# Effective Permeability of Multi Air Gap Ferrite Core 3-Phase Medium Frequency Transformer in Isolated DC-DC Converters

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{ds}of the MOSFET (or other power semiconductor switch). It should ensure the magnetizing current sufficient to charge and discharge the C

_{ds}during the dead time of a VSC leg. In the dual active bridge (DAB) converter [2,22], the magnetizing inductance should not increase the VSC current and it should be considered at low operating power.

- Determination of the equivalent B(H) and the equivalent magnetic permeability in a three-phase multi air gap ferrite core MFT.
- Demonstration that the equivalent magnetic permeability and the average air gap length of the multi air gap ferrite core MFT are nonlinear functions of the number of air gaps.
- Proposal of an exponential scaling function, enabling a rapid estimation of the magnetizing inductance based on the ferrite material datasheet only.

## 2. High Power Medium Frequency Transformer

#### 2.1. MFT Prototypes

#### 2.2. Magnetic Core

## 3. Equivalent B(H) Measurement

#### 3.1. Measurement Setup

_{1}and i

_{2}are the current of the first and second excitation winding respectively, N

_{exc}is the number of turns of each excitation winding, l

_{m}is the average magnetic circuit length (visualized in Table 2), u

_{aux}is the voltage of the auxiliary coil placed on the yoke, T is the period of the excitation voltage, Φ is the core magnetic flux, N

_{aux}is the number of turns of the auxiliary coil, and A

_{c}is the average cross-section of the core.

#### 3.2. Measurement Results

_{aux}correspond to the main magnetic flux in two side columns and two yokes. The Φ

_{0}corresponds to the magnetic flux in the central column. It is observed that the magnetic flux in the central column is below 5% of the main flux so it seems fair to neglect it.

_{c}and remanent flux density B

_{r}can be captured.

#### 3.3. Synthesis of Equivalent B(H) Measurement

_{r}(H) are presented in Figure 7. The 3C90 datasheet curves [57] are plotted for comparison. As expected, a significant difference between the datasheet and the measurement is observed. There is a difference between T1 and T2 since they have a different core assembly, T1 having more parasitic air gaps than T2 (see Figure 3). For each MFT, the equivalent B(H) differs slightly for different measurement circuits. This proves that the parasitic air gaps are randomly distributed in the core assembly. For each transformer, the authors arbitrarily select the solid line curve (CA) as the reference B(H) for the whole core.

## 4. Finite Element Simulation

#### 4.1. Finite Element Model

_{1}domain is the volume of the windings, the Ω

_{2}domain is the volume of the core, and the Ω

_{3}domain consists of the air surrounding the MFT. In this model, it is assumed that the magnetic core is homogenized. It means that the core components: ferrite, air gaps and also glue, impregnation resin, etc. form a homogenous material. In a similar manner, the winding is also homogenized.

_{c}= 0.25 S/m (at 25 °C) and μ

_{c}= dB/dH are defined in the previous section (Figure 7b, curve T2 CA). In Ansys Maxwell, the material conductivity enables the calculation of eddy current effects. However, it can be noticed that the ferrite conductivity is low so the eddy current effects do not have a significant impact on the magnetic field and core power loss.

#### 4.2. Magnetic Simulations

_{1}, L

_{2}and L

_{3}, which correspond to the primary winding. The voltage sources model the VSC square output voltage, and R

_{p}is the primary winding resistance.

_{0}H) in the central and the right column can be observed due to the nonlinearity of the B(H) curve.

## 5. Experimental Verifications

#### 5.1. Converter Test Bench

#### 5.2. No Load Test Experimental Results

## 6. Scaling of Relative Permeability

_{r}

_{0}= 5300, is read from Figure 15. Thus, the equivalent relative permeability ratio K

_{µ}of the multi air gap core can be calculated with:

_{r}is the equivalent relative permeability defined in Figure 15 for T1 or T2. The equivalent relative permeability ratio is plotted in Figure 16 as a function of a number of parasitic air gaps.

_{m}is the average magnetic circuit length, l

_{I}is the length of the I-core, l

_{a}is the average air gap length, and A

_{c}is the average cross-section of the core. Assuming that the average magnetic circuit length l

_{m}is equal to n∙l

_{I}, then it can be found the relative average air gap length l

_{a}/l

_{m}defined as:

_{a}equals n times the known individual air gap length l

_{g}, the relative average air gap length l

_{a}/l

_{m}is a linear function of n:

_{g}changes between prototypes. However, considering the proposed exponential interpolation (11), the effective relative average air gap length is a nonlinear function of n as presented in Figure 17. This is due to the fact that the I-core is not an ideal rectangular cuboid and its dimensions vary from one sample to another. As a consequence, the mechanical assembly of the core gets more difficult when a large number of I-cores is assembled.

## 7. Conclusions

## Author Contributions

## Funding

^{2}Laboratory at the Gdansk University of Technology.

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Two assemblies of 4 randomly selected 3C90 ferrite I-cores showing the perpendicular parasitic air gap and the longitudinal parasitic air gap measuring up to about 0.5 mm.

## Appendix B

**Figure A2.**Single-phase multi air gap MFT core assembly: MFT4 with 4 air gaps (

**left**) and MFT6 with 6 air gaps (

**right**).

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**Figure 1.**Three-phase isolated dc-dc converter circuit diagram; C

_{r}is the optional resonant capacitor.

**Figure 2.**Medium frequency transformer prototype T2 showing primary winding terminals: 1*-1, 2*-2, 3*-3, secondary winding terminals: 4*-4, 5*-5, 6*-6, three columns A, B, C, and additional auxiliary coils AUX1 and AUX2 for flux measurement (blue wire around the yoke).

**Figure 3.**Medium frequency transformer core assembly composed of elementary I-cores: T1 (

**left**) and T2 (

**right**).

**Figure 4.**Circuit diagram of the equivalent B(H) measurement setup where the windings C and A are supplied.

**Figure 5.**Waveforms of the T2 supplied with C and A windings: (

**a**) measured supply voltage u

_{s}, excitation currents i

_{1}(C) and i

_{2}(A), auxiliary coil voltage u

_{aux}(AUX1) and zero coil voltage u

_{0}(B); (

**b**) magnetic flux of the auxiliary coil Φ

_{aux}(AUX1) and magnetic flux of the zero coil Φ

_{0}(B).

**Figure 6.**Measured equivalent B(H) of the T2 supplied with C and A windings: upward curve (red), downward curve (green) and interpolated anhysteretic curve (blue).

**Figure 7.**Synthesis of equivalent B(H) measurement: (

**a**) equivalent anhysteretic B(H); (

**b**) equivalent relative permeability µ

_{r}; curves based on 3C90 datasheet (black) and measurement: T2 supply of A and B windings (red), T2 supply of B and C windings (green), T2 supply of C and A windings (blue)—the same as in Figure 6, T1 supply of A and B windings (cyan), T1 supply of B and C windings (yellow), T1 supply of C and A windings (magenta).

**Figure 8.**3D MFT model divided into three computational domains: Ω

_{1}volume of the windings (orange), Ω

_{2}volume of the homogenized core (grey) and Ω

_{3}air surrounding the MFT (white).

**Figure 9.**MFT no load test equivalent circuit model coupled with the finite element model through the nonlinear inductances L

_{1}, L

_{2}and L

_{3}.

**Figure 10.**MFT no load test magnetic transient simulation result: primary phase voltage (

**top**) and primary current (

**bottom**); the dashed vertical line indicates the time instant for the magnetostatic simulation.

**Figure 11.**Magnetic flux density B magnitude on the core surface with the current excitation i

_{1}= −2.76 A, i

_{2}= −1.93 A, i

_{3}= 4.69 A; the dashed line indicates the magnetic flux path in the centre of the core.

**Figure 12.**Magnetic field strength H and magnetic flux density B along the path in the centre of the core passing through the central column, top yoke, right column, and bottom yoke; the values of static permeability B/(µ

_{0}H) are presented.

**Figure 14.**MFT T2 no load test primary current: experimental result (solid line), magnetic transient simulation result (dashed line).

**Figure 15.**Equivalent anhysteretic B(H): datasheet and measurement (solid line), linear interpolation (dashed line).

**Figure 16.**Equivalent relative permeability ratio K

_{µ}in the function of a number of parasitic air gaps n: datasheet, T2 and T1 measurement (stars), exponential interpolation (red dashed line), and single-phase multi air gap transformer MAG4 and MAG6 measurement (circles).

**Figure 17.**Relative average air gap length l

_{a}/l

_{m}in the function of a number of parasitic air gaps n: T2, T1, MAG4 and MAG6 measurement (stars/circle), the corresponding idealized reluctance model (solid lines), and the relative average air gap length calculated based on the proposed exponential interpolation (red dashed line).

**Table 1.**Specification of the medium frequency transformer prototypes for the nominal operating conditions.

Parameter | T1 Dd | T1 Yy | T2 Yy |
---|---|---|---|

Phase voltage (V) | 980 | 566 | 566 |

Phase current (A) | 36 | 65 | 65 |

Core flux density (T) | 0.22 | 0.15 | 0.27 |

Winding current density (A/mm^{2}) | 1.2 | 2.1 | 2.1 |

Dimensions of active parts (cm) | 67 × 20 × 35 | 45 × 20 × 30 | |

Total weight (kg) | 57 | 36 |

u_{s} | u_{aux} | u_{0} | Magnetic Flux Path |
---|---|---|---|

A + B | AUX1 | C | |

B + C | AUX2 | A | |

C + A | AUX1 or AUX2 | B |

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**MDPI and ACS Style**

Dworakowski, P.; Wilk, A.; Michna, M.; Lefebvre, B.; Sixdenier, F.; Mermet-Guyennet, M.
Effective Permeability of Multi Air Gap Ferrite Core 3-Phase Medium Frequency Transformer in Isolated DC-DC Converters. *Energies* **2020**, *13*, 1352.
https://doi.org/10.3390/en13061352

**AMA Style**

Dworakowski P, Wilk A, Michna M, Lefebvre B, Sixdenier F, Mermet-Guyennet M.
Effective Permeability of Multi Air Gap Ferrite Core 3-Phase Medium Frequency Transformer in Isolated DC-DC Converters. *Energies*. 2020; 13(6):1352.
https://doi.org/10.3390/en13061352

**Chicago/Turabian Style**

Dworakowski, Piotr, Andrzej Wilk, Michal Michna, Bruno Lefebvre, Fabien Sixdenier, and Michel Mermet-Guyennet.
2020. "Effective Permeability of Multi Air Gap Ferrite Core 3-Phase Medium Frequency Transformer in Isolated DC-DC Converters" *Energies* 13, no. 6: 1352.
https://doi.org/10.3390/en13061352