Analysis and Design of a Dual-Frequency Capacitive Power Transfer System to Reduce Coupler Voltage Stress

Abstract

In a capacitive power transfer (CPT) system, the coupling capacitance formed between the coupling plates is very small only in the pF or nF range, which leads to high voltage stress among the coupling plates during energy transmission, which increases the risk of an electrical breakdown between the coupled plates. To solve this problem, a novel dual-frequency CPT system is proposed in this paper, which uses the “peak clipping” effect caused by the superposition of the fundamental wave and third harmonic wave to reduce the voltage stress of the coupled plates. Through the detailed analysis of the working principle of the CPT system, it is shown that the dual-frequency CPT system can indeed reduce the high voltage stress among the coupled plate to 84.3% of the equivalent single-frequency system and can also reduce the inverter conduction losses to 90%. A 200 W prototype is designed with the proposed scheme, and the experimental results confirm the correctness of the theoretical derivation.

1. Introduction

As a new non-contact power supply technology, wireless power transfer (WPT) technology removes the shackle of power lines and greatly improves the flexibility of power supply equipment. Meanwhile, this technology can avoid the potential safety problems while charging caused by traditional contact charging, since it has no contact sparks and is not affected by rain, snow, or other harsh environmental factors [1]. WPT technology is widely used because of its power supply safety, flexibility, and reliability [2,3,4,5,6]. At present, the transmission modes of WPT are mainly divided into inductive power transfer (IPT) and capacitive power transfer (CPT). An IPT system uses a high-frequency magnetic field generated by coupling coils for power transmission, while CPT technology uses a high-frequency electric field generated by coupling plates for power transmission [7,8]. IPT has been proposed for many years and has made good achievements in the field of WPT, with transmission power up to 1 MW [9], and system optimization can increase the efficiency to 96% [10]. However, an IPT system requires the use of a large number of magnetic cores and expensive Litz wires to construct a high-frequency magnetic field as a medium for power transmission, resulting in a high cost and the heavy coupling mechanism of the IPT system. In addition, the high-frequency magnetic field will cause eddy current loss in the surrounding metals, which increases the power loss of the system and reduces the efficiency of the system [11]. Moreover, magnetic field radiation also poses a threat to the personal safety of organisms [12]. In contrast, a CPT system uses a high-frequency electric field to transmit power without magnetic field radiation, so it can be applied to the wireless charging of implantable biomedical devices. A CPT system is proposed that uses a displacement current to realize the wireless charging of the subcutaneous sensors in [13]. With the human tissue layer as the transmission medium, the corresponding CPT system’s coupling mechanism model is established, which realizes the research and development of a wireless power supply system suitable for the human implantable medical devices in [14,15,16]. In addition, the CPT system uses lower-cost and lighter-weight aluminum plates as the coupling mechanism, so it is suitable for electric locomotive charging, with its strict cost and weight restrictions. Refs. [17,18,19] show that a lot of research has been accomplished on the coupling mechanism model and compensation topology of the static-charging CPT systems of electric vehicles. In [20], the coupling capacitance is formed by transforming the tire and the ground to realize a dynamic wireless power supply for electric vehicles. Ref. [21] proposes a coupling mechanism model suitable for rail transit, which can provide electric energy for moving locomotives. In addition, a CPT system is not sensitive to metals and has good environmental compatibility. In [22], a wireless power supply for refrigerated containers made of metal is realized. In [23], a CPT system is proposed for the internal winding of a motor. Therefore, CPT has been widely studies by scholars in China and around the globe because of its lower cost, better environmental compatibility, and negligible eddy current loss.

However, CPT systems usually have very high voltage stresses between their coupled plates due to the very small coupling capacitance that is only in the pF or nF range. The high voltage at both ends of the coupled plate lead to the emission of a strong electric field and even the risk of dielectric breakdown [10]. Therefore, reducing the voltage stress on the coupled plate is a key problem. The voltage on the coupling plate can be reduced by increasing the switching frequency of the inverter (that is, adopting a larger system frequency), but the system efficiency is reduced due to the increase in switching loss [24], and the higher frequency requires the higher performance of the device itself. In addition, in [25], the coupling voltage stress can also be reduced by increasing the coupling capacitance formed between the coupling plates. However, in some specific scenarios, due to the limitations of the external environment, the capacitance value formed by the coupling plates has already been determined, or it is difficult to increase the rating value required by the system. In [21], the voltage on both sides of the coupling plate is changed by adjusting the resonant network and the isolation transformer to reduce the voltage between the coupling plates. Ref. [10] proposes a parameter design method for a two-sided LCLC compensation CPT system based on the pre-designed voltage limit of the coupled plate. Ref. [26] redistributes the voltage stress on all the compensation components and coupling plates through a mathematical calculation based on a double-sided LC compensation circuit and reduces the voltage stress of the coupling plate by optimizing the compensation network and system parameters in a single-frequency system.

In this paper, a novel voltage stress optimization approach is proposed to reduce the voltage stress of the coupled plate by introducing the voltage component of the third harmonic frequency. As shown in Figure 1, the superposition of the primary and tertiary frequency voltages can produce an obvious “peak clipping” effect. A multi-frequency system has been studied for IPT. Ref. [27] makes a detailed performance analysis of a multi-frequency power transmission system for IPT. Refs. [28,29,30] realizes multi-channel and multi-load energy transmission by using a multi-frequency IPT system. However, compared with IPT, CPT has a different coupling structure and compensation topology, so the analysis of a multi-frequency power transmission system in an IPT system cannot be directly applied to a CPT system.

In this paper, a new dual-frequency CPT system is proposed. Due to its special compensation network design, both the primary and tertiary frequency voltage components of the inverter output voltage can be utilized to transmit energy at two frequencies. In addition, the voltage stress of the coupling plate can be reduced by controlling the ratio of the transmitted energy of the two frequencies, while the conduction loss of the inverter can be reduced. The contributions of this paper are as follows. (a) A CPT system capable of dual-frequency power transmission is proposed. Compared with the equivalent single-frequency system, a dual-frequency system can reduce the voltage stress amplitude on the coupled plate and also reduce the conduction loss of the inverter. (b) The working principle of the dual-frequency CPT system is analyzed in detail, and the control strategy of the energy transmission ratio between the primary and tertiary frequency channels is proposed. (c) According to the theoretical derivation scheme, a test prototype is designed to verify the correctness of the proposed theory.

The rest of this paper is organized as follows. In the Section 2, the proposed circuit model is introduced systematically, including the dual-power equivalent model and the resonant unit module equivalent model. Then, the circuit analysis of the proposed dual-frequency system is carried out to obtain the control method of the power transmission ratio at different frequencies. In the Section 3, the equivalent fundamental single-frequency circuit is introduced, and the single-frequency circuit and the dual-frequency circuit are compared and analyzed. The influence of the introduction of the third harmonic frequency voltage component on the voltage stress among the coupled plates and the inverter conduction losses is explained in detail. Then, the system parameter design process is given. In Section 4, the correctness of the proposed theory is verified by simulation and experiment. The Section 5 summarizes the full text.

2. System Description and Working Principle

2.1. System Description

The dual-frequency CPT system is shown in Figure 2a. The system has two power sources that provide power at two frequencies from the fundamental and third harmonic frequency voltage components of the full-bridge inverter output voltage. Assuming that the duty ratio of the full-bridge inverter is 50%, the rectangular wave voltage $v_{in}\left(t\right)$ on the AC side can be decomposed into a Fourier series composed of odd harmonics.

$$\begin{array}{c}{v}_{in}\left(t\right)=\frac{4V_{dc}}{\pi }{\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}\frac{\sin \left(\left(2n-1\right)\omega t\right)}{2n-1}, n=1,2,3,\cdots \end{array}$$

(1)

For different frequencies, we can define each parameter separately and express it with corresponding subscripts (i.e., 1 represents the fundamental wave, and 3 represents the third harmonic wave). The root mean square value of the fundamental and third harmonic voltage can be expressed as

$$\begin{array}{c}{V}_{in,1}=\frac{2\sqrt{2}{V}_{dc}}{\pi }, V_{in,3}=\frac{2\sqrt{2}{V}_{dc}}{3\pi }\end{array}$$

(2)

The resonant compensation structure adopts a topological structure similar to that of a double-sided LC, though different from the general double-sided LC compensation. Due to the need to make the system reach the resonant condition at the same time under the dual frequencies of the fundamental and third harmonic, we need to specifically design the resonant compensation module. In Figure 2b, the original single impedance element is replaced by a combination of $L_i$ and $C_i$ connected in parallel in series $C_i^{\prime }$ (or $L_j$ and $C_j$ connected in parallel in series $L_j^{\prime }$) [30], so the equivalent impedance is represented by $Z_i$ or $Z_j$; then,

$$\begin{gathered} Z_i=\frac{j\omega L_i}{1-{\omega }^2{L}_i{C}_i}+\frac{1}{j\omega C_i^{\prime }} i=1,4 \\ \begin{array}{c}{Z}_j=\frac{j\omega L_j}{1-{\omega }^2{L}_j{C}_j}+j\omega L_j^{\prime } j=2,3\end{array} \end{gathered}$$

(3)

The two capacitors formed by the coupled plates are a series structure. The equivalent capacitance can be represented by $C_X$, and the equivalent impedance can be represented by $Z_X$, namely,

$$\begin{array}{c}{C}_X=\frac{C_m}{2}, Z_X=\frac{1}{j\omega C_X}=\frac{2}{j\omega C_m}\end{array}$$

(4)

The equivalent Π-type circuit model of the dual-frequency CPT system is shown in Figure 2c. The full-bridge inverter generates high-frequency AC voltage to supply power to the primary side resonant network. The high voltage of the fundamental and third harmonic frequency voltages, $V_{in,1}$ and $V_{in,3}$, respectively, raised by the resonant network generates a displacement current between the coupling plate and then supplies power to the load end after the secondary resonant network depressurizes. The key condition for the normal operation of the dual-frequency system is that the input impedance of the system is very low at both frequencies to realize the dual-frequency wireless power transmission.

2.2. Working Principle

Figure 2d shows the equivalent T-type circuit model transformed from the Π-type circuit model of the dual-frequency CPT system. So

$$\begin{array}{c}{Z}_{2X}=\frac{Z_2{Z}_X}{Z_2+Z_3+Z_X},Z_{3X}=\frac{Z_3{Z}_X}{Z_2+Z_3+Z_X}, Z_{23}=\frac{Z_2{Z}_3}{Z_2+Z_3+Z_X}\end{array}$$

(5)

The relation between the input current $I_{in}$, output current $I_{out}$, and input voltage $V_{in}$ can be expressed as

$$\begin{array}{c}\left[\begin{array}{c}{V}_{in}\\ 0\end{array}\right]=\left[\begin{array}{cc}{Z}_1+Z_{2X}+Z_{23}& -Z_{23}\\ -Z_{23}& Z_{23}+Z_{3X}+Z_4+R_L\end{array}\right]·\left[\begin{array}{c}{I}_{in}\\ I_{out}\end{array}\right]\end{array}$$

(6)

In order to achieve the resonant state of the system to achieve the zero phase angle (ZPA) characteristic,

$$\begin{array}{c}{Z}_1+Z_{2X}+Z_{23}=0, Z_{23}+Z_{3X}+Z_4=0\end{array}$$

(7)

According to Equations (6) and (7), $I_{in}$ and $I_{out}$ can be expressed as

$$\begin{array}{c}{I}_{in}=-\frac{V_{in}{R}_L}{Z_{23}^2}, I_{out}=-\frac{V_{in}}{Z_{23}} \end{array}$$

(8)

According to Formula (6), $Z_{23}$ determines the values of the input current $I_{in}$ and output current $I_{out}$ when $V_{in}$ and $R_L$ are constant. Therefore, we can control the power transmission ratio at different frequencies by controlling the value of $Z_{23}$ at different frequencies.

3. System Design Considering Coupler Voltage Stress and Inverter Conduction Losses

A constant, $k$, is used to represent the ratio of power transmission at different frequencies. $P_{out}$ is the total transmission power; $P_{out,1}$ and $P_{out,3}$ represent the transmission power under the fundamental and third harmonic frequencies, respectively.

$$\begin{array}{c}{P}_{out,1}=k{P}_{out}, P_{out,3}=\left(1-k\right)P_{out}\end{array}$$

(9)

In the ideal state without considering the system loss, the transmission power of the system can be expressed as

$$\begin{array}{c}{P}_{out,1}=I_{out,1}^2{R}_L={\left(\frac{V_{in,1}}{Z_{23,1}}\right)}^2{R}_L, P_{out,3}=I_{out,3}^2{R}_L={\left(\frac{V_{in,3}}{Z_{23,3}}\right)}^2{R}_L\end{array}$$

(10)

Here, we assume an equivalent fundamental single-frequency system that transmits the same amount of power as the dual-frequency system. In the following derivation, we define the single-frequency system and the dual-frequency system with the subscripts s and d, respectively, so

$$\begin{array}{c}{P}_{out,s}=I_{out,s}^2{R}_L={\left(\frac{V_{in,1}}{Z_{23,s}}\right)}^2{R}_L\end{array}$$

(11)

By combining Equations (9)–(11), Equation (12) can be obtained.

$$\begin{array}{c}\frac{1}{Z_{23,1}^2}=\frac{k}{Z_{23,s}^2} , \frac{1}{Z_{23,3}^2}=\frac{9\left(1-k\right)}{Z_{23,s}^2}\end{array}$$

(12)

We take $Z_{23,s}$ as the base value, so the unit values $Z_{23,1}^{*}$ and $Z_{23,3}^{*}$ can be expressed as

$$\begin{array}{c}{Z}_{23,1}^{*}=\sqrt{\frac{1}{k}}, Z_{23,3}^{*}=\sqrt{\frac{1}{9\left(1-k\right)}}\end{array}$$

(13)

3.1. Analysis of Coupler Voltage Stress

According to Figure 2c, the equations of KVL are

$$\begin{array}{c}\left[\begin{array}{c}{V}_{in}\\ 0\\ 0\end{array}\right]=\left[\begin{array}{ccc}{Z}_1+Z_2& -Z_2& 0\\ -Z_2& Z_2+Z_X+Z_3& -Z_3\\ 0& -Z_3& Z_2+Z_4+R_L\end{array}\right]·\left[\begin{array}{c}{I}_{in}\\ I_m\\ I_{out}\end{array}\right]\end{array}$$

(14)

By combining Equations (5), (7), (8) and (14), Equation (15) can be obtained.

$$\begin{array}{c}{I}_m=\frac{-V_{in}\left[\left(Z_2+Z_X+Z_3\right)R_L+Z_3^2\right]}{Z_2{Z}_3^2}\end{array}$$

(15)

Then, the coupler voltage $V_m$ can be represented as

$$\begin{array}{c}{V}_m=\frac{-V_{in}{Z}_m\left[\left(Z_2+Z_X+Z_3\right)R_L+Z_3^2\right]}{Z_2{Z}_3^2}=\frac{-V_{in}{Z}_m}{Z_2}·\left[1+\frac{\left(Z_2+Z_X+Z_3\right)R_L}{Z_3^2}\right]\end{array}$$

(16)

Although dual-frequency power transmission results in the existence of two power transmission “channels” that can themselves be controlled independently by the corresponding parameters at their respective frequencies, the relative positioning of the voltage components at the two frequencies is still very important, especially when considering the voltage stress of the coupled plate. Since the relative positioning between two signals at different frequencies cannot be defined by using the phase shift angle, we select a special position: inverter output terminal voltage $v_{in}\left(t\right)$. Here, there is no phase shift when the fundamental and third harmonic voltage components refer to a common origin, namely,

$$\begin{array}{c}{v}_{in}\left(t\right)=\sqrt{2}{V}_{in,1}\sin \left(\omega t\right)+\sqrt{2}{V}_{in,3}\sin \left(3\omega t\right).\end{array}$$

(17)

As shown in Figure 1, the voltage components $v_{m,1}\left(t\right)=V_{m,1}\sin \left(\omega t+{\theta }_1\right)$ and $v_{m,3}\left(t\right)=V_{m,3}\sin \left(3\omega t+{\theta }_3\right)$ of the coupled plate voltage $v_m\left(t\right)$ at the two frequencies need to meet the following conditions in order to achieve the desired “peak clipping” effect.

$$\begin{array}{c}{\theta }_3-3{\theta }_1=2n\pi , n=0,\pm 1,\pm 2\cdots \end{array}$$

(18)

When Equation (19) is true, Equation (18) is satisfied.

$$\begin{array}{c}\left|Z_3^2\right|\gg \left|\left(Z_2+Z_X+Z_3\right)R_L\right|\end{array}$$

(19)

Equation (20) can be obtained by simplifying Equation (16).

$$\begin{array}{c}{V}_m=-\frac{V_{in}{Z}_m}{Z_2}\end{array}$$

(20)

Equation (19) is not difficult to realize, so take one of the cases, Equation (21), to simplify the operation.

$$\begin{array}{c}{Z}_{23}=\frac{Z_2{Z}_3}{Z_2+Z_X+Z_3}=-Z_2\end{array}$$

(21)

So

$$\begin{array}{c}{Z}_X+Z_2+2Z_3=0\end{array}$$

(22)

$$\begin{array}{c}{V}_m=\frac{V_{in}{Z}_m}{Z_{23}}\end{array}$$

(23)

It should be noted that Equation (19) should still be met in the subsequent parameter design. With $V_{m,s}$ as the basic value, the normalized value of the fundamental and the third harmonic frequency components ($V_{m,1}^{*}$ and $V_{m,3}^{*}$) in the plate voltage of the dual-frequency system can be expressed as

$$\begin{array}{c}{V}_{m,1}^{*}=\frac{Z_{m,1}^{*}}{Z_{23,1}^{*}}=\sqrt{k}, V_{m,3}^{*}=\frac{Z_{m,3}^{*}}{3Z_{23,3}^{*}}=\frac{\sqrt{1-k}}{3}\end{array}$$

(24)

Then, the voltage $v_{m,d}\left(t\right)$ on the coupled plate of the dual-frequency system can be expressed as

$$v_{m,d}\left(t\right)=\sqrt{2}{V}_{m,s}\left[\sqrt{k}\sin \left(\omega t\right)+\frac{\sqrt{1-k}}{3}\sin \left(3\omega t\right)\right]$$

(25)

Taking the voltage amplitude $V_{m,s,max}=\sqrt{2}{V}_{m,s}$ as the basic value, the normalized coupler voltage amplitude $V_{m,d,max}^{*}$ of the dual-frequency system can be expressed as

$$\begin{array}{c}{V}_{m,d,max}^{*}=max\left[\sqrt{k}\sin \left(\omega t\right)+\frac{\sqrt{1-k}}{3}\sin \left(3\omega t\right)\right]\end{array}$$

(26)

According to Equation (26), the relationship between the normalized coupler voltage amplitude $V_{m,d,max}^{*}$ and power-sharing ratio $k$ in the dual-frequency system can be obtained, as shown in Figure 3. The higher the proportion of energy transmitted through the third harmonic frequency “channel” is, the lower the voltage stress among the coupled plates is. When the energy is transmitted only through the third harmonic frequency “channel”, the voltage stress on the coupled plate is 33.3% of the equivalent single-frequency system. This shows that it is viable to reduce the voltage stress among the coupled plates by simultaneously transmitting energy at the fundamental and third harmonic frequencies. Of course, the power-sharing ratio $k$ value should also pay attention to the influence of other factors.

3.2. Analysis of Inverter Conduction Losses

According to Equation (8), the inverter currents of the equivalent single-frequency system and the dual-frequency system can be expressed as

$$\begin{array}{c}{I}_{in,s}=-\frac{V_{in,1}{R}_L}{Z_{23,s}^2}, I_{in,1}=-\frac{V_{in,1}{R}_L}{Z_{23,1}^2}, I_{in,3}=-\frac{V_{in,3}{R}_L}{Z_{23,3}^2}\end{array}$$

(27)

Further, it can be concluded that when compared with the equivalent single-frequency system, the normalized inverter conduction losses $P_{inloss,d}^{*}$ of the dual-frequency system can be expressed as

$$\begin{array}{c}{P}_{inloss,d}^{*}=\frac{P_{inloss,d}}{P_{inloss,s}}=\frac{I_{in,1}^2+I_{in,3}^2}{I_{in,s}^2}=k^2+9{\left(1-k\right)}^2\end{array}$$

(28)

According to Equation (28), the relationship between the normalized inverter conduction losses $P_{inloss,d}^{*}$ and power-sharing ratio $k$ in the dual-frequency system can be obtained, as shown in Figure 4. When the energy is transmitted only through the third harmonic frequency “channel”, the inverter conduction losses are nine times that of the equivalent single-frequency system. When $k=0.9$, the inverter conduction losses $P_{inloss,d}$ of the dual-frequency system are the smallest: 90% of the equivalent single-frequency system with the same transmission power.

3.3. System Parameter Design

The coupler voltage amplitude $V_{m,d,max}$ and inverter conduction losses $P_{inloss,d}$ of the system under different conditions can be obtained, as shown in

Table 1. $V_{m,d,max}$ and $P_{inloss,d}$ of the system under different conditions.

$\mathit{k}$	${\mathit{V}}_{\mathit{m},\mathit{d},\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{P}}_{\mathit{i}\mathit{n}\mathit{l}\mathit{o}\mathit{s}\mathit{s},\mathit{d}}$
0	$0.333·\sqrt{2}{V}_{m,s}$	$9P_{inloss,s}$
0.2	$0.594·\sqrt{2}{V}_{m,s}$	$5.8P_{inloss,s}$
0.4	$0.632·\sqrt{2}{V}_{m,s}$	$3.4P_{inloss,s}$
0.6	$0.7·\sqrt{2}{V}_{m,s}$	$1.8P_{inloss,s}$
0.8	$0.775·\sqrt{2}{V}_{m,s}$	$P_{inloss,s}$
0.9	$0.843·\sqrt{2}{V}_{m,s}$	$0.9P_{inloss,s}$
1	$\sqrt{2}{V}_{m,s}$	$P_{inloss,s}$

. By controlling the power-sharing ratio $k=0.9$, the voltage stress among the coupled plates and inverter conduction losses of the dual-frequency system are better than those of the single-frequency system.

To reduce the voltage stress among the coupled plates, a parameter design method based on the dual-frequency CPT system is proposed as follows.

First, determine the pre-designed system parameters $V_{in}$, $f$, and $C_m$ and total output power $P_L$, where $C_m$ is directly measured by measuring equipment. Then, the appropriate power-sharing ratio $k$ is selected. In this paper, $k=0.9$ is finally selected through the analysis of the system. Second, based on the selected system parameters, the appropriate $Z_{23}$($Z_{23,1}$ and $Z_{23,3}$) is selected according to Equation (13). To obtain the value of each element, we must calculate the different values of each compensation block $Z_i$ ($i=1, 4$) and $Z_j$ ($j=2, 3$) at two frequencies. Based on the existing analytical derivation, $Z_2$ and $Z_3$ can be obtained from Equations (21) and (22), respectively. Third, based on the Π-type to T-type transformation relationship, $Z_{2X}$ and $Z_{3X}$ can be obtained according to Equations (5) and (7), respectively. To realize the load-independent constant current (CC) output and input zero phase angle (ZPA), we can obtain $Z_1$ and $Z_4$ according to Equation (7). Finally, based on the different values of $Z_1$, $Z_2$, $Z_3$, and $Z_4$ at both frequencies, the parameter values of each inductor and capacitor element can be flexibly designed according to Equation (3). The system design process of the proposed dual-frequency CPT system is shown in Figure 5.

4. Experimental Verification

To verify the above derivation, a test prototype was designed, as shown in Figure 6. The parameters of each component of the system were designed according to Figure 5 and recorded in

Table 2. Parameters of the experimental setup.

Parameter	Value	Parameter	Value
Input DC Voltage	50 V	$C_m$	1.28 nF
Switching Frequency	500 kHz	$L_3$	21.6 µH
$L_1$	3 µH	$C_3$	787 pF
$C_1$	11.64 nF	$L_3^{\prime }$	50 µH
$C_1^{\prime }$	8.8 nF	$L_4$	29.15 µH
$L_2$	2.1 µH	$C_4$	7.12 nF
$C_2$	18.19 nF	$C_4^{\prime }$	816 pF
$L_2^{\prime }$	3 µH	$R_L$	40 Ω

. In the actual experimental operation, we used the precision LCR meter in the laboratory to ensure that the actual value of each parameter is as close to the expected value as possible. Coupled plates adopted PCB board design, and Cm = 1.28 nF. The switching frequency of the inverter was selected as 500 kHz. To reduce the influence of the skin effect, a Litz wire containing 1200 strands with a diameter of 0.05 mm was selected. WIMA high-frequency thin-film capacitors were selected to reduce power losses. The inverter was made of CREE silicon carbide (SiC) MOSFETs, and the rectifier was made of Infineon’s SiC diodes.

Figure 7 shows the bode plots of system input admittance. It can be seen that the CPT system can realize the input ZPA characteristics at the two frequencies, 500 kHz and 1500 kHz, and the input admittance is about 0.1 s, meeting the requirements for dual-frequency power transmission.

Figure 8 shows the timing diagram of the system input voltage $v_{in}$, input current $i_{in}$, coupler voltage $v_m$, and output current $i_{out}$, where (a) is for the simulation test, and (b) is for the experimental test. The waveforms of the input current $i_{in}$ and coupler voltage $v_m$ all show the “peak clipping” effect, which is consistent with the previous theoretical proof and proves the feasibility of using additional third harmonic frequency transmission “channel” to reduce the voltage stress among the coupled plates and the inverter conduction losses.

Figure 9 shows the simulation waveforms of (a) the coupler voltage $v_m$ and (b) inverter output current $i_{in}$ between the proposed dual-frequency system and the equivalent single-frequency system. By comparison, we can see the feasibility of using a dual-frequency system to reduce the voltage stress among the coupled plates and the inverter conduction losses.

Figure 10 shows the output power and efficiency under different inverter input voltages. When the output power is higher than 200 W, the efficiency of the CPT system reaches 83.71%.

Figure 11 shows the waveforms after adding the rectifier structure. We found that the inverter output current waveform appeared distorted, and the system deviated from the resonant state.

As shown in Figure 12, enhanced multiple harmonic analysis (eMHA), a load equivalent method where $Z_{Leq,n}=R_{Leq,n}+j{X}_{Leq,n}$, is proposed in [31]. The rectified CPT system is converted into a series of linear systems with complex load impedance, which explains the waveform distortion. In addition, it can be seen from Equation (23) that the voltage stress among the coupled plates of the dual-frequency CPT system is load-independent, so the voltage waveform of the plate is not affected before or after the addition of the rectifier bridge. Therefore, we just need to modify the parameters of $Z_4$ to bring the system back to the resonant state.

Table 3. Parameters of the corrected system.

Parameter	${\mathit{L}}_4$	${\mathit{C}}_4$	${\mathit{C}}_4^{\prime }$
Original Value	29.15 µH	7.12 nF	816 pF
Corrected Value	3.3 µH	12.7 nF	626 pF

shows the corrected parameter values.

5. Conclusions

A novel dual-frequency CPT system is proposed in this paper. The goal of reducing the voltage stress among the coupler plates is realized by introducing additional third harmonic voltage components. By controlling the energy transfer ratio of 9:1 ($k=0.9$) at the fundamental and third harmonic frequencies, the coupler voltage stress is reduced to 84.3%, and the inverter conduction losses are reduced to 90% of the equivalent single-frequency system. An experimental prototype with a transmission power of 200 W is designed to verify the effectiveness of the proposed dual-frequency CPT system at a system frequency of 500 kHz. The results of the simulation and experiment verify the correctness of the theoretical derivation. In addition, we correct the parameters of $Z_4$ to make the system return to the resonant state according to eMHA.