Mô hình hóa quá trình tự đốt nóng của cuộn cảm để nghiên cứu sự trao đổi điện - từ - nhiệt

TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 1 MODELING OF THE SELF-HEATING PROCESS OF AN INDUCTANCE TO STUDY THERMAL - MAGNETIC ELECTRIC EXCHANGES MÔ HÌNH HÓA QUÁ TRÌNH TỰ ĐỐT NÓNG CỦA CUỘN CẢM ĐỂ NGHIÊN CỨU SỰ TRAO ĐỔI ĐIỆN - TỪ - NHIỆT Anh Tuan Bui - Tuan Anh Kieu Electric Power University Abstract: This paper focuses on thermal stresses on magnetic materials under Curie temperature. The aim of this article is to study the influence of temperature on all standard static magnetic properties. The Jiles-Atherton model and “flux tube” model are used in order to reproduce static and dynamic hysteresis loops for MnZn N30 (Epsco) alloy. For each temperature, the six model parameters are optimized from measurements. The model parameters variations are also discussed. Finally, the electromagnetic model is associated with a simple thermal model to simulate energy exchanges among the three thermal - magnetic - electric areas towards self-heating process of an inductance. The simulation outcomes will be compared with experimental results. Keywords: Magnetic hysteresis; Magnetic materials; Modeling; Magneto-thermal coupling. Tóm tắt: Bài viết này tập trung vào các ứng suất nhiệt trên vật liệu từ dưới nhiệt độ Curie. Nghiên cứu ảnh hưởng của nhiệt độ đến tất cả các thuộc tính từ tính của vật liệu từ. Mô hình Jiles-Atherton và mô hình "ống từ thông" được sử dụng để mô phỏng các đường cong từ trễ ở chế độ ổn định tĩnh và chế độ ổn định động của vật liệu từ ferit MnZn N30 (Epsco). Đối với mỗi nhiệt độ, sáu thông số của hai mô hình mô phỏng trên được tối ưu hóa từ các phép đo. Sự thay đổi các thông số trong hai mô hình mô phỏng sẽ được tìm hiểu. Cuối cùng, mô hình điện từ được kết hợp với một mô hình nhiệt đơn giản mô phỏng quá trình tự trao đổi năng lượng giữa ba lĩnh vực: điện - từ - nhiệt đối với hiện tượng tự đốt nóng của một cuộn cảm. Kết quả mô phỏng sẽ được so sánh với các kết quả thực nghiệm. Từ khóa: Từ trễ, vật liệu từ, mô hình hóa, liên kết từ - nhiệt.1 1 Ngày nhận bài: 30/07/2015; Ngày chấp nhận: 03/08/2015; Phản biện: TS Nguyễn Đức Huy. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 2 1. INTRODUCTION The magnetic circuit in the electromagnetic system is a key element of an efficient energy conversion. The optimization of the magnetic circuit geometry, the control of energy efficiency through the use of powerful magnetic materials and a thorough knowledge of their behavior, especially under high stress as temperatures and high frequencies that are meet more today. The temperature at which occurs the disappearance of spontaneous magnetization is called the Curie temperature. The effect is not as brutal as it seems. The temperature increase leads to an evolution of the saturation magnetization, coercive field, remanent flux density, resistivity and magnetic losses, etc [4], [5]. The objective of this study is to build a model as complete as possible to cover a wide class of samples of magnetic materials. This model must take into account several aspects of the phenomena as the initial magnetization curve and the major loop. The model should allow further integration of the evolution of the hysteresis loop based on temperature and frequency. Finally, it must be fast enough for inclusion in design and simulation software. The modeling of magnetic materials plays an important role in modeling systems in electromagnetism. Many studies have shown that the mechanisms at the origin of the phenomenon of magnetization depends on many factors [4]: the material, the excitation field, the external conditions,... From an experimental point of view, two operating regimes can be distinguished: the quasi-static and the dynamic one. Below certain frequencies, the hysteresis loop does not depend on frequency. The material is in a quasi-static mode. Several models are proposed to describe this mode [1], [6]. To meet out our objectives, we must have a model with a basic mathematical and physical enough flexibility and a complete implementation for the integration of additional parameters that take into account the temperature and frequency. One of these models is characterized by a physical basis and theoretical particularly comprehensive. This is the Jiles- Atherton model [1], [2]. In dynamic regime, the hysteresis loop expands with the frequency increase that is the energy loss is high in dynamic mode. This paper presents first the static and dynamic behaviors when the temperature increases. It also presents the static hysteresis model and the dynamic model that can modelize the hysteresis characteristics of magnetic materials as a function of temperature. The “flux tube” model [6] is used to model the dynamic behavior. The MnZn N30 (Epcos) magnetic material is used here because this material has a low Curie temperature (around 1300C), so we can clearly see the change of factors: power loss, the magnetization, temperature, resistance. In addition, this material is widely used in the fields of electrical, electronic,... Finally, this material is used on self - heating inductor to achieve a coupling TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 3 k MM dH dM irran e irr )(  k MM dH dM irran e irr )(  between three areas: electric - magnetic - thermal. 2. THE “FLUX TUBE” MODEL The Jiles-Atherton model, based on physical considerations, is able to describe the quasi-static hysteresis loops. It assumes that the exchange energy per unit volume is equal to the exchange of magnetostatic energy added by hysteresis loss. The magnetization M is separated into two components: the reversible component Mrev and the irreversible component Mirr. The irreversible component can be written as follows [1]: where the constant k is related to the average energy density of Bloch walls. The parameter δ takes the value 1 when dH/dt >0 and the value -1 when dH/dt <0. Jiles and Atherton show that the reversible magnetization is proportional to the difference (Mirr-Man): with c is a coefficient of reversibility as c [0,1]. So the total magnetization is the sum of components reversible and irreversible [3]: The following differential equation is obtained: with Equation (4) describes the behavior law M(H). The five parameters c, a, k, α and Ms are determined from measurements (magnetization curve and major loop) and by using an optimization algorithm [6]. When the frequency increases, several dynamic effects appear inside the material, the eddy currents are increasing. This increase is illustrated by an expansion of the B (H) loop. The "flux tube" model [6] is build by considering the material as an homogeneous flux tube. This can be expressed in terms of flows through the tube and parameter γ can be identified by a first order ordinary differential equation (6): Hdyn is the excitation field, Hstat is a fictitious field function of the flux density, γ is a coefficient depending on the material magnetic and electrical properties (resistivity, permeability,...). Its value may be calculated approximately by the equation:     edH irrdMc edH andMc edH andMc edH irrdMc dH dM    11 1  )( irranrev MMcM  )( irranirrirrrev MMcMMMM  dt dB BHH statdyn  )( 12 . 2d   (1) (2) (3) (4) (5) (6) (7) TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 4 with δ is the conductivity and d is the sample thickness. The model has the advantage of being simple because it requires the identification of a single dynamic parameter and have a very fast computation time. The "flux tube" model can use the Jiles- Atherton model in order evaluate Hstat(B). Equation (4) expresses the static model as a relation B(Hstat). It may equally well be placed under Hstat(B) form which is done to solve the equation (4). The coefficient γ is optimized by comparison between the measured and simulated hysteresis loops. The “flux tube” model therefore needs the identification of six parameters (five static parameters and one dynamic parameter). The “flux tube” model (6) has been implemented in the Matlab Simulink simulation software to test its accuracy according to several criteria. The Simulink scheme describing the model is given in Fig.1. Fig.1. Simulink diagram for the “flux tube” model 3. MEASUREMENTS AND SIMULATIONS In order to get a well suited hysteresis model for various ferromagnetic materials, preparation and knowledge of measurement techniques are important to have accurate baseline data. A magnetic material characterization bench has been developed for quickly measuring hysteresis loops B(H) with high accuracy. For our purpose, we need to measure the B(H) loops for several temperature values. Fig.2 shows a scheme of the test bench used for these measures. The samples are placed in an oven that increases the temperature (maximum around 2500C). The samples are placed in an aluminum box to obtain the temperature stability on the sample measurement (below 10C) after two hours. We have used two thermocouples to control the stability and the homogenization of the temperature, one is placed in the aluminum box space and the other is fixed to the sample. The process of measurement is realized when the temperature of both thermocouples is the same. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 5 Thanks to Ampere and Maxwell laws, H and B are determined by the following formulas: Fig 2. Schematic of bench measurement The temperature changes the magnetic materials properties mainly by 2 processes: either by an irreversible evolution of their local composition (aging) or by reversible changes of their electromagnetic constant with temperature. The Fig.3 expresses the evolution of hysteresis loops until the Curie temperature. This clearly shows that as the temperature increases, the saturation induction density, the coercive field density and the remanent induction density decrease, as does the lower hysteresis losses. Testing of the material beyond the Curie temperature (135°C) gave rise to complete thermal demagnetization as expected for this material. The material is excited by a very low frequency sinusoidal excitation field in the static regime. In a first step, the static model is identified and validated at 1Hz. At each temperature value, the five Jiles- Atherton model parameters are optimized. The Fig.4 show good agreement between the B(H) loops obtained by the model with those obtained by measurements for the same input signal and each temperature. Fig.3. Evolution of B(H) loop as a function of temperature in statique regime (1 Hz) The variation of each Jiles-Atherton model parameter versus temperature is shown on Fig.5. They tend to decrease unless the parameter c, it tends to          edtSNBdt d Ne I L N H R U I mshunt . 1 . 2 2 1  (8) TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 6 increase for the MnZn N30 material when the temperature increases [6]. Fig.4. B (H) loop measured and simulated at 230C and 1000C, 1Hz Fig.5. Evolution of five parameters of Jiles-Atherton model as a function temperature When the frequency increases, several dynamic effects appear inside the material. The most visible effect is an expansion of the B (H) loop. The “flux tube" model is used to model this behavior. This model has the advantage of being simple and having a computation time very fast. The parameter γ is optimized for the maximum excitation frequency (here 10 kHz) until the error between measured and simulated on iron losses is below 10% for each temperature. Fig.6. B (H) measured and simulated loops at 230C and 1000C, 10 kHz The Fig.6 show good agreement between measured and simulated loops at 10 kHz for each temperature. Once calibrated parameter γ, we have all the comparison criteria to estimate the performance of this method. Then, the value of γ will be used for other frequencies (lower). The γ parameter variation versus temperature is shown on Fig.7. Fig.7. Evolution of the parameter γ as a function of temperature The γ parameter tends to decrease when the temperature rises to 85°C. From this temperature, it tend to increase, we believe, to compensate the error made by the static model (OF1_115°C ≈ 2.5* TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 7 OF1_23° C and ∆Bs_115°C≈ 6.75*∆Bs_23°C) (Tab.1). The simulation quality is estimated by comparing B(H) measured loops and simulated ones for the same input signals. The criteria are the relative error between maximum measured and simulated induction, loops area and the signal quality obtained for the same input signal (H). These criteria give a quality estimation of the model. The signal quality is estimated by the normalized mean square error (MSE) between the measured and simulated inductions [6]: with N, the number of points in each of the two vectors; Bmes and Bsim are the measured and simulated inductions respectively; max (Bmes) is the maximum induction value obtained by measurement. In static regime, the quality of the simulation is estimated via the relative error and the square error. The results are expressed by Tab.1. Within the measurement interval, we obtain for any temperature; the maximum induction relative error is less than 0.7% and the mean square error OF1 is less than 0.03%. These results represent a good performance of the static model because it is a wide range of temperature variation. Moreover, the error is almost constant over the entire temperature range. In dynamic regime, the quality of the simulation is estimated by the maximum induction relative error, the mean square error OF1 and the relative error between the measured and simulated loops area. The model performance is summarized by Tab.1. For the maximum measured frequency, we get a mean relative error for the iron losses of 0.12% and the mean square error is 0.056%. Tab.1. Models performance θ(°C) Static regime Dynamic regime 10 kHz 5 kHz ∆Bs (%) OF1*10-4 ∆P (%) OF1*10-4 ∆P (%) OF1*10-4 23 0.36 2.1 1.8 3.9 8.8 4.3 65 0.29 1.9 5.7 7.1 0.8 5.6 85 0.98 2.1 5.9 4.3 2.9 2.6 100 0.51 2.7 8.9 6.6 6.9 4. 115 2.43 5.3 2.7 5.8 4.9 6.2 4. MODELING OF SELF - HEATING OF AN INDUCTANCE We use the previous simulation results to achieve a coupling of the fields: electric - magnetic - thermal of self - heating of an inductance. The magnetic material of the 2 1 1 )max( )()(1             N j mes simmes B jBjB N OF (9) TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 8 magnetic circuit is MnZn N30 material. The magnetic component is thermally insulated (carton box + foam insulation). A simple thermal model is first proposed to estimate the operating temperature of the transient component from Joule losses and iron losses. 4.1. Development of a thermal model Many approaches are used to describe heat transfer and to achieve a satisfactory estimate of operating temperatures. Some approaches lead to a temperature mapping, computed at any point of the component (numerical methods). Others can only give the calculated temperature in some parts of component (conventional analytical methods, nodal method. In our work we use the nodal method to model the transient heat transfer. This method involves fixing insulated areas, each zone forming a node. Several simplifying assumptions are adopted:  Homogeneity of temperature inside the magnetic core and copper winding. Under these conditions, each element (core and winding) corresponds to a node and 2 thermocouple;  Capacity of thermal insulation neglected due to its low mass;  Natural convection on the surface of the box neglected, resulting in surface temperature assumed equal to ambient temperature. We checked that increasing the temperature did not exceed 2°C. These assumptions allow us to define two thermal zones (Fig.8) corresponding to the magnetic material on the one hand, and the primary winding, on the other. Both areas are home to Joule heating due to losses in the copper (Pj) and iron losses in the torus (Pf). We assign the center of gravity of each area and a source node representing losses.  Thermal capacity Cth1 and Cth2 correspond to thermal energy storage: Cth1 for the magnetic material and Cth2 for copper;  Rth1: between the core and winding, which reflects the sum of the resistances of conduction through the CT (resistance between the center of the ferrite and the surface), contact the torus - primary winding and the winding (resistance between the periphery and center conductor);  Rth2: between the coil and the ambient air, which reflects the sum of the resistances of the contacts winding conduction - insulator and insulator - outer surface;  Rth3: between the core and the surrounding air, which reflects the sum of the resistances of conduction contacts torus - insulation and insulation - exterior surface. The parameters of the thermal equivalent circuit are determined in two steps:  Identification of thermal resistance from the steady;  identification of thermal capacity with the transitional regime. We obtain the following results: Rth1 = 8.43882°C/W; Rth2 = 80.685°C/W; Rth3=45.2542°C/W; Cth1= 104 J/°C.kg and Cth2 = 1.5 J/°C.kg. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 9 Fig.8. Schematic of thermal model of the magnetic component studied 4.2. Algorithm coupled electro - magneto - thermal The corresponding coupling algorithm is shown in Fig.9. At each change in temperature Δθ, the model determines the electromagnetic iron losses and Joule losses. The convergence of our model is not very dependent on temperature Δθ, and in particular as regards the last iteration. After several tests, not adopted Δθ = 1°C gives the best trade-precision computation time. Two models, electromagnetic and thermal, are implemented in the Matlab environment. 4.3. Model validation We validate our work by comparing the results of measurements and simulations in different conditions: sinusoidal voltage sources and non-sinusoidal, various frequencies. To quantify the precision, the following criteria are used:  The square error between measured and simulated temperatures (OF1): where:  θmes, θsim: temperatures measured and simulated;  N: number of measurement points in time and for θmes, θsim;  max (θmes): maximum temperature reached. The maximum relative error: (11) 2 1 1 )max( )()(1           N j mes simmes jj N OF   (10)         .100 )max( (j)(j) max(%)Δ mes simmes max    TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 10 Fig.9. Coupling algorithm electro - magneto – thermal Fig.10 - Fig.12 show the variations of measured and simulated temperatures for different excitation sources: sinusoidal, rectangular and triangular at 10 kHz. These results also show good correspondence between measurement and simulation, and confirm the performance of our model (Tab.2). Fig.10. Temperatures measured and simulated for a sinusoidal source at 10 kHz Fig.11. Temperatures measured and simulated for a rectangular source at 10 kHz Fig.12. Temperatures measured and simulated for a triangular source at 10 kHz 11 Số 9 - tháng 10 năm 2015 Criteria Excitation sources Sinusoidal Rectangular Triangular OF1_material 2.19E-4 4.71E-5 1.08E-4 OF1_winding 3.56E-5 1.77E-4 3.73E-4 Δθmax_material (%) 0.02 1.23 1.78 Δθmax_winding (%) 0.02 2.63 5.31 Computation time (s) 136 153 142 Tab.2. Performance of coupled models Our model satisfies for different sources of tension: the maximum squared error is less than 3*10-4 for the material, and 4*10-4 for the winding. The maximum relative error is less than 2% for the material and 6% for winding. The other advantage of our model is short time calculate. 5. CONCLUSION The results on the MnZn N30 material depending on the temperature are very encouraging. The model contains the following benefits: rapid calculation time, easy implementation. These benefits provide users with a simple model. Moreover, this model with the input H and output B, easily invertible, allows easy integration for modeling electrical engineering systems. The parameter γ in the “flux tube” model is easily determined. We realized a thermal-electromagnetic coupling to study self-heating of another single component, a coil with magnetic core. We developed a simple thermal model capable of estimating the operating temperature of the magnetic component from Joule losses and iron losses. The performance of the “flux tube” model coupled with the thermal model allows to determine with accuracy quite satisfactory self- heating of different parts of component (coil + magnetic circuit). REFERENCES [1] D.C. Jiles and D.L. Atherton, Ferromagnetic Hysteresis. IEEE Transactions on Magnetics, Vol. 19, No 5, Sep 1983, 2183-2185. [2] Jacek Izydorczyk, A New Algorithm for Extraction of Parameters Jiles and Atherton Hysteresis Model. IEEE Transactions on Magnetics, pp. 3132-3134, Vol.42, No.10, 11/2006. TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC (ISSN: 1859 - 4557) Số 9 - tháng 10 năm 2015 12 [3] DC.JILES, D.L ATHERTON, Theory of ferromagnetic hysteresis. Journal of magnetism and magnetic materials, 1986, Vol.61, p48-60. [4] Pierre Brissonneau, Magnétisme et matériaux magnétiques. Edition Hermès, pp86-87, 1997. [5] Richard Lebourgebois, Ferrites doux pour l’électronique de puissance. Techniques de l’Ingénieur, N 3 260. [6] A.T. Bui, N. Burais, L. Morel, F. Sixdenier, Y. Zitouni, Characterization and modelling of temperature influence on “flux tube” magnetic properties. 19th Soft Magnetic Materials Conference, Turin : Italy (2009). Biography: Anh Tuan Bui, was born on 01/9/1978. Lecturer Faculty of Electrical system Power University. Graduated from Hanoi University of Technology in 2001, majoring in electrical systems. Completion of the Master's program in 2006 with the same major. From 2007 to 2011, the authors PhD student in the lab Ampere University Claude Bernard Lyon 1 with the electrical materials field.