Ferrites and Nanocrystalline Alloys Applied to DC–DC Converters for Renewable Energies

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To get new DC–DC converters with higher power density.

Abstract

The medium frequency transformer (MFT) with nanocrystalline alloys is quintessential in new DC–DC converters involved in various front-end applications. The center piece to achieve high-performance, efficient MFTs is the core. There are various options of core materials; however, no deep information is available about which material characteristics and design procedure combo are best to get high performance MFTs while operating at maximal power density. To provide new insights about interrelation between the selection of the core material with the compliance technical specifications, differently to other proposals, this research work aims to design and build, with the same methodology, two MFT prototypes at 20 kHz, with nanocrystalline and ferrite cores, to highlight power density, and overall performance and cost, as matching design criteria. As the experimental results show, a nanocrystalline core has the highest power density (36.91 kW/L), designed at 0.8 T to obtain low losses at 20 kHz, achieving an efficiency of 99.7%. The power density in the ferrite MFT is 56.4% lower than in the nanocrystalline MFT. However, regarding construction cost, the ferrite MFT is 46% lower, providing this a trend towards low-cost DC–DC converters. Finally, high power density in MFTs increases the power density of power DC–DC converters, which have relevant applications in fuel cell-supplied systems, renewable energies, electric vehicles, and solid-state transformers.

1. Introduction

1.1. Motivation and Incitement

Medium frequency transformer, MFTs, are fundamental to the development of newly advanced DC–DC converters [1,2,3,4]. These converters are becoming key player in building Smart-Grid environments [5], in interfacing renewable energy to medium power grids [6,7,8,9], in constructing the emerging solid-state transformer technology [5,10], and in electric vehicles, as shown in Figure 1. Better performance MTFs are needed for improving such converters. Therefore, an important point is the selection of the core material in the pursuit of achieving high power density and excellent efficiency of the MFT [11,12,13,14,15].

1.2. Literature Review

The design of MFTs (600 to 20 kHz) [9,10,16,17] always involve a core material in the procedure. The core materials can be nanocrystalline alloys [16,17], ferrites [18], or silicon steel [9,10]. Each of these materials is mainly selected for a specific frequency range. Silicon steel is used between 600 Hz [9], and 1 kHz [10]. Nanocrystalline alloys are used between 1 kHz [19] and 5 kHz [16,17] and ferrites from 20 kHz and up [18]. At its respective frequency, each material has high efficiency (>98%). Besides efficiency, each material has additional interesting characteristics. For instance, silicon steel is a low-cost material, it is highly available in the market, and there is plenty of information available for designers for it. In the case of ferrite, this is also a low-cost material, but also, it has high power density at high frequencies. Different from the previous materials, nanocrystalline alloys are a high-cost material (considering per unit of volume) compared to ferrites and silicon steel. However, it has higher permeability, lower core losses, and a higher design flux density in the 1 to 5 kHz range [16,17,19].

MFT designers rely on the latest information from the most recent research works in order to select the most suitable core material for obtaining the best behavior at given frequency and power levels of MFTs [20].

Related to cutting-edge information, nanocrystalline alloys and silicon steel MFTs are compared at 1 kHz [19]. In this situation, the MFT with nanocrystalline core shows higher power density and 7.8 times less weight. Nanocrystalline alloys cores are designed for 0.9 T flux density and silicon steel for 0.1 T. Although the cost per cm³ of the nanocrystalline material (87.62 USD/kg) is higher than silicon steel (11.71 USD/kg), the final total cost of both MFTs is similar because of smaller size of the MFT obtained with nanocrystalline alloys, USD 395 for the silicon steel MFT, and USD 375 for the nanocrystalline one.

Silicon steel is not commonly considered for MFT designs at frequencies higher than 1 kHz because its core losses sharply increase beyond this frequency. As a consequence, the power density decreases, and the size of the MFT increases due a lower flux density. On the other hand, in the case of nanocrystalline cores, MFTs are mostly designed at medium frequency range (5 kHz) [16,17]. At this range, both high-efficiency and high power densities are obtained. To the best of the Author’s knowledge, currently, there are no research works presenting any kind of information about physical prototypes of nanocrystalline alloys MFTs designed and operating at 20 kHz.

Table 1. Comparison of medium frequency transformers with nanocrystalline core.

Reference	Frequency (kHz)	Bac (T)	Core Material	Power (kVA)	Efficiency (%)	Power Density (kW/L)
[19]	1	0.9	Nanocrystalline	1	80.2	2.50
[21]	10	0.46	Nanocrystalline	200	99.4	8.00
[17]	5	0.9	Nanocrystalline	50	99.5	11.50
[16]	5	0.9	Nanocrystalline	1	99.4	15.01
[15]	10	0.3	Nanocrystalline	5	96	19.90
[22]	10	0.3	Nanocrystalline	35	99.2	23.30
[14]	10	0.3	Nanocrystalline	35	98	23.30
This proposal	20	0.8	Nanocrystalline	1	99.7	36.91

shows a comparison of transformers with a nanocrystalline core in the medium frequency range. Therefore, it is important to compare the MFT prototypes of nanocrystalline alloys and ferrites at 20 kHz, in order to find a high power density in MFTs with high efficiency. This article presents the design, the experimental results of prototypes, and the comparison of results between these two MFTs, as well as a discussion about the power density of the MFT obtained in this document and the power density obtained in previous works.

Ferrites can manage flux densities of 0.2–0.3 T. Although these are low flux densities values compared to those of nanocrystalline alloys, ferrites have generally been used at 10 kHz and up while nanocrystalline alloys usually at 5 kHz with power density 1.5 times higher [16]. Therefore, from the point of view of research, the design and construction of nanocrystalline alloys MFTs at 20 kHz, as well as comparing the performance of this MFT with that of the ferrite MFT, at such frequency, are of great interest. The strong and weak points of the nanocrystalline MFTs at such a frequency is unknown.

1.3. Contribution and Paper Organization

The results of this research work provide clear information to MFT designers to obtain more efficient (>98%), lower cost, and higher power density transformers, applied to DC–DC converters.

This paper is organized as follows: Section 2 presents the description of magnetic materials, and the MFTs design. Section 3 presents the core dimensions, core losses, winding losses, and calculation of the dispersion inductance. Section 4 presents the simulations of the behavior of both transformers, with nanocrystalline alloys and with ferrites. Section 5 presents the experimental prototypes of the two transformers tested with the Full-Bridge converter. Finally, Section 6 presents a discussion of the results, and Section 7 has our conclusions.

2. MFT Design and Magnetic Materials

2.1. MFTs Design

Currently, there are several MFT design methods [9,10,16,17,18]. Each method has a specific contribution and main emphasis on specific characteristics of the MFT, such as temperature, core losses, core cost, and power density, among others. Each method is commonly verified, implementing and testing an experimental prototype designed with a specific material (nanocrystalline alloys, ferrites, or silicon steel).

For this research work, the design method is based on [16], in order to obtain a high power density, and therefore increase the power density of DC–DC converters that use MFTs, in addition to increasing the power density of systems that use this type of converter, such as photovoltaic and wind systems, electric vehicles, and solid state transformers. Figure 2 presents the flow diagram of the design procedure used in this work.

Two MFTs are designed, one with nanocrystalline alloy core, and another with a ferrite core both at to 20 kHz, 1 kVA, and 120/240 V.

2.2. Magnetic Materials

Magnetic materials of cores are key in defining the performance of MFTs, its losses, and its nominal frequency of operation. Parameters and characteristics of magnetic materials such as permeability and flux density are two main elements to be considered by MFT designers. In the present work, two core materials suitable for the medium frequency range, nanocrystalline alloys, and ferrites, are used for designing and implementing MFTs. The main characteristics of nanocrystalline alloys and ferrites important for the design of MFTs are presented in

Table 2. Characteristics of ferrites and nanocrystalline alloys for MFT design.

Parameter	Material 1	Material 2
Material	Ferrite (3C90)	Nanocrystalline (Vitroperm 500 F)
Core	Block core	Laminate (0.02 mm)
Maximum flux density	0.35 T	1.2 T
Permeability	2000–4000	15,000–150,000
Cost	Low	High

.

As presented in Table 2, both the maximum flux density and the permeability are higher in nanocrystalline alloys than in ferrites, but the latter has lower cost per unit volume (USD/cm³). It should also be noted that ferrites have flux densities of 0.2–0.3 T at 20 kHz [18], while nanocrystalline alloys are 0.9 T at 5 kHz [16,17]. Then, a question arises, is the performance, efficiency, cost, and size of a nanocrystalline alloy MFT better than a ferrite MFT at 20 kHz. Although nanocrystalline alloy cores tend to achieve higher power density, the low cost of ferrites gives these materials a chance to a lower-cost MFT prototype.

3. Core Dimensions, Core Losses, and Winding Losses

The core dimensions, core losses, winding losses, and dispersion inductances of each MFT are calculated in this section. This inductance is a crucial parameter for power flow control in Dual-Active-Bridge (DAB) DC–DC converter.

Afterward, in Section 5, the experimental results from both laboratory prototypes are compared. This comparison provides MFT designers with useful, cutting-edge information to better select the most suitable core material, and obtain new, improved DC–DC converters with higher power density, lower cost, and smaller size.

3.1. Geometric Dimensions

The geometric dimensions of the core obtained with each material can be used to estimate the final power density in the MFT. The calculation of the core dimensions is obtained based on the input power, turns ratio, MFT frequency, design flux density, and the minimum required efficiency. These parameters are reflected in the transversal section area of the MFT (A_c). This area confines the magnetic flux of both windings, its precise calculation represents a tendency to obtain an MFT with high performance, favorable temperature conditions, high power density, and high efficiency for a specific frequency. In addition, from this result, the amount of material required in the core for the initial design power and frequency is obtained [16].

$$A_c=\frac{\left(P_{in}+\frac{P_{in}}{{\eta }_1}\right)·{10}^4}{k_f{k}_{u}Jf{W}_a{B}_{ac}}$$

(1)

where P_in is the input power in the MFT, ${\eta }_1$ is the minimum efficiency required to obtain, K_f is the waveform factor (4.44 for sine waves, 4.0 for square waves), K_u is the window utilization factor (0.4), J is the current density, f is the frequency, B_ac is the design flux density, and W_a the window area.

For B_ac, two conditions must be met. The first condition is that B_ac < B_max (B_max, maximum flux density of the material). The second condition is that given a B_ac and frequency the specific power losses (w/kg) are obtained, which must meet the following condition.

$$\frac{P_{fe1}}{kg}·\left(Weight\right)<1\%Pin$$

(2)

where P_fe1 are the specific power losses calculated and weight are the kg obtained from the MFT after Equation (1). From the initial condition of efficiency greater than 98%, due to this, a maximum of 1% is designated for losses in the core and 1% for losses in the winding.

Figure 3 shows the dimensions of the nanocrystalline alloy and ferrite cores.

The nanocrystalline core obtained is 73.26% smaller than the ferrite core. The reason for this is the different value the flux density of each material.

The cores are toroidal type because they are readily available for both materials, and also because its lower dispersed flux associated with the geometry compared to the CC or EE core types. Higher flux density means higher power density in the MFT [19]. The calculation of the number of turns in the MFT of the primary winding (N_p) and secondary winding (N_s) is carried out employing Equations (3) and (4), respectively.

Table 3. Volume and weigh of both MFTs.

Parameter	MFT with Ferrite Core	MFT with Nanocrystallyne Core
Vcore	22.03 cm3	5.89 cm3
Vwinding	39.95 cm3	21.20 cm3
Vtotal	61.98 cm3	27.09 cm3
Wcore	100 grs	32.9 grs
WMFT	238 grs	102 grs

presents a comparison of the winding volume (V_winding), core volume (V_core), core weight (W_core), and total weight (W_MFT) of each MFT designed.

$$N_p=\frac{V_{in}·{10}^4}{k_f{B}_{ac}f{A}_c}$$

(3)

$$N_s=\frac{N_p·V_0}{V_{in}}$$

(4)

where V_in is the input voltage, and V_o is the input voltage.

The winding volume of the nanocrystalline MFT is 46.93% lower than the ferrite MFT. The total calculated volume of the MFT with a nanocrystalline core is 56.29% lower than the MFT with a ferrite core.

3.2. Core Losses and Winding Losses

At medium frequencies, high permeability is always desirable. According to Table 2, nanocrystalline alloys have a higher flux density and higher permeability.

Table 4. Power density in kW/L of the MFTs.

MFT	Flux Density	Power Density (kW/L)
Ferrite core	0.2 T	16.13
Nanocrystalline core	0.8 T	36.91

shows the power density in kW/L for the flux density of the MFTs designed for each material.

Core losses and winding losses of both MFTs are calculated using the method presented in [16].

Winding losses:

$$L_{\omega }=L_p+L_s$$

(5)

$$L_p=\left(I_{in}^2\right)·R_1$$

(6)

$$L_s=\left(I_o^2\right)·R_2$$

(7)

where L_ω are the winding losses in watts, L_p losses in the primary winding, L_s losses in the secondary winding, I_in primary winding current, I_o secondary winding current, R₁ primary winding resistance, and R₂ secondary winding resistance.

R₁ y R₂ are calculated with Equations (8) and (9), respectively.

$$R_1=\left(ML{T}_1·N_p·\mu \mathsf{\Omega }/c{m}_1\right)·{10}^{-6}$$

(8)

$$R_2=\left(ML{T}_2·N_s·\mu \mathsf{\Omega }/c{m}_2\right)·{10}^{-6}$$

(9)

where MLT₁ and MLT₂ are the mean lengths of the primary and secondary windings, respectively; $\mu \mathsf{\Omega }/c{m}_1$ and $\mu \mathsf{\Omega }/c{m}_2$ are the resistances per centimeter of the primary and secondary winding conductors, respectively.

The MFT with ferrite core features an R₁ = 0.0300 Ω, R₂ = 0.1222 Ω, L_p = 2.52 W, L_s = 2.31 W, and an L_ω = 4.83 W. The MFT with nanocrystalline core has an R₁ = 0.0114 Ω, R₂ = 0.0465 Ω, L_p = 0.88 W, L_s = 0.81 W, and L_ω = 1.69 W.

Core losses:

Figure 4 shows core losses (W/kg) versus flux density for the nanocrystalline material used in this paper (Vitroperm 500F), for a frequency of 20 kHz [23]. In [23], a more detailed characterization of these nanocrystalline materials is presented for a wider range of frequencies.

The core losses of the MFT are calculated by Equation (10).

$$P_{fe}=(P_{fe1})·W_{fe}$$

(10)

where P_fe1 are the material losses (W/kg) and W_fe the core weight (kg). In this case for nanocrystalline alloys, P_fe1 is 40 W/kg for 20 kHz), and W_fe = 0.033 kg, this specified for 1 kVA. It can be notice that P_fe is directly proportional to the frequency and the flux density.

These results are subsequently verified with experimental testing in Section 5.

Table 5. Losses calculated of the MFTs.

MFT	MFT with Ferrite Core	MFT with Nanocrystalline Core
Lω	4.83 W	1.69 W
Pfe	2.29 W	1.32 W
Ltot	7.12 W	3.01 W
Efficiency	99.28%	99.69%

shows the core losses (P_fe), the winding losses (L_ω), and the total losses (L_tot) of both MFTs at 20 kHz and 1 kVA.

The MFT with nanocrystalline core has 65% fewer winding losses than the ferrite MFT. Regarding core losses, the nanocrystalline MFT has almost 42% fewer losses than the ferrite MFT. Finally, the total losses of nanocrystalline MFT are 3.01 W for an efficiency of 99.69%. In the case of the ferrite, MFT total losses are 7.12 W, and efficiency is 99.28%. The total losses and efficiency obtained in these designs are experimentally verified in Section 5.

3.3. Temperature Rise

The temperature rise in MFTs is crucial for avoid overheating and damage to the MFT. The temperature rise is calculated using the following equation:

$$T_t=450{\left(\frac{P_{tot}}{A_t}\right)}^{0.826}$$

(11)

where P_tot are the total losses in watts, A_t is the surface area of the transformer in cm², and T_t is the temperature rise in Celsius (°C) [16].

3.4. Calculation of the Dispersion Inductance

The magnitude of the dispersion inductance (L_d) in the MFT is crucial determining the control and operation range of the DC–DC converters with DAB [24].

The dispersion inductance represents the dispersed flow of the transformer observed from an equivalent circuit. In the case of MFTs, a low dispersion inductance represents a lower dispersed flow, which results in a tendency to obtain high efficiency. The calculation of this parameter is presented and verified using the finite element method in [24] to assure high precision in the calculation. The classical equation to calculate L_d and a new equation presented in [24] are shown in Equations (12) and (13), respectively. Equation (13) presents a higher precision [24]. In this document, L_d is calculated with Equation (13).

$$L_d={\mu }_{0}ML{T}_{pri}\frac{m_1^2{N}_{l1}^2}{h_{\omega }}\left[d_{iso}+\frac{m_1{d}_{pri}+\left(m_1-1\right)d_{pri}+m_2{d}_{sec}+\left(m_2-1\right)d_{ins2}}{3}\right]$$

(12)

$$\begin{array}{cc}\hfill L_d={\mu }_0\frac{N_{L1}^2}{h_{\omega }}{m}_1 [& ML{T}_{iso}{m}_1{d}_{iso}+ML{T}_{pri}\frac{\left(m_1-1\right)\left(2m_1-1\right)}{6}{d}_{ins1}+ML{T}_{sec}\frac{m_1\left(m_2-1\right)\left(2m_2-1\right)}{6m_2}{d}_{ins2}\hfill \\ & +ML{T}_{pri}\frac{\sin \left(\frac{2{∆}_1}{\alpha \delta }\right)4\alpha {\delta }^2\left(m_1^2-1\right)+4d_{pri}\left(2m_1^2+1\right)}{24{\left(\sin \frac{2{∆}_1}{\alpha \delta }\right)}^2}\hfill \\ & -ML{T}_{pri}\frac{\alpha {\delta }^2\left(\frac{4{∆}_1}{\alpha \delta }\right)\left(2m_1^2+1\right)-8d_{pri}\left(1-m_1^2\right)\cos \left(\frac{2{∆}_1}{\alpha \delta }\right)}{24{\left(\sin \frac{2{∆}_1}{\alpha \delta }\right)}^2}\hfill \\ & +ML{T}_{sec}\frac{m_1}{m_2}\frac{\sin \left(\frac{2{∆}_2}{\alpha \delta }\right)4\alpha {\delta }^2\left(m_2^2-1\right)+4d_{sec}\left(2m_2^2+1\right)}{24{\left(\sin \frac{2{∆}_2}{\alpha \delta }\right)}^2}\hfill \\ & -ML{T}_{sec}\frac{m_1}{m_2}\frac{\alpha {\delta }^2\sin \left(\frac{4{∆}_2}{\alpha \delta }\right)\left(2m_2^2+1\right)}{24{\left(\sin \frac{2{∆}_2}{\alpha \delta }\right)}^2}+ML{T}_{sec}\frac{m_1}{m_2}\frac{8d_{sec}\left(1-m_2^2\right)\cos \left(\frac{2{∆}_2}{\alpha \delta }\right)}{24{\left(\sin \frac{2{∆}_2}{\alpha \delta }\right)}^2}]\hfill \end{array}$$

(13)

where:

${\mu }_0$= vacuum permeability	diso = isolation distance
dins1 = insulation distance between the layers of the primary	NL1 = turns per layer
dins2 = insulation distance between the layers of the secondary	hw = winding height
m1 = number of layers in the primary	dpri = thickness of the primary
m2 = number of layers in the secondary	dsec = thickness of the secondary
MLTiso = mean length of the isolation distance	∆1 = penetration ratio of the primary, ${∆}_1=\frac{d_{pri}}{\delta }$
MLTpri = mean length turns of primary portion	∆2 = penetration ratio of the primary, ${∆}_2=\frac{d_{sec}}{\delta }$
MLTsec = mean length turns of secondary portion	$\alpha =\frac{1+j}{\delta }$ where $\delta$ is the skin depth

Table 6. Dispersion inductance calculated in both MFTs.

MFT	MFT with Ferrite Core	MFT with Nanocrystallyne Core
Ld	2.06 µH	1.01 µH

shows the calculated values of L_d for the two transformers. Figure 5 show the curves of L_d values in a range from 600 Hz a 20 kHz for both MFTs, they were obtained with the “Model 891, 300 kHz Bench LCR Meter–B&K Precision”.

According to Table 6, the MFT with a nanocrystalline core has 51% lower L_d than the MFT with ferrites. The precise calculation of the L_d is essential to obtain a representative equivalent circuit of the MFT, and thus verify their operation in simulation, as presented in Section 4.

4. Simulation

The equivalent circuit of the MFT is implemented in Simulink-MATLAB. Figure 6 shows the Full-Bridge converter and the equivalent circuit of the MFT. Both MFTs are tested with square waves.

The parameters, input voltages, operating frequency, and load for the simulations are presented in

Table 7. Parameters of the simulation model.

MFT	MFT with Ferrite Core	MFT with Nanocrystallyne Core
Vin	120 V	120 V
f	20 kHz	20 kHz
Load	60 Ω	60 Ω
Ld1	2.06 µH	1.01 µH
Ld2	9.09 µH	4.06 µH
R1	0.0300 Ω	0.0114 Ω
R2	0.1222 Ω	0.0465 Ω
Lm	10.2 mH	28.7 mH
Rm	6288 Ω	10,909 Ω

. In the equivalent circuit, R₁ and R₂ are the resistances of the primary and secondary winding, respectively, L_m is the magnetization inductance, R_m is the magnetization branch resistance, and L_d1 and L_d2 are the primary and secondary dispersion inductances.

Figure 7a,b shows the input and output voltages and currents of the nanocrystalline and ferrite MFTs, respectively.

Table 8. Input and output voltages and currents for MFTs.

MFT	MFT with Ferrite Core	MFT with Nanocrystallyne Core
Vin	120 V	120 V
Vo	235.4 V	237.8 V
Iin	7.852 A	7.936 A
Io	3.924 A	3.963 A
Efficiency	98.03%	98.96%

shows the values of each input/output signal in both MFTs as well as the efficiency obtained in each case.

Figure 7 and Table 8 evidence that the nanocrystalline MFT has better performance and efficiency than the ferrite MFT.

The high efficiency obtained from both MFTs (>98%) opens the possibility of using any of both materials in high-performance applications, but comparing parameters of high impacts, such as the power density and construction cost, makes the difference, as shown in Section 5.

5. Experimental Results

The values of the parameters for both MFT prototypes are obtained from the methodology presented in [16] and are listed in

Table 9. Construction parameters of the MFTs.

MFT	MFT with Ferrite Core	MFT with Nanocrystallyne Core
Number of phases	1-phase	1-phase
Core type	Toroidal (3C90)	Toroidal (Vitroperm 500 F, W514)
Core dimensions	5.1 × 3.2 × 1.9 cm	3 × 2 × 1.5 cm
Number of turns of primary winding	42 turns	22 turns
Number of turns of secondary winding	85 turns	45 turns
Primary winding caliber	13 AWG	13 AWG
Secondary winding caliber	16 AWG	16 AWG
Flux density	0.2 T	0.8 T
Permeability	3456	53,355

.

The flux density obtained is higher in the nanocrystalline MFT, 0.8 T, than in the ferrite MFT, 0.2 T. This latter MFT has larger dimensions core and lower total power density. The total number of turns of the windings of the nanocrystalline MFT is 47.7% lower compared to the ferrite MFT. The nanocrystalline and ferrite cores are shown in Figure 8. The MFT prototypes are shown in Figure 9.

Both MFTs are tested at 120 V/1 kVA/20 kHz with the typical square waves presented in the DC–DC converters, using a full H-Bridge converter. The equipment used is a DC variable source from 0 to 120 V, DSP, 12 V_DC Source, MFT, load, and oscilloscope, as shown in Figure 10. In Figure 11, a block diagram is shown that represents the experimental configuration of Figure 10. Figure 12, Figure 13 and Figure 14 show experimental results at 6%, 25%, and 50% of the nominal power. The input and output voltages and currents resulted from experimentation with both MFTs are shown in Figure 15 at full load.

The comparison between voltages and currents of both MFTs prototypes at full load (Figure 15) are presented in

Table 10. Input and output voltages and currents, and efficiency obtained at full load.

MFT	MFT with Ferrite Core	MFT with Nanocrystallyne Core
Vin	119 V	120 V
Vo	247 V	245 V
Iin	7.95 A	7.44 A
Io	3.75 A	3.64 A
Efficiency	97.9%	99.8%

, along with the corresponding efficiency values.

Table 11. Efficiencies obtained at 50%, 25%, and 6% of the nominal power.

Pnom	MFT with Ferrite Core	MFT with Nanocrystallyne Core
6%	93.5%	98.4%
25%	98.1%	99.8%
50%	97.7%	99.6%

shows the efficiencies obtained at 6%, 25%, and 50% of the nominal power. Figure 16 shows a thermography of the MFT prototype, taken with a thermal camera (Milwaukee M12TM 7.8 KP). In this case, the maximum temperature was 73.7 °C for ferrite, and 46.4 °C for nanocrystalline alloys. The test was realized at 28 °C room temperature for 1 h.

The high efficiency of the nanocrystalline MFT is 99.8%, and 97.9% of the ferrite MFT. The other two parameters of great interest are the power density and the construction cost of each MFT. A comparison of these parameters is shown in

Table 12. MFTs construction cost.

MFT	MFT with Ferrite Core	MFT with Nanocrystallyne Core
Power density	16.1 kW/L	36.9 kW/L
Core cost	USD 13.86	USD 28.24
Winding cost	USD 1.86	USD 0.99
Total cost of MFT	USD 15.72	USD 29.23

.

The power density in the ferrite MTF is 56.4% lower than in the nanocrystalline MFT. Regarding cost, the ferrite core is 50.82% lower compared to the nanocrystalline core, and a total transformer cost of 46.12% lower. Therefore, from the point of view of power density, for the upper corner of the medium frequency range, 20 kHz, it is higher in the nanocrystalline MFTs. However, regarding construction cost, the ferrite MFT is lower. This comparison result gives a new guideline to MFT designers useful to select the core material most suitable according to the application requirements, for instance, to obtain novel DC–DC converters of higher power density or low construction cost for fuel cell supplied systems, solid state transformers, electric vehicles, wind systems, and photovoltaic systems.

6. Discussion

Table 13. MFTs with high power density within the mid-frequency range (600–20 kHz).

Reference	Material	Bac (T)	Frequency (kHz)	Power (kVA)	Efficiency (%)	Year	Power Density (kW/L)
[9]	Silicon Steel	0.6	0.6	0.8	99	2017	1.29
[19]	Nanoc./Silic. Steel	0.9/0.1	1	1	80.2/99.1	2019	2.50/0.25
[10]	Silicon Steel	0.5	1	35	99.4	2017	2.96
[21]	Nanocrystalline	0.46	10	200	99.4	2020	8.00
[18]	Ferrite	0.35	20	10	99.2	2017	9.25
[17]	Nanocrystalline	0.9	5	50	99.5	2016	11.50
[16]	Nanocrystalline	0.9	5	1	99.4	2018	15.01
[15]	Nanocrystalline	0.3	10	5	96	2017	19.90
[22]	Nanoc./Ferrite	0.3/0.2	10	35	99.2/99.5	2016	23.30/11.7
[14]	Nanocrystalline	0.3	10	35	98	2017	23.30
This proposal	Nanocrystalline	0.8	20	1	99.8	2021	36.91

presents relevant recent research focused on MFTs with high power density and in the mid-frequency range (600–20 kHz).

The flux density in the silicon steel MFTs at 1 kHz [10], 0.5 T, is higher than the one in the ferrite MFT at 20 kHz [18], 0.2 T; therefore, the former would be 2.5 times smaller in dimensions than the latter, if both were designed at the same frequency. Nevertheless, ferrites have a relative advantage compared to silicon steel in this case, higher flux density at higher frequency (>10 kHz); however, the former is preferable for designs in the medium frequency range, around 20 kHz.

Collaterally, a relatively higher power density can be reached with the ferrite MFT, as shown in Table 13, as a result of designing the MFT at a higher frequency. In [19], a comparison of two MFTs, designed for the same frequency (1 kHz), one with nanocrystalline alloys and silicon steel the other, the first MFT evidences better characteristics in terms of power density, total cost, and efficiency. Moreover, in the 5 kHz range, the design option is also nanocrystalline alloys [16,17].

However, around the 20 kHz range, ferrite cores [18] have been the standard option. In addition to the main objective of obtaining a high power density MFT, the work presented in this paper challenges and tackles this presumption by designing and building MFTs with nanocrystalline cores at 20 kHz, and comparing performances with a ferrite MFT.

The analysis and results presented in this paper highlight the better performance, 56.4% higher in terms of the power density, for instance, of the nanocrystalline MFT compared to the ferrite MFT. However, in terms of construction cost, the ferrite MFT is 46% lower than nanocrystalline MFT. Regarding efficiency, both transformers have an efficiency higher than 98% at 20 kHz and 1 kVA.

On the other hand, nanocrystalline alloys have also been designed at a frequency of 10 kHz [14,15,21,22], obtaining high efficiencies (>98%) [14,21,22], and power densities of up to 23.3 kW/L [14,22]. As shown in Table 13, in the present proposal a transformer efficiency of 99.8% and a power density of 36.91 kW/L are obtained. The present power density obtained from the MFT contributes to the development of a higher power density in DC–DC converters. These converters are key in applications such as fuel cell supplied systems, electric vehicles, solid state transformers, and photovoltaic systems.

7. Conclusions

The analysis and construction of MFTs are key to developing new advanced DC–DC converters in terms of maximal power density, low cost, and greater efficiency. The core materials and the design process essential in defining the characteristics and performance of MFTs.

The nanocrystalline MFT has 2.3 times higher power density than the ferrite MFT. Regarding construction costs, the ferrite-core MFT costs are half that of the nanocrystalline ones. Both transformers have high efficiency (>98%). The nanocrystalline-core MFT in this work surpasses those presented in cutting-edge literature in terms of power density, 36.91 kW/L, and efficiency, 99.7%. Therefore, this works contributes to putting a step forward in obtaining DC–DC converters better suited to modern day applications such as fuel cell supplied systems, photovoltaic systems, electric vehicles, and power electronic transformer (solid-state transformers).