Extended Model of Flux Displacement in Rectangular Iron Sheets and Geometry-Dependent Hysteresis Loss considering Saturation

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1 4 7th International Conference on Electrical Machines and Systems (ICEMS),Oct. -5, 4, Hangzhou, China Extended Model of Flux Displacement in Rectangular Iron Sheets and Geometry-Dependent Hysteresis Loss considering Saturation Florian Bachheibl, Dieter Gerling Chair of Electrical Drives and Actuators, University of Federal Defense Munich, Germany Abstract Iron Loss is calculated including the effect of flux displacement towards the edges of a magnetic conductor. Saturation effects are considered and calculations under DC-Bias are compared to measurement results. A good agreement is found. I. INTRODUCTION Iron loss of magnetic conductors is usually calculated using the Steinmetz Equation or its modifications. Losses are separated, subdividing per-cycle-losses into hysteresis loss, classical eddy current loss and excess loss []. In the design of electrical machines, it is very common to obtain calculation results for iron loss that differ from measurement by as much as -% []. In [3], the authors have shown that flux displacement and the resulting increase of flux density in the edges of a rectangular iron sheet may be one of the causes for that effect. The predictions drawn from those results are however limited to a sufficiently linear operating point, where the field distribution is very much similar to the distribution of high-frequency AC-current in rectangular conductors, as presented in [4]. It has also been shown that this linear operating point is as low as. T for the material 3EX, if subjected to a sinusoidal field with a frequency of 3 Hz. The equations including saturation, as presented in this paper, can no longer be calculated analytically. However, they can also be applied to the loss analysis of DC-Biased Inductors. Calculation results are compared to measurements performed in [5]. II. MOTIVATION Reference [3] reveals a distribution of flux density as shown in Fig.. This is only a theoretical figure, as saturation reduces the occurrence of eddy currents and therefore prevents an effect at that scale. This paper tries to overcome those limitations by introducing saturation into the formulae. III. DERIVATION OF FORMULAE The sheet of iron under investigation and the employed coordinate system are shown in Fig.. The width w and height h of the sheet cross-section are assumed to be given by machine geometry and material thickness respectively, whereas infinite length is premised along the z-axis. The derivation presented in [3] is based on a constant permeability µ of the magnetic material in the law of induction: B H rot( E) ( ) t t In this paper, this constant interrelationship will be replaced by the Langevin-Function, which is often used to model saturation, as in the Jiles-Atherton model of hysteresis [6]: B a,b,h k coth k abs H e H H k abs H Where k and k are material constants and e H is the unity vector parallel to the applied field intensity. Inserting ( ) into ( ) yields: rot(e) k coth k abs H e H H t ( 3 ) k abs H ( ) y h x w x Fig. : Distribution of magnetic flux density for f=3 Hz Fig. : Sheet of iron under investigation 345

2 For simplicity, the applied field is assumed to only consist of a z-component: H(x, y, t) H (x, y, t) e z z ( 4 ) which results in all eddy currents being bound to the x-yplane. Therefore, ( 3 ) yields: with E E B x y dt y x z ( 5 ) B z kh z kcsch bhz H z t kh ( 6 ) z Equation ( 6 ) is not defined for H z =, but can be continuously substituted at this point. Choosing a very small value close to the pole instead yields a satisfying solution. Using the quasistatic form of Ampère s law: we obtain: dd rot(h) j dt Hz Ex y Hz E x y E ( 7 ) ( 8. ) ( 8. ) Table : Coefficients for Matlab s PDE Toolbox Coefficient Value a c - kh z k csch khz H z kh z d f Assuming that the excitation of the sheet under test is applied in the form of a homogenous external field intensity, this boundary condition follows obviously. IV. MODELLING SATURATION The measurement results used to validate the findings in this paper are drawn from [5]. The authors used the material 3Q4 from WSPC for their experiments, which is equivalent to M4-3S according to the manufacturer. Material data for M4-3S was obtained from a steel datasheet [7]. The parameters necessary for the presented model are the parameters k and k for Langevin s equation ( ) and the loss parameters for hysteresis and eddy-currentloss. In order to obtain the fitted parameters for Langevin s equation, a nonlinear least squares algorithm was used. Since a good representation of the material characteristics is most important near the saturation region, the objective function was weighted with the polarization. Fig. 3 displays the vendor-supplied material curve and the fitted J-H-curve. The constants were obtained as follows: k =.97 and k =.35. and therefore, inserting ( 8. ) and ( 8. ) into ( 5 ): x y t Hz Hz Bz ( 9 ) Equation ( 9 ) is a nonlinear parabolic differential equation, which can be solved numerically by Matlab s PDE Toolbox. Matlab s template for parabolic differential equations is as follows: d u div cgrad u a u f ( ) The toolbox allows the definition of functions for each variable in ( ) which may be dependent on the solution and its derivatives. Therefore, the values in Table are chosen as coefficients. Using this formulation, excitation has to be applied as boundary condition. The law of continuity states that the tangential component of H must remain constant at material borders. Polarization [T] J vs. H Fitted Model Field Intensity [A/m] Fig. 3: Fitted J-H-Curve 345

3 V. MODELLING IRON LOSS Despite many attempts [8]-[] to replace the Steinmetz model [3] extended by loss-separation [], [4] for nonsinusoidal waveforms, it is still the standard in characterizing materials. One reason for this is, according to [5], that measurements performed under the IEC644- standard using an Epstein frame [6] have the great advantage that all investigators obtain nearly identical results. However, the obtained parameters can now be used to parameterize models evolved from Steinmetz s equation such as GSE, IGSE, MSE or the equivalent elliptical loop [7]. While the first three models work best in post-processing, the latter can be used online. The authors therefore consider it best-suited for the model presented above and for future work. The equivalent elliptical loop was presented in [7] and a very useful way of implementing it for loss-calculation of non-sinusoidal waveforms was presented in [8]. The model treats hysteresis, eddy-current and excess loss, but here it is only used for calculating hysteresis loss as the presented FE-model itself provides a better way to calculate eddy current loss. Solving the model yields a distribution of magnetic field intensity varying over time. Using ( 8. ) and ( 8. ), it is possible to calculate the electric field inside the domain and therefore, using P E da eddy A ( ) it is possible to calculate the eddy-current-loss in the iron sheet. Usually, loss separation relies on loss data for a range of frequencies in order to differentiate between linear dependency on the frequency, which is the case for hysteresis loss, and square dependency on frequency which is approximately the case for eddy-current-loss. Excess loss is intentionally not considered in this paper but shall be included in further investigations where it can be calculated from own measurement data. In order to obtain the parameter for hysteresis loss, eddy current loss was calculated using ( ) for several values of polarization. From these values, the eddy current loss parameter was calculated to be k C =49.68* -6. Eddy-current loss was then subtracted from the total loss given by the manufacturer [7] to obtain the hysteresis loss. Subsequently, hysteresis loss was fitted onto ( ) using a nonlinear least squares algorithm. H H P k f B ( ) The loss parameter for hysteresis loss k H was calculated for a frequency of 5 Hz and obtained as k H = Hysteresis Loss Manufacturer Data Fitted Model Peak Polarization [T] Fig. 4: Fitted loss-model VI. SIMULATION RESULTS As excitation is applied by using an external field intensity vector as boundary condition, it is necessary to calculate the required excitation for each value of polarization. The measurements in [5] were performed with an excitation winding and a search coil that registered the change of flux as a voltage waveform. Using the Langevin-function ( ) without regarding the vacuum flux-density μ H, it is possible to calculate the field intensity for which 95% of the maximum polarization in the material is reached. The obtained value of 9. A/m lies well beneath some of the DC-bias amplitudes applied in [5] which extend to as much as 45 A/m. Therefore, to still obtain an AC-amplitude of flux density of up to.7 T as in the measurements, it is necessary to apply very high field intensities such that the DC-Bias plus the ACcomponent partially yield a negative result: exc DC AC H H H sin( t) ( 3 ) In other words, the AC-oscillation of flux density is not symmetrical to its DC-part and therefore, its mean value is also lower than the DC-value. In order to obtain the necessary value for the excitation, the following equation was solved for H AC using a nonlinear solver (trust-region-dogleg): BH Dc H AC BHDcH AC B AC,Set ( 4 ) Here, B AC,Set is the chosen amplitude of the AC oscillation which is superimposed to the DC-bias. The obtained excitation amplitudes were then applied to the FEM-model. As already mentioned in [3], the field intensity inside the iron sheet is reduced by eddy currents forming, according to Lenz s law, in a way to suppress the force driving them. Therefore, the highest values of both field intensity and flux 3453

4 density are observed at the edges of the region, which coincides with the boundary region, where excitations are applied. In consequence, the desired flux density cannot be reached by using a field intensity calculated from ( 4 ). The deviations between the calculated value and the value obtained from simulation are displayed in Fig. 5. To make this point more clear, Fig. 6 shows the distribution of flux density at the lower inflection point of the excitation, and Fig. 7 displays the same quantity at the maximum excitation field. Simulated Peak Flux Density [T].5.5 Reference Bias A/m Bias 5 A/m Bias 5 A/m Bias 45 A/m.5.5 Set Peak Flux Density [T] Fig. 5: Set Flux Density compared with Simulation result Fig. 7: Distribution of Flux Density at the point of maximum Excitation To overcome this issue, the values obtained from ( 4 ) were used as an initial point for an iterative optimization, resulting in a new set of field intensities as shown in Fig. 8. It is obvious that higher excitation amplitudes are necessary when a higher DC-premagnetization is present. This finding is further elucidated in Fig. 9, where the B-H-curves are plotted for a constant AC-amplitude of the flux of B AC =.5T and different DC-biases of, 5, 5 and 45 A/m. It is worth noting that although these curves resemble DC hysteresiscurves, hysteresis is not yet regarded in the calculations. The widening observed in the curves is solely caused by a phase shift which originates from eddy-currents. 5 AC Field Intensity [A/m] 4 3 Bias 45 A/m Bias 5 A/m Bias 5 A/m Bias A/m Fig. 6: Distribution of Flux Density at the point of minimum Excitation.5.5 AC-Amplitude of Flux Density [T] Fig. 8: Field Intensity versus AC flux density 3454

5 Flux Density [T].5.5 Bias A/m -.5 Bias 5 A/m Bias 5 A/m Bias 45 A/m Field Intensity [A/m] Fig. 9: B-H-Curves for B AC =.5 T and different DC-biases In Fig. 9, the curves for DC-biases of 5 A/m and 45 A/m overlap largely, which is the case because both extend over the point of saturation described earlier. However, the curve for H DC =45 A/m is very distorted in the time domain, as the flux density only changes when the applied AC-field becomes negative and thus counters the DC-field, thus forcing the material out of saturation. This effect can be observed in Fig. and also explains why the green curve in Fig. 9 is a little bit wider than the black curve. The db/dt values for a waveform as shown in Fig. reach very high values in the small dents, causing strong eddy currents which in turn widen the B-H-curve. Field Intensity [A/m] Phase Angle [ ] Fig. : Excitation field for sinusoidal flux density control of the current waveform. This was implemented in the model by calculating the field intensity required to achieve a sinusoidal flux in every timestep. The field intensity waveform applied is shown in Fig. for a DC-bias of 5 A/m and an AC-amplitude of.5 T. Losses were calculated for different values of DC-Bias and different amplitudes for the AC-component of flux density. The results obtained from simulation are plotted in Fig. and the measurement results for a non-biased measurement and for a DC-bias of 45 A/m taken from [5] are summarized in Table. Flux Density [T] Core Loss [W/kg].5.5 DC-Bias 45 A/m DC-Bias 5 A/m DC-Bias 5 A/m DC-Bias A/m Phase Angle [ ] Fig. : Flux density versus phase angle for HDC=45 A/m.5.5 Peak Flux Density [T] Fig. : Core Loss Versus Peak Flux Density VII. CALCULATION OF CORE LOSSES The standard method for measuring core loss according to [6] demands that the induced voltage and therefore the average flux be sinusoidal, which requires a closed loop Table : Measurement Results taken from [6].5 T T.5 T A/m. W/kg.4 W/kg W/kg 45 A/m.38 W/kg.78 W/kg.5 W/kg 3455

6 VIII. CONCLUSION A model for the field distribution inside a conducting magnetic sheet was developed based on Maxwell s Equations using a direct approach without utilizing vector potential. Saturation was modelled with a Langevin-Function and hysteresis losses were described using the Equivalent Elliptical Loop model. The theory that flux displacement towards the material edges increases overall loss could be confirmed. The measured values and the results from simulation are in good agreement. The increase in core loss for DC-biased samples could therefore partially be explained. This effect will be further investigated using own measurement data for an E-core sample. Including hysteresis in the field calculation could improve the quality of results especially in the DC-biased case as it causes a further phase shift between flux and external field, resulting in a different eddy-current characteristic. REFERENCES [] G. Bertotti, "General properties of power losses in soft ferromagnetic materials" in IEEE Transactions On Magnetics 4, no. (January 988): [] J. Reinert, A. Brockmeyer and R. De Doncker, "Calculation of losses in ferro- and ferrimagnetic materials based on the modified Steinmetz equation" in IEEE Transactions On Industry Applications 37, no. 4 (July-August ): [3] F. Bachheibl and D. Gerling, Flux Displacement in Rectangular Iron Sheets and Geometry-Dependent Hysteresis Loss in International Conference on Electrical Machines, 4. ICEM 4, Berlin (September 4). In print. [4] D. Gerling, "Approximate analytical calculation of the skin effect in rectangular wires" in 9 International Conference On Electrical Machines & Systems (January 9) [5] Z. Zhao et al., "Magnetic Property Modelling of Laminated Silicon Steel Sheets under DC-Biasing Magnetization" in Sixth International Conference On Electromagnetic Field Problems & Applications (June ): -4. [6] D.C. Jiles and D.L. Atherton, Ferromagnetic hyteresis in IEEE Transactions on Magnetics 9, no.5 (September 983): [7] Datasheet Orb Steel. Unisil M4-3S, Orb Electrical Steels, Cogent Power (). [8] J. Muhlethaler, J. Biela, J.W. Kolar and A. Ecklebe, "Core Losses Under the DC Bias Condition Based on Steinmetz Parameters" in IEEE Transactions On Power Electronics 7, no. (February ): [9] R. Kaczmarek, M. Amar and F. Protat, "Iron loss under PWM voltage supply on Epstein frame and in induction motor core" in IEEE Transactions On Magnetics 3, no. (January 996): [] S. Iyasu, T. Shimizu and K. Ishii "A novel iron loss calculation method on power converters based on dynamic minor loop" in 5 European Conference on Power Electronics and Applications, (September 5): 6-. [] W.A. Roshen "A Practical, Accurate and Very General Core Loss Model for Nonsinusoidal Waveforms" in IEEE Transactions On Power Electronics, no. (January 7): 3-4. [] M. Jiong, L. Hongyang, M. Hao and B. Zhihong, "A novel hysteresis loss calculation method of filter inductor in PWM inverters" in IECON 3-39Th Annual Conference Of The IEEE Industrial Electronics Society (November 3): [3] C.P. Steinmetz, "On the law of hysteresis" in Proceedings Of The IEEE 7, no. (January 984): 97-. [4] H. Jordan, Die ferromagnetischen Konstanten für schwache Wechselfelder in Elektr. Nachr. Tech. I (94): 7-9. [5] S. Tumański, Handbook of Magnetic Measurements. st ed. Boca Raton, FL: CRC Press,. [6] Magnetic Materials Part : Methods of measurement of the magnetic properties of electrical sheet and strip by means of an Epstein frame, IEC644- (996). [7] D. Lin, P. Zhou, W.N. Fu, Z. Badics and Z.J. Cendes, "A dynamic core loss model for soft ferromagnetic and power ferrite materials in transient finite element analysis" in IEEE Transactions On Magnetics 4, no. (March 4): [8] M. Wasekura, C.-M. Wang, Y. Maeda and R.D. Lorenz, "A transient core loss calculation algorithm for soft magnetic composite material" in 3 IEEE Energy Conversion Congress & Exposition (3):