# Direct Consideration of Eddy Current Losses in Laminated Magnetic Cores in Finite Element Method (FEM) Calculations Using the Laplace Transform

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Computational Method

#### 2.1. Direct Synthesis of Equivalent Magnetic Permeability of Laminated Sheets in the Form of a Transfer Function of an IIR Filter

#### 2.2. Inclusion of Eddy Current Losses in the Laminated Magnetic Circuit in Transient Finite Element Method Calculations

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Synthesis of the equivalent magnetic permeability, $\widehat{\mu}$, in the form of an R,C system.

**Figure 2.**The synthesis of the equivalent magnetic reluctivity $\widehat{\nu}=\frac{1}{\widehat{\mu}}$ in the form of an R,L system.

**Figure 4.**The dependence of the magnetic magnetic flux density B on the strength of the magnetic field H at frequency f = 1500 Hz; 1—the curve including losses of eddy current and hysteresis, 2—the curve including only eddy current losses, 3—the primary magnetization curve (10).

**Figure 5.**The diagram presents general dependency of the changeability of polynomial coefficients that comprise the numerator of the developed IIR filter in the domain of Laplace variable s, in the function of ${B}^{2}$.

**Figure 6.**The diagram presents the general dependency of the changeability of polynomial coefficients that comprise the denominator of the developed IIR filter in the domain of the Laplace variable, s, in the function of ${B}^{2}$.

**Figure 7.**The cross-section x–y of a coil wound on a magnetic core; red—cross-section of the magnetic core, green—cross-section through the coil.

**Figure 8.**The distribution of the magnetic flux density vector in the cross-sectional area of the choke for angular frequencies $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$ and time t = 0.1 s.

**Figure 9.**The distribution of the magnetic field intensity vector $\overrightarrow{H}$ corresponding to the magnetic flux density $\overrightarrow{B}$ from Figure 8 $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$ and time $t=0.1$ s.

**Figure 10.**The waveform of the choke current ${I}_{c}(*100)$ and voltage supplying the choke ${E}_{w}$ for angular frequencies $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$ after switching it on.

**Figure 11.**The waveform of the power supply, energy loss in iron, energy loss of resistance, and a zero energy balance for angular frequencies $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$.

**Figure 12.**The momentary power waveform in leakage inductance (${L}_{sr}$), power in the air area (air), power in the winding area (winding), which all increased 100 times, and the power loss in laminated sheets for angular frequencies $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$.

**Figure 13.**The supply energy waveform, energy losses in iron, energy of losses of resistance, and zeroing energy balance for the angular frequencies $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$.

**Figure 14.**The instantaneous power waveform in leakage inductance (${L}_{rs}$) of the power in the air space (air), power in the winding space (winding), all magnified 100 times, and the power of losses in laminated sheets for the angular frequencies $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$.

**Figure 15.**The instantaneous power waveform in leakage inductance (${L}_{rs}$), in air space (pow), winding space power (uzw), all magnified 100 times, and the power of losses in laminated sheets for the angular frequencies $\omega =3141.6\frac{\mathrm{rad}}{\mathrm{s}}$.

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**MDPI and ACS Style**

Gołębiowski, M.; Gołębiowski, L.; Smoleń, A.; Mazur, D.
Direct Consideration of Eddy Current Losses in Laminated Magnetic Cores in Finite Element Method (FEM) Calculations Using the Laplace Transform. *Energies* **2020**, *13*, 1174.
https://doi.org/10.3390/en13051174

**AMA Style**

Gołębiowski M, Gołębiowski L, Smoleń A, Mazur D.
Direct Consideration of Eddy Current Losses in Laminated Magnetic Cores in Finite Element Method (FEM) Calculations Using the Laplace Transform. *Energies*. 2020; 13(5):1174.
https://doi.org/10.3390/en13051174

**Chicago/Turabian Style**

Gołębiowski, Marek, Lesław Gołębiowski, Andrzej Smoleń, and Damian Mazur.
2020. "Direct Consideration of Eddy Current Losses in Laminated Magnetic Cores in Finite Element Method (FEM) Calculations Using the Laplace Transform" *Energies* 13, no. 5: 1174.
https://doi.org/10.3390/en13051174