Extended-Distance Capacitive Wireless Power Transfer System Based on Generalized Parity–Time Symmetry

Abstract

A capacitive wireless power transfer (CPT) system based on parity–time (PT) symmetry achieves constant output characteristics under distance variation without additionally increasing the system complexity of the control strategy, where the concept of PT symmetry is derived from quantum mechanics, and the systems satisfying PT symmetry are invariant under space and time inversion. However, the exact PT-symmetric region (i.e., strong coupling region) of the general system is limited by the symmetry of the structure and parameters. To overcome this limitation, a novel generalized parity–time (GPT)-symmetric CPT system is proposed in this article. According to the equivalent circuit method, the circuit model of the proposed system is built, and the transfer characteristics are analyzed. Furthermore, a prototype is implemented to verify the feasibility of the proposed CPT system. The results show that the PT-symmetric region is extended by 169.23% compared with the traditional PT-based CPT system, and a constant output power of 21.5 W is transferred with a constant transfer efficiency of 90%.

1. Introduction

Capacitive wireless power transfer (CPT) technology is a wireless charging technology that achieves energy transfer through the electric field coupling between metal plates [1,2,3]. It distinguishes itself in the realm of wireless power transfer due to its cost-effectiveness, lack of eddy current effects, light weight, and insensitivity to metal obstacles. Nowadays, CPT technology has found extensive applications across various fields, such as underwater equipment, implantable medical equipment, and electric vehicle charging, showcasing its robust potential for practical use [4,5,6,7,8,9].

It is still a crucial challenge for existing CPT technology to achieve constant output characteristics, such as constant current (CC), constant voltage (CV), constant output power, and invariable transfer efficiency. Therefore, when the transfer distance changes, ref. [10] proposes that a bilateral communication dual-loop control CPT system should be used to reduce the effect of coupling misalignment and load variation, which track the input and output voltage to realize frequency tuning. The authors of [11] present a reconfigurable capacitive coupler, which can be reconfigured according to the landing positions and directions and achieve constant output when misalignment tolerance is encountered. The authors of [12] adopt the method of the use of a dual transmitter and single receiver to achieve higher than 85% transfer efficiency under misaligned conditions, but this increases the topological complexity of the system. Meanwhile, [13] utilizes a fractional-order approach to achieve the strong robustness of the CPT system when the coupling coefficient changes from 0.45 to 0.75 and the resonant frequency detuning is within 5%, but the application of a fractional order increases the complexity of control under certain circumstances. The authors of [14] employ the LCLC-S/LCLC-M topology to achieve CC/CV output when misalignment happens in the x- (y-) or θ-direction, but this method increases the complexity of topology and control in certain circumstances and also increases cost. In addition, ref. [15] introduces a multiplate capacitive coupler structure that utilizes the characteristics of a six-plate structure to reduce the geometric sensitivity caused by the misalignment of the coupling plates, achieving a constant output power of 2.0 kW, but the design of the capacitive coupler is complex. The authors of [16] propose a frequency-agile mode and use frequency bifurcation to realize the constant output characteristics of the system at the secondary resonance, but this requires a more complex circuit topology and control strategy. The authors of [17] propose an LCLC-LC compensation structure; when the system load changes, the system operating frequency is switched to achieve constant output current or voltage, but when the system transmission distance changes, the constant transmission characteristics of the system may not be achieved. The authors of [18] present a class-E zero-voltage switching parallel resonant inverter and a class-D rectifier to realize constant output voltage without any feedback control loops, but when the load resistance is large, the transfer efficiency of the system is low and may not be applicable when the system is misaligned. The authors of [19] propose a 6.78 MHz class-E power amplifier to realize quasi-constant current output, but when the system transmission distance changes, the constant transmission characteristics of the system may not be achieved. Consequently, the above methods are not conducive to engineering implementation to a certain extent.

The concept of PT symmetry is derived from quantum mechanics, and the PT-symmetric system is invariant under parity (P) and time (T) reversal, where P-reversal means spatial inversion; T-reversal is time inversion. In 2017, Assawaworrarit introduced PT symmetry to the WPT system for the first time, which achieves constant transfer characteristics when the system structure and parameters satisfy the condition of PT symmetry [20]. This system does not increase the complexity of control or topologies. The authors of [21] introduce the parity–time (PT) symmetry principle into the CPT system, achieving constant output characteristics, which extend the application of PT-symmetric wireless power transfer systems from the magnetic field to the electric field.

Nevertheless, the existing PT-based CPT systems demand rigorous structural and parametric symmetry, and transfer distance is restricted within the exact PT-symmetric region [21,22]. Generalized parity–time (GPT) symmetry is a general form of PT symmetry, which has the characteristics of PT symmetry under asymmetric structures and parameters [23].

To increase the transfer distance of the constant output characteristics (i.e., exact PT-symmetric region) and flexibility of the PT-based CPT system, a novel GPT-based CPT system is proposed in this article. The proposed system is composed of an asymmetric series–parallel series (S–PS) structure, which can extend the exact PT-symmetric region and adjust the output power and transfer efficiency by varying the capacitive distribution ratio.

The rest of this article is organized as follows. In Section 2, the proposed GPT system is modeled by circuit theory, and the circuit equivalent principle is used to reduce the order of the system. The output characteristics of the system are analyzed, and the output characteristics of different capacitance and inductance ratios are plotted. The appropriate inductance and capacitance ratio is selected according to the output characteristic curve of the system and compared with the original PT-CPT system. In Section 3, a 21.5 w prototype is built to verify the feasibility of the proposed system. The output characteristics of the system and input and output waveforms are given and analyzed. In Section 4, some conclusions are given by combining theoretical analysis and experimental verification.

2. System Structure and Modeling

2.1. Capacitive Coupler

The capacitive coupler is the core component of the CPT system, and its equivalent circuit is shown in Figure 1a. Both C_P1 and C_P2 are constructed entirely from metal plates. The transmitting (Tx) and receiving (Rx) sides are coupled by the capacitances C_P1 and C_P2. Then, the mutual capacitance can be expressed as C_M = C_P1C_P2/(C_P1 + C_P2) [24,25], and the further equivalent π-type circuit is shown Figure 1b.

2.2. GPT-Based CPT System with S-PS Structure

The GPT-based CPT system with S-PS structure topology is shown in Figure 2. Here, the power of the system is provided by the negative resistance −R_N, which is constructed by the sampling circuit and full-bridge inverter on the Tx side, and the inverter is constructed by four MOSFETs (S₁–S₄). L_T and L_R are the resonant inductors on the Tx and Rx sides, respectively. C_T and C_R are the tuning capacitors on the Tx and Rx sides, respectively. C_S1 and C_S2 are series and parallel tuning capacitors on the Rx side. C_M is the mutual capacitor between the Tx and Rx sides. r_T and r_R represent the internal resistances of the Tx and Rx sides, respectively. R_L is the load resistance of the system. L_T and L_R are the inductances of the Tx and Rx sides, respectively. i_T, i_CM, i_R, and i_L are the currents through the Tx side, mutual capacitor, Rx side, and load, respectively. u_T and u_L are the input and output voltages of the system, respectively. U_DC is the DC input voltage of the full-bridge inverter.

Due to the fact that the quality factor of the CPT system is usually high, the current of the steady-state system can be considered as a sine wave. Therefore, the steady-state currents are analyzed by the fundamental wave method. i_T, i_CM, i_R, and i_L can be expressed as İ_T, İ_CM, İ_R, and İ_L by the vector method.

Then, the parallel capacitance at the Rx side increases the order of the system, which also brings great inconvenience to the modeling and calculation of the system. However, the equivalent circuit theory provides a theoretical method for reducing the order of a system, and the PS structure network at the Rx side can be equivalent to the S structure network, as shown in Figure 3.

The parameters of the equivalent S compensation topology are as follows:

$$\left\{\begin{array}{l}{R}_{\mathrm{Leq}}=R_{\mathrm{L}}{\displaystyle \frac{C_{\mathrm{S}1}^2}{{\omega }^2{R}_{\mathrm{L}}^2{C}_{\mathrm{S}2}^2{C}_{\mathrm{S}1}^2+{\left(C_{\mathrm{S}2}+C_{\mathrm{S}1}\right)}^2}}\\ C_{\mathrm{S}}={\displaystyle \frac{{\omega }^2{R}_{\mathrm{L}}^2{C}_{\mathrm{S}1}^2{C}_{\mathrm{S}2}^2+{\left(C_{\mathrm{S}2}+C_{\mathrm{S}1}\right)}^2}{{\omega }^2{R}_{\mathrm{L}}^2{C}_{\mathrm{S}1}^2{C}_{\mathrm{S}2}+\left(C_{\mathrm{S}2}+C_{\mathrm{S}1}\right)}}\end{array}\right.$$

(1)

According to the parameters of the general CPT system, ${\omega }^2{R}_{\mathrm{L}}^2{C}_{\mathrm{S}1}^2{C}_{\mathrm{S}2}^2\ll {\left(C_{\mathrm{S}1}+C_{\mathrm{S}2}\right)}^2$ and ${\omega }^2{R}_{\mathrm{L}}^2{C}_{\mathrm{S}1}^2{C}_{\mathrm{S}2}\ll C_{\mathrm{S}1}+C_{\mathrm{S}2}$ are satisfied, so (1) can be approximated as

$$\left\{\begin{array}{l}{R}_{\mathrm{Leq}}=R_{\mathrm{L}}{\left({\displaystyle \frac{C_{\mathrm{S}1}}{C_{\mathrm{S}1}+C_{\mathrm{S}2}}}\right)}^2\\ C_{\mathrm{S}}=C_{\mathrm{S}2}+C_{\mathrm{S}1}\end{array}\right.$$

(2)

Here, α is defined as the distribution ratio of equivalent capacitance to parallel capacitance, that is

$$\left\{\begin{array}{l}{C}_{\mathrm{S}1}=\alpha C_{\mathrm{S}}\\ C_{\mathrm{S}2}=\left(1-\alpha \right)C_{\mathrm{S}}\end{array}\right.$$

(3)

Then, the equivalent load resistance can be expressed as

$$R_{\mathrm{Leq}}={\alpha }^2{R}_{\mathrm{L}}$$

(4)

Therefore, the GPT-based CPT system with the S-PS compensation structure can be equivalent to the SS compensation structure, as shown in Figure 4. In order to quantify the relationship between system characteristics and circuit parameters, circuit theory (CT) is used to model the system, and the equivalent circuit equation can be written as follows:

$$\left\{\begin{array}{l}\left(-R_{\mathrm{N}}+r_{\mathrm{T}}+j\omega L_{\mathrm{T}}+{\displaystyle \frac{C_{\mathrm{R}}+C_{\mathrm{M}}}{j\omega \mu }}\right){\dot{I}}_{\mathrm{T}}+{\displaystyle \frac{C_{\mathrm{M}}}{j\omega \mu }}{\dot{I}}_{\mathrm{R}}=0\\ {\displaystyle \frac{C_{\mathrm{M}}}{j\omega \mu }}{\dot{I}}_{\mathrm{T}}+\left(R_{\mathrm{Leq}}+r_{\mathrm{R}}+j\omega L_{\mathrm{R}}+{\displaystyle \frac{1}{j\omega C_{\mathrm{S}}}}+{\displaystyle \frac{C_{\mathrm{R}}+C_{\mathrm{M}}}{j\omega \mu }}\right){\dot{I}}_{\mathrm{R}}=0\end{array}\right.$$

(5)

where $\mu =C_{\mathrm{T}}{C}_{\mathrm{M}}+C_{\mathrm{T}}{C}_{\mathrm{R}}+C_{\mathrm{M}}{C}_{\mathrm{R}}$, ω is the operating angular frequency, and ${\omega }_{\mathrm{T}}=1/\sqrt{L_{\mathrm{T}}{C}_{\mathrm{T}}}$ and ${\omega }_{\mathrm{R}}=\sqrt{\left(1/C_{\mathrm{S}}+1/C_{\mathrm{R}}\right)/L_{\mathrm{R}}}$ are the natural resonant frequencies of the Tx and Rx sides. $k=C_{\mathrm{M}}/\sqrt{C_{\mathrm{T}}{C}_{\mathrm{R}}}$ is the coupling coefficient between the Tx and Rx plates. It is noted that the proposed system needs to ensure that the natural resonant frequencies satisfy the GPT-symmetric conditions, that is, ${\omega }_{\mathrm{T}}={\omega }_{\mathrm{R}}={\omega }_0$. Moreover, in a general CPT system, C_T on the Tx side and C_R on the Rx side are generally equal.

Defining $\gamma =L_{\mathrm{R}}/L_{\mathrm{T}}$ as the inductance ratio between the Tx and Rx sides, the load side capacitance can be expressed as

$$C_{\mathrm{S}}={\displaystyle \frac{C_{\mathrm{T}}{C}_{\mathrm{R}}}{\gamma C_{\mathrm{R}}-C_{\mathrm{T}}}}$$

(6)

Then, (5) can be rewritten as

$$\left[\begin{array}{cc}{Q}_{\mathrm{T}}^{-1}+j\left(\lambda -{\displaystyle \frac{k+1}{\lambda \left(2k+1\right)}}\right)& -j{\displaystyle \frac{k}{\lambda \left(2k+1\right)}}\\ -j{\displaystyle \frac{k}{\gamma \lambda \left(2k+1\right)}}& Q_{\mathrm{R}}^{-1}+j\left(\lambda -{\displaystyle \frac{\left(2\gamma -1\right)k+\gamma }{\gamma \lambda \left(2k+1\right)}}\right)\end{array}\right]\left[\begin{array}{c}{\dot{I}}_{\mathrm{T}}\\ {\dot{I}}_{\mathrm{R}}\end{array}\right]=0$$

(7)

where $Q_{\mathrm{R}}={\omega }_0{L}_{\mathrm{R}}/\left(R_{\mathrm{Leq}}+r_{\mathrm{R}}\right)$ and $Q_{\mathrm{T}}={\omega }_0{L}_{\mathrm{T}}/\left(-R_{\mathrm{N}}+r_{\mathrm{T}}\right)$ are the quality factors of the Rx and Tx sides. $\lambda =\omega /{\omega }_0$ is the ratio of the operating angular frequency to the resonant angular frequency.

From (7), the fundamental characteristics of the system can be obtained as

$$\left[Q_{\mathrm{T}}^{-1}+j\left(\lambda -{\displaystyle \frac{1}{\lambda }}{\displaystyle \frac{k+1}{2k+1}}\right)\right]\left[Q_{\mathrm{R}}^{-1}+j\left(\lambda -{\displaystyle \frac{\left(2\gamma -1\right)k+\gamma }{\gamma \lambda \left(2k+1\right)}}\right)\right]+{\displaystyle \frac{1}{\gamma }}{\left({\displaystyle \frac{k}{\lambda \left(2k+1\right)}}\right)}^2=0$$

(8)

Separating the real and imaginary parts in the characteristic equation, (8) can be written as

$$\left\{\begin{array}{l}{Q}_{\mathrm{T}}^{-1}{Q}_{\mathrm{R}}^{-1}-{\left(\lambda -{\displaystyle \frac{1}{\lambda }}\right)}^2+\left({\displaystyle \frac{1}{{\lambda }^2}}-1\right)\left({\displaystyle \frac{1}{\gamma }}+1\right){\displaystyle \frac{k}{2k+1}}=0\\ Q_{\mathrm{T}}^{-1}\left(\lambda -{\displaystyle \frac{\left(2\gamma -1\right)k+\gamma }{\gamma \lambda \left(2k+1\right)}}\right)+Q_{\mathrm{R}}^{-1}\left(\lambda -{\displaystyle \frac{k+1}{\lambda \left(2k+1\right)}}\right)=0\end{array}\right.$$

(9)

According to (9), the operating angular frequency can be derived as

$$\omega =\left\{\begin{array}{l}{\omega }_0\sqrt{{\displaystyle \frac{k+1}{2k+1}}}, k<k_{\mathrm{C}}\\ {\omega }_0\sqrt{{\displaystyle \frac{2-\Delta \pm \sqrt{{\Delta }^2-4Q_{\mathrm{R}}^{-2}}}{2}}}, k_{\mathrm{C}}\le k<1\end{array}\right.$$

(10)

where $\Delta =Q_{\mathrm{R}}^{-2}+k\left(1+\gamma \right)/\left[\left(1+2k\right)/\gamma \right]$, and $k_{\mathrm{C}}$ is the critical coupling coefficient, which can be expressed as

$$k_{\mathrm{C}}={\displaystyle \frac{\gamma \left(2Q_{\mathrm{R}}^{-1}-Q_{\mathrm{R}}^{-2}\right)}{1+\gamma -2\gamma Q_{\mathrm{R}}^{-1}\left(2-Q_{\mathrm{R}}^{-1}\right)}}$$

(11)

As can be observed from (10) and (11), the operating range of the proposed system can be divided into the exact PT-symmetric region with frequency splitting $\left(k_{\mathrm{C}}\le k\le 1\right)$ and the broken PT-symmetric region with frequency unity $\left(k<k_{\mathrm{C}}\right)$. Therefore, according to the range of the coupling coefficient k, the transfer characteristics of the system can be discussed as follows.

2.2.1. In the Exact PT-Symmetric Region $\left(k_{\mathrm{C}}\le k\le 1\right)$

When the system coupling coefficient k between the Tx side and Rx side exceeds the critical coupling coefficient k_C, the system works in the exact PT-symmetric region with frequency splitting (i.e., strong coupling region). In this case, the GPT-based CPT system can automatically select the high-frequency branch f₁ or low-frequency branch f₂ to operate. The operating frequency is

$$f_{1,2}=f_0\sqrt{{\displaystyle \frac{2-\Delta \pm \sqrt{{\Delta }^2-4Q_{\mathrm{R}}^{-2}}}{2}}}$$

(12)

According to (9), when the system operates in the exact PT-symmetric region,

$$Q_{\mathrm{T}}^{-1}+Q_{\mathrm{R}}^{-1}=0$$

(13)

Then, the negative resistance can be expressed as

$$-R_{\mathrm{N}}=-{\displaystyle \frac{{\alpha }^2{R}_{\mathrm{L}}+r_{\mathrm{R}}}{\gamma }}-r_{\mathrm{T}}$$

(14)

Submitting (13) into (5), the current ratio of the Rx and Tx sides can be described as

$${\displaystyle \frac{I_{\mathrm{R}}}{I_{\mathrm{T}}}}={\displaystyle \frac{1}{\gamma }}$$

(15)

According to (15) and Kirchhoff’s Current Law, the current ratio of the load and Rx side can be obtained as

$${\displaystyle \frac{I_{\mathrm{L}}}{I_{\mathrm{R}}}}=\alpha$$

(16)

Then, according to (15) and (16), the current ratio of the load and Tx side is

$${\displaystyle \frac{I_{\mathrm{L}}}{I_{\mathrm{T}}}}={\displaystyle \frac{\alpha }{\gamma }}$$

(17)

Since $U_{\mathrm{L}}=R_{\mathrm{L}}{I}_{\mathrm{L}}$ and $U_{\mathrm{T}}=R_{\mathrm{N}}{I}_{\mathrm{T}}$, the relationship between load voltage and input voltage can be expressed as

$${\displaystyle \frac{U_{\mathrm{L}}}{U_{\mathrm{T}}}}={\displaystyle \frac{I_{\mathrm{L}}{R}_{\mathrm{L}}}{I_{\mathrm{T}}{R}_{\mathrm{N}}}}={\displaystyle \frac{\alpha R_{\mathrm{L}}}{{\alpha }^2{R}_{\mathrm{L}}+r_{\mathrm{R}}+\gamma r_{\mathrm{T}}}}$$

(18)

Therefore, the output power and transfer efficiency can be expressed as follows:

$$P={\displaystyle \frac{U_{\mathrm{L}}^2}{R_{\mathrm{L}}}}={\displaystyle \frac{{\alpha }^2{U}_{\mathrm{T}}^2{R}_{\mathrm{L}}}{{\left({\alpha }^2{R}_{\mathrm{L}}+r_{\mathrm{R}}+\gamma r_{\mathrm{T}}\right)}^2}}$$

(19)

where $U_{\mathrm{T}}=2\sqrt{2}{U}_{\mathrm{DC}}/\pi$.

$$\eta ={\displaystyle \frac{R_{\mathrm{L}}{I}_{\mathrm{L}}^2}{R_{\mathrm{L}}{I}_{\mathrm{L}}^2+r_{\mathrm{T}}{I}_{\mathrm{T}}^2+r_{\mathrm{R}}{I}_{\mathrm{R}}^2}}={\displaystyle \frac{{\alpha }^2{R}_{\mathrm{L}}}{{\alpha }^2{R}_{\mathrm{L}}+{\gamma }^2{r}_{\mathrm{T}}+r_{\mathrm{R}}}}$$

(20)

By employing the aforementioned equation, it is evidently perceived that the current ratio, voltage ratio, output power, and transfer efficiency in the exact PT-symmetric region are independent of the coupling coefficient k.

Consequently, the maintenance of constant output characteristics independent of the coupling coefficient can be achieved even if the Rx side moves in the strong coupling region.

2.2.2. In the Broken PT-Symmetric Region $\left(k<k_{\mathrm{C}}\right)$

When the system coupling coefficient k between the Tx side and Rx side is less than the critical coupling coefficient k_C, the system works in the broken PT-symmetric region with frequency unity (i.e., weak coupling region). The GPT-based CPT system operates at the resonant frequency f₃, which can be derived as

$$f_3=f_0\sqrt{{\displaystyle \frac{k+1}{2k+1}}}$$

(21)

When the system operates in the weak coupling region, the negative resistance can be deduced as

$$-R_{\mathrm{N}}=-{\displaystyle \frac{1}{\left(2k+1\right)\left(k+1\right)}}\left({\displaystyle \frac{k^2{\omega }_0^2{L}_{\mathrm{T}}^2}{r_{\mathrm{R}}+{\alpha }^2{R}_{\mathrm{L}}}}\right)-r_{\mathrm{T}}$$

(22)

Based on (5), the current ratio between the load and Tx side can be presented as

$${\displaystyle \frac{I_{\mathrm{L}}}{I_{\mathrm{T}}}}={\displaystyle \frac{1}{\sqrt{\left(k+1\right)\left(2k+1\right)}}}{\displaystyle \frac{\alpha k{\omega }_0{L}_{\mathrm{T}}}{r_{\mathrm{R}}+{\alpha }^2{R}_{\mathrm{L}}}}$$

(23)

According to (22) and (23), the voltage ratio between the load and Tx side can be expressed as

$${\displaystyle \frac{U_{\mathrm{L}}}{U_{\mathrm{T}}}}={\displaystyle \frac{I_{\mathrm{L}}{R}_{\mathrm{L}}}{I_{\mathrm{T}}{R}_{\mathrm{N}}}}={\displaystyle \frac{\alpha k{\omega }_0{L}_{\mathrm{T}}{R}_{\mathrm{L}}\sqrt{\left(2k+1\right)\left(k+1\right)}}{k^2{\omega }_0^2{L}_{\mathrm{T}}^2+r_{\mathrm{T}}\left(r_{\mathrm{R}}+{\alpha }^2{R}_{\mathrm{L}}\right)\sqrt{\left(2k+1\right)\left(k+1\right)}}}$$

(24)

Subsequently, the output power and transfer efficiency can be represented as follows:

$$P={\displaystyle \frac{U_{\mathrm{L}}^2}{R_{\mathrm{L}}}}={\displaystyle \frac{{\left(\alpha k{\omega }_0{L}_{\mathrm{T}}\sqrt{\left(2k+1\right)\left(k+1\right)}{U}_{\mathrm{T}}\right)}^2{R}_{\mathrm{L}}}{{\left(k^2{\omega }_0^2{L}_{\mathrm{T}}^2+r_{\mathrm{T}}\left(r_{\mathrm{R}}+{\alpha }^2{R}_{\mathrm{L}}\right)\sqrt{\left(2k+1\right)\left(k+1\right)}\right)}^2}}$$

(25)

$$\eta ={\displaystyle \frac{{\alpha }^2{\left(k{\omega }_0{L}_{\mathrm{T}}\right)}^2{R}_{\mathrm{L}}}{\left({\alpha }^2{R}_{\mathrm{L}}+r_{\mathrm{R}}\right)\left[{\left(k{\omega }_0{L}_{\mathrm{T}}\right)}^2+r_{\mathrm{T}}\left(k+1\right)\left(2k+1\right)\left({\alpha }^2{R}_{\mathrm{L}}+r_{\mathrm{R}}\right)\right]}}$$

(26)

From (25) and (26), it can be seen that in the weak coupling region, the output power and transfer efficiency are dependent on the coupling coefficient k, and the system cannot achieve constant output characteristics.

As a result, in practical applications, it is advisable to avoid the occurrence of the system coupling coefficient k between the Tx side and Rx side being less than the critical coupling coefficient k_c to obtain stable output characteristics.

2.3. Parameter Selection and Comparison

Based on (11) and (20), the critical coupling coefficient k_C and transfer efficiency η are related to the capacitance distribution ratio α and inductance coefficient ratio γ. The transfer characteristic curves are drawn in Figure 5, and the circuit parameters are listed in

Table 1. The parameters of the experimental prototype.

Symbols	Values	Symbols	Values
CT	606.641 pF	RL	15 Ω
CR	606.035 pF	rT	682.137 mΩ
CS1	11.518 nF	rR	714.613 mΩ
CS2	606.406 pF	fT	1.292 MHz
LT	25.0061 μH	fR	1.293 MHz
LR	26.2493 μH	UDC	21 V

. It can be seen that k_C and η increase with the increase in α and decrease with the increase in γ, but the output power P decreases with the increase in α and γ.

Figure 6 depicts the relationship between the system’s operating frequency and coupling coefficient, from which the effects of different capacitance distribution ratios on the system operating frequency and critical coupling coefficient can be compared more intuitively.

From the above analysis, it can be seen that the method of PS resonant capacitors on the load side can reduce the critical coupling coefficient and improve the output power of the system, but this will sacrifice transfer efficiency. Similarly, the inductance coefficient ratio has a similar impact on the system’s transfer characteristics. Therefore, through a comprehensive analysis of Figure 5 and Figure 6, it can be found that the selection of α = 0.95 and γ = 1.05 can extend the PT-symmetric region while maintaining 90% transfer efficiency.

The transfer characteristics of the original SS compensation topology and proposed S-PS compensation topology are compared in Figure 7; it can be seen that under the same circuit parameters, the S-PS structure can reduce the critical coupling coefficient from 0.0873 of the original SS structure to 0.0764.

3. Experimental Verification

The experimental prototype is shown in Figure 8, and the circuit parameters are listed in Table 1. The transmitter and receiver are constructed from cylindrical coppers with a height of 150 mm and diameters of 95 and 85 mm, respectively.

The relationship between the coupling coefficient and the insertion depth of the capacitive coupler was measured by a precise impedance analyzer. Here, we add a plastic gasket with a height of 10 mm and a diameter of 85 mm to the bottom of the internal barrel metal plate and measure the depth of the plate insertion by adding the number of gaskets. The measured data are shown in Figure 9. The PT-symmetric region of the original PT-based CPT system with SS topology is the blue region. The system widening region proposed in this paper is the light red region, and the PT-symmetric region of the proposed GPT-based CPT system is the light red region plus the blue region.

Figure 10 shows the experimental waveforms of voltages u_T and u_L and currents i_T and i_L when the insertion depth is 150 mm. From Figure 10a, it can be found that the phase difference between the input voltage u_T and transmitter current i_T is 0 in the reference direction, which means that negative resistance is realized, and the experimental measured data agree well with the theoretical calculated values.

More experimental measured data of output power and transfer efficiency are given in Figure 11; it can be seen that in the exact PT-symmetric region, output power stabilizes around 21.5 W, while the transfer efficiency is 90%.

In addition, the critical coupling coefficient is k_C = 0.0764, which is significantly reduced compared to the k_C = 0.0873 of the original SS topology. Therefore, the experimental results are basically consistent with the theoretical analysis.

4. Conclusions

In this paper, a novel GPT-based CPT system with S-PS compensation topology is proposed, which extends the insertion depth while maintaining constant output characteristics. The theoretical and experimental results show that the critical coupling coefficient of the proposed system decreases from 0.0873 to 0.0764, the insertion depth is extended by approximately 18 mm, and the exact PT-symmetric region is expanded by 169.23% compared with the original PT-based CPT system. Moreover, the proposed system can maintain stable current and voltage ratios, as well as a constant output power of 21.5 W and a constant transfer efficiency of 90%. In a word, the proposed system effectively enhances the system’s insertion depth without increasing the control complexity, thus expanding the practicability of the PT-based CPT system.