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

<jats:p>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.</jats:p>

section_content: 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]. 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 cm 3 of the nanocrystalline
section_name: Introduction 1.Motivation and Incitement

section_content: 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 cm 3 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 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.
section_name: Literature Review

section_content: 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.
section_name: Contribution and Paper Organization

section_name: MFT Design and Magnetic Materials

section_content: 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.
section_name: MFTs Design

section_content: 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.
section_name: MFTs Design

section_content: 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. 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.
section_name: Magnetic Materials

section_content: 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. 

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/cm 3 ). 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. 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.
section_name: Magnetic Materials

section_content: 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]. 

where P in is the input power in the MFT, η 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. 

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 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. 

(3)

where V in is the input voltage, and V o is the input voltage. 

Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 18 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 (Np) and secondary winding (Ns) is carried out employing Equations ( 3 ) and ( 4 ), respectively. Table 3 presents a comparison of the winding volume (Vwinding), core volume (Vcore), core weight (Wcore), and total weight (WMFT) of each MFT designed. 

where Vin is the input voltage, and Vo 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.
section_name: Geometric Dimensions

section_content: At medium frequencies, high permeability is always desirable. According to Table 2, nanocrystalline alloys have a higher flux density and higher permeability. Table 4 shows the power density in kW/L for the flux density of the MFTs designed for each material. 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.
section_name: Core Losses and Winding Losses

section_content: At medium frequencies, high permeability is always desirable. According to Table 2, nanocrystalline alloys have a higher flux density and higher permeability. Table 4 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:

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, R 1 primary winding resistance, and R 2 secondary winding resistance. R 1 y R 2 are calculated with Equations ( 8 ) and ( 9 ), respectively. 

where MLT 1 and MLT 2 are the mean lengths of the primary and secondary windings, respectively; µΩ/cm 1 and µΩ/cm 2 are the resistances per centimeter of the primary and secondary winding conductors, respectively. The MFT with ferrite core features an R 1 = 0. 0300 Ω, R 2 = 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 R 1 = 0. 0114 Ω, R 2 = 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. 

sented in [16]. 

Winding losses:

= ( ) • (7) where Lω are the winding losses in watts, Lp losses in the primary winding, Ls losses in the secondary winding, Iin primary winding current, Io secondary winding current, R1 primary winding resistance, and R2 secondary winding resistance. R1 y R2 are calculated with Equations ( 8 ) and ( 9 ), respectively. 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). The core losses of the MFT are calculated by Equation (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 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.
section_name: Core Losses and Winding Losses

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

where P tot are the total losses in watts, A t is the surface area of the transformer in cm 2, and T t is the temperature rise in Celsius ( • C) [16].
section_name: Temperature Rise

section_content: 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). 

where:

µ 0 = vacuum permeability d iso = isolation distance d ins1 = insulation distance between the layers of the primary N L1 = turns per layer d ins2 = insulation distance between the layers of the secondary h w = winding height m 1 = number of layers in the primary d pri = thickness of the primary m 2 = number of layers in the secondary d sec = thickness of the secondary MLT iso = mean length of the isolation distance ∆ 1 = penetration ratio of the primary, ∆ 1 = d pri δ

MLT pri = mean length turns of primary portion ∆ 2 = penetration ratio of the primary, ∆ Table 6 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". = 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, ∆ = MLTpri = mean length turns of primary portion ∆2 = penetration ratio of the primary, ∆ = MLTsec = mean length turns of secondary portion = where is the skin depth Table 6 shows the calculated values of Ld for the two transformers. Figure 5 show the curves of Ld 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 Ld than the MFT with ferrites. The precise calculation of the Ld is essential to obtain a representative equivalent circuit of the MFT, and thus verify their operation in simulation, as presented in Section 4.
section_name: Calculation of the Dispersion Inductance

section_content: 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. In the equivalent circuit, R1 and R2 are the resistances of the primary and secondary winding, respectively, Lm is the magnetization inductance, Rm is the magnetization branch resistance, and Ld1 and Ld2 are the primary and secondary dispersion inductances. 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.
section_name: Simulation

section_content: 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. In the equivalent circuit, R 1 and R 2 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. 

tested with square waves. 

The parameters, input voltages, operating frequency, and load for the simulations are presented in Table 7. In the equivalent circuit, R1 and R2 are the resistances of the primary and secondary winding, respectively, Lm is the magnetization inductance, Rm is the magnetization branch resistance, and Ld1 and Ld2 are the primary and secondary dispersion inductances. 8 shows the values of each input/output signal in both MFTs as well as the efficiency obtained in each case. Figure 7a,b shows the input and output voltages and currents of the nanocrystalline and ferrite MFTs, respectively. Table 8 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.
section_name: Simulation

section_content: The values of the parameters for both MFT prototypes are obtained from the methodology presented in [16] and are listed in Table 9. 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. Figures 12 13 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 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 VDC 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. Figures 12 13 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, along with the corresponding efficiency values. Table 11 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 comparison between voltages and currents of both MFTs prototypes at full load (Figure 15 ) are presented in Table 10, along with the corresponding efficiency values. Table 11 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 comparison between voltages and currents of both MFTs prototypes at full load (Figure 15 ) are presented in Table 10, along with the corresponding efficiency values. Table 11 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. 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. 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. 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. 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.
section_name: Experimental Results

section_content: Table 13 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.
section_name: Discussion

section_content: 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).
section_name: Conclusions

section_name: Acknowledgments:

section_content: The authors thank the UMSNH (Universidad Michoacán de San Nicolas de Hidalgo), the ENES Juriquilla-UNAM ( Escuela Nacional de Estudios Superiores Unidad Juriquilla, Universidad Nacional Autónoma de México) and CONACYT for supporting our research and projects leading to the writing of the present paper.
section_name: Acknowledgments:

section_name: Data Availability Statement: Not applicable.

section_content: Appl. Sci. 2022, 12, 709
section_name: Data Availability Statement: Not applicable.

section_name: Conflicts of Interest:

section_content: The authors declare no conflict of interest.
section_name: Conflicts of Interest: