*5.2. No Load Test Experimental Results*

The measured waveform of the MFT T2 no load current is presented in Figure 14. The simulated no load current from the previous section is plotted for the comparison. Generally, quite a good fit between the simulation and the measurement is observed. Some minor differences are discussed hereafter.

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

There are some high frequency oscillations present in the measurement. They are due to the parasitic capacitance of the windings that have not been modelled. This could be improved by adding the winding self and mutual capacitances into the model. However, the simulation time would increase significantly.

There are some differences in the current amplitude of different phases. As it has been presented in Figure 7, the *B*(*H*) is not strictly the same for the whole core. Since in the simulation, the authors have assumed a single equivalent *B*(*H*), then it seems normal to observe some differences in the measured currents.

Moreover, there might be some differences due to the fact that the simulation model assumes the anhysteretic *B*(*H*). In Figure 6, one can see that the measured equivalent *B*(*H*) is hysteretic, thus it may influence the shape of the current waveform, in particular, the corresponding ascending and descending slopes of the current.

Finally, the RMS current error is within 10% and the authors consider this acceptable. If the datasheet *B*(*H*) was used (Figure 7), then the RMS current error would reach approximately 500%. This experimental result proves the validity of the measured equivalent *B*(*H*).

#### **6. Scaling of Relative Permeability**

The approach presented in the previous paragraphs has limited usage in the MFT design process since it is based on the measurement on a physical device. This limits the practical usage to post-manufacturing analysis or to a new design of a similar transformer. In this section, an approach based on a simple count of perpendicular parasitic air gaps is proposed.

In the MFT design process from scratch, when evaluating the performance of isolated dc-dc converters, one is usually interested in the magnetizing inductance at the nominal *B*(*H*) operating point. This is usually below the *B*(*H*) saturation, so the anhysteretic curve from Figure 7 can be linearized as presented in Figure 15.

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

From Figure 3, we can count the number of perpendicular parasitic air gaps along the magnetic path. This equals to 10 and 14 for T2 and T1 respectively. The core used for the datasheet measurement had zero air gaps. The value of datasheet linearized relative permeability, which equals μ*r*<sup>0</sup> = 5300, is read from Figure 15. Thus, the equivalent relative permeability ratio *K*μ of the multi air gap core can be calculated with:

$$\mathcal{K}\_{\mu} = \frac{\mu\_r}{\mu\_{r0}} \tag{10}$$

where μ*<sup>r</sup>* 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.

**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).

In addition, an exponential interpolation is proposed allowing to estimate the equivalent relative permeability for any high power ferrite core MFT with a similar core assembly. The exponential interpolation function is defined as:

$$\mathcal{K}\_{\mu}(n) = \mathcal{e}^{-0.155u} \tag{11}$$

where *n* is the number of perpendicular parasitic air gaps along the magnetic flux path.

This function was validated with the experimental *B*(*H*) measurement on two single-phase multi air gap (MAG) transformers presented in Appendix B. The MAG4 transformer has four air gaps and MAG6 has six air gaps. Both use the same I-cores as T1 and T2. The resulting ratios are displayed in Figure 16 and it can be seen that for MAG4 the ratio is slightly higher than the exponential interpolation. This is normal because for this transformer the I-cores were carefully selected to minimize the parasitic air gaps and the core assembly is simpler compared to the three-phase MFT. However, a general trend of the equivalent relative permeability ratio is clearly observed even if the four MFT prototypes involve different technologies and different manufacturers.

Furthermore, a simple reluctance model of the magnetic core neglecting the fringing effect is considered according to [35]. The total magnetic circuit reluctance can be related to the sum of the I-core and air gap reluctances as:

$$\frac{l\_m}{\mu\_0 \mu\_r A\_c} = n \frac{l\_I}{\mu\_0 \mu\_0 \mu\_c} + \frac{l\_a}{\mu\_0 A\_c} \tag{12}$$

where *lm* is the average magnetic circuit length, *lI* is the length of the I-core, *la* is the average air gap length, and *Ac* is the average cross-section of the core. Assuming that the average magnetic circuit length *lm* is equal to *n*·*lI*, then it can be found the relative average air gap length *la*/*lm* defined as:

$$\frac{l\_a}{l\_m} = \frac{1}{\mu\_r} - \frac{1}{\mu\_{r0}}\tag{13}$$

Considering an ideal core assembly, where the average air gap length *la* equals *n* times the known individual air gap length *lg*, the relative average air gap length *la*/*lm* is a linear function of *n:*

$$\frac{l\_a}{l\_m} = \frac{l\_\mathcal{\mathcal{G}}}{l\_m} n \tag{14}$$

In Figure 17, these linear functions are presented for four transformers T2, T1, MAG4, and MAG6. It was verified that the individual air gap length *lg* 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.

**Figure 17.** Relative average air gap length *la*/*lm* 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).

The proposed approach can be used in scaling the datasheet *B*(*H*) for a finite element simulation, in the rapid estimation of transformer magnetizing inductance or in evaluating the size of the average air gap length. The magnetizing inductance can be estimated based on the magnetic reluctance model according to:

$$L\_m = K\_\mu(n) \frac{\mu\_0 \mu\_{r0} N^2 A\_c}{l\_m} \tag{15}$$

where *N* is the primary/secondary number of turns. It shall be mentioned that the proposed estimation is meant to provide an order of magnitude of the magnetizing inductance. This shall be sufficient when evaluating the performance of isolated dc-dc converters. However, 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.
