Single-Wire Electric-Field Coupling Power Transmission Using Nonlinear Parity-Time-Symmetric Model with Coupled-Mode Theory

Abstract

The output power and transmission efficiency of the traditional single-wire electric-field coupling power transmission (ECPT) system will drop sharply with the increase of the distance between transmitter and receiver, thus, in order to solve the above problem, in this paper, a new nonlinear parity-time (PT)-symmetric model for single-wire ECPT system based on coupled-mode theory (CMT) is proposed. The proposed model for single-wire ECPT system not only achieves constant output power but also obtains a high constant transmission efficiency against variable distance, and the steady-state characteristics of the single-wire ECPT system are analyzed. Based on the theoretical analysis and circuit simulation, it shows that the transmission efficiency with constant output power remains 60% over a transmission distance of approximately 34 m without the need for any tuning. Furthermore, the application of a nonlinear PT-symmetric circuit based on CMT enables robust electric power transfer to moving devices or vehicles.

1. Introduction

Since the discovery of electricity, the power transmission has mainly depended on the wires, and the way of power transmission through wires requires at least two wires to provide conduction paths for conduction current, which may be difficult to install but also has a large number of safety issues, such as the risk of fire and electric shock caused by short circuits. However, in the early 20th century, Nikola Tesla first proposed the concept of wireless power transmission (WPT) [1], which is a novel technology that can transfer power through air without wires. His imagination about global power transmission without wires also shocked the world. Although Tesla’s experiment was not completed due to funding problems [2], it caused numerous experts and scholars to start exploration and research on this technology which can provide human beings with more convenience of production and lifestyles.

Currently, WPT technology is mainly divided into three categories [3], namely, non-radiative WPT, radiative WPT, and electromechanical WPT. Among them, the most extensively studied are inductively coupling wireless power transmission (ICPT) [4], magnetic resonant coupling wireless power transmission (MRWPT) [5], microwave wireless power transmission (MWPT) [6], and laser wireless power transmission (LWPT) [7]. The concept of ICPT was first proposed by Professor J. T. Boys of the University of Auckland, New Zealand. His research team started to study this technology as early as the 1990s, and achieved a great deal of success [8,9,10]. In the theoretical analysis, a lot of studies have been done on the basic principle, characteristic analysis, compensation circuit analysis, frequency splitting and stability of ICPT [8,9,10]. In practical application, inductive wireless charging for electric vehicles, mobile robots, and implanted devices were studied [11,12,13]. Professor Marin of MIT (Massachusetts Institute of Technology) proposed a new MRWPT technology and published it in the magazine Science in 2007 [5]. Until now, great progress has been made in the modeling of the system, impedance matching, electromagnetic safety, and standard setting [14,15]; in the application, Haier developed the tailless television in 2010, the KAIST (Korea Advanced Institute of Science and Technology) established a 12-km demonstration route of dynamic power supply for electric vehicles in Gumi city, South Korea in 2015, and the company of Zhonghui Chuangzhi in China applied MRWPT technology to charge a mine equipment monitoring system. As early as 1964, Willian C. Brown of Raytheon succeeded in using the MWPT technology to power a helicopter platform [16]. Subsequently, many scientists continued to study this technology, however, the bottleneck of transmission efficiency higher than 10% cannot be broken through. The technology of LWPT is still in the experimental stage and has not been put into practical application, Japan started experimenting on LWPT in 1997 [17], and used laser energy to power robots, kites, and helicopters [7,18]; scientists from Germany and the United States conducted experiments about LWPT in succession [19,20]. The first two WPT technologies have more extensive application and development prospects since the transmission efficiency of the latter two kinds of WPT technology is very low and the sensitivity of the direction is extremely high. The transmission distance of inductively coupling WPT is relatively close, in the range of tens of centimeters, and the distance of magnetic resonant coupling WPT is relatively far, in the range of a few meters, while its output power is not high, only suitable for low-power applications, such as mobile phones and other portable devices [21]. In addition, the output power and transmission efficiency of these two WPT system both decrease rapidly with the increase of the transmission distance [22,23]. To overcome the danger of two-wire or three-wire power transmission modes and improve the transmission distance at the same time, a single-wire power transmission system was proposed [24]. Dmitry S. Strebkov of the All-Russia Institute of Agricultural Electrification produced single-wire power transmission systems of different voltage levels with a total capacity of 1 kW and high transmission efficiency in 2000 [25]. In 2011, Steve Jackson gave a presentation about single-wire power transmission at McMaster University in Canada, with a transmission distance of up to 20 m [26].

The single-wire power transmission system can transfer high energy to the load with only one wire, simultaneously the transmission distance can be up to tens of meters. Whereas keeping constant output power while maintaining high constant transmission efficiency against variable distance is also a significant challenge for single-wire ECPT systems.

Furthermore, the concept of PT-symmetry originated in the optical system, and was first applied to the related fields of WPT in 2017 [27]. A PT-symmetric system supports two phases, including an unbroken and broken phase, only the characteristics of unbroken phase were used in WPT system. In the unbroken phase, eigenfrequencies of the system remain real and the energy is equally stored in transmitting and receiving resonators [28]. Despite its many advantages, the nonlinear PT-symmetric model is not applied in actual WPT systems, especially the single-wire ECPT system.

In order to make the output power and efficiency of the single-wire ECPT system robust to the transmission distance, in this paper, a novel nonlinear PT-symmetric model based on CMT for single-wire ECPT system is proposed. Firstly, a theoretical model of single-wire ECPT system is established based on CMT, then, the PT-symmetric model of the system is obtained, and the transmission characteristics of the system is analyzed. Finally, the results by the analysis of circuit simulation show that the transmission efficiency with constant output power remains near 60% within the critical distance of 34 m.

2. System Structure and Modeling

In general, single-wire ECPT systems mainly contain a source, transmitter, receiver, and load, as shown in Figure 1. The source is usually achieved by a converter, such as half-bridge, full-bridge, and others, which serves as power supply for the whole system. The transmitter includes a step-up transformer T₁ and a metal conductor Q₁ that obtains the power from the source and transfers it to the receiver through coupling electric field. The receiver comprises a metal conductor Q₂ and step-down transformer T₂ that provides power for a resistant load. Particularly, the coupling capacitance between Q₁ and Q₂ can be expressed as $C_{12}=4\pi {\epsilon }_0{r}^2/d$ when both Q₁ and Q₂ are spherical metal conductors of radius r, ${\epsilon }_0$ is the permittivity of vacuum.

2.1. Discrete State Space Model

As illustrated in Figure 1, the transformer T₁ on the transmitting side only plays a role in boosting the input voltage, similarly, the transformer T₂ on the receiving side only serves to reduce the output voltage, the source reflected to the high-voltage coil on the transmitting side is proportional to the original voltage source or current source, and the load reflected to the low-voltage coil on the receiving side is also proportional to the actual load. Thus, the system shown in Figure 1 can be simplified to a two-coil equivalent circuit, as shown in Figure 2, it is a detailed schematic of the entire single-wire ECPT system based on the nonlinear PT model, which includes a half-bridge converter, coil inductance L₁ and L₂, internal resistance r₁ and r₂, parasitic capacitance C₁ and C₂, coupling capacitor C₁₂, and load resistance R_L. It is important to note that the half-bridge converter is used to realize a negative resistor with the characteristics of high power nonlinear gain saturation, thus attaining a PT-symmetric mode of the entire single-wire ECPT system. Besides, the implementation of the negative resistor is not limited to half-bridge converter.

The dynamics of this nonlinear PT-symmetric circuit in Figure 2 can be fully described by the discrete state space equations as follows

$$\{\begin{array}{l}\frac{d{i}_1}{dt}=-\frac{r_1}{L_1}{i}_1-\frac{1}{L_1}{u}_1+\frac{1}{L_1}{u}_{\mathrm{in}}\\ \frac{d{u}_1}{dt}=\frac{\left(C_{12}+C_2\right)}{\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}{i}_1+\frac{C_{12}}{\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}{i}_2\\ \frac{d{i}_2}{dt}=-\frac{\left(r_2+R_L\right)}{L_2}{i}_2-\frac{1}{L_2}{u}_2\\ \frac{d{u}_2}{dt}=\frac{C_{12}}{\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}{i}_1+\frac{\left(C_{12}+C_1\right)}{\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}{i}_2\end{array}$$

(1)

where i₁ and i₂ are currents flowing through the inductors, u₁ and u₂ are voltages across the capacitors, u_in is the output voltage of the half-bridge converter.

Equation (1) can be solved immediately when u_in is given. Figure 3 shows the waveform of u_in at two different operation cases, as well as i₁ and the gate drive signals of S₁ and S₂ (v_{g_S1} and v_{g_S2}). To simplify the analysis, the delay time and dead time is not considered in modeling. As shown in Figure 3, the voltage u_in depends on current waveform i₁ across the LC (inductance and capacitance) tank of the transmitter [29]. Therefore, assuming the direction of current i₁ shown in Figure 2 is positive, u_in is defined as

$$u_{\mathrm{in}}=\frac{\mathrm{sgn}\left(i_1\right)+1}{2}{V}_{\mathrm{in}}$$

(2)

where V_in is the input dc voltage of the half-bridge converter.

2.2. Dynamic Modeling by CMT

The state space averaging method is difficult to apply in the steady-state analysis and transient-state analysis of the system based on circuit theory because of the fast varying state variables, i_n and u_n, n = 1, 2. To overcome these problems, a dynamic modeling method based on coupled modes is proposed [29].

The voltages and currents of coupled resonators can be represented by coupled modes

$$a_{\mathrm{n}}=\sqrt{\frac{C_{\mathrm{n}}}{2}}{u}_{\mathrm{n}}+j\sqrt{\frac{L_{\mathrm{n}}}{2}}{i}_{\mathrm{n}}=A_{\mathrm{n}}{e}^{-j\left(\omega t+{\theta }_{\mathrm{n}}\right)},n=1,2$$

(3)

where the subscript 1 (or 2) denotes the transmitter (or receiver), the variables a_n are defined so that the energy contained in resonators n is ${\left|a_{\mathrm{n}}\right|}^2$, ω is the operating angular frequency of system, the variables A_n represent the amplitudes of the modes while the variables θ_n refers to the phase of the modes. Both A_n and θ_n vary slowly with time.

To determine the dynamic equations of the coupled modes, representing the currents flowing through inductors and the voltage across capacitors by Equation (3), and substituting them and Equation (2) into Equation (1), the dynamic equations of the coupled modes can be derived out from the discrete state space model. Then, by assuming the slowly varying variables A_n and θ_n are constant during a switching period, and eliminating the high-frequency terms by averaging method [30], while remaining the low-frequency characteristics, thus, the time-invariant averaged model can be described by

$$\{\begin{array}{l}\frac{d{A}_1}{dt}=-\frac{r_1}{2L_1}{A}_1-\frac{C_{12}\sqrt{C_1{C}_2}}{2\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}\frac{1}{\sqrt{L_2{C}_2}}{A}_2\sin \left({\theta }_2-{\theta }_1\right)-\frac{1}{\sqrt{2L_1}}\frac{1}{\pi }{V}_{\mathrm{in}}\\ \frac{d{\theta }_1}{dt}=-\omega +\frac{1}{2}\frac{1}{\sqrt{L_1{C}_1}}+\frac{\left(C_{12}+C_2\right)C_1}{2\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}\frac{1}{\sqrt{L_1{C}_1}}+\frac{C_{12}\sqrt{C_1{C}_2}}{2\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}\frac{1}{\sqrt{L_2{C}_2}}\frac{A_2}{A_1}\cos \left({\theta }_2-{\theta }_1\right)\\ \frac{d{A}_2}{dt}=-\frac{\left(r_2+R_{\mathrm{L}}\right)}{2L_2}{A}_2+\frac{C_{12}\sqrt{C_1{C}_2}}{2\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}\frac{1}{\sqrt{L_1{C}_1}}{A}_1\sin \left({\theta }_2-{\theta }_1\right)\\ \frac{d{\theta }_2}{dt}=-\omega +\frac{1}{2}\frac{1}{\sqrt{L_2{C}_2}}+\frac{C_{12}\sqrt{C_1{C}_2}}{2\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}\frac{1}{\sqrt{L_1{C}_1}}\frac{A_1}{A_2}\cos \left({\theta }_2-{\theta }_1\right)+\frac{\left(C_{12}+C_1\right)C_2}{2\left(C_{12}{C}_1+C_{12}{C}_2+C_1{C}_2\right)}\frac{1}{\sqrt{L_2{C}_2}}\end{array}$$

(4)

The derivative of Equation (3) over time can be defined as

$$\frac{d{a}_{\mathrm{n}}}{dt}=\frac{d{A}_{\mathrm{n}}}{dt}{e}^{-j\left(\omega t+{\theta }_{\mathrm{n}}\right)}-j\left(\omega +\frac{d{\theta }_{\mathrm{n}}}{dt}\right)A_n{e}^{-j\left(\omega t+{\theta }_{\mathrm{n}}\right)},\mathrm{n}=1,2$$

(5)

Substituting Equation (4) into Equation (5), the dynamic equations of coupled modes for single-wire ECPT system can be derived as

$$\{\begin{array}{l}\frac{d{a}_1}{dt}=\left[-j{\omega }_1\frac{1}{2}\left(1+\frac{1+\frac{1}{k_{12}}\sqrt{\frac{C_2}{C_1}}}{1+\frac{1}{k_{12}}\sqrt{\frac{C_2}{C_1}}+\frac{C_2}{C_1}}\right)+g_0-{\tau }_{10}\right]a_1-j{\omega }_2\frac{k_{12}}{2\left(\frac{C_1+C_2}{\sqrt{C_1{C}_2}}{k}_{12}+1\right)}{a}_2\\ \frac{d{a}_2}{dt}=\left[-j{\omega }_2\frac{1}{2}\left(1+\frac{1+\frac{1}{k_{12}}\sqrt{\frac{C_1}{C_2}}}{1+\frac{1}{k_{12}}\sqrt{\frac{C_1}{C_2}}+\frac{C_1}{C_2}}\right)-{\tau }_{20}-{\tau }_{\mathrm{L}}\right]a_2-j{\omega }_1\frac{k_{12}}{2\left(\frac{C_1+C_2}{\sqrt{C_1{C}_2}}{k}_{12}+1\right)}{a}_1\end{array}$$

(6)

where the natural resonant frequencies are defined as: ${\omega }_1=1/\sqrt{L_1{C}_1}$ and ${\omega }_2=1/\sqrt{L_2{C}_2}$. The loss rates are denoted as: ${\tau }_{10}=r_1/2L_1$, ${\tau }_{20}=r_2/2L_2$, and ${\tau }_{\mathrm{L}}=R_{\mathrm{L}}/2L_2$. The electric-field coupling coefficient between transmitter and receiver is $k_{12}=C_{12}/\sqrt{C_1{C}_2}$, and the gain rate is $g_0=V_{\mathrm{in}}/\left(\pi \sqrt{2L_1}\left|a_1\right|\right)$.

For a long-range ECPT system, natural resonant frequencies of two LC tanks are usually close to each other (${\omega }_1\approx {\omega }_2$). Besides, a PT-symmetric system requires that the two natural resonant frequencies are the same (${\omega }_1={\omega }_2={\omega }_0$). The term C₁₂ is for the coupling capacitor that is generated by the displacement current between the metal conductors on the transmitting and receiving sides, which is much smaller than C₁ and C₂ in general. In addition, the values of ${\tau }_{10}$, ${\tau }_{20}$, ${\tau }_L$, $g_0\left|a_1\right|$ and ω₀k₁₂ are relatively close in weak coupling region. Therefore, to simplify Equation (6), the simplified coupled-mode equations can be derived as follows

$$\{\begin{array}{l}\frac{d{a}_1}{dt}=\left(-j{\omega }_0+g_0-{\tau }_{10}\right)a_1-j\kappa a_2\\ \frac{d{a}_2}{dt}=\left(-j{\omega }_0-{\tau }_{20}-{\tau }_{\mathrm{L}}\right)a_2-j\kappa a_1\end{array}$$

(7)

where $\kappa ={\omega }_0{k}_{12}/2$ is defined as the coupling coefficient between two modes a₁ and a₂.

3. Analysis of Transmission Characteristics

The transmission performance, such as the power delivered to the load P_L, transmission efficiency η and the operating frequency ω, can be easily derived from the steady solutions of the coupled-mode Equation (7).

Defining $a_{1,2}\propto e^{-j\omega t}$, the operating frequency in the steady state can be obtained by characteristic equation

$$\left[-j\left(\omega -{\omega }_0\right)-g_0+{\tau }_{10}\right]\left[-j\left(\omega -{\omega }_0\right)+{\tau }_{20}+{\tau }_{\mathrm{L}}\right]+{\kappa }^2=0$$

(8)

Separating the real and imaginary parts of Equation (8), it can be see that there are two equations to be satisfied, as follows

$$\{\begin{array}{l}{\left(\omega -{\omega }_0\right)}^2+\left(g_0-{\tau }_{10}\right)\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)-{\kappa }^2=0\\ \left(\omega -{\omega }_0\right)\left(g_0-{\tau }_{10}-{\tau }_{20}-{\tau }_{\mathrm{L}}\right)=0\end{array}$$

(9)

By solving the above Equation (9), it can be found that there are two regions containing solutions of Equation (9), depending on the relative values of $\kappa$ and ${\tau }_{20}+{\tau }_{\mathrm{L}}$, as shown in Figure 4. In the over-coupled region ($\kappa \ge {\tau }_{20}+{\tau }_{\mathrm{L}}$), which is called PT-symmetric phase, the system supports two modes with two solutions of frequency $\omega ={\omega }_0+\sqrt{{\kappa }^2-{\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)}^2}$ and $\omega ={\omega }_0-\sqrt{{\kappa }^2-{\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)}^2}$, the gain coefficient is exactly balanced with all the losses of system, that is, $g_0={\tau }_{10}+{\tau }_{20}+{\tau }_{\mathrm{L}}$. The mode amplitudes of two modes are equal, that is, $\left|a_2/a_1\right|=1$, which means $\left|a_2\right|=V_{\mathrm{in}}/\left[\pi \sqrt{2L_1}\left({\tau }_{10}+{\tau }_{20}+{\tau }_{\mathrm{L}}\right)\right]$. In the under-coupled region ($\kappa <{\tau }_{20}+{\tau }_{\mathrm{L}}$), which is called PT broken phase, only one mode is located at $\omega ={\omega }_0$, the corresponding gain coefficient is $g_0={\tau }_{10}+{\kappa }^2/\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)$, and the ratio of these two mode amplitudes is $\left|a_2/a_1\right|=\kappa /\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)$, that is, $\left|a_2\right|=\kappa V_{\mathrm{in}}/\left\{\pi \sqrt{2L_1}\left[{\tau }_{10}\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)+{\kappa }^2\right]\right\}$. Moreover, $\kappa ={\tau }_{20}+{\tau }_{\mathrm{L}}$ is the critical coupling coefficient.

Based on the above analysis of multiple solutions of the system, the power delivered to the load can be expressed as

$$P_{\mathrm{L}}=2{\tau }_{\mathrm{L}}{\left|a_2\right|}^2=\{\begin{array}{cc}\frac{{\tau }_{\mathrm{L}}{V}_{\mathrm{in}}^2}{{\pi }^2{L}_1{\left({\tau }_{10}+{\tau }_{20}+{\tau }_{\mathrm{L}}\right)}^2}& ,\kappa \ge {\tau }_{20}+{\tau }_{\mathrm{L}}\\ \frac{{\tau }_{\mathrm{L}}{\kappa }^2{V}_{\mathrm{in}}^2}{{\pi }^2{L}_1{\left[{\tau }_{10}\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)+{\kappa }^2\right]}^2}& ,\kappa <{\tau }_{20}+{\tau }_{\mathrm{L}}\end{array}$$

(10)

and the transmission efficiency can be represented as

$$\eta =\frac{2{\tau }_{\mathrm{L}}{\left|a_2\right|}^2}{2{\tau }_{10}{\left|a_1\right|}^2+2\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right){\left|a_2\right|}^2}\times 100\%=\{\begin{array}{cc}\frac{{\tau }_{\mathrm{L}}}{{\tau }_{10}+{\tau }_{20}+{\tau }_{\mathrm{L}}}& ,\kappa \ge {\tau }_{20}+{\tau }_{\mathrm{L}}\\ \frac{{\kappa }^2{\tau }_{\mathrm{L}}}{\left[{\tau }_{10}\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)+{\kappa }^2\right]\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)}& ,\kappa <{\tau }_{20}+{\tau }_{\mathrm{L}}\end{array}$$

(11)

From Equations (10) and (11), it can be see that the output power and transmission efficiency of the system remain constant in the over-coupled region.

4. Comparison of Theoretical Analysis and Simulation

To show the wonderful characteristics of the actual single-wire ECPT system, two spherical metal conductors of radius 22.5 cm are adopted to generate the electric field to transfer the energy obtained from the transmitter to the receiver. The metal balls placed on the top of the PVC tube are connected to the earth through a large inductance L₁ and L₂ winding around the PVC tube, and the grounding side of the inductors is connected with a single metal wire to form a conductive closed loop with the upper metal balls. The capacitance between the metal balls is expressed by the coupling capacitance C₁₂, the parasitic capacitance of the metal balls relative to the single metal wire is described by C₁ and C₂. The specific parameters is shown in

Table 1. Parameters of the single-wire ECPT system.

Parameters	Values
Vin	200 V
L1	63.4 mH
L2	63.4 mH
C1	35 pF
C2	35 pF
r1	50 Ω
r2	50 Ω
RL	150 Ω
r	22.5 cm
f0	106.8 kHz

.

Combining the specific expression of coupling capacitance C₁₂ with the detailed expression of critical coupling coefficient $\kappa ={\tau }_{20}+{\tau }_L$, and substituting the parameters of Table 1 into it, the critical distance can be obtained

$$d_{\mathrm{c}}=\frac{111.2r^2{\omega }_0{L}_2}{\left(r_2+R_{\mathrm{L}}\right)C_1}\times {10}^{-12}\approx 34.23\mathrm{m}$$

(12)

To verify the results of theoretical analysis, a circuit simulation model is built. By substituting the parameters of Table 1 into Equations (10) and (11), and combining with circuit simulation, the comparison of the results of theoretical analysis and circuit simulation are shown in Figure 5.

As illustrated in Figure 5, the circuit simulation results are consistent with the theoretical analysis. In the over-coupled region ($d\le d_{\mathrm{c}}$), the system operates in low-frequency mode, that is, the operating frequency automatically meet the condition of $\omega ={\omega }_0-\sqrt{{\kappa }^2-{\left({\tau }_{20}+{\tau }_{\mathrm{L}}\right)}^2}$ without any tuning, the output power is constant near 19 W, and the transmission efficiency is near stable at 60% over a distance of 34 m with a deviation less than 5%. The internal resistances of the inductor is 50 Ω in our simulation, the efficiency of the system can reach 90% if the internal resistance is further reduced. In addition, in the under-coupled region, the operating frequency is fixed at natural resonant frequency, $f_0=106.8\mathrm{kHz}$.

5. Discussion and Conclusions

This paper proposes a new nonlinear PT-symmetric model for single-wire ECPT system based on CMT, which is an extraordinary progress compared to traditional ECPT system. The proposed model can be used to achieve robust power transmission via electric field with constant output power and high constant transmission efficiency over a wide range of transmission distance. Based on the proposed model, the negative resistor, which is used to realize a gain rate g₀, is achieved by a half-bridge converter, and the control of half-bridge converter depends only on the information of current flowing through the resonant inductor of transmitter, which is very simple and feasible. Moreover, the simulation results are consistent with the theoretical analysis, the load can obtain stable power of 19 W with constant efficiency of 60% over distances in excess of 34 m.