Analysis and Design of Wireless Power Transfer Systems Applied to Electrical Vehicle Supercapacitor Charge Using Variable-Resistance-Based Method

: Supercapacitors (SCs) are widely used as energy storage devices in many practical applications of wireless power transfer. However, the modeling and analysis of a wireless power transfer system are seldom based on SC load; thus, the effects of the charging process on the wireless power transfer system cannot be analyzed clearly. In this paper, a variable-resistance-based method is proposed for the modeling and analysis of the process of constant current charging. First, how to make an SC equivalent in variable resistance is described, and the relationship between SC and variable resistance is considered. Next, the charging process, including charging current, voltage, power and transfer efﬁciency, is analyzed in detail. Furthermore, the effects of transmitting side voltage and frequency offset on this system are studied, and the optimal design method for an SC-load WPT system is provided on a preliminary basis. Finally, the theoretical derivation and analysis are veriﬁed by means of simulations and experiments.


Introduction
Wireless power transfer (WPT) technology has been developed in recent years as a new technology for electric power transmission.WPT technology avoids direct electrical connection, and, thus, brings many advantages, such as the elimination of sparking and risk of electric shock, safe and convenient charging, and ease of achieving an automatic power supply [1][2][3].Due to these obvious advantages, WPT has been widely tested and used in many fields, such as rail transportation vehicles, household appliances, biomedical equipment and other applications.WPT technology is conducive to reducing energy storage devices and extending mileage for practical rail transportation vehicles.Combined with energy storage devices, WPT technology can promote the development of electric vehicles [4][5][6].
In some practical applications of WPT technology, there are various types of loads, such as battery, supercapacitor (SC), and electric motor [7][8][9].In the application of rail transit vehicles, an SC is considered suitable as an on-board energy storage device, because of the following characteristics: (1) high power density; (2) accurate measurement of state of charge; (3) long cycle-life and low maintenance cost; and (4) lack of use of heavy-metal chemical materials and, thus, low environmental pollution [10,11].With these advantages, SC can be chosen as a WPT system load for power supply.For such a WPT system applied in electric vehicles, a three-phase rectifier is used to convert the grid alternating current (AC) voltage to the direct current (DC) bus voltage.Typically, the transmitter is fed with a high frequency AC current which is converted via a single-phase inverter and generates an AC magnetic flux linkage; then, the receiver induces an AC current for supplying power to the load.As the SC load requires a small DC current ripple, a filter is typically connected between the AC/DC rectifier and SC, as shown in Figure 1.When the load and system parameters are designed reasonably and remain constant, a constant-load-resistance WPT system can easily work stably and effectively.However, the output voltage and power of such an SC-load WPT system will vary during charging under a constant charging current, which will lead to changes in the system characteristics.This is not favorable in terms of the efficient and stable operation of the system.
Energies 2022, 15, x FOR PEER REVIEW 2 of 16 current for supplying power to the load.As the SC load requires a small DC current ripple, a filter is typically connected between the AC/DC rectifier and SC, as shown in Figure 1.
When the load and system parameters are designed reasonably and remain constant, a constant-load-resistance WPT system can easily work stably and effectively.However, the output voltage and power of such an SC-load WPT system will vary during charging under a constant charging current, which will lead to changes in the system characteristics.This is not favorable in terms of the efficient and stable operation of the system.At present, the analysis of WPT systems is typically based on constant load resistance.The rectifier and load in the receiver side are generally considered as a constant load resistance [12][13][14].Similarly, when studying only the variation of the coupling coils and system frequency, the change in load is rarely considered [15][16][17].Furthermore, [9,18] used a constant-voltage load model to analyze the battery load.These studies were based on the constant value of a load, which is not fully suitable for WPT systems which use the SC as the final load.The application of SC in a WPT system can be found in [19][20][21][22][23][24].However, [19] presents a technique for the suppression of voltage transients through use of dynamically reconfigurable SC banks, in which SC is not a load, but, instead, a parallel energy system.The authors of [20] considered an SC optimal control method in order to satisfy the requirements of efficiency and power; however, they did not analyze the charging characteristics.For the modeling and analysis of an SC-load WPT system, [21,22] used a classical SC model as the WPT system load, but only established the relationship between the voltage, the current of the SC and the equivalent resistance of the WPT system, but did not consider the whole WPT system for analysis.The authors of [23,24] attempted to develop an SC as the WPT system load for relevant applications, but did not analyze system models.To solve the problem, the present paper proposes a variable-resistancebased (VRB) method which can describe the variation in the charging process of SCs.The analytical derivations and analysis could serve as a basis for developing WPT systems using SCs as on-board energy storage devices.
The remainder of this paper is organized as follows: In Section 2, the configuration of the SC-load WPT system is shown.A variable-resistance-based method is proposed for analyzing such a system, and the method is explained.The voltages and currents of the transmitter and receiver are derived according to the VRB method.In addition, the relationship between the output power and transfer efficiency with the voltage of an SC are calculated.In Section 3, the variations in the charging process with time are emphasized, At present, the analysis of WPT systems is typically based on constant load resistance.The rectifier and load in the receiver side are generally considered as a constant load resistance [12][13][14].Similarly, when studying only the variation of the coupling coils and system frequency, the change in load is rarely considered [15][16][17].Furthermore, refs.[9,18] used a constant-voltage load model to analyze the battery load.These studies were based on the constant value of a load, which is not fully suitable for WPT systems which use the SC as the final load.The application of SC in a WPT system can be found in [19][20][21][22][23][24].However, ref. [19] presents a technique for the suppression of voltage transients through use of dynamically reconfigurable SC banks, in which SC is not a load, but, instead, a parallel energy system.The authors of [20] considered an SC optimal control method in order to satisfy the requirements of efficiency and power; however, they did not analyze the charging characteristics.For the modeling and analysis of an SC-load WPT system, refs.[21,22] used a classical SC model as the WPT system load, but only established the relationship between the voltage, the current of the SC and the equivalent resistance of the WPT system, but did not consider the whole WPT system for analysis.The authors of [23,24] attempted to develop an SC as the WPT system load for relevant applications, but did not analyze system models.To solve the problem, the present paper proposes a variable-resistance-based (VRB) method which can describe the variation in the charging process of SCs.The analytical derivations and analysis could serve as a basis for developing WPT systems using SCs as on-board energy storage devices.
The remainder of this paper is organized as follows: In Section 2, the configuration of the SC-load WPT system is shown.A variable-resistance-based method is proposed for analyzing such a system, and the method is explained.The voltages and currents of the transmitter and receiver are derived according to the VRB method.In addition, the relationship between the output power and transfer efficiency with the voltage of an SC are calculated.In Section 3, the variations in the charging process with time are emphasized, including the voltage of an SC, the output power of the WPT system, and the transfer efficiency.The effect of the transmitter voltage on transfer efficiency is analyzed.Furthermore, the effects of frequency offset on the WPT system, including the effects on voltage gain and current gain, are studied, and then the design process for the SC-load WPT system is described.In Section 4, a simulation model and experimental platform are built to verify the theoretical analysis and calculation.Finally, the conclusions are drawn in Section 5.

WPT System Model Using
The topology of the SC-load WPT system in Figure 1 can be modeled by circuit theory, as shown in Figure 2. The system usually consists of a DC bus voltage, an inverter (working at tens of kHz), transmitting and receiving coils, a rectifier, a filter and an SCload, where U bus is the DC bus voltage, U SC is the SC charging voltage, S1-S4 are the full-controlled semiconductor switches, D 1 -D 4 are the diodes, R 1 (R 2 ), L 1 (L 2 ) and C 1 (C 2 ) are the transmitting (receiving) side coil resistance, inductance and series compensation capacitor, M is the mutual inductance of the transmitting and receiving side coils, C is an SC equivalent capacitor, C d and L d are a DC side filter capacitor and inductor, respectively, I 1 and I 2 are the root-mean-square (RMS) value of the transmitter current and the receiver current, respectively, and I SC and U SC are the RMS values of the SC charging current and voltage, respectively.
Energies 2022, 15, x FOR PEER REVIEW 3 of 16 including the voltage of an SC, the output power of the WPT system, and the transfer efficiency.The effect of the transmitter voltage on transfer efficiency is analyzed.Furthermore, the effects of frequency offset on the WPT system, including the effects on voltage gain and current gain, are studied, and then the design process for the SC-load WPT system is described.In Section 4, a simulation model and experimental platform are built to verify the theoretical analysis and calculation.Finally, the conclusions are drawn in Section 5.

Variable-Resistance-Based WPT System Model
The topology of the SC-load WPT system in Figure 1 can be modeled by circuit theory, as shown in Figure 2. The system usually consists of a DC bus voltage, an inverter (working at tens of kHz), transmitting and receiving coils, a rectifier, a filter and an SCload, where Ubus is the DC bus voltage, USC is the SC charging voltage, S1-S4 are the fullcontrolled semiconductor switches, D1-D4 are the diodes, R1 (R2), L1 (L2) and C1 (C2) are the transmitting (receiving) side coil resistance, inductance and series compensation capacitor, M is the mutual inductance of the transmitting and receiving side coils, C is an SC equivalent capacitor, Cd and Ld are a DC side filter capacitor and inductor, respectively, I1 and I2 are the root-mean-square (RMS) value of the transmitter current and the receiver current, respectively, and ISC and USC are the RMS values of the SC charging current and voltage, respectively.The full-bridge inverter transforms DC power into high-frequency AC power, and the receiving side rectifier converts AC power to DC power.Therefore, the AC voltages of the inverter and rectifier are square waves with amplitudes of Ubus and USC, respectively.According to the Fourier decomposition, only the fundamental frequency component is considered because the high-order harmonic components are so small, and these harmonics can hardly be transmitted.The equivalent circuit is demonstrated in Figure 3, where U1 and U2 are the fundamental RMS values of the AC side voltages of the inverter and the rectifier, I1 and I2 are the transmitter current and receiver current, and Rv is the equivalent resistance load for the receiving coil.The full-bridge inverter transforms DC power into high-frequency AC power, and the receiving side rectifier converts AC power to DC power.Therefore, the AC voltages of the inverter and rectifier are square waves with amplitudes of U bus and U SC , respectively.According to the Fourier decomposition, only the fundamental frequency component is considered because the high-order harmonic components are so small, and these harmonics can hardly be transmitted.The equivalent circuit is demonstrated in Figure 3, where U 1 and U 2 are the fundamental RMS values of the AC side voltages of the inverter and the rectifier, I 1 and I 2 are the transmitter current and receiver current, and R v is the equivalent resistance load for the receiving coil.
including the voltage of an SC, the output power of the WPT system, and the transfer efficiency.The effect of the transmitter voltage on transfer efficiency is analyzed.Furthermore, the effects of frequency offset on the WPT system, including the effects on voltage gain and current gain, are studied, and then the design process for the SC-load WPT system is described.In Section 4, a simulation model and experimental platform are built to verify the theoretical analysis and calculation.Finally, the conclusions are drawn in Section 5.

Variable-Resistance-Based WPT System Model
The topology of the SC-load WPT system in Figure 1 can be modeled by circuit theory, as shown in Figure 2. The system usually consists of a DC bus voltage, an inverter (working at tens of kHz), transmitting and receiving coils, a rectifier, a filter and an SCload, where Ubus is the DC bus voltage, USC is the SC charging voltage, S1-S4 are the fullcontrolled semiconductor switches, D1-D4 are the diodes, R1 (R2), L1 (L2) and C1 (C2) are the transmitting (receiving) side coil resistance, inductance and series compensation capacitor, M is the mutual inductance of the transmitting and receiving side coils, C is an SC equivalent capacitor, Cd and Ld are a DC side filter capacitor and inductor, respectively, I1 and I2 are the root-mean-square (RMS) value of the transmitter current and the receiver current, respectively, and ISC and USC are the RMS values of the SC charging current and voltage, respectively.The full-bridge inverter transforms DC power into high-frequency AC power, and the receiving side rectifier converts AC power to DC power.Therefore, the AC voltages of the inverter and rectifier are square waves with amplitudes of Ubus and USC, respectively.According to the Fourier decomposition, only the fundamental frequency component is considered because the high-order harmonic components are so small, and these harmonics can hardly be transmitted.The equivalent circuit is demonstrated in Figure 3, where U1 and U2 are the fundamental RMS values of the AC side voltages of the inverter and the rectifier, I1 and I2 are the transmitter current and receiver current, and Rv is the equivalent resistance load for the receiving coil.Based on the Fourier decomposition of the square wave, U 1 and U 2 are expressed as follows: Considering a WPT system charging for an SC with constant I SC , and according to the characteristics of an SC, the terminal voltage will increase, which will lead to the equivalent resistance of SC varying, since its equivalent resistance R SC can be defined as U SC divided by I SC .To consider the influence of using the SC as a load, the equivalent transformations of the rectifier and SC are adopted.
First, let the equivalent impedance seen by the receiver be a resistive load R v , as shown in Figure 3.The key is to determine the relationship between the SC and R v ; this method is known as the variable-resistance-based method.
R v can be calculated as: Based on the above description, the AC impedance method can be applied to approximately analyze the steady-state performance, as shown in Figure 3.According to Kirchhoff's voltage law (KVL), the transmitter and receiver voltages and currents in the equivalent circuit model are given as follows: .
where ω is the real operating angular frequency, and Z 1 and Z 2 are the transmitter and receiver coil impedance, as follows: The I 1 , I 2 , output power (P OUT ) and transfer efficiency (η trn ) are solved as follows: .

Variable-Resistance Load Derivations
Typically, a transmitter and receiver are required under resonance to enhance the system transfer capability and efficiency.The compensation is a series-series method because the values of the compensation capacitor are not affected by load, and it is easily achieved.In addition, the series-series compensation has the characteristics of a constant output current, which is suitable for SC charging.
The resonant angular frequencies of the transmitter and receiver are obtained as follows: In this case, Equation ( 5) can be simplified as follows: .
If the self-resistance of the coil is ignored, the I 2 is solved as follows: Energies 2022, 15, 5867 5 of 15 It can be found that, if U 1 , ω and M of the WPT system are all constant, I 2 can remain constant.Therefore, the SC charging current is also constant: In this case, U SC can be calculated.Now u SC is used to replace U SC , because the voltage is varying at the process of charging; thus, u SC can also be expressed as follows: where t char is the charging time, and U SC_0 is the initial voltage of SC.Substituting (11) into (12), u SC can be rewritten as follows: Therefore, the charging current and voltage of the SC power supplied by the WPT system are determined.Furthermore, substituting (11) into (2), R v can be simplified as follows: R v will change when u SC increases; it is also determined by U 1 , ω and M.
The power level of the system determines the choice of devices and the operating frequency ω of the system.Therefore, it is necessary to calculate and analyze the power level as follows: As shown in Figure 3, the transfer efficiency is defined as the power consumed by R v divided by the power generated by the source.Taking the transmitter and receiver power losses into consideration, the transfer efficiency is derived as follows: Considering the charging process of SC causing the variation of R v , the efficiency of the WPT system using the SC as load can be obtained as follows: Using the VRB method, it is easy to relate u SC to the WTP system parameters.

Variation in Charging Process with Time
In the process of charging for SC, R v , u SC , P OUT and η trn vary with the charging time.First, the charging time will cause a linear change of u SC under the constant current.Then, R v will increase linearly, which will result in I 1 changing, as shown in Equation (10).Next, U 2 will increase, then P OUT will increase with the charging time.In practical conditions, the variations will occur at the same time.Based on the above deviations and analysis, the relationship between these key parameters is shown as Figure 4.
In the process of charging for SC, Rv, uSC, POUT and ηtrn vary with the charging time.First, the charging time will cause a linear change of uSC under the constant current.Then, Rv will increase linearly, which will result in I1 changing, as shown in Equation (10).Next, U2 will increase, then POUT will increase with the charging time.In practical conditions, the variations will occur at the same time.Based on the above deviations and analysis, the relationship between these key parameters is shown as Figure 4.The SC voltage with charging time is shown as (13).Similarly, Rv and POUT with charging time are calculated as follows: In practical applications, especially in high-power applications, such as rail transportation, WPT system efficiency is particularly significant.A miniscule decline in efficiency will cause much energy loss.In addition, the transfer efficiency of the SC-load WPT system is different from other types of load and is variable over a wide range.This is why it is necessary to analyze the transfer efficiency, which is shown as (17), and the transfer efficiency is maximized when: Then, the optimal value (uSC_OPT_η) of uSC can be solved as follows: The maximum transfer efficiency (ηtrn_max) is derived as follows: The SC voltage with charging time is shown as (13).Similarly, R v and P OUT with charging time are calculated as follows: In practical applications, especially in high-power applications, such as rail transportation, WPT system efficiency is particularly significant.A miniscule decline in efficiency will cause much energy loss.In addition, the transfer efficiency of the SC-load WPT system is different from other types of load and is variable over a wide range.This is why it is necessary to analyze the transfer efficiency, which is shown as (17), and the transfer efficiency is maximized when: Then, the optimal value (u SC_OPT_η ) of u SC can be solved as follows: The maximum transfer efficiency (η trn_max ) is derived as follows: The transfer coil quality factor (T Q ) [9] is defined as follows: Considering that T Q varies from 10 to 400, the theoretical result is shown as Figure 5.It can be seen that higher T Q is beneficial to η trn_max .From (33), it can be seen that high ω, large M and small R 1 /R 2 can improve the maximum transfer efficiency.

Effects of Transmitter Voltage on Efficiency
According to Equation (17), if the coupling coil is designed well, based on (23), a high TQ can enhance the transfer efficiency, and the transfer efficiency can be determined by U1.In addition, U1 is converted via an inverter.Typically, the phase-shift control is adopted to adjust U1.According to the Fourier decomposition, the fundamental harmonic value of the transmitter voltage U1 can be expressed as follows: where α is the phase-shift angle.Substituting Equation ( 25) into ( 17), the efficiency can be rewritten as follows: If the transmitter and receiver are almost the same, R1 is equal to R2.Then, uSC_OPT_η can be rewritten as follows: The experimental prototype of the SC-load WPT system parameters are tabulated in Table 1.For this platform, its T Q is equal to 35.47; thus, the maximum efficiency is about 0.9452.However, as discussed above, the will vary over a wide range.In addition, the variation in transfer efficiency must be analyzed.Substituting the system parameters into (17), the transfer efficiency vs. u SC is plotted in Figure 6.When using the WPT system charging an SC, u SC increases from 0 V to 100 V, and the transfer efficiency tends to rise, then fall.When u SC is up to u SC_OPT_η , η trn_max can be obtained.Therefore, in practical applications, a suitable work voltage range of the SC should be chosen to obtain high efficiency.

Effects of Transmitter Voltage on Efficiency
According to Equation ( 17), if the coupling coil is designed well, based on ( 23), a high TQ can enhance the transfer efficiency, and the transfer efficiency can be determined by U1.In addition, U1 is converted via an inverter.Typically, the phase-shift control is adopted to adjust U1.According to the Fourier decomposition, the fundamental harmonic value of the transmitter voltage U1 can be expressed as follows:

Effects of Transmitter Voltage on Efficiency
According to Equation (17), if the coupling coil is designed well, based on (23), a high T Q can enhance the transfer efficiency, and the transfer efficiency can be determined by U 1 .In addition, U 1 is converted via an inverter.Typically, the phase-shift control is adopted to adjust U 1 .According to the Fourier decomposition, the fundamental harmonic value of the transmitter voltage U 1 can be expressed as follows: where α is the phase-shift angle.Substituting Equation ( 25) into ( 17), the efficiency can be rewritten as follows: If the transmitter and receiver are almost the same, R 1 is equal to R 2 .Then, u SC_OPT_η can be rewritten as follows: First, only U bus is considered, the phase-shift angle is assumed as zero, then (27) can be simplified as u SC_OPT = U bus .For such a WPT system, if we seek to maintain high efficiency, the work range of SC should be close to U bus , or U bus should be controlled to match the SC charging voltage.
Next, the effect of the diffident DC bus voltage on the transfer efficiency is considered.Letting U bus of Equation ( 26) vary from 10 V to 100 V, the results are shown as Figure 7.When U bus increases, the maximum transfer efficiency can be obtained when U bus is the same as u SC_OPT_η , as the red line shows.It can be found from the surface that, if the SC-load WPT system is intended to maintain a high transfer efficiency in the process of charging, then U bus should be optimally designed.For example, based on the Table 1, if u SC operates on 30 V to 50 V, a 40 V DC bus voltage can be chosen.First, only Ubus is considered, the phase-shift angle is assumed as zero, then (27) can be simplified as uSC_OPT = Ubus.For such a WPT system, if we seek to maintain high efficiency, the work range of SC should be close to Ubus, or Ubus should be controlled to match the SC charging voltage.
Next, the effect of the diffident DC bus voltage on the transfer efficiency is considered.Letting Ubus of Equation ( 26) vary from 10 V to 100 V, the results are shown as Figure 7.When Ubus increases, the maximum transfer efficiency can be obtained when Ubus is the same as uSC_OPT_η, as the red line shows.It can be found from the surface that, if the SCload WPT system is intended to maintain a high transfer efficiency in the process of charging, then Ubus should be optimally designed.For example, based on the Table 1, if uSC operates on 30 V to 50 V, a 40 V DC bus voltage can be chosen.Then, the phase-shift angle requires to be considered.Assuming that Ubus is a constant 60 V and R1 is equal to R2, α is normally controlled in the range of 0° to 180°, and the radian is about 0-3, then the results obtained are those plotted in Figure 8.The results show that the transfer efficiency can be adjusted by the value of the phase-shift angle.When Equation ( 27) is satisfied, the maximum transfer efficiency can be obtained, as shown by the red line.Then, the phase-shift angle requires to be considered.Assuming that U bus is a constant 60 V and R 1 is equal to 2 , α is normally controlled in the range of 0 • to 180 • , and the radian is about 0-3, then the results obtained are those plotted in Figure 8.The results show that the transfer efficiency can be adjusted by the value of the phase-shift angle.When Equation ( 27) is satisfied, the maximum transfer efficiency can be obtained, as shown by the red line.
Then, the phase-shift angle requires to be considered.Assuming that Ubus is a constant 60 V and R1 is equal to R2, α is normally controlled in the range of 0° to 180°, and the radian is about 0-3, then the results obtained are those plotted in Figure 8.The results show that the transfer efficiency can be adjusted by the value of the phase-shift angle.When Equation ( 27) is satisfied, the maximum transfer efficiency can obtained, as shown by the red line.The transmitter voltage is decided by both the DC bus voltage and the phase-shift angle, and these two factors further affect the transfer efficiency.

Effects of Frequency Offset on WPT System
In practical applications of the WPT system, the value of the compensation capacitor cannot be selected continuously, and the operating frequency may need to be adjusted to achieve the soft-switching of the inverter.Therefore, it is possible to cause the operating frequency to deviate from the resonant frequency, so it is necessary to analyze the effects of frequency offset on the SC-load WPT system.The transmitter voltage is decided by both the DC bus voltage and the phase-shift angle, and these two factors further affect the transfer efficiency.

Effects of Frequency Offset on WPT System
In practical applications of the WPT system, the value of the compensation capacitor cannot be selected continuously, and the operating frequency may need to be adjusted to achieve the soft-switching of the inverter.Therefore, it is possible to cause the operating frequency to deviate from the resonant frequency, so it is necessary to analyze the effects of frequency offset on the SC-load WPT system.
The requirements for SC charging are constant current and fast charging.These can be satisfied under resonance, as shown in the above analysis.It is easy to determine how the frequency deviation will result in the receiver current varying from the R v .Then I SC cannot remain constant; thus, the characteristics of the SC-load WPT system will be different.Under the non-resonance of the transmitter and receiver, I 2 can be obtained according to (5), and I SC can be calculated as follows: It can be observed that I SC will vary with the change in R v , and that R v is affected by I SC .Therefore, the variation in R v cannot be calculated as shown in (14).However, R v can be calculated based on (2), by testing the voltage and current of SC, or by another calculation method, as shown below.
Assuming I SC can remain constant at a slight time delay (t), the initial value of I SC (I SC_0 ) can be obtained from (28), when R v = 0 and the initial SC voltage (u SC_0 ) is zero.Therefore, the SC voltage (u SC_k ) and current (i SC_k ) at every moment can be calculated as follows: where k represents the moment of SC charging.
Then the variable resistance R v can be calculated as: R v has two variables, u SC_k and i SC_k , compared with the constant charging current, which can be expressed as a variable u SC .We can use a variable (R SC ) instead of two variables (u SC_k and i SC_k ); then the characteristics of the SC-load WPT system under frequency deviation can be analyzed based on the VRB method, and the difference can be used simultaneously to measure the voltage and the current to determine the variable resistance R v .
The voltage gain (G V ) refers to the ratio of the output voltage to the input voltage, so G V can show the change in receiver voltage when U 1 is constant.The current gain (G I ) is analyzed similarly.The analysis below and calculation are based on the equivalent circuit model shown in Figure 3.
Here, the angular frequency offset (ω n ) is defined as the operating angular frequency (ω) divided the resonant angular frequency (ω n ): G V is defined as: where After ignoring the coil self-resistances, and substituting the parameters into (32), G V can be obtained as: Considering the above formula, it can be further deduced as where k = M/ √ L 1 L 2 is the coupling coefficient of the transmitter and receiver coils, and n = √ L 1 /L 2 is the effective turn ratio of the transmitter and receiver coils.Similarly, G I can be calculated as follows Taking Table 1 into consideration, the results can be plotted as shown in Figures 9  and 10.At the point ω n = 1, it is consistent with the above analysis.The voltage gain is a linear rise, and the U 2 and SC voltage will increase linearly if the U 1 is constant.However, if ω n is shifted, G I will decrease when R SC is larger.In contrast, G U will increase and have two peak points when R SC is smaller, as shown in Figure 9.
Energies 2022, 15, x FOR PEER REVIEW 11 of 16 Considering the above formula, it can be further deduced as = / k M L L is the coupling coefficient of the transmitter and receiver coils, and L L is the effective turn ratio of the transmitter and receiver coils.
Similarly, GI can be calculated as follows Taking Table 1 into consideration, the results can be plotted as shown in Figures 9  and 10.At the point ωn = 1, it is consistent with the above analysis.The voltage gain is a linear rise, and the U2 and SC voltage will increase linearly if the U1 is constant.However, if ωn is shifted, GI will decrease when RSC is larger.In contrast, GU will increase and have two peak points when RSC is smaller, as shown in Figure 9.
As for the current gain shown in Figure 10, I2 will remain constant, and I1 will increase linearly under the resonant frequency; thus, the curve of GI is an inverse proportion function when ωn = 1.When the operating frequency is shifted, GI will decrease, especially when RSC is smaller.Based on this analysis, it can be concluded that a small RSC is not favorable, and the system will demonstrate better characteristics when RSC is chosen as the suitable value.As for the current gain shown in Figure 10, I 2 will remain constant, and I 1 will increase linearly under the resonant frequency; thus, the curve of G I is an inverse proportion function when ω n = 1.When the operating frequency is shifted, G I will decrease, especially when R SC is smaller.Based on this analysis, it can be concluded that a small R SC is not favorable, and the system will demonstrate better characteristics when R SC is chosen as the suitable value.

Design Process of SC-Load WPT System
Based on the above analysis, the characteristics of the SC load WPT system can be determined, and the optimal design method for the SC-load WPT system can be preliminarily given.First, the key parameters of the SC load need to be determined according to the application.Considering the constant current charging for the SC, and the range of the operating voltage and time of SC requirements, the capacitance value of the SC can be calculated.Furthermore, the parameters of the WPT system can be designed.According to the SC maximum charging power, the higher frequency of the switching devices is conducive to improving the transfer efficiency.Next, the DC bus voltage can be set suitably, and, if it is limited, the phase-shift angle can be designed according to (27).Then, the mutual inductance can be determined.Finally, the coupling coils should be designed to meet the requirements of the mutual inductance and achieve the smallest equivalent resistances of the transmitter and receiver coils.The design procedure is shown as Figure 11.

Design Process of SC-Load WPT System
Based on the above analysis, the characteristics of the SC load WPT system can be determined, and the optimal design method for the SC-load WPT system can be preliminarily given.First, the key parameters of the SC load need to be determined according to the application.Considering the constant current charging for the SC, and the range of the operating voltage and time of SC requirements, the capacitance value of the SC can be calculated.Furthermore, the parameters of the WPT system can be designed.According to the SC maximum charging power, the higher frequency of the switching devices is conducive to improving the transfer efficiency.Next, the DC bus voltage can be set suitably, and, if it is limited, the phase-shift angle can be designed according to (27).Then, the mutual inductance can be determined.Finally, the coupling coils should be designed to meet the requirements of the mutual inductance and achieve the smallest equivalent resistances of the transmitter and receiver coils.The design procedure is shown as Figure 11.

Simulation and Experimental Setup
To verify the theoretical derivations and analysis, an experimental prototype was fabricated as shown in Figure 12.The topology of the WPT system using SC as a load shared the same configuration with that in Figure 1.The parameters were consistent with

Simulation and Experimental Setup
To verify the theoretical derivations and analysis, an experimental prototype was fabricated as shown in Figure 12.The topology of the WPT system using SC as a load shared the same configuration with that in Figure 1.The parameters were consistent with Table 1.A SanRex three-phase bridge DF100AA160 was used to convert the grid AC to DC.A CREE SiC MOSFET C2M0080120D was adopted as the active switch of the inverter (DC/AC).Since the value of the compensation capacitor is limited, the inductance of the coupling coils is resonant with the compensation capacitor on the 58 kHz switching frequency.Therefore, this switching frequency was chosen as the system operating frequency.The transmitter and the receiver inductors were circular coils with a single layer and a turn number of 18, composed of tightly-wound litz wires with a diameter of 3 mm.The external diameter of the coil was 40 cm and the gap between the transmitter and receiver was 15 cm.The inductor and capacitor were chosen under the resonance for both the transmitter and receiver, and their resonant frequencies were the same as the operating frequency to improve the efficiency and reduce the VA rating of the WPT system.The SC voltage, charging current, transmitter and receiver AC currents and voltages were tested using a Tektronic DPO3034 digital phosphor oscilloscope.The series experiments were conducted in which the DC bus voltages were 40 V, 60 V and 80 V, respectively, with α = 45° under Ubus = 60 V, together with the corresponding simulations in PSIM, as displayed in Figure 13.For each series experiment, the SC was charged from non-energy (0 V) to 100 V under constant current charging.
Under the 40 V DC bus voltage and phase-shift (0° and 45°), the recorded experimental waveforms of the transmitter input voltage (u1), transmitter input current (i1), receiver output voltage (u2), and receiver output current (i2) are shown in Figure 14.The inductor and capacitor were chosen under the resonance for both the transmitter and receiver, and their resonant frequencies were the same as the operating frequency to improve the efficiency and reduce the VA rating of the WPT system.The SC voltage, charging current, transmitter and receiver AC currents and voltages were tested using a Tektronic DPO3034 digital phosphor oscilloscope.The series experiments were conducted in which the DC bus voltages were 40 V, 60 V and 80 V, respectively, with α = 45 • under U bus = 60 V, together with the corresponding simulations in PSIM, as displayed in Figure 13.For each series experiment, the SC was charged from non-energy (0 V) to 100 V under constant current charging.The inductor and capacitor were chosen under the resonance for both the transmitter and receiver, and their resonant frequencies were the same as the operating frequency to improve the efficiency and reduce the VA rating of the WPT system.The SC voltage, charging current, transmitter and receiver AC currents and voltages were tested using a Tektronic DPO3034 digital phosphor oscilloscope.The series experiments were conducted in which the DC bus voltages were 40 V, 60 V and 80 V, respectively, with α = 45° under Ubus = 60 V, together with the corresponding simulations in PSIM, as displayed in Figure 13.For each series experiment, the SC was charged from non-energy (0 V) to 100 V under constant current charging.

Results and Discussions
Under the 40 V DC bus voltage and phase-shift (0° and 45°), the recorded experimental waveforms of the transmitter input voltage (u1), transmitter input current (i1), receiver output voltage (u2), and receiver output current (i2) are shown in Figure 14.

Results and Discussions
When using the WPT system charging for the SC from non-energy to 100V under a DC bus voltage of 60 V, the charging current and the output power are illustrated in Figure 15.The calculations of I SC are constant, because the equivalent resistances of the transmitter and receiver (R 1 and R 2 ) are ignored.However, in the simulations and experiments, due to the existence of the R 1 and R 2 , the charging current and output power decreased slightly, which was to be expected; thus, it can be concluded that the simulation and experimental results agree with the calculations, verifying the validation of the VRB method.
Energies 2022, 15, x FOR PEER REVIEW 14 of 16 When using the WPT system charging for the SC from non-energy to 100V under a DC bus voltage of 60 V, the charging current and the output power are illustrated in Figure 15.The calculations of ISC are constant, because the equivalent resistances of the transmitter and receiver (R1 and R2) are ignored.However, in the simulations and experiments, due to the existence of the R1 and R2, the charging current and output power decreased slightly, which was to be expected; thus, it can be concluded that the simulation and experimental results agree with the calculations, verifying the validation of the VRB method.
For adjustment of the DC bus voltages to 40 V, 60 V and 80 V, the calculations, simulations and experiments for transfer efficiency vs. uSC are shown in Figure 16.The transfer efficiency ηtrn increased then decreased, as in the analysis, and the maximum transfer efficiency point was where uSC = Ubus.When Ubus increased, the corresponding optimal SC voltage increased, as shown in Figure 16a-c.The mismatches of the calculations, simulations and the experiments were generally caused by inaccuracy in the parameters.For Ubus = 60 V, the phase-shift angle was set at α = 45°, and uSC_OPT was equal to 55 V, as shown in Figure 16d.The existence of the phase-shift angle resulted in the waveform of the transmitter U1 distortion, as shown in Figure 14, and when uSC increased the voltage shock increased.Due to the limitations of the experimental prototype, uSC was only charged to 60 V, but it was verified that the optimal SC voltage dropped compared with α = 0°.For adjustment of the DC bus voltages to 40 V, 60 V and 80 V, the calculations, simulations and experiments for transfer efficiency vs. u SC are shown in Figure 16.The transfer efficiency η trn increased then decreased, as in the analysis, and the maximum transfer efficiency point was where u SC = U bus .When U bus increased, the corresponding optimal SC voltage increased, as shown in Figure 16a-c.The mismatches of the calculations, simulations and the experiments were generally caused by inaccuracy in the parameters.For U bus = 60 V, the phase-shift angle was set at α = 45 • , and u SC_OPT was equal to 55 V, as shown in Figure 16d.The existence of the phase-shift angle resulted in the waveform of the transmitter U 1 distortion, as shown in Figure 14, and when u SC increased the voltage shock increased.Due to the limitations of the experimental prototype, u SC was only charged to 60 V, but it was verified that the optimal SC voltage dropped compared with α = 0

Conclusions
In existing WPT system model analysis methods, the equivalent resistance is often used to analyze various loads.For an SC load, the equivalent load will change in real time during the charging process, and the conventional model cannot accurately establish the relationship between the load and the system.To address this problem, a novel variableresistance-based method was proposed to describe the relationship between the WPT system parameters (including equivalent load, output current, system power and transmission efficiency) and the SC parameters (including charging voltage, time and capacitance).Based on the method, a mathematical model of the WPT system using SC as a load was built.Additionally, the effects of the transmitter voltage on the transfer efficiency and the effects of frequency offset on the voltage and current gain were investigated.Based on the results, an optimal design process for the SC-load WPT system was provided to guide the system design.Finally, an experimental prototype and simulation model were constructed and the correctness of the theoretical analysis verified.

Figure 1 .
Figure 1.WPT system configuration in rail transit vehicle.

Figure 1 .
Figure 1.WPT system configuration in rail transit vehicle.

Figure 3 .
Figure 3. Equivalent circuit of VRB transmitter and receiver.

Figure 3 .
Figure 3. Equivalent circuit of VRB transmitter and receiver.Figure 3. Equivalent circuit of VRB transmitter and receiver.

Figure 3 .
Figure 3. Equivalent circuit of VRB transmitter and receiver.Figure 3. Equivalent circuit of VRB transmitter and receiver.

Figure 4 .
Figure 4. Relationship between some key parameters.

Figure 4 .
Figure 4. Relationship between some key parameters.

Figure 5 .
Figure 5. Curve of the maximum efficiency vs. T Q .

Figure 6 .
Figure 6.Curve of the transfer efficiency vs. u SC .

Figure 7 .
Figure 7. Transfer efficiency vs. U bus and u SC .

Figure 9 .
Figure 9. Voltage gain GV vs. RSC and ωn.Figure 9. Voltage gain G V vs. R SC and ω n .

Figure 9 .
Figure 9. Voltage gain GV vs. RSC and ωn.Figure 9. Voltage gain G V vs. R SC and ω n .

Figure 10 .
Figure 10.Current gain GI vs. RSC and ωn.Figure 10.Current gain G I vs. R SC and ω n .

Figure 10 .
Figure 10.Current gain GI vs. RSC and ωn.Figure 10.Current gain G I vs. R SC and ω n .

Figure 15 .
Figure 15.Calculations, simulations, and experiments of ISC, POUT.Figure 15.Calculations, simulations, and experiments of I SC , P OUT .

Figure 15 .
Figure 15.Calculations, simulations, and experiments of ISC, POUT.Figure 15.Calculations, simulations, and experiments of I SC , P OUT .