**5. Experimental Results**

With the methodology thus created, a full-bridge RI with ED, used for a contactless charging station of a motoped, was designed and studied. It is shown in Figure 7 and consisted of the following main blocks: an electronic converter (inverter), together with the control system and drivers; a power supply; a measuring system; and a load composed of a transmitter and receiver. The power circuit of the inverter was implemented according to Figure 1.

**Figure 7.** Block diagram of the experimental setup.

During the implementation of the experiment, the following input data were used: output power P = 2.5 kW, operating frequency of the inverter f = 30 kHz, power supply voltage E = 300 V, resonant capacitor *CR* = 0.95 μF, resonant inductor *LR* = 16.31 μH, *W*1*/W*<sup>2</sup> = 1, *ω*0*/ω* = 1.2–1.4 and equivalent load *R* = 8.9 Ω.

Figure 8 shows a photo of the stand on which the experimental results were obtained.

**Figure 8.** A PI with ED that was used for the realization of a contactless charging station of a motoped.

ϑ

Furthermore, simulation studies of the system were performed with these data. In Table 4, the results obtained from the stand are compared with those from a program for the simulation of power electronic devices. The table shows that RI actually had dosing properties, which was determined by the average value of the consumed current *I*0, equal to the input current *Iin = P/E =* 8.3 A, which was practically the same value from the analysis and the computer experiment.

**Table 4.** Results from the design, the computer experiment and the practical research.


This basic property of the circuit was also confirmed by the load voltage *Uoutm*, which also coincided with the calculations and the experiment. Obtaining the set power could be shown by the following result of the table, corresponding to the physical processes in RI with ED, namely, that the difference between the currents of the transistors and diodes, i.e., *IVT* and *IVD*, were equal to *Iin* and corresponded to the energy consumption of energy in the interval *ϑ = 0* ÷ *ϑ<sup>d</sup>* and a short circuit of the supply DC bus in the interval *ϑ* = *ϑ<sup>d</sup>* ÷ *π* − *ϕ0*. In addition, compliance with the boundary conditions for switching and periodicity could be used as an additional criterion for the reliability of the obtained results. In this case, they were expressed as obtaining the set operating mode of the RI in which the PT turned on and off at zero current (ZCS) (see Figure 9a). It should also be noted that in the calculations, the parameters of the current pulse through the key devices were obtained with great reliability, as the difference with the measured values did not exceed 5%.

**Figure 9.** Graphical results obtained from the operation of the stand: (**a**) current through the resonant inductor (43.6 A maximum value) and (**b**) voltage on the resonant capacitor (maximum value 150 V).

Figure 9 presents the results of the following measurements: (a) current through the resonant inductor and (b) voltage on the resonant capacitor, performed with the help of the experimental bench under the conditions described above and operating frequency 30 kHz.

The presented experimental results fully supported the conclusions made in the analysis of the power scheme and confirmed the validity of the design methodology.

#### **6. Discussion**

The main results obtained from the theoretical and applied scientific problems developed in this article were as follows:

(1) A methodology for designing an RI with ED was developed that provided switching of semiconductor devices at ZCS and ZVS and was characterized by a satisfactory accuracy of not less than 5%, as shown by computer and practical experiments.

(2) A study of the current pulse of the RI was made and its parameters were determined in order to obtain the minimum installed reactive power in the alternating current circuit and electrical loads and losses in the transistors.

(3) The property of the RI with ED was shown to maintain a constant power in the load, regardless of the change of its parameters, when switching the semiconductor devices at ZCS and ZVS. This adaptability and self-matching properties make it very flexible and convenient to use as a wide-range power source for charging stations, including contactless charging.

From the comparisons made with studies of power circuits used for the realization of charging stations presented in [27,28,30–32], it is necessary to conclude that the main alternative of the considered converters with energy dosing was the resonant inverters with reverse diodes and voltage-fed inverters. In order to work with soft commutations, the latter requires the addition of additional resonant circuits (so-called quasi-resonant), and in the case of reverse diode circuits, it is possible to work both in soft commutation mode and to limit the maximum current of the transistors [33]. Unfortunately, in inverters with reverse diodes, power maintenance is achieved through the synthesis of complex control algorithms and controllers. On the other hand, the challenges and limitations associated with the implementation of an RI with ED are related to the fact that no energy is consumed from the power source during the whole period, the lack of limitation of the maximum current through the transistors and the need for sufficient time to dissipate the energy in the resonant inductor when working with high-resistance loads and low power. Regarding this aspect, the achievement of electromagnetic compatibility standards requires the addition of additional modules and ancillary devices.

#### **7. Conclusions**

The manuscript presents a study of a charging station for electric vehicles based on a resonant converter with energy dosing. Based on the analysis performed in the established mode of operation of the power circuit, the main relations were derived, through which, the values of the circuit elements were determined. On the basis of the analytical expressions, computer simulations and experiments, the advantages of this class of schemes for the realization of charging stations with different capacities and applications were shown. Regarding a continuation of the current research, the combination of a design with techniques of mathematical modeling and computational mathematics in order to determine optimal values of circuit elements for different objective functions, such as minimum losses, maximum efficiency and minimum dimensions, should be undertaken. In addition, future research could conduct experiments at higher capacities, and in order to achieve greater flexibility, they will be built on a modular principle, as well as the dynamic performance of the system, in order to achieve an aperiodic transition process with minimum duration.

The proposed scheme was also successfully used for contactless dynamic charging of electric vehicles [34]. The main advantage of energy dosing schemes is that the power does not depend on the size of the load. Therefore, during driving, when the equivalent load is constantly changing (due to the coefficient of magnetic coupling), the power transferred to the vehicle is constant.

**Author Contributions:** N.M. and N.H. were involved in the full process of producing this paper, including conceptualization, methodology, modeling, validation, visualization and preparing the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Bulgarian National Scientific Fund, grant number KΠ-06- H37/25/18.12.2019, and the APC was funded by KΠ-06-H37/25/18.12.2019.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This research was carried out within the framework of the project "Optimal design and management of electrical energy storage systems", KΠ-06-H37/25/18.12.2019, Bulgar-ian National Scientific Fund.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

