A New Approach in Analytical Analysis of Eddy Currents in Laminated Core

Transcription

1 J. Basic. Appl. Sci. Res., (7) , 1 1, TextRoa Publication ISSN Journal of Basic an Applie Scientific Research A New Approach in Analtical Analsis of E Currents in Laminate Core Jaber Merrikhi Ahangarkolaei *, Yaser Merrikhi Ahangarkolaei an Mir Maji Temoori Department of Electrical Engineering, Firoozkooh Branch, Islamic Aza Universit, Firoozkooh, IRAN ABSTRACT Conventional methos of e-currents moeling are base on solving iffusion equation, which its variable is the magnetic fiel intensit; an then calculating e currents istribution b using the Curl of magnetic fiel intensit. Using the Curl operator necessitates calculating the erivative of fiel intensit, which is obtaine b some numerical fiel analsis methos such as Finite Elements Metho (FEM). Taking erivative causes man isasters, such as iscontinuit in the current ensit istribution. In this paper, for the first time, iffusion equation is written for the electric fiel strength as the main variable. Base on the analtical formula erive from the propose metho, the same results are achieve. Simplicit of the propose metho enables us to emplo it in numerical analsis such as finite element metho. KEY WORDS: e current, laminate core, power loss, skin effect. INTRODUCTION Ferromagnetic materials, such as iron, nickel an their allos are wiel use for builing cores of magnetic evices [1]. In the esign of ac machines an rives, the preiction of power losses in magnetic cores is important []. The etermination of e currents in a given magnetic configuration is alwas a complicate task. As Depicte in Fig.1, the cores are usuall laminate to reuce e-current loss, however this loss can still be significant, especiall at the higher frequencies prouce b pulse-with moulation waveforms in tpical ac rives [1]. Fig.1: Laminate magnetic core Design engineers usuall preict lamination loss using two methos []. One wa is to use manufacturer s atasheets. Usuall in the atasheet of the core, a cluster of curves can be foun, which show the core loss per unit volume uner sine-wave excitation versus the maximum flux ensit for ifferent values of frequenc, but such curves are often not available or in practical applications are often ifferent from those uner which the atasheets ha been obtaine. The other metho is to use a simple formula for e-current loss as a function of lamination thickness, but it is onl vali for frequencies of about 6 Hz or less. In the case of most ac rives, significant frequenc content in the kilohertz an megahertz regions is require. At such high frequencies, e-current loss is much greater than hsteresis loss an, thus, core loss is almost entirel ue to e currents []. The subject of core losses in AC machines an in particular laminations has been a subject of research since the as of Steinmetz [3]. Steinmetz propose approximate formulas for the harmonic case which are quite similar to those currentl use toa. His work was empiricall erive an supporte from a theoretical viewpoint b Bertotti [4]. It is assume in [3] that the magnetic flux istribution in the sheet is uniform, making the metho applicable onl in the frequenc range where the skin effect is negligible or to the magnetic cores with low conuctivities or thin sheets [5]. This latter formulation i not prove accurac enough which motivate Bertotti to propose an aitional term, the excess loss term which accounte for the ifference between measurements an theor [6]. * Corresponing Author: Jaber Merrikhi Ahangarkolaei, Department of Electrical Engineering, Firoozkooh Branch, Islamic Aza Universit, Firoozkooh, IRAN. [jmerrikhi@iaufb.ac.ir] 741

2 Ahangarkolaei et al., 1 Moreover, in the last ecaes, a great number of papers have been evote to obtain approximate e-current losses. Conventional methos of moeling e-currents are base on solving iffusion equation for magnetic fiel intensit. The e currents istribution is then obtaine b taking the Curl of magnetic fiel intensit. Using the Curl operator necessitates calculating the erivative of fiel intensit. Taking erivative causes man problems, such as iscontinuit in the current ensit istribution an amplification of the errors [7]. The objective of this paper is to present a new analtical metho of e current ensit an e current core power loss from Integral form of Faraa's law (Eq.1), that can be use in numerical analsis. E. l (1) t Where H, J, B, E are the magnetic fiel strength, current ensit, magnetic flux ensit, an electric fiel strength respectivel. 1.Calculation of e current in a single sheet For a lamination with thickness an height l as shown in Fig., an assuming constant magnetic permeabilit an conuctivit, electrical fiel onl has the z irection component as a function of x. Fig.: Single lamination an the coorinate sstem. Neglecting the isplacement current, Maxwell s equations an Ohm s law becomes: ( x, t ) ( x, t ) () t x Letting E(t) be sinusoial an assuming the complex exponential form: j t ( x, t) Re( ( x) e ) (3) Where ω is the angular frequenc. Combining the (Eq.3), (Eq.4) equations, we have : (, ) E (, ) z x t j x t (4) x Stu of Fig. will show that, bounar conitions exist for thin sheets with an axiall applie fiel: E E (5) x (6) z z (7) ( x ) E (8) ( x ) E (9) Eq. 4 is a stanar secon orer ifferential equation, which has a general solution an it is convenient to write the solution of ifferential equation in term of hperbolic functions. A sinh( Tx ) B cosh( Tx ) (1) Where T is constant an is equal to: T (1 j) f j (11) Which A an B being arbitrar constant values. Base on the bounar conition, E iels sinh( Tx) E (1) sinh( T ) 74

3 J. Basic. Appl. Sci. Res., (7) , 1 There remains onl E to be evaluate. Using Faraa s line integral an integrating aroun the path ABCD in Fig., iel: E. l (13) t Where ϕ is the total flux within the line ABCD. If the height is much greater than the thickness of sheet, the electrical fiel E in complex notation form is: j E (14) l In the case shown in Fig., The magnetic fiel intensit is escribe b an orinar secon-orer ifferential equation, which is the Helmholtz equation, H ( x, t) H ( x, t) (15) t x For a sinusoial applie magnetic strength, Eq.5 becomes. H ( x, t ) j H ( x, t ) (16) x Solution of Eq.17 can be foun in previous articles an becomes [8]: cosh( Tx) H ( x) H (17) cosh( T ) Where, H is the bounar conition. H( ) H( ) H (18) The magnetic fiel intensit as a function of x is shown in Fig.3 for H =1(A/m), σ= 1.4e6 an µ r =3 at f=1, an 5 khz. It can be seen that the magnetic fiel intensit ecreases with increasing frequenc f in the center of the lamination. Fig.3: The magnetic fiel intensit as a function of x at f=1, an 5 khz The magnetic flux flowing through the cross section of a single lamination perpenicular to the xz plane is: / / l lh H ( x) s ( ) ( ) cosh( ) H x x l H x x Tx x s / cosh( T ) / 743

4 Ahangarkolaei et al., 1 T T lh sinh( ) sinh( ) lh T tanh( ) (19) T T cosh( ) T Substitution of (Eq.) an (Eq.15) into (Eq.13) an J E prouces: j H sinh( ) sinh( ) T Tx Tx J z ( x) tanh( ) TH () T sinh( T ) cosh( T ) The magnitue of the e current ensit as a function of x is shown in Fig.4, for H =1(A/m), σ= 1.4e6 an µ r =3 at f=1, an 5 khz. Fig. 4: The magnitue of the e current ensit as a function of x at f=1, an 5 khz.e current power loss The e current power loss ensit istribution in a conucting core average with respect of time is: 3 P ( x ) E. J J ( W / m ) (1) Substitution of (Eq.) into (Eq.1) prouces: 1 sinh( Tx) P( x) TH () cosh( T ) The average power loss ensit over the lamination with is the power loss per unit volume, calle the specific core loss. Its average value in the sheet is: / / 1 1 sinh( Tx) TH sinh( T ) T Pv P( x) x TH x (3) / T cosh( T ) cosh ( ) Next, to emonstrate the valiit of Eq.3, the result of Eq.3 is compare with the e current loss from the Curl of magnetic fiel intensit. RESULTS Recall that Eq. 3 is obtaine from integral form of Maxwell's equations. In the latest research Instea of Eq.3, in [8] Bermúez proposes Eq.4 that obtaine from Curl of magnetic fiel intensit. 744

5 J. Basic. Appl. Sci. Res., (7) , 1 H sinh( ) sin( ) Pv W m cosh( ) cos( ) 3 ( / ) Where f. Bermúez in [8] showe that Eq.4 is exact for an frequenc. In Fig. 5, we show the losses given b the above analtical formulas versus frequenc in a last given lamination. Can be seen, the results of Eq.3 is completel consistent with the results of Eq. 4. (4) Conclusion Fig. 5: Average e-current loss versus f(hz) In this paper, a new analtical consieration of e current loss is evaluate. For this purpose, Integral form of Faraa s law is emploe to calculate e current ensit an e current core power loss. Utilization Steinmetz an Bertotti theoretical methos, a comparison between this metho an their methos are accomplishe. Base on the analtical formula erive from the propose metho, the same results are achieve. Simplicit of the propose metho enables us to emplo it in numerical analsis such as finite element metho. Acknowlegment This research work has been accomplishe at Department of electrical engineering, Firoozkooh branch, Islamic Aza Universit, Firoozkooh, IRAN. Authors woul like to thanks from its financial support. REFERENCES 1. J. Xu, A. Lakhsasi, Z. Yao an V. Rajagopalan,1996. A Practical Moeling Metho for E-Current Losses Computation in Laminate Magnetic Cores. IEEE Inustr Applications Conference : John R. Brauer, Zoltan J. Cenes, Bruce C. Beihoff, an Kenneth P. Phillips,. Laminate Steel E-Current Loss versus Frequenc Compute Using Finite Elements. IEEE Transactions On Inustr Applications, Vol. 36, No. 4 : C. P. Steinmetz,1984. On the law of hsteresis.proc. IEEE, vol. 7, no. : G. Bertotti, General properties of power losses in soft ferromagnetic materials. IEEE Trans. Magn. Vol. 4 : Ema Dlala, 8. A Simplifie Iron Loss Moel for Laminate Magnetic Cores. IEEE Transactions On Magnetics, Vol. 44, No. 11: Yu Zhang, Rui Guan, Pragasen Pilla an Ming-C. Cheng, 9. General Core Loss Moels on a Magnetic Lamination. IEMDC '9. IEEE International Electric Machines an Drives Conference, 9. : J.Merrikhi, J.S.Moghani, E.Fallah, 8. A New Approach in Finite Element Analsis of E Current with Flux Skin Effect. Amirkabir International Journal of Science an Technolog : Alfreo Bermúez, Dolores Gómez, an Pilar Salgao, 8. E-Current Losses in Laminate Cores an the Computation of an Equivalent Conuctivit. IEEE Transactions On Magnetics, Vol. 44, No. 1: