Loss Simulation by Finite-Element Magnetic Field Analysis Considering Dielectric Effect and Magnetic Hysteresis in EI-Shaped Mn–Zn Ferrite Core

Power electronic devices such as inductors and transformers are required to be driven with high frequency according to downsizing. Mn-Zn ferrite is one of the high-frequency magnetic materials. The dimensional resonance occurs in Mn-Zn cores due to the increase of the dielectric constant and significantly affects the eddy current loss [1]. The equivalent RC circuit of Mn-Zn ferrite was modeled by the grains and their boundary layers and can explain the effective dielectric property by the contribution of the capacitance [2]. The boundary layers with high-resistance suppress the eddy current in the grains at low frequencies, while as frequency increases the suppression of the eddy current decreases by charge accumulation on the surface of the grains. The calculating method of the frequency dependent dielectric property by the capacitance was proposed and the dimensional resonance was reproduced by applying the method to the magnetic field equations of linear magnetic materials by using a cylindrical approximation [3]. In order to analyze the eddy current loss of complex shaped inductors at high frequencies, we apply the dielectric effect to the A- $\varphi$ method of the finite element magnetic field analysis. On the other hand, the magnetic hysteresis loss increases according to the increase of the magnetic flux density in the core. Therefore, we used the play model [4] to express the magnetic hysteresis for finite amplitude of magnetic flux. For confirming the calculation accuracy, core losses of an EI-shaped inductor was calculated and the frequency dependent loss was compared with experimental results. In the simulation, the core loss was divided into hysteresis loss in DC, eddy current loss and excess loss, and their contributions were analyzed. II. METHOD The current density of the grains is $j_{1}$ as follows [3]. $j_{1} = \sigma_{1}( {E-rq/}\varepsilon )$ (1) Where, $\sigma_{1}$ is the electrical conductivity, E is the average electric field in the grains, q is the charge density accumulated in the surface of the grains, \varepsilon is a dielectric constant in the boundary layers and r is the ratio of the thickness of the boundary layers to the diameter of the grains. Since the volume of the grains is much larger than the boundary layers, the $j_{1}$ can be approximated as the current density of the whole magnetic material. Therefore, as shown in the equations (2)-(3), the $j_{1}$ can be substituted into the magnetic field equations of the A- \varphi method. Where, the magnetic field H(B) depending on the flux density is calculated by the play model, $c_{\beta }$ is a coefficient of excess loss, $\mu _{0}$ is the vacuum permeability and $J_{0}$ is an exciting current. $\sigma_{1} ( \partial A/ \partial t+ \varphi + {rq/}\varepsilon )+ \times H(B)+ \times \times ( c_{\beta }/ \mu _{0}) ( \partial A/ \partial t) = J_{0}(2) \sigma _{1} ( \partial A/ \partial t+ \varphi + {rq/}\varepsilon ) =0(3)$ The total loss $P_{c}$ can be expressed by the sum of hysteresis loss in DC, eddy current loss and excess loss respectively as shown in the equation (4). $P_{c} = V_{EI}$ %HdB $+V_{EI}{\int}(j_{1}^{2}/\sigma _{1}+ {r} j_{2}^{2}/\sigma_{2} ) dt +V_{EI}$% $( c_{\beta }/ \mu _{0}) ( \partial B/ \partial t) dB(4)$ Here, $j_{2}$ is the current density in the grain boundary layers, $\sigma_{2}$ is the electrical conductivity and $V_{EI}$ is volume of the EI-shaped core. The integration by magnetic flux density is calculated in one loop and the time integration is calculated in one cycle. III. RESULTS For the simulation, we initially fitted the material parameters using several measurement results. The $\sigma_{1}$, $\sigma_{2}, r, \varepsilon$ were derived from the frequency dependent conductivity of the thin plate. In addition, the hysteresis parameter of the play model [4] is from the B-H curve at the low frequency (10 kHz) and the $c_{\beta }$ is from the fitting the loss of high frequency (1000 kHz) using the toroidal core. In the simulation, we used the EI-shaped core shown in Fig.1. The average magnetic flux density in the core region wounded by the coil was controlled to be constant while varying the frequency. Figure 2(a) shows the frequency dependence of the core loss at the flux density 50mT and 100mT. The simulation results of the core loss agree well with measurements. Figures 2(b) and Fig. 2(c) show the simulation results of the core losses of different contributions at 50mT and 100mT respectively. At the low frequencies, the loss rate of the hysteresis is large, for example, it is 54% at 50 kHz in Fig. 2(c). The excess losses are the largest and are approximately proportional to the frequency. The eddy currents are very small at low frequencies, but their loss gradient in the frequency increase up to 1000 kHz. It should be noted that such sharp increase in the eddy current loss as a function of frequency cannot be reproduced by a simulation of homogeneous material without the dielectric effect [5]. Since the frequency of the dimensional resonance is about 1200 kHz, the macroscopic phenomenon that the resistance of the boundary layers seems to be short-circuited by charge accumulation in the capacitance results in the increase of eddy current at the high frequencies. It was found that the finite element magnetic field analysis which considers the dielectric effect and the magnetic hysteresis can accurately predict the losses of the EI-shaped Mn-Zn ferrite core and analyze the loss divided into different kinds.