calculation of eddy current losses using the electrodynamic ... - Komel

Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. I 111Bronisław Tomczuk, Dariusz KoterasOpole University of Technology, Dept. of Industrial Electrical EngineeringCALCULATION OF EDDY CURRENT LOSSES USING THEELECTRODYNAMIC SIMILARITY LAWSOBLICZANIE STRAT WIROPRĄDOWYCH Z WYKORZYSTANIEM PRAWPODOBIEŃSTWA ELKTRODYNAMICZNEGOAbstract: The results of the eddy currents losses calculations with using electrodynamics scaling werepresented in this paper. Scaling rules were used for obtain the same values of the eddy currents losses. For thecalculations Finite Element Method was used. Numerical calculations were verified by measurements and agood agreement was obtained.Streszczenie: W artykule przedstawiono wyniki obliczeń strat wiroprądowych z wykorzystaniem skalowaniaelektrodynamicznego. W modelowaniu matematycznym wykorzystano zasady skalowania obiektów w celuotrzymania jednakowych strat. Do obliczeń zastosowano Metodę Elementów Skończonych. Obliczenia zostałyzweryfikowane pomiarowo na obiekcie rzeczywistym i otrzymano dobrą zgodność obliczeń z pomiarami.Słowa kluczowe: analiza polowa, skalowanie elektrodynamiczne, straty wiroprądoweKeywords: field analysis, electrodynamics scaling, eddy current losses1. IntroductionBig dimensions of the electrical devices (forexample power transformers) in present dayscause difficulties in measurement verificationof their field analysis. On the other side, manyelectrical appliances have small dimensions, sothat the testing probes cannot be placed insideof their construction. Taking into account thedifficulties, one wants to build the scaledphysical models. The manufactured scaledphysical objects are building with similarityrules. Their testing allows to determine the fullscaled object characteristics [3]. The similarityrule depends on quantity, which must be holdfor scaled and original model, as well. In ourwork we have modeled the eddy current lossesin the magnetic core. Therefore the same valueof the scaled model eddy current losses shouldgovern the losses of the original object.2. Similarity rules used in this work2.1 General assumptions Dorgorg Horg orgEorgtorg Borgorg Eorg torg(1)The quantities in Maxwell’s equations whichdescribe the electromagnetic fields for theoriginal, have subscript “org” in the paper.Similarly to the equations above the quantitiesconcern the scaled models have subscript “scl”.A scale factor, as a ratio of a scale-modelquantity (X scl ) to the original quantity (X or ), hasbeen defined in this work for. Theelectromagnetic scale factors, for the mostimportant quantities are given in Eq. 2.HEsclsclmH , mE ,H Emmfffsclorgorgsclorgtt, msclorg, msclorgorgsclorg,l, mllsclorg(2)Identity of the Maxwell’s equations for originaland scaled model [2, 3, 5] arises the expressions3a and 3b.m2 fm mm l1(3a)m m 2 m fm l1(3b)For materials with linear characteristics, whichhave been assumed in this work, we obtain theexpression below.m m m 1(4)

112 Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. IThus, the magnetic field strength H current density J can be scaledmJmlmI 1m m mHHland the(5)2.2 The eddy current losses balanceLosses which are generating inside theconductors and magnetically soft iron elementsinfluence heating of all electromagneticdevices. Thus, the computer aided design(CAD) should allow us to calculate the losses asexactly as possible. Using the field analysis, wehave computed the losses in original magneticcircuit, and inside its scaled duplicate. We havedetermined the similarity rules to define scaledobject eddy currents losses.In this paper the conductivity and magneticpermeability factors m =m =1 were assumed.Based on the expression (3a) the frequencyscale-factor can be given as m f =1/m 2 l . Thus, therelationship, between scale factors for magneticfield strength m H and geometric dimensions m l ,is equal to1mH (6)mlIn our mathematical models we assumed thesame number of turns N in the original andscaled model. Thus, the values of the excitationcurrents for scaled models is givenlaminated amorphous toroid which create a partof the magnetic leg. Due to the solid part, themagnetic flux density in all magnetic circuit isrelatively low. Thus, the circuit can be assumedas magnetically linear. We assumed the relativepermeability of the magnetic parts from theamorphous ribbon: r =20173 (for the yokes)and r =2067 (for the legs). Because the flux inthe legs is perpendicular to wounding directionof the amorphous ribbon, the small permeabilityvalue is assumed. For the solid hollow cylinder,made from casting steel, the relativepermeability r =150 is assumed, and electricalconductivity =2.5·10 6 S/m have been included.a)Iscl m m I m I(7)Hlorglorg3. Analysed object and its mathematicalmodel3.1 Analysed objectWe have studied the modular construction ofthe amorphous, 1-phase transformer. Outlineand cross-section along YZ plane of theanalysed object are presented in figures 1a) and1b). Moreover, in this figures the maindimensions and Cartesian system are given.Owing to modular construction of theamorphous transformer, we determined theeddy current losses in the toroidal solidelement. It has been placed in the magneticcircuit as one part of its core. The core part wasmade from casting steel and situated in the rightside leg of the amorphous magnetic circuit (Fig.1b). Eddy current losses inside the solid toroidare several times higher than the ones inside theb)Fig. 1. Amorphous 1-phase transformer(original)a) Outline of the transformerb) Cross-section along YZ plane.

Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. I 113The excitation winding has been wounded withN=116 turns and placed on the laminated (left)leg, Fig. 1b.3.2 Mathematical modelFor numerical calculations the Finite ElementMethod (FEM) has been implemented inmodule Elektra of the commercial packageOPERA 3D [1]. This module enables us toanalyse the electromagnetic fields taking intoaccount the effects of eddy currents. It can bedone using total A tand reduced A rmagneticvector potentials [1, 4]. B A t(8) B 0H S A r(9)In our mathematical models, we assumedseveral values of rms values for the sinusoidalwaves of the excitation current: (1.42 to 9.50)A.The frequency of the supplying signal wasassumed to be f=50 Hz. Due to significantcurrent losses inside the steel solid toroid thelosses in the amorphous parts, can be neglected.Moreover, in our calculations we neglected thelosses appeared in the excitation coil. It can bepossible due to small cross section of its wires.4. Calculation results and measurementverificationIn this paper two variants of the field analysishave been carried out. The first variant W 1concerns the model two times smaller than theoriginal object (m l =0.5). The second variant W 2relates to model which is two times bigger(m l =2) than the original. The assumedfrequency for W 1 variant was f scl =200 Hz. Thus,the scale factor for the frequency amounts m f =4.The rms values of the excitation current, has bechanged (I scl = 1.01÷6.74)A for consideredcases. In W 2 model the frequency f scl =12.5 Hzand its scale factor (m f =0.25) were arranged.Using Eq. 7 the rms current values forsinusoidal waves were changed from I scl = 2.0 toI scl = 13.3A, for the modeled objects.a)b)c)

114 Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. ICalculated values of the eddy current losses fordifferent excitations in both scaled variants andoriginal object are placed in the table 1.Table 1. Calculated values of the eddy currentlosses inside the solid toroidI orgCalculationsm l =0.5 original m l =2[A] [W] [W] [W]1.42 5.44 5.85 6.003.06 24.90 27.29 28.005.25 73.05 80.07 82.167.23 138.40 151.70 155.669.50 239.2 262.56 269.42Fig. 2. Eddy current density distributions on thesolid toroid surface and magnetic flux densityon plane YZ for excited current (originalobject) I org =5.25 Aa) original modelb) model with scaled factor m l =0.5c) model with scaled factor m l =2In figures 2a÷2c the eddy current densitiesdistributions on the surface of the casting steeli.e. solid toroid, for both original and scaled(two variants) numerical models, are presented.The calculations have been carried out for rmsvalue of the excitation current I org =5.25 A. Thepresented distributions are similar to each other.Despite different dimensions and excitationcurrent frequencies, the magnetic fields insidethe analyzed magnetic circuits are notsignificantly different. For original object themaximal value of the eddy current density isslightly lower than 3 MA/m 2 , whereas for thescaled model with dimension factor m l =0.5(variant W 1 ) the maximum values reach almost8 MA/m 2 . In the case of two times bigger object(m l =2), this value is about 1.28 MA/m 2 .Eddy currents significantly influence the corelosses, naturally. In this paper the losses of thesolid steel toroid were calculated using theexpression below2 J P eddV (10)V The relative differences between obtainedvalues for both scaled (variants W 1 and W 2 ) andoriginal models are lower than 10 %. Thus thecalculations carried out in this work confirmcorrectness of the scaling rules. In Fig. 3 thecalculated and measured values of the corelosses inside of the solid toroid, for original andscaled objects, were given. As we can see, thepresented values for the two objects come tonear.Fig. 3. The calculated and measured values ofthe eddy current losses inside the solid toroidfor the original and scaled objects.5. ConclusionsIn this paper the eddy current losses inside thesolid toroid, placed in the modular 1-phasemagnetic circuit from amorphous ribbon, werecalculated. The electromagnetic similarity ruleshave been analysed and used. The similarityrules were assumed to hold the same values ofthe eddy current losses. Objects for two scaledfactors m l =0.5 and m l =2 have been modeled.

Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. I 115Obtained values of the losses (table 1) confirmthe correctness of the similarity rules. Thus, bycombining the scaled physical model and themathematical field problem solution, thedesigner can create, without expensiveprototyping, the accurate geometry of anelectromagnetic device.6. References[1] OPERA Manager User Guide Version 13,Cobham Technical Services, Oxford, England,2009.[2] Zakrzewski K., Tomczuk B., Waindok A.,Nonlinear scaled models in 3-d calculation oftransformer magnetic circuits, XVIIISymposium Electromagnetic Phenomena inNonlinear Circuits, Poznan, Poland, 28-30 VI,2004, pp. 31-32.[3] Turowski J., Elektrodynamika Techniczna, WNTWarszawa 1993r.[4] Xu E.X., Simkin J., Total/Reduced magneticvector potential and electrical scalar potentialfor eddy current calculation, Compumag’03,Saratoga Springs, 13-17 VII 2003, pp. I / 8 – 9.[5] Zakrzewski K., Modelowanie pólelektromagnetycznych w projektowaniutransformatorów, Przegląd Elektrotechniczny, nr3, 2002, s. 59-63.D. Koteras was born in Swidnica, Poland in1972. He received the MS and PhD degrees inElectrical Engineering from Opole Universityof Technology in 1998 and 2006, respectively.Since 1998, he has been employed at theuniversity. Presently, he is an adjunct.ReviewerProf. dr hab. inż. Wojciech Szeląg7. AcknowledgementsThis paper is partially supported by theNational Science Centre under grant no. NN510 533739.8. AuthorsProf. Bronislaw Tomczuk, (Ph.D., D.Sc.),Department of Electrical Engineering, OpoleUniversity of Technology, ul. Proszkowska 76,45-758 Opole.PhD Eng. Dariusz Koteras, as above.B. Tomczuk was born in Poland in 1953. Hereceived his PhD and DSc degrees from theTechnical University of Lodz (Poland) in 1985and 1995, respectively. Prior to that, hereceived the MSc degree in ElectricalEngineering with honours in 1977. Since 1978,he has been on the staff of the TechnicalUniversity of Opole, Poland. From 1996, he hasbeen within the professor staff of Faculty ofElectrical Engineering, Automatic Control andInformatics. He obtained his full professor titlein 2007. At present, he is the Head of theDepartment of Industrial Electrical Engineeringat the Opole University of Technology.

116 Zeszyty problemowe – Maszyny Elektryczne Nr 100/2013 cz. I