Zeszyty probemowe Maszyny Eektryczne Nr 100/2013 cz. I 111 Bronisław Tomczuk, Dariusz Koteras Opoe University of Technoogy, Dept. of Industria Eectrica Engineering CALCULATION OF EDDY CURRENT LOSSES USING TE ELECTRODYNAMIC SIMILARITY LAWS OBLICZANIE STRAT WIROPRĄDOWYC Z WYKORZYSTANIEM PRAW PODOBIEŃSTWA ELKTRODYNAMICZNEGO Abstract: The resuts of the eddy currents osses cacuations with using eectrodynamics scaing were presented in this paper. Scaing rues were used for obtain the same vaues of the eddy currents osses. For the cacuations Finite Eement Method was used. Numerica cacuations were verified by measurements and a good agreement was obtained. Streszczenie: W artykue przedstawiono wyniki obiczeń strat wiroprądowych z wykorzystaniem skaowania eektrodynamicznego. W modeowaniu matematycznym wykorzystano zasady skaowania obiektów w ceu otrzymania jednakowych strat. Do obiczeń zastosowano Metodę Eementów Skończonych. Obiczenia zostały zweryfikowane pomiarowo na obiekcie rzeczywistym i otrzymano dobrą zgodność obiczeń z pomiarami. Słowa kuczowe: anaiza poowa, skaowanie eektrodynamiczne, straty wiroprądowe Keywords: fied anaysis, eectrodynamics scaing, eddy current osses 1. Introduction Big dimensions of the eectrica devices (for exampe power transformers) in present days cause difficuties in measurement verification of their fied anaysis. On the other side, many eectrica appiances have sma dimensions, so that the testing probes cannot be paced inside of their construction. Taking into account the difficuties, one wants to buid the scaed physica modes. The manufactured scaed physica objects are buiding with simiarity rues. Their testing aows to determine the fu scaed object characteristics [3]. The simiarity rue depends on quantity, which must be hod for scaed and origina mode, as we. In our work we have modeed the eddy current osses in the magnetic core. Therefore the same vaue of the scaed mode eddy current osses shoud govern the osses of the origina object. 2. Simiarity rues used in this work 2.1 Genera assumptions D E t B E t (1) The quantities in Maxwe s equations which describe the eectromagnetic fieds for the origina, have subscript in the paper. Simiary to the equations above the quantities concern the scaed modes have subscript. A scae factor, as a ratio of a scae-mode quantity (X ) to the origina quantity (X or ), has been defined in this work for. The eectromagnetic scae factors, for the most important quantities are given in Eq. 2. E m, me, E m m f f f t t, m, m,, m (2) Identity of the Maxwe s equations for origina and scaed mode [2, 3, 5] arises the expressions 3a and 3b. m 2 f m mm 1 (3a) m m 2 m f m 1 (3b) For materias with inear characteristics, which have been assumed in this work, we obtain the expression beow. m m m 1 (4)
112 Zeszyty probemowe Maszyny Eektryczne Nr 100/2013 cz. I Thus, the magnetic fied strength current density J can be scaed mj m mi 1 m m m and the (5) 2.2 The eddy current osses baance Losses which are generating inside the conductors and magneticay soft iron eements infuence heating of a eectromagnetic devices. Thus, the computer aided design (CAD) shoud aow us to cacuate the osses as exacty as possibe. Using the fied anaysis, we have computed the osses in origina magnetic circuit, and inside its scaed dupicate. We have determined the simiarity rues to define scaed object eddy currents osses. In this paper the conductivity and magnetic permeabiity factors m =m =1 were assumed. Based on the expression (3a) the frequency scae-factor can be given as m f =1/m 2. Thus, the reationship, between scae factors for magnetic fied strength m and geometric dimensions m, is equa to 1 m (6) m In our mathematica modes we assumed the same number of turns N in the origina and scaed mode. Thus, the vaues of the excitation currents for scaed modes is given aminated amorphous toroid which create a part of the magnetic eg. Due to the soid part, the magnetic fux density in a magnetic circuit is reativey ow. Thus, the circuit can be assumed as magneticay inear. We assumed the reative permeabiity of the magnetic parts from the amorphous ribbon: r =20173 (for the yokes) and r =2067 (for the egs). Because the fux in the egs is perpendicuar to wounding direction of the amorphous ribbon, the sma permeabiity vaue is assumed. For the soid hoow cyinder, made from casting stee, the reative permeabiity r =150 is assumed, and eectrica conductivity =2.5 10 6 S/m have been incuded. a) I m m I m I (7) 3. Anaysed object and its mathematica mode 3.1 Anaysed object We have studied the moduar construction of the amorphous, 1-phase transformer. Outine and cross-section aong YZ pane of the anaysed object are presented in figures 1a) and 1b). Moreover, in this figures the main dimensions and Cartesian system are given. Owing to moduar construction of the amorphous transformer, we determined the eddy current osses in the toroida soid eement. It has been paced in the magnetic circuit as one part of its core. The core part was made from casting stee and situated in the right side eg of the amorphous magnetic circuit (Fig. 1b). Eddy current osses inside the soid toroid are severa times higher than the ones inside the b) Fig. 1. Amorphous 1-phase transformer (origina) a) Outine of the transformer b) Cross-section aong YZ pane.
Zeszyty probemowe Maszyny Eektryczne Nr 100/2013 cz. I 113 The excitation winding has been wounded with N=116 turns and paced on the aminated (eft) eg, Fig. 1b. 3.2 Mathematica mode For numerica cacuations the Finite Eement Method (FEM) has been impemented in modue Eektra of the commercia package OPERA 3D [1]. This modue enabes us to anayse the eectromagnetic fieds taking into account the effects of eddy currents. It can be done using tota A t and reduced A r magnetic vector potentias [1, 4]. B A t (8) B 0 S A r (9) In our mathematica modes, we assumed severa vaues of rms vaues for the sinusoida waves of the excitation current: (1.42 to 9.50)A. The frequency of the suppying signa was assumed to be f=50 z. Due to significant current osses inside the stee soid toroid the osses in the amorphous parts, can be negected. Moreover, in our cacuations we negected the osses appeared in the excitation coi. It can be possibe due to sma cross section of its wires. 4. Cacuation resuts and measurement verification In this paper two variants of the fied anaysis have been carried out. The first variant W 1 concerns the mode two times smaer than the origina object (m =0.5). The second variant W 2 reates to mode which is two times bigger (m =2) than the origina. The assumed frequency for W 1 variant was f =200 z. Thus, the scae factor for the frequency amounts m f =4. The rms vaues of the excitation current, has be changed (I = 1.01 6.74)A for considered cases. In W 2 mode the frequency f =12.5 z and its scae factor (m f =0.25) were arranged. Using Eq. 7 the rms current vaues for sinusoida waves were changed from I = 2.0 to I = 13.3A, for the modeed objects. a) b) c)
114 Zeszyty probemowe Maszyny Eektryczne Nr 100/2013 cz. I Cacuated vaues of the eddy current osses for different excitations in both scaed variants and origina object are paced in the tabe 1. Tabe 1. Cacuated vaues of the eddy current osses inside the soid toroid I Cacuations m =0.5 origina m =2 [A] [W] [W] [W] 1.42 5.44 5.85 6.00 3.06 24.90 27.29 28.00 5.25 73.05 80.07 82.16 7.23 138.40 151.70 155.66 9.50 239.2 262.56 269.42 Fig. 2. Eddy current density distributions on the soid toroid surface and magnetic fux density on pane YZ for excited current (origina object) I =5.25 A a) origina mode b) mode with scaed factor m =0.5 c) mode with scaed factor m =2 In figures 2a 2c the eddy current densities distributions on the surface of the casting stee i.e. soid toroid, for both origina and scaed (two variants) numerica modes, are presented. The cacuations have been carried out for rms vaue of the excitation current I =5.25 A. The presented distributions are simiar to each other. Despite different dimensions and excitation current frequencies, the magnetic fieds inside the anayzed magnetic circuits are not significanty different. For origina object the maxima vaue of the eddy current density is sighty ower than 3 MA/m 2, whereas for the scaed mode with dimension factor m =0.5 (variant W 1 ) the maximum vaues reach amost 8 MA/m 2. In the case of two times bigger object (m =2), this vaue is about 1.28 MA/m 2. Eddy currents significanty infuence the core osses, naturay. In this paper the osses of the soid stee toroid were cacuated using the expression beow 2 J P ed dv (10) V The reative differences between obtained vaues for both scaed (variants W 1 and W 2 ) and origina modes are ower than 10 %. Thus the cacuations carried out in this work confirm correctness of the scaing rues. In Fig. 3 the cacuated and measured vaues of the core osses inside of the soid toroid, for origina and scaed objects, were given. As we can see, the presented vaues for the two objects come to near. Fig. 3. The cacuated and measured vaues of the eddy current osses inside the soid toroid for the origina and scaed objects. 5. Concusions In this paper the eddy current osses inside the soid toroid, paced in the moduar 1-phase magnetic circuit from amorphous ribbon, were cacuated. The eectromagnetic simiarity rues have been anaysed and used. The simiarity rues were assumed to hod the same vaues of the eddy current osses. Objects for two scaed factors m =0.5 and m =2 have been modeed.
Zeszyty probemowe Maszyny Eektryczne Nr 100/2013 cz. I 115 Obtained vaues of the osses (tabe 1) confirm the correctness of the simiarity rues. Thus, by combining the scaed physica mode and the mathematica fied probem soution, the designer can create, without expensive prototyping, the accurate geometry of an eectromagnetic device. 6. References [1] OPERA Manager User Guide Version 13, Cobham Technica Services, Oxford, Engand, 2009. [2] Zakrzewski K., Tomczuk B., Waindok A., Noninear scaed modes in 3-d cacuation of transformer magnetic circuits, XVIII Symposium Eectromagnetic Phenomena in Noninear Circuits, Poznan, Poand, 28-30 VI, 2004, pp. 31-32. [3] Turowski J., Eektrodynamika Techniczna, WNT Warszawa 1993r. [4] Xu E.X., Simkin J., Tota/Reduced magnetic vector potentia and eectrica scaar potentia for eddy current cacuation, Compumag 03, Saratoga Springs, 13-17 VII 2003, pp. I / 8 9. [5] Zakrzewski K., Modeowanie pó eektromagnetycznych w projektowaniu transformatorów, Przegąd Eektrotechniczny, nr 3, 2002, s. 59-63. D. Koteras was born in Swidnica, Poand in 1972. e received the MS and PhD degrees in Eectrica Engineering from Opoe University of Technoogy in 1998 and 2006, respectivey. Since 1998, he has been empoyed at the university. Presenty, he is an adjunct. Reviewer Prof. dr hab. inż. Wojciech Szeąg 7. Acknowedgements This paper is partiay supported by the Nationa Science Centre under grant no. N N510 533739. 8. Authors Prof. Bronisaw Tomczuk, (Ph.D., D.Sc.), Department of Eectrica Engineering, Opoe University of Technoogy, u. Proszkowska 76, 45-758 Opoe. PhD Eng. Dariusz Koteras, as above. B. Tomczuk was born in Poand in 1953. e received his PhD and DSc degrees from the Technica University of Lodz (Poand) in 1985 and 1995, respectivey. Prior to that, he received the MSc degree in Eectrica Engineering with honours in 1977. Since 1978, he has been on the staff of the Technica University of Opoe, Poand. From 1996, he has been within the professor staff of Facuty of Eectrica Engineering, Automatic Contro and Informatics. e obtained his fu professor tite in 2007. At present, he is the ead of the Department of Industria Eectrica Engineering at the Opoe University of Technoogy.
116 Zeszyty probemowe Maszyny Eektryczne Nr 100/2013 cz. I