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

: 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.


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, Energies 2018, 11, 532 3 of 10 transformer T 1 and a metal conductor Q 1 that obtains the power from the source and transfers it to the receiver through coupling electric field. The receiver comprises a metal conductor Q 2 and step-down transformer T 2 that provides power for a resistant load. Particularly, the coupling capacitance between Q 1 and Q 2 can be expressed as C 12 = 4πε 0 r 2 /d when both Q 1 and Q 2 are spherical metal conductors of radius r, ε 0 is the permittivity of vacuum.

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 RL. 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.

Discrete State Space Model
As illustrated in Figure 1, the transformer T 1 on the transmitting side only plays a role in boosting the input voltage, similarly, the transformer T 2 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 L 1 and L 2 , internal resistance r 1 and r 2 , parasitic capacitance C 1 and C 2 , coupling capacitor C 12 , 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.
Energies 2018, 11, x FOR PEER REVIEW 3 of 10 down transformer T2 that provides power for a resistant load. Particularly, the coupling capacitance between Q1 and Q2 can be expressed as

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 RL. 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 where i 1 and i 2 are currents flowing through the inductors, u 1 and u 2 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 i 1 and the gate drive signals of S 1 and S 2 (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 i 1 across the LC (inductance and capacitance) tank of the transmitter [29]. Therefore, assuming the direction of current i 1 shown in Figure 2 is positive, u in is defined as where V in is the input dc voltage of the 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 where i1 and i2 are currents flowing through the inductors, u1 and u2 are voltages across the capacitors, uin is the output voltage of the half-bridge converter.
Equation (1) can be solved immediately when uin is given. Figure 3 shows the waveform of uin at two different operation cases, as well as i1 and the gate drive signals of S1 and S2 (vg_S1 and vg_S2). To simplify the analysis, the delay time and dead time is not considered in modeling. As shown in Figure  3, the voltage uin 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, uin is defined as where Vin is the input dc voltage of the half-bridge converter.

Dynamic Modeling by CMT
The state space averaging method is difficult to apply in the steady-state analysis and transientstate analysis of the system based on circuit theory because of the fast varying state variables, in and un, 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   n n n n n n n n 1,2 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 n = C n 2 u n + j L n 2 i n = A n e −j(ωt+θ n ) , n = 1, 2 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 |a n | 2 , ω is the operating angular frequency of system, the variables Energies 2018, 11, 532

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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 The derivative of Equation (3) over time can be defined as Substituting Equation (4) into Equation (5), the dynamic equations of coupled modes for single-wire ECPT system can be derived as where the natural resonant frequencies are defined as: ω 1 = 1/ √ L 1 C 1 and ω 2 = 1/ √ L 2 C 2 . The loss rates are denoted as: τ 10 = r 1 /2L 1 , τ 20 = r 2 /2L 2 , and τ L = R L /2L 2 . The electric-field coupling coefficient between transmitter and receiver is k 12 = C 12 / √ C 1 C 2 , and the gain rate is g 0 = V in / π √ 2L 1 |a 1 | . For a long-range ECPT system, natural resonant frequencies of two LC tanks are usually close to each other (ω 1 ≈ ω 2 ). Besides, a PT-symmetric system requires that the two natural resonant frequencies are the same (ω 1 = ω 2 = ω 0 ). The term C 12 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 C 1 and C 2 in general. In addition, the values of τ 10 , τ 20 , τ L , g 0 |a 1 | and ω 0 k 12 are relatively close in weak coupling region. Therefore, to simplify Equation (6), the simplified coupled-mode equations can be derived as follows where κ = ω 0 k 12 /2 is defined as the coupling coefficient between two modes a 1 and a 2 .

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).
is defined as the coupling coefficient between two modes a1 and a2.

Analysis of Transmission Characteristics
The transmission performance, such as the power delivered to the load PL, transmission efficiency η and the operating frequency ω, can be easily derived from the steady solutions of the coupled-mode Equation (7).
, the operating frequency in the steady state can be obtained by characteristic Separating the real and imaginary parts of Equation (8), it can be see that there are two equations to be satisfied, as follows 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  and 20 L    , as shown in Figure 4. In the over- , which is called PT-symmetric phase, the system supports two modes with two solutions of frequency , the gain coefficient is exactly balanced with all the losses of system, that is, 0 amplitudes of two modes are equal, that is, 2 1 1 a a  , which means In the under-coupled region ( , which is called PT broken phase, only one mode is located at 0    , the corresponding gain coefficient is , and the ratio of these two mode amplitudes is 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 Energies 2018, 11, 532 7 of 10 and the transmission efficiency can be represented as 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.

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 L 1 and L 2 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 C 12 , the parasitic capacitance of the metal balls relative to the single metal wire is described by C 1 and C 2 . The specific parameters is shown in Table 1. Combining the specific expression of coupling capacitance C 12 with the detailed expression of critical coupling coefficient κ = τ 20 + τ L , and substituting the parameters of Table 1 into it, the critical distance can be obtained 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 ≤ d c ), the system operates in low-frequency mode, that is, the operating frequency automatically meet the condition of ω = ω 0 − κ 2 − (τ 20 + τ L ) 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 kHz.

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 g 0 , 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.