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Flux Displacement in Rectangular Iron Sheets and Geometry-Dependent Hysteresis Loss F. Bachheibl, D. Gerling Abstract – This paper suggests a possible explanation to the geometry-dependence of core loss by solving Maxwell’s equations for the flux distribution inside a magnetic conductor carrying eddy currents. The solution reveals a displacement of flux towards the edges of the magnetic sheet. Core loss with and without the displacement effect are compared for the tooth of a highly excited and high-frequency electric machine. A significant difference from standard calculation methods is obtained. Index Terms -- Analytical Calculation, Core Loss, Excess Loss, Flux Displacement, Magnetic Skin Effect, Rectangular Cross Section derivation of the flux distribution is obtained by solving the dual problem to the distribution of current in rectangular conductors, as investigated in [1]. y x h w E I. INTRODUCTION LECTRIC MACHINES are subject to various sources of loss, the most prominent being conduction loss and iron loss. Conduction loss is reliably quantifiable, whereas iron loss is still subject to great uncertainty when designing an electrical machine. As conduction loss is the major loss source in electrical machines, up to now, iron loss has been of minor importance, resulting in an inaccuracy in calculation of as much as up to 2000% in extreme cases [8]. The striving for efficiency and compactness of electrical machines now puts an increasing focus on iron losses, that is to say eddy current loss and hysteresis loss. Minimization of eddy current loss in electrical machines is achieved by lamination of the flux-conducting parts. This measure reduces the impact of two loss-mechanisms: First of all, eddy currents produce conduction losses in the magnetic core. Secondly, the field is displaced out of the center of the magnetic body towards the edges of the material, resulting in a locally higher flux density and therefore a higher overall hysteresis loss. This paper investigates the second effect and tries to establish that it might be one of the causes for the aforementioned inaccuracies which should be taken into consideration for future design of electric machines and for the estimation of losses. II. Fig. 1. Sheet of iron under investigation In Maxwell’s equation (1) for the electric field, the permeability µ is considered a constant in order to facilitate the derivation: rot ( E ) E E x (x, y, t) ex E y (x, y, t) ey (2) whereas the magnetic flux and field only have a nonzero component in z-direction: H(x, y, t) H z (x, y, t) e z (3) Additionally, it is assumed that field quantities are independent of z, which is consistent to the assumption of infinite length along that axis. From (1), (2) and (3), the following conclusions can be drawn: E y H x 0 dt z H y E x 0 dt z E y Ex Hz dt x y A. Derivation of the field distribution The sheet of transformer steel under investigation is depicted in Fig. 1. Its geometric properties are the width w in direction of the x-axis and the height h following the yaxis. It is assumed to be of infinite length in z-direction. The M.Sc. Florian Bachheibl is with University of Federal Defence, Chair for electrical Drives and Actuators, D-85577 Neubiberg, Germany (E-mail: [email protected]) ©2014 IEEE 978-1-4799-4775-1/14/$31.00 (1) It is assumed that the electric Field E, and therefore the current density, only exist in the x-y-plane, FLUX DISPLACEMENT IN RECTANGULAR IRON CORES This work was supported by SIEMENS AG, Industry Sector, Large Drives H B t t ( 4.1 ) ( 4.2 ) ( 4.3 ) From Maxwell’s Equation (5) for the magnetic flux, disregarding the displacement current: 1187 Prof. Dr.-Ing. Dieter Gerling is with University of Federal Defence, Chair for electrical Drives and Actuators, D-85577 Neubiberg, Germany (E-mail: [email protected]) dD rot(H) j E dt (5) and (2) and (3), we obtain the following conditions: H z Ex y ( 6.1 ) H z E y x ( 6.2 ) Inserting equations (6.1) and (6.2) into (4.3), we get: 2 H z 2 H z H z 2 2 x y dt (x≥0, y≥0) is regarded in the following to calculate k and l . In order to obtain a complete enclosure to the Volume V, the covers in the drawing plane would have to be considered, but can be neglected due to the assumption of infinite length. Additionally, Volume V is enclosed by the boundaries of the region under investigation. These boundaries towards the environment may be disregarded for the solution of (12) as they cannot be penetrated by current and therefore (14) must apply: E A j A 0 (7) y Which is identical to the result obtained in [1] for the electric field intensity in rectangular conductors. B. Determination of constants As derived in [1] for the electric field, the magnetic field intensity on the sheet is described by (8): H z (x, y, t) C cosh kx cosh ly e jt ( 14 ) A1 z h/2 A2 x (8) w/2 where Fig. 2: Current flow in upper right quarter of the domain 2 2 k l j (9) Inserting (6.1) and (6.2) into (13) yields the following result: h w cosh l cosh k 2 2 C can be obtained by assuming that the flux passing through the area be independent of the frequency. Therefore, it must be guaranteed that: f (t) w /2 h /2 H z (x, y, t)dydx f 0 (t) And therefore: k l w h w h sinh k sinh l 2 2 ( 11 ) is to be chosen identical to the DCThereby, the value for B case with a homogenous distribution of flux. In order to obtain k and l , Maxwell’s equation for the electric field is applied to the problem: E 0 E dA E dA E dA A1 C. Characteristic Distribution of Fields The highest induction in electrical machines appears in the teeth and therefore one section of electrical steel located in the tooth of an electrical machine is chosen to demonstrate the effect of flux displacement due to eddy currents. Table 1 contains all relevant information on an electric turbocharger motor developed at the Institute for Electrical Drives and Actuators, discussed in [6], which is selected for its high supply frequency. Table 1: Machine Data ( 12 ) Here, ρ is the density of volume charge in the region. Equation (12) in its integral form reads: V ( 16 ) Notably, for square geometries with h=w, we obtain that k=l, as also found by [1]. Equations (9) and (16) easily allow the calculation of l and k . Solving (10) for C yields: B̂ 4 l h k w ( 10 ) w /2 h /2 C ( 15 ) A2 ( 13 ) Rated power Rated torque Tooth width Tooth induction Maximum speed Maximum supply Frequency Sheet material Sheet conductivity (150°C) Sheet relative permeability (1.2 T) 230 W 0.12 Nm 8.5 mm 1.2 T 18000 min-1 3200 Hz 30 EX 1200 1.2195·106 Sm-1 5970 The areas A1 and A2 represent the areas in Fig. 2 colored in1188Material data for 30 EX 1200 was obtained from a steel black and blue, respectively, which cut the region of Fig. 1 datasheet [10].The slice of material considered below has into four identical pieces. One fourth of the iron sheet the tooth width set as width w and the material thickness set as height h. As Fig. 3 indicates, the operating point of flux density at 1.2 T is situated just at the saturation point of the material. The formulae developed above lose their validity beyond this point of saturation, as constant permeability had been assumed. The effect of flux displacement causes local field intensity and flux density to increase to values well above the homogenous case. Figures 6 to 11 therefore only convey qualitative information on the effect flux-displacement has on the distribution of magnetic field. Fig. 5: Magnetic field intensity for f=50 Hz at t=0.55/ω Fig. 3: B-H-Curve of 30 EX 1200 (Manufacturer Data) Figures 4 to 7 show the distribution of magnetic field intensity, assuming a flux corresponding to a flux density of 1.2 T. The distribution of field and flux density has been evaluated at different time steps for 50 and 3200 Hz, respectively. There is an obvious phase shift in the field intensity between the edges and the center of the conductor, resulting from a behavior similar to a wave travelling from the edges towards the inside of the conductor. Figures 8 and 9 illustrate the electric field intensity for t=1/ω and for t=0.4 ω for a frequency of 3200 Hz. Fig. 6: Magnetic field intensity for f=3200 Hz at t=1/ω Fig. 7: Magnetic field intensity for f=3200 Hz at t=0.55/ω Fig. 4: Magnetic field intensity for f=50 Hz at t=1/ω 1189 Fig. 8: Electric field intensity for f=3200 Hz at t=1/ω As already mentioned, field intensities as obtained for 3200 Hz excite the material well into saturation. In order to calculate the flux density, the B-H-curve shown in Fig. 3 is fitted by a Langevin-Function, which is very suitable for modelling saturation and also used in the Jiles-Atherton Model [3]: 1 B a, b, H a coth b H bH ( 17 ) The parameter fit is performed by a nonlinear least square algorithm and obtains the following values for a and b: Table 2: Coefficients for Langevin-Function Fig. 9: Electric field intensity for f=3200 Hz at t=0.4 ω Figures 10 and 11 display the maximum values of field intensity over time, as attained by each point in the domain. Fig. 10: Distribution of peak magnetic field intensity for f=50 Hz a b 1.379 0.06431 Using (17), the flux densities corresponding to the field intensities shown in Fig. 6 and Fig. 11 can be calculated. The resulting flux density distribution and flux density at t=1/ω are shown in Figures 13 to 15. The flux densities shown do result from eddy currents and their respective fields as displayed in Figures 4 to 11. However, it needs to be said that eddy currents are driven by a change in flux density and not in field intensity. This change, being subject to saturation effects, is too small for an effect at that scale. The chain of effects shown in Fig. 12 elucidates these dependencies which take effect simultaneously, according to Maxwell’s equations in general and their application in this paper in particular. An externally excited magnetic field causes flux inside the material, which is subject to saturation. Changes in flux density lead to an electric field and therefore to eddy currents. Those cause an internal magnetic field that adds to the external, homogenous magnetic field as fields superimpose by simple addition. The nonlinear dependency of flux and field intensity determines the flux excited by external and internal magnetic field, thus closing the loop of Fig. 12. In Section E, the implications of saturation as well as a possible remedy are discussed in order to enlarge the scope of applicability of the formulae developed. Fig. 11: Distribution of peak magnetic field intensity for f=3200 Hz Fig. 12: Chain of effects 1190 Fig. 15: Total core loss versus flux density ANSYS Maxwell’s material library provides a convenient tool to calculate coefficients for loss separation using core loss data, conductivity and thickness of the material. The following coefficients can thus be obtained for the steel in use: Fig. 13: Distribution of peak magnetic flux density for f=3200 Hz Table 3: Loss coefficients for 30 EX 1200, scaled for W/kg Hysteresis loss coefficient kH Eddy-current loss coefficient kE 0.0157 22.9*10-6 Using these coefficients, it is possible to calculate the total core loss by ( 20 ): 2 k f B Pv k H f B E Fig. 14: Magnetic Flux Density for f=3200 Hz at t=1/ω D. Increase in Hysteresis Loss As discussed in Section C, the maximum flux density in the iron sheet is higher than in the homogenous case, which leads to an increase in hysteresis loss. Material loss data for electrical steel is generally provided to fit the Steinmetz loss equation presented in [7]: p v Cm f B ( 18 ) Where Cm, α and β are experimentally obtained constants. Figure Fig. 15 contains manufacturer data for core loss at different peak polarizations [10]for the material 30 EX 1200 at 1 kHz. 2 ( 20 ) Hysteresis loss for one loop is independent of the speed of magnetization and therefore, it is assumed to be only linearly dependent of the frequency, whereas eddy-current loss is squarely proportional to frequency. The results presented in section C, however suggest a different behavior. Flux is displaced from the center of the iron sheet to its corners, resulting in a significantly higher flux density and therefore also in a higher hysteresis loss overall. As indicated in Section A, µ was considered to be constant for the derivation of formulae. Therefore it is crucial for the following investigation of hysteresis loss not to exceed the range of validity of this assumption. Referring to the B-HCurve in Fig. 3, it can be concluded that saturation occurs at 1.2 T and that the material behaves linearly beneath that point. That being said, every conclusion drawn from a setup of which no point experiences a peak flux density higher than 1.2 T is valid on the grounds of satisfying the assumptions made. For frequencies up to 3200 Hz, homogenous flux density is therefore limited to 0.01 T to avoid saturation in the edges. The results shown in Fig. 16 have been calculated for flux densities from 0.001 T to 0.01 T. It has been found that the flux density has no effect on the relative increase in hysteresis loss as long as no saturation occurs. The relative increase ΔP has been calculated according to: Usually, loss curves are used to obtain factors for hysteresis, eddy current and excess loss for calculation purposes, an approach introduced by Jordan [9] and Bertotti [2]. Despite many efforts in the domain of loss modelling, according to [8], “Many papers report that calculation errors between 200%-2000% can occur” when calculating Iron Losses in electrical machines. In the following section, the influence of flux-crowding on hysteresis loss is investigated and it is shown that it can have a significant effect which may 1191 account for some of the prediction errors. P PDisplacement PHomogenous PHomogenous ( 19 ) The same holds true for all effects induced by this change, notably eddy currents and electric field changes in the electrical steel sheet and its isolation. In summary, displacement currents may be neglected for the investigation of flux displacement. III. CONCLUSION Fig. 16: Relative increase in hysteresis loss vs. frequency E. Scope of validity and future work The assumptions limiting the scope of applicability of equations (8), (9), (11) and (16) are the following: i.)µ is considered to be constant ii.)Displacement currents inside and outside of the sheet are neglected As shown in Section D, disregarding saturation (assumption i) restricts the range of validity, as local flux densities increase so drastically that even flux densities of 0.01 T for the homogenous case lead close to saturation in the edges. In future work, an attempt will be made at substituting the constant interrelationship between H and B by a Langevin-type dependency, which may create the need to solve the system numerically. Neglecting displacement currents (assumption ii) is necessary to obtain an integral formulation for H. The effect of displacement currents is dependent on the permittivity of the insulation material and on the rate of change of the electric field. The permittivity ε0εr is of order 10-11 for resin insulation [4]. As also the rate of change of the field plays a role in the magnitude of displacement currents, it has to be investigated in the light of inverter-feed. Voltage rise times of IGBT-based inverters are typically close to 50 ns as in [5], resulting in dU/dt-Values of 12V/ns for a 600V switch. This rate of change of order 1010 elevates displacement current to a non-negligible quantity in conductors. Yet, these rates of change have only minimal effect on the field: t 2 U t Switch U DC ( 21 ) being the rate of voltage rise and the integration With U interval being equal to the switch time of 50ns, the change of flux linkage caused by a switching event is of order 10-9, as the right side of (21) simplifies to: Switch U DC t Switch ( 22 ) The flux distribution in a conductive magnetic sheet subject to sinusoidal external magnetic fields has been derived from Maxwell’s formulae. Hysteresis loss has been calculated for the obtained distorted and the uniform distribution of flux density. It was shown that a significant increase of hysteresis loss occurs for high frequencies, caused by Flux Displacement towards the Edges. The assumption of constant permeability, however, demands that the maximum flux density in the iron sheet not exceed the linear range, thus limiting the scope of applicability of this work. Therefore, a numerical model including nonlinearities is under development and will soon be presented. IV. REFERENCES D. Gerling. (2009) "Approximate analytical calculation of the skin effect in rectangular wires", Electrical Machines and Systems, 2009. ICEMS 2009. IEEE International Conference on, 15-18 Nov. 2009 [2] G. Bertotti. (1988) “General properties of power losses in soft ferromagnetic materials” Magnetics, IEEE Transactions on, 24(1), pp. 621-630 [3] D. C. Jiles and D. L. Atherton. (1983) “Ferromagnetic hysteresis”, Magnetics, IEEE Transactions on, 19(5), pp. 2183-2185 [4] A. Helgeson. (2000) “Analysis of dielectric response measurement methods and dielectric properties of resin-rich insulation during processing”, Ph.D. dissertation, Department for Electric Power Systems, KTH Stockholm, 2000 [5] Datasheet Infineon. (2011), "SIGC223T120R2C", Infineon Technologies [6] D. Gerling and G. Dajaku. (2004) “Comparison of Different Calculation Methods for the Induction Motor with Multilayer Rotor Structure”, Electrical Machines, 2004. ICEM 2004. International Conference on, 5-8 Sept. 2004 [7] C. Steinmetz. (1984) "On the law of hysteresis." Proceedings of the IEEE. 72(2) pp. 197-221 [8] J. Reinert, A. Brockmeyer and R. De Doncker. (2001) “Calculation of losses in ferro- and ferrimagnetic materials based on the modified Steinmetz equation”, Industry Applications, IEEE Transactions on [9] H. Jordan. (1924) “Die ferromagnetischen Konstanten für schwache Wechselfelder”, Elektr. Nach. Techn., Berlin [10] Datasheet NKK. (1998) “NK SuperE-Core Magnetic Property Curves”, NKK Corp. [1] V. M.Sc. Florian Bachheibl graduated with a Master of Science in Mathematical Engineering from the University of Federal Defense Munich in 2011. He is a research assistant at the Institute of Electrical Drives and Actuators at the University of Federal Defence in Munich, led by Prof. Dr.Ing. D. Gerling. Prof. Dr.-Ing. Dieter Gerling received his diploma and Ph.D. degrees in Electrical Engineering from the Technical University of Aachen, Germany in 1986 and 1992, respectively. From 1986 to 1999 he was with Philips Research Laboratories in Aachen, Germany as a Research Scientist and later as Senior Scientist. In 1999 Dr. Gerling joined Robert Bosch GmbH in Bühl, Germany as Director. Since 2001, he is full Professor and Head of the Institute of Electrical Drives at the University of Federal Defense Munich, Germany. 1192 Powered by TCPDF (www.tcpdf.org) BIOGRAPHIES