**1. Introduction**

Mobility is one of the most important characteristics of a modern and smart society. The dynamics and nature of human relations in recent years are unthinkable without the constant movement of people, goods and capital. Environmental problems and challenges have necessitated a new concept for the provision of transport connections and systems through the implementation and growing dominance of electric vehicles. Regarding this aspect, there are several main obstacles to the distribution of electric vehicles: the provision of the necessary electricity, the underdeveloped infrastructure of the electricity transmission network with its insufficient capacity, minimizing the impact of the charging infrastructure on the energy transmission network and the development of a new class of power electronic devices and systems designed to charge energy storage elements [1–3]. Despite the diversity of these research problems, what they have in common is that their sustainable and effective solution is related to the development of power electronics. The aim of the present work was to present the capabilities of a class of power circuits of electronic energy converters, known as resonant converters with energy dosing, which, due to their unique properties and characteristics, are very suitable for the realization of charging stations [4,5].

**Citation:** Madzharov, N.; Hinov, N. High-Performance Power Converter for Charging Electric Vehicles. *Energies* **2021**, *14*, 8569. https:// doi.org/10.3390/en14248569

Academic Editor: Massimiliano Luna

Received: 3 November 2021 Accepted: 15 December 2021 Published: 19 December 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

### **2. Overview of Different Methods and Power Converters for the Realization of Charging Stations**

The development of power electronics has led to a wide variety of applications and, accordingly, topologies and operating modes of power electronic converters and systems. In particular, the development of power electronic devices with applications for charging electric vehicles can be divided into two main groups: direct and contactless charging [6–12].

Characteristic of both large groups is the use of resonant transducers in the process of converting electricity. This is due to some of their main advantages over other types of converters, namely [13,14]:


On the other hand, the use of resonant converters is associated with several difficulties and challenges, such as:


Addressing these challenges is usually done in two ways: either by improving the methods of and synthesis of power electronic systems control, including the use of artificial intelligence techniques [15–19] or by proposing power schemes that are weakly dependent on the operating modes of the load changes and have possibilities for self-adjustment of the converter to the requirements of the specific application [20–24].

This study proposed the use of resonant converters with improved characteristics that eliminate the most significant part of these shortcomings through the use of voltagelimiting circuits (fully or partially) on the resonant capacitor. These schemes have gained wide popularity in the specialized literature, such as energy dosing schemes. The present research involved the development of the long-term work of the authors in this field and was aimed at the implementation of this class of converters for the purpose of charging stations.

#### **3. Basic Relations in the Analysis of Resonant Converters with Energy Dosing**

The method of energy dosing (ED) has been used in several applications of AC and DC power supplies, where their use initially began in industrial technologies based on induction heating [4,5,24,25]. The loads of the converters in these applications are characterized by highly variable parameters, which can often vary from idle to shortcircuiting. These properties of ED converters make them very suitable for the development of contactless charging stations for static and dynamic charging of electric vehicles. Their stable operation is obtained by fulfilling the following condition:

$$k\_S = (d\mathcal{U}\_T / d\mathcal{I}\_T - d\mathcal{U} / d\mathcal{I}) > 0\tag{1}$$

where *kS* is the coefficient of stability of the system, while *dUT*/*dIT* and *dU*/*dI* are the dynamic resistances of the load and the DC-DC converter, respectively.

Figure 1 shows a power scheme of a half-bridge DC-DC converter with energy dosing. It consists of a half-bridge resonant inverter with energy dosing (RI with ED) without reverse diodes, a high-frequency matching transformer and an output rectifier with a capacitive filter and an equivalent load.

Of the possible operating modes, the most suitable when using the circuit is the mode when the operating frequency is less than the resonant frequency of the alternating current (AC) circuit of RI with ED, i.e., *f<f* 0. Figure 2 shows the time diagrams of the current through the resonant inductor *LR*, the voltage of the resonant capacitor *CR*, the output current and the voltage of the transistor *VT*1, which clarifies the principle of operation. It should be noted that the transistors operate with zero on and off current (ZCS).

**Figure 2.** Basic timing diagrams of a half-bridge DC–DC converter with energy dosing.

The indicated time diagrams show that in the operation of the power circuit during each half-period the following intervals are distinguished: 0 ÷ *ϑd*, *ϑ<sup>d</sup>* ÷ *ϑ*0, *ϑ*<sup>0</sup> ÷ *ϑ*<sup>01</sup> and *ϑ*<sup>01</sup> ÷ *π*. On the other hand, for the analysis of the resonant inverter, it is important to distinguish between the intervals (0 ÷ *ϑd*) in which energy is consumed from the power source and those in which the energy accumulated in the resonant elements supplies the output of the circuit. The operation of the considered power scheme for one of the two half-periods of its operation (0 ÷ *π*) is explained in detail using a modal diagram, which is shown in Figure 3. In order to achieve convenient ratios, the following assumptions are made: the active and passive elements are ideal, and the times for their commutation are neglected. In addition, the processes in the scheme are considered after reaching a set mode of operation when there is periodicity and repeatability of the state variables.

**Figure 3.** Modal diagram of a half-bridge DC–DC converter with energy dosing.

In the analysis of the circuit, the secondary circuit is brought to the primary circuit; as a result of which, the output capacitor, the load and the transformer are replaced by a voltage source *nUOUT*, where *n* is the transformation coefficient (*n* = *W*1*/W*2). The behavior of the converter is determined mainly by the values of the elements *LR* and *CR*, and more generally, by the following parameters [5,24,25]:


$$Z\_0 = \sqrt{L\_R / C\_R} \tag{2}$$


$$
\omega\_0 = \sqrt{1/L\_R C\_R} \tag{3}
$$


$$T\_0 = 2\pi/\omega\_0\tag{4}$$


$$
\gamma = T\_0 / 2T \tag{5}
$$

If the losses in the elements of the converter are ignored, then its input power will be equal to the output. Taking this assumption into account, the following electrical ratios for the input current and output voltage can be recorded as follows:

$$I\_{IN} = I\_{OUT}/2\pi \qquad \qquad \mathcal{U}\_{\text{OUT}} = E/2\pi \tag{6}$$

Diodes *VD*<sup>1</sup> and *VD*<sup>2</sup> start to conduct when the maximum and minimum values of the voltage on the capacitor *CR* becomes equal to +*E*/2 or −*E*/2. If the load resistance or operating frequency has values that are significantly different from the nominal ones, the capacitor *CR* will not be recharged from −*E*/2 to +*E*/2 and the diodes *VD*<sup>1</sup> and *VD*<sup>2</sup>

will not turn on. In this case, the voltage to which the capacitor will be recharged can be determined using the expressions from Equations (2) to (6).

$$dL\_{\mathbb{CR}} = \mathbb{E}/4 + \pi \,\mathrm{E}Z\_0/8\gamma \,\mathrm{R}n^2 \tag{7}$$

The latter expression has a very important practical meaning. It displays the ratio for the values of the load resistance at which the energy dosing mode is performed, i.e., limiting the voltage on the capacitor *CR* to ±*E*/2 by *VD*<sup>1</sup> and *VD*2, namely,

$$
\mathcal{R} \le \pi \, Z\_0 / 2\gamma \, n^2 \tag{8}
$$

This mode, which is considered optimal, is characterized by two main intervals: the consumption of energy from the power supply and the short circuit of the alternating current circuit when energy from the power supply is not consumed.

*3.1. Electromagnetic Analysis of the Converter at the Time of Energy Consumption from the Power Source 0* ÷ *ϑ<sup>d</sup>*

During the interval 0 ÷ *ϑ<sup>d</sup>* (Figure 2), only the transistor *VT*<sup>2</sup> is unlocked and the equivalent circuit consists of the power supply connected in series, the reduced load circuit and the resonant inductor and capacitor. The natural frequency of the AC circuit (a-b in Figure 2) must be higher than the control frequency, i.e., ω0/ω > 1, in order to obtain a tendency for the current to drop to zero at the beginning of the interval *ϕ*0. The following frequency ratio is valid for the considered operating mode:

$$\frac{\omega\_0}{\omega} = \frac{\pi}{\pi - \varrho\_0 - (0.65 \div 1.\mathcal{T})\varrho\_0} > 1\tag{9}$$

$$\text{i.e.}\ \omega\_0/\omega = 1.2 \div 1.4\tag{10}$$

Taking into account the expression for the current through the resonant inductor:

$$\dot{q}\_{L\_R}(\theta) = \frac{E}{\omega\_0 L\_R} \exp^{-\theta/2Q} \sin \frac{\omega\_0}{\omega} \theta \tag{11}$$

and the voltage across the resonant capacitor *CR*:

$$
\mu\_{\rm CR}(\theta\_d) = \frac{1}{\omega\_0 \mathbb{C}\_R} \int\_0^{\theta\_d} i(\theta) = \frac{E}{2} \tag{12}
$$

as determined at the end of the interval of energy consumption from the power source, the moment that corresponds to the angle *ϑ<sup>d</sup>* is

$$\theta\_d = \left[\pi - \operatorname{arctg}(2\mathbb{Q}\omega\_0/\omega)\right]/(\omega\_0/\omega) \tag{13}$$

where *Q* = *ωLR*/*RE* is the quality factor of the resonant circuit, *RE* is the equivalent resistance of the AC circuit between points b and c of Figure 1 and *ϑ* = *ωt*.
