*3.2. Modes of Operation*

*Mode I [t0* < *t* ≤ *t1]***:** This time interval corresponds to the first half of the voltage boosting mode with the duty cycle of *Db*/2. A positive voltage is applied to the transformer primary winding by turning on the switches *S*1 and *S*4 and the secondary winding voltage equals *nVIN*. For better understanding, the equivalent circuit of this interval is given in Figure 6a. The resonance capacitor voltage has a minimum value of - Δ *VCr*/2 at the time instant *t0*. Also, the initial resonance current is nearly zero prior to turning on the switch *Q*1 that will be turned on at nearly zero current switching (ZCS) conditions. The voltage of the resonant capacitor assists the secondary winding voltage to accelerate the current charging of the resonant inductor, similar to the conventional boost converter. On the state-plane, the voltage of the resonant capacitor moves from point *A* to point *B* during this time interval with roughly sinusoidal shape, as shown in Figure 5. The state variable equations for the resonance current and voltage can be expressed in time domain as in Equations (4) and (5), respectively.

$$i\_{Llk}(t) = \frac{r\_1}{Z\_T} \sin(\pi - \omega\_r(t - t\_0)),\tag{4}$$

$$v\_{\rm Cr}(t) = nV\_{\rm IN} + r\_1 \cos(\pi - \omega\_r (t - t\_0)),\tag{5}$$

$$r\_1 = nV\_{IN} + \frac{\Delta V\_{Gr}}{2},\tag{6}$$

where *r*1 refers to the radius of the trajectory arc segmen<sup>t</sup> with center at (*nVIN*; 0), ω*r* = 2π *fr* is the angular resonance frequency in rad/s.

**Figure 6.** Equivalent circuit of the resonant tank operation during: (**a**) Mode I and (**b**) Mode II.

*Mode II [t1* < *t* ≤ *t2]***:** The switch *Q*1 is switched o ff and the bidirectional switch current *Iac* drops to zero. The stored energy in the resonant inductor is releasing directly to the load, as shown in Figure 6b. The diode *D*1 begins to conduct as it has a forward-biased state. The current of the resonant inductance is still flowing in the same direction with reversed voltage polarity, that is, it equals ( *VOUT*/2 − *nVIN*) at the instant *t*1. During this interval, the resonant inductor resonates with the resonant capacitor, and therefore, the resonance capacitor voltage moves from point B to point C along the trajectory curve. The length of this path is represented by the angle β (in rad). The capacitor voltage reaches its maximum value at Δ *VCr*/2, and the resonant current reaches zero at the instant *t*2. Equations (7)–(9) describe the converter operation in this mode before the resonant current drops to zero.

$$i\_{Llk}(t) = \frac{r\_2}{Z\_r} \sin(\beta - \omega\_r(t - t\_1)),\tag{7}$$

$$v\_{\rm Cr}(t) = nV\_{\rm IN} - \frac{V\_{\rm OUT}}{2} + r\_2 \cos(\beta - \omega\_7 (t - t\_1)),\tag{8}$$

$$r\_2 = -nV\_{IN} + \frac{V\_{OUT}}{2} + \frac{\Delta V\_{Cr}}{2},\tag{9}$$

where *r*2 refers to the radius of the arc trajectory segmen<sup>t</sup> with center at ((*nVIN* − *VOUT*/2); 0).

*Mode III [t2* < *t* ≤ *t3]***:** The resonant current equals zero as all the stored energy is released into the load in the previous mode, and the diode *D*1 is turned o ff at ZCS. Consequently, the converter enters the discontinuous conduction mode (DCM) and no energy is transferred to the load during this mode. The capacitor voltage remains constant at its maximum value at Δ *VCr*/2 until the end of this period at *TSW*/2.

*Mode IV [t3* < *t* ≤ *t4]***:** This refers to the dead-time interval and the primary switches (*S*1, *S*4) are turned o ff at *t*3. The magnetizing current denotes the circulating current referred to the primary winding to charge/discharge the parasitic output capacitance of switches (*S*1, *S*4) and (*S*2, *S*3), respectively. Therefore, the voltage across the switches (*S*2, *S*3) equals zero before the time instant *t*4. The values of the dead time and the magnetizing inductance are interdependent and must ensure full discharging of the switch parasitic capacitances. This allows the primary switches to be turned on at zero voltage switching (ZVS) conditions.

*Mode V [t4* < *t* ≤ *t0]***:** This represents the negative half-cycle of the switching period. The converter operates similar to that during time interval [*t*0; *t*4].

## *3.3. DC Voltage Gain Derivation*

The segments of the state-plane trajectory of the resonant tank that correspond to the positive and negative half-cycles are symmetric. Therefore, only the trajectory segmen<sup>t</sup> A-B-C is considered to derive the DC voltage gain expression for the proposed converter. The general expressions for the circle radius in Modes I and II are given by Equations (10) and (11), respectively. The two circles intersect at point B, which results in (12).

$$r\_1^2 = (v\_{\rm Cr} - nV\_{\rm IN})^2 + (Z\_r i\_{\rm Llk})^2,\tag{10}$$

$$r\_2^2 = \left(v\_{\rm Cr} - nV\_{\rm IN} + \frac{V\_{\rm OUT}}{2}\right)^2 + \left(Z\_{\rm r}i\_{\rm Llk}\right)^2,\tag{11}$$

$$\left(v\_{\rm Cr}(t\_1) - nV\_{\rm IN}\right)^2 + \left(Z\_r i\_{\rm Llk}(t\_1)\right)^2 - r\_1^2 = \left(v\_{\rm Cr}(t\_1) - nV\_{\rm IN} + \frac{V\_{\rm OUT}}{2}\right)^2 + \left(Z\_r i\_{\rm Llk}(t\_1)\right)^2 - r\_2^2.\tag{12}$$

The resonant inductor current and resonant capacitor voltage at the instant *t*1 can be given as:

$$i\_{Llk}(t\_1) = \frac{r\_1}{Z\_r} \sin(\omega\_{l'} t\_1) \tag{13}$$

$$v\_{\mathbb{C}r}(t\_1) = nV\_{IN} + r\_1 \cos(\omega\_r t\_1) \tag{14}$$

The resonance path angle β is derived from Equations (7) and (13) as follows:

$$\beta = \pi - \sin^{-1}\left(\frac{r\_1}{r\_2}\sin(\omega\_r t\_1)\right). \tag{15}$$

Then, by substituting *t*1 = *DbTSW*/2 in Equations (13)–(15), the cumulative duty cycle of the converter can be expressed as:

$$D\_b = \frac{2\cos^{-1}\left(\frac{\frac{T\_{SW}P\_{OUT}}{4C\_r}(4-G\_n) + nV\_{IN}V\_{OUT}}{nV\_{IN}V\_{OUT} + \frac{T\_{SW}P\_{OUT}}{4C\_r}G\_n}\right)}{\alpha\_r T\_{SW}}\tag{16}$$

where *Gn* = *VOUT nVIN* is the normalized DC voltage gain.

It follows from Equation (16) that the duty cycle depends not only on the level of the converter output power but also on the resonance tank parameters.

For the topology in [18], the duty cycle *Db* is given in Equation (17).

$$D\_b = \frac{2L\_{lk}\sqrt{\frac{2P\_{OUT}T\_{SW}}{V\_{OUT}C\_r}(V\_{OUT} - nV\_{IN})}}{Z\_r n V\_{IN}T\_{SW}}.\tag{17}$$

#### *3.4. Comparison of DC Voltage Gain and Input Operating Range*

This section provides a comparison between the proposed and the baseline [18] topologies. The main feature of the proposed topology is the input voltage regulation in a wide voltage and power range. The superiority over the baseline SRC topology is achieved as the proposed converter can provide much higher DC voltage gain at the same duty cycle *Db*, as shown in Figure 7a. Moreover, the proposed converter is much less sensitive to the value of the resonant inductor, while the baseline SRC topology shows a strong dependence of the available voltage and power regulation range on the resonant inductance value. Based on Equation (16), the operating range limit is shown in Figure 7b where it is compared to the target operating range defined by the maximum input current *IINm* = 20 A, the maximum input power *PINm* = 300 W, and the maximum input voltage *VINm* = 30 V, while the minimum input voltage is limited to 10 V to limit the converter power loss. The given target operating range is typical for module-level PV applications, as the interface converter should be capable of operating under partial shading, when the global maximum power point can occur at voltage as low as 10 V. The area highlighted with yellow color shows the region where the baseline SRC topology from [18] cannot operate as the limiting line is drawn theoretically for a critical case of *Db* = 0.8. In ractice, the duty cycle value is always below unity *Db* < 1, which means that experimental regulation range of the baseline topology will be even more limited than that in Figure 7b.

**Figure 7.** Comparison between the proposed topology and [18] in terms of (**a**) the DC voltage at *PIN* = 300 W and (**b**) target and feasible input voltage and power ranges for *Lr* = 100 μH.
