*2.2. Magnetic Core*

The magnetic core of the MFT prototypes is made of MnZn ferrite 3C90 from Ferroxcube. The core is assembled with I-cores measuring 25 mm × 25 mm × 100 mm each. The core assembly is presented in Figure 3. In this core design, the I-cores are not interleaved. It can be seen that the core involves multiple parasitic air gaps. Moreover, due to manufacturing tolerances, the I-core is not an ideal rectangular cuboid and its dimensions vary from one sample to another. This causes the non-uniform parasitic air gaps in the core. There are at least two types of parasitic air gaps: perpendicular and longitudinal to the axis of the magnetic flux path. The authors claim that the parasitic air gap size is unpredictable at the industrial design stage and that it cannot be modelled precisely. In Appendix A, some example views of the ferrite core assembly are presented. It can be seen that the air gap length varies from almost zero to about 0.5 mm. Consequently, the use of material datasheet in the calculation of effective magnetizing inductance leads to significant errors. However, the magnetizing inductance or the equivalent *B*(*H*) can be measured on the transformer prototype. Such a measurement can be helpful in a new transformer design with a similar core assembly.

**Figure 3.** Medium frequency transformer core assembly composed of elementary I-cores: T1 (**left**) and T2 (**right**).

#### **3. Equivalent B(H) Measurement**

### *3.1. Measurement Setup*

The nonlinear magnetic properties of core material are represented by the magnetic permeability, which relates the magnetic flux density *B* with the magnetic field strength *H*. The nonlinear magnetic properties of a transformer core can be described by the current-dependent flux linkage characteristics Ψ(*i*) using the experimental approach. From the flux linkage characteristics, the *B*(*H*) curve can be determined under certain simplifying assumptions. The measurement of Ψ(*i*) hysteretic characteristics for inherently asymmetric three-phase transformer with three columns was proposed in [56]. In this approach to determine Ψ(*i*) characteristics for each winding, only two phases are excited in a special manner.

A dedicated static *B*(*H*) measurement setup was developed as presented in Figure 4. It is composed of a high current AC power supply, oscilloscope and probes. The primary and secondary windings of each phase were connected in series in order to achieve a high magnetomotive force (MMF). The windings of two columns were connected in anti-parallel so that their MMFs add together. Two additional auxiliary coils (AUX1 and AUX2) were placed on the yoke allowing the measurement of the magnetic flux in the core (see the blue wire in Figure 2) and minimizing the magnetic coupling in the air. The voltage of the remaining winding (so-called zero-coil) is measured in order to verify that the magnetic flux coupled with this winding is close to zero.

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

For each MFT prototype, three measurements were performed according to the winding configurations presented in Table 2. The frequency of the power supply in the static *B*(*H*) measurement setup was set to 100 Hz. This value was considered in order to minimize the effect of eddy currents (considering a high frequency material as ferrite) and to achieve good performance of the available power supply.


**Table 2.** Winding configurations of the equivalent *B*(*H*) mearement circuits.

The waveforms of the magnetic flux density *B*(*t*) and the magnetic field strength *H*(*t*) were calculated with:

$$H(t) = \frac{N\_{\text{exc}}[i\_1(t) + i\_2(t)]}{l\_{\text{mf}}} \tag{1}$$

$$\mathcal{O}(t) = \int\_0^T u\_{\text{max}}(t)dt\tag{2}$$

$$B(t) = \frac{\Phi(t)}{N\_{\text{aux}} A\_{\text{c}}}.\tag{3}$$

where *i*<sup>1</sup> and *i*<sup>2</sup> are the current of the first and second excitation winding respectively, *Nexc* is the number of turns of each excitation winding, *lm* is the average magnetic circuit length (visualized in Table 2), *uaux* 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, *Naux* is the number of turns of the auxiliary coil, and *Ac* is the average cross-section of the core.

#### *3.2. Measurement Results*

The measured waveforms for the example case where the C and A windings of T2 are supplied are presented in Figure 5a. The measurement was performed with the transformer temperature equal to ambient at 25 ◦C. It can be observed that the supply voltage is close to sinusoidal. The currents in two excitation windings show the core saturation. Their amplitudes are slightly different due to a difference in winding impedance. The amplitude of the zero-coil voltage is relatively low.

**Figure 5.** Waveforms of the T2 supplied with C and A windings: (**a**) measured supply voltage *us*, excitation currents *i*<sup>1</sup> (C) and *i*<sup>2</sup> (A), auxiliary coil voltage *uaux* (AUX1) and zero coil voltage *u*<sup>0</sup> (B); (**b**) magnetic flux of the auxiliary coil Φ*aux* (AUX1) and magnetic flux of the zero coil Φ<sup>0</sup> (B).

Figure 5b presents the waveforms of the magnetic flux calculated according to (2). The Φ*aux* correspond to the main magnetic flux in two side columns and two yokes. The Φ<sup>0</sup> 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.

Thanks to (1) and (3), the magnetic field strength *H* and the magnetic flux density *B* are calculated. In Figure 6, the resulting *B*(*H*) is plotted for the positive values of *H*. The *B*(*H*) is separated into the upward and downward curves, which are then interpolated with piecewise linear functions in order to facilitate the data analysis. The anhysteretic *B*(*H*) curve is calculated as the average of the interpolated upward and downward curves and further filtered to achieve a smooth curve adequate for further processing. Moreover, the coercive field *Hc* and remanent flux density *Br* can be captured.

**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).

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

The measurement process presented in the previous section was repeated for the MFT T1 and T2 for the cases with the supply of windings: A and B, B and C, and C and A, according to Table 2. The measured equivalent anhysteretic *B*(*H*) and relative permeability μ*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.

**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).

#### **4. Finite Element Simulation**

#### *4.1. Finite Element Model*

A 3D MFT T2 model was developed in Ansys Maxwell. A simplified transformer geometry was considered. The model was divided into three computational domains as shown in Figure 8. The Ω<sup>1</sup> domain is the volume of the windings, the Ω<sup>2</sup> domain is the volume of the core, and the Ω<sup>3</sup> 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.

The Maxwell's equations for the defined domains have the form:

$$\nabla \times \mathbf{H} = \begin{cases} \mathbf{j} \text{ in } \Omega\_1 \\ \overset{\ast}{\sigma} \mathbf{E} \text{ in } \Omega\_2 \\ 0 \text{ in } \Omega\_3 \end{cases} \tag{4}$$

$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}; \nabla \bullet \mathbf{B} = 0; \mathbf{B} = \nabla \times \mathbf{A} \tag{5}
$$

where <sup>↔</sup> σ is the electrical conductivity tensor:

$$
\stackrel{\leftrightarrow}{\sigma} = \begin{bmatrix}
\sigma\_{xx}(\mathbf{x}, \mathbf{y}, z) & 0 & 0 \\
0 & \sigma\_{yy}(\mathbf{x}, \mathbf{y}, z) & 0 \\
0 & 0 & \sigma\_{zz}(\mathbf{x}, \mathbf{y}, z)
\end{bmatrix} \tag{6}
$$

The permeability tensor, which for nonlinear properties describes the relation between d*B* and d*H* in the constitutive equation, can be expressed as:

$$
\begin{aligned}
\xleftrightarrow{\mu} &= \begin{cases}
\mu\_0 \text{ in } \Omega\_1 \\
\overset{\leftrightarrow}{\mu}\_{\text{core}} \text{ in } \Omega\_2 \\
\mu\_0 \text{ in } \Omega\_3
\end{cases}
\end{aligned}
\tag{7}
$$

where <sup>↔</sup> μ*core* is the magnetic permeability tensor:

$$
\stackrel{\leftrightarrow}{\boldsymbol{\mu}}\_{\text{core}} = \begin{bmatrix}
\mu\_{xx}(\mathbf{x}, \mathbf{y}, \mathbf{z}) & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mu\_{yy}(\mathbf{x}, \mathbf{y}, \mathbf{z}) & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mu\_{zz}(\mathbf{x}, \mathbf{y}, \mathbf{z})
\end{bmatrix} \tag{8}
$$

It was assumed that the ferrite core has isotropic electrical and magnetic properties. Hence, the electrical conductivity and magnetic permeability tensors have the form:

$$
\stackrel{\leftrightarrow}{\sigma} = \begin{bmatrix} \sigma\_{\mathcal{L}} & 0 & 0 \\ 0 & \sigma\_{\mathcal{L}} & 0 \\ 0 & 0 & \sigma\_{\mathcal{L}} \end{bmatrix} \stackrel{\leftrightarrow}{\mathbf{u}}\_{\text{core}} = \begin{bmatrix} \mu\_{\mathcal{L}} & 0 & 0 \\ 0 & \mu\_{\mathcal{L}} & 0 \\ 0 & 0 & \mu\_{\mathcal{L}} \end{bmatrix} \tag{9}
$$

where σ*<sup>c</sup>* = 0.25 S/m (at 25 ◦C) and μ*<sup>c</sup>* = d*B*/d*H* 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.

**Figure 8.** 3D MFT model divided into three computational domains: Ω<sup>1</sup> volume of the windings (orange), Ω<sup>2</sup> volume of the homogenized core (grey) and Ω<sup>3</sup> air surrounding the MFT (white).

#### *4.2. Magnetic Simulations*

In order to perform a magnetic transient simulation, the finite element model was coupled with an equivalent circuit model. A no load test was considered, as presented in Figure 9. The coupling between the finite element model and the equivalent circuit model is done through the nonlinear inductances *L*1, *L*<sup>2</sup> and *L*3, which correspond to the primary winding. The voltage sources model the VSC square output voltage, and *Rp* is the primary winding resistance.

**Figure 9.** MFT no load test equivalent circuit model coupled with the finite element model through the nonlinear inductances *L*1, *L*<sup>2</sup> and *L*3.

The magnetic transient simulation result is presented in Figure 10. The MFT phase voltage is presented, being a typical VSC output voltage waveform. The MFT primary current is presented in steady-state. This result will be further used to validate the measured equivalent *B*(*H*).

**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.

In Figure 11, the magnetostatic simulation result corresponding to the time instant defined by the dashed line in Figure 10 is presented. The magnitude of the flux density is plotted on the core surface and the maximum value of 0.27 T is observed, as expected. In Figure 12, the magnetic field strength and the magnetic flux density are plotted along the path defined by the dashed line in Figure 11. The different values of quotient *B*/(μ0*H*) in the central and the right column can be observed due to the nonlinearity of the *B*(*H*) curve.

**Figure 11.** Magnetic flux density *B* magnitude on the core surface with the current excitation *i*<sup>1</sup> = −2.76 A, *i*<sup>2</sup> = −1.93 A, *i*<sup>3</sup> = 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*/(μ0*H*) are presented.

#### **5. Experimental Verifications**

#### *5.1. Converter Test Bench*

The power converter test bench was developed for the 100 kW dc-dc converter, as presented in Figure 13. A MFT no load test was considered in order to evaluate the magnetizing inductance. In the no load test, the VSC1 operates normally with 1200 Vdc input voltage and the AC terminals of the VSC2 are disconnected. The circuit diagram of the experimental setup is equivalent to the one used in the simulation that is presented in Figure 9. The test was performed at an ambient temperature of 25 ◦C. The MFT temperature was measured nearly equal to the ambient as the test lasts for a few minutes only.

**Figure 13.** 100 kW three-phase isolated dc-dc converter test bench implementation.
