1-D FEM-Based Approach for Extracting Dimension-Independent Material Properties of Mn-Zn Toroidal Ferrite Cores

This article presents a finite element (FE) method (FEM)-based approach for identifying the dimension-independent magnetic and dielectric properties of Mn-Zn toroidal ferrite cores. Two 1-D axisymmetric FE models are used for solving the full-wave electromagnetic field equations in a toroidal ferrite core using either the electric field strength (<inline-formula> <tex-math notation="LaTeX">$\boldsymbol {E}$ </tex-math></inline-formula>) or the magnetic field strength (<inline-formula> <tex-math notation="LaTeX">$\boldsymbol {H}$ </tex-math></inline-formula>) as the variable. Impedance measurements are performed for the toroidal cores over a frequency range of 10 kHz–10 MHz. The FE models and the impedance measurements are combined with a non-linear least-squares algorithm for solving the inverse problem for extracting the dimension-independent values of complex permeability and complex permittivity. The extracted permeability is validated against the datasheet of the measured samples.


I. INTRODUCTION
M ANGANESE-ZINC ferrites have been extensively used in high-frequency electromagnetic devices because of their low energy losses [1]. Still, the loss mechanisms in ferrites have not been fully revealed, mainly due to the lack of knowledge about their intrinsic magnetic and electrical properties. Although the permeability is usually the main parameter for designing ferrite cores, the complex permittivity plays a big role in core losses, especially when the operating frequency is in the MHz range [2]. Thus, complex permittivity should be considered during the design of high-frequency power electronic devices based on ferrite cores.
Ferrite suppliers do not usually provide any information about the conductivity and permittivity of ferrite cores. Therefore, modeling and measuring ferrite cores have been popular research topics in the past few decades. Due to the combination of high permeability and high permittivity, ferrite cores are subject to dimensional resonance effects at high frequency. The cause of such effects is the occurrence of standing electromagnetic waves whose wavelength is of the same order of magnitude as the size of the core [3]. Because of these dimensional effects, direct measurement of the dimension-independent or "intrinsic" material properties is a difficult task. Thus, inverse approaches need to be developed for extracting the intrinsic parameters based on measurements.
A common approach for modeling ferrite cores is based on solving the wave equation for the electric and/or the magnetic field in the ferrite core and extracting the dimension-independent complex permeability μ core = (μ r − j μ r )μ 0 and complex permittivity ε core = (ε r − j ε r )ε 0 by fitting the calculated impedance to the measured values [4]- [7]. Huang  is based on solving analytically the electric field in a cuboid ferrite capacitor [5]. The impedance measurements and the analytical model were combined with the Newton-Raphson method to identify the dimension-independent values of μ core and ε core of the samples. Stadler et al. proposed a methodology similar to Huang's work for measuring the AC conductivity and the dielectric constant of Mn-Zn toroidal ferrite cores at different temperatures [7]. However, it was not clear how μ core is considered in the inverse approach presented in [7]. Another drawback of the previous models is that only one of the extracted parameters, i.e., either μ core or ε core , is validated against the magnetic or dielectric measurements used in the inverse algorithm, respectively. The main objective of this work is to develop a finite element (FE) method (FEM)-based approach for extracting the dimension-independent electromagnetic material parameters, i.e., μ core and ε core of toroidal Mn-Zn ferrite cores and address the aforementioned drawbacks. The extracted μ core and ε core are validated against the supplier datasheet and dielectric measurements carried out by the authors, respectively. The measurement setup and the impedance measurements for the toroidal samples are discussed first in Section II, and the computational models are derived later in the same section. The fitting results and the extracted dimension-independent electromagnetic parameters are presented in Section III. Section IV is devoted to discussion and conclusion.

A. Impedance Measurements
Impedance measurements for three toroidal cores are carried out at room temperature in the frequency range of 10 kHz-10 MHz with Novocontrol Alpha-A Modular dielectric/impedance measurement system in combination with ZG4 extension test interface and standard sample cell BDS1200 [8].  two electrodes, a sinusoidal voltage is imposed across the sample, and the total current through the sample is measured according to Fig. 1 (left). The samples were provided by Magnetics ® , and their data are shown in Table I [9]. To get rid of the contact resistance, the top and bottom surfaces of the samples were coated with two layers of nickel and gold of thicknesses of 0.01 and 0.1 μm, respectively, using an electron beam evaporation setup. Moreover, before carrying out any measurements for each core, open-circuit, short-circuit, and standard load calibrations of the measurement setup were performed to minimize the measurement errors.
If ferrite cores have the same chemical composition and preparation process, it is reasonable to assume that they share the same intrinsic material properties, at least with an accuracy related to the repeatability of the manufacturing process. According to the equivalent circuit approach which models the ferrite core using constant phase elements [10], [11], the phase angles of samples with identical material properties should be equal until the occurrence of the dimensional resonance. From the phase angles of the measured samples shown in Fig. 2, it is clear that the samples do not share exactly the same material properties. Even if all the samples were in the same delivery and, according to the supplier, they belong to the same F-material category with initial permeability μ r = 3000±20% at 10 kHz, we can see from the data in Table I that the samples do not come from the same lot and do not even have the same country of origin. If the samples are excited by electric field, the phase angle first decreases when the frequency increases, since the samples behave as imperfect capacitors. However, the significance of the magnetic field increases with frequency, and it can be noticed from Fig. 2 that the phase angle changes direction at some point, e.g., at about 250 kHz for T36/23/10. The dimensional resonance behavior occurs when the phase angle is zero. T36/23/10 shows the dimensional resonance at about 900 kHz.

B. Computational Models
The electric and magnetic fields in the measured samples are investigated by two different formulations. Due to the circumferential symmetry of the measurement setup [12], [13], the electromagnetic field distributions on any cross section along the circumference are the same. Consider a toroidal ferrite core with a rectangular cross section with outer radius R 2 , inner radius R 1 , and height d as depicted in Fig. 1 (right). Cylindrical coordinates r, ϕ, and z and frequency domain analysis are adopted. Assuming homogeneous and isotropic material, both the electric field strength ( E) and magnetic field strength ( H) are only functions of the radius r and independent of ϕ and z: E = E(r )ẑ and H = H (r )φ, whereẑ andφ are the unit vectors in the z and ϕ directions, respectively. Therefore, the computational models are based on solving the Maxwell equations in the core in a 1-D axisymmetric setting. Small-signal variation is assumed, so that μ core and ε core can be assumed to be independent of the fields. From the Maxwell equations, the governing equations for the electromagnetic fields can be derived in terms of either E or H. The formulations for both the Eand H-based models are written as where ω = 2π f is the angular frequency and σ is the conductivity which is obtained by DC measurement. The Ebased model is supplied by the measured voltage U and the H-based model by the measured current I . The boundary conditions are written as where r denotes the left boundary of the model and can be set to r = R 1 to model only the core or to r = 0 to account also for the air at the center of the core. Let (ω) = [μ core , ε core ] be a vector which contains the unknown material parameters. The impedance Z c (ω, ) can be calculated by where = core when only the core is modeled or = air ∪ core when also the air is considered. Standard Galerkin 1-D FEM is used for solving (1)- (3).
For extracting the dimension-independent material parameters μ core and ε core , we formulate the inverse problem as an optimization problem The FE models are combined with a non-linear least-squares algorithm for solving (4) and fitting μ core and ε core at each frequency ω i so that the calculated impedances match with the measured values [14].

III. RESULTS
The Eor H-based model presented in Section II can be combined with the impedance measurements of the three toroidal cores for extracting the dimension-independent material parameters. It was observed that considering the air at the center of the core has negligible effect on the results in both models, which implies that it is enough to consider only the core with boundary conditions (2) at r = R 1 . The measured and fitted impedances of the toroidal cores used for extracting the dimension-independent μ core and ε core are shown in Fig. 3. The dimension-independent μ core is traced in the frequency range of 250 kHz-10 MHz, and it is comparable to the values provided in the datasheet as shown in Fig. 4 [9]. The permeability could not be traced below 250 kHz since it has negligible effect on the field solution at lower frequencies.
The dimension-independent ε core is traced in the frequency range of 10 kHz-10 MHz as shown in Fig. 5. ε r becomes negative around 1 MHz. There is no physical explanation for the negative values, and this failure might be attributed to a few reasons, i.e., the samples do not have identical material properties, the sensitivity of the inverse solver to the measurement data, and the noise level.
The distributions of the Eand H-fields along the radial position for T22/14/6 in the case that the air is considered at    432 kHz are shown in Fig. 6. It can be seen that the Neumann and Dirichlet conditions (2) for the Eand H-based models at r = R 1 are reasonable when only the core is modeled. Moreover, the extracted parameters shown in Figs. 4 and 5 were used with the Eand H-based models to simulate the impedance of T22/14/6 in both cases, with and without considering the air at the center of the core. It is shown in Fig. 7 that the impedances calculated with the Eand H-based models match in both cases.

IV. DISCUSSION AND CONCLUSION
The Eand H-based formulations are presented in this work to solve the full-wave electromagnetic field equations in a toroidal ferrite core. The air at the center of the core has negligible effect on the field distributions in the core. The complex permeability cannot be traced well at low frequencies since the permeability does not have a significant effect on the field distribution. The extracted permeability agrees with the datasheet of the measured samples. The complex permittivity is traced for the whole frequency range, but the imaginary part has unreasonable negative values around 1 MHz.