**1. Introduction**

In recent years, benefiting from advances in switching devices, magnetic materials, and control methods, power conversion systems based on MFTs have gradually increased. With the increase of operating frequency, the volume and weight of the transformer can be reduced, thus improving the power density of the converter. Therefore, the MFT has broad application prospects in many fields—such as flexible DC transmission, renewable energy grid connection, and electrified transportation [1–3]. A typical application occasion is vessel integrated power system (IPS) [4], and a modular multi-level converter-bidirectional DC/DC converter (MMC-BDC) for IPS is shown in Figure 1 [5]. The converter adopts multiple DAB converters to realize DC/DC energy conversion and bidirectional flow, which can serve DC loads or energy storage elements.

In the topology of DAB, MFT provides voltage matching and galvanic isolation. The phase-shifting inductance (PSI) is determined by the power capacity of the DAB converter and provides the instantaneous energy storage in the conversion process [6]. In the reported high-power DAB prototypes, an auxiliary inductor and an MFT is usually used. The volume and weight of the auxiliary inductor may even be close to the MFT [3]. In [7], the integrated/non-integrated MFT designs are compared. Although the integrated design has obvious advantages in power density, the non-integrated design is finally chosen for the sake of heat dissipation and installation flexibility. In [8–10], the design method for

**Citation:** Zhang, X.; Xiao, F.; Wang, R.; Kang, W.; Yang, B. Modeling and Design of High-Power Enhanced Leakage-Inductance-Integrated Medium-Frequency Transformers for DAB Converters. *Energies* **2022**, *15*, 1361. https://doi.org/10.3390/ en15041361

Academic Editors: Alon Kuperman and Alessandro Lampasi

Received: 27 January 2022 Accepted: 10 February 2022 Published: 14 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

leakage-inductance-integrated (LII) MFT has been investigated. However, in order to achieve a larger leakage inductance, the LII MFT requires more winding turns or a greater primary-secondary distance, and the unconstrained leakage flux will cause additional losses in tape wound cores. Therefore, it is hard for the LII MFT to meet the requirements of the high inductance and the high power.

**Figure 1.** The modular multilevel converter–bidirectional DC–DC converter.

Thermal performance is one of the decisive factors for the success of MFT design. Natural air cooling is the most common heat dissipation method for MFTs. In [11] and [12], the thermal network modeling method for natural air-cooled MFT is comprehensively described. In order to increase power density, water cooling is adopted in [13] and [14], and forced air cooling is adopted in [3,15]. In [3], the forced convection formula for circular tubes is adopted for thermal modeling without considering the structural characteristics of the winding air channel.

Many achievements have emerged in the optimal design of MFT in recent years, which further pushes the MFT to large-scale industrial applications. In [16,17], the brute force grid search method is adopted to optimize the design of a 100 kW/10 kHz shell-type MFT. The sensitivity analysis of design parameters is carried out based on the large-scaled calculation data. In [18], the researchers attempt to establish a unified mathematical model for transformer/inductor optimal design based on existing classical parameter models and carried out the optimal design with the goal of minimum core and minimum loss. A 200 kVA/10 kHz shell-type MFT is designed based on the genetic algorithm in [10], and the optimal design of the heat sink is embedded in the design process. The above work is carried out for typical core-type or shell-type MFT. The design of inductor-integrated MFT is more challenging due to the parameter coupling and the integration structure.

Compared with the previous work, the main contributions of the work in this paper include the following: (1) Different from the conventional leakage-inductance-integrated (LII) MFT, a concentric-winding enhanced leakage-inductance-integrated (CW ELII) MFT, which includes an additional core, is proposed in this paper. Moreover, a developed example of 200 kW, 4 kHz, 200 μH MFT with CW ELII structure is presented specifically. (2) The air channel in the forced air-cooled winding is simplified to the parallel-plate structure, and the thermal resistance model is established based on classic models. (3) For different leakage inductance requirements, the power density of the CW LII and CW ELII solutions are quantitatively compared.

This paper is organized as follows. Section 2 discusses different integration structures and focuses on the magnetic circuit analysis of the CW ELII MFT in DAB. Section 3 introduces the calculation method of the loss and the leakage inductance. Section 4 presents the thermal modeling method of air–water combined cooled MFT. In Section 5, the design method and quantitative comparison of the CW LII and CW ELII MFT are presented. The developed 200 kW, 4 kHz, 200 μH MFT prototype, and experimental results are also presented in Section 5. Finally, Section 6 lays out the conclusions and future work possibilities.

#### **2. Integration Structures and Operating Mode**

#### *2.1. Integration Structures*

The inductor-integrated MFT can be divided into three categories: detached-inductor integrated (DII), leakage-inductance-integrated (LII), and enhanced leakage-inductanceintegrated (ELII) MFT. The characteristic of DII MFT is that it has independent inductor windings. The inductor core and the transformer core are integrated or just packaged together. The DII MFT will not be focused on in this paper.

LII MFT features no additional winding or core. The target leakage inductance is achieved only through the design of the winding structure. Figure 2a, b show typical separate-winding (SW) and concentric-winding (CW) structures. It is usually easier to achieve a greater leakage inductance by the SW structure [3]. However, since the magnetic motive force (MMF) on the core limb of the SW MFT is unbalanced, the leakage flux is widely distributed in the external space, which may lead to eddy losses in the adjacent metal structure. Therefore, the SW MFT should be used in compact high-power converters with caution. No matter the SW or the CW structure, part of the leakage flux will pass through the core tape vertically and cause apparent eddy current loss of the wound core [19].

**Figure 2.** Comparison of different integration structures. (**a**) SW LII, (**b**) CW LII, (**c**) SW ELII, (**d**) CW ELII.

The characteristic of ELII MFT is that the additional core is used instead of the air in the leakage flux path of the LII MFT. Due to the higher equivalent permeability of the additional core, it is possible to achieve a higher power density. Figure 2c shows the SW ELII structure, which will cause the external leakage flux like the SW LII structure. The flux from the main core to the additional core limb will pass through the core tape vertically, which will cause local additional core loss [19]. Figure 2d shows the CW ELII structure, in which the additional independent core restricts the flow path of the leakage flux. Therefore, the CW ELII structure may be more suitable for the requirements of the high power and high inductance.

### *2.2. CW ELII MFT Operating Mode in DAB*

The circuit topology of DAB is shown in Figure 3. The MFT is excited by H-bridges on both sides simultaneously, and the operating condition is determined by the duty cycle and the phase-shifting duty of the two bridges. In order to analyze the operating mode of the CW ELII MFT under typical DAB excitation, the magnetic circuit model is established. As shown in Figure 4, the voltage excitations *u*<sup>p</sup> and *u*<sup>s</sup> at the primary and the secondary sides work as flux sources *Φ*<sup>p</sup> and *Φ*<sup>s</sup> in the same direction in the magnetic circuit.

$$\begin{cases} N\_{\rm P} \Phi\_{\rm P}(t) = \int u\_{\rm P}(t) \, \mathrm{d}t \\\ N\_{\rm s} \Phi\_{\rm s}(t) = \int u\_{\rm s}(t) \, \mathrm{d}t \end{cases} \tag{1}$$

where *N*p and *N*s are the number of turns of the primary and secondary windings.

**Figure 3.** DAB circuit topology.

**Figure 4.** Magnetic circuit model of CW ELII MFT.

In Figure 4, *R*m-cl and *R*m-cy are respectively the reluctance of the core limb and core yoke; *R*m-σ<sup>p</sup> and *R*m-σ<sup>s</sup> are respectively the air leakage reluctance of the primary winding and the secondary winding; *R*m-σ<sup>a</sup> is the reluctance of the additional core. The magnetic circuit equation is given as

$$\begin{cases} \Phi\_{\sigma\mathfrak{a}}R\_{\mathfrak{m}-\sigma\mathfrak{a}} - 2\Phi\_{\sigma\mathfrak{p}}R\_{\mathfrak{m}-\sigma\mathfrak{p}} + \Phi\_{\mathfrak{c}}R\_{\mathfrak{m}-\mathfrak{c}\mathfrak{y}} = 0\\ \Phi\_{\sigma\mathfrak{a}}R\_{\mathfrak{m}-\sigma\mathfrak{a}} - 2\Phi\_{\mathfrak{c}}R\_{\mathfrak{m}-\mathfrak{c}\mathfrak{l}} + 2\Phi\_{\sigma\mathfrak{s}}R\_{\mathfrak{m}-\sigma\mathfrak{s}} - \Phi\_{\mathfrak{c}}R\_{\mathfrak{m}-\mathfrak{c}\mathfrak{y}} = 0\\ \Phi\_{\sigma\mathfrak{a}} + \Phi\_{\sigma\mathfrak{p}} + \Phi\_{\mathfrak{c}} = \Phi\_{\mathbb{P}}\\ \Phi\_{\sigma\mathfrak{s}} + \Phi\_{\mathfrak{c}} = \Phi\_{\mathfrak{s}} \end{cases} \tag{2}$$

According to the reluctances, the additional leakage inductance *L*σa, primary leakage inductance *L*σp, and secondary leakage inductance *L*'σs (in primary) can be defined as

$$\begin{cases} L\_{\sigma \text{a}} = N\_{\text{P}}^2 / R\_{\text{m-}\sigma \text{a}} \\\ L\_{\sigma \text{p}} = N\_{\text{P}}^2 / 2R\_{\text{m-}\sigma \text{p}} \\\ L\_{\sigma \text{s}}' = N\_{\text{P}}^2 / 2R\_{\text{m-}\sigma \text{s}} \end{cases} \tag{3}$$

Since the main core is without air gaps, the reluctances *R*m-cl and *R*m-cy are much smaller than those of the air or the additional core and can be assumed to be zero. Thus, according to (1) and (2), the magnetic fluxes can be obtained.

$$\begin{cases} \begin{aligned} \Phi\_{\mathsf{C}} &= \frac{L\_{\mathsf{os}}' \Phi\_{\mathsf{P}} + \left(L\_{\mathsf{op}} + L\_{\mathsf{os}}\right) \Phi\_{\mathsf{s}}}{L\_{\mathsf{os}}' + L\_{\mathsf{op}} + L\_{\mathsf{op}}}\\ \Phi\_{\mathsf{o}\mathsf{a}} &= \frac{L\_{\mathsf{os}} \left(\Phi\_{\mathsf{P}} - \Phi\_{\mathsf{s}}\right)}{L\_{\mathsf{os}}' + L\_{\mathsf{op}} + L\_{\mathsf{op}}}\\ \Phi\_{\mathsf{o}\mathsf{p}} &= \frac{L\_{\mathsf{op}} \left(\Phi\_{\mathsf{P}} - \Phi\_{\mathsf{s}}\right)}{L\_{\mathsf{os}}' + L\_{\mathsf{op}} + L\_{\mathsf{op}}}\\ \Phi\_{\mathsf{o}\mathsf{s}} &= \frac{-L\_{\mathsf{os}}' \left(\Phi\_{\mathsf{P}} - \Phi\_{\mathsf{s}}\right)}{L\_{\mathsf{os}}' + L\_{\mathsf{op}} + L\_{\mathsf{op}}} \end{aligned} \tag{4}$$

Take typical DAB excitation as an example: the duty of two H-bridges *D*<sup>h</sup> = 0.5; phaseshifting duty 0 < *D*s < 0.5. Then the voltage excitations of MFT, *u*p and *u*s, are given as

$$\begin{aligned} u\_{\mathbb{P}}(t) &= \begin{cases} \, \, \, \, \mathcal{U}\_{\mathrm{d1}\prime} \, [0, 0.5T] \\\, \, \, \, \, \, \, \, \, [0.5T, T] \\\, u\_{\mathrm{s}}(t) &= \frac{\mathcal{U}\_{\mathrm{d1}\prime}}{\mathcal{U}\_{\mathrm{d1}}} u\_{\mathbb{P}}(t - D\_{\mathrm{s}}T) \end{cases} \end{aligned} \tag{5}$$

where *U*d1 is the DC bus voltage of the primary side, and *U*d2 is the DC bus voltage of the secondary side.

The induced electromotive force *e*c(*t*) of the main core and *e*σa(*t*) of the additional core can be calculated according to Equations (1), (4), and (5).

$$-\boldsymbol{\varepsilon\_{c}}(t) = \begin{cases} \frac{L\_{\text{on}}' - \text{L}\_{\text{op}} - \text{L}\_{\text{on}}}{L\_{\text{on}}' + L\_{\text{op}} + L\_{\text{on}}} \text{Id}\_{\text{d1}}, [0, D\_{\text{s}}T] \\\\ \frac{-L\_{\text{on}}' + L\_{\text{on}} + L\_{\text{on}}}{L\_{\text{on}}' + L\_{\text{on}} + L\_{\text{on}}} \text{Id}\_{\text{d1}}, [0.5T, (0.5 + D\_{\text{s}})T] \\\\ -\text{Id}\_{\text{d1}}, [(0.5 + D\_{\text{s}})T, T] \\\\ -\text{e}\_{\text{ca}}(t) = \begin{cases} \frac{2L\_{\text{on}}}{L\_{\text{on}}' + L\_{\text{on}} + L\_{\text{on}}} \text{Id}\_{\text{d1}}, [0, D\_{\text{s}}T] \\\\ 0, [D\_{\text{s}}T, 0.5T] \\ -\frac{2L\_{\text{on}}}{L\_{\text{on}}' + L\_{\text{on}} + L\_{\text{on}}} \text{Id}\_{\text{d1}}, [0.5T, (0.5 + D\_{\text{s}})T] \\\\ 0, [(0.5 + D\_{\text{s}})T, T] \end{cases} \end{cases} \tag{7}$$

For example, the turns ratio of the MFT satisfies *N*p:*N*<sup>s</sup> = *U*d1:*U*d2, and the leakage inductances satisfies *L*σa + *L*σp + *L*'σs = 5*L*σp = 5*L*'σs. Then the voltage and flux waveforms with the phase-shifting duty of 0 and 0.25 are shown in Figure 5.

**Figure 5.** Voltage, induced electromotive force, and flux waveforms of CW ELII MFT under typical DAB excitation. (**a**) *D*<sup>h</sup> = 0.5, *D*<sup>s</sup> = 0; (**b**) *D*<sup>h</sup> = 0.5, *D*<sup>s</sup> = 0.25.

#### **3. Loss and Leakage Inductance Calculation**

Loss and inductance are important electrical parameters of MFT. In view of the structure particularity of the CW ELII MFT, this section discusses the calculation methods of the loss and the leakage inductance based on the magnetic circuit analysis above.

#### *3.1. Core Loss of CW ELII MFT*

The core loss of CW ELII MFT contains the main core loss and the additional core loss. Core loss under non-sinusoidal excitation can be calculated by the improved generalized Steinmetz equation (iGSE) [20], in which the peak value and the differential of the flux density are needed. Based on the magnetic circuit analysis and the assumption that *L*σ<sup>p</sup> = *L*'σs, the peak value and the differential of the flux densities can be derived from (6) and (7) as

$$B\_{\rm m} = \frac{\mathcal{U}\_{\rm d1}}{2N\_{\rm p}f A\_{\rm cv}} \left[ \frac{1}{2} - \frac{L\_{\rm crw}}{L\_{\rm int}} D\_{\rm s} \right] \tag{8}$$

$$B\_{\rm ma} = \frac{\mathcal{U}\_{\rm d1}}{N\_{\rm p} f A\_{\rm ncv}} \frac{L\_{\rm ora}}{L\_{\rm int}} D\_{\rm s} \tag{9}$$

$$\begin{aligned} \left| \frac{\mathrm{d}B\_{\mathrm{m}}}{\mathrm{d}t} \right| &= \begin{cases} \frac{I\_{\mathrm{Lo}}}{I\_{\mathrm{mi}}} \frac{\mathrm{d}I\_{\mathrm{d}}}{\mathrm{N}\_{\mathrm{P}}A\_{\mathrm{ev}}} \left[ 0 \, \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!$$

Based on the iGSE model [20], the engineering calculation model for the core losses under DAB excitation (*D*<sup>h</sup> =0.5) is given as

$$P\_{\rm c} = M\_{\rm c} k\_{\rm i} f^{\rm c-\beta} \left(\frac{lI\_{\rm d1}}{N\_{\rm P} A\_{\rm cv}}\right)^{\beta} \left(\frac{1}{2} - \frac{L\_{\rm crw} D\_{\rm s}}{L\_{\rm int}}\right)^{\beta - \alpha} \left[1 + 2D\_{\rm s} \left(\frac{L\_{\rm cra}^{\rm a}}{L\_{\rm int}^{\rm a}} - 1\right)\right] \tag{12}$$

$$P\_{\rm ca} = M\_{\rm ac} k\_{\rm i} 2^{\delta + 1} f^{a - \beta} \left( \frac{\mathcal{U}\_{\rm d1}}{N\_{\rm P} A\_{\rm acv}} \frac{L\_{\rm cra}}{L\_{\rm int}} \right)^{\beta} D\_{\rm s}^{1 - a + \beta} \tag{13}$$

$$k\_{\rm i} = \frac{k}{\left(2\pi\right)^{a-1} \int\_0^{2\pi} |\cos\theta|^a 2^{\beta - a} d\theta} \tag{14}$$

where *P*c is the loss of the main core; *P*ca is the loss of the additional core; *M*c is the effective weight of the main core; *M*ac is the effective weight of the additional core; *f* is frequency, *k*<sup>i</sup> is the iGSE coefficient. *k*, *α*, and *β* are Steinmetz coefficients, which are extracted from sinusoidal measurement data of a gapped nanocrystalline core sample, which is shown in Figure 6.

The experiment is carried out with the core sample, and the calculated value and the measured value are compared in Figure 6, verifying the effectiveness of the model.

**Figure 6.** Comparison of experimental and calculated losses of the main core (*D*<sup>h</sup> = 0.5, *D*<sup>s</sup> = 0).

#### *3.2. Winding Loss*

In order to calculate the winding losses under non-sinusoidal excitation, the current is usually decomposed by Fourier transform, and the winding loss is given as

$$P\_{\rm Cu} = \sum\_{i=1}^{\infty} I\_{\rm RMS-i}^2 R\_{\rm AC-i} = \sum\_{i=1}^{\infty} I\_{\rm RMS-i}^2 F\_{\rm R-i} R\_{\rm DC} \tag{15}$$

where *I*RMS-*<sup>i</sup>* is the *i*th harmonic components of the load current; the ac resistance coefficient *F*<sup>R</sup> can be obtained by the Dowell model [21] as

$$F\_{\mathbb{R}} = \Delta \left[ \frac{\sin \text{h} 2\Delta + \sin 2\Delta}{\cos \text{h} 2\Delta - \cos 2\Delta} + \frac{2\left(m^2 - 1\right)}{3} \frac{\sin \text{h} \Delta - \sin \Delta}{\cos \text{h} \Delta + \cos \Delta} \right] \tag{16}$$

$$
\Delta = d\_{\rm c} \sqrt{\pi f \sigma \mu\_0 \frac{nh\_{\rm c}}{h\_{\rm wd}}} \tag{17}
$$

where, *m* is the number of winding layers, *d*c is the thickness of conductor, *h*c is the height of single conductor, Δ is the ratio of conductor thickness to penetrated depth; *σ* is conductivity, *f* is frequency, *μ*<sup>0</sup> is the vacuum permeability, *n* is the number of conductors per layer, and *h*wd is the window height.
