*3.2. Operation of the Converter in the Short Circuit Interval of the Alternating Current Circuit ϑ<sup>d</sup>* ÷ *ϑ<sup>0</sup>*

At the moment corresponding to the angle *ϑd*, the resonant capacitor *CR* is charged to voltage −*E*/2 and the diode *VD*<sup>2</sup> is turned on. In essence, the electromagnetic processes in the second interval *ϑ<sup>d</sup>* ÷ *ϑ*<sup>0</sup> are aperiodic. The expression for the current through the resonant inductor is of the form

$$i\_{LR}(\theta) = i\_{LR}(\theta\_d) \cdot \exp^{-\frac{R}{\alpha \cdot l\_R} \theta} \tag{14}$$

where *R=RE*.

It is very important from a practical point of view to determine the value of the current at the end of the interval (moment corresponding to an angle *π* − *ϕ*0) because the level of switching losses in the transistors depends on it. It is calculated using the current expression *iLR (ϑ)* at *ϑ = π* − *ϕ*0. In this regard, additional information about the processes in the circuit is given by the expressions for the voltages across the resonant inductor and the equivalent load:

$$
\mu\_{LR} = -Ldi\_{LR}/dt\tag{15}
$$

$$u\_{OUT} = R\left[n i\_{LR}(\theta\_d) \exp^{-\theta/Q}\right] \tag{16}$$

From the presented relations, it is clear that if at the moment corresponding to an angle *π* − *ϕ*0, the resonant current has zero value, then the voltages *ULR* and *UOUT* will also have zero values. At this point, the voltage on the transistors will also have a zero value, i.e., they will turn on at zero current and will turn off at zero current and zero voltage.

#### *3.3. Stabilization and Regulation of the Output Power and Voltage of the Converter*

The energy of the capacitor when recharging from −*E*/2 to +*E*/2 is equal to

$$\mathcal{W} = \mathbb{C}\_{\mathbb{R}} \mathbb{E}^2 / 2 \tag{17}$$

When transistors *VT*<sup>1</sup> or *VT*<sup>2</sup> are turned on, this energy is transmitted to the load, i.e.,

$$
\mathbb{C}\_{\mathbb{R}}\mathbb{E}^2/2 = \mathbb{U}\_{\text{OUT}} \, I\_{\text{OUT}} \, T/2. \tag{18}
$$

In this case, the power of the converter *P* for one period is expressed by the ratio

$$P = E^2 \, f \, \mathcal{C}\_{\mathbb{R}} = E \, I\_d = \mathcal{U}\_{\text{OUT}} \, I\_{\text{OUT}} = const,\tag{19}$$

The first conclusion that can be drawn from the expressions for *W* and *P* is that at a constant operating frequency, supply voltage and capacitance of the resonant capacitor, the transmitted power in the load is constant and does not depend on its parameters. Maintaining a constant power means that the output DC voltage *UOUT* is self-aligning with the load parameters.

To set the power level in accordance with Equation (19), the value of the capacitor *CR* is most often changed. For this purpose, the developed electronic keys are used, for which there are author's claims regarding their use for similar purposes. They consist of only one transistor (IGBT or MOSFET) with a reverse diode and a series-connected capacitor *CR*, the capacitance of which is in accordance with the desired power (Equation (19)). Figure 4 shows the schematic diagram of four electronic switches, which participate in the circuit of the converter of Figure 1 and are connected to points a and c. The presented circuit contains half as many elements as the traditional electronic switch, while filled with two oppositely connected transistors with reverse diodes and, therefore, has less static losses compared to other embodiments [24,25].

**Figure 4.** Electronic semiconductor switches for setting the power level.

The combinations between the turned on and turned off capacitors make it possible to set fifteen power values in the range 0 ÷ *PNOM*. No galvanically separated signals are required to control the individual transistors.

It is important to note that the switching of the capacitors *CR1–CR4* from Figure 4 can be done during the operation of the converter without receiving current and voltage overloads from the transistors.

The second characteristic feature of the converter is obtained by substituting the expression for the load current *IOUT* = *UOUT*/*R* into Equation (19). In this way, a ratio is obtained that shows the relationship between the input and output voltage:

$$\mathcal{U}\_{\text{OUT}} = E \sqrt{\mathcal{C}\_R \mathcal{R} / T} \tag{20}$$

The conclusion to be drawn from this expression is that by changing the operating frequency, the output voltage can be invariably maintained when the value of the load and/or the input voltage changes. Figure 5 shows the dependence of the output voltage in relative units as a function of frequency at different loads. The information from this characteristic is used in the design of the converter because it takes into account, on the one hand, the relationship between the value of the load and the capacitance of the capacitor *CR* setting the power and, on the other, the dependence of output voltage on frequency and input voltage.

**Figure 5.** Control characteristics of a converter with energy dosing.

The regulation or stabilization of the output voltage is carried out via feedback, changing the operating frequency of the converter. The analytical dependence of the regulation law can be deduced by differentiating the output power expression with respect to the control frequency *f*. After some transformations using the expressions:

$$\mathcal{U}\_{\text{OUT}} \ I\_{\text{OUT}} = \mathbb{C}\_{\text{R}} E^2 f\_{\prime} \tag{21}$$

$$d\, \mathcal{U}\_{\text{OUT}} \, d\dot{\mathbf{u}}\_{\text{OUT}} + I\_{\text{OUT}} \, d\boldsymbol{u}\_{\text{OUT}} = \mathcal{C}\_{\text{R}} \boldsymbol{E}^{2} \, d\boldsymbol{f},\tag{22}$$

$$d\mu\_{\rm OUT} = d i\_{\rm OUT}(\mathbb{R}/\langle 1/\omega|\mathcal{C}\_{\rm F}) \tag{23}$$

$$I\_{\rm OUT} = \mathcal{U}\_{\rm OUT} / \mathcal{R} \tag{24}$$

$$\text{It is found that } \frac{du\_{OUT}}{df} = \frac{E}{2} \sqrt{\frac{\mathbb{C}\_R R}{f}} \frac{1}{1 + \omega \mathbb{C}\_F R/2}. \tag{25}$$

This expression represents the transfer function of the control system and represents the law according to which the operating frequency must be changed when the load parameters change in order to stabilize the constant output voltage.
