكارگيري انرژي الكترومغناطيسي روش موثري براي محاسبه اندوكتانس پراكندگي ترانسفورماتور اراي ه شده است. اين روش در مقايسه با ساير روشها از جمله المان شار و روش تصاوير موثرتر و دقيقتر است. اين موضوع از طريق شبيه سازي تاييد مي شود. براي نشان دادن دقت روش محاسبه انرژي نتيجه محاسبات با نتايج ا زمايشهاي اتصال كوتاه براي يك ترانسفورماتور تك فاز و ترانسفورماتورهاي توزيع جفت فاز و يك ترانسفورماتور فشار قوي مقايسه شده است.. TRODUCTO Determinatin f transfrmer leakage reactance using magnetic cres has lng been an area f interest t engineers invlve in the esign f pwer an istributin transfrmers. This is require fr preicting the perfrmance f transfrmers befre actual assembly f the transfrmers. A meth has been presente fr estimating the leakage reactance by flu tube in rer t be inclue in an electric circuit mel f transfrmer []. Cmputer-base numerical slutin techniques using finite elements analysis, bunary element meth an bunary-integral meth are accurate an frm an imprtant part f the esign prceures. Hwever, they require rather elabrate cmputer resurces an a smewhat lengthy setup befre a slutin can be btaine. Als a clsefrm slutin ften prvies mre insight abut critical physical parameters than a cmputer-base numerical slutin. Finite-element meth is use fr calculatin f leakage inuctances f electric machineries nwaays, but it nees special sftware an ample time [-4]. A superpsitin f -D analysis results fr apprimate eterminatin f the flu ensity istributin is presente in Reference. The FEM is als use t etermine the leakage inuctance f synchrnus machines []. Dwell has als presente an analytical thery t calculate the hmic lsses f transfrmers but there has nt been invlve the reactance [5]. The reverse esign meth is als applie t partial cre transfrmers; but the meth is use fr special type f transfrmers [6]. The winings shul be wrappe arun the cre with primary wining insie the secnary wining an the JE Transactins B: Applicatins Vl. 6,., April 00-4

2 ykes an limbs that usually frm the rest f the cre in full-cre transfrmers, were nt presente [6]. n this paper a clse frm slutin technique applicable t the leakage reactance calculatins fr transfrmers is presente. An emphasis is n the evelpment f a simple meth t characterize the leakage reactance f the transfrmers. Leakage reactance calculatins play an imprtant rle in esigning gemetry f transfrmers. The esign parameters may be varie as such that the require shrt circuit impeance is etermine. A D representatin prves t be satisfactry in etermining the leakage reactance. Final epressins are evelpe n a per-unit-flength basis fr the thir imensin. Certain assumptins have been mae in this calculatin. En effects intruce by the terminatins in D cnfiguratins are nt evaluate here. There are ifferent techniques fr the leakage reactance evaluatin in transfrmers. The mst cmmn technique is the use f the flu leakage elements an estimatin f the flu in ifferent parts f the transfrmer [,7-]. The images technique can als be use. The base f this meth is cnsiering the image f every turn f the wining where the magnetic ptential vectr [,] is emplye t cmpute the mutual an leakage inuctances. Althugh the technique is effective, the cmputatin result epens n the current f the image cnuctr []. This paper presents a nvel technique fr calculatin f the leakage inuctance in ifferent parts f the transfrmer using the electrmagnetic stre energy.. COMPUTATO USG FLUX ELEMET TECHQUE n rer t cmpute the leakage reactance analytically, sme apprimatin is require t achieve a clse-lp slutin []. The assumptins are:. The leakage flu istributin in the wining an the space between the winings must be in the irectin f the wining aial.. The leakage flu is unifrmly istribute alng the length f the winings.. The leakage flu in the space f tw winings is ivie equally between them. The leakage flu fr each wining fr a tw-wining transfrmer, base n the abve assumptins is []: s λ = µ 0 L ( + ) / mt L C Using the fllwing equatin: X = π fλ an reflecting leakage reactance between HV winings t the primary sie yiels: X=X +( / ) X () Equatin will be as fllws: X s = πfµ (L mt ( + ) + L mt L c s ( + )) () () (4) f it is assume that L mt = L mt (meaning that the length f each turn f HV an LV winings are equal). Equatin can be simplifie as fllws: X + = π fµ 0 L mt ( + s ) L C (5) This is the cnventinal equatin use in References 8, 9 an 0..COMPUTATO USG MAGE METHOD This meth is base n cnsiering an mage cnuctr t the cre fr each lp, because the surface f the cre is an equiptent fr the magnetic scalar ptential, s the cre surface will be a mirrr fr magnetic fiel []. Using the magnetic vectr ptential fr a 4 - Vl. 6,., April 00 JE Transactins B: Applicatins

3 circular filament fr tw cnuctrs, the leakage inuctance is []: f L leakage = f r, r, A(r ), A image (r ), A(r -r ), A image (r -r ), i, i, i image, i image (6) Where r an r are the raiuses f cnuctrs, A an A image are the magnetic vectr ptentials with respect t actual an image cnuctrs that are epenent n elliptic integral f first an secn kins [,]. The current f image cnuctr has a value ifferent frm the real cnuctr an it shul be ajuste fr each image cnuctr. S the result f the leakage inuctance changes accring t the efault f image cnuctr currents. There are several recmmenatins fr this parameter t evaluate the best result f mage Meth but there is nt a fie rule fr it an the errr frm calculatin an test results may nt be ptimize. CORE r r ra r s LV WDG s HV WDG 4. COMPUTATO USG EERGY TECHQUE The electrmagnetic energy stre in the winings an the space between them can be use t calculate the inuctance between the winings an the leakage inuctance. The previus assumptins are cnsiere here in rer t btain a clse-frm slutin. Cnsier the path F in Figure, mmf fr the path having istance frm the beginning f LV wining is [7]: F = (7) Rising frm 0 t increases the magnetic fiel intensity an appraches its peak value /L c. A vlume ifferential frm LV wining as shwn in Figure, with height, thickness an raius r+s + is cnsiere. The electrmagnetic energy stre in this element is [4]: w = µ H v (8) an the ttal energy is: Figure. The vlume element f LV wining fr energy cmputatin. πµ W = w = ( r s πµ r + s ( + ) 4 + ) = (9) Similarly the stre energy in HV wining can be JE Transactins B: Applicatins Vl. 6,., April 00-4

4 etermine: W πµ ( r + s + s = ) (0) With a cnstant magnetic fiel intensity between the winings, W a the electrmagnetic energy stre between them is: Va W a = µ 0H a () Hence: πµ W a = + s ( r + s + ) s () The stre energy fr this tw-wining transfrmer is: W a = Leq = W + W + W () Using Equatins 9-, the inuctance will be as fllws: The fllwing simplificatin is applie: L L mt mt + L mt = πr = ra(π ) ave, L mt = πr ave (8) Equatin 7 is cnverte int Equatin 5. t means that the flu element meth is an apprimatin f the energy meth. 5. SMULATO 5. A Single-Phase Transfrmer A small single-phase transfrmer with specificatins given in Table is simulate []. The image meth result f this reference is use t cmpare the accuracy f ifferent meths. Using the previus results f the transfrmer frm Reference, the result f ifferent meths can be cmpare. As shwn in Table, the errr using the energy meth is lwer than that f the πµ r Leq = [( + r ( ) + ras] 4 ) 4 r, r an r a are efine as fllws: r = r + s r a = r + s + + r = r + s + + s s + + (4) (5) (6) TABLE. Specificatins f the Prpse Single-Phase Transfrmer. Dimensins are in mm. Pwer kva / 8/8 Vltage 0 v L c 98 S 5 S r 60 A ntable pint is that if term (r /+ /4) is substitute by r ave / an term (r /- /4) in replace by r ave /, L eq will be: πµ Xeq = πf rave + r ave ( + ras ) (7) TABLE. Cmputatins Results Using Different Meths. Meth nuctance (mh) Errr in respect t the test (%) Data sheet Flu meth mage meth Energy meth Test results 0.45 *** 44 - Vl. 6,., April 00 JE Transactins B: Applicatins

5 TABLE. Specificatins f the Prpse Transfrmers. Pwer kva D mm D mm D mm V/ Zsc % image meth. Als the flu element meth has larger errr cmpare t the ther tw meths. 5. Three-Phase Distributin Transfrmers Three types f three-phase istributin transfrmers with vltages 0/0.4 kv an cnnectin grup YZn5 with specificatin given in Table are cnsiere. Figure shws the imensins f the 5,50 an 00kVA transfrmers. The last clumn f Table shws the result f shrt-circuit test f transfrmers in the factry. The specificatins an the test results (last clumn) in Table are measure fr mre than 00 istributin transfrmers. The shrt-circuit impeance has been Figure. A 50 kv, 500 kva test transfrmer esigne an manufacture in ran. D D D calculate using the abve-mentine meths. The results fr these transfrmers have been summarize in Table 4. The result shws that energy meth has the best accuracy cmpare t ther meths fr all f these istributin transfrmers. D : iameter f cre, D: iameter f primary wining D: iameter f secnary wining Figure. A simple gemetrical schematic Dimensins f the transfrmer. 5. A Single-phase High Vltage Test Transfrmer The result f simulatin f a 500 kva, 50kV test transfrmer, illustrate in Figure an esigne by the authrs an manufacture in ran, is shwn in Table 5. The schematic presentatin f winings are in Figure 4. JE Transactins B: Applicatins Vl. 6,., April 00-45

6 TABLE 4. Cmputatin Results fr Transfrmers. Pwer kva Meth mpeance (%) Errr (%) t is nticeable that the transfrmer has winings: lw vltage, high vltage an cupling (fr energy transmissin t upper steps) Equatin (5) Flu Meth mage Meth Energy Meth Test result 4. **** Equatin (5) Flu Meth magemeth Energy Meth Test result 4.09 **** Equatin (5) Flu Meth mage Meth Energy Meth Test result 4.48 **** 6. COCLUSOS Different analytical meths fr the leakage inuctance f transfrmer calculatin have been cmpare. t has been shwn that the energy meth is the mst accurate ne. Althugh the image meth is accurate an cnvenient, it epens n the current f the image cnuctrs. These results are cmpare by practical measurements fr three types f transfrmers: small single-phase transfrmer, three-phase istributin transfrmers an a high vltage test transfrmer. S by calculating stre electrmagnetic energy in winings an istance between them leakage reactance will be calculate simply an accurately cmparing t test result. 7. APPEDX Calculatin f Leakage Reactance f Test Transfrmer The reactance between winings calculate tw by tw an the etail f calculatin is given here. The result shws that energy meth is als reliable in this case. Because f trapezi shape f HV wining f test transfrmer accring t Figure 5, the frmula f 0 will be change: W = π µ w [ (r + k L C ) (LC + LC ) -LV Wining - HV Wining -Cupling Wining k LC (r +.5 k LC ) (LC + LC (r + k L C ) LC LC ln ] LC L C LC + LC LC ) + Figure 4. Schematic f winings f the test transfrmer. (A) 46 - Vl. 6,., April 00 JE Transactins B: Applicatins

7 TABLE 5. Result f Meths fr the Test Transfrmer. Where k is a cnstant. LV HV LV- Cuple HV- Cuple Flu Met mage Met Energy Met Test Result λ L mt L c s s,s 8. LST OF SYMBOLS Leakage flu. f turn f wining current Mean length f ne turn f wining length f winw f cre With f wining Distance between cre an LV winings Distance between LV an HV winings r ave, r ave Mean raius f HV an LV wining X F W L eq H respectively Reactance f transfrmer Frequency f current f winings Electrmagnetic energy stre in active part Equivalent inuctance f transfrmer Magnetic fiel intensity in a istance f first layer f primary wining 9. REFERECES Figure 5. Dimensins f winings f the test transfrmer. Cnsiering Leakage reactance between LV an HV: Xeq LC w = (r k ( L C 4 π + klc + L C. f µ L C + LC LC r w ( + 4 ) (LC + LC (r + ) L C + ras + k LC ) L C L C L C ) w (r klc ) L C ln L C (A)..Karsai, K., Kerenyi, D. an Kiss, L., "Large Pwer Transfrmers", Hungary, Elsevier, (987).. Shima, Kazumasa an Miyshi Takahshi, Finite- Element Calculatin f Leakage nuctances f a Suture Salient-Ple Synchrnus Machine with Damper Circuits, EEE Transactin n Energy Cnversin, Vlume 7, umber 4, (December 00), Branislaw Tmczuk, Analysis f -D Magnetic Fiels in High Leakage Reactance Transfrmers, EEE Transactin n Magnetics, Vl.,.5, (September 994), Guemes Alns an Antni, J., A ew Meth fr Calculating f Leakage Reactances an rn Lsses in Transfrmers, 5 th nternatinal Cnference n Electrical Machines an Systems, (CEMS 00), Vl., Freeric Rbert an Pierre Mathys, Ohmic Lsses Calculatin in SMPS Transfrmers: umerical Stuy f Dwell s Apprach Accuracy, EEE Transactin n Magnetics, Vl. 4,.4, (July 998), Liew, M. C. an Bger, P. S., Partial Transfrmer Design Using Reverse Meling Techniques, EE Prc. Electrical Pwer Applicatin, Vl. 48,. 6, (vember 00), JE Transactins B: Applicatins Vl. 6,., April 00-47

8 7. Agrawal, R. K., "Principles f Electrical Machines Design", Thir Eitin, nia, (997). 8. Turwski, J., Turwski, M. an Kpec, M., "Meths f Three-Dimensinal etwrk Slutin f Leakage Fiel f Three-Phase Transfrmers", EEE Transactin n Magnetics, Vl. 6,. 5, (September 990), Blume, L. F., et al., "Transfrmer Engineering", Secn Eitin, Jhn Wily an Sns, (95). 0. Allan, D. J. an Harrisn, T. H., "Design fr Reliability f High Vltage Pwer Transfrmers an Reactrs", GEC Rev., Vl., (985).. Smith, J., Transfrmer Reactance Calculatins with Digital Cmputers, AEE Trans., (956), DeLen, F. an Semlyen, A., "Efficient Calculatin f Elementary Parameters f Transfrmers", EEE Trans. n Pwer Delivery, Vl. 7,., (January 99), Wilc, D. J., Hurley, W. G. an Cnln, M., "Calculatin f Self an Mutual mpeances Between Sectin f Transfrmer Winings", EE Prceeing, Vlume 6, Part C, umber 5, (September 989), Kruse, Paul C., "Analysis f Electric Machinery ", McGraw-Hill, (986) Vl. 6,., April 00 JE Transactins B: Applicatins

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