**7. Conclusions**

The analysis of the effective permeability and average air gap length in multi air gap ferrite core three-phase medium frequency transformer was presented. The calculation of the magnetizing inductance in multi air gap ferrite core MFT based on core material datasheet leads to significant errors. This may impact the design of isolated dc-dc converters as the magnetizing inductance influences their performance.

The measurement of the equivalent *B*(*H*) and the equivalent permeability for two three-phase MFT prototypes was presented. The measured equivalent *B*(*H*) was used in a finite element simulation, giving good results when compared to a 100 kW dc-dc converter no load operation. The use of the anhysteretic *B*(*H*) gives satisfactory results within 10% error compared to the experiment.

This article demonstrates 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. An empirical scaling function is proposed for the rapid estimation of the magnetizing inductance in the multi air gap MFT. In fact, the relative average air gap length increases with the number of parasitic air gaps due to the increasing difficulty in mechanical assembly of the core. The proposed scaling function can be used in the design of isolated dc-dc converters using 25 mm × 25 mm × 100 mm I-cores or similar, based on the core material datasheet and a number of parasitic air gaps.

The measured or scaled equivalent *B*(*H*) can also be used in the equivalent circuit simulation instead of finite element simulation. This would allow more convenient simulations as well including the winding capacitance. The measured *B*(*H*) can be further utilized in the simulations taking into account the hysteresis. This work shall be further extended taking into account the influence of the temperature since the ferrite relative permeability depends on the temperature. The proposed scaling function could be further validated with a large number of MFT prototypes with different types of I-cores and a different number of parasitic air gaps in order to determine the uncertainty range. The experimental validation of the influence of the magnetizing inductance on the performance of three-phase isolated dc-dc converters is recommended.

**Author Contributions:** Conceptualization, P.D. and A.W.; methodology, P.D., A.W. and M.M.; software, P.D.; validation, P.D.; formal analysis, P.D., A.W. and M.M.; investigation, P.D., A.W. and M.M.; resources, P.D., A.W. and M.M.; data curation, P.D.; writing—original draft preparation, P.D., A.W. and M.M; writing—review and editing, P.D., A.W., M.M. and F.S.; visualization, P.D., A.W. and M.M; supervision, A.W., M.M., B.L., M.M.-G.; project administration, B.L. and M.M.-G.; funding acquisition, M.M.-G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" Program (ANE-ITE-002-01) and the LINTE2 Laboratory at the Gdansk University of Technology.

**Acknowledgments:** The authors would like to thank Caroline Stackler, Martin Guillet and Alexis Fouineau for their valuable support in writing this article.

**Conflicts of Interest:** The authors declare no conflict of interest.
