Investigating the Impact of Uneven Magnetic Flux Density Distribution on Core Loss Estimation

Transcription

1 IEEE PEDS 2017, Honolulu, USA December 2017 Investigating the Impact of Uneven Magnetic Flux Density Distribution on Core Loss Estimation Farideh Javidi N. 1, Morten Nymand 2 The Maersk Mc-Kinney Moller Institute University of Southern Denmark Odense, Denmark { 1 javidi, 2 mny}@mmmi.sdu.dk Yiren Wang 3, Andrew J. Forsyth 4 School of Electrical and Electronic Engineering The University of Manchester Manchester, United Kingdom { 3 yiren.wang, 4 andrew.forsyth}@manchester.ac.uk Abstract There are several approaches for loss estimation in magnetic cores, and all these approaches highly rely on accurate information about flux density distribution in the cores. It is often assumed that the magnetic flux density evenly distributes throughout the core and the overall core loss is calculated according to an effective flux density value and the macroscopic dimensions of the cores. However, the flux distribution in the core can alter by core shapes and/or operating conditions due to nonlinear material properties. This paper studies the element-wise estimation of the loss in magnetic cores. FEM has been used to investigate the flux density distribution in the core and the loss has been estimated considering this distribution. Finally, comparative results are shown between the classical macroscopic core loss estimation using effective dimensions and the element-wise loss estimation. The presented work in this paper has been carried out for two common excitation waveforms in power electronics applications, sinusoid and square-wave and for two different core shapes, toroid and E-cores. Results show that ±10% discrepancy should be expected in loss estimation of the core using effective dimensions under both excitation waveforms. Keywords magnetic flux density; core loss estimation; finite element modelling. I. INTRODUCTION Loss estimation of magnetic cores basically relies on the loss density information and the core volume. The core loss density depends on the magnetic characteristics of the core material as well as the operating frequency and the magnetic field density. The core manufacturer usually provides loss information of each core material in a wide range of operating points. Moreover, each specific core datasheet (with specific core shape) provides the loss values at few certain operating points. In general, Steinmetz equation is the common well-known formula for characterizing the loss of a magnetic core exciting with sinusoid waveforms. Steinmetz parameters are achievable with curve fitting of the loss information provided by the core material manufacturer. On the other hand, in many power electronics applications, inductors/transformers are exposed to square waveforms. Due to non-linear behavior of magnetic cores, the harmonic analysis is not accurate and the core loss should be estimated specifically for square-wave excitation [1], [2]. Improved Generalized Steinmetz equation, igse, is the further development of SE for square-wave excitations. igse has promisingly shown good agreements of core loss estimation with measurements and it is very convenient to use, since it uses Steinmetz parameters from the core material datasheet [1], [2]. In either of the mentioned popular core loss estimation methods, the magnetic field density in a core plays an important role. These methods basically give the loss density value based on a single value of flux density. And the total core loss is then obtained by multiplying the volumetric loss density by the effective core volume. Magnetic field density usually distributes non-uniformly in a core, since the magnetic path length differs among the core volume. Moreover, the non-linear characteristic of the magnetic permeability of the material leads to redistributing the magnetic field density among the core volume (further discussion of this phenomena is presented in section II). The non-uniform distribution of the magnetic field density indeed impacts the core loss prediction [3]-[6]. In this paper, to increase the accuracy of the loss estimation and to better understand the core loss distribution, the magnetic core will be discretized into smaller sections and the loss will be estimated in an element-wise manner. II. POWER LOSS ESTIMATION OF A MAGNETIC CORE A. Magnetic Flux Density Distribution Magnetic flux density in a core, B, relies on the magnetic field intensity, H, with a factor µ which is the magnetic permeability of the core material (B=µH). The magnetic field intensity, H=Ni/l, depends on N, coil number of turns; l, the magnetic path length; and I, the excitation current. Due to the fact that l varies among the core volume, H has different values at different core volume sections. And so, B will have non-uniform distribution through the core volume. Fig. 1 shows the flux lines for two different core shapes. It is evident that each of the flux lines has different magnetic path lengths and therefore, H accordingly will have different values in the cross-section area of the core /17/$ European Union 567

2 (a) Fig. 1. Flux lines have different magnetic path length in two different cores, (a) toroid, and E-core Fig. 1 (a) shows the flux lines in a toroid. Due to the fact that l varies inversely proportional to the radius of the toroid, H distributes non-uniformly through the core cross-section. In Fig. 1, it can be seen that there is a flux crowding in the sharp corners of the E-core while the outer parts of the core are not fully utilized. Furthermore, the nonlinearity of µ leads to redistribution of magnetic flux density regardless of the H distribution. This characteristic of the core material may also lead to local saturation of the core which leads to locally exceeding the core loss value [7]-[9]. The common method for calculating H and B in a core is to consider the effective dimensions and suppose that B distributes uniformly in the core volume with effective B value. The core loss is then estimated with effective B value. The effective dimensions, however, are derived at very low excitation levels (where the material may be assumed to obey the Rayleigh and Peterson relations) [10], but they are often used in practice at higher excitation levels for the merit of convenience. In case of magnetic cores with more complex geometric shapes, the effective dimension consideration may neglect the effects of the local saturation. Therefore, the estimated loss value might become unreliable. B. Core Loss Estimation Methods The common loss estimation method for magnetic cores excited with sinusoid, SE, is stated as follows: α ΔB P = kf v (1) 2 where ΔB is peak to peak flux density with frequency f, P v is the volumetric power loss density, and k, α and are Steinmetz parameters available from the datasheet. Eq. (1) shows that the loss estimation in SE requires the amplitude information of flux density distributed in the core. The other common excitation waveform in many power electronics applications is square-wave. Under square-wave, igse, is a precise method for loss estimation. In igse, the magnetic core loss is estimated as follows [1]: α 1 T db α P = k ( B) dt v Δ, (2) i 0 T dt ( k = i k ) 2π cosθ 2 dθ ( ) α 1 2π α α 0 where T is the time cycle of the periodic square-wave excitation. igse is very convenient to use, since it needs Steinmetz parameters for loss estimation which is usually provided by the magnetic cores manufacturer. Equation (2) states that the peak flux density and the instantaneous time variation of flux density, ΔB and db/dt, are used for loss estimation under square-wave. For triangle flux density waveform, shown in Fig. 2, db/dt is constant in each time interval of (0, DT) and (DT, T). Thus, (2) can be simplifies and restated as: 1 ( ) ( 1 ) 1 α α α P = k Δ B f D + D v i Equation (3) clearly shows that to estimate the loss in magnetic materials under square-wave excitation, only the amplitude information of the flux density is required (similar to SE). Thus, clear knowledge about the distribution of B in the core is highly required. (11) and (3) show that the magnetic field distribution is required for loss estimation of the core excited with either sinusoidal or square-wave excitations. Fig. 3 shows an inductor with a toroid core in 3D COMSOL and the magnetic field density distribution is demonstrated. It can be seen that the flux density distribute inversely proportional to the radius of the toroid, though it distributes consistently along with the height of the core. As a result, a 2D finite element analysis will be sufficiently accurate for the investigation of B distribution. III. ELEMENT-WISE CORE LOSS ESTIMATION METHOD Effective dimensions are usually used to estimate the core loss in magnetic cores. And so, the geometric effect as well as the nonlinearity of the core permeability might not be fully considered. The nonlinearity of the material permeability may lead to local saturation in sharp corners in many of the cores [7], [8]. Core loss estimation in element scale enables to consider the B values in each specific element and will lead to more accurate estimation of the loss in magnetic materials. Fig. 2. Core voltage and magnetic flux density waveforms under squarewave excitation (3) 568

3 B. Introducing the Loss Factor In order to investigate the B distribution impact on loss estimation of a core, a loss factor parameter has been introduced. The loss factor basically includes the B distribution information as well as the core volume. In the traditional core loss prediction methods, which are addressed as macroscopic method in this paper, the loss factor is introduced as follows: B = Δ 2 v M macroscopic core (4) Fig. 3. Magnetic flux density distribution in the core volume, 3D FEM modelling of an inductor with Kool Mu toroid core, 77439A7, and 32 turns winding and I = 2 In element-wise method, core loss will be estimated in element scale rather than macroscopic volume using finite element method. In this method, B distribution will be found in each specific element and the core loss is estimated using the true B value in each particular element. The estimated loss in the core will be the summation of the estimated core loss in each element. A. Descritizing the Core Volume Regardless of the excitation waveforms, the B distribution should be found specifically in each element. In this work, 2D COMSOL has been used to discretize the geometry of the core into many small elements. The B distribution is considered homogenous in each of the element and it is the mean absolute value of B among the whole surface of the element (see Fig. 4 (a)) 1. Due to the consistency of the B value orienting the in-plane direction (indicated in Fig. 3), the discretized elements can be extended to 3D with considering the consistent depth of the core, d, and the surface area of the element, A elem, as shown in Fig. 4. To realize an accurate estimation in this method, the elements should be considered as small as possible. However, very small elements lead to increasing the number of elements and complexity of the model which results in increasing the computational time. Therefore, a compromise has to be found between a good mesh quality (and so an accurate loss estimation) and a reasonable computational time. COMSOL itself has a useful feature for monitoring the mesh quality. In the modellings of this work, the minimum mesh quality is above 0.1 and the average mesh quality is around 0.8 which are the adequate mesh quality to obtain an accurate result [11]. where B is acquired with considering the effective magnetic path length and effective core cross-section area, and v core is the effective core volume. In element-wise core loss estimation method, however, the B distribution information is considered in each volumetric element. Therefore, the loss factor will be restated as follows: ΔB da 2 (5) M element wise = elem where ΔB is the peak-to-peak flux density in the element, d is the element (and core) depth and A elem is the surface area of each element (see Fig. 4 ). Once the loss factor for each element has been found, the overall loss factor of the core will be the summation of the loss factor values in each element. C. Element-wise Method vs.macroscopicmethod Using the introduced loss factor, the loss in a magnetic core will be estimated as follows: P = α Mkf α α Mk f D ( D) 1 α + i sinusoidal (6) square-wave (a) 1 The mean absolute B in each element can be estated as follows: B (N ) m a B(N ) + b B(N ) + c B(N ) = a + b + c Fig. 4. (a) Discretizing the core surface area into different 2D elements, volumetric elements considering the depth of the core, d 569

4 Regarding which method is used, macroscopic or element-wise method, the associated loss factor (respectively (4) or (5)) should be used in (6). In the element-wise method, the B distribution has been considered specifically in each of the volumetric elements. The corresponding loss factor, in this method, is calculated using the specific B value in each element times the element volume. Finally, the core loss will be the total loss factor of the core times the operating frequency and the parameter specifications, k. In the macroscopic method, however, the effective dimensions l e and A e are used to find H e and consequently B e. Then, B e is presumed to be consistent all over the core volume and the loss factor is calculated using B e value and the effective volume. The final core loss value, similarly to the element-wise method, is the multiplication of the calculated loss factor and the operating frequency and the material specifications, k. IV. RESULTS AND DISCUSSION COMSOL Multiphysics 5.3 has been used to study the magnetic field density impact on core loss estimation in different magnetic cores with different core material and shapes (Table1). The core materials have been selected considering the availability of the material specifications in COMSOL library. First, the magnetic flux density distribution in the core has been studied. Due to the simplicity of the toroid geometry (uniform cross-section area along the magnetic path), B distributes inversely proportional to the radius. In more complex core geometries such as E-core in this work, there is a high possibility of local saturation in the inner corners while the outer corners of the core are not fully utilized. This usually happens, due to the flux lines tendency to shorten the magnetic path length and flattening the bends. So, the magnetic flux regularly crowds at the inner corners of the core and they rarely reach to the outer corners. Fig. 5 shows the magnetic field distribution of two different inductors with the same core material, Kool Mu, µ = 60. Using the effective dimension, B e is 840 mt, which indicates no saturation is happening in the core (B s = 1200 mt). However, the localized magnetic flux density in the inner part of the core as well as the sharp corners of the core dissipates significant loss, and so it generates considerable heat (B 1000 mt). This is very likely to lowering the saturation level and creates local saturation. In continue, element-wise core loss estimation has been applied for the 6 selected cores. The introduced loss factor has been obtained in the macroscopic method and elementwise method, M macroscopic and M element-wise at different excitation levels. The excitation levels are the provided effective magnetic flux density in the core at 0.3, 0.5 and 0.7 of B sat of the core material. Table1. Core information under study [12], [13] Kool-Mµ µ r=60, B sat = 1200 mt C µ r=2300, B sat = 380 mt C µ r=3000, B sat = 350 mt Toroid 77439A7 ZR43615TC 0F43615TC E-Core K4022E060 0R45530EC 0F45530EC B (mt) Bmin Bmax (a) Fig. 5. Magnetic flux density distribution for two Kool Mu inductors magnetized with B e = 0.7B s = 840 mt, (a) toroid core, 77439A7 with 32 turns winding, B min = 746 mt, B max = 925 mt, k4022e060 with 8 turns winding, B min = 33.9 mt, B max = 1070 mt 570

5 The discrepancy between the obtained loss factors in the two approaches at different excitation levels are shown in Fig. 6. The results show that a discrepancy around ±10% should be expected between the loss estimations using the macroscopic method and the element-wise method. Depending on the core material, this discrepancy varies noticeably. In fact, Steinmetz parameter,, as a specification of the core material determines the contribution of magnetic flux density in the loss estimation (see (1), (3)). The study has been done for 3 different core materials. It can be seen that for Kool Mµ material, due to the lowest value compared to other two ferrite materials, in both toroid and E- core, the discrepancy in loss factor is the smallest. However, for the other two Ferrite materials, due to the larger value for F materials, FerriteF cores have higher discrepancies. The core shape, in this study, does not significantly impact on the loss prediction in either of methods. In macroscopic loss prediction method, it is supposed that the core volume is fully magnetized with B e. In E-cores, particularly, the loss exceed in some areas where flux crowding occurs. However, there are some local parts where are not fully magnetized. Thus, there is a balance in loss prediction considering uniform magnetic field density distribution in macroscopic method and the tolerance with element-wise method stays around ±10%. Moreover, at low excitations, it is very likely that the whole core volume is not magnetized properly. Considerable discrepancies in Fig. 6 at low excitation indicate very well this explanation. V. CONCLUSIONS This paper studies the magnetic flux density distribution in two common core shapes, toroid and e-core. The change of magnetic path length among the core volume as well as the nonlinearity of the magnetic permeability leads to an uneven distribution of flux density in the core. This introduces local saturation or not fully magnetization of the core. It has been demonstrated that magnetic flux density distribute inversely proportional to the radius of the toroid cores. In E-cores, particularly, there is flux crowding in the inner corners and the outer corners of the core might not be magnetized. Furthermore, it has been analytically demonstrated that the flux density distribution in the core contributes similarly in loss estimation under sinusoidal and square-wave excitation. The impact of uneven distribution of magnetic flux density on the core loss estimation has been investigated. The classic macroscopic core loss estimation considers a homogenous effective magnetic flux density all over the core volume. In this work, an element-wise method using FEM has been presented where the core volume is discretized into many small elements with a relatively homogenous flux density. The obtained results from this method indicate around ±10% discrepancies with the macroscopic core loss estimation. Core material properties (Steinmetz parameter, ) impact significantly in the magnetic field density contribution to the core loss estimation and it leads to creating higher discrepancies. Absolute Loss Factor Discrepancy (%) 10% 8% 6% 4% 2% 0.3B sat (a) K4022E060 0R45530EC 0F45530EC 0.5B sat 0.7B sat Effective magnetic field density, B e Fig. 6. Absolute loss factor discrepancy between the macroscopic method and the element-wise method for core loss estimation of the selected cores, (a) toroid cores, E-cores ACKNOWLEDGMENT This project is sponsored by Green Power Electronics Test Lab. REFERENCES [1] J. Muhlethaler, J. Biela, J. W. Kolar and A. Ecklebe, "Improved Core- Loss Calculation for Magnetic Components Employed in Power Electronic Systems," in IEEE Transactions on Power Electronics, vol. 27, no. 2, pp , Feb [2] 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, vol. 27, no. 2, pp , Feb

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