A Detailed Analysis and Gain Derivation of Reconfigurable Voltage Rectifier-Based LLC Converter

Abstract

In this paper, a complete analysis of an LLC resonant converter with a customized rectifier structure is presented. The converter is intended for wide, low-input, high-output voltage DC bus applications. The performance of the converter is assessed using comprehensive time-domain and fundamental harmonic approximation (FHA), which demonstrates its capacity to operate across an ample range of voltages by precisely adjusting the rectifier structure. The converter’s capability is illustrated by deriving and discussing detailed mode operation, steady-state analysis, and DC gain equations. In order to verify the theoretical analysis, a prototype with a power output of 250 watts is constructed and subjected to testing. The results of the testing demonstrate that the converter is both feasible and effective. The experimental findings illustrate its capacity to manage vast voltage ranges while upholding high efficiency. In addition, the converter utilizes a frequency switching modulation (FSM) to connect with a photovoltaic (PV) panel and control the high output voltage. This demonstrates its adaptability in renewable energy applications. The validation is in accordance with theoretical predictions, demonstrating the converter’s high-efficiency performance and versatility.

1. Introduction

With the ongoing increase in global energy consumption, primarily due to industrial expansion and higher load needs, the drawbacks of traditional grid systems that rely on fossil fuels have become more evident. Around three billion people worldwide have limited access to energy, and more than one billion have no access to electricity [1]. Given that more than $80\%$ of the world’s energy is now obtained from finite and polluting fossil fuels, the demand for sustainable alternatives is exceptionally urgent [2]. The primary objective of the International Renewable Energy Agency (IRENA) is to achieve a significant increase in renewable energy generation, specifically from solar and wind power, to reach $85\%$ by the year 2050 [3]. This transition is additionally reinforced by the increasing embrace of hybrid renewable energy systems (HRES), which integrate photovoltaic (PV) panels, wind turbines, and fuel cells. These systems mitigate the environmental issues linked to fossil fuels and provide a dependable and cost-efficient solution for supplying electricity to isolated and rural regions where expanding the traditional power grid is difficult. Incorporating these sustainable sources is crucial for fulfilling the increasing energy requirements while reducing environmental harm [4,5,6].

To address electricity access issues, photovoltaic (PV) energy systems optimize energy storage, control systems, and DC–DC converters for microinverters to maximize energy collection and consumption. PV wattrouters need to track voltage and current waveforms to prevent surplus electricity from returning to the grid to manage residential power flow [7,8,9,10]. Furthermore, the PV tilt and azimuth angles have a significant impact on the amount of solar radiation that reaches the PV modules, thus directly affecting energy output [11]. PV systems can be categorized as grid-tied, grid-interactive, or off-grid systems, with more than $90\%$ of them being connected to the electrical grid [12]. Microinverters have several advantages over other types of inverters. They can operate effectively even when there is partial shade, give flexibility, and provide better safety. Nevertheless, the sporadic characteristics of PV systems need the use of energy storage devices [13,14]. The integration of PV panels involves the utilization of DC–DC converters such as the Dual Active Bridge (DAB) converter. This converter is employed to transform the low and fluctuating voltage generated by PV panels into a high and stable voltage. Although DAB converters offer advantages, they encounter efficiency obstacles in large voltage range applications because of limited soft-switching ranges and substantial turn-off losses [15,16,17]. The LLC resonant converter, renowned for its Zero-Voltage Switching (ZVS) and Zero-Current Switching (ZCS) characteristics, provides a straightforward, high-amplification, and high-power-density solution for a broad range of voltage applications in PV systems and high-voltage batteries [18,19,20].

The effectiveness and dependability of DC–DC converters in power electronic transformers (PETs) are significantly impacted by the design and performance of the preceding multilevel converter. An efficiently designed multilevel converter improves the quality of the input voltage, decreases harmonic distortion, and enables better control of voltage and current. These factors collectively enhance the dynamic response and stability of the system, even when the load conditions change. The multilevel converter increases efficiency and improves operational effectiveness and power transfer capacity by decreasing stress on components and enabling soft switching. This makes it an essential component in high-power applications and contributes to the overall performance of PET systems [21,22,23].

Numerous LLC resonant topologies have been described in [19,24,25,26,27,28,29,30] to improve efficiency and overcome the constraints of traditional LLC converters in applications with a wide range of voltages. Ren et al. have introduced a Sigma DC/DC converter that integrates a conventional LLC converter as a DC transformer with two multi-resonant converters for voltage regulation. This architectural design effortlessly switches between DCX and D2D modes based on input voltage range, optimizing efficiency and adaptability [24]. In [19], the authors examine an LLC converter circuit specifically designed to achieve ZVS. This circuit includes a voltage doubler rectifier and a variable transformer turn-ratio, which allows for three configurations depending on the input voltage levels. The topology proposed by Elezab et al. in [25] uses a split resonant capacitor and adjustable secondary component, allowing for parallel mode for low output voltages and series mode for high output voltages. This design keeps operation near to the resonance frequency, reducing strain on the components and improving efficiency. In [26], the author presents a modified two-channel LLC resonant converter with two primary power switches, resonant inductors, and capacitors. This design maintains high voltage gain and performance regardless of input circumstances. This circuit’s layout provides low input impedance at low voltages for high gain and high input impedance at high volts for low gain. The author in [27] introduces a bidirectional three-level T-type LLC resonant isolated DC–DC converter that utilizes a new three-degrees-of-freedom (3DOF) modulation technique. The converter employs phase-shift control, variable frequency modulation, and pulse width modulation to perform soft switching over a broad voltage ratio range and under low loads, thereby greatly improving efficiency compared to conventional converters. The paper in [28] presents a new multi-mode rectifier-based LLC resonant converter that includes a new rectifier construction capable of operating in three different modes. The rectifier architecture enables a substantial increase in voltage amplification while simultaneously restricting the range of switching frequencies, resulting in enhanced efficiency. The converter described in [29] has two HB switching networks on the primary side. On the secondary side, there is a rectification stage that uses two secondary-side windings of transformers connected in series. The converter accomplishes ZVS and ZCS by employing fixed-frequency operation with phase-shift control, thereby eliminating losses. In [30], a dual-bridge LLC resonant converter incorporates a buck-boost converter in the front stage to accommodate higher input voltage. This, in turn, enhances the transformer turns ratio and decreases primary conduction losses. In addition, the converter employs full-bridge and half-bridge structures to efficiently function with different input voltages. The auxiliary switches are employed to regulate the functioning of the converter, facilitating the transfer from full-bridge to half-bridge modes.

The paper in [31] presents a novel LLC converter that incorporates a reconfigurable rectifier. The converter is specifically built to cater to applications that require wide input voltage ranges and high output voltage. The converter effectively addresses the efficiency challenges of conventional wide-range converters by incorporating three distinct rectifier structures: voltage-doubler (VDR), voltage-tripler (VTR), and voltage fifth-folder (VFR). These structures adapt based on input voltage levels, narrowing the switching frequency range. The design incorporates advanced features such as zero-voltage switching (ZVS) in power MOSFETs and zero-current switching (ZCS) in power diodes. These features effectively minimize voltage stress on rectifier components and promote overall efficiency.

This paper extends the work presented in [31] and introduces several key contributions. The time-domain approach is utilized to provide a comprehensive examination of voltage gain. In addition, it illustrates the waveforms during mode transitions. Furthermore, it examines the voltage amplification of the reconfigurable voltage rectifier-based LLC converter using time-domain analysis and compares it with frequency harmonic analysis (FHA) approaches. Theoretical method validation is achieved by the presentation of experimental results, which showcase the time-domain modes and the two operational stages. Furthermore, the transition mode is demonstrated capable of maintaining a constant output voltage during the presence of fluctuations in the input voltage. The paper additionally provides a comprehensive analysis of converter efficiency across various input voltages, emphasizing the benefits of adjusting between modes.

The paper is organized in the following manner: Section 2 introduces the time-domain analysis and explores the potential signal shapes for the converter. Section 3 provides a detailed analysis of each stage, including a mathematical model for each. Next, in Section 4, the DC gain equations and characteristic curves of the LLC Converter based on a reconfigurable voltage rectifier are derived. The experimental results are presented in Section 5, while the conclusions are provided in Section 6.

2. Time-Domain Analysis

The proposed converter, shown in Figure 1, is an LLC converter that is equipped with a reconfigurable rectifier. It is specifically designed to work effectively across a wide variety of input and output voltages, while also retaining a limited spectrum of switching frequencies [31]. The system incorporates three types of rectifiers: a voltage-doubler rectifier (VDR), a voltage-tripler rectifier (VTR), and a voltage fifth-folder rectifier (VFR). These rectifiers can be switched automatically depending on the input voltage, resulting in a significant increase in voltage and a reduction in voltage stress on the components [31].

With the intention of simplifying the study of the reconfigurable voltage rectifier-based LLC Converter shown in Figure 1, we will assume that the four primary switches are ideal and will disregard the dead time instead. Moreover, the symbol T denotes the specific number that characterizes the rectifier circuit. The VDR configuration corresponds to a rectifier of type 2, the VTR configuration corresponds to a rectifier of type 3, and the VFR configuration corresponds to a rectifier of type 5. Figure 2, Figure 3 and Figure 4 present an analysis of the waveforms of the converter when it is operating in a steady state. The converter that is being presented can function in five different modes, which are depicted in Figure 2, Figure 3 and Figure 4 and are described in more depth below:

In P-mode, the tank voltage is equal to the input voltage, and the voltage across the magnetizing inductance is positively clamped to the output voltage. This is determined by the turns ratio and rectifier type, as shown in Figure 5. As a consequence, a sinusoidal resonant current with a positive value is produced.

$P_N$-mode occurs when the tank voltage becomes negative and the voltage across the magnetizing inductance is clamped to the output voltage in a negative manner. This is determined by factors such as the turns ratio and rectifier type. In $P_N$-mode, the resonant current takes the shape of a negative sine wave, as depicted in Figure 6.

In N-mode, shown in Figure 7, there is a resonant tank that is similar to P-mode. However, in N-mode, the voltage of the magnetizing inductance is clamped negatively to the output voltage. This is influenced by the turns ratio and the type of rectifier used.

O-mode refers to the operational state of the converter when the output power becomes zero, resulting in the elimination of the impact of the output voltage on the magnetizing inductance, as depicted in Figure 8. In addition, the voltage of the tank is clamped to the input voltage in a positive manner.

$O_N$-mode occurs when the input voltage of the resonant tank in O-mode becomes negative, causing a negative voltage across the input, as shown in Figure 9.

The steady-state waveforms, shown in Figure 2, Figure 3 and Figure 4, of the reconfigurable converter can be classified into two shapes: $OPO{P}_N$ and $PN{P}_N{O}_N$ based on the switching frequency of $Q_1$ and $Q_2$. The following is a description of these waveform shapes:

-

The waveform shape $OPO{P}_N$ happens when the switching frequency matches the resonant frequency. This shape is represented in Figure 2.

-

The waveform shape $PN{P}_N{O}_N$ is depicted in Figure 3 for VTR and Figure 4 for VFR, showing the steady-state waveforms. The steady-state waveforms of converter exhibit the $PN{P}_N{O}_N$ shape when the switching frequency exceeds the resonant.

3. Stage by Stage Analysis

To determine the gain occurring in the converter, a step-by-step examination is necessary. The study presupposes that the principal switches are in their optimal state and disregards any delay in the system. Each waveform, namely $OPO{P}_N$ and $PN{P}_N{O}_N$, consists of four primary stages (I–IV) during a whole cycle, as depicted in Figure 2, Figure 3 and Figure 4. The subsequent paragraph provides a concise description of the four stages that occur during a switching cycle when the converter runs at a resonant frequency $\left(OPO{P}_N\right)$. Furthermore, each stage description includes the equations for the resonant inductor current $\left(i_{L_r}\right)$, resonant capacitor voltage $\left(v_{c_r}\right)$, and magnetizing inductor current $\left(i_{L_m}\right)$.

Stage I: [$t_0$, $t_1$). The switching cycle initiates with the O-mode due to the fact that the magnetizing voltage does not first attain the value of $n{V}_o/t$. At this stage, the resonant frequency is equal to the value of the magnetizing current. The resonant tank is subjected to a positive voltage. In addition, the secondary side is disconnected, resulting in all rectifier diode currents being zero.

$$v_c\left(t\right)=i_{L_r}\left(t_0\right)Z_{m}sin\left[{\omega }_m(t-t_0)\right]+\left(V_c\left(0\right)-V_{in}\right)cos\left[{\omega }_m(t-t_0)\right]+V_{in}$$

(1)

$$i_{L_r}\left(t\right)=i_m\left(t\right)=i_{L_r}\left(t_0\right)cos\left[{\omega }_m(t-t_0)\right]-{\displaystyle \frac{V_c\left(0\right)-V_{in}}{Z_m}}sin\left[{\omega }_m(t-t_0)\right]$$

(2)

where $i_{L_r}\left(t_0\right)$, $V_c\left(t_0\right)$, and $i_m\left(t_0\right)$ are the initial inductor current, capacitor voltage, and magnetizing current, respectively, ${\omega }_m=1/\sqrt{\left(L_r+L_m\right)C_r}$ and $Z_m=\sqrt{\left(L_r+L_m\right)/C_r}$.

Stage II: [$t_1$, $t_2$). Once the magnetizing voltage reaches its maximum, the converter is capable of working in P mode, causing the resonant current signal to transform to a sinusoidal waveform. The resonant tank maintains a positive voltage while the magnetizing current remains below the resonant current. During this stage, the secondary side of the circuit is connected, resulting in the conduction of the rectifier diodes. In addition, the currents reach their maximum value as they pass through $D_2$ and $D_7$.

$$v_c\left(t\right)=i_{L_r}\left(t_1\right)Z_{r}sin\left[{\omega }_r(t-t_1)\right]+\left({\displaystyle \frac{n{V}_o}{T}}+V_c\left(t_1\right)-V_{in}\right)cos\left[{\omega }_r(t-t_1)\right]+V_{in}-{\displaystyle \frac{n{V}_o}{T}}$$

(3)

$$i_{L_r}\left(t\right)=i_{L_r}\left(t_1\right)cos\left[{\omega }_r(t-t_1)\right]-{\displaystyle \frac{\frac{n{V}_o}{T}+V_c\left(t_1\right)-V_{in}}{Z_r}}sin\left[{\omega }_r(t-t_1)\right]$$

(4)

$$i_m\left(t\right)=\int {\displaystyle \frac{V_m}{L_m}}dt={\displaystyle \frac{n{V}_o}{T{L}_m}}(t-t_1)+i_m\left(t_1\right)$$

(5)

Stage III: [$t_2$, $t_3$). Due to the insufficient reversal of the magnetizing voltage, the converter must operate in O mode once more. As a result, when the rectifier diodes are turned off, the secondary circuit is disconnected from the primary circuit.

$$v_c\left(t\right)=i_{L_r}\left(t_2\right)Z_{m}sin\left[{\omega }_m(t-t_2)\right]+\left(V_c\left(t_2\right)-V_{in}\right)cos\left[{\omega }_m(t-t_2)\right]+V_{in}$$

(6)

$$i_{L_r}\left(t\right)=i_m\left(t\right)=i_{L_r}\left(t_2\right)cos\left[{\omega }_m(t-t_2)\right]-{\displaystyle \frac{V_c\left(t_2\right)-V_{in}}{Z_m}}sin\left[{\omega }_m(t-t_2)\right]$$

(7)

Stage IV: [$t_3$, $t_4$). During the second half of the switching cycle, the converter mainly functions in $P_n$ mode due to the ability of the magnetizing voltage to sustain its negative value, $-nVo/t$. At this stage, the voltage of the resonant tank is adjusted to match the negative input voltage. On the secondary side, $D_1$ and $D_6$ begin to conduct current and reach their maximum value in the middle of this stage until they turn off at the end of the P mode.

$$v_c\left(t\right)=i_{L_r}\left(t_3\right)Z_{r}sin\left[{\omega }_r(t-t_3)\right]+\left({\displaystyle \frac{-n{V}_o}{T}}+V_c\left(t_3\right)+V_{in}\right)cos\left[{\omega }_r(t-t_3)\right]-V_{in}+{\displaystyle \frac{n{V}_o}{T}}$$

(8)

$$i_{L_r}\left(t\right)=i_{L_r}\left(t_3\right)cos\left[{\omega }_r(t-t_3)\right]-{\displaystyle \frac{\frac{-n{V}_o}{T}+V_c\left(t_3\right)+V_{in}}{Z_r}}sin\left[{\omega }_r(t-t_3)\right]$$

(9)

$$i_m\left(t\right)=\int {\displaystyle \frac{V_m}{L_m}}dt={\displaystyle \frac{-n{V}_o}{T{L}_m}}(t-t_3)+i_m\left(t_3\right)$$

(10)

The following paragraph presents the equations for the current in the resonant inductor $\left(i_{L_r}\right)$, the voltage across the resonant capacitor $\left(v_{c_r}\right)$, and the current in the magnetizing inductor $\left(i_{L_m}\right)$ for every stage in $PN{P}_N{O}_N$. Furthermore, a detailed explanation of the behavior of the primary voltages and current during this switching cycle.

Stage I: [$t_0$, $t_1$). The start of this stage occurs when switches $Q_1$ and $Q_4$ are activated using soft switching. The amplitude of the resonant current exceeds that of the magnetizing current, resulting in the formation of a sinusoidal waveform. The resonant tank’s input voltage reaches its maximum, which is equivalent to the input voltage, while the voltage across the magnetizing inductor is equal to the total output voltage on the primary side. During VTR mode, the currents pass through diodes $D_2$, $D_3$, and $D_7$ on the secondary side. In VFR mode, the currents pass through diodes $D_3$, $D_5$, and $D_7$ on the secondary side. The cycle concludes when the tank voltage switches to a negative input voltage, resulting in the deactivation of $D_2$, $D_3$, $D_5$, and $D_7$.

$$v_c\left(t\right)=i_{L_r}\left(t_0\right)Z_{r}sin\left[{\omega }_r(t-t_0)\right]+\left({\displaystyle \frac{n{V}_o}{T}}+V_c\left(0\right)-V_{in}\right)cos\left[{\omega }_r(t-t_0)\right]+V_{in}-{\displaystyle \frac{n{V}_o}{T}}$$

(11)

$$i_{L_r}\left(t\right)=i_{L_r}\left(t_0\right)cos\left[{\omega }_r(t-t_0)\right]-{\displaystyle \frac{\frac{n{V}_o}{T}+V_c\left(0\right)-V_{in}}{Z_r}}sin\left[{\omega }_r(t-t_0)\right]$$

(12)

$$i_m\left(t\right)=\int {\displaystyle \frac{V_m}{L_m}}dt={\displaystyle \frac{n{V}_o}{T{L}_m}}(t-t_0)+i_m\left(t_o\right)$$

(13)

Stage II: [$t_1$, $t_2$). Due to the fact that the duration of switches $Q_1$ and $Q_4$ are less than half of the resonant time, Stage I is not fully completed, resulting in the resonant current being unequal to the magnetizing current at $t_1$. Resonant current begins to diminish at Stage II. The input voltage of the resonant tank is negative, while the magnetizing voltage remains constant. The currents that are flowing through $D_6$ for VTR mode and $D_4$ and $D_6$ for VFR mode are gradually intensifying.

$$v_c\left(t\right)=i_{L_r}\left(t_1\right)Z_{r}sin\left[{\omega }_r(t-t_1)\right]+\left({\displaystyle \frac{n{V}_o}{T}}+V_c\left(t_1\right)+V_{in}\right)cos\left[{\omega }_r(t-t_1)\right]-V_{in}-{\displaystyle \frac{n{V}_o}{T}}$$

(14)

$$i_{L_r}\left(t\right)=i_{L_r}\left(t_1\right)cos\left[{\omega }_r(t-t_1)\right]-{\displaystyle \frac{\frac{n{V}_o}{T}+V_c\left(t_1\right)+V_{in}}{Z_r}}sin\left[{\omega }_r(t-t_1)\right]$$

(15)

$$i_m\left(t\right)=\int {\displaystyle \frac{V_m}{L_m}}dt={\displaystyle \frac{n{V}_o}{T{L}_m}}(t-t_1)+i_m\left(t_1\right)$$

(16)

Stage III: [$t_2$, $t_3$). This stage is analogous to Stage I, with the distinction that the resonant current reaches its entire cycle and becomes equal to the magnetizing current by the end of this stage. Furthermore, the input voltage that resonates remains negative, but the voltage across the magnetizing inductor undergoes an opposite change. The currents flowing through $D_6$ for VTR mode and $D_4$ and $D_6$ for VFR mode reach their peak values at the midpoint of the stage.

$$v_c\left(t\right)=i_{L_r}\left(t_2\right)Z_{r}sin\left[{\omega }_r(t-t_2)\right]+\left({\displaystyle \frac{-n{V}_o}{T}}+V_c\left(t_2\right)+V_{in}\right)cos\left[{\omega }_r(t-t_2)\right]-V_{in}+{\displaystyle \frac{n{V}_o}{T}}$$

(17)

$$i_{L_r}\left(t\right)=i_{L_r}\left(t_2\right)cos\left[{\omega }_r(t-t_2)\right]-{\displaystyle \frac{\frac{-n{V}_o}{T}+V_c\left(t_2\right)+V_{in}}{Z_r}}sin\left[{\omega }_r(t-t_2)\right]$$

(18)

$$i_m\left(t\right)=\int {\displaystyle \frac{V_m}{L_m}}dt={\displaystyle \frac{-n{V}_o}{T{L}_m}}(t-t_2)+i_m\left(t_2\right)$$

(19)

Stage IV: [$t_3$, $t_4$). At this stage, the rectifier diodes are deactivated since the resonant current is equivalent to the magnetizing current. The converter switches to a mode that limits the output current from flowing through the primary side.

$$v_c\left(t\right)=i_{L_r}\left(t_3\right)Z_{m}sin\left[{\omega }_m(t-t_3)\right]+\left(V_c\left(t_3\right)+V_{in}\right)cos\left[{\omega }_m(t-t_3)\right]-V_{in}$$

(20)

$$i_{L_r}\left(t\right)=i_m\left(t\right)=i_{L_r}\left(t_3\right)cos\left[{\omega }_m(t-t_3)\right]-{\displaystyle \frac{V_c\left(t_3\right)+V_{in}}{Z_m}}sin\left[{\omega }_m(t-t_3)\right]$$

(21)

4. Voltage Gain Analysis

Various techniques can be used to calculate the voltage gain (G), and each technique has its own advantages and disadvantages. Fundamental harmonic approximation (FHA) and time-domain analysis are commonly employed techniques to manage the gain of the LLC converter. The FHA approach, as described and examined in article [31], eliminates the higher harmonics of the input tank voltage ($v_{i_{ac}}$) and the output tank voltage ($v_{o_{ac}}$) in order to generate a pure sine wave. This paper will examine and discuss the process of determining the gain by time-domain analysis and thereafter compare it to the FHA technique.

The most precise approach for determining the voltage gains of the converter is through time-domain analysis. This approach is renowned for its accuracy since it examines the resonant inductor current, resonant capacitor voltage, and magnetizing inductor current in every mode of operation. Three assumptions are taken into account when studying the voltage gains in time-domain analysis: The dead time between switches $Q_1$ and $Q_2$ and $Q_3$ and $Q_4$ is disregarded; the output capacitors in VDR, VTR and VFR structures are excessively big to uphold a consistent output voltage; the passive and active components are assumed to be excellent. The output rectifier current can be obtained by using the following method.

$$I_o={\displaystyle \frac{n}{t_4-t_0}}{\int }_{t_0}^{t_4}\left(i_{L_r}\left(t\right)-i_{L_m}\left(t\right)\right)dt={\displaystyle \frac{T{V}_o}{R_o}}$$

(22)

The Equation (22) mentioned above is applicable only to the P, $P_N$, and N stages. This is because the resonant current is equal to the magnetizing current in the O and $O_N$ stages.

The time intervals in the steady-state waveforms of the converter in VDR, VTR, and VFR can be computed based on the notion of P, $P_N$, N, O and $O_N$ stages. The P stage begins at time $t_1$ and finishes at time $t_2$ for VDR mode, while it begins at time $t_0$ and ends at time $t_1$ for VTR and VFR modes. The $P_N$ stage exists during the last time interval from $t_3$ to $t_4$ in VDR mode, and from $t_2$ to $t_3$ in VTR and VFR modes. Alternatively, the O stage occurs throughout the time intervals $t_0$ to $t_1$ and $t_2$ to $t_3$ in VDR mode. In VTR and VFR mode, the $O_N$ stage is present from $t_3$ to $t_4$. The converter transitions to stage N in both VTR and VFR mode between time $t_1$ and $t_2$. The time interval following $t_4$ in all modes exhibits symmetrical characteristics to the time interval between $t_0$ and $t_4$. This property yields the following conditions:

$$\left\{\begin{array}{c}{i}_{L_m}\left(t_4\right)=i_{L_m}\left(t_0\right)\hfill \\ i_{L_r}\left(t_2\right)=i_{L_m}\left(t_2\right)\hfill \\ v_{C_r}\left(t_4\right)=-v_{C_r}\left(t_0\right)\hfill \\ i_{L_r}\left(t_4\right)=i_{L_m}\left(t_0\right)\hfill \end{array}\right.$$

(23)

Analyses are used to determine the initial values of the resonant inductor current, resonant capacitor voltage, and magnetizing inductor by exploiting the continuity between the modes of operation. The initial currents $i_{L_r}\left(t_0\right)$ and $i_{L_m}\left(t_0\right)$ can be computed using the following equations.

$$i_{L_r}\left(t_0\right)=i_{L_m}\left(t_0\right)=-{\displaystyle \frac{n{t}_r{V}_o}{4L_{m}T}}$$

(24)

The initial voltage of the capacitor, $v_{C_r}\left(t_0\right)$, can be calculated by substituting the formulae for the resonant current and magnetizing current into the equation for the output current, Equation (22).

$$v_{C_r}\left(t_0\right)=-{\displaystyle \frac{\frac{t_r(n{V}_o-T{V}_{in})}{T}+\frac{Z_r{t}_s\pi T{V}_o}{2n{R}_o}}{t_r}}$$

(25)

To get the voltage gain of the converter (G), one can substitute the beginning values of $i_{L_r}\left(t_0\right)$, $i_{L_m}\left(t_0\right)$, and $v_{C_r}\left(t_0\right)$ into the operating stage equations of P, $P_N$, N, O and $O_N$ stages as follows:

$$G={\displaystyle \frac{2}{cos\left[w_m\left(\frac{t_r}{2}-\frac{t_s}{2}\right)\right]\left(\frac{2n}{T}-\frac{Z_r\pi (\frac{T{t}_s}{2n{R}_o}+\frac{n{t}_r}{T{Z}_r\pi })}{t_r}\right)+\frac{Z_r\pi (\frac{T{t}_s}{2n{R}_o}+\frac{n{t}_r}{T{Z}_r\pi })}{t_r}+\frac{n{t}_r{Z}_{r}sin\left[w_m\left(\frac{t_r}{2}-\frac{t_s}{2}\right)\right]\sqrt{m+1}}{4L_{m}T}}}$$

(26)

The voltage gain of the converter (G) with different rectifier structures can be illustrated in Figure 10 by utilizing Equation (26).

In the analysis of FHA, the quality factor (Q) indicates the extent to which the voltage gain is increased or decreased. The AC equivalent resistance $R_{a{c}_T}$, which has an inverse relationship with Q, fluctuates depending on the rectifier construction. Therefore, the Q values for the three rectifier designs can be computed in the following manner:

$$Q={\displaystyle \frac{2\pi L_r{f}_r}{R_{a{c}_T}}}=\left\{\begin{array}{cc}{\displaystyle \frac{{\pi }^3{L}_r{f}_r}{n^2{R}_o}}\left(\mathrm{VDR}\mathrm{mode}\right)\hfill & \\ {\displaystyle \frac{3{\pi }^3{L}_r{f}_r}{2n^2{R}_o}}\left(\mathrm{VTR}\mathrm{mode}\right)\hfill \\ {\displaystyle \frac{5{\pi }^3{L}_r{f}_r}{2n^2{R}_o}}\left(\mathrm{VFR}\mathrm{mode}\right)\hfill \end{array}\right.$$

(27)

For a comparison between the time-domain method and FHA method, the voltage gain of the converter is depicted in Figure 11 using both approaches. The time-domain technique can provide precise gain when the converter runs within the frequency range of $f_r$ and $f_m$. Nevertheless, the accuracy of the time-domain technique declines as the converter operates beyond resonance and enters the capacitive zone. Conversely, the FHA method is a method that provides an estimation of the voltage gain of the LLC resonant converter. Thus, the accuracy of the FHA technique is much inferior to that of the time-domain method.

5. Experimental Results

An experimental prototype with a power output of 250 W has been created to verify the functioning and efficiency of the reconfigurable rectifier LLC converter for PV microinverters designed for high DC bus applications.

Table 1. Specifications of the executed prototype.

Parameter	Value/Description
Input voltage	$V_{in}=$ 25–100 V
Output Voltage	$V_o=500$ V
Rated Power	$P_o=250$ W
Resonant frequency	$f_r=100$ kHz
Switching frequency	$f_s=$ 69–113 kHz
Resonant inductor	$L_r=18$ μH
Resonant capacitor	$c_r=144$ nF
Magnetizing inductor	$L_m=60$ μH
Output capacitor	100 nF
Transformer	$n=37:100$
Primary MOSFET	IPB320N20N3
Secondary MOSFET	FCB070N65
Rectifier Diode	VS8ETH03

provides a concise overview of the primary specifications and design parameters. The four main switches are regulated by FSM signals with a duty cycle of $50\%$ and a dead time of 100 nanoseconds. The prototype employs the $STM32F334K8T6$ microprocessor to regulate the PWM signals for $Q_1–Q_4$ and the gate signals for $Q_5$ and $Q_6$. Figure 12 depicts the laboratory prototype.

Figure 13, Figure 14 and Figure 15 display the measurement waveforms of the output voltage ($v_o$), resonant current ($i_{L_r}$), gate-source voltage ($v_{gs}$), and drain-source voltage ($v_{ds}$) for $Q_1$ and $Q_2$ under various situations. As seen in Section 2, switches $Q_1$ and $Q_4$ have the same switching cycle and are complementary to switches $Q_2$ and $Q_3$.

Figure 13 illustrates the tank’s performance when it works in close proximity to resonance under typical load circumstances. In this case, the converter functions in VDR mode, where the output voltage on the rectifier side can be doubled. The resonant current exhibits a waveform shape known as an $OPO{P}_N$. When $Q_1$ is activated, the O mode initiates and becomes visible prior to the P-mode if the magnitizing voltage is insufficiently clamped to the load at the start of the half cycle. Following that, the P mode commences with a duration that is shorter than half of the switching frequency. Subsequently, the O mode reappears due to the absence of the voltage required to achieve completion of the P mode. During the second half of the cycle, the resonant tank can only operate in $P_N$ mode since the magnetizing voltage is sufficient to maintain the required voltage for completing the $P_N$ mode.

When the converter switches to VTR mode, the main parameters are displayed in Figure 14. Here, the resonant current initiates in the P mode when $Q_1$ and $Q_4$ are activated, and the magnetizing voltage is constrained to its highest voltage level. Subsequently, the N mode commences as the P mode cannot be accomplished due to the deactivation of $V_{gs}$. Furthermore, the $P_N$ mode begins anew, this time with a negative input voltage. The waveform concludes in the $O_N$ mode due to the inability of the magnetizing voltage to sustain its negative voltage.

Figure 15 shows that the switching time is generally shorter than the resonant time in VFR mode. As a result, the P mode cannot complete its stage and the resonant current does not match the magnetizing current when the $V_{gs}$ is deactivated. The second halves of all waveforms are being generated while the tank voltage is negative as a result of $V_{g{s}_2}$ and $V_{g{s}_3}$ being activated. The $P_N$ mode is capable of completing its cycle since it has enough time available. At the conclusion of the cycle, the $O_N$ mode occurs, while the magnetizing voltage cannot be clamped to a completely positive voltage.

The experimental results are used to draw the gain characteristics curve, which serves to verify the gain equations derived in Equation (26). Figure 16 displays the gain curves for three modes, namely VDR, VTR, and VFR modes, both experimentally and theoretically, at two different power levels: 100 W and 250 W. The figure demonstrates that the experimental findings closely align with the gain equations across a broad range, both below and above resonance. It is important to acknowledge that the precision of the gain equation decreases slightly toward the second resonant frequency, $f_m$. Overall, the resulting gain equations provide an effective indicator of the tank behavior for the reconfigurable rectifier-based LLC converter.

Figure 17 illustrates the gate signals of transistors $Q_5$ and $Q_6$, as well as the output voltage ($V_o$) and input voltage ($V_{in}$). The reconfigurable rectifier-based LLC converter adjusts the rectifier structure by selectively activating and deactivating $Q_5$ and $Q_6$ in response to variations in the input, with the goal of maintaining a stable output voltage of 500 V.

The efficiency curve for various input voltages is shown in Figure 18, Figure 19 and Figure 20. At an input voltage of 80 V, the converter functions in VDR mode. Figure 18 illustrates the effectiveness when the input voltage is set at 80. Figure 19 depicts an input voltage of 62 V and the converter is functioning in VTR mode. Nevertheless, the graph illustrates how the efficiency might be diminished when the converter runs in VDR mode at that specific input voltage. When the input voltage is 40, the efficiency decreases in VDR and VTR modes compared to VFR mode, as illustrated in Figure 20. Efficiency curves demonstrate the advantages of the converter by transitioning to a different mode as the input voltage begins to decline, resulting in excellent efficiency throughout a broad range of input voltages.

6. Conclusions

This study presents a thorough investigation of the steady-state behavior of the Reconfigurable Voltage Rectifier-Based LLC Converter when connected to a PV panel for applications requiring high DC voltage. The primary objective of the paper is to analyze the influence of switching frequency and rectifier structure on the converter. At first, there are five modes of functioning that are explained, providing specific information about the situations in which they occur. Next, a comprehensive step-by-step analysis is provided for the scenarios in which switches $Q_1$ and $Q_4$ are activated, as well as when switches $Q_2$ and $Q_3$ are activated. In addition, the gain equation for the converter is constructed, encompassing all essential equations, such as the initial voltage and current equations, output current, and symmetrical conditions. This paper presents a discussion on the accuracy differences between two gain derivation methods, aiming to provide a clear comparison between them. Moreover, the gain curve is examined across different parameters to comprehend the impact of each parameter on the converter’s gain. In order to verify the theoretical calculations, a prototype of the converter with a power rating of 250 watts is constructed. The experimental waveforms exhibit a high degree of similarity to the analysis offered, and the experimental gain curves correspond closely to the gain equations generated. Therefore, the developed gain equations can precisely forecast the gain characteristics of the converter. The gain characteristics and their variable dependence will provide significant insights for future studies when choosing design parameters.