**1. Introduction**

With the progress of the power electronics system, more and more attention is paid to the development of wireless power transmission (WPT) [1,2]. Nowadays, many types of new and e ffective WPT systems can be found in the open literature [3–5]. Particularly popular today are inductive power transfer (IPT) systems, the operation principle of which is based on the transmission of electricity through an alternating magnetic field and the use of the phenomenon of electromagnetic induction. Unfortunately, due to the relatively high value of the leakage flux [6], deteriorating e fficiency of the system and problems related to electromagnetic compatibility (EMC), IPT systems are mostly used in low power applications like mobile phones wireless charging [7]. However, the research is underway on the applications of IPT systems for charging the batteries of high-power devices like, for example, electric vehicles [8,9]. Currently, in the research concerning WPT, the capacitive power transmission (CPT) systems are often considered in the open literature as an e ffective alternative solution to IPT [10,11]. The most common structure of the CPT system consists of four separated galvanic conductive plates and a circuit with appropriate parameters [12], the energy transfer is achieved through the coupling capacitances of the plates working at a resonance condition forced by the supply and receiver circuits. The indisputable advantage of CPT systems is the ability to transfer a relatively large amount of energy over considerable distances while maintaining high transmission efficiency [13]. This is possible due to the usage of dedicated compensation systems, which are made

up of a set of inductors and capacitors in appropriate configuration and allow to increase the voltage between capacitor plates [14,15]. The third group of WPT systems are newly developed systems combining the features of both IPT and CPT approaches. An example proposition of such a system has been discussed by Y. Achour and J. Starzy ´nski in [16]. This kind of system can make a significant contribution to the development of high-frequency WPT systems.

Analyzing the features of the systems discussed above, it could be noticed that all of them reach the maximum e fficiency at fixed relative positions between the receiver and transmitter, while maintaining the same distance between them. Usually, the change of this relative position leads to deterioration of the transmission e fficiency. Unfortunately, it is di fficult to find in the open literature systems in which a change of the position of the receiver relative to the transmitter does not result in a significant decrease in transmitting power and transmission e fficiency. In the group of IPT systems in which energy transmission proceeds due to capacitive coupling, i.e., CPT systems, the proposition presented in [17] is particularly interesting. Authors of the work propose an application of two types of electrodes that di ffer in length. This approach enables the possibility of movement of the receiving electrodes relative to the transmitting ones without a coupling capacitance change. However, it should be noted, that in this type of solution the receiving electrodes can be moved only along the length of the transmitting electrodes. In the case of a change of the arrangemen<sup>t</sup> of electrodes along the width, the decrease in capacitance can be observed, which directly results in a significant decrease of the e fficiency of the system.

In the present paper, a novel concept of a wireless capacitive power transfer system with sliding receiver has been proposed. The system is characterized by a high value of transmission e fficiency for any (even accidental) arrangemen<sup>t</sup> of transmitting electrodes. Therefore, the proposed system can work properly in any position of the receiving electrode relative to the transmitting electrode, i.e., for any position both along the *x*- and *y*-axis. It also allows the receiving plate (electrode) to be rotated relative to the transmitting one by any angle (Section 2). Moreover, it should also be noted, that contrary to many solutions presented in open literature, the proposed CPT system contains of only two plates (electrodes). Undoubtedly, this solution will be especially valued wherever a highly e fficient power transmission is required and the proper arrangemen<sup>t</sup> of the transmitting electrodes of the CPT system is di fficult to implement. The system discussed here has been designed primarily for charging mobile devices such as smartphones and battery powered cordless screwdrivers. The basic configuration of the elaborated CPT system with sliding receiver has been presented in Figure 1. The parameters of equivalent CPT circuits were calculated on the basis of the field model of the capacitive power transfer. The model as well as calculations were realized in ANSYS Maxwell. Results of field model calculations have been then implemented in the authored software to analyze operating states of the CPT system co-working with an E-class inverter [18]. In the paper, authors have limited themselves to discuss and present the analysis of the impact of the plate arrangemen<sup>t</sup> on the value of transmitted power and transmission e fficiency. The results of simulation calculations for the selected operating state of the proposed CPT system with sliding receiver are given. An analysis of the content of higher harmonics was also carried out for the obtained current waveforms, i.e., supply current and receiver current. The results of simulation calculations were compared with the results of the measurements obtained for the prototype CPT system. In the paper, the influence of the rotation of electrodes relative to each other is not taken into account.

**Figure 1.** The configuration of elaborated capacitive power transmission (CPT) system. **Figure1.**Theconfigurationofelaborated capacitivepowertransmission(CPT)system.**Figure 1.** The configuration of elaborated capacitive power transmission (CPT) system.

#### **2. The Field Model of the Elaborated CPT System 2. The Field Model of the Elaborated CPT System 2. The Field Model of the Elaborated CPT System**

receiving plate.

In order to study the proposed capacitive power transmission system, the performance of the field model has been developed in the professional finite element method (FEM) package ANSYS Maxwell. The structure of the considered system has been shown in Figure 2. In order to study the proposed capacitive power transmission system, the performance of the field model has been developed in the professional finite element method (FEM) package ANSYS Maxwell.The structure of the considered system has been shown in Figure 2. In order to study the proposed capacitive power transmission system, the performance of the field model has been developed in the professional finite element method (FEM) package ANSYS Maxwell. The structure of the considered system has been shown in Figure 2. 

**Figure 2.** The structural views of the proposed CPT system, (**a**) the bottom view of the transmitting board; (**b**) the isometric view of the CPT system; (**c**) the top view of the CPT system; (**d**) with rotating **Figure 2.** The structural views of the proposed CPT system, (**a**) the bottom view of the transmitting board; (**b**) the isometric view of the CPT system; (**c**) the top view of the CPT system; (**d**) with rotating receiving plate. **Figure 2.** The structural views of the proposed CPT system, (**a**) the bottom view of the transmitting board; (**b**) the isometric view of the CPT system; (**c**) the top view of the CPT system; (**d**) with rotating receiving plate.

The CPT system consists of two parallel transmission plates (A) and (B) i.e., a transmitting (transmitter) and receiving plate (receiver), respectively, see Figure 2a,b. Both boards have been made as double-layer printed circuit board (PCB) technology. Referring to the transmission plate (A), its bottom layer made of copper, see Figure 2, is the cover of the transmission capacitor. Whereas the upper layer covered by the path made of copper (Figure 2b) is the ground of the system (GND). The receiving plate is also made as a two-layer plate, in which its bottom side with the The CPT system consists of two parallel transmission plates (A) and (B) i.e., a transmitting (transmitter) and receiving plate (receiver), respectively, see Figure 2a,b. Both boards have been made as double-layer printed circuit board (PCB) technology. Referring to the transmission plate (A), its bottom layer made of copper, see Figure 2, is the cover of the transmission capacitor. Whereas the upper layer covered by the path made of copper (Figure 2b) is the ground of the system (GND). The receiving plate is also made as a two-layer plate, in which its bottom side with the The CPT system consists of two parallel transmission plates (A) and (B) i.e., a transmitting (transmitter) and receiving plate (receiver), respectively, see Figure 2a,b. Both boards have been made as double-layer printed circuit board (PCB) technology. Referring to the transmission plate (A), its bottom layer made of copper, see Figure 2, is the cover of the transmission capacitor. Whereas the upper layer covered by the path made of copper (Figure 2b) is the ground of the system (GND). The receiving plate is also made as a two-layer plate, in which its bottom side with the conductive

path is the GND layer, while the upper side of the plate (B) is the second of the transmission capacitor linings. The dimensions of the transmitting plate (A) are 330 mm × 230 mm while the receiving plate (B) is 130 mm× 110 mm. The bottom view of the elaborated model is shown in Figure 3. conductive path is the GND layer, while the upper side of the plate (B) is the second of the transmission capacitor linings. The dimensions of the transmitting plate (A) are 330 mm × 230 mm while the receiving plate (B) is 130 mm× 110 mm. The bottom view of the elaborated model is shown in Figure 3. 

**Figure 3.** View of the bottom side of the receiver board. **Figure 3.** View of the bottom side of the receiver board.

The developed system was designed to supply receivers of arbitrary position of the receiver relative to the transmitter. In order to maximize the transmission efficiency and the resultant capacity value between the plates, it was decided to use a contact connection of layers constituting The developed system was designed to supply receivers of arbitrary position of the receiver relative to the transmitter. In order to maximize the transmission efficiency and the resultant capacity value between the plates, it was decided to use a contact connection of layers constituting the ground of the system (GND).

the ground of the system (GND). In the work, as mentioned earlier, to determine the value of the resultant capacitance *Cs* between the plates of the considered system and the capacitances constituting the parasitic capacitances *Cp*, i.e., the capacitances between the linings of the transmission capacitor and the layers (linings) constituting the GND layer of the system, professional software was used in which to analyze the electrostatic field the popular FE method was implemented using the formulation of the electric potential *V*. The capacitance values obtained in the software as a function of the position of the receiver relative to the transmitter are given in Section 4. In the work, as mentioned earlier, to determine the value of the resultant capacitance *Cs* between the plates of the considered system and the capacitances constituting the parasitic capacitances *Cp*, i.e., the capacitances between the linings of the transmission capacitor and the layers (linings) constituting the GND layer of the system, professional software was used in which to analyze the electrostatic field the popular FE method was implemented using the formulation of the electric potential *V*. The capacitance values obtained in the software as a function of the position of the receiver relative to the transmitter are given in Section 4.

#### **3. The Circuit Model of the Elaborated CPT System**

**3. The Circuit Model of the Elaborated CPT System** 

For the purposes of analysis of the operating states of the capacitive power transmission system, a circuit model was developed. Due to the resonance operation characteristics of the CPT system, an E-type inverter was chosen to supply the system. The power supply system together with the power transmission circuit and the receiver is shown in Figure 4. The power supply system consists of the *Ud* voltage source, *Ld* reactor, *T*1 switching transistor, *D*1 return diode and output capacitor *CT*. The transmission circuit, the field model which has been presented in Section 2, consists of the main coupled capacity *Cg*, two parasitic capacities *Cp*1 and *Cp*2, as well as the contact surface *S*1. The *Lr* inductor together with the transmission system and the receiver form a resonant For the purposes of analysis of the operating states of the capacitive power transmission system, a circuit model was developed. Due to the resonance operation characteristics of the CPT system, an E-type inverter was chosen to supply the system. The power supply system together with the power transmission circuit and the receiver is shown in Figure 4. The power supply system consists of the *Ud* voltage source, *Ld* reactor, *T*1 switching transistor, *D*1 return diode and output capacitor *CT*. The transmission circuit, the field model which has been presented in Section 2, consists of the main coupled capacity *Cg*, two parasitic capacities *Cp*1 and *Cp*2, as well as the contact surface *S*1. The *Lr Electronics* inductor together with the transmission system and the receiver form a resonant circuit. **2020**, *9*, x FOR PEER REVIEW 5 of 15 

**Figure 4.** The structural schema of the considered CPT system. **Figure 4.** The structural schema of the considered CPT system.

The presence of *Cp*1 and *Cp*2 parasitic capacitances negatively affects the efficiency of energy transmission. Total leveling of these capacitances is unfortunately impossible. By using solutions such as limiting parallel surfaces or reducing the width of layers constituting the ground of the CPT system, it was only possible to limit the values of the considered capacitances to a minimum. The simulation that has been carried out as a par<sup>t</sup> of the study takes into account the presence of parasitic 264

All capacitances included in the transmission circuit are variable due to fluctuations caused by the changeofthepositionofthereceiversystemboard relativetothetransmitterboard.**Figure 4.** The structural schema of the considered CPT system. 

The presence of *Cp*1 and *Cp*2 parasitic capacitances negatively a ffects the e fficiency of energy transmission. Total leveling of these capacitances is unfortunately impossible. By using solutions such as limiting parallel surfaces or reducing the width of layers constituting the ground of the CPT system, it was only possible to limit the values of the considered capacitances to a minimum. The simulation that has been carried out as a part of the study takes into account the presence of parasitic capacities of the transmission system, as well as their variability depending on the position of the receiver relative to the transmitter. The operation of the system is based on the resonance phenomenon, which enables compensation of the coupling capacitance of the system. Formulas that allow to determine the value of resonance parameters are presented in Section 4. The presence of *Cp*1 and *Cp*2 parasitic capacitances negatively affects the efficiency of energy transmission. Total leveling of these capacitances is unfortunately impossible. By using solutions such as limiting parallel surfaces or reducing the width of layers constituting the ground of the CPT system, it was only possible to limit the values of the considered capacitances to a minimum. The simulation that has been carried out as a part of the study takes into account the presence of parasitic capacities of the transmission system, as well as their variability depending on the position of the receiver relative to the transmitter. The operation of the system is based on the resonance phenomenon, which enables compensation of the coupling capacitance of the system. Formulas that allow to determine the value of resonance parameters are presented in Section 4. 

#### **4. Determining the Resonance Circuit's Parameters 4. Determining the Resonance Circuit's Parameters**

For the purpose of calculation of individual parameters of the supply system, the circuit from Figure 4 has been brought to the simpler form and shown in Figure 5. For the purpose of calculation of individual parameters of the supply system, the circuit from Figure 4 has been brought to the simpler form and shown in Figure 5. 

**Figure 5.** Simplified structural circuit of the mobile CPT system. **Figure 5.** Simplified structural circuit of the mobile CPT system.

In order to obtain resonance state between *Lr* and *Cs* it is necessary to perform the following calculations. According to the circuit theory, the resultant coupling capacitance *Cs* value has been obtained by using Equation (1): In order to obtain resonance state between *Lr* and *Cs* it is necessary to perform the following calculations. According to the circuit theory, the resultant coupling capacitance *Cs* value has been obtained by using Equation (1):

$$\begin{array}{l} \mathbf{Z} = \mathbf{R} + j\mathbf{X}\_{Lr} - j\mathbf{X}\_{Cs} = j\mathbf{X}\_{Lr} + (\frac{-j\mathbf{X}\_{Csp1}}{j\mathbf{X}\_{Csp1}}) \cdot (\frac{-j\mathbf{X}\_{Csp}}{-j\mathbf{X}\_{Csp2}\mathbf{R}\mathbf{R}\_o - \mu\mathbf{X}\_{Csp2}}{-j\mathbf{X}\_{Csp2}\mathbf{R}\mathbf{R}\_o - \mu\mathbf{X}\_{Csp1}})\\ \mathbf{Z} = \mathbf{R} + jX\_{Lr} - jX\_{Cr} = jX\_{Lr} + \frac{-jX\_{Cp1}\mathbf{R}\mathbf{R}\_o - \mu\mathbf{X}\_{Csp2}}{-j\mathbf{X}\_{Csp1}\mathbf{R}\mathbf{R}\_o - \mu\mathbf{X}\_{Csp2}} \end{array} \tag{1}$$

*o* −

$$\begin{aligned} \mathbf{x}\_r - jX\_{\odot} &= jX\_{Lr} + \frac{jX\_{\odot,\mathbf{y}}}{-jX\_{\odot}} + \frac{j}{R} \mathbf{X}\_{\odot p}^{\top X} \mathbf{\mathcal{R}}\_o^2 - jX\_{\odot} \mathbf{\hat{n}}^{\top \mathbf{C} \mathbf{p} \mathbf{1}} \\ \text{llows:} & \mathbf{y} = \frac{jX\_{\odot}}{R\_o - jX\_{\odot p2}} - jX\_{\odot p1} \end{aligned} \tag{1}$$

in which *R* can be calculated as follows:

$$\begin{aligned} \text{in which } R \text{ can be calculated as follows:}\\ R = \frac{X\_{\text{Cp1}}^2 X\_{\text{Cp2}}^2 R\_o}{R\_o^2 \left(X\_{\text{Cp1}} + X\_{\text{Cp2}} + X\_{\text{Cg}}\right)^2 + \left(X\_{\text{Cp1}} X\_{\text{Cp2}} + X\_{\text{Cg}} X\_{\text{Cg2}}\right)^2} \end{aligned} \tag{2a}$$

whereas *XCs* can be calculated as follows:

$$X\_{\mathbb{C}s} = \frac{1}{a\mathbb{C}\_s} = \frac{X\_{\mathbb{C}p1}\Big(R\_o^2 \Big(X\_{\mathbb{C}p1}\Big(X\_{\mathbb{C}g} + X\_{\mathbb{C}p2}\Big) + \Big(X\_{\mathbb{C}g}^2 + X\_{\mathbb{C}p2}^2\Big)\Big) + X\_{\mathbb{C}p2}X\_{\mathbb{C}g1}\Big(X\_{\mathbb{C}p1}X\_{\mathbb{C}p2} + X\_{\mathbb{C}p2}X\_{\mathbb{C}g2} + 2R\_o^2\Big)\Big)}{R\_o^2 \Big(X\_{\mathbb{C}p1} + X\_{\mathbb{C}p2} + X\_{\mathbb{C}g}\Big)^2 + \Big(X\_{\mathbb{C}p1}X\_{\mathbb{C}p2} + X\_{\mathbb{C}g}X\_{\mathbb{C}p2}\Big)^2} \tag{2b}$$

where *XLr* is the reactance of the resonance inductor *Lr*, *XCs* is the reactance of the resultant capacitor *Cs*, *XCp*1 and *XCp*2 are the reactances of the parasitic capacitors *Cp*1 and *Cp*2, respectively, *XCg* is the reactance of the coupling capacitor *Cg*, *R* is the resistance of the equivalent circuit (i.e., the resistance seen by the inverter resulting from bringing the circuit in Figure 4 into equivalent circuit as shown in Figure 5) and ω is the pulsation of the resonance circuit.

The inappropriate value of load resistance results in non-optimal conditions of the transistor switching circuit. We can distinguish three fundamental working states of the circuit: optimal, sub-optimal and non-optimal [18]. Due to fluctuations of the resultant capacitance *Cs* in case of movement of the receiver board, the considered circuit has been designed to operate in sub-optimal and optimal states. The value of the optimal resistance value has been calculated as follow:

$$R\_0 = \frac{1}{\eta \cdot \omega \cdot \mathbf{C}\_s \cdot \left(Q - \frac{n \cdot (\pi^2 - 4)}{16}\right)}\tag{3}$$

In Equation (3), η is the estimated value of system transmission efficiency adopted for the purpose of the design (in the design process the value η was assumed to be 0.95) and *Q* is the loaded quality factor at the operating frequency of circuit. In the work, it has been assumed that the value of the *Q*L coefficient is equal to 10 [19–21]. According to [21] the choice of value of *QL* ≈ 10 enables to construct the inverter, which is characterized by low value of power losses. Moreover, due to application of the condition of *QL* > 7, for switch-on duty ratio value *D* = 0.5; the waveform of the load current *Io* is similar in shape to the sine wave that results in a low value of higher harmonic distortion [20,21].

The value of transmitted power to the receiver can be calculated by usage of previously calculated optimal values of the load resistance and following relation [22]

$$P\_o = \frac{8 \cdot \mathcal{U}\_d^2}{(\pi^2 + 4) \cdot \mathcal{R}\_o} \tag{4}$$

The value of the inductance for the resonance circuit can be calculated by following formula:

$$L\_r = \frac{Q \cdot R}{\omega} \tag{5}$$

Besides the parameters of the resonance and transmitting circuit, it is necessary to specify the parameters of the supplying inverter. The Equations (6)–(8) enable to determine the values of inductance for the choke *Ld.* and the output capacitance *CT* of the transistor *T*1, i.e.,

$$T = \frac{1}{f} \tag{6}$$

$$L\_d = \frac{\mathcal{U}\_d \cdot T}{2 \cdot \Delta I\_d} \tag{7}$$

$$\mathbb{C}\_{T} = \frac{P\_o}{\pi \cdot \mathcal{U}\_d^2 \cdot \omega} \cdot \eta \tag{8}$$

where Δ*Id* is the maximum pulsation of the input current *Id*, *T* is the switching period of the inverter and *f* is the switching frequency of the inverter.

In order to simplify the calculation and shorten the convergence time, additional values of the output capacitance and the series resistance of transistors, inductors and also capacitors can be neglected. In this work, due to the high complexity of the issue, the equivalent series resistance (ESR) of designed transmitting capacitor has not been considered.

(9)

(9)

*Electronics* **2020**, *9*, 841 In order to simplify the calculation and shorten the convergence time, additional values of the output capacitance and the series resistance of transistors, inductors and also capacitors can be neglected. In this work, due to the high complexity of the issue, the equivalent series resistance (ESR) of designed transmitting capacitor has not been considered. In order to simplify the calculation and shorten the convergence time, additional values of the output capacitance and the series resistance of transistors, inductors and also capacitors can be neglected. In this work, due to the high complexity of the issue, the equivalent series resistance (ESR) of designed transmitting capacitor has not been considered. 
