**1. Introduction**

The medium frequency transformer (MFT) is one of the key components in the isolated dc-dc converters [1–4] related to: smart grids [5], photovoltaic power plants [6], wind power plants [7], and electric vehicle charging [8,9]. The three-phase topology is considered for high power applications where the high power density and high efficiency are required. In [10,11], an analytical approach was proposed to compare multi-phase dc-dc topologies. In [12], the single-phase and three-phase topologies were compared. A 10 kVA 1 kHz three-phase MFT prototype was reported in [13], and a 2 kVA 100 kHz three-phase MFT was reported in [14]. A 5 MW three-phase converter was presented in [15] but using three single-phase MFTs. The general circuit diagram of the three-phase isolated dc-dc converter is composed of two voltage source converters (VSC), and an MFT is presented in Figure 1.

**Figure 1.** Three-phase isolated dc-dc converter circuit diagram; *Cr* is the optional resonant capacitor.

The performance of the converter highly depends on the MFT and its equivalent circuit parameters. The leakage inductance has a significant influence on the operation of the converter and the specified value is usually well achieved in the MFT development process. In the LLC resonant dc-dc (LLC) converter [1,16,17], the magnetizing inductance has a significant effect on the zero voltage switching (ZVS) [18–20], but it may be difficult to achieve within reasonable error margins [21]. The maximum value of the magnetizing inductance should take into account the drain-source capacitance *Cds* of the MOSFET (or other power semiconductor switch). It should ensure the magnetizing current sufficient to charge and discharge the *Cds* 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.

The operating frequency of the 100 kilowatt class isolated dc-dc converters is considered in the range from few kilohertz to tens of kilohertz [23–25]. The voltage and current fundamentals and harmonics influence the design of the MFT magnetic core and windings. The choice of MFT magnetic core material should be done according to the material properties and cost. The performance factor, which is defined as a product of the frequency and flux density at a specified core loss density, is used to compare different types of core materials [26,27]. The amorphous and especially nanocrystalline materials are preferred in the low and medium frequencies due to the high flux density [28,29]. On the other hand, the main advantage of ferrite cores is their low power loss, which makes them an attractive material for the construction of medium and high frequency transformers [30,31]. The ferrite also offers low cost in terms of material and transformer assembly. In [32], the ferrite core MFT was considered for an optimized dc-dc converter operating at a few kHz. Finally, the ferrite seems as a good candidate for the short term industrialization of the high power three-phase MFT. However, the construction of a ferrite magnetic core for high power MFT requires an assembly of type "I" cores since the C-cores or E-cores do not exist for large transformers. This results in a multi air gap structure of the magnetic core.

The influence of the air gap on the transformer magnetic properties in LLC converters was analysed in [33,34]. It was assumed that the air gap length was known and controlled in the MFT design process. The considered air gaps had relatively large size in order to reduce the slope of the *B*(*H*) curve and to minimize the influence of magnetic saturation on the magnetizing inductance value. The influence of the air gap length on the equivalent magnetic permeability, magnetic reluctance and magnetizing inductance in ferrite core transformers was analysed in [35–38]. The influence on the core and winding power loss was studied in [39–41]. All the analysed cases considered a single and uniform air gap of a known length. The analysis of a single but non-uniform air gap in toroidal cores was presented in [42,43]. The influence of the number of uniform air gaps with a controlled size on the magnetic properties and transients in a current transducer was considered in [44].

In the transformer core structure characterized by a construction periodicity (ferromagnetic material—air gap, ferromagnetic material—diamagnetic material, etc.), it is possible to utilize the homogenization technique or multiscale methods in the description of magnetic properties (reluctance of homogenized core, equivalent magnetic permeability, equivalent *B*(*H*), etc.). The use of the homogenization technique in the finite element method (FEM) analysis of step-lap joints in steel sheet transformers was proposed in [45]. The homogenization technique was further developed in 2D FEM of steel sheet cores [46–48] and amorphous cores [49]. The multiscale methods were proposed in the analysis of the magnetic properties of transformer cores in [50]. In order to increase the accuracy of magnetic computations, a higher order FEM [51] and a step-wise method were proposed [52].

In all the presented references, it was assumed that the air gap length or the diamagnetic material dimensions were known. However, during the core assembly, the core experiences different mechanical constraints, which are required to ensure its integrity. This impacts the magnetic properties [53] and changes the core structure near the air gaps. In many cases, these changes are difficult to determine, especially once the core is assembled.

According to the authors' knowledge, a study of multiple air gaps in the ferrite core transformers, enabling an efficient MFT design for the isolated dc-dc converters, has not been reported. In this article, it is proposed the analysis of the number of air gaps on the equivalent *B*(*H*) and the equivalent magnetic permeability. It is considered that different MFTs have a similar probability distribution of the average air gap length. The authors propose an experimental approach to the determination of the equivalent *B*(*H*), implying that the physical phenomena as: nonlinearity, fringing effect, structure dissymmetry, technological aspects, etc. are taken into account.

The novel aspects of this work includes:


The multi air gap medium frequency transformer prototype is presented in Section 2. The measurement of the magnetic flux in the function of the magnetizing current and the calculation of the equivalent *B*(*H*) and the equivalent magnetic permeability are presented in Section 3. The finite element simulation of the MFT no load test, using the measured equivalent *B*(*H*), is presented in Section 4. The FEM simulation result is compared with an experimental measurement on a 100 kW 1.2 kV 20 kHz dc-dc converter in Section 5. The results are analysed and discussed in Section 6, where the influence of the number of air gaps on the equivalent permeability and the average air gap length are presented. A scaling function enabling a rapid estimation of the magnetizing inductance is proposed.

#### **2. High Power Medium Frequency Transformer**

#### *2.1. MFT Prototypes*

The authors have developed two three-phase MFT prototypes for a 100 kW 1.2 kV 20 kHz dc-dc converter. The dc-dc converter is presented in details in [54]. The three-phase structure is still novel in the MFT applications with very little demonstrators. The specifications of two MFT prototypes T1 and T2 are presented in Table 1. The MFT T1 can operate in delta and star vector groups whereas the T2 in star only. The winding of both transformers is made of the same litz wire composed of 3870 strands of 0.1 mm diameter.


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

The MFT T2 is presented in Figure 2 and its design is detailed in [55]. In particular, a significant difference between the calculated and measured magnetizing inductance is highlighted. This shows the important influence of the parasitic air gaps on the magnetizing inductance.

**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).
