**5.SelectedResults5. Selected Results**

 **5. Selected Results** 

In this work, to determine values of parameters as well as to conduct the analysis of discussed CPT system, the field model presented in Section 2 was used. At first, the calculations of values of coupling capacitances and parasitic capacitances were performed for different locations of receiving board relative to the transmitting board, starting from "0" (see Figure 2c). According to Equation (9) resultant parasitic capacitance *Cp* is equal to the sum of parasitic capacitances *Cp*1 and *Cp*2. In this work, to determine values of parameters as well as to conduct the analysis of discussed CPT system, the field model presented in Section 2 was used. At first, the calculations of values of coupling capacitances and parasitic capacitances were performed for different locations of receiving board relative to the transmitting board, starting from "0" (see Figure 2c). According to Equation (9) resultant parasitic capacitance *Cp* is equal to the sum of parasitic capacitances *Cp*1 and *Cp*2. In this work, to determine values of parameters as well as to conduct the analysis of discussed CPT system, the field model presented in Section 2 was used. At first, the calculations of values of coupling capacitances and parasitic capacitances were performed for different locations of receiving board relative to the transmitting board, starting from "0" (see Figure 2c). According to Equation (9) resultant parasitic capacitance *Cp* is equal to the sum of parasitic capacitances *Cp*1 and *Cp*2. *CC*+*C*(9)

$$
\stackrel{\rho}{\mathcal{L}} = \stackrel{\rho}{\mathcal{C}}\_{\mathcal{V}^{\mathcal{A}\_1}\_{\mathcal{I}}} + \stackrel{\rho}{\mathcal{C}}\_{\mathcal{V}^{\mathcal{Q}\_2}\_{\mathcal{I}}} \qquad \stackrel{\cdots}{\mathcal{C}} \qquad \stackrel{\cdots}{\cdots} \qquad \stackrel{\cdots}{\cdots} \qquad \cdots
$$

Obtained results in the form of 3D plots of coupling capacitance *Cg* and resultant parasitic capacitance *Cp* are presented in Figures 6 and 7, respectively. Obtained results in the form of 3D plots of coupling capacitance *Cg* and resultant parasitic capacitance *Cp* are presented in Figures 6 and 7, respectively. Obtained results in the form of 3D plots of coupling capacitance *Cg* and resultant parasitic capacitance *Cp* are presented in Figures 6 and 7, respectively. 

*Cp*

*C p*

 =

**Figure 6.** The 3D plot of the coupling capacitance *Cg* as a function of the relative position. **Figure 6.** The 3D plot of the coupling capacitance *Cg* as a function of the relative position. **Figure 6.** The 3D plot of the coupling capacitance *Cg* as a function of the relative position. 

**Figure 7.** The 3D plot of the resultant parasitic capacitance *Cp* as a function of the relative position.

In Figure 6 the relation between the coupling capacitance *Cg* and the relative position of receiving board in relation to transmitter has been given. By analyzing Figures 6 and 7 it can be concluded that the extremum for coupling capacitance *Cg* is located in the middle of the transmitting board, whereas

**Figure 7.** The 3D plot of the resultant parasitic capacitance *Cp* as a function of the relative position.

*Electronics* **2020**, *9*, 841 for the resultant parasitic capacitance *Cp* it is on the borders of the transmitting board. The range of capacitance changes is equal to 12.4 pF (2.4%) and 27.7 pF (24.2%) for coupling capacitance *Cg* and resultant parasitic capacitance *Cp,* respectively. The equivalent capacitance *Cs* describing the resultant capacitance of the transmitting system has been shown in Figure 8. In Figure 6 the relation between the coupling capacitance *Cg* and the relative position of receiving board in relation to transmitter has been given. By analyzing Figures 6 and 7 it can be concluded that the extremum for coupling capacitance *Cg* is located in the middle of the transmitting board, whereas for the resultant parasitic capacitance *Cp* it is on the borders of the transmitting board. The range of capacitance changes is equal to 12.4 pF (2.4%) and 27.7 pF (24.2%) for coupling capacitance *Cg* and resultant parasitic capacitance *Cp,* respectively*.* The equivalent capacitance *Cs* describing the resultant capacitance of the transmitting system has been shown in Figure 8. 

**Figure 8.** The 3D plot of the resultant coupling capacitance *Cs* as a function of the relative position. **Figure 8.** The 3D plot of the resultant coupling capacitance *Cs* as a function of the relative position.

The relationship presented in Figure 8 shows that the highest values of the resultant capacity Cs are obtained for positions where the center of the receiving system board is located at a distance corresponding to approximately 1/4 of the length of the transmitter from the outer borders in the *y* direction for the adopted reference system and half of the length of the plate in the *x* direction of the reference system. The relationship presented in Figure 8 shows that the highest values of the resultant capacity *Cs* are obtained for positions where the center of the receiving system board is located at a distance corresponding to approximately 1/4 of the length of the transmitter from the outer borders in the *y* direction for the adopted reference system and half of the length of the plate in the *x* direction of the reference system.

In the next step of studies, the analysis of operating states of the CPT system co-working with an E-class inverter has been performed. The results in form of 3D plots of transmitted power and transmission efficiency have been shown in Figures 9 and 10. The calculation of the transmitted power *PT*, the input electrical power *Pd* supplied to the CPT system and total efficiency *<sup>t</sup>* as a function of receiver position have been calculated using the formulas: In the next step of studies, the analysis of operating states of the CPT system co-working with an E-class inverter has been performed. The results in form of 3D plots of transmitted power and transmission efficiency have been shown in Figures 9 and 10. The calculation of the transmitted power *PT*, the input electrical power *Pd* supplied to the CPT system and total efficiency η*t* as a function of receiver position have been calculated using the formulas:

$$P\_T^{P\_r} \stackrel{\text{def}}{=} \underset{T}{\stackrel{\text{I}}{\text{I}}} \stackrel{\text{P}}{\stackrel{\text{I}}{\text{R}}} \stackrel{\text{I}}{\stackrel{\text{I}}{\text{R}}} \stackrel{\text{I}}{I} \stackrel{\text{I}}{\text{I}} \stackrel{\text{I}}{\text{t}} \stackrel{\text{t}}{\text{t}} \text{dt}\tag{408}$$

$$P\_d = \frac{1}{T} \begin{bmatrix} U\_d \ I\_d \ \mathbf{t} \end{bmatrix} \tag{10b}$$
 
$$\begin{array}{c} \text{(10b)} \\ \text{(10c)} \end{array} \tag{10b}$$

$$P\_d = \frac{1}{T} \int\_{\frac{P\_s}{P}} \mathbb{L}\_d I\_d(t) dt \tag{10b}$$

$$
\rho \rho = \frac{\cdot}{P\_{\text{ref}}} \tag{10c}
$$

$$
\eta\_t = \frac{P\_t}{P\_d} \tag{10c}
$$

**Figure 9.** The 3D plot of the transmitted power as a function of the relative position of the receiver. **Figure 9.** The 3D plot of the transmitted power as a function of the relative position of the receiver. **Figure 9.** The 3D plot of the transmitted power as a function of the relative position of the receiver.

**Figure 10.** The 3D plot of the total efficiency as a function of the relative position of the receiver. **Figure 10.** The 3D plot of the total efficiency as a function of the relative position of the receiver. **Figure 10.** The 3D plot of the total efficiency as a function of the relative position of the receiver.

Changes in the values of transmitted power and transmission efficiency result directly from fluctuation in the values of individual system capacitances. It can be seen that decreases in efficiency of the CPT system (about 2%) are observed for the central location of the receiving board. However, in this position it can be observed that the system obtains the maximum power values that can be transferred between the transmitter and the receiver, obtaining values of 20.5 W. The highest values of the system efficiency, as could be expected, are obtained for the positions of the receiver located at one quarter of the distance from the outer borders of the transmitter for the adopted direction of the reference system. The last step of the research was to conduct a simulation in order to verify the current and voltage waveforms on individual elements of the power supply circuit from Figure 4. Simulation has been performed for the optimal working point, located approximately a quarter of the length of the transmitter from the outer borders in *y* direction for the adopted reference system and half of the length of the plate in the *x* direction of the reference system. Parameters of the system for that point are as follows: *Ro* = 31.56 Ǎ, *Lr* = 50 μH, *CT* = 1 nF, *Ld* = 1.15 mH, *Cg* = 511.25 pF, *Cp*1 = 89.07 pF and *Cp*2 = Changes in the values of transmitted power and transmission efficiency result directly from fluctuation in the values of individual system capacitances. It can be seen that decreases in efficiency of the CPT system (about 2%) are observed for the central location of the receiving board. However, in this position it can be observed that the system obtains the maximum power values that can be transferred between the transmitter and the receiver, obtaining values of 20.5 W. The highest values of the system efficiency, as could be expected, are obtained for the positions of the receiver located at one quarter of the distance from the outer borders of the transmitter for the adopted direction of the reference system. The last step of the research was to conduct a simulation in order to verify the current and voltage waveforms on individual elements of the power supply circuit from Figure 4. Simulation has been performed for the optimal working point, located approximately a quarter of the length of the transmitter from the outer borders in *y* direction for the adopted reference system and half of the length of the plate in the *x* direction of the reference system. Parameters of the system for that point are as follows: *Ro* = 31.56 Ǎ, *Lr* = 50 μH, *CT* = 1 nF, *Ld* = 1.15 mH, *Cg* = 511.25 pF, *Cp*1 = 89.07 pF and *Cp*2 = Changes in the values of transmitted power and transmission efficiency result directly from fluctuation in the values of individual system capacitances. It can be seen that decreases in efficiency of the CPT system (about 2%) are observed for the central location of the receiving board. However, in this position it can be observed that the system obtains the maximum power values that can be transferred between the transmitter and the receiver, obtaining values of 20.5 W. The highest values of the system efficiency, as could be expected, are obtained for the positions of the receiver located at one quarter of the distance from the outer borders of the transmitter for the adopted direction of the reference system. The last step of the research was to conduct a simulation in order to verify the current and voltage waveforms on individual elements of the power supply circuit from Figure 4. Simulation has been performed for the optimal working point, located approximately a quarter of the length of the transmitter from the outer borders in *y* direction for the adopted reference system and half of the length of the plate in the *x* direction of the reference system. Parameters of the system for that point are as follows: *Ro* = 31.56 Ω, *Lr* = 50 μH, *CT* = 1 nF, *Ld* = 1.15 mH, *Cg* = 511.25 pF, *Cp*1 = 89.07 pF and *Cp*2= 14.77pF.Thesimulationhasbeencarriedoutunderfollowingconditions:Δ*Id*= 20mA,Q = 10,

14.77 pF. The simulation has been carried out under following conditions: ̇*Id* = 20 mA, Q ൌ 10, *Ud* <sup>=</sup> 40 V and *f* = 1 MHz. Figures 11–13 show the current and voltage waveforms corresponded to the elements marked in Figure 4. 14.77 pF. The simulation has been carried out under following conditions: ̇*Id* = 20 mA, Q ൌ 10, *Ud* <sup>=</sup>40 V and *f* = 1 MHz. Figures 11–13 show the current and voltage waveforms corresponded to the elements marked in Figure 4. 

*Ud* = 40 V and *f* = 1 MHz. Figures 11–13 show the current and voltage waveforms corresponded to the elements marked in Figure 4. *Electronics* **2020**, *9*, x FOR PEER REVIEW 10 of 15 *Electronics* **2020**, *9*, x FOR PEER REVIEW 10 of 15 

The waveforms in Figure 11 confirm the validity of performed calculations presented in Section 4. By analyzing the waveforms of current *IT* and voltage *UT* on the switching transistor *T*1, it can be clearly stated that the system worked in a sub-optimal/near-optimal stage. Due to the fluctuation of circuit parameters related to the change of location of the receiving board, it is impossible to obtain optimal operating conditions for each position of the receiver. However, by maintaining the sub-optimal working conditions in the whole working area, it is possible to obtain high parameters of energy transmission regardless of the fluctuation of the values of individual capacities. The waveforms in Figure 11 confirm the validity of performed calculations presented in Section 4. By analyzing the waveforms of current *IT* and voltage *UT* on the switching transistor *T*1, it can be clearly stated that the system worked in a sub-optimal/near-optimal stage. Due to the fluctuation of circuit parameters related to the change of location of the receiving board, it is impossible to obtain optimal operating conditions for each position of the receiver. However, by maintaining the sub-optimal working conditions in the whole working area, it is possible to obtain high parameters of energy transmission regardless of the fluctuation of the values of individual capacities. The waveforms in Figure 11 confirm the validity of performed calculations presented in Section 4. By analyzing the waveforms of current *IT* and voltage *UT* on the switching transistor *T*1, it can be clearly stated that the system worked in a sub-optimal/near-optimal stage. Due to the fluctuation of circuit parameters related to the change of location of the receiving board, it is impossible to obtain optimal operating conditions for each position of the receiver. However, by maintaining the sub-optimal working conditions in the whole working area, it is possible to obtain high parameters of energy transmission regardless of the fluctuation of the values of individual capacities. 

Figures 12 and 13 present the waveforms of input current *Id* and output current (receiver current) *Io*, as well as voltage drops on the resonant elements of the system *UL* and *UCg* respectively. First of all, attention should be paid to the noticeable presence of zero order harmonic in the output current *Io* and in the voltage waveforms *UL* and *UCg* This phenomenon is characteristic of systems supplied by E-class inverters, due to the necessity of separating two resonance systems of different parameters depending on the state of the transistor's operation. Figures 12 and 13 present the waveforms of input current *Id* and output current *(*receiver current*) Io*, as well as voltage drops on the resonant elements of the system *UL* and *UCg* respectively. First of all, attention should be paid to the noticeable presence of zero order harmonic in the output current *Io* and in the voltage waveforms *UL* and *UCg* This phenomenon is characteristic of systems supplied by E-class inverters, due to the necessity of separating two resonance systems of different parameters depending on the state of the transistor's operation. Figures 12 and 13 present waveforms of input current *Id* and output current *(*receiver current*) Io*, as well as voltage drops the resonant elements of the system *UL* and *UCg* respectively. First of all, attention should be to noticeable presence of zero order harmonic in the output current *Io* and in the voltage waveforms *UL* and *UCg* This phenomenon is characteristic of systems supplied by E-class inverters, due to the necessity of separating two resonance systems of different parameters depending on the state of the transistor's operation. 

**Figure 11.** Waveforms of: (**a**) the pulse width modulation (PWM) signal controlling the gate of the transistor *T*1; (**b**) the voltage drop *UT* on transistor *T*1; (**c**) the current *It* flowing through the transistor *<sup>T</sup>*1; (**d**) the current *ICT* flowing through the output capacitor *CT*. **Figure 11.** Waveforms of: (**a**) the pulse width modulation (PWM) signal controlling the gate of the transistor *T*1; (**b**) the voltage drop *UT* on transistor *T*1; (**c**) the current *It* flowing through the transistor *T*1; (**d**) the current *ICT* flowing through the output capacitor *CT*. **Figure 11.** Waveforms of: (**a**) the pulse width modulation (PWM) signal controlling the gate of the transistor *T*1; (**b**) the voltage drop *UT* on transistor *T*1; (**c**) the current *It* flowing through the transistor *<sup>T</sup>*1; (**d**) the current *ICT* flowing through the output capacitor *CT*. 

**Figure 12.** Waveforms of (**a**) input current *Id* and (**b**) output current *Io*.

**Figure 13.** Waveforms of (**a**) voltage drop on inductor *UL* and *(***b**) voltage drop on capacitor *UCg*. **Figure 13.** Waveforms of (**a**) voltage drop on inductor *UL* and (**b**) voltage drop on capacitor *UCg*. **Figure 13.** Waveforms of (**a**) voltage drop on inductor *UL* and *(***b**) voltage drop on capacitor *UCg*. 

In order to verify simulation results, the experimental stand of elaborated CPT system was set up and appropriate measurements were conducted. The laboratory stand is shown in Figure 14. The values of parameters of individual elements that have been used in the laboratory system as well as the position of the receiving plate relative to the transmitting one are corresponded to ones used in the simulation. During tests, values of voltage and current on individual elements of the system have been measured and summarized in the form of waveforms shown in Figures 15 and 16. In order to verify simulation results, the experimental stand of elaborated CPT system was set up and appropriate measurements were conducted. The laboratory stand is shown in Figure 14. The values of parameters of individual elements that have been used in the laboratory system as well as the position of the receiving plate relative to the transmitting one are corresponded to ones used in the simulation. During tests, values of voltage and current on individual elements of the system have been measured and summarized in the form of waveforms shown in Figures 15 and 16. In order to verify simulation results, the experimental stand of elaborated CPT system was set up and appropriate measurements were conducted. The laboratory stand is shown in Figure 14. The values of parameters of individual elements that have been used in the laboratory system as well as the position of the receiving plate relative to the transmitting one are corresponded to ones used in the simulation. During tests, values of voltage and current on individual elements of the system have been measured and summarized in the form of waveforms shown in Figures 15 and 16. 

**Figure 14.** The experimental stand of the elaborated CPT system. **Figure 14. Figure 14.** The experimental stand of the elaborated CPT system. The experimental stand of the elaborated CPT system.

**Figure 15.** The comparative waveforms of simulation and experimental results: (**a**) output current *Io* and (**b**) input current *Id*. **Figure 15.** The comparative waveforms of simulation and experimental results: (**a**) output current *Io* and (**b**) input current *Id*. **Figure 15.** The comparative waveforms of simulation and experimental results: (**a**) output current *Io* and (**b**) input current *Id*. 

**Figure 16.** The comparative waveforms of simulation and experimental results: (**a**) transistor voltage **Figure 16.** The comparative waveforms of simulation and experimental results: (**a**) transistor voltage drop *UT* and (**b**) transmitted power *PT*. **Figure 16.** The comparative waveforms of simulation and experimental results: (**a**) transistor voltage drop *UT* and (**b**) transmitted power *PT*.

drop *UT* and (**b**) transmitted power *PT*. The quality of the obtained waveforms was verified by means of Fast. Fourier Transform (FFT) analysis. The obtained results are presented in Figure 17. By analyzing the results presented in Figure 15, it can be noticed that in the receiver current, apart from the first harmonic, the influence of the zero and second harmonic can be also observed. The presence of these harmonics is directly related to the change in configuration of the resonant circuit resulting from the change of operation states of the transistor [22]. The quality of the obtained waveforms was verified by means of Fast. Fourier Transform (FFT) analysis. The obtained results are presented in Figure 17. By analyzing the results presented in Figure 15, it can be noticed that in the receiver current, apart from the first harmonic, the influence of the zero and second harmonic can be also observed. The presence of these harmonics is directly related to the change in configuration of the resonant circuit resulting from the change of operation states of the transistor [22]. The quality of the obtained waveforms was verified by means of Fast. Fourier Transform (FFT) analysis. The obtained results are presented in Figure 17. By analyzing the results presented in Figure 15, it can be noticed that in the receiver current, apart from the first harmonic, the influence of the zero and second harmonic can be also observed. The presence of these harmonics is directly related to the change in configuration of the resonant circuit resulting from the change of operation states of the transistor [22].

**Figure 17.** The FFT analyses of: (**a**) receiver current waveform *Io* obtained from the simulation and (**b**) receiver current waveform *Io* obtained from the measurements. **Figure 17.** The FFT analyses of: (**a**) receiver current waveform *Io* obtained from the simulation and (**b**) receiver current waveform *Io* obtained from the measurements.

Analyzing the waveforms presented in Figures 15–17, the convergence between the results obtained on the basis of the computed simulation and the ones obtained from measurements of the system can be clearly noticed. Undoubtedly, it proves the validity of the assumptions made and the reliability of the elaborated model. However, it should be noted, that there are differences between the efficiency value obtained by simulation (ca. 92%) and measurements (ca. 84%) that result from the negligence of values of coupled capacitor series resistances ESR in simulation process. Therefore, it can be stated that the ESR resistance of the coupled capacitor contributes to a significant decrease in the transmission efficiency. During the tests some observations were made. Both pressure and equilibrium of the receiving plate (i.e., its distribution) is particularly important and have significant impact on operation of the system. Uneven distribution of the receiving plate contributes to the fluctuation of the coupled capacitance value due to formation of the irregular air gap between the transmitting and receiving plates. Analyzing the waveforms presented in Figures 15–17, the convergence between the results obtained on the basis of the computed simulation and the ones obtained from measurements of the system can be clearly noticed. Undoubtedly, it proves the validity of the assumptions made and the reliability of the elaborated model. However, it should be noted, that there are di fferences between the efficiency value obtained by simulation (ca. 92%) and measurements (ca. 84%) that result from the negligence of values of coupled capacitor series resistances ESR in simulation process. Therefore, it can be stated that the ESR resistance of the coupled capacitor contributes to a significant decrease in the transmission e fficiency. During the tests some observations were made. Both pressure and equilibrium of the receiving plate (i.e., its distribution) is particularly important and have significant impact on operation of the system. Uneven distribution of the receiving plate contributes to the fluctuation of the coupled capacitance value due to formation of the irregular air gap between the transmitting and receiving plates.
