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Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method

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Published/Copyright: May 24, 2018
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Open Physics
From the journal Open Physics Volume 16 Issue 1

Abstract

In this work, we highlight the influence of the material uncertainties (magnetic permeability, electric conductivity of a Mn-Zn ferrite core, and electric permittivity of wire insulation) on the RLC parameters of a wound inductor extracted from the finite element method. To that end, the finite element method is embedded in a Monte Carlo simulation. We show that considering mentioned different material properties as real random variables, leads to significant variations in the distributions of the RLC parameters.

Keywords: ferrite core; finite element; material uncertainties; Monte Carlo simulation; wound magnetic components
PACS: 02.70.Dh; 87.10.Mn; 85.70.Ge; 85.70.Ay

1 Introduction

In most cases in finite element (FE) modeling, some electromagnetic properties of materials are not sufficiently known due to the lack of certain information (i.e. uncertainties on data presented in the specification sheets). This is the case of wound magnetic components. Such components are typically used in switch-mode power supplies where the high switching frequencies yields small values of the volume and weight. On the other hand, the fast switching operation also implies the occurrence of undesirable phenomena (skin and proximity effect, influence of the parasitic capacitances, higher magnetic losses, etc.), still challenging to model. Indeed, one of the main limitations in frequency increase is the energy dissipation by losses in ferrite that produces heating of the electronic circuits. Due to their complex structure and related complex electrical behavior, there are few data on the conductivity properties of the sintered Mn-Zn ferrite causing difficulties in correctly modeling the eddy current phenomenon [1]. Also the errors arising in the characterization of soft magnetic materials from measurements, are directly reflected in the mathematical models of material and so resulting in inaccuracies in numerical simulations. Despite recent efforts oriented in this direction [2, 3], some differences are still observed between numerical simulations (through extracted RLC equivalent circuits) and experimental measurements conducted on high-frequency wound magnetic components (see e.g. [4]). To some extent, such discrepancies are caused by geometric and material uncertainties. For instance, the magnetic permeability and the electric conductivity of the ferrite magnetic core are usually not known with accuracy less than 20% and 5%, respectively. In addition, the wire diameter may also vary due to the crushing of the surrounding insulating material, and uncertainties in the winding configuration due to the wounding procedure may also arise [5].

The influence of the variability of input parameters of the numerical model on its outputs must be considered. The uses of perturbation method, Monte Carlo simulation, polynomial chaos, etc. are usually performed for this purpose [6, 7]. The perturbation method is based on a Taylor series development of random quantities in the numerical model around their mean value. The calculation of derivatives can be quite complicated and costly, which limits the use of this method only to order 1 and for models having certain regularity. According to the literature, the theory of polynomial chaos is advantageous in the case where several random entries have to be taken into account. On the other hand the Monte Carlo method is simpler to implement but expensive in computing time because it needs several thousand iterations to tend towards a better estimate.

In the present paper, we study the influence of material uncertainties on the RLC parameters of wound inductors using the Finite element method. Since there are only three main random inputs (magnetic permeability and electrical conductivity of the ferrite core, and electrical permittivity of the insulation) to be considered in this analysis, we will use the Monte Carlo method to analyze their influences on the RLC parameters of the wound inductor. We present the applied methodology including the stochastic approach and the FE numerical model used to extract the RLC parameters in Section 2. In Section 3, we will describe the different used models for the random inputs including uniform uncertainties distributions along the frequency. Thereafter in the Section 4, the simulation results on a test case will be presented and discussed. Finally in the conclusion, the strengths and the weaknesses of the proposed methodology will be discussed and some perspective works will be proposed.

2 Proposed methodology

In order to obtain the distribution the RLC parameters of the wound inductor due to the propagation of material uncertainties through the numerical model, the Monte Carlo simulation is well known as simple and easy to implement [5, 6, 7]. In this stochastic method, the considered input parameters are the magnetic permeability and the electrical conductivity of the ferrite core and the electrical permittivity of the wire insulating material. Both permeability and permittivity are modeled as complex functions of the frequency with the same accuracy along the frequency while the electrical conductivity is taken as real number function of the frequency too. They are all supposed to be uniformly distributed between two limit values computed from the manufacturer’s data [8, 9]. These random inputs are propagated through the deterministic numerical model to the outputs parameters (RLC). The implemented FE 2D model of wound inductor is based on two uncoupled problems for computing the RLC parameters. A magneto-dynamic a-v formulation with circuit coupling is employed to extract the RL parameters of the inductor [10]. The generated electric potentials of the conductors are then used in an electrostatics problem unlike in many other works (in which a linear repartition of the electric potential along the conductors is a priori assumed). In that way, a weak coupling between the electric and magnetic fields is considered. Besides, massive conductors are considered in order to take into account the skin and proximity effects in conductors. The stray capacitance of the inductor is obtained from the electrostatic energy computed by solving the electrostatics problem in the non conductive materials.

3 Modeling material uncertainties

3.1 Magnetic permeability of the ferrite core

In this study, the ferrite core is considered as magnetic dielectric characterized by its permeability and permittivity. The global approach for modelling different losses in ferrite, consists in representing the electrical permittivity and the magnetic permeability as complex quantities function of the frequency.

μ(ω)=μ′(ω)−iμ″(ω)(1)
ξ(ω)=ξ′(ω)−iξ″(ω)(2)

whith ω, the angular pulsation; μ′, ξ′ and μ”, ξ” the real and the imaginary parts of the magnetic permeability and the electrical permittivity respectively.

The real parts represent the magnetic and electric energy storage terms while the imaginary parts, highlight the magnetic and the electric losses respectively. The magnetic losses include the hysteresis and residual losses while the electric losses include Joule losses (due to eddy currents, in the ferrite core for instance) and the dielectric losses associated to the rotation of dielectric dipoles due to the alternative electric field [11].

For the frequency dependence of the complex forms, the Debye relaxation model [12] is sufficient for modelling both permeability and permittivity, of the ferrite core (see (3) and (4)).

μ(f)=μs/(1+i(f/frμ))(3)
ξ(f)=ξinf+(ξs−ξinf)/(1+i(f/frξ))(4)

with μs, ξs, the static (i.e. at zero frequency) permeability and permittivity respectively; ξinf, the permittivity at infinite frequency; frμ and frξ, the relaxation frequencies of the material properties. For these analytic models, static parameters are considered as random variables for our deterministic model while the relaxation frequencies are derived from data sheet as in other references [13]. The Figure 2 illustrates the evolutions as frequency functions of the real and imaginary parts of the relative magnetic permeability of the 3F35 ferrite core, with a relaxation frequency located around 5.5e + 6 Hz. One can deduce that in a limited frequency range (up to 100 kHz), the magnetic permeability could be considered as a constant real parameter.

Figure 1 Complex permeability of a Mn-Zn ferrite core as a function of frequency
Figure 1

Complex permeability of a Mn-Zn ferrite core as a function of frequency

Figure 2 Complex permittivity of polyurethane as a function of frequency
Figure 2

Complex permittivity of polyurethane as a function of frequency

3.2 Electric permittivity of wire insulating

For the electrical permittivity of the wire insulating material (i.e. polyurethane), the Havriliak-Negami relaxation model (empirical modification of Debye relaxation) is the well adapted in this study from the experimental point of view [14, 15]. The real and imaginary parts of the permittivity are taken from the following expression:

ξ(ω)=ξinf+(ξs−ξinf)/[1+(iωτ0)(1−α)]β(5)

where ξs and ξinf are static (at zero frequency) and infinite frequency permittivity respectively; ω the angular pulsation; τ0 (= 1/(2πfmax)) the characteristic time associated to the relaxation frequency fmax; and α and β, the shape parameters defining the symmetrical and asymmetrical broadening of the dielectric dispersion curve. From Figure 3, one can see the variation with the frequency of real and imaginary parts of the polyurethane permittivity adopted in this paper. One can notify that decreases monotonically with the frequency and that could be considered as constant on a limited frequency band.

Figure 3 Conductivity versus frequency for ferrite core
Figure 3

Conductivity versus frequency for ferrite core

3.3 Electrical conductivity of the ferrite core

The variation of the conductivity with the frequency can be deduced from the imaginary part of the electrical permittivity [12] of the material (either for ferrite or polyurethane) as presented below.

σ(f)=σdc+σac(f)(6)

with σdc, the electric conductivity at zero frequency or dc conductivity; and σac, the conductivity at hi frequency.

This latter is linked to the imaginary part of the permittivity by the following expression:

σac=2πfξ0ξ″(f)(7)

where f is the frequency and ξ0 (= 8.854187e – 12 F/m), the vacuum permittivity. The variation with frequency of the conductivity of the polyurethane is depicted on the Figure 4. One can notice that significant Joule losses may appear at hi frequencies, but can be neglected at low frequency.

Figure 4 Distribution of Joule losses in the wound inductor at 1 MHz
Figure 4

Distribution of Joule losses in the wound inductor at 1 MHz

4 Numerical results

The stochastic study is applied on a 2D axisymmetric model of a wound inductor made of 81-turns and of 0.315mm wire diameter. The pattern of conductors in the winding window is chosen so as to reduce the parasitic capacitances of the system. Hence the hexagonal arrangement of conductors in the winding window and the discontinuous type arrangement are then chosen [16]. The evolution of the extracted RLC parameters with frequency, from the Monte Carlo simulation, is depicted in Figure 6. This is the result of Monte Carlo 2350 iterations for which the three random inputs parameters are frequency-dependent and uniformly distributed between two limits computed from data sheets. The increasing dispersion of R with the frequency is due to the skin and proximity effects in conductors and in the ferrite core. The quasi-constant dispersion of inductance along the frequency is due to the fact that the deep down eddy currents (in the ferrite core, see Figure 4), which are shifted in phase and could affect the inductance, are exponentially smaller and so their effects are small or negligible on the distribution of L. While the eddy currents which flow at the exterior surface of the core is in phase with the coil current and then can significantly affect the real part of coil impedance. For cons, the stray capacitances show an increasing dispersion with frequency. This is due to the dispersion of the uncertainties on the insulating material properties (which are taken into account in the capacitance computation) with the frequency.

Figure 5 Histograms of RLC parameters at 1 MHz
Figure 5

Histograms of RLC parameters at 1 MHz

Figure 6 Extracted RLC parameters against frequency
Figure 6

Extracted RLC parameters against frequency

The distributions of RLC parameters at 1 MHz are depicted in the Figure 5. It may be mentioned that they are uniformly distributed around their mean values as the random inputs parameters. For convergence analysis, the adapted mathematical expectations of RLC parameters at 1 MHz with their 95% confidence intervals are depicted in Figure 7. It can be observed here that Monte Carlo simulation achieves a good convergence from 1500 iterations thanks to the used stochastic models of materials uncertainties.

Figure 7 Convergence of RLC parameters at 1 MHz with their confidence intervals
Figure 7

Convergence of RLC parameters at 1 MHz with their confidence intervals

5 Conclusion

In this paper, we have analyzed the influence of material uncertainties (permeability and conductivity of the ferrite core and the permittivity of the wire insulating material) on the extracted RLC parameters of a wound inductor through the FE method. The proposed methodology using MC simulation has been applied successfully to a 2D model including random parameters (material uncertainties only). However, in case of the consideration of geometrical uncertainties (winding pattern and air gap core for example) in addition to the material uncertainties (permeability and conductivity of ferrite core), a mixed stochastic approach implementing for instance chaos polynomials expansions and Monte Carlo simulation, could be interesting in order to benefit from the advantages of both techniques. Another interesting work would be to take into account the material uncertainties in nonlinear behavior laws (e.g. B-H curve of ferromagnetic materials) for a sensitivity analysis on the performances of electrical machines for instance. In future works, we plan to compare the impedances computed from the numerical models and ones resulting from experimental measurements.

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Received: 2017-11-02
Accepted: 2017-12-18
Published Online: 2018-05-24

© 2018 G. Lossa et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2018-0033
Keywords for this article
ferrite core; finite element; material uncertainties; Monte Carlo simulation; wound magnetic components
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BY-NC-ND 4.0

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  44. On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem
  45. Ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons
  46. Analysis of projectile motion in view of conformable derivative
  47. Computing multiple ABC index and multiple GA index of some grid graphs
  48. Terahertz pulse imaging: A novel denoising method by combing the ant colony algorithm with the compressive sensing
  49. Characteristics of microscopic pore-throat structure of tight oil reservoirs in Sichuan Basin measured by rate-controlled mercury injection
  50. An activity window model for social interaction structure on Twitter
  51. Transient thermal regime trough the constitutive matrix applied to asynchronous electrical machine using the cell method
  52. On the zagreb polynomials of benzenoid systems
  53. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  54. The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method
  55. Quantifying the global solar radiation received in Pietermaritzburg, KwaZulu-Natal to motivate the consumption of solar technologies
  56. Sturm-Liouville difference equations having Bessel and hydrogen atom potential type
  57. Study on the response characteristics of oil wells after deep profile control in low permeability fractured reservoirs
  58. Depiction and analysis of a modified theta shaped double negative metamaterial for satellite application
  59. An attempt to geometrize electromagnetism
  60. Structure of traveling wave solutions for some nonlinear models via modified mathematical method
  61. Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation
  62. Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations
  63. Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
  64. Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
  65. Shooting method analysis in wire coating withdrawing from a bath of Oldroyd 8-constant fluid with temperature dependent viscosity
  66. Rank correlation between centrality metrics in complex networks: an empirical study
  67. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
  68. Modeling of electric and heat processes in spot resistance welding of cross-wire steel bars
  69. Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
  70. Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
  71. Thermal constitutive matrix applied to asynchronous electrical machine using the cell method
  72. Temperature distribution around thin electroconductive layers created on composite textile substrates
  73. Model of the multipolar engine with decreased cogging torque by asymmetrical distribution of the magnets
  74. Analysis of spatial thermal field in a magnetic bearing
  75. Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
  76. Assessment of short/long term electric field strength measurements for a pilot district
  77. Simulation study and experimental results for detection and classification of the transient capacitor inrush current using discrete wavelet transform and artificial intelligence
  78. Magnetic transmission gear finite element simulation with iron pole hysteresis
  79. Pulsed excitation terahertz tomography – multiparametric approach
  80. Low and high frequency model of three phase transformer by frequency response analysis measurement
  81. Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
  82. Optimal design of a for middle-low-speed maglev trains
  83. Eddy current modeling in linear and nonlinear multifilamentary composite materials
  84. The visual attention saliency map for movie retrospection
  85. AC/DC current ratio in a current superimposition variable flux reluctance machine
  86. Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
  87. Cogging force reduction in linear tubular flux switching permanent-magnet machines
  88. Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
  89. Electro-magneto-hydrodynamic lubrication
  90. 3-D Electromagnetic field analysis of wireless power transfer system using K computer
  91. Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
  92. Design, fabrication and testing of electroadhesive interdigital electrodes
  93. A method to reduce partial discharges in motor windings fed by PWM inverter
  94. Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
  95. Special Issue Applications of Nonlinear Dynamics
  96. Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
  97. Microblog topic evolution computing based on LDA algorithm
  98. Modeling the creep damage effect on the creep crack growth behavior of rotor steel
  99. Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
  100. Chinese open information extraction based on DBMCSS in the field of national information resources
  101. 10.1515/phys-2018-0079
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  103. Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
  104. Temperature detection technology of power equipment based on Fiber Bragg Grating
  105. Research on a rotational speed control strategy of the mandrel in a rotary steering system
  106. Dynamic load balancing algorithm for large data flow in distributed complex networks
  107. Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
  108. Fractal-based techniques for physiological time series: An updated approach
  109. Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
  110. Application of modified culture Kalman filter in bearing fault diagnosis
  111. Exact solutions and conservation laws for the modified equal width-Burgers equation
  112. On topological properties of block shift and hierarchical hypercube networks
  113. Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
  122. Fast recognition method of moving video images based on BP neural networks
  123. Compressed sensing image restoration algorithm based on improved SURF operator
  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
  125. Face recognition method based on GA-BP neural network algorithm
  126. Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
  127. Localization and recognition algorithm for fuzzy anomaly data in big data networks
  128. Urban road traffic flow control under incidental congestion as a function of accident duration
  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
  133. A text-Image feature mapping algorithm based on transfer learning
  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
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