Power Losses of the High-Voltage High-Frequency Coaxial Cable Energy Transfer System

Abstract

The paper shows the construction and operation of the High-Voltage High-Frequency Coaxial Cable Energy Transfer System dedicated to a three-phase 500 V mining grid with an ungrounded neutral point. The correct operation of the model was verified through simulation and experiments. This paper focuses on the overall system efficiency and the power loss analysis of its components. Based on these measurements, it is concluded that the presented system is suitable for mining applications, where high energy conversion efficiency is essential due to the difficulty of dissipating heat to the environment.

1. Introduction

Power electronic converters are employed in a wide range of applications, including industrial systems, energy systems, transportation, and consumer electronics. In all these cases, their operation involves converting electrical energy from one form to another. This conversion is subject to power losses. To ensure proper operation with limited converter internal temperatures, it is necessary to use sufficiently efficient cooling. This is most often carried out using forced air fans. In high-power converters, a more expensive solution is also used, i.e., liquid cooling with deionized water or other liquids [1,2]. However, in some applications, implementing forced cooling systems is exceptionally challenging, such as in underground mining. This is due to the presence of an explosive or dusty atmosphere, high humidity, and strong vibrations, which can lead to leaks in liquid-cooling systems. The solution to such a problem is the need to build highly efficient converters that achieve lower power losses, resulting in lower temperatures inside the converter and, in turn, extending their service life.

In underground mining applications, electrical energy is often transmitted over long distances while maintaining relatively low power losses. This can be achieved using low- or high-voltage three-phase systems operating at 50/60 Hz, or through direct-voltage transmission, e.g., LVDC or HVDC [3,4]. The use of electric systems is also associated with the integration of energy storage, typically in the form of batteries. The adoption of battery-powered electric systems for underground mine transportation helps to reduce emissions, noise levels, and ventilation requirements [5,6].

Currently, most power transmission systems are composed of multiple interconnected subsystems. In such architectures, voltage conversion is frequently required, along with galvanic isolation to enable series interconnection of subsystems and to ensure proper fault decoupling between them. These functionalities are typically achieved using converters equipped with transformers, most often operating at medium frequencies inside isolated DC-DC converters. Numerous studies demonstrate the integration of isolated DC-DC converters with photovoltaic (PV) installations [7,8], battery energy storage systems (BESS) [9,10], and wind energy conversion systems (WECS) [11,12]. Hybrid configurations combining, for example, PV systems with energy storage units have also been widely reported [13,14,15]. Publications on modular multilevel converters (MMC) designed as HVDC interfaces for utility-scale PV systems emphasize that such systems typically incorporate isolated DC–DC conversion stages or medium-frequency transformers to provide galvanic isolation, voltage adaptation, and fault decoupling, even though the MMC itself is not an isolated converter topology [15]. Although the present work focuses on a single-phase high-voltage, high-frequency cable-based energy transfer system intended for underground mining applications, the challenges related to electrical isolation, fault decoupling, and efficient power transfer are conceptually aligned with those encountered in modern HVDC systems interfacing renewable energy sources.

This paper analyzes an electric energy transmission system using a single-phase high-voltage cable operating at 10 kHz [16,17]. This solution is economically justified by the lower cost of a high-voltage cable compared to a three-phase or HVDC cable, as well as its much lower weight. Reducing the weight of the cable significantly facilitates its transport and installation in mining applications.

The use of a high-voltage, high-frequency cable requires converters that convert energy from the 50 Hz circuit to the high-frequency circuit and from the high-frequency circuit to either direct voltage or a 50 Hz alternating voltage circuit. Another obstacle when using high-frequency cables is the occurrence of radiated disturbances. These disturbances can be effectively eliminated by using an appropriate cable, e.g., a coaxial cable [18]. Using a high-voltage cable reduces the active cross-section of the conductor; however, this is achieved at the expense of strengthening the cable insulation. However, this action has a more beneficial effect on the insulation-to-copper weight ratio. This means the high-voltage cable will have a lower specific weight (given in kg/km).

A power transmission system was developed based on a concept similar to the Single Wire Energy Transfer (SWET) principle [19,20]. Such systems are characterized by a simplified structure, a reduced number of conductors, and high efficiency resulting from resonant operation at a high frequency. Additional advantages include lower voltage drops and improved operational safety, as high-frequency current is less hazardous to humans compared to a 50 Hz mains current. However, since the use of the earth as a return path is not permitted in mining environments due to safety requirements and the protection of electrical equipment, a coaxial cable was selected for energy transmission, while maintaining high-voltage and high-frequency operating parameters.

The energy transfer system using a coaxial cable can employ resonant inverters with various resonant-circuit topologies [21]. In the developed system, a resonant converter topology was selected, consisting of two series-resonant circuits separated by the capacitance representing the coaxial cable. This converter is referred to in the paper as a dual resonant converter.

The choice of this resonant circuit topology was motivated by its relative simplicity and its ability to maintain a stable output voltage transfer function regardless of load power variations. The designed resonant circuit was selected solely because its resonant frequency was approximately equal to the switching frequency of 10 kHz.

The main contributions of this work are as follows:

(1)

A complete modeling framework for the medium-frequency energy-transfer system, including all converters and passive components, enabling system-level analysis.

(2)

A comprehensive loss analysis covering semiconductors, inductors, transformers, and the medium-frequency cable, identifying dominant loss mechanisms and their impact on natural-cooling operation constraints.

The paper is organized as follows: Section 2 provides a general description of a high-voltage, high-frequency coaxial cable energy transfer system divided into its component converters. Section 3 presents an analytical model of power losses for all component converters in the analyzed energy transfer system, providing a detailed description. Section 4 will present a summary of the results obtained for the entire High-Voltage High-Frequency Coaxial Cable Energy Transfer system, whereas Section 4 will present the conclusions.

2. Description of the System

The High-Voltage High-Frequency Coaxial Cable Energy Transfer (HVHFCCET) and Wireless Energy Transfer (WET) systems, together with their main electrical parameters, are presented in Figure 1.

The High-Voltage High-Frequency Coaxial Cable Energy Transfer is composed of the following components:

• REC1 is a grid-tied three-phase converter operating as a rectifier with a power factor close to unity. The converter is equipped with an LCL-type AC filter, whose general design principles for grid-connected applications are described in [22,23], while the detailed design methodology of the specific LCL filter used in this work is presented in publication [24]. The converter operates at a switching frequency of 50 kHz. For a grid line-to-line voltage of 500 V, the output DC-link voltage is set to 760 V and can be increased up to 870 V.
• INV1 is a converter that operates as a resonant inverter to deliver the energy to High-Voltage High-Frequency Coaxial Cable (HVHFCC) through a resonant tank and the Tr₁ transformer. The switching frequency is set to 10 kHz, and the HVHFCC operating voltage is 3.5 kV. The transformer Tr₁ is a step-up transformer with the turns ratio N_prim/N_sec = 0.2288.
• A 100 m coaxial cable RG11/U was used as the HVHFCC. The RG11/U cable meets the US Military specification MIL-C-17, with a nominal peak voltage of 5.2 kV and a characteristic impedance of Z = 75 Ω.
• Transformer TR2 with the second resonant tank and REC2 rectifier converts energy from the coaxial cable to a DC output, at which the voltage is varied between 700 V and 870 V. The combination of converters INV1 and REC2, two transformers Tr₁ and Tr₂, a coaxial cable, and two resonant circuits forms a converter known as the dual-series resonant converter. The transformer TR2 is a step-down transformer with the turns ratio N_sec/N_prim = 4.3703.
• INV2 is a three-phase inverter. Converter INV2 is structurally identical to REC1; however, the direction of power transfer is opposite. In REC1, power flows from the AC side to the DC side, while in INV2, it flows from the DC side to the AC side. The inverter produces a three-phase 500 V, 50 Hz supply. An LCL filter with the same topology as that used in REC1 is connected to the INV2 output. INV2 is controlled using PWM with a switching frequency of 50 kHz, and third-harmonic injection is applied to the modulation signals to increase the available modulation range.

The Wireless Energy Transfer is used to supply the battery charger of the suspended drivetrain. This system is composed of the following components:

• INV3 is a resonant inverter, which operates as a high-frequency, high-voltage AC source to deliver the energy to the REC4 converter through the TR3 transformer, a capacitive coupler, and TR4. The switching frequency of INV3 is equal to 300 kHz.
• Transformer TR4 with the REC4 is mounted on the moving part of the suspended monorail and converts energy from the capacitive coupler to a 48 V DC voltage. This voltage feeds the battery charger.

The Wireless Energy Transfer has been thoroughly covered in refs. [16,17] and is beyond the scope of the present paper. The rated power of the entire system (HVHFCCET with WET) is 7 kW, of which 2 kW is associated with the WET section, while the remaining 5 kW is transferred to the three-phase output of the INV2 converter. Since the INV3 and REC4 converter stage is not analyzed in this work, it is assumed that the total output power at the output of the INV2 converter may vary from 0 to 7 kW.

Figure 2 illustrates the physical arrangement of the main components of the HVHFCCET system within two explosion-proof enclosures. The first enclosure (Figure 2a) houses the REC1 and INV1 converters together with the input-side resonant circuit. The second enclosure (Figure 2b) contains the output-side resonant circuit as well as the INV2 and REC2 converters.

3. Power Loss Analysis

Power loss calculations and measurements were performed for three parts of the converter system. In the first part, the power losses of the AC grid converter, comprising the REC1 converter and grid filters, are presented. In the second part, the power losses of a dual resonant converter composed of the INV1 resonant converter, transformers with the coaxial cable, and the REC2 converter are depicted. In the third part, the power losses of the INV2 converter with filters are shown.

3.1. AC Grid Converter Losses

Figure 3 shows the connection of the REC1 converter to the three-phase 500 V grid. The REC1 is a two-level SiC MOSFET converter switching at a frequency of f_S = 50 kHz. Between the grid and the converter, a common-mode filter and an LCL filter are placed. The common-mode filter suppresses 50 kHz disturbances and their harmonics.

Table 1. Parameters of components of the AC grid converter with its filters.

Parameter	Symbol	Value
AC grid voltage, phase-to-phase rms value	Vg	500 V
REC1 converter DC voltage	Vdc	760 V
Common-mode choke inductance	Lcm	29 mH
Common-mode filter capacitance	Ccm	100 nF
Common-mode choke winding resistance	RLcmw	10 mΩ
LCL filter inductance Lf1	Lf1	500 μH
LCL filter inductance Lf2	Lf2	65 μH
LCL filter capacitance Cf	Cf	1 μF
Winding resistance of inductor Lf1	RLf1w	35 mΩ
Winding resistance of inductor Lf2	RLf2w	16 mΩ
Series resistance in the capacitor path RCf	RCf	1 Ω
The series equivalent resistance of the DC-link capacitors	RCREC1	26 mΩ
DC-link capacitances	CINV1, CREC2	705 μF

presents the key parameters of the REC1 converter and the filters, including the resistive components where power losses occur. The parameters listed in Table 1 are a combination of design values (e.g., voltages) and experimentally measured characteristics of the passive filter elements, including their inductances and capacitances. The winding resistances of the inductors were determined for the highest winding temperature expected during the rated power transfer of 7 kW.

3.1.1. REC1 Converter Losses

The power loss calculations begin with the REC1 converter, which, when operating in conjunction with the power grid, exhibits sinusoidal phase currents. The transistors switch at 50 kHz. In the REC1 converter, losses P_T,REC1 occur primarily in the transistors and consist of conduction and switching losses. The purpose of the REC1 converter is to maintain a constant DC-link voltage of V_dc = 760 V.

Conduction losses depend on the phase current and are described by (1):

$$P_{\mathrm{Tcon}}\left(I_{\mathrm{O}}\right)=3\cdot R_{\mathrm{DSon}}{I_{\mathrm{O}}}^2$$

(1)

where R_DSon is the on-state resistance of a single MOSFET and equals 23 mΩ (for appropriate gate voltage and temperature values), and factor 3 accounts for the number of converter phases. Equation (1) is valid due to the switched operation of the converter; each phase current flows through the one turned-on MOSFET. This is true when the dead time is relatively low, and its effect is neglected.

MOSFET switching losses are calculated based on the switching energy characteristics provided in the transistor’s datasheet [25]. These energies depend on the switched currents, the DC-link voltage, the gate resistance, and the temperature. The energy characteristics are given as functions of the MOSFET current, i_T, as (2).

$$e_{\mathrm{on}}\left(i_{\mathrm{T}},V_{\mathrm{dc}}\right)=\left(a_{\mathrm{on}}{i}_{\mathrm{T}}+b_{\mathrm{on}}\right){\displaystyle \frac{V_{\mathrm{dc}}}{V_{\mathrm{dc}}^{\prime }}}; e_{\mathrm{off}}\left(i_{\mathrm{T}},V_{\mathrm{dc}}\right)=\left(a_{\mathrm{off}}{i}_{\mathrm{T}}+b_{\mathrm{off}}\right){\displaystyle \frac{V_{\mathrm{dc}}}{V_{\mathrm{dc}}^{\prime }}}$$

(2)

where V_dc is the DC-link voltage, and V’_dc = 600 V is the voltage at which the power loss measurements described in the datasheet were conducted. In this formula, a linear relationship between the DC voltage and the power losses is assumed. Switching energies are obtained from the datasheet, considering the gate resistance influence; thus, the following coefficients are obtained: a_on = 24.84 μJ/A, b_on = 227.7 μJ, a_off = 0, b_off = 111.4 μJ/A.

Assuming grid currents are sinusoidal, the switching power losses in all six MOSFETs of the REC1 converter are given by (3).

$$P_{\mathrm{Tsw}}\left(I_{\mathrm{O}},V_{\mathrm{dc}}\right)=6f_{\mathrm{S}}\left({\displaystyle \frac{a_{\mathrm{on}}+a_{\mathrm{off}}}{\mathsf{\pi }}}\sqrt{2}{I}_{\mathrm{O}}+{\displaystyle \frac{b_{\mathrm{on}}+b_{\mathrm{off}}}{2}}\right){\displaystyle \frac{V_{\mathrm{dc}}}{V_{\mathrm{dc}}^{\prime }}}$$

(3)

where the DC-link voltage is constant at V_dc = 760 V.

Total transistor power losses of the REC1 converter, P_T,REC1, which are the sum of the switching and conduction power losses, are presented as functions of the output power P_out in Figure 4. It is assumed that the phase currents I_O are symmetrical and with the power factor close to unity, the corresponding relationship is given in (4).

$$I_{\mathrm{O}}={\displaystyle \frac{P_{\mathrm{O}}}{\sqrt{3}{V}_{\mathrm{g}}}}$$

(4)

Figure 4 shows that, for the selected switching frequency of 50 kHz, the switching losses P_Tsw are significantly higher than the conduction losses P_Tcon. This behaviour results from operating SiC MOSFETs at a relatively high switching frequency, which was chosen to meet the required filter and converter performance constraints.

It should be noted that, in the analysis of REC1 power losses, the output power P_out refers to the power at the DC-link of this converter, which differs from the other cases presented in this paper, where the output power is consistently defined at the end of the subsystem rather than at the INV2 output. Only in the final part of the paper are the power losses of the entire converter system reported with respect to the unified output power P_out at the INV2 output.

3.1.2. Power Losses Generated in Passive Filters of the AC Grid Converter

In the later stages of the REC1 converter power loss calculations, all losses in the LCL and common-mode filter resistances, PCB trace resistances, and the wires connecting the filters to the transistors are considered. For each phase, these resistances are equal to R_PCB = 5 mΩ and R_wire = 3 mΩ. The total resistance of the filters, PCB traces, and wires is R_f = R_Lf1w + R_Lf2w + R_Lf1w + R_Lcmw + R_PCB + R_wire = 69 mΩ, and power losses across these resistances are calculated from (5).

$$P_{\mathrm{fw}}=3\cdot R_{\mathrm{f}}{\left({\displaystyle \frac{P_{\mathrm{O}}}{\sqrt{3}{V}_{\mathrm{g}}\mathsf{\eta }}}\right)}^2$$

(5)

where η is the efficiency of the converter, assumed to be 98%, is taken into account due to the phase current increase caused by the addition of the converter’s power losses.

In addition to conduction losses in the filters, there are also losses in the series damping resistors of the LCL filter and core losses in the inductors. In the L_f1 inductors, the core losses for each core are P_Lf1core = 0.6 W, mainly caused by the high-frequency current component. In the L_f2 inductors, the core losses are almost 0 W, since the phase current is nearly sinusoidal. This occurs because the high-frequency current components flow through the L_f1 inductors and the C_f capacitors.

The currents flowing through the C_f capacitors are, in simplified terms, triangular with a frequency of f_S = 50 kHz, and their RMS value is practically independent of the output power of the REC1 converter. Based on measurements and simulations, the RMS value of these currents was determined to be I_Cf = 0.78 A, resulting in power losses in the R_Cf filter resistor of P_RCf = 0.6 W.

Another source of power losses in the supply path of the REC1 converter, which are not conduction losses, are the core losses in the common-mode choke. It should be noted that this choke compensates common-mode currents at a level of 50 mA [24]. This causes a temperature rise in the choke of approximately 40 °C, allowing the core losses to be estimated at P_Lcmcore = 4 W. Due to the operation of the REC1 converter with the AC grid, it operates with an almost constant modulation index. Therefore, the common-mode current value does not depend on the output power. Power losses in the AC-grid converter filters, which are independent of the load power, are given in (6).

$$P_{\mathrm{fc}}=3P_{\mathrm{Lf}1\mathrm{core}}+3P_{\mathrm{RCf}}+P_{\mathrm{Lcmcore}}=3\cdot 0.6 \mathrm{W}+3\cdot 0.6 \mathrm{W}+4 \mathrm{W}=7.6 \mathrm{W}$$

(6)

Figure 5 shows the total power losses P_f,REC1 inside the filters of the AC grid converter, along with their components: P_fw and P_fc.

3.1.3. Power Losses Generated in the AC Grid Converter

The sum of all power losses generated inside the transistors and the passive filter of the REC1 converter is shown in Figure 6. These losses are compared with the power losses measured during the efficiency measurement of the converter. The output power was limited to 8 operating points.

Figure 6 shows that the calculated power losses closely match the measured losses. Based on this, it can be concluded that the presented power loss analysis accurately reflects the actual power losses in the REC1 converter.

The simulation analysis has been performed to reveal the power losses in the DC-link capacitors, showing that these losses depend on the output power and at maximum power are equal to 1 W. These power losses, which make a small contribution to the total power losses, are not considered in the analysis presented here.

3.2. Dual Resonant Converter Losses

The second part of the High-Voltage High-Frequency Coaxial Cable Energy Transfer System is the dual resonant converter shown in Figure 7. This converter is composed of the INV1 converter, the input L_r1C_r1 resonant tank, transformer Tr₁, coaxial cable, transformer Tr₂, the output resonant tank L_r2C_r2, and diode rectifier REC2. The parameter values of the converter components are shown in

Table 2. Parameters of components of the dual resonant converter.

Parameter	Symbol	Value
INV1 converter DC voltage	Vdcin1	760 V
Resonant tank inductance	Lr1, Lr2	1.04 mH
Resonant tank capacitance	Cr1, Cr2	238 nF
Winding resistance of inductor Lr1 @20 °C	RLr1w, RLr2w	25.5 mΩ
Transformer Tr1, Tr2 primary side voltage	Vprim	800 V
Transformer Tr1, Tr2 secondary side voltage	Vsec	3500 V
Transformer Tr1, Tr2 numbers of turns	Nprim/Nsec	27/118
Transformer Tr1, Tr2 primary side winding resistance @20 °C	RTr1wp, RTr2wp	100 mΩ
Transformer Tr1, Tr2 secondary side winding resistance @20 °C	RTr1ws, RTr2ws	1.2 Ω
Transformer Tr1, Tr2 magnetizing inductance	LTr1m, LTr2m	35 mH
Transformer Tr1, Tr2 primary side leakage inductance	LTr1σp, LTr2σp	26 μH
Transformer Tr1, Tr2 secondary side leakage inductance	LTr1σs, LTr2σs	497 μH
Coaxial cable length	lcoax	100 m
Coaxial cable wire resistance	Rcoaxw	2.3 Ω
Coaxial cable capacitance	Ccoax	6.7 nF
Input and output capacitances	CINV1, CREC2	705 μF

Parameters not listed in the table but presented on the schematic in Figure 7 are discussed in more detail in the following subsections.

. The switching frequency is f_S = 10 kHz.

Power losses are generated in both semiconductor and passive components. Power losses of passive components are represented in Figure 7 by resistors. More specifically, these are losses in the following components:

• Conduction and switching losses in INV1 converter transistors, P_INV1Tcon, P_INV1Tsw.
• Winding losses in L_r1 and L_r2 inductors, P_Lr1w, P_Lr2w.
• Core losses in L_r1 and L_r2 inductors, P_Lr1c and P_Lr2c.
• Primary winding losses in Tr₁ and Tr₂ transformers, P_Tr1wp, P_Tr2wp.
• Core losses in Tr₁ and Tr₂ transformers, P_Tr1c, P_Tr2c.
• Secondary winding losses in Tr₁ and Tr₂ transformers, P_Tr1ws, P_Tr2ws.
• Wire and dielectric losses in the coaxial cable, P_coaxw, P_coaxd.
• Conduction and switching losses in REC2 converter diodes, P_REC2Dcon, P_REC2Dsw.

For calculating power losses, the operating conditions of the dual-series resonant converter are presented as waveforms of the resonant inductor currents i_Lr1 and i_Lr2, and transformer Tr₁ primary-side voltage v_Tr1p, along with their harmonic spectra for output powers ranging from 1 kW to 7 kW. These waveforms and harmonics are obtained in the GeckoCIRCUITS simulator (Gecko-Simulations AG, Zurich, Switzerland) and presented in Figure 8.

The RMS values of the resonant inductor current can be used to calculate conduction power losses in the inductor and transformer windings, the coaxial cable resistance, and conduction losses in the transistors of the INV1 converter and the diodes of the REC2 converter.

Based on the harmonic analysis of the currents i_Lr1 and i_Lr2, it is evident that in the current of the resonant inductor L_r1, there is a significant difference between the amplitude of the fundamental harmonic current and the amplitude of the active component. Such a difference does not occur in the current of the inductor L_r2. This difference is because the current in inductor L_r1 contains a capacitive component of the coaxial cable current. This component has a constant amplitude of approximately 7 A. Still, as the output power P_out increases, the active component becomes dominant, causing the amplitude of the fundamental harmonic and the active component to have similar values.

Both resonant circuits (L_r1-C_r1, L_r2-C_r2) have been designed to be tuned to a resonance frequency close to the switching frequency f_S = 10 kHz (7).

$$f_{\mathrm{r}}={\displaystyle \frac{1}{2\mathsf{\pi }\sqrt{L_{\mathrm{r}1}{C}_{\mathrm{r}1}}}}={\displaystyle \frac{1}{2\mathsf{\pi }\sqrt{L_{\mathrm{r}2}{C}_{\mathrm{r}2}}}}=10.116 \mathrm{kHz}$$

(7)

Due to the presence of the coaxial cable capacitance, the resonance frequency shifts to the value determined by (8).

$$f_{\mathrm{rsh}}={\displaystyle \frac{1}{2\mathsf{\pi }\sqrt{\left(L_{\mathrm{r}1}+L_{\mathrm{Tr}1\mathsf{\sigma }\mathrm{p}}+{\left(\frac{N_1}{N_2}\right)}^2{L}_{\mathrm{Tr}1\mathsf{\sigma }\mathrm{s}}\right)\frac{C_{\mathrm{r}1}\cdot {\left(\frac{N_2}{N_1}\right)}^2{C}_{\mathrm{coax}}}{C_{\mathrm{r}1}+{\left(\frac{N_2}{N_1}\right)}^2{C}_{\mathrm{coax}}}}}}=16.695 \mathrm{kHz}$$

(8)

The occurrence of the resonance frequency f_rsh above the switching frequency f_S causes the passive circuit between the INV1 and REC2 converters to exhibit a capacitive character for the fundamental harmonic and an inductive character for the higher-order harmonics present in the output voltage of the INV1 converter, such as the 3rd, 5th, 7th, and so on.

In Figure 8, the primary-side voltage of the transformer is shown to change relative to the output voltage of the INV1 converter. This change is due to phase shifts in the higher harmonics. It can be explained by the input impedance characteristics as a function of frequency, as shown in Figure 9a. The input impedance Z_in(ω) is measured at the output terminals of the INV1 converter with the REC2 converter disconnected. The voltage at the transformer terminals can be expressed using the following transfer function (9).

$${\displaystyle \frac{V_{\mathrm{Tr}1}\left(\mathsf{\omega }\right)}{V_{\mathrm{INV}1}\left(\mathsf{\omega }\right)}}={\displaystyle \frac{Z_{\mathrm{in}}\left(\mathsf{\omega }\right)-R_{\mathrm{Lr}1\mathrm{w}}-j\mathsf{\omega }{L}_{\mathrm{r}1}+j\frac{1}{\mathsf{\omega }{C}_{\mathrm{r}1}}-R_{\mathrm{Tr}1\mathrm{wp}}-j\mathsf{\omega }{L}_{\mathrm{Tr}1\mathsf{\sigma }\mathrm{p}}}{Z_{\mathrm{in}}\left(\mathsf{\omega }\right)}}$$

(9)

The transfer function (9) is presented as a frequency response in Figure 9b. The output voltage waveform of the INV1 converter, V_INV1, is a rectangular waveform consisting of odd harmonics. Based on the transfer function (9), it can be seen that the transformer voltage V_Tr1 contains the fundamental harmonic with the same amplitude as the inverter voltage V_INV1, with the same phase shift. Higher harmonics, such as the 3rd, 5th, and 7th, have smaller amplitudes relative to the amplitudes of the harmonics in the voltage V_INV1, and their phase shift angles are −180°. The voltage waveform at the transformer terminals is needed to calculate power losses in the transformer core and in the coaxial cable dielectric. It is also worth noting that the presented impedance characteristics were obtained under the assumption that the passive circuit between the INV1 and REC2 converters is unloaded. The analysis presented in the article is based on simulation waveforms that account for the output power consumption on the DC side of the REC2 converter.

3.2.1. Power Losses in Resonant Inductor L_r1

The losses in the resonant inductor are generated inside its winding and magnetic core. The specifications of this inductor are shown in

Table 3. Resonant inductor L_r1 core and winding specification.

Parameter	Symbol	Value
Inductance	Lr1	1.04 mH
Type of the core	---	E65
Number of stacked cores	Nc	6
Number of turns	N	21
Cross-sectional area of magnetic cores	Ac	32.4 cm2
Core material	---	3C94
Core size (all cores)	wm × h × l	65 mm × 65.6 mm × 164.4 mm
Core volume (all cores)	Vc	474 cm3
Winding resistance @20 °C	RLr1w	25.5 mΩ

.

Winding losses depend on the rms value of the inductor current I_Lr1rms, winding resistance, and temperature according to (10).

$$P_{\mathrm{Lr}1\mathrm{w}}\left(T\right)={I_{\mathrm{Lr}1\mathrm{rms}}}^2{R}_{\mathrm{Lr}1\mathrm{w}}\left(1+{\mathsf{\alpha }}_{\mathrm{Cu}}\left(T-20°\mathrm{C}\right)\right)$$

(10)

where α_Cu is the temperature coefficient for copper, α_Cu = 0.0039 1/K, T is the temperature given in °C, and R_Lr1w is the inductor winding resistance at 20 °C.

For example, for an output power of P_out = 7 kW, the rms current value is I_Lr1rms = 11.0 A. Assuming that the inductor heats to 60 °C, the power losses calculated from (10) are 3.56 W. The exact temperature to which the resonant inductor heats up will be calculated after the introduction of its thermal model and the calculation of core power losses.

The Generalized Steinmetz Equation [26,27] is used to calculate power losses in the magnetic core of the resonant inductor. This method uses the ferrite material coefficients for the Steinmetz Equation with sinusoidal excitation. These coefficients for material 3C94 are as follows: k_c = 3.53 W/m³, α = 1.42, β = 2.885. Power losses in the inductor core using the Generalized Steinmetz Equation are given as (11). These power losses are given for the set temperature T = 100 °C.

$$P_{\mathrm{Lr}1\mathrm{c},100°C}=V_{\mathrm{c}}{\displaystyle \frac{1}{T_{\mathrm{S}}}}{{\displaystyle \underset{0}{\overset{T_{\mathrm{S}}}{\int }}{k}_1\left|\frac{\mathrm{d}B\left(t\right)}{\mathrm{d}t}\right|}}^{\mathsf{\alpha }}{\left|B\left(t\right)\right|}^{\mathsf{\beta }-\mathsf{\alpha }}\mathrm{d}t$$

(11)

where V_c is the core volume, T_S is the switching period, T_S = 1/f_S = 100 μs, k₁ is given as (12).

$$k_1={\displaystyle \frac{k_{\mathrm{c}}}{{\left(2\mathsf{\pi }\right)}^{\mathsf{\alpha }-1}{\displaystyle \underset{0}{\overset{2\mathsf{\pi }}{\int }}{\left|\cos \left(\mathsf{\theta }\right)\right|}^{\mathsf{\alpha }}}{\left|\sin \left(\mathsf{\theta }\right)\right|}^{\mathsf{\beta }-\mathsf{\alpha }}\mathrm{d}\mathsf{\theta }}}=1.251{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^3}}$$

(12)

B(t) is the magnetic flux density in the core of the inductor and is given as (13).

$$B\left(t\right)={\displaystyle \frac{L_{\mathrm{r}1}{i}_{\mathrm{Lr}1}\left(t\right)}{N{A}_{\mathrm{c}}}}$$

(13)

By representing the inductor current as a harmonic series, determining the magnetic flux density B(t) and its derivative dB(t)/dt is relatively simple. The Fourier series consists of odd harmonics, i.e., 1st, 3rd, 5th, and 7th. As an example, for an output power P_out = 7 kW, both magnetic flux density and its derivative are shown in Figure 10.

Substituting the relationships (12) and (13) into formula (11), the obtained power losses are for a temperature of 100 °C. At other temperatures, the losses in the magnetic core may be higher, and the exact relationship between temperature and losses is described for the 3C94 material with (14).

$$Y_{3C94}\left(T\right)= 1.25359\cdot {10}^{-4}{T}^2-0.02226T+1.97278$$

(14)

Multiplying core losses (11) by the function Y_3C94(T), temperature-dependent losses are obtained (15).

$$P_{\mathrm{Lr}1\mathrm{c}}\left(T\right)=P_{\mathrm{Lr}1\mathrm{c},100°C}{Y}_{3C94}\left(T\right)$$

(15)

Total power losses in the inductor L_r1 are the sum of the winding (10) and core losses (15) as given in (16) and are presented as functions of temperature in Figure 11a.

$$P_{\mathrm{Lr}1}\left(T\right)=P_{\mathrm{Lr}1\mathrm{w}}\left(T\right)+P_{\mathrm{Lr}1\mathrm{c}}\left(T\right)$$

(16)

The exact temperature value to which the inductor will heat up can be determined by the heat transfer relation P_diss(T). It is the relationship describing two heat transfer phenomena between the inductor and its surroundings at ambient temperature T_a. These phenomena are radiation P_rad(T) and convection P_conv(T) as given in (17)–(19) [28]. The amount of heat transferred from the inductor depends on the inductor temperature T, which is assumed to be uniform at its surface, the surface A from which the heat is dissipated, the ambient temperature, which is assumed to be T_a = 20 °C, the emissivity of the inductor surface E (assumed to be 0.9), and inductor height h.

$$P_{\mathrm{diss}}\left(T\right)=P_{\mathrm{rad}}\left(T\right)+P_{\mathrm{conv}}\left(T\right)$$

(17)

$$P_{\mathrm{rad}}\left(T\right)=\mathsf{\sigma }EA\cdot \left[{\left(T+273.15 °\mathrm{C}\right)}^4-{\left(T_{\mathrm{a}}+273.15°\mathrm{C}\right)}^4\right]$$

(18)

$$P_{\mathrm{conv}}\left(T\right)=1.34\cdot A\cdot {\displaystyle \frac{{\left(T+T_{\mathrm{a}}\right)}^{1.25}}{h^{0.25}}}$$

(19)

where σ is the Stefan-Boltzmann constant equal to 5.7 × 10⁻⁸ Wm⁻²K⁻⁴, the areas A in (18) and (19) for the inductor L_r1 are different because heat is not released from the bottom surface of the inductor via the convection process. Area A in (18) is 514.7 cm^2, and 407.8 cm² in (19). The characteristic of heat transfer P_diss is shown in Figure 11a together with the characteristics of the power losses of the inductor, P_Lr1. At the intersection of these characteristics, the exact value of power loss and inductor temperature is obtained. The resultant inductor power loss and temperature characteristics given for the specific output power P_out are shown in Figure 11b.

From Figure 11a, it can be seen that the power losses in the L_r1 inductor, as a function of temperature T, decrease with increasing T. This is because the dominant losses in the inductor are core losses. The function Y(T) for 3C94 material has a minimum near T = 100 °C. In all cases, power losses P_Lr1 increase with increasing output power P_out. The total power dissipation for inductor L_r1 at an output of 7 kW is 23.8 W at a fixed temperature of T_Lr1 = 60 °C.

3.2.2. Power Losses in Transformer Tr₁

Transformer Tr₁ is intended to step up the voltage in the coaxial cable relative to the voltage of converter INV1. The voltage in the coaxial cable line is 3.5 kV. The specifications of transformer Tr₁ are shown in

Table 4. Transformer Tr₁ core and winding specification.

Parameter	Symbol	Value
Magnetizing inductance	LTr1m	35 mH
Number of turns	Nprim/Nsec	27/118
Core size (all cores)	wm × h × l	80 mm × 76.2 mm × 198 mm
Primary winding resistance @20 °C	RTr1wp	144 mΩ
Secondary winding resistance @20 °C	RTr1ws	1.74 Ω

. As the transformer was custom-designed, not all of its parameters were available to the authors. Winding losses are evaluated from the measured winding resistances at 10 kHz, while core losses are determined from no-load loss measurements conducted under laboratory conditions, outside the complete system.

The magnetizing inductance L_Tr1m listed in Table 1 and Table 4 was obtained from no-load measurements performed at the actual operating voltage and frequency of the converters. This method provides more reliable values than small-signal impedance analysis, which does not reproduce the real magnetic operating point of medium-frequency transformers. Since both transformers Tr₁ and Tr₂ share identical geometry and core material, the same measurement procedure is applied to both.

The primary winding current is equal to the resonant inductor current. In the analysis, the magnetizing current of the transformer is neglected. The primary winding losses depend on the rms value of the winding current I_Lr1rms, winding resistance, and temperature T according to (20).

$$P_{\mathrm{Tr}1\mathrm{wp}}\left(T\right)={I_{\mathrm{Lr}1\mathrm{rms}}}^2{R}_{\mathrm{Tr}1\mathrm{wp}}\left(1+{\mathsf{\alpha }}_{\mathrm{Cu}}\left(T-20°\mathrm{C}\right)\right)$$

(20)

For example, for an output power of P_out = 7 kW, the rms current value is I_Lr1rms = 11.0 A. Assuming that the transformer heats up to 60 °C, the power losses calculated from (20) are 20.14 W.

Because the magnetizing current is assumed to be zero, the secondary winding current is given as (N_prim/N_sec)i_Lr1. Thus, the secondary winding power losses can be expressed as (21).

$$P_{\mathrm{Tr}1\mathrm{ws}}\left(T\right)={\left({\displaystyle \frac{N_{\mathrm{prim}}}{N_{\sec }}}{I}_{\mathrm{Lr}1\mathrm{rms}}\right)}^2{R}_{\mathrm{Tr}1\mathrm{ws}}\left(1+{\mathsf{\alpha }}_{\mathrm{Cu}}\left(T-20°\mathrm{C}\right)\right)$$

(21)

At an output power of P_out = 7 kW and a temperature of 60 °C, the power losses on the secondary winding (21) are 12.74 W.

The core losses of the transformer were measured during its no-load operation in the test setup shown in Figure 12. The resonant circuit, consisting of an inductor and a capacitor connected in parallel to the primary side of the transformer, has a resonant frequency of 10 kHz and a high quality factor. As a result, the voltage on the primary side of the transformer during the no-load loss measurements was practically sinusoidal.

In the test setup shown in Figure 12, the amplitude of the transformer primary-side voltage was controlled by the DC voltage of inverter INV1. To reproduce the transformer voltage conditions, it was assumed that the sinusoidal voltage amplitude during the no-load test would equal the supply system peak voltage. The peak transformer voltage is obtained as the algebraic sum of the harmonic voltage amplitudes (22).

$$V_{\mathrm{Tr}1\mathrm{m}}=V_{\mathrm{Tr}1,1\mathrm{h}}+V_{\mathrm{Tr}1,3\mathrm{h}}-V_{\mathrm{Tr}1,5\mathrm{h}}+V_{\mathrm{Tr}1,7\mathrm{h}}=1157 \mathrm{V}$$

(22)

where the amplitudes of the 1st, 3rd, 5th, and 7th harmonics of the transformer voltage, V_Tr1,1h, V_Tr1,3h, V_Tr1,5h, and V_Tr1,7h are equal to 965 V, 210 V, 25 V, and 7 V, respectively, the minus sign of the 5th harmonic follows from the fact that in the square-wave voltage of inverter INV1, the 1st and 5th harmonics reach their maximum at the same instant, while the other harmonics are negative. Due to the −180° phase shift in the transfer function (9) for the 3rd–7th harmonics, their signs are inverted.

In the no-load test of the transformer (Figure 9), the measured power losses were 32.5 W. These losses were attributed entirely to the core and assumed to be constant, independent of output power and temperature. No significant variation in no-load losses with temperature was observed. Although both output power and temperature affect core losses, these effects were neglected for simplicity. This assumption is justified because the no-load losses remain essentially constant, as the fundamental voltage component applied to the transformer does not vary with load and is the dominant contributor to core losses.

Total power losses in the transformer Tr₁ are the sum of losses of both windings (20), (21), and core losses P_Tr1c as given in (23) and are presented as functions of temperature in Figure 13a.

$$P_{\mathrm{Tr}1}\left(T\right)=P_{\mathrm{Tr}1\mathrm{wp}}\left(T\right)+P_{\mathrm{Tr}1\mathrm{ws}}\left(T\right)+P_{\mathrm{Tr}1\mathrm{c}}$$

(23)

For calculating the heat transfer from the transformer Tr₁, a similar approach is used to that applied to the inductor (17)–(19). It is assumed that the temperature of both transformer windings and the core is the same and uniformly distributed on the surface of the transformer. Another assumption is that the transformer is a cuboid without considering the dimensions of the winding extending beyond the core outline.

By analyzing the power loss characteristics shown in Figure 13a, it can be observed that the losses are less dependent on temperature compared to those in the L_r1 inductor. This is due to the behavior of core losses in the transformer, which are higher than in the inductor. As a result, the transformer losses do not drop below 39 W and increase up to 70 W at an output power of 7 kW. At this point, the transformer temperature reaches 90 °C.

The black and orange curves in Figure 13a correspond to 1 kW and 2 kW operation, respectively. Their close overlap results from dominant core losses, which are largely independent of the transferred power, and from nearly identical RMS current values, with the power increase mainly related to a change in the power factor (see Figure 8b).

In the developed HVHFCCET system, the transformer Tr₁ was installed inside an aluminum enclosure with proper electrical insulation, which allowed effective dissipation of the heat generated in the transformer, reducing its temperature by more than 20 °C.

In the HVHFCCET system, a second transformer Tr₂, with the exact specifications as Tr₁ is used. The voltage on the HV side of Tr₂ is assumed equal to the secondary voltage of Tr₁, since the voltage drop across the coaxial cable is negligible. Consequently, the core losses of Tr₂ are considered the same as those of Tr₁. Moreover, because the capacitive current of the coaxial cable has little effect on the RMS current of Tr₁, both transformers are assumed to operate under the same current conditions and thus exhibit equal power losses.

3.2.3. Power Losses in Coaxial Cable

In a coaxial cable, two factors cause power losses. The first is conduction losses in the copper centre conductor and braid shield, and the second is due to dielectric dissipation in a polyethylene insulator. The coaxial cable specifications are shown in

Table 5. Coaxial cable specification.

Parameter	Symbol	Value
Type of coaxial cable	---	RG11/U
Cable length	lcoax	100 m
Characteristic impedance	Z	75 Ω
Cable capacitance	Ccoax	6.7 nF
Cable resistance	Rcoax	2.26 Ω

. The inductance of the coaxial cable is much smaller than the leakage inductances at the high-voltage side of transformers and is neglected in the analysis.

The resistance R_coax listed in Table 5 includes contributions of the inner conductor and the shield, and is specified at 20 °C. The current in the coaxial cable is determined by the transformer primary side current i_Lr1 multiplied by the transformer turns ratio. It is assumed that this current flows through the entire coaxial cable, resulting in conduction losses according to (24). This means that, under this assumption, the capacitive current component of the coaxial cable does not spread evenly along its length; it flows unattenuated to the end of the cable. This assumption avoids using a distributed coaxial cable model and simplifies the analysis without causing significant errors.

$$P_{\mathrm{Coaxw}}\left(T\right)={\left({\displaystyle \frac{N_{\mathrm{prim}}}{N_{\sec }}}{I}_{\mathrm{Lr}1}\right)}^2{R}_{\mathrm{coax}}\left(1+{\mathsf{\alpha }}_{\mathrm{Cu}}\left(T-20°\mathrm{C}\right)\right)$$

(24)

The exact temperature T at which the coaxial cable heats up will be calculated after introducing its thermal model and analyzing power losses in the cable dielectric.

The cause of power losses in the dielectric of the coaxial cable is its operation at a high peak voltage (up to 5.06 kV), at a relatively high frequency (f_S = 10 kHz), with the presence of harmonics. The coaxial cable dielectric is made from polyethylene, which has a well-known relative permittivity ε_r = 2.3; however, its dissipation factor tan(δ) varies due to the value of the electric field strength applied, frequency, and temperature [29,30]. Knowing the dissipation factor, tan(δ), the applied voltage, which is the voltage at the secondary side of the transformer Tr₁, the operating frequency, and the capacitance of the coaxial cable, C_coax, the dielectric power loss can be determined by (25).

$$P_{\mathrm{Coaxd}}={\left({\displaystyle \frac{N_{\sec }}{N_{\mathrm{prim}}}}{\displaystyle \frac{V_{\mathrm{Tr}1\mathrm{m}}}{\sqrt{2}}}\right)}^2\mathsf{\omega }{C}_{\mathrm{coax}}\tan \left(\mathsf{\delta }\right)$$

(25)

In the literature, dissipation factor values are reported in the range of 10⁻⁴ to 10⁻². Such a wide spread results from the fact that these values are usually specified either at the power frequency (e.g., 50 Hz) or at different electric field strengths [31]. In the analyzed case, the electric field in the coaxial cable exceeds 3 kV/mm. The presence of higher harmonics in the voltage also causes a change in the dissipation factor [32].

Due to the difficulty in determining the loss tangent in a coaxial cable under high-voltage, high-frequency conditions, it was decided to perform two no-load measurement tests with the dual-resonant converter. In test 2 (Figure 14), a transformer Tr₂ was connected to the coaxial cable output, with its low-voltage side disconnected from the rest of the circuit. On the primary (low-voltage) side of transformer Tr₁, the power losses P_test2 (26) were measured. P_test2 corresponds to the no-load power losses of both transformers Tr₁ and Tr₂, as well as the dielectric losses in the coaxial cable, P_Coaxd. This approach is justified because the turn ratios of both transformers are the same, and under the assumption that the coaxial cable does not distort the voltage at the high-voltage sides of the transformers.

In the third test (Figure 15), the power losses were measured with the second transformer Tr_2, disconnected from the coaxial cable. In this case, the measured power P_test3 (27) corresponds to the sum of the no-load losses of a single transformer and the dielectric losses in the coaxial cable. By rearranging (26) and (27), the dielectric power losses (28) can be determined, which are assumed to be independent of the output power of the converter system.

$$P_{\mathrm{Test}2}= 2P_{\mathrm{Tr}}+P_{\mathrm{Coaxd}}$$

(26)

$$P_{\mathrm{Test}3}=P_{\mathrm{Tr}}+P_{\mathrm{Coaxd}}$$

(27)

$$P_{\mathrm{Coaxd}}=2P_{\mathrm{Test}3}-P_{\mathrm{Test}2}$$

(28)

The losses in both tests were measured using an impedance analyzer on the primary side of transformer Tr₁, while ensuring that the maximum voltage was close to the rated value V_Tr1m = 1157 V. In Test 2, the measured power was P_Test2 = 77.8 W. In contrast, in Test 3, it was P_Test3 = 44.9 W. Based on (28), the coaxial dielectric losses were determined as P_coaxd = 12.0 W.

Similarly to transformer core losses, it was assumed that dielectric losses are independent of the power transmitted through the coaxial cable and of the temperature, since no variations in the cable temperature were observed during the tests.

By substituting into (25) the values of voltage and power losses P_coaxd, and taking into account the cable capacitance C_coax = 6.7 nF, it can be determined that the dissipation factor tan(δ) = 0.0022.

3.2.4. Power Losses in Converter INV1

The INV1 converter is a resonant full-bridge converter that generates a rectangular output voltage at 10 kHz. The same MOSFET devices as in the REC1 converter are employed in its design. The DC-link voltage is also identical to that of REC1, Vdc = 760 V. Therefore, the same loss calculation method as for the transistors of REC1 can be applied, except that the output current is non-sinusoidal. The transistors may operate under ZVS conditions. Additionally, in this case, the operating frequency is 10 kHz, and the number of transistors is four. The conduction losses are expressed by (29).

$$P_{\mathrm{Tcon}}=2R_{\mathrm{DSon}}{I_{\mathrm{Lr}1\mathrm{rms}}}^2$$

(29)

The factor of 2 in (29) arises from the fact that, in the INV1 converter, two transistors are always conducting. Knowing the rms value of the output current, which is also the current of the resonant inductor L_r1 and is shown in Figure 8, the conduction loss characteristic as a function of the output power can be determined, as presented in Figure 16.

It is worth emphasizing that the conduction- and switching-loss models used in this work remain consistent across REC1, INV1, and INV2 despite the different modulation strategies applied in these converters. This results from the use of MOSFET-based 2L-VSI topologies, whose loss characteristics are fundamentally simpler than those of IGBT counterparts. Unlike IGBTs, MOSFETs exhibit symmetric bidirectional conduction through the channel without threshold-voltage–dependent conduction behavior. As a consequence, the analytical expressions for conduction losses (1), (29) differ only in the number of conducting converter legs. Similarly, the influence of PWM type (SPWM, third-harmonic injection, or square-wave operation in the case of INV1) on switching losses is limited, provided that the current waveforms at the switching instants remain comparable. For these reasons, a unified MOSFET loss model is used throughout the analysis, and factors such as rms currents, the MOSFET on-state resistance, and the number of converter legs differentiate the resulting loss profiles. This also explains why the results obtained here cannot be directly compared with those from IGBT-based studies [33], where the modulation strategy has a much greater impact on the analytical formulations of both conduction and switching losses.

The switching power losses in the INV1 converter are caused by nonzero current values at the switching instants. Since the input impedance Z_in(ω) at the switching frequency f_S = 10 kHz is capacitive (Figure 9), ZVS conditions are not achieved at low load power. Only for P_out > 3.5 kW does the circuit characteristic become inductive, enabling ZVS operation. An example waveform of the output current of the INV1 converter, i_Lr1, at P_out = 4.5 kW is shown in Figure 17.

Under ZVS conditions, power losses are generated during transistor turn-off, whereas at low output power P_out, when ZVS conditions are not satisfied, losses occur during turn-on. Based on the current waveforms obtained from simulation (Figure 8), the current values were determined as I_on at t = 0, I_on = i_Lr1(0), and I_off = i_Lr1(1/(2f_S)). Since the INV1 converter has four transistors, the switching losses are expressed by (30).

$$P_{\mathrm{Tsw}}=4f_{\mathrm{S}}\left(e_{\mathrm{on}}\left(I_{\mathrm{on}}\right)+e_{\mathrm{off}}\left(I_{\mathrm{off}}\right)\right){\displaystyle \frac{V_{\mathrm{dc}}}{V_{\mathrm{dc}}^{\prime }}}$$

(30)

In (30), the switching frequency is f_S = 10 kHz, the DC-link voltage is V_dc = 760 V, and the voltage at which the datasheet switching losses were measured is V’_dc = 600 V.

The energy values e_on and e_off were taken from the datasheet with correction factors applied to account for the influence of the gate resistance on switching energy losses. Figure 16 presents the switching energy loss characteristic as a function of the output power.

It can be observed that higher switching power losses occur at low output power, where no ZVS conditions exist, whereas above 3.5 kW, the losses stabilize at a constant level. This is mainly because the applied MOSFET devices exhibit higher turn-on than turn-off losses, resulting in a significant reduction in switching power losses under ZVS operation. Furthermore, it should be noted that the lower switching power losses in the INV1 converter compared to the REC1 converter are due to the INV1 converter’s resonant nature and a switching frequency that is 5 times lower than that of the REC1 converter.

3.2.5. Power Losses in Converter REC2

The REC2 rectifier is built using 1200 V SiC Schottky barrier diodes [34] due to the output characteristics of the diodes (Figure 18a), conduction losses will occur as expressed by (31). Although the conduction characteristic is temperature-dependent, in this analysis, it is assumed that the diodes operate at a constant temperature of 50 °C. This corresponds to the maximum diode temperature observed during the tests, thereby simplifying the power-loss analysis.

$$P_{\mathrm{Dcon}}=4{\displaystyle \frac{1}{T_{\mathrm{S}}}}{\displaystyle \underset{0}{\overset{T_{\mathrm{S}}}{\int }}{v}_{\mathrm{D}}\left(i_{\mathrm{Lr}2}\right)\cdot i_{\mathrm{Lr}2}} \mathrm{d}t$$

(31)

In (31), the integration is carried out over the whole switching period T_S = 1/f_S = 100 μs, even though the diodes conduct current for roughly half of it. This approach is motivated by oscillations in the rectifier input current waveforms at low current levels, which make the diode conduction time ambiguous. Such a definition allows the conduction losses in the diodes to be properly determined, since the diode voltage drop v_D(i_Lr2) is equal to zero for negative values of i_Lr2, as shown in Figure 18b,c. One can see that oscillations of the i_Lr2 at P_out = 2 kW do not influence the diode’s instantaneous power.

The conduction power losses characteristic of all four diodes in REC2 as a function of the output power P_out is shown in Figure 19.

SiC Schottky diodes have zero reverse-recovery charge, but their junction capacitance still causes switching losses [35,36]. The switching losses, described by (32), scale linearly with the rectifier dc-link voltage. It should also be noted that during the tests, the output voltage v_dc2 depended on the output power and varied linearly, taking the values v_dc21 = 788 V at P_out1 = 2.2 kW, and v_dc22 = 711 V at P_out2 = 5.8 kW.

$$P_{\mathrm{Dsw}}=4f_{\mathrm{S}}{Q}_{\mathrm{D}}\left(v_{\mathrm{dc}2}\right)v_{\mathrm{dc}2}$$

(32)

By considering both linear functions in (32), the switching power loss characteristic of the REC2 rectifier diodes can be determined. This characteristic, together with the total converter losses, is presented in Figure 19.

Based on Figure 19, the total power losses in the REC2 converter at the maximum output power of P_out = 7 kW are comparable to those in the INV2 converter. The switching losses decrease with increasing P_out, due to the reduction in v_dc2.

3.2.6. Power Losses in Dual Resonant Converter Tracks and Wires

The last factor contributing to power losses in the dual resonant converter is the resistance of the wires, PCB traces, and the inductors connecting the DC circuits of REC1 to INV2 and REC2 to INV2. The total connection resistance is 62 mΩ, consisting of 20 mΩ from PCB traces, 10 mΩ from wires, and 32 mΩ from the dc-link inductors. It is assumed, as a simplification, that the inductor current i_Lr1 flows through all additional resistances. At an output power of P_out = 7 kW, the rms current is 11 A, resulting in a total power loss of 7.5 W in the wires and traces. Due to their negligible influence, the series resistances of the DC-link capacitors are not considered in the power loss calculation.

3.2.7. Power Losses Generated in the Dual Resonant Converter

The total power losses in the components of the dual resonant converter are shown in Figure 20. These losses are compared with the power losses measured during the converter’s efficiency test. The output power was limited to 8 operating points.

Figure 20 shows that the calculated power losses closely match the measured losses. Based on this, it can be concluded that the presented power loss analysis accurately reflects the actual power losses in the dual resonant converter.

3.3. Output Converter Losses

In the output stage, the INV2 converter operates as a three-phase inverter supplying the load. Its operation is analogous to that of the REC1 converter (Figure 3), except that the phase current direction is reversed. The INV2 converter uses the same SiC MOSFET devices as in the REC1 converter. The passive filters employed in INV2 share the same structure and parameters as those of the grid-side converter.

An additional distinction, beyond the reversed direction of phase currents, is the observed decrease in the dc-link voltage between converters REC2 and INV2 during system-level testing. Variations in the dc-link voltage, v_dc2, are compensated at the INV2 converter output by adjusting the modulation index. For power loss analysis, the same relationships applied to the REC1 converter can be extended to the INV2 converter; nevertheless, the reduction in DC-link voltage results in lower switching losses. The total power losses of the INV2 converter are presented as a function of the output power in Figure 21.

4. Conclusions

This paper presents an analysis of power losses in a converter system employing a high-voltage, high-frequency coaxial cable energy transfer. All significant sources of losses have been identified, and the loss characteristics are presented as functions of output power and, in some cases, temperature. Temperature-dependent loss characteristics are essential for modeling the losses in inductive components.

Figure 22 illustrates the distribution of power losses across the individual components of the power transmission system. The power losses for this example were calculated based on the approximation of the characteristics of all components of the converter system and are presented for an output power of 7 kW.

It can be observed that the highest power losses occur in the dual resonant converter, P_INV1REC2 = 267.4 W. This is primarily due to the large number of components in this converter, the presence of two voltage conversion stages, and the inclusion of both the INV1 and REC2 converters. The other converters exhibit lower losses, with the lowest observed in the INV2 converter. This reduction is attributed to its operation at a lower DC-link voltage compared to the REC1 converter. The lower losses are mainly due to the reduction in switching losses, which are the dominant loss component in both REC1 and INV2 converters.

The total power losses in the HVHFCCET system at an output power of 7 kW amount to 509 W, yielding an efficiency of 93.2%. The most considerable losses are observed in the resonant transformers. In subsequent research carried out by the authors, the transformers will be subjected to detailed analysis and optimized for efficiency.

It is worth emphasizing that the investigated coaxial cable line had a length of 100 m, for which the measured cable losses were 26.5 W at the maximum output power of 7 kW. Assuming linear scaling of resistive losses, extending the cable length to 1 km would result in losses of approximately 265 W.

For comparison, an equivalent conventional three-phase cable line operating at 500 V and 50 Hz, realized using a three-core copper cable with a conductor cross-section of 4 mm², would exhibit losses of approximately 907 W at the same transmitted power. Consequently, for a 1 km transmission distance, cable losses in the three-phase system are more than 3.4 times higher than those of the coaxial cable.

When total system losses are considered, the HVHFCCET system with a 1 km coaxial cable would exhibit overall losses of approximately 747 W, which are lower than the losses in the three-phase cable alone. The crossover point, at which the total losses of both systems become equal, occurs at a cable length of approximately 750 m. Below this distance, conventional three-phase transmission remains more efficient, as confirmed by the experimental 100 m setup, for which the HVHFCCET system losses reached 509 W compared to calculated 90.7 W in the three-phase cable.

An additional advantage of the HVHFCCET approach is the significantly lower mass of the transmission line. The mass of the RG-11/U coaxial cable is approximately 140 kg/km, compared to about 252 kg/km for a non-shielded three-core cable. Considering that armored cables are typically required in mining applications, the proposed coaxial-based solution can be up to two times lighter than conventional three-phase cable systems.

The developed power transmission system was designed for underground coal mines and, due to the need to limit earth-return currents, employed a coaxial cable. In future research, the authors plan to use the developed transmission system for Single Wire Energy Transfer (SWET), including overhead lines. It is anticipated that the resonant circuits will need to be redesigned to maximize energy transfer through the ground, i.e., to make the system independent of the unpredictable ground impedance, which varies with line location and weather conditions.