The Application of Transformers with High-Temperature Superconducting Windings Considering the Skin Effect in Mobile Power Supply Systems

Abstract

The active and passive components of transformer electrical equipment have reached their limits regarding modernization and optimization, leading to the implementation of innovative approaches. This is particularly relevant for mobile and autonomous energy complexes due to the introduction of increased frequency, which can be advantageous, especially in geoengineering, where the energy efficiency of electrical equipment is crucial. The new design of transformer equipment utilizing cryogenic technologies incorporates high-temperature superconducting (HTS) windings, a dielectric filler made of liquid nitrogen, and a three-dimensional magnetic system based on amorphous alloys. The finite element method showed that the skin effect does not impact HTS windings compared to conventional designs when the frequency increases. The analysis and synthesis of the parameters of the magnetic system made from amorphous iron and HTS windings in an HTS transformer with a dielectric medium of liquid nitrogen at a temperature of 77 K were performed, significantly reducing the mass and size characteristics of the HTS transformer compared to traditional counterparts while eliminating environmental and fire hazards. Based on these studies, an experimental prototype of an industrial HTS transformer with a capacity of 25 kVA was designed and manufactured.

1. Introduction

When designing power supply systems for autonomous and mobile objects of electric transportation and logistics and expeditions of geological survey works, the energy efficiency and mass and dimensions parameters of transformer power equipment and energy storage are urgent. The use of semiconductor voltage converters is a promising development direction for replacing electrical machines based on the physical phenomenon of electromagnetic induction. However, at the moment, semiconductor devices are costly and unreliable, leading to power supply failures and even accidents under specific circumstances. The priority device for voltage transformation at a given frequency is transformer power equipment based on magnetically coupled inductive coils (inductors).

A transformer, a highly reliable device of simple construction, has a low specific power per unit mass, approximately 180 W/kg. Since transformer power equipment is included in the mobile power supply system, its energy efficiency and mass and dimensions parameters significantly influence the ability of mobile objects to move during geological explorations, particularly in the northern regions of Russia, including the polar (Arctic) circle.

The most effective way to increase the specific power of transformer electrical equipment is to increase the frequency of electric current generation. Still, various technical problems arise related to the physical properties of materials and the design of electrical systems. This increases reactance, dielectric losses, corona discharge, and thermal effects on equipment elements.

The special interest of the scientific community is concentrated on the problem of the skin effect [1] arising from the electromagnetic effect of increased frequency on the conductors of the windings and the magnetic core of the transformer. The authors of [2,3,4] analyzed the distribution of the thermal field of a system of several parallel conductors, considering the skin effect. Taking into account the fact that inverters with pulse width modulation power many electric machines, the impact of small magnetization reversal cycles causes a profound skin effect [5,6] in electrical steels, even the thinnest plates with a thickness of 0.1–0.2 mm; at the same time, capacitive effects generate common-mode currents that cause electromagnetic interference to the equipment.

The use of an increased frequency poses serious problems for inventors concerning the thermal balance of the magnetic circuit, which manifests itself with losses due to magnetization reversal and the occurrence of eddy currents. Earlier, a practical investigation of a high-frequency pulse power transformer with a mixed core at 200 kHz was carried out [7]. The numerical simulation of the skin effect is performed in [8] with the determination of its influence on the reactance and comparison with experimental measurements of copper wires at 0.1–10⁴ kHz. Another paper [9] describes the anisotropy of the critical current in a superconducting air-core transformer at 2.2 kHz. The influence of the cryogenic environment on the characteristics of electric and amorphous iron magnetic cores is shown in [10,11].

It is worth considering that reducing the overall characteristics of electric machines has now become possible with the advent of HTS conductors, which are widely used in various fields of activity [12,13].

The solution proposed in the article is intended for mobile and autonomous systems that do not have access to a centralized power supply system. At the same time, renewable energy sources such as wind power plants and solar panels can be used as an energy carrier, which when combined with high-power transformers with a dielectric medium in the form of liquid nitrogen, will lead to the operation of an environmentally friendly energy system. In addition, the possibility of using traditional sources of electricity, such as diesel fuel and natural gas, is not excluded. These sources can be useful as backup or additional capacities, especially during peak periods or when there is a shortage of renewable energy. However, given the environmental aspects, their use should be kept to a minimum in order to minimize the emissions of carbon dioxide and other pollutants (Figure 1).

The study’s scientific novelty lies in the development and comprehensive analysis of a high-power superconducting transformer device adapted to operate under conditions with mobile and autonomous power supply systems while increasing the frequency of the generation source up to 800 Hz, considering the skin effect.

The purpose of the study is to develop a new design of transformer power equipment based on cryogenic technologies with improved electrical, technical, and economic characteristics using liquid nitrogen at 77 K as a cryogenic dielectric medium.

• In practice, several methods are used to reduce the mass and dimensional parameters of transformer power equipment, including the following:
• The application of magnetic core material with high magnetic characteristics at an increase in frequency up to 800 Hz;
• The application of materials with high current density for transformer windings;
• The increase in the operating frequency in the electrical system.

2. Materials and Methods

Currently, most transformer magnetic cores are made of electrical steels and ferrites. The maximum operating induction at which a transformer has acceptable magnetic core losses within the technical specifications is in the range of B_c = 1.5–1.9 T. As the core reaches saturation, the leakage flux related to the high-voltage winding increases [14]. A further increase in the operating induction will cause a decrease in the magnetic permeability, thus lowering the winding inductances, which results in the rise in magnetic core losses. For this reason, increasing induction is not advisable with available soft magnetic materials. A reduction in transformer dimensions by increasing the current density is a promising direction due to the development of HTS wires of the 2nd generation [15,16,17,18].

The design of transformer windings from a superconducting wire, which has zero AC resistance at the boiling temperature of liquid nitrogen (77 K/−198 °C), allows for eliminating resistance losses [11]. The current density in a superconducting wire can reach 500 A/mm² compared to 2.8 A/mm² for copper. The difference of 250 times significantly reduces the volume and weight of transformer windings [19,20,21,22].

The most effective method to reduce the dimensions and weight of the transformer is related to the increase in the operating current frequency (1) but requires a careful approach to the selection of applied materials and transformer design [23,24,25]:

$$d=0.507\sqrt[4]{{\displaystyle \frac{P^{\prime }\beta a_r{k}_r}{f{u}_r{B}^2{K}_c^2}}}$$

(1)

where $P^{\prime }$—single-phase power, W; $\beta =\pi d/l$—the value determining the ratio between the diameter ($d$) and the height ($l$) of the winding; $a_r$—the effective width of the dissipation field; $k_r$—the coefficient of reducing the ideal dissipation field to the actual one (Rogowski coefficient); $f$—the network frequency, Hz; $u_r$—the reactive component of the short circuit voltage, %; $B$—the maximum induction in the core leg, T; and $k_c$—the coefficient of filling the circle area with steel.

As follows from (1), the AC frequency and the core diameter are inversely proportional. Consequently, the magnetic core weight of a superconducting transformer is significantly reduced. When increasing the frequency, the number of turns of the transformer windings decreases. It should be noted that when the winding diameter decreases, the distance between the core legs also decreases. This results in a reduction in the window area of the magnetic core. Consequently, the volume and weight of the core iron decreases.

$$W={\displaystyle \frac{U}{4.44fBS}}$$

(2)

where $U$—voltage, V and $S$—the active electrical steel area of the magnetic core, m².

The use of high frequencies always results in the current concentration near the outer part of the conductor, which is called the “skin effect” [26,27].

At the industrial frequency of 50 or 60 Hz used in real-world practice, the resistance to the skin effect is negligible and is not usually considered. The effect is caused by the magnetic flux in the internal part of the conductor, which is induced by the operating current in the wire. The current induced by the internal magnetic flux is in the opposite direction to the operating current I when closer to the center of the wire and in the same direction for the outer part of the conductor. The skin effect results in an exponential decrease in the current density with depth.

The non-uniform current distribution along the cross-section of the conductor can be clarified by theoretical analysis using Bessel functions, which consider the presence of internal induction in the conductor to be several orders of magnitude less than the external induction. For complex amplitudes of the current density and the magnetic field strength, Equations (3) and (4) are applied as follows:

$${\displaystyle \frac{d^2{J}_m}{d{r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{d{J}_m}{dr}}=j\omega \mu \sigma J_m$$

(3)

$${\displaystyle \frac{d^2{H}_m}{d{r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{d{H}_m}{dr}}-{\displaystyle \frac{H_m}{r^2}}=j\omega \mu \sigma H_m$$

(4)

where $J_m$—the complex amplitude of the current density, A/m²; $H_m$—magnetic field strength, A/m; $r$—resistance, Om; $\sigma$—conductivity of the conductor; Cm—magnetic permeability; and $\omega$—angular frequency.

The impedance of a cylindrical conductor is determined by the sum of the internal impedance and the external impedance of the conductor (5), (6).

$$Z=Z_{in}+Z_{ext}$$

(5)

The internal impedance is defined by Equation (6):

$$Z_{in}=R_0\cdot {\displaystyle \frac{u_a}{2}}{\displaystyle \frac{J_0\left(u_a\right)}{J_1\left(u_a\right)}}=\mathit{Re}\left(Z_{in}\right)+I_m\left(Z_{in}\right)$$

(6)

where $R_0$—DC resistance of the conductor under steady-state conditions, Ω; $J_0\left(u_a\right)$ and $J_0\left(u_a\right)$—Bessel functions of the zero and first order, respectively; $u_a=j^{2/3}\sqrt{2x}$, $x=a/\delta$, $a$—the conductor radius, m; $\delta$—the surface layer depth, mm; $\mathit{Re}\left(Z_{in}\right)$—the actual component of the internal impedance, Ω; $I_m\left(Z_{in}\right)$—the complex inductive component of the internal impedance, jΩ.

The external impedance is defined by Equation (7):

$$Z_{ext}=j{X}_{ext}=j2\pi f{L}_{ext}$$

(7)

where $L_{ext}$—external inductance.

The series decomposition of the Bessel function is given by (8):

$${\displaystyle \frac{u_a}{2}}{\displaystyle \frac{J_0\left(u_a\right)}{J_1\left(u_a\right)}}=1-{\displaystyle \frac{{u_a}^2}{8}}-{\displaystyle \frac{{u_a}^4}{192}}-{\displaystyle \frac{{u_a}^6}{3074}}-\dots$$

(8)

In the Bessel function form of the solution, the voltage developed along the wire due to an alternating current $I=I_0{e}^{i\omega t}$ with amplitude $I_0$ given by (9):

$$V_{emf}=R_{0}I[1-\sum _{n=1}^{\infty }{S}_{2n}{\left[\right(-j\omega {\mu }_0\mu l/\left(R_0\pi \right)\left)\right]}^n+j\omega L_{ext}I$$

(9)

where S₂ = 1/8, S₄ = 1/192, and S₆ = 1/3074; $l$—conductor length; and $L_{ext}$—external inductance.

The electromagnetic surface depth $\delta$ (9) at which the current density is 1/e (≈37%) of its surface density is determined by the following equation [28]:

$$\delta =\sqrt{{\displaystyle \frac{1}{\pi \sigma f\mu }}}$$

(10)

where $\sigma$—conductivity of the conductor, S; $\mu$—permeability of the conductor.

From the surface to the center of the conductor, the current density $J$ decreases according to the exponential dependence (10):

$$J_Z=J_0\cdot e^{-{\displaystyle \frac{z}{\delta }}}$$

(11)

where $J_0$—the current density at the surface, A/mm²; $z$—the depth of the calculation, mm.

It follows from (6) and (7) that the closer to the center of the conductor, the lower the current density. Thus, a considerable area of the wire cross-section is not involved in the electricity transmission (Figure 2).

The above analysis shows that it is necessary to consider the skin effect when designing a superconducting transformer operated at high frequencies. Using ELCUT software (professional version 6.6) [29,30], the simulation of current concentration near the surface was carried out by a finite element method for a copper conductor of 5.7 mm diameter and 2.4 A/mm² current density at AC frequencies of 200, 400, and 800 Hz.

Figure 3 shows a color diagram of the current concentration near the conductor surface (skin effect illustration). A finite element mesh with 1306 nodes was obtained in the cross-section of the 25.5 mm² copper conductor with a sampling step of 0.05 mm. This fully illustrates the phenomenon when the skin effect becomes more evident at 400 Hz and above. Copper is diamagnetic in its magnetic properties, but its magnetic permeability is greater than zero, being equal to $\mu$ = 0.999990. This results in the penetration of the magnetic field into the conductor depth and the formation of the magnetic induction of a particular strength, leading to the appearance of eddy currents. The increase in eddy currents, in turn, causes more intensive current forcing to the conductor surface. In this regard, a rise in heat generation and voltage drop are observed.

Superconductors are absolute diamagnetics [31] whose magnetic permeability is zero. This means the magnetic field is eliminated from the conductor volume under the superconducting state. In a superconductor, electrons form pairs called Cooper pairs, thus providing the superconducting state. They have zero spin and can move without interaction with the crystal lattice, causing zero resistance in the material. This is confirmed by the effect of expelling the external magnetic field from the superconductor body due to internal eddy currents (Meissner effect) in superconductors. HTS conductors have zero resistance at 77 K; hence, applying Equation (9) is difficult.

Investigation of HTS Transformers at High Frequencies

For the investigation, the properties and characteristics of the transformer magnetic core made of the soft magnetic fast-quenched 1CP amorphous alloy with the 1B AMET magnetic core were analyzed at different network frequencies [32].

Figure 4 shows the hysteresis loop of the amorphous core obtained experimentally using an oscilloscope. The loop’s rectangular shape is a feature of amorphous alloys due to which the value of the coercive force corresponds to the operating value of the magnetic field strength.

The 1CP tape has the following alloying elements: B, Si, P, C, Co, and Ba. Obtained by quenching from the liquid state due to the high cooling rate (˃1000 K/s), the metal passes into the passive state that provides high corrosion resistance for various aggressive media. These properties of amorphous alloys make them an excellent analog of conventional electrical steel for designing power transformers with high energy efficiency [33,34,35].

Table 1. Initial technical characteristics of the HTS transformer.

Parameter	Value
Rated power, kVA	100
Number of phases	3
Winding connection	Δ/Yn-0
Cryostat material	Expanded polystyrene
Dielectric medium	Liquid nitrogen
Operating temperature, K	77
Current frequency, Hz	50, 200, 400, 800
Winding parameters
Winding parameter	HV winding	LV winding
Rated voltage, V	10,000	400
HTS tape width, mm	4 × 0.1	12 × 0.1
Insulation	Polyamide varnish
Rated current, A	3.7	162.5
Current density (плoтнocть тoкa), A/mm2	9.25	135
Magnetic core parameters
	Amorphous magnetic core 1B
Alloying elements	B, Si, P, C, Co, Ba
Saturation induction, Bm	1.57

shows the technical characteristics of the HTS transformer.

The investigation focuses on the relationship between the transformer core remagnetization frequency and the change in the magnetic core dimensions, hysteresis loop losses, and eddy current losses.

The remagnetization losses and eddy current losses are calculated according to the Steinmetz equation [36]:

$$P_h=nf{B}^2{G}_{st}$$

(12)

$$P_{ed}=y{f}^2{B}^2{G}_{st}$$

(13)

where $y$ and $n$—the coefficients characterizing the used ferrimagnetic material; $G_{st}$—the transformer core weight.

The coefficients n and y were determined from the parameters of the hysteresis loop area to the magnetic core volume at 50 Hz (Figure 3) and specific losses in the 1B amorphous alloy.

The obtained dependences of the magnetic core losses with the frequency increase are shown in Figure 5.

According to Steinmetz Equations (11) and (12) and Figure 4, there is a linear dependence of hysteresis losses and an exponential dependence of eddy current losses on the frequency increase. Modern electrical steels have relatively high magnetic saturation induction up to 2 T [37] but low magnetic permeability, which in turn leads to a considerable increase in the magnetic field strength to achieve the required induction in the core. This results in an enlarged hysteresis loop area that illustrates remagnetization losses.

Using amorphous alloys to reduce the losses in the magnetic core associated with the core remagnetization is reasonable. Such alloys lack strict periodicity and long-range order in the arrangement of atoms inherent in the crystal structure of magnetically soft electrical steels, which do not have inter-domain boundaries (Figure 6). Due to high values of magnetic permeability ($\mu =\mathrm{50,000}-\mathrm{70,000}$), amorphous alloys are preferably used in power transformers at high frequencies [38].

For modern electrical steel used in a transformer with plate thicknesses of 0.17–0.5 mm and depending on the magnetic core design, the ratio $P_B/P_h$ can vary in the range of 0.2–7. In the case of amorphous iron, the ratio $P_B/P_h$ is in the range of 0.17–2.9 at high network frequencies of 50–800 Hz. This is due to the high resistivity of 100–300 µOhm·cm, which is slightly higher than the resistance of cold-rolled steel, being in the range of 40–80 µOhm·cm [39,40,41].

The next vital point is reducing the core volume and weight according to (1). The dependence f and the magnetic core weight are inversely proportional, as presented in Figure 7.

In the paper, the dependence illustrating the influence of increasing frequency on the EMF (13) increase was obtained directly affects the number of turns of transformer windings (presented in Figure 8). The found characteristics of the HTS transformer at the frequency increase are summarized in

Table 2. Technical parameters of the HTS transformer.

f (network frequency), Hz	50	200	400	800
d (leg diameter), m	0.135	0.095	0.08	0.067
Gst (magnetic core weight), kg	432	184	123	81
Et (turn EMF), V	3.382	6.701	9.623	13.32
wLV (number of secondary turns)	68	34	24	17
wHV (number of primary turns)	2945	1473	1040	737
Φm (magnetic flow), Wb	0.015	0.008	0.005	0.004
Ph (hysteresis losses), W	171	297	396	532
Ped (eddy current losses), W	25	223	594	1595
Pid (core losses), W	196	521	990	2127

.

$$E=4.44fBS$$

(14)

3. HTS Transformer Prototype

The presented paper considers a model of an HTS transformer with a spatial design (i.e., legs in different planes) of the magnetic system (Figure 9). This design is characterized by the presence of the so-called “warm” magnetic core. It means there is no direct contact between the core and the cryogenic medium. This technical solution imposes certain limitations on the design and production of the cryostat.

One of the key aspects of this design is the reduction in heat fluxes from te magnet core into the liquid nitrogen. Since the core is not directly located in the cryogenic circuit, heat losses caused by magnet core heating are minimized (Figure 10). This will avoid losses of cryogenic liquid that positively affect the transformer efficiency and diminish the need for the frequent replenishment of liquid nitrogen [42].

The simultaneous application of the spatial magnetic system (i.e., legs in different planes) reduces the core yoke size by half, considerably decreasing the core weight. In this case, the complete symmetry of the magnetic system is maintained. As a consequence, the no-load current of the transformer and the magnetic core losses are significantly reduced.

By integrating an enhanced magnetic system and high current density in HTS, there is a significant opportunity to boost the specific energy of the transformer. While this value stands at approximately 180 W/kg for a conventional oil-filled transformer, it rises to 500 W/kg in the case of an HTS transformer operating at a frequency of 50 Hz. This enhancement results in a marked reduction in the transformer’s size and weight, which is crucial for enhancing the efficiency and mobility of modern energy systems. Figure 11 compares the dimensional characteristics of HTS transformers at various frequencies.

Despite the size reduction, the cost of HTS transformers is still significantly higher than that of their traditional counterparts. This higher cost is attributed to the complexity of manufacturing HTS materials, which requires expensive rare earth metals such as yttrium, lanthanum, and neodymium. The elevated prices of these materials and the specialized processing methods needed to produce superconducting components lead to increased production costs for HTS transformers.

Due to the high cost, special attention should be given to applying HTS transformers in electrical power systems. Losses in the magnetic core primarily contribute to losses in an HTS transformer; to achieve maximum efficiency, HTS transformers must be operated at the highest possible load factor. Consequently, the payback period of such a transformer when the frequency increases relative to the transformer at 50 Hz largely depends on the load level, as shown in Figure 12.

Reducing operating costs for an HTS transformer also removes the need for the periodic chromatography of transformer oil, which serves as a dielectric medium and cools the transformer’s active components. In this case, liquid nitrogen is used as a cryogenic and dielectric medium, significantly reducing costs. An adsorption-based liquid nitrogen station facilitates the acquisition of the dielectric medium at various locations and under any conditions.

Dielectric Cryogenic Medium in the Form of Liquid Nitrogen

Liquid nitrogen is one of the most widely used cryogenic dielectrics. With a boiling temperature of 77 K, it has high thermal conductivity and low viscosity, making it ideal for HTS cooling systems. Unlike many other dielectrics, polymers, and some electrical oils, liquid nitrogen is not subject to aging processes that ensure the stability of its characteristics over time. Its simple molecular structure (N₂) eliminates the formation of decomposition products that also provide the reliability of systems using liquid nitrogen as a cooling agent [43].

Liquid nitrogen’s electrical breakdown is possible in the presence of impurities, including unstable atoms, ions, and insulation material deposits. These impurities can significantly reduce the electrical strength of the dielectric, so their presence should be strictly controlled. To prevent such problems, nitrogen is purified and filtered before being used in critical applications. The physical properties of liquid nitrogen are summarized in

Table 3. Physical properties of liquid nitrogen at 77.3 K and 0.10 MPa.

Parameter	Value
Molar mass, g·mol−1	28.01
Liquid phase density at saturation, kg·m−3	807.4
Gas phase density at saturation, kg·m−3	4 604
Speed of sound in liquid phase, m·s−1	860
Volumetric expansion of liquid (77.3 K, 0.10 MPa) into gas (293 K, 0.10 MPa)	1:694
Relative permittivity of liquid nitrogen	1.46
Relative permittivity of gaseous nitrogen	1.00
Electrical resistivity, Ohm·m	>1 × 1016
Surface tension, N·m−1	8.9 × 10−3
Dynamic viscosity, Pa·s	1.65 × 10−4
Thermal conductivity, W·m−1·K−1	0.14
Heat capacity, J·g−1·K−1	2.04
Enthalpy of vaporization, J·g−1	199.3
Critical point at 3.35 MPa, K	126.21
Triple point at 0.0125 MPa, K	63.1

.

Low reactivity helps to reduce or eliminate the influence of liquid nitrogen on active (windings, magnetic core) and passive (insulating materials, cryostat) transformer elements.

4. Discussion of Results

The investigations confirmed the practicability of increasing the AC network frequency up to 800 Hz. This is especially important for the considerable reduction in dimensional and weight parameters of electrical installations in mobile, autonomous, and local power supply systems, as well as for solving geoengineering tasks when changing the location of geo-surveying works [44].

In particular, in many power supply systems of this kind at river and sea vessels and aircraft, the transition to the higher frequency has already been performed. For example, 200 Hz frequency is used for hydrofoils in Russia, while 400 Hz frequency is used in the defense industry of some countries. However, with the possibility of reducing the resistance to zero in superconductors, it is reasonable to increase the frequency to 800 Hz. This will allow for the duplication of energy efficiency for electrical installations. In this case, energy efficiency is still estimated as a considerable reduction in consumables or initial investments, a decrease in active power losses during operation, and a reduction in maintenance work because there is no need to conduct oil chromatography two times a year. Since energy efficiency is a system concept, it will be the subject of a separate paper. But it is pretty evident since in remote power supply systems, for example, a geological expedition with variable location, the weight characteristics and dimensions are very important while moving. The more important advantage is that the autonomous generation source used in such cases can have a minor generation power, thus also reducing their cost. Finally, the dielectric medium (i.e., liquid nitrogen) can be replaced at any geolocation point by the engineering team or expedition from the air using a turboexpander using the technology of Academician P. Kapitsa.

The influence of the frequency increase on the cross-section of winding conductors caused by the skin effect was shown. In addition, it affects the magnetic core characteristics, which is why it is reasonable to use amorphous iron and, in general, the mass and dimensions parameters of the transformer [45,46].

The obtained dependences presented in Figure 4, Figure 6 and Figure 7 show that the synthesis of such parameters as the frequency increases and the amorphous iron magnetic core results in the exponential rise in heat losses in the magnetic core based on Steinmetz Equations (11) and (12). This requires the decrease in the magnetic core induction with increasing electric current frequency that, in the end, influences the number of winding turns and the core weight. The presence of the cryostat requires 10–20 mm of wall thickness (Figure 8) to maintain a cryogenic temperature of 77 K. The insulating layers between the windings and the magnetic core generally compensate for this.

For example, the mass and dimensions parameters of an HTS transformer with the windings immersed in a cryostat, liquid nitrogen as a dielectric medium, and a warm, magnetic core are 432 kg at 50 Hz (industrial frequency), 123 kg at 400 Hz (i.e., 3.5 times less), and 86 kg at 800 Hz (i.e., more than five times less than at the industrial frequency).

It should be noted that an HTS transformer at the same frequency is more than 2–2.5 times smaller in dimensions than a conventional transformer [47,48,49]. Considering the frequency increase, its mass and dimensions parameters are almost 10 times less than a conventional transformer at the industrial frequency. Therefore, it is more energy efficient than a traditional transformer in order of magnitude.

In recent years, the tendency to increase the operating frequency has become one of the main vectors of power industry development in distributed generation due to improved technical and economic costs. In addition, the performance characteristics of sea and airborne transportation facilities are enhanced. However, this tendency is strongly limited by the phenomenon of current concentration near the conductor surface. This so-called skin effect increases the resistance of conductors and causes active power losses, thus reducing the energy efficiency of power equipment. For example, the resistance rises approximately 2.5–3 times at 400 Hz compared to the resistance at the industrial frequency of 50 Hz. Consequently, active power losses in transformer windings increase proportionally to the resistance. In the case of windings made of superconductor materials, it is reasonable to use the frequency of 800 and 1000 Hz. The use of HTS conductors in high-frequency power equipment has a synergistic effect, since HTS is an absolute diamagnetic in the superconducting state that prevents the penetration of the magnetic field into the conductor depth and the formation of a surface layer.

The phenomenon of superconductivity and the possibility of ensuring it at the temperature of liquid nitrogen (77 K) opens almost unlimited directions for the use of higher frequencies since the resistance is zero at any frequency; therefore, the problem of frequency limitation is eliminated. At present, the power engineering community has not yet realized this. In this paper, we have attempted to show that the mass and dimensions parameters of power equipment are reduced by 5–6 times at the frequency of 400 Hz applied for mobile systems when using superconducting windings and by 9–12 times at the frequency of 800 Hz, which we reasonably suggest to investigate and implement.

In [7], the author concluded that the combination of core layers of amorphous, ferrite, and electrical iron resulted in a decrease in the hot spot temperature that increased the rated power of the transformer. In addition to the amorphous magnetic core, the following are used in our work: in addition to the amorphous magnetic core, superconducting windings are used in our work, which leads to a positive correlation of the thermal regime of the core, a decrease in overall parameters, and an increase in efficiency relative to a similar transformer with conventional conductors. In [8,9], a practical experiment of an HTS transformer with an air core was carried out. This solution raises doubts about the prospects for successful application, since the absence of a magnetic circuit leads to a significant decrease in coil induction and, as a result, to the considerable consumption of high-current conductors through conductive windings, which leads to a positive correlation of the thermal regime of the core, a decrease in overall parameters, and an increase in efficiency relative to a similar transformer with conventional conductors. In [8,9], a practical experiment of a high-power transformer with an air core was carried out. This solution raises doubts about the prospects of successful application, since the absence of a magnetic circuit leads to a significant decrease in coil induction and, as a result, to considerable consumption of the HTS conductor. Research conducted with transformers with amorphous cores and HTS windings operating at elevated frequencies has shown significant potential for use in pulse power converter systems. These devices feature single-phase configurations and were tested at approximately 2.2 kHz and higher without integrating cryostats designed for industrial applications. It is also worth noting the actual use of a superconducting substation with an autonomous cryogenic liquid nitrogen supply system, which includes an HTS transformer and a 0.4 kV power cable using HTS. At the same time, it was possible to achieve energy efficiency indicators, which allowed us to draw final conclusions about the high energy efficiency of using superconductivity technology in the electric power industry. [50]. The current study aims to develop a three-phase power transformer that can operate within the 400–800 Hz range, removing the need for inverters. As part of this project, we explored the combined effects of an amorphous core and HTS windings on the skin effect, leading to the development of a prototype that includes a cryostat with a warm core, where the core is not directly submerged in liquid nitrogen [51,52].

A significant advantage of HTS transformers is the ability to self-limit short-circuit currents after leaving the superconducting state by a transformer, which is detailed in [53,54,55,56]. These studies relate to industrial-frequency centralized power supply systems and do not take into account the effects of the high-frequency magnetic scattering field on the critical value of the high-frequency current HTS windings.

5. Conclusions

1. The study examined the feasibility of applying transformers with HT windings and a cryogenic dielectric medium in liquid nitrogen. Considering the autonomy of power systems, a mathematical model was developed to account for the skin effect, aiming to increase operational frequencies up to 800 Hz. However, the current maximum frequency is 400 Hz. Upon transitioning to the superconductive state, the disappearance of active resistance also eliminates the skin effect, which previously hindered frequency escalation. At the same time, increasing the frequency is justified by significant reductions in equipment size and installation costs, ultimately allowing for an increase in functional load capacity.

2. The practicability of the frequency increase up to 800 Hz instead of the industrial frequency of 50 Hz for mobile, autonomous, and local power supply systems has been proved. In this case, active power losses in transformer windings are reduced to zero, which allows for increasing the current density from 2.4 to 500 A/mm² in HTS transformer windings. Then, it decreases the mass and dimensions parameters of HTS transformers and generation sources operated at the object.

3. It can be stated that the resistance of superconducting winding wires is reduced to zero when using a cryogenic design and in the dielectric medium in the form of liquid nitrogen (77 K). This removes the negative effect of forcing the current to the conductor surface with the frequency increase (the skin effect), in which the resistance increases by 2.5–3 times at 800 Hz. It is equivalent to removing the power consumer from the generation source at a three times more significant distance with the corresponding lengthening of the transmission line. However, this effect is not observed in superconductors, giving new opportunities for increasing the AC frequency.

4. A novel structural design for transformers featuring HTS windings and a spatial magnetic system based on an amorphous core has been proposed. In some cases, the high-voltage winding may be made using non-superconducting conductors. Because high voltages involve low currents and small conductor cross-sections, immersion in liquid nitrogen significantly improves current density. This design provides more uniform cooling of the windings, as verified by a patent.

5. It has also been emphasized that transformers with HTS windings and a cryogenic dielectric medium like liquid nitrogen are inherently fire-resistant, explosion-proof, and environmentally friendly. These features offer a significant advantage compared to alternative dielectric media such as mineral oil, synthetic middle oil, and sulfur hexafluoride. This aspect is vital for mobile and autonomous power supply systems and remote geological exploration and engineering facilities in extreme environments, such as the far north.