A General Parameter Optimization Method for a Capacitive Power Transfer System with an Asymmetrical Structure

Capacitive power transfer (CPT) is an attractive wireless power transfer (WPT) technology and it has been widely studied in many applications. Symmetrical structures and high-order compensation networks are always produced as optimization results and common configurations for high-efficiency CPT systems. However, in space-limited scenarios, an asymmetric structure tends to be a better choice. The related large number of high-order asymmetric system parameters is a key problem in parameter design. In this study, a general parameter design method that is based on reactive power optimization is proposed for an electric field resonance-based CPT system with an asymmetric six-plate coupler. The reactive power in the compensation network was analyzed and optimized under the constraint of transferred power. With equal reactive power, the optimization complexity was significantly reduced and the optimized system parameters were provided. To validate the effectiveness of the proposed method, a 1 MHz, 3.2 kW asymmetric CPT protype with 100 mm gap distance was implemented. The results indicate that, with the optimized parameters, high system efficiency can be achieved when the system’s volume is reduced. At the rated power, about 95% DC–DC overall efficiency was achieved through a 6-pF coupling capacitor.

Inductive wireless power transfer is one of the most promising WPT technologies and extensive studies have been implemented on the IPT system. In the IPT system, the magnetic coupler is a key component that always consists of transmission coils with a symmetric or asymmetric structure [26]. Since the magnetic couplers are always loosely coupled [27], heavy ferrite cores are needed in order to redistribute the magnetic field in the coupler and enhance the magnetic coupling of the transmission coils. Besides this, shielding plates are required to prevent the magnetic field from leaking into the surrounding environment. To improve the transferred power and efficiency, compensation between the transmitting plates or between the receiving plates. The equivalent circuit model with four coupling capacitors is shown in Figure 2a [48]. Due to the fact that the capacitors C S1 and C S2 are connected in series, the coupling capacitor C S can be further simplified, where C S = C S1 · C S2 C S1 + C S2 (1) As described in [50], a symmetric CPT system with an identical transmitter and receiver is verified to be an efficient configuration. However, the shielding distances d 1 and d 2 tend to have large values (usually in the range of 50 mm to 100 mm) in order to achieve high system efficiency. For space-limited scenarios, such as EV wireless charging applications, the limited space in the receiving side determines that the shielding distance d 2 cannot be very large. Therefore, an asymmetric coupler with different distances d 1 and d 2 (where d 1 > d 2 ) is more practical in EV wireless charging applications. According to the different plate lengths l 1 and l 2 , the asymmetric structure can be divided into three categories: l 1 = l 2 , l 1 > l 2 , and l 1 < l 2 , as shown in Figure 3.
Electronics 2022, 11, x FOR PEER REVIEW 4 of 25 CS1, CS2, C2, and C3. CS1 and CS2 represent the equivalent mutual capacitors between the transmitting plates and receiving plates; C2 and C3 represent the equivalent self-capacitors between the transmitting plates or between the receiving plates. The equivalent circuit model with four coupling capacitors is shown in Figure 2a [48]. Due to the fact that the capacitors CS1 and CS2 are connected in series, the coupling capacitor CS can be further simplified, where 1 2 As described in [50], a symmetric CPT system with an identical transmitter and receiver is verified to be an efficient configuration. However, the shielding distances d1 and d2 tend to have large values (usually in the range of 50 mm to 100 mm) in order to achieve high system efficiency. For space-limited scenarios, such as EV wireless charging applications, the limited space in the receiving side determines that the shielding distance d2 cannot be very large. Therefore, an asymmetric coupler with different distances d1 and d2 (where d1 > d2) is more practical in EV wireless charging applications. According to the different plate lengths l1 and l2, the asymmetric structure can be divided into three categories: l1 = l2, l1 > l2, and l1 < l2, as shown in Figure 3. (a)   Electronics 2022, 11, x FOR PEER REVIEW 4 of 25 CS1, CS2, C2, and C3. CS1 and CS2 represent the equivalent mutual capacitors between the transmitting plates and receiving plates; C2 and C3 represent the equivalent self-capacitors between the transmitting plates or between the receiving plates. The equivalent circuit model with four coupling capacitors is shown in Figure 2a [48]. Due to the fact that the capacitors CS1 and CS2 are connected in series, the coupling capacitor CS can be further simplified, where 1 2 As described in [50], a symmetric CPT system with an identical transmitter and receiver is verified to be an efficient configuration. However, the shielding distances d1 and d2 tend to have large values (usually in the range of 50 mm to 100 mm) in order to achieve high system efficiency. For space-limited scenarios, such as EV wireless charging applications, the limited space in the receiving side determines that the shielding distance d2 cannot be very large. Therefore, an asymmetric coupler with different distances d1 and d2 (where d1 > d2) is more practical in EV wireless charging applications. According to the different plate lengths l1 and l2, the asymmetric structure can be divided into three categories: l1 = l2, l1 > l2, and l1 < l2, as shown in Figure 3.

Receiving Plates
Transmitting Plates

Shielding Plates
Shielding Plates (a) As described in [50], a symmetric CPT system with an identical transmitter and receiver is verified to be an efficient configuration. However, the shielding distances d1 and d2 tend to have large values (usually in the range of 50 mm to 100 mm) in order to achieve high system efficiency. For space-limited scenarios, such as EV wireless charging applications, the limited space in the receiving side determines that the shielding distance d2 cannot be very large. Therefore, an asymmetric coupler with different distances d1 and d2 (where d1 > d2) is more practical in EV wireless charging applications. According to the different plate lengths l1 and l2, the asymmetric structure can be divided into three categories: l1 = l2, l1 > l2, and l1 < l2, as shown in Figure 3. (a) Electronics 2022, 11, 922 5 of 25 In the capacitive coupler, the coupling capacitors C 2 , C 3 , and C S are determined by the airgap distances d 1 , d 2 , and d 12 and plate lengths l 1 and l 2 . An asymmetric coupler structure always results in asymmetric circuit parameters. In Figure 3a and c, l 1 is not greater than l 2 and d 1 is greater than d 2 , as a result the capacitor value of C 2 is less than that of C 3 . In Figure 3b, the relationship between C 2 and C 3 is determined by the asymmetric distance and plate length and the equivalent C 2 and C 3 can be obtained by appropriate configuration. When the plate lengths l 1 and l 2 are not equal, as shown in Figure 3b,c, a small horizontal misalignment has little effect on the coupling capacitance. In the cases in which l 1 ≥ l 2 , as shown in Figure 3a,b, the leaked electric field that results from the small misalignment conditions are mostly exposed to the receiving side. As a comparison, the leaked electric field that results from small misalignment conditions for the asymmetric coupler that is shown in Figure 3c are mostly exposed to the transmitting side. Considering this, the asymmetric structure that is shown in Figure 3c was selected in order to form the asymmetric CPT system in this study.

CPT System Model
In this study, an EFR compensation network was used. With the asymmetric capacitive coupler that is shown in Figure 3c, the equivalent circuit model of the asymmetric CPT system is shown in Figure 4, where C 3 is greater than C 2 . The EFR compensation circuit was formed by L 1 , C 1 , L 2 , C 2 , L 3 , C 3 , L 4 , and C 4 . M 12 and M 34 represent the mutual inductances between L 1 and L 2 and between L 3 and L 4 , respectively. C S represents the mutual capacitance in the capacitive coupler. V in and V out are the input and output direct current (DC) voltage sources, respectively. The capacitors C 2 , C 3 , and C S achieve EFR in the compensation circuit and the EFR is resonant with the inductors L 2 and L 3 at an angular frequency of ω 0 . Besides this, C 1 and C 4 are resonant with L 1 and L 4 at an angular frequency of ω 0 , respectively.
In the capacitive coupler, the coupling capacitors C2, C3, and CS are determined by the airgap distances d1, d2, and d12 and plate lengths l1 and l2. An asymmetric coupler structure always results in asymmetric circuit parameters. In Figure 3a and c, l1 is not greater than l2 and d1 is greater than d2, as a result the capacitor value of C2 is less than that of C3. In Figure 3b, the relationship between C2 and C3 is determined by the asymmetric distance and plate length and the equivalent C2 and C3 can be obtained by appropriate configuration. When the plate lengths l1 and l2 are not equal, as shown in Figure 3b,c, a small horizontal misalignment has little effect on the coupling capacitance. In the cases in which l1 ≥ l2, as shown in Figure 3a,b, the leaked electric field that results from the small misalignment conditions are mostly exposed to the receiving side. As a comparison, the leaked electric field that results from small misalignment conditions for the asymmetric coupler that is shown in Figure 3c are mostly exposed to the transmitting side. Considering this, the asymmetric structure that is shown in Figure 3c was selected in order to form the asymmetric CPT system in this study.

CPT System Model
In this study, an EFR compensation network was used. With the asymmetric capacitive coupler that is shown in Figure 3c, the equivalent circuit model of the asymmetric CPT system is shown in Figure 4, where C3 is greater than C2. The EFR compensation circuit was formed by L1, C1, L2, C2, L3, C3, L4, and C4. M12 and M34 represent the mutual inductances between L1 and L2 and between L3 and L4, respectively. CS represents the mutual capacitance in the capacitive coupler. Vin and Vout are the input and output direct current (DC) voltage sources, respectively. The capacitors C2, C3, and CS achieve EFR in the compensation circuit and the EFR is resonant with the inductors L2 and L3 at an angular frequency of ω0. Besides this, C1 and C4 are resonant with L1 and L4 at an angular frequency of ω0, respectively.  As shown in Figure 5, the equivalent circuit model of an EFR-based CPT system can be obtained according to the method of fundamental harmonics approximation. The internal resistances of all of the circuit components are ignored. U21, U12, U43, and U34 represent the induced voltage sources that are generated in the coupling inductors L1, L2, L3, and L4. UAB and Uab represent the input and output AC voltages, respectively, where By using UAB as the reference, the voltage and current phasors can then be obtained based on the superposition theorem as follows [50]:  Figure 5, the equivalent circuit model of an EFR-based CPT system can be obtained according to the method of fundamental harmonics approximation. The internal resistances of all of the circuit components are ignored. U 21 , U 12 , U 43 , and U 34 represent the induced voltage sources that are generated in the coupling inductors L 1 , L 2 , L 3 , and L 4 . U AB and U ab represent the input and output AC voltages, respectively, where

As shown in
Electronics 2022, 11, 922 6 of 25 By using U AB as the reference, the voltage and current phasors can then be obtained based on the superposition theorem as follows [50]: We can see from Equation (3) that the current and voltage phasors on the input side are in phase, which means zero phase angle is achieved on the input side. Since all of the internal resistances are not considered, the output power is equal to the input power, which can be described as The mutual inductances M 12 and M 34 that are featured in Equation (4) can be described as where K 12 and K 34 are the coupling coefficients. By substituting Equation (5) into Equation (4), the transferred power can be further described as Electronics 2022, 11, x FOR PEER REVIEW 6 of 25 We can see from Equation (3) that the current and voltage phasors on the input side are in phase, which means zero phase angle is achieved on the input side. Since all of the internal resistances are not considered, the output power is equal to the input power, which can be described as ( ) The mutual inductances M12 and M34 that are featured in Equation (4) can be described as ( ) ( ) where K12 and K34 are the coupling coefficients. By substituting Equation (5) into Equation (4), the transferred power can be further described as

Reactive Power on a Compensation Network
According to Equations (3) and (5), the capacitive reactive power in the system that is shown in Figure 5 can be obtained as follows: Here, Q C is used to represent the total capacitive reactive power, where According to Equations (3) and (5), the reactive power on inductors l 1 , l 2 , l 3 , and l 4 can be calculated as follows: The total inductive reactive power Q L can be described as Since the CPT system is in a resonant state, the total capacitive reactive power and total inductive reactive power are equal. We can see from Equations (7)-(10) that the reactive power is excited by the voltage sources U AB and U ab . Since the expressions of the reactive power are complex, the optimization of the reactive power is very difficult.

Analysis of System Efficiency
The efficiency of the CPT system is determined by the transferred power and power losses in the circuit components. Increasing the transferred power while decreasing the losses can improve the transmission efficiency. Typically, the power losses are divided into four main parts: rectifier loss P loss,rec , inverter loss P loss,inv , capacitor loss P loss,C , and inductor loss P loss,L . Considering these losses, the system's efficiency η can be expressed as η = P P + P loss,rec + P loss,inv + P loss,C + P loss,L where P loss,C = ∑ P loss,Ci P loss,L = ∑ P loss,Lj i = S, 1, 2, 3, 4; j = 1, 2, 3, 4 Electronics 2022, 11, 922 8 of 25 In Equation (11), the rectifier loss and inverter loss include the switching losses and conduction losses, which are determined by the switching devices and switching states; the losses in the inductors and capacitors are mainly determined by the reactive power that is in the passive components and the quality factors of the components. Therefore, the applications of low conduction-resistance power devices and high quality factor passive components and the realization of a ZVS state are usually efficient methods for the CPT system [40,56]. When the switching devices and transferred power are determined, we assume that the rectifier loss and inverter loss are constants. In this case, the system efficiency can be improved by reducing the losses in the passive components.
Here, the reactive power of the passive components can be represented as Q Ci and Q Lj ; the quality factors can be represented as Q * Ci and Q * Lj . The power losses in the passive components can be further described as The system efficiency can be obtained from Equations (11)-(13) as In this study, the inductors were wound with Litz wire and the capacitors usually had low dissipation factors. To simplify the analysis, we assumed that all of the capacitors (inductors) had the same quality factors, which can be represented as Q * C (Q * L ). Given these facts, Equation (14) can be changed to where Q C and Q L represent the total capacitive or inductive reactive power in the CPT system. When the transferred power is determined, the efficiency in Equation (15) can be improved by decreasing the reactive power or increasing the quality factors. When the quality factors are determined, the transmission efficiency is mainly dependent on the reactive power. Therefore, the reactive power in the CPT system should be optimized for efficiency improvement. The cases in which the passive components have different quality factors are described in Appendix A.

General Optimization Method
As described above, under the constraint of transferred power, the reactive power should be optimized in order to improve the system's efficiency. In this case, we set the ratio of reactive power to transferred power as the optimization objective. Based on the properties of a linear CPT system, a general optimization method was proposed in order to reduce the optimization's complexity. By applying the equal reactive power to the optimization objective, the general circuit conditions under which the circuit parameters should be satisfied were obtained. Based on the circuit conditions, the system model can be simplified and all of the system parameters can be optimized.

General Optimization Method Based on Equal Reactive Power
To minimize the optimization objective, the transferred power should be considered while minimizing the reactive power. Therefore, the transferred power can be seen as a constraint for the reduction of the reactive power. Since the total inductive reactive power was equal to the total capacitive reactive power, it was sufficient to minimize the inductive reactive power in this study. In the CPT system, the reactive power is excited by the voltage sources U AB and U ab , which can be represented as Q AB and Q ab , respectively. From Equation (10), Q AB and Q ab can be expressed as From Equations (10) and (16), the inductive reactive power can then be expressed as In this study, the system parameters ω 0 , U AB , U ab , K 12 , K 34 , C 1 , C 4 , C 2 , C 3 , and C S are independent; they can be adjusted independently. Since the reactive powers Q AB and Q ab in the linear system are excited by the independent voltage sources U AB and U ab , Q AB and Q ab are independent. The relationship between the transferred power P and the reactive power Q AB and Q ab can then be expressed as We can see from Equation (18) that when the transferred power is determined, the product of Q AB and Q ab has a minimum value. For each given product value of Q AB and Q ab , the total inductive reactive power has a minimum value, where The equal sign in Equation (18) is achieved when Q AB and Q ab are equal. Under the constraint of transferred power, it can be verified that equal Q AB and Q ab can be achieved. Here, we set the equal reactive powers Q AB and Q ab as a circuit condition. According to the equal reactive powers Q AB and Q ab , the total inductive reactive power can then be expressed as where Considering the transferred power, the optimization objective can then be derived from Equations (6) and (20) as Furthermore, Equation (22) can be expressed as Since K 12 and K 34 are independent, their values can be adjusted independently. For each given value of K 12 and K 34 , the optimization objective in Equation (23) can be further decreased when K 12 and K 34 are equal. In this case, Equation (23) can be changed to where By substituting Equation (25) into Equation (21), the relationship between C 1 and C 4 can be obtained as In this study, an asymmetric six-plate capacitive coupler was used, where C 2 < C 3 . To simplify the analysis, the coefficients a and b are used to describe the relationship between capacitors C 2 , C 3 and C S , where From Equations (27) and (24), the optimization objective can be derived as When the value of b is determined, the minimized Q L /P can be obtained. At the minimum value point, the value of a can be expressed as We can see from Equation (30) that the minimum value of Q L /P is just related to the coupling coefficients; by increasing the values of K 12 and K 34 , the value of Q L /P can be further decreased.
By substituting Equations (25) and (26) into Equation (6), the transferred power can be described as According to Equation (2), Equation (31) can be changed to From the above analysis, a general parameter optimization method that is based on equal reactive power can be obtained. Figure 6 shows the design flowchart. Considering the actual application scenarios, the system requirements are given and defined as inputs. The optimized coefficients K 12 , K 34 , a and b can be obtained by Equations (25), (28) and (29). Based on Equation (27), the relationship between the coupling capacitors can be derived. With the given airgap distances d 12 and d 2 , an appropriate capacitive coupler with optimized coupling capacitors can be determined. According to Equations (26) and (27), the optimized C 1 and C 4 can be obtained. The resonant inductors can then be obtained by using the resonant conditions. Finally, the parameters can be adjusted iteratively depending on the available commercial inductors and capacitors.
Electronics 2022, 11, x FOR PEER REVIEW 11 From the above analysis, a general parameter optimization method that is base equal reactive power can be obtained. Figure 6 shows the design flowchart. Conside the actual application scenarios, the system requirements are given and defined as inp The optimized coefficients K12, K34, a and b can be obtained by Equations (25), (28) and Based on Equation (27), the relationship between the coupling capacitors can be der With the given airgap distances d12 and d2, an appropriate capacitive coupler with mized coupling capacitors can be determined. According to Equations (26) and (27) optimized C1 and C4 can be obtained. The resonant inductors can then be obtaine using the resonant conditions. Finally, the parameters can be adjusted iteratively dep ing on the available commercial inductors and capacitors.

Comparation with the General Optimization Method
Both the two-stage optimization method [50] and the general optimization me that is proposed in this study aim to increase the system's efficiency by optimizing reactive power in the CPT system. The optimization objectives and optimization proce

Comparation with the General Optimization Method
Both the two-stage optimization method [50] and the general optimization method that is proposed in this study aim to increase the system's efficiency by optimizing the reactive power in the CPT system. The optimization objectives and optimization processes in the two methods are similar. The difference is that in the two-stage optimization method the capacitive coupler is first optimized and then the symmetric coupling parameters are verified to be an efficient configuration for the CPT system. The two-stage method is suitable for a CPT system with symmetric circuit parameters. In order to further the research, this paper studies the asymmetric CPT system and its general optimization method. Based on the properties of a linear CPT system, equal reactive power was used to optimize the reactive power in the CPT system. This method is suitable for both symmetric and asymmetric CPT systems. Besides this, when the coupling capacitors C 2 and C 3 were equal, the optimization results in the two studies were same.

Parameter Design and Implementation
Considering the actual requirements in EV wireless charging applications, a CPT prototype with an asymmetric coupling structure was implemented in order to validate the proposed method.

System Requirements
As shown in Table 1, the system requirements in an actual CPT system are given. To satisfy the requirements of the charging distance and passability, the transferred distance d 12 was set to 100 mm. Since the installation space under the vehicle chassis was limited, the airgap distance d 2 on the vehicle's side was set to 20 mm. The operating frequency of the CPT system was 1 MHz. The transferred power was set to 3.2 kW and both the input and output voltages were set to 450 V. We can see from Equations (28) and (30) that the optimization objective can be reduced by increasing the value of the coupling coefficients K 12 and K 34 . However, studies have shown that a large third harmonic can be induced in the inverter when K 12 and K 34 are greater than 0.4 [62]. Therefore, the values of K 12 and K 34 were both set to 0.4 in this study.

Coupler Design
In this study, the asymmetric six-plate capacitive coupler was formed by 2 mm-thick aluminum plates. The structure is shown in Figure 7 and the dimensions of the designed coupler are shown in Table 2. The transmitting plates and receiving plates are square and centrosymmetric. On the receiving side, the plate length l 2 was 600 mm, the shielding edges l e3 and l e4 were both 50 mm, and the plate separation l s2 was 100 mm. The distance d 2 was 20 mm and the transmission distance d 12 was 100 mm. The remaining dimensions of the transmitting side were plate length l 1 and distance d 1 .  In order to obtain the appropriate plate length l1 and shielding distance d1, ANSYS Maxwell was used to simulate the coupling capacitors under different l1 and d1 values. According to Equation (24), the optimization objective under different values of l1 and d1 can then be obtained. The results are shown in Figure 8, where the ratio values decrease with increasing plate length l1 and airgap distance d1. Without a loss of generality, the plate length was set to 300 mm in this study in order to form an asymmetric capacitive coupler. With the defined l1, the plate separation ls2 and the shielding edges le1 and le2 could be obtained.  In order to obtain the appropriate plate length l 1 and shielding distance d 1 , ANSYS Maxwell was used to simulate the coupling capacitors under different l 1 and d 1 values. According to Equation (24), the optimization objective under different values of l 1 and d 1 can then be obtained. The results are shown in Figure 8, where the ratio values decrease with increasing plate length l 1 and airgap distance d 1 . Without a loss of generality, the plate length was set to 300 mm in this study in order to form an asymmetric capacitive coupler. With the defined l 1 , the plate separation l s2 and the shielding edges l e1 and l e2 could be obtained.  In order to obtain the appropriate plate length l1 and shielding distance d1, ANSYS Maxwell was used to simulate the coupling capacitors under different l1 and d1 values. According to Equation (24), the optimization objective under different values of l1 and d1 can then be obtained. The results are shown in Figure 8, where the ratio values decrease with increasing plate length l1 and airgap distance d1. Without a loss of generality, the plate length was set to 300 mm in this study in order to form an asymmetric capacitive coupler. With the defined l1, the plate separation ls2 and the shielding edges le1 and le2 could be obtained.  The only parameter that needed to be seriously considered was the gap distance d 1 . When other parameters are determined, the variation of d 1 can affect the values of the coupling capacitances, which can be simulated by using ANSYS Maxwell. Figure 9 shows the simulated values of C 2 , C 3 , and C S and coefficients a and b under different values of airgap distance d 1 .
We can see from Figure 9a that the capacitance of C 3 is greater than that of C 2 and C S . With an increasing airgap distance d 1 , the values of C 2 and C 3 decrease, while C S increases slightly. As a result, both a and b decrease with increasing d 1 , as shown in Figure 9b. In Figure 10, the optimization objective under different distance d 1 is calculated. To reduce the value of Q L /P, the value of the gap distance d 1 tends to be large. However, with a large d 1 , the value of C 2 is very small, which enlarges the inductance values and makes the CPT system more sensitive to surrounding disturbances. Therefore, both Q L /P and C 2 should be considered in the selection of d 1 . To achieve an appropriate Q L /P and avoid C 2 being too small, d 1 was set to 50 mm. Correspondingly, Q L /P was set to 16.5, a was set to 2.06, b was set to 17.06, C 2 was set to 12.25 pF, C 3 was set to 101.5 pF, and C S was set to 5.95 pF. The only parameter that needed to be seriously considered was the gap distance d1. When other parameters are determined, the variation of d1 can affect the values of the coupling capacitances, which can be simulated by using ANSYS Maxwell. Figure 9 shows the simulated values of C2, C3, and CS and coefficients a and b under different values of airgap distance d1.
We can see from Figure 9a that the capacitance of C3 is greater than that of C2 and CS. With an increasing airgap distance d1, the values of C2 and C3 decrease, while CS increases slightly. As a result, both a and b decrease with increasing d1, as shown in Figure 9b. In Figure 10, the optimization objective under different distance d1 is calculated. To reduce the value of QL/P, the value of the gap distance d1 tends to be large. However, with a large d1, the value of C2 is very small, which enlarges the inductance values and makes the CPT system more sensitive to surrounding disturbances. Therefore, both QL/P and C2 should be considered in the selection of d1. To achieve an appropriate QL/P and avoid C2 being too small, d1 was set to 50 mm. Correspondingly, QL/P was set to 16.5, a was set to 2.06, b was set to 17.06, C2 was set to 12.25 pF, C3 was set to 101.5 pF, and CS was set to 5.95 pF.  It should be noted that when b is given, the theoretical minimum QL/P can tained from Equations (29) and (30). If we assume that the value of b does not chan the variation of a, the relationship between the ratio of QL/P and the coefficient a i in Figure 11. The minimum QL/P value of 10 can be achieved when the coefficient a To make the coefficient a higher than 0, the value of b should be less than 5.25. Via the finite element simulation software ANASYS Maxwell, the performan capacitive coupler with the given parameters was simulated. Figure 12 shows th lated capacitances C2 and CS under different X and Y misalignments. With an inc misalignment condition, the simulated C2 values increase while the CS values dec It should be noted that when b is given, the theoretical minimum Q L /P can be obtained from Equations (29) and (30). If we assume that the value of b does not change with the variation of a, the relationship between the ratio of Q L /P and the coefficient a is shown in Figure 11. The minimum Q L /P value of 10 can be achieved when the coefficient a is −0.65. To make the coefficient a higher than 0, the value of b should be less than 5.25. It should be noted that when b is given, the theoretical minimum QL tained from Equations (29) and (30). If we assume that the value of b does no the variation of a, the relationship between the ratio of QL/P and the coefficie in Figure 11. The minimum QL/P value of 10 can be achieved when the coeffic To make the coefficient a higher than 0, the value of b should be less than 5. Via the finite element simulation software ANASYS Maxwell, the perfo capacitive coupler with the given parameters was simulated. Figure 12 sho lated capacitances C2 and CS under different X and Y misalignments. With misalignment condition, the simulated C2 values increase while the CS value Via the finite element simulation software ANASYS Maxwell, the performance of the capacitive coupler with the given parameters was simulated. Figure 12 shows the simulated capacitances C 2 and C S under different X and Y misalignments. With an increasing misalignment condition, the simulated C 2 values increase while the C S values decrease.

Resonant Parameter Design
With the given system parameters, the other resonant parameters can then tained. Considering the power losses in an actual system, the transferred power used to calculate the circuit parameters should be set slightly higher than the rated Based on Equations (6) and (31) The resonant inductors L1, L2, L3, and L4 can be obtained based on the followin nant relationship: The designed system's parameters are shown in Table 3. It should be noted th parasitic capacitance between the windings of the inductors is inevitable and shou be eliminated in this study. The structure of the actual inductor is shown in Figu Generally, the parallel-connected parasitic capacitance in the inductors is very smal

Resonant Parameter Design
With the given system parameters, the other resonant parameters can then be obtained. Considering the power losses in an actual system, the transferred power that is used to calculate the circuit parameters should be set slightly higher than the rated value. Based on Equations (6) and (31), the capacitances C 1 and C 4 can be calculated as.
The resonant inductors L 1 , L 2 , L 3 , and L 4 can be obtained based on the following resonant relationship: The designed system's parameters are shown in Table 3. It should be noted that the parasitic capacitance between the windings of the inductors is inevitable and should not be eliminated in this study. The structure of the actual inductor is shown in Figure 13.
Generally, the parallel-connected parasitic capacitance in the inductors is very small (usually in the range of several picofarads), which is close to the capacitance values of C 2 and C S . As a result, the resonant inductors L 2 and L 3 deviated from the design values. Besides this, the metal plates were deformed due to the influence of gravity, resulting in a small variation of the coupling capacitance C S . Considering the actual value of the ceramic capacitors and the effect of parasitic capacitances in the inductors, C 1 and C 4 were adjusted to 4.2 nF. To avoid measurement errors, the parameters were tuned by using a network analyzer. capacitors and the effect of parasitic capacitances in the inductors, C1 and C4 were adjusted to 4.2 nF. To avoid measurement errors, the parameters were tuned by using a network analyzer.

Simulation Results
In this study, the system's performance was simulated by using LTspice. The circuit that is shown in Figure 4 was used to model the proposed CPT system. The simulation's results are shown in Figures 14-16. The waveforms of the voltages UC2 and UC3, input current, and voltage (I1 and UAB) are shown in Figure 14. Due to the asymmetric configuration of the CPT system, the rms values of UC2 are greater than those of UC3. The phase difference between UC2 and UC3 is about 78°. The phases of the current and voltage on the input side are almost identical, which allowed the system to operate at high-power factor conditions.
The simulated values of QL and QL/P under different values of the coefficient a are shown in Figure 15. The results show that the reactive power increases with increasing output power and coefficient a value and the ratio of QL/P increases with increasing coefficient a value and decreases with increasing output power. In this study, the coefficient a was set to 2, and the value of QL/P at the rated power was approximately 17, which agrees well with the designed value.
The system efficiency under different values of the coefficient a was simulated and the results are displayed in Figure 16. The results show that the system efficiency increased with increasing output power and it decreased with increasing values of the coefficient a. The trend of system efficiency in Figure 15 is opposite to that of QL/P in Figure  15, which agrees well with the analysis.

Simulation Results
In this study, the system's performance was simulated by using LTspice. The circuit that is shown in Figure 4 was used to model the proposed CPT system. The simulation's results are shown in Figures 14-16. The waveforms of the voltages U C2 and U C3 , input current, and voltage (I 1 and U AB ) are shown in Figure 14. Due to the asymmetric configuration of the CPT system, the rms values of U C2 are greater than those of U C3 . The phase difference between U C2 and U C3 is about 78 • . The phases of the current and voltage on the input side are almost identical, which allowed the system to operate at high-power factor conditions.
The simulated values of Q L and Q L /P under different values of the coefficient a are shown in Figure 15. The results show that the reactive power increases with increasing output power and coefficient a value and the ratio of Q L /P increases with increasing coefficient a value and decreases with increasing output power. In this study, the coefficient a was set to 2, and the value of Q L /P at the rated power was approximately 17, which agrees well with the designed value.
The system efficiency under different values of the coefficient a was simulated and the results are displayed in Figure 16. The results show that the system efficiency increased with increasing output power and it decreased with increasing values of the coefficient a. The trend of system efficiency in Figure 15 is opposite to that of Q L /P in Figure 15, which agrees well with the analysis.          The voltages between the plates in the CPT system with different values of coefficient a were also analyzed. By using the proposed method, the system parameters under different values of a can be obtained. The voltages between the plates at the rated power can then be calculated as shown in Table 4. The voltages between the shielding plates (P 5 and P 6 ) were almost 0, which shows a good shielding effect in this design. Since the circuit parameters in designed CPT system were asymmetric, the voltages between the plates in the primary and secondary side were different. The voltages between the transmitting plates (P 1 and P 2 ) were higher than those which were observed between the receiving plates (P 3 and P 4 ). When the value of a increased from 1 to 10, the voltages between the plates in the transmitting side (P 1 , P 2 , and P 5 ) decreased, while in the receiving side (P 3 , P 4 , and P 6 ) the voltages increased. Besides this, the voltages between the transmission plates (P 1 and P 3 ) decreased with the increasing value of a. In this study, the voltage between P 1 and P 5 was approximately 7.56 kV. There was no risk of arcing [48] because the airgap distance between the plates was large.

Experimental Verification
To verify the proposed method, a 3.2-kW CPT prototype was built as shown in Figure 17. The asymmetric capacitive coupler with a large receiver and small transmitter was made up of six aluminum plates. The plates were held by PVC tubes and ceramic insulators and the outside shielding plates were floating. Under the capacitive coupler, a black steel plate was used as the ground. The mutual capacitance was 6.75 pF and the transferred distance was 100 mm. Just like the schematic circuit that is shown in Figure 4, a DC power source was used in order to provide a DC voltage to the inverter on the transmitting side of this prototype. An electric DC load, together with several parallelconnected resistors, was used as the load and connected to the rectifier on the receiving side. The power converters were formed by SiC devices IMZ120R045M1 and IDW40G120C5B. The inductors L 1 -L 4 were wound onto the PVC tubes with 1200-strand Litz wire with a diameter of 0.04 mm, which allowed easy adjustment of the coupling coefficients K 12 and K 34 . The compensation capacitors consisted of high voltage, low dissipation factor ceramic capacitors.
From the designed prototype, the experimental results at the rated power are shown in Figures 18 and 19. The plates in the coupler were well aligned. In order to reduce the switching losses and improve the system's efficiency, zero-voltage turn-on should be achieved for the switching devices. By slightly reducing the value of inductor L 2 , the ZVS condition was realized, as shown in Figure 18a,b. The waveforms of the input voltage and current were almost in phase and the ZVS condition was achieved by a phase shift of the input current. At the rated input and output DC voltages, about 3.2 kW power was delivered from the DC source to the electric DC load and resistors with an efficiency of 95%. The transmission efficiency under different output power is shown in the Figure 19. With different output voltages, the maximum system efficiencies were both higher than 94.5%.     The comparison between the presently designed CPT system and the previou lished CPT systems that have a large airgap distance and kilowatt-scale power is in Table 5. The DC-DC overall efficiency is considered. As shown in Table 5, the and coupling area that were used in this study are small, resulting in a lower volum coupling capacitance. When the element numbers of the systems are similar, the de CPT system can achieve high transmission efficiency with a relatively low coupl pacitance. The power loss distribution of the presently designed system was analyzed. A ing to the calculation method that was proposed in [44,63,64], the power losses MOSFETs, diodes, inductors, and capacitors were obtained. Based on the on-res and switching-off current, the losses in the switching devices could then be calcula losses in the inductors and capacitors were obtained from the measured quality The remaining losses are from within the coupling plates. As shown in Figure 20, ductors and capacitive coupler dissipate 39.2% and 33.6% of the total loss, respe which makes up the main losses of the system. The comparison between the presently designed CPT system and the previous published CPT systems that have a large airgap distance and kilowatt-scale power is shown in Table 5. The DC-DC overall efficiency is considered. As shown in Table 5, the airgap and coupling area that were used in this study are small, resulting in a lower volume and coupling capacitance. When the element numbers of the systems are similar, the designed CPT system can achieve high transmission efficiency with a relatively low coupling capacitance. The power loss distribution of the presently designed system was analyzed. According to the calculation method that was proposed in [44,63,64], the power losses in the MOSFETs, diodes, inductors, and capacitors were obtained. Based on the on-resistance and switchingoff current, the losses in the switching devices could then be calculated; the losses in the inductors and capacitors were obtained from the measured quality factors. The remaining losses are from within the coupling plates. As shown in Figure 20, the inductors and capacitive coupler dissipate 39.2% and 33.6% of the total loss, respectively, which makes up the main losses of the system.

Conclusions
A general parameter optimization method for an asymmetric CPT system is p posed in this article. A six-plate capacitive coupler with an asymmetric structure was u and compensated by an EFR compensation circuit. The asymmetric coupler structure w analyzed and the whole system was modeled. The relationship between the transfer power and reactive power was analyzed. Based on the properties of a linear CPT syst an optimization method was proposed. By using the equal reactive power, the circuit c ditions that are required in order to optimize the reactive power were obtained. With th circuit conditions, the reactive power was simplified and the circuit parameters were timized. Based on the optimization process, a general parameter optimization method w proposed and implemented on a 3.2-kW, 1 MHz CPT protype. Both the simulated a experimental results show that, with the proposed method, high system efficiency can achieved for an asymmetric CPT system with low coupling capacitance. About 3.2 kW power was transferred with a DC-DC efficiency of 95%. Although the special compen tion topology and an asymmetric coupler were used in this study, the proposed meth is not limited to this topology and it is suitable for both symmetric and asymmetric C systems.

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

Appendix A
The conditions under which CPT systems have different quality factors are analy in Appendix A. According to the analysis that was presented in Section 2.4, when ot parameters are determined, the transmission efficiency of the CPT system is depend on the reactive power and quality factors of the circuit components. With unequal qua factors applied on the circuit components, the optimization objective expression in study can then be changed to

Conclusions
A general parameter optimization method for an asymmetric CPT system is proposed in this article. A six-plate capacitive coupler with an asymmetric structure was used and compensated by an EFR compensation circuit. The asymmetric coupler structure was analyzed and the whole system was modeled. The relationship between the transferred power and reactive power was analyzed. Based on the properties of a linear CPT system, an optimization method was proposed. By using the equal reactive power, the circuit conditions that are required in order to optimize the reactive power were obtained. With these circuit conditions, the reactive power was simplified and the circuit parameters were optimized. Based on the optimization process, a general parameter optimization method was proposed and implemented on a 3.2-kW, 1 MHz CPT protype. Both the simulated and experimental results show that, with the proposed method, high system efficiency can be achieved for an asymmetric CPT system with low coupling capacitance. About 3.2 kW of power was transferred with a DC-DC efficiency of 95%. Although the special compensation topology and an asymmetric coupler were used in this study, the proposed method is not limited to this topology and it is suitable for both symmetric and asymmetric CPT systems.

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

Appendix A
The conditions under which CPT systems have different quality factors are analyzed in Appendix A. According to the analysis that was presented in Section 2.4, when other parameters are determined, the transmission efficiency of the CPT system is dependent on the reactive power and quality factors of the circuit components. With unequal quality factors applied on the circuit components, the optimization objective expression in the study can then be changed to ∑ Q Lj Q * Lj P + ∑ Q Ci Q * Ci P (j = 1, 2, 3, 4; i = S, 1, 2, 3, 4). (A1) When a CPT system is built, the quality factors of the circuit components are determined. Here, the quality factors can be seen as independent from each other. Since the quality factors do not affect the independence of the system parameters, the properties of linear CPT system can also be used to reduce the complexity of the optimization objective. The difference is that, with unequal quality factors, the circuit conditions that are obtained to simplify the optimization objective contain unequal quality factors.
The parameter optimization design process for a CPT system with unequal quality factors is similar to that of one with equal quality factors. Therefore, we just demonstrate the optimization of a CPT system with equal quality factors in this article.