Eddy-Current Losses in Rectifier Transformers

Transcription

1 IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS89, NO. 7, SEPTEMBER/OCTOBER qantities of engineering interest. In the majority of past work, the great blk of compter time has been consmed in the third step, and considerable effort has been expended by varios investigators to find methods of accelerating the convergence of iterative soltion methods. The athors have been motivated primarily by the view that the best way of redcing compter time is to redce the size of the system of eqations that needs to be solved, hence directing attention to the second step in the general procedre. The athors cannot emphasize strongly enogh that, in'their opinion, it is beside the point to ask whether they can solve 5000 nonlinear eqations faster than anyone else; their prpose is precisely to avoid having the 5000 eqations in the first place. It is believed that the reslts presented show that the qaiitities sally desired by designerscrrent waveforms, indctance vales, and the likeare determined to sfficient accracies by relatively few eqations. Many thosands of nodal eqations are therefore nnecessary except for the prpose of modeling the complicated bondary shapes encontered in electromagnetic devices. Using topologically nrestricted trianglar elements, the nmber of nodes actally needed is sally less than onetenth of the nmber necessary to constrct a finite difference mesh of fixed lines; for example, one pole pitch of an electric machine cross section can be adeqately modeled by abot 100 to 300 nodes. A proper economic comparison therefore shold be based on soltion of the bondaryvale problem, not on the time taken for a particlar nmber of nonlinear algebraic eqations. On this basis, present indications are that the finite element method is strongly competitive. Another point raised by the discssers in the qestion of programming ease. It is of corse tre that in the method presented matrix assembly and soltion will form separate phases, nlike in finite difference relaxation schemes. FORTRAN listings of comparable programs therefore are likely to be abot eqally long. However, it is important to note that no problemdependent information need be incorporated in the finite element program, so that the same program may be sed to solve varios widely diverse problems with no changes whatsoever. Almost no knowledge of the internal mathematical strctre is reqired of the ser, a point of no small importance. By contrast, the athors know of no nonlinear relaxation program in whch acceleration parameters are atomatically optimized. In closing, the athors wish to correct one misconception. If a finite difference mesh covering a homogeneos region with Dirichlet bondaries is diagonally sbdivided into triangles, and these triangles are regarded as finite elements, the system of eqations that reslts for Laplace's eqation (or its nonlinear conterpart) is exactly the same. However, the same is not tre for Poisson's eqation. In other words, the lefthand side of (15) is identical in the two approaches, whereas the righthand side is not. The reasons for the difference, and their effect, are discssed in [11]. Where Nemann bondaries or material inhomogeneities occr, the reslting systems of eqations are rarely, if ever, similar. EddyCrrent Losses in Rectifier Transformers SERGIO CREPAZ AbstractTwo methods are developed for evalating the eddycrrent losses occrring in a rectifier transformer dring operation, i.e., when carrying plsating crrents. The first is based on the analysis of the whole waveshape; the second starts from the harmonic composition of crrent crves. Each system is sefl from a different point of view, and both achieve the same reslts. Experimental measrements confirm the theoretical assmption, i.e., that eddycrrent losses with plsating crrents can be some mltiples higher than those evalated by the tests prescribed by standards. Eddycrrent loss evalation is made for two, six, and twelveplse systems. INTRODUCTION PRACTICE HAS shown, that real losses in rectifier transformers carrying plsating crrents are higher than those evalated on the basis of the tests prescribed by standards. It is also known that this increase of losses is de to a rise in eddycrrent loss. This phenomenon becomes particlarly conspicos in electrolytic plant transformers where eddycrrent 'losses even with sinsoidal crrents are high becase of the Paper 70 TP 131PWR, recommended and approved by the Transformers Committee of the IEEE Power Grop for presentation at the IEEE Winter Power Meeting, New York, N. Y., Janary 2530, Manscript sbmitted September 9, 1969; made available for printing November 26, The athor is with the Department of Electrical Engineering, Polytechnic Institte of Milan, Milan, Italy large condctor sections (2030 percent RI2) and in siliconcontrolled rectifier power circits where high vales of delay angle determine very fast commtations. BASIC FORMULA FOR EDDYCURRENT Loss EVALUATION Eddycrrent loss occrring in a transformer winding in the case of sinsoidal crrent is generally evalated by where Pe, sin 5_ PQ 45 = av/(7rfio/p) (hr/h). Approximation of this formla is discssed in Appendix I; symbols are listed in Appendix III. Use of (1) assmes that eddycrrent loss is proportional to the sqare of the freqency. Beginning from (1) we can reach the evalation of the eddycrrent loss with plsating crrent in two ways: 1) finding a formla similar to (1) bt correct for a crrent of arbitrary shape and applying it to the crrent flowing in rectifier circits; 2) determining the harmonic composition of plsating crrents and, by means of (1) applied to single harmonics, evalating the eddycrrent loss as sm of several contribtions. We shall follow both systems becase the first gives a very simple and general conclsive expression, while the second can (1)

2 1652 IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, SEPTEMBER/OCTOBER 1970 not only frnish sefl information on the contribtion of the varios harmonics, bt can make eddycrrent loss evalation possible on the basis of experimental measrements. EDDYCURRENT Loss EVALUATION BY CURRENT WAVESHAPE EddyCrrent Loss Formla for Crrent of Arbitrary Shape Eqation (1) can be rewritten as Pe, sin 5m2 1 h.2a4 (42f2) a A) Po 45 h2 4 The three factors on the righthand side represent, respectively: l) inflence of geometric dimensions of winding where eddycrrent loss occrs, 2) inflence of variation of the crrent with time, 3) physical parameters characterizing the condctor material. In particlar, factor 2) signifies meansqare vale of derivative of crrent meansqare vale of crrent Now let i(t) be a periodic crrent of arbitrary shape, I being its rms vale; let also i'(t) = d/dt i(t),i' being the rms vale of i'(t). We can write the eddycrrent loss expression in a transformer winding carrying a crrent of arbitrary shape: Pe, sin 5m2_ 1 a4 I'2//.2a\ PR 45 h2 4 F2 (3) This expression can be rigorosly jstified by sing the same procedre adopted to obtain (1); it has to be regarded as correct at least p to the nineteenth harmonic, assming as fndamental the freqency of 50 or 60 Hz (see Appendix I). When 112/12 is evalated, (3) gives the soltion to the proposed problem. Evalation of 12 in Rectifier Circits Normally all standards define the basis of crrent rating on instantaneos commtation, i.e., with crrent waves of rectanglar shape. The reasons are clearly stated in [6] and [7]: "Owing to the fact that the rms vales of the crrents... are somewhat smaller than those in the test, a positive error is encontered... This positive error is assmed to be compensated for by the negative error reslting from the fact that the additional stray losses cased by the harmonics... are disregarded." A more accrate evalation can be made nder the assmptions that 1) the dc crrent spplied to the load is withot ripple (indctive load), and 2) the commtation loop comprises only indctance. In simpler cases (twoplse systems and some sixplse systems) the crrent has the shape of Fig. 1(a) where front and tail (corresponding to the commtation with the preceding and the following phase) are mathematically described by the fnction of commtation fc: rise: i(t) = fmd (2) c in Appendix II. For a crrent crve as in Fig. 1 (a) and theoretical condction angle y we compte I2 a+u +au 12 = Id f[2dt+ (y U) + (1 fc)2dwt and let (owt i (t) (a) = 12 V 21 _ Jf 7 a Y We finally obtain Fwt (fcfc2)d(ot] 1 B+ 'I'(a,) = (fc f2) dwot. T J I2 = I,2 [1 2r 7r v (4),U) where Id2 y/ir is the sqare of the rms vale with instantaneos commtation and the remaining righthand factor is the correction for commtation, a fnction of a and. More complex crrent crves concerning six and twelveplse systems can be treated in a similar manner and will lead to analogos reslts. Vales referring to the more freqently sed rectifier circits are fond in the first and second colmns of Tables I111; vales of T(a, ) are given in Table IV. Evalation of I12 in Rectifier Circits Referring to the crrent shape of Fig. l(a), we immediately recognize that, independent of the vale of v, the derivative of the crrent is different from zero only when crrent is rising and decaying; its crve is shown in Fig. 1(b). The rise expression is i'(t) = fc l= sin ct dt COfcoscOs a (a + ) and the decay expression is Fig. 1. i'(t) (1MId = coid_ sin cot. dt cos acos (a +) Hence by integration in halfcycle we have (b) i'(t) [sin 2a sin 2(a + )] + (4) decay: i(t) = (1 fc)id cos a Cos a Cos cot cos (a + ) where wt is separately defined for rise and decay. The meanings of a and and their relations to the circit elements are shown and if 7r [Cos Cos (a )2 1 [sin2a sin2(a+)] r(a,) = [cos a cos (a + )]

3 CREPAZ: EDDYCURRENT LOSSES IN RECTIFIER TRANSFORMERS 1653 TABLE I TWOPULSE SYSTEMS I with Correction Factor: Instantaneos Correction for Pe. pls Commtation Commtation 1/2 Pe, sin Singleway 2Id2 1 2ip(a,) W2Id2 X (a,) 8 (a,) &(a,) Dobleway Id2 1 4#(a,) 4 Cw2Id2 X ~(a,u) 4 (ae) 7r 7r 1 4t,&(a,) TABLE II SIXPULSE SYSTEMS Ia with Correction Factors: Instantaneos Correction for Pe, pls Commtation Commtation 1/2 Pe. sin 2 Singleway Id P(a,) X (a, 6 3 W1d 7r 134t(a,) (a,) ) Dobleway (a,) Y Connection 3 d 34t'(a,) W2d2(X(r ~ 3(aU) A Connection Id2 34(a,) 2 W3Id 7r I34(a,) TABLE III TWELVEPULSE SYSTEMS I2 with Correction Factor: Instantaneos Correction for Pe. pl. Commtation Commtation J/2 Pe, sin Y or A Primary Id t(a,) Connection ) Id2 X (a,u) 7r 11.66(a,) 4+2V Y and A Secondary 4+ / rwi 6 r(a,) Connection I3 =1 1.64,(a.) r 134(a,) _~~~~~~~~~~~~~~~~~~~7 Commtation TABLE IV FUNCTION Vl(a,U) X 100 PhaseControl Angle c Angle 400s500o600o700n * goo

4 1654 IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, SEPTEMBER/OCTOBER 1970 TABLE V FUNCTION r (a, ) Commtation Angle PhaseControl Angle ac we can conclde that I 2 = 2 CW2Id2r(a,U). Ir More complex crrent crves concerning six and twelveplse systems can be treated in a similar manner and will lead to analogos reslts; vales referring to the more freqently sed rectifier circits are fond in the third colmns of Tables IIII; vales of t(a, ) are given in Table V. It mst be pointed ot that I12 vales are the same in single and dobleway circits, since in both cases all windings of the same phase ndergo the same magnetomotive force variations. Conclsive Formla and Correction Factor of (1) for Plsating Crrents The ratio between (5) and (4) allows the elimination of Id2 and the rebilding of an expression similar to (1) containing a correction factor that represents the ratio between eddycrrent loss with plsating crrents and eddycrrent loss with sinsoidal crrents. The vales of the varios correction factors are given in the forth colmns of Tables IIII. Since t(a,o) = o, eddycrrent loss tends to be infinite when tends to zero. EDDYCURRENT Loss EVALUATION BY HARMONIC ANALYSIS Basic Relation From the definition of rms vale of a crrent of arbitrary shape in terms of harmonics and hence * P =Pl + + Pn + * That is, loss de to a crrent of arbitrary shape is given by the sm of losses de to the single harmonics. Harmonic Analysis and Total Losses with Instantaneos Commtation Harmonic analysis of a rectanglar wave (twoplse system) and of the thleoretical waves of six and twelveplse systems (5) brings the following reslts: II rms vale of fndamental I rms vale of crrent (22= 0.90, twoplse 7r 3 = = 0.955, sixplse?r 6(\/+ 1) 0.988, twelveplse The harmonic composition is only of order nq 4± 1 and the amplitde of the nth harmonic is 11/n. RI2 loss: EddyCrrent Loss: Under the assmption that eddycrrent loss is proportional to the sqare of the freqency, if ki = Pe, s in /IP at the fndamental, at the nth harmonic it will be kn = n2ki. This leads to Pe, pls. =kip, + + n2k + = EnklPl = o. We find the reslt of the preceding paragraph; i.e., that eddycrrent loss tends to be infinite when the commtation angle tends to zero. Harmonic Analysis and Total Losses with Real Commtation The reslts achieved above are still correct; the rms vales of each of the varios harmonics in respect the theoretical vales are redced to a factor rn(a,). Sch factors, which are rather difficlt to calclate, are generally available from charts contained in rectifier standards [6]. I2R Loss: P =Pi + ***+ rn2( n2 ) Pi+**P1 n

5 CREPAZ: EDDYCURRENT LOSSES IN RECTIFIER TRANSFORMERS 1655 Pe.pts. Pe,sin '\\\ Fig. 2. EddyCrrent Loss: Pe, pl=k ipi+ + n2klrn2(a,) Pi + = k1pi(l + + rn2(a,) + Correction Factor of (1) for Plsating Crrents It is now possible to evalate the ratio between eddycrrent loss with plsating crrent (identified by a and ) and eddycrrent loss with sinsoidal crrent, which, at the same rms crrent, are Pe, sin = kp r +2(a,) n2 Pe, pls (1 + * + rn2(ce,u) + Pe, sin (1 + ' ' + [rn2(a,)]/n2 + (1 + * + 2(e/ rn ) +.. )) (6) (1/I1) 2 This reslt can be applied to any rectifier circit if only really occrring harmonics are considered; it is particlarly helpfl when it is possible to analyze the crrent experimentally. Agreement Between Relations of Table IIII and Eqation (6) The comparison between the correction factors obtained from crrent crve analysis and from harmonic analysis shows a satisfactory agreement between the two methods. In Fig. 2 both vales (for a = 0, = 1535', threephase bridge circit) are plotted: correction factors dedcted from harmonic analysis (stopped at the nineteenth harmonic) are of corse lower than those obtained by crrent crve analysis; the difference never reaches 10 percent. EXPERIMENTAL RESULTS Direct measrements of the losses in the phases of a transformer dring the operation in a rectifier circit were made by the circit of Fig. 3, where the feeding voltage and the load resistance were tned in order to get the proper vales of the commtation angle. Becase real waveshapes are slightly different from the theoretical ones, the comparison between calclation and measrement has been made sing (6), determining the harmonic content by an analyzer. Of corse the instrments also mst give, with a high harmonic content, rms vales of crrents and average vales of power. The circit has been sed in the following way. 1) Switch A open, switch B closedwith singlephase feeding and resistive load, the transformer carries sinsoidal crrents; the measred losses are eqal to the shortcircit losses. 2) Switch A open, switch B openwith singlephase feeding and with indctive load, the transformer carries plsating Watts Fig. 3. comm. angle Fig. 4. Wiatts 10 ' o _I l Ios l comm.angle Fig. 5. crrents with theoretical condition angle 7r. Measred losses, as a fnction of commtation angle, are plotted in Fig. 4, crve 1; crve 2 shows the loss evalation by harmonic analysis. 3) Switch A closed, switch B openwith threephase feeding and indctive load, the transformer carries plsating crrents with theoretical condction angle 27r/3. Measred losses, as a fnction of commtation angle, are plotted in Fig. 5, crve 1; crve 2 shows the losses evalated by harmonic analysis. It

6 1656 IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, SEPTEMBER/OCTOBER 1970 has to be pointed ot that measred losses, however slightly, are systematically lower than the ones calclated by means of (6); this difference increases at low vales of commtation angle. CONCLUSION Two systems for the evalation of eddycrrent loss in transformers carrying plsating crrents have been reported. In both cases the ratio eddycrrent loss with plsating crrents eddycrrent loss with sinsoidal crrents has been determined, either as a fnction of delay angle a and of commtation angle, or as a fnction of the harmonic content of crrents. We may conclde the following. 1) Eddycrrent loss may reach some mltiples of that occrring with sinsoidal crrents. 2) Eddycrrent loss increases when the commtating angle decreases. 3) Phase control redces the commtation angle and also cases an increase of eddycrrent loss. 4) Most of the contribtions to eddycrrent loss are given by harmonics p to the nineteenth. 5) It is possible to evalate eddycrrent loss dring operation by means of the harmonic analysis of the crrent. APPENDIX I COMPLETE FORMULA FOR EDDYCURRENT Loss EVALUATION The complete expression for evalation of eddycrrent loss with sinsoidal crrents is where 1+ Pe, B A=+ ) + (t), A =m l Pc2 3 sinh 20 + sin 24 cosh 2 cos 2t t t t sinh t sin t cosh t + cos t _ It may be seen that if t < 1.2 the terms of series developments with exponents higher than 4 may be omitted; i.e., when t < 1.2 eddycrrent losses are proportional to the sqare of the freqency. If hjlh = 0.8 (normal vale in transformer design) and t = 1.2 at 50 Hz, then a = 13.5 mm. However, since a in transformers seldom exceeds 3 mm, it is clear that the freqency that makes t = 1.2 is nearly 1000 Hz. APPENDIX II RECTIFIER CIRCUIT FORMULAS The following formlas give the vales of the main qantities occrring in a rectifier circit: 1) phase control (delay) angle ca 2) commtation angle Ea COS ae = 1 Edo COS (a + ) = cos a X6PIdpU = Cos a Ez EdO 3) indctive voltage drop (per nit vale) Ex XcpIdp Edo 2 Edo Ea Id Idp i(t) I it(t) It Il, In xcp Pot Pe, sin Pe, pls k = Pe, sin/pc2 a v q n f Ao p m a h APPENDIX II1 SYMBOLS average vale of direct voltage at no load redction of average vale of direct voltage cased by phase control redction of average vale of direct voltage cased by commtation average vale of direct crrent in load circit Id, per nit periodic crrent of arbitrary shape flowing throgh transformer winding rms vale of i(t) derivative of i(t) rms vale of i'(t) rms vale of first, nth harmonics of I commtating reactance, per nit 12R loss eddycrrent loss with sinsoidal crrent eddycrrent loss with plsating crrent eddycrrent loss ratio for sinsoidal crrent commtating angle (angle of overlap) phase control angle theoretic angle of commtation nmber of plses per cycle order of harmonic spply freqency (27rf = c = anglar freqency) X 101 henries/meter condctor resistivity nmber of radial layers radial width of each layer total height of coil in direction of leakage flx net height of condctors in direction of leakage flx. REFERENCES [1] R. Bean, N. Chackan, H. Moore, and E. Wentz, Transformers for the Electric Power Indstry. New York: McGrawHill, 1959, pp [2] H. B. Dwight, Electrical Coils and Condctors: Their Electrical Characteristics and Theory. New York: McGrawHill, 1945, pp [3] J. Schaefer, Rectifier Circits: Theory and Design. New York: Wiley, 1965, pp , [41 R. Richter, Elektrische Maschinen, pt. I. Berlin: Springer, 1924, pp [5] ANSI Standard C57.18, "Poolcathode mercryarc rectifier transformers, " [6] International Electrotechnical Commission, "Recommendations for mercryarc converters," IEC pbl. 84, sec. 342, [7] "Monocrystalline semicondctor rectifier cells, stack assemblies, and eqipment," IEC Pbl. 146, sec. 342, 1963.