A Low Q Three-Phase Series Resonant Converter for PV Applications

Abstract

In this paper, a soft-switched three-phase resonant converter with a low quality factor (Q) design is proposed for Photovoltaic (PV) energy harvesting systems with a very wide range of operating conditions. Due to the low quality factor (Q ≤ 1), the voltage stress across the capacitors is drastically reduced. However, a major challenge of using a low Q design is the wide switching frequency range required for operation over a wide range of load and input conditions. In order to reduce the switching frequency range, this paper introduces a hybrid modulation technique employing asymmetric pulse width modulation (APWM) along with variable frequency modulation. The proposed modulation scheme also substantially extends the soft-switching range of the converter from full-load upto 10–35% load condition over a wide range of line voltages. To sustain soft-switching in the entire operating region of the converter, the converter is operated in a hybrid structure with single-phase and burst modes at light load conditions. A comprehensive time-domain analysis of the proposed converter is presented, which greatly improves accuracy over conventional frequency-domain modeling. Experimental results from a 1 kW prototype are presented to verify the performance of the converter and validate the theoretical analysis.

1. Introduction

DC-DC converters are used in PV applications for the extraction of maximum power from PV panels by maintaining their operation at a maximum power point (MPP). As the characteristics of the PV panels are highly dependent on the environmental conditions, the DC-DC stage should be capable of operating efficiently over a wide range of line voltages and load conditions. Resonant power converters provide high performance for PV applications due to their capability to offer soft-switching over such a wide range of operating conditions [1,2,3,4].

Single-phase resonant converter topologies with various modulation techniques have extensively been used for PV applications [5,6,7,8,9,10,11,12]. Recent advancements in photovoltaic technology have increased the power generation capability of PV panels [13] and the ongoing research on different photovoltaic materials promises to improve the efficiency of PV cells [14,15]. The PV panels are often connected in parallel and series configurations for higher power output. As a result, the PV systems require converters with increased power handling capability. Single-phase resonant converters face severe component stresses when used at power levels exceeding 1 kW. Compared to single-phase resonant converters, three-phase resonant converters become a viable alternative since they offer reduced stress at high power levels. The three-phase converters also have reduced filter size requirements due to the higher frequency ripple in the input and output current. This results in increased power density of the converter.

Extensive research on three-phase resonant DC-DC converters focusing on different topologies and modulation techniques has been reported in the literature [16,17,18,19,20,21,22,23,24,25,26,27,28,29]. A three-phase LCC topology and LLC topology were discussed in [17,18], respectively, where variable frequency modulation was used for regulation. In both cases, a high-quality factor design was implemented. While a high Q design allows the converter to operate over a narrow switching frequency range for different operating conditions, it has the adverse effect of increased stress across the resonant components. Reference [19] implements a three-phase LLC topology with a low-quality factor; however, it requires two additional diodes to achieve wide-range operation. A fixed-frequency LLC converter with two modules was proposed in [20] where the output power is controlled by the phase-shift between the modules. This converter has the advantage of fixed-frequency operation allowing optimization of magnetic components but requires an additional boost transformer which makes the converter bulky. A three-phase three-level series resonant converter with asymmetric duty cycle control was proposed in [22]. However, ZVS for all the switches was maintained only till 40% load. At light load conditions, to maintain soft-switching and to improve efficiency three-phase resonant converters may also be operated in burst mode [24,30] or switched to single-phase operation [31].

Resonant converters can be modeled using frequency-domain or time-domain analysis methods. Frequency-domain analysis method has been predominantly used to model three-phase resonant converters [17,18,19,20,21]. In this method, a single-phase equivalent circuit of the converter is derived to carry out the analysis. Fundamental harmonic approximation (FHA) was used to model a three-phase LLC converter in [18]. This method simplifies the analysis and gives good results for converters with high quality factors. When the quality factor of the system is reduced, the converter waveforms become non-sinusoidal. For such systems, the analysis may consider multiple harmonics on the reflected load side and the inverter side to improve accuracy [17]. However, the frequency-domain technique fails to model the converter accurately in discontinuous conduction modes, as the reflected voltage across the transformer can no longer be defined properly. The accuracy of this method may be further affected when applied to three-phase resonant converters since to derive a single-phase equivalent circuit, transformer configurations in $Y-\Delta$ or $\Delta -Y$ have to be converted into $Y-Y$ configuration [21]. By doing so, the waveforms actually present in the converter including any phase shift between the secondary and the primary side waveforms are neglected in the analysis. In comparison, the time-domain analysis method while being more computation-intensive is capable of modeling resonant converters with high accuracy [29,32,33,34]. Modeling of a three-phase LLC converter using the time-domain method was attempted in [28]; however, closed-form steady-state solutions were not provided. Reference [29] provided time-domain solutions for a three-phase series resonant converter; however, all possible modes of operation were not discussed. This article provides a comprehensive time-domain model for a three-phase series resonant converter with a detailed analysis of the different operating modes of the converter.

In this paper, a three-phase series resonant converter with a low Q design is proposed for the DC-DC conversion stage in PV applications as shown in Figure 1. Compared to the state-of-the-art single-phase resonant converters, the three-phase resonant converter offers reduced component stress making it more suitable for high-power applications. In addition, the low Q design reduces the voltage stress across the resonant capacitors. To reduce the required switching frequency range for the low Q converter, a hybrid modulation technique is proposed. Under the proposed modulation, the converter operates with soft-switching over a wide range of load and input voltage conditions. By operating the converter in a hybrid structure, soft-switching is realized in the entire operating range of the converter. Figure 2 shows a diagram of the research flow.

2. Three-Phase Series Resonant Converter

Figure 1 shows the circuit diagram of a three-phase series resonant converter. The converter comprises six switches—$S_1$ to $S_6$ on the primary side of the transformer and a three-phase diode bridge ($D_1$ to $D_6$) on the secondary side along with a capacitive output filter. The three-phase HF transformer is connected in a $\Delta -Y$ configuration as the converter is utilized for a step-up application. This topology uses two-element LC series resonant tank which is simple to design and has almost constant efficiency from full load to light load conditions. However, the main drawback of this topology is the wide switching frequency range required to maintain regulation under a wide range of operating conditions when a low Q design is implemented. This paper uses a hybrid modulation strategy using asymmetric pulse width modulation along with variable frequency control to reduce the required frequency range. The switching frequency $f_{sw}$ of the converter is maintained above the resonant frequency $f_r$ to achieve zero voltage switching at turn-on. A phase shift of 120 degrees is maintained between the inverter legs for a balanced flow of currents in the individual phases. The upper switches $S_1,S_3,S_5$ are operated with a duty cycle $D\le 0.5$ while the remaining switches are operated with a complementary duty cycle $D^{\prime }=(1-D)$.

3. Steady-State Analysis of Converter

With a low-quality factor, the converter exhibits non-sinusoidal waveforms and the time-domain modeling is chosen to analyze the converter. The assumptions made for the analysis are detailed below.

• The semiconductor devices are considered to have ideal characteristics.
• The magnetic components and capacitors are considered to be ideal and hence their resistances are neglected in the analysis.
• The impact of dead time is neglected.

An impact of neglecting the resistance of the resonant components, the semiconductor device is a slight offset (<1%) between the calculated and observed gain as the voltage drops are neglected. However, this allows the gain equation to be much more simplified. Hence, the assumptions made allow the analysis to be simplified to a great extent without significant loss of accuracy.

3.1. Equivalent Circuit of the Converter

An equivalent circuit of the converter shown in Figure 3 is considered for the time-domain analysis. The three inverter legs are operated with identical duty cycles phase-shifted by 120°. As a result, the resonant currents through the three phases and the voltages across the resonant capacitors are identical and shifted in phase by 120°.

The circuit in Figure 3 can be described by the differential equations given below.

$$\begin{array}{cc}\hfill i_L\left(t\right)& =C_s\frac{d{v}_c\left(t\right)}{dt}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill L_s\frac{d{i}_L\left(t\right)}{dt}+v_c\left(t\right)& =\frac{1}{3}(2(v_1-v_3)+v_2-v_4)\hfill \end{array}$$

(2)

The resonant current $i_L\left(t\right)$ and the voltage across the capacitor $v_c\left(t\right)$ are found from Equations (1) and (2) as given below.

$$i_L\left(t\right)=i_L\left(t_i\right)\cos \left(w_{r}t\right)+\frac{1}{3Z_o}(2(v_1-v_3)+v_2-v_4-3v_c\left(t_i\right))\sin \left(w_{r}t\right)$$

(3)

$$v_c\left(t\right)=\frac{1}{3}(2(v_1-v_3)+v_2-v_4)(1-\cos \left(w_{r}t\right))+v_c\left(t_i\right)\cos \left(w_{r}t\right)+Z_o{i}_L\left(t_i\right)\sin \left(w_{r}t\right)$$

(4)

where $w_r=2\pi f_r$, $Z_o=\sqrt{\frac{L_s}{C_s}}$, and $t_i$ marks the time at the start of the specific interval.

Let $X\left(t\right)=\left[\begin{array}{c}{i}_L\left(t\right)\\ v_c\left(t\right)\end{array}\right]$ be the state vector, then from (3) and (4), we get the following expression.

$$X\left(t\right)=A(t-t_i)X\left(t_i\right)+\frac{1}{3}(2(v_1-v_3)+v_2-v_4)B(t-t_i)$$

(5)

where $A\left(t\right)=\left[\begin{array}{cc}\cos \left(w_{r}t\right)& \frac{-1}{Z_o}\sin \left(w_{r}t\right)\\ Z_o\sin \left(w_{r}t\right)& \cos \left(w_{r}t\right)\end{array}\right]$ and $B\left(t\right)=\left[\begin{array}{c}\frac{1}{Z_o}\sin \left(w_{r}t\right)\\ 1-\cos \left(w_{r}t\right)\end{array}\right]$.

Depending on the duty cycle, frequency and load, the converter may exhibit different modes of conduction. The following sections describe these modes in detail. The secondary currents of the converter can be found as $i_{s1}=\frac{n(i_{pr1}-i_{pr2})}{3},i_{s2}=\frac{n(i_{pr2}-i_{pr3})}{3}$ and $i_{s3}=\frac{n(i_{pr3}-i_{pr1})}{3}$ with the transformer connected in $\Delta -Y$ configuration. The primary current $i_{pr1}$ is designated as $i_L\left(t\right)$ in the analysis and the remaining currents can be derived from it.

3.2. Continuous Conduction Modes

In continuous conduction modes (CCM), both the primary and the secondary currents of the converter are continuous. The converter may operate in seven possible continuous conduction modes (CCM1 to CCM7) depending on the duty cycle and time instants $t_1$ and $t_2$, where $t_1$ is the zero crossing instant of $i_{s1}$ as it transitions from negative to positive polarity and $t_2$ is the zero crossing instant of $i_{s3}$ transitioning from positive to negative polarity.

The conditions for which the converter exhibits CCM1 mode are given as follows: $D>0.33$, $0<t_1<T_{sw}(D-\frac{1}{3})$, and $T_{sw}(D-\frac{1}{3})<t_2<\frac{T_{sw}}{3}$. The voltage and current waveforms of the converter in this mode are presented in Figure 4. It can be observed that there are 12 different stages in each switching period.

3.2.1. Stage 1 $[0$ to $t_1]$

This stage starts when the switch $S_1$ is turned on. As seen from Figure 5a, the inverter switches $S_1,S_4$ and $S_5$ are in conduction along with the diodes $D_2,D_4$ and $D_5$. This stage ends at the zero-crossing of the secondary current $i_{s1}$ at $t=t_1$. During this interval, $v_1\left(t\right)=V_{in},v_2\left(t\right)=-V_{in},v_3\left(t\right)=-\frac{V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{-V_o^{\prime }}{3}$ where $V_o^{\prime }=n{V}_o$ with n denoting the transformer turns ratio. From (5), the state vector is given as

$$X\left(t\right)=A\left(t\right)X\left(0\right)+\frac{1}{3}(V_{in}+V_o^{\prime })B\left(t\right)$$

(6)

where $X\left(0\right)=\left[\begin{array}{c}{i}_L\left(0\right)\\ v_c\left(0\right)\end{array}\right]$.

3.2.2. Stage 2 [$t_1$ to $T_{sw}(D-\frac{1}{3})$]

During this interval, the inverter switches $S_1,S_4$ and $S_5$ and the diodes $D_1,D_4$ and $D_5$ are in conduction as seen in Figure 5b. In this duration, the voltages $v_1\left(t\right)=V_{in},v_2\left(t\right)=-V_{in},v_3\left(t\right)=\frac{V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{-2V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-t_1)X\left(t_1\right)+\frac{1}{3}\left(V_{in}\right)B(t-t_1)$$

(7)

where $X\left(t_1\right)$ can be found by replacing t with $t_1$ in (6).

3.2.3. Stage 3 [$T_{sw}(D-\frac{1}{3})$ to $t_2$]

This stage starts when switch $S_6$ is turned on with switch $S_5$ turned off. The devices in conduction for this interval are switches $S_1,S_4$ and $S_6$ and the diodes $D_1,D_4$ and $D_5$ as shown in Figure 5c. This stage ends at the zero-crossing of the secondary current $i_{s3}$ at $t=t_2$. In this interval, $v_1\left(t\right)=V_{in},v_2\left(t\right)=0,v_3\left(t\right)=\frac{V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{-2V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-T_{sw}(D-\frac{1}{3}))X\left(T_{sw}(D-\frac{1}{3})\right)+\frac{1}{3}\left(2V_{in}\right)B(t-T_{sw}(D-\frac{1}{3}))$$

(8)

3.2.4. Stage 4 [$t_2$ to $\frac{T_{sw}}{3}$]

In this stage, the inverter switches $S_1,S_4$ and $S_6$ and the diodes $D_1,D_4$ and $D_6$ are conducting as seen in Figure 5d. This stage ends when the switch $S_3$ is turned on and $S_4$ stops conducting at $t=\frac{T_{sw}}{3}$. During this interval, $v_1\left(t\right)=V_{in},v_2\left(t\right)=0,v_3\left(t\right)=\frac{2V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{-V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-t_2)X\left(t_2\right)+\frac{1}{3}(2V_{in}-V_o^{\prime })B(t-t_2))$$

(9)

3.2.5. Stage 5 [$\frac{T_{sw}}{3}$ to $(t_1+\frac{T_{sw}}{3})$]

In this stage, the inverter switches $S_1,S_3$ and $S_6$ and the diodes $D_1,D_4$ and $D_6$ are conducting as seen in Figure 5e. This stage ends at the zero-crossing of the secondary current $i_{s2}$ at $t=t_1+\frac{T_{sw}}{3}$. During this interval, $v_1\left(t\right)=0,v_2\left(t\right)=V_{in},v_3\left(t\right)=\frac{2V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{-V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-\frac{T_{sw}}{3})X\left(\frac{T_{sw}}{3}\right)+\frac{1}{3}(V_{in}-V_o^{\prime })B(t-\frac{T_{sw}}{3}))$$

(10)

3.2.6. Stage 6 [$(t_1+\frac{T_{sw}}{3})$ to $D{T}_{sw}$]

The circuit conditions in this interval are shown in Figure 5f with the switches $S_1,S_3$ and $S_6$ and the diodes $D_1,D_3$ and $D_6$ in conduction. Equation (10) still provides the solution for the state vector in this interval as the voltage $\frac{1}{3}(2(v_1-v_3)+v_2-v_4)$ remains the same as the previous interval.

3.2.7. Stage 7 [$D{T}_{sw}$ to $(t_2+\frac{T_{sw}}{3})$]

This interval begins when switch $S_1$ is turned off and $S_2$ starts conduction at $t=D{T}_{sw}$. The devices in conduction during this interval are the switches $S_2,S_3$ and $S_6$, and the diodes $D_1$, $D_3$, and $D_6$ as seen in Figure 5g. This stage ends at the zero-crossing of $i_{s1}$ when $t=t_2+\frac{T_{sw}}{3}$. During this interval, $v_1\left(t\right)=-V_{in},v_2\left(t\right)=V_{in},v_3\left(t\right)=\frac{V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-D{T}_{sw})X\left(D{T}_{sw}\right)+\frac{1}{3}(-V_{in}-V_o^{\prime })B(t-D{T}_{sw})$$

(11)

3.2.8. Stage 8 [$(t_2+\frac{T_{sw}}{3})$ to $\frac{2T_{sw}}{3}$]

As seen in Figure 5h, in this interval, the inverter switches $S_2,S_3$ and $S_6$ and the diodes $D_2,D_3$ and $D_6$ are conducting. At $t=\frac{2T_{sw}}{3}$ when switch $S_5$ is turned on and $S_6$ stops conduction, this interval ends. In this duration, $v_1\left(t\right)=-V_{in},v_2\left(t\right)=V_{in},v_3\left(t\right)=\frac{-V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{2V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-(t_2+\frac{T_{sw}}{3}))X(t_2+\frac{T_{sw}}{3})+\frac{1}{3}(-V_{in})B(t-(t_2+\frac{T_{sw}}{3}))$$

(12)

3.2.9. Stage 9 [$\frac{2T_{sw}}{3}$ to $(t_1+\frac{2T_{sw}}{3})$]

During this interval, the inverter switches $S_2,S_3$ and $S_5$ and the diodes $D_2,D_3$ and $D_6$ are conducting as seen in Figure 5i. This stage ends at $t=t_1+\frac{2T_{sw}}{3}$ when the zero-crossing of $i_{s3}$ occurs. In this duration, $v_1\left(t\right)=-V_{in},v_2\left(t\right)=0,v_3\left(t\right)=\frac{-V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{2V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-\frac{2T_{sw}}{3})X(\frac{2T_{sw}}{3})+\frac{1}{3}(-2V_{in})B(t-\frac{2T_{sw}}{3})$$

(13)

3.2.10. Stage 10 [$(t_1+\frac{2T_{sw}}{3})$ to $T_{sw}(D+\frac{1}{3})$]

In this stage, the inverter switches $S_2,S_3$ and $S_5$ and the diodes $D_2,D_3$ and $D_5$ are conducting as seen in Figure 5j. During this time $v_1\left(t\right)=-V_{in},v_2\left(t\right)=0,v_3\left(t\right)=\frac{-2V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-(t_1+\frac{2T_{sw}}{3}))X(t_1+\frac{2T_{sw}}{3})+\frac{1}{3}(-2V_{in}+V_o^{\prime })B(t-(t_1+\frac{2T_{sw}}{3}))$$

(14)

This interval ends when switch $S_3$ is turned off and $S_4$ starts conduction at $t=T_{sw}(D+\frac{1}{3})$.

3.2.11. Stage 11 [$T_{sw}(D+\frac{1}{3})$ to $(t_2+\frac{2T_{sw}}{3})$]

The devices in conduction for this interval are the switches $S_2,S_4$ and $S_5$ and the diodes $D_2,D_3$ and $D_5$ as seen in Figure 5k. This stage ends at the zero-crossing of $i_{s2}$ when $t=t_2+\frac{2T_{sw}}{3}$. In this interval $v_1\left(t\right)=0,v_2\left(t\right)=-V_{in},v_3\left(t\right)=\frac{-2V_o^{\prime }}{3}$ and $v_4\left(t\right)=\frac{V_o^{\prime }}{3}$ and

$$X\left(t\right)=A(t-T_{sw}(D+\frac{1}{3}))X\left(T_{sw}(D+\frac{1}{3})\right)+\frac{1}{3}(-V_{in}+V_o^{\prime })B(t-T_{sw}(D+\frac{1}{3}))$$

(15)

3.2.12. Stage 12 [$(t_2+\frac{2T_{sw}}{3})$ to $T_{sw}$]

The devices in conduction for this interval are the inverter switches $S_2,S_4$ and $S_5$ and the diodes $D_2,D_4$ and $D_5$ as seen in Figure 5l. This stage ends when switch $S_1$ is turned on and $S_2$ stops conduction at $t=T_{sw}$. Equation (15) gives the state vector in this interval as well since the effective voltage across the resonant tank $\frac{1}{3}(2(v_1-v_3)+v_2-v_4)$ remains the same as the previous interval.

The initial values of the state vector are found by solving $X\left(0\right)=X\left(T_{sw}\right)$ and are given below.

$$X\left(0\right)=\left[\begin{array}{c}{i}_L\left(0\right)\\ v_c\left(0\right)\end{array}\right]={\alpha }_1{V}_o^{\prime }+{\beta }_1{V}_{in}$$

(16)

where ${\alpha }_1$ is a function of $t_1$ and $t_2$ and is defined along with ${\beta }_1$ in Appendix A. At $t=t_1$, the secondary current $i_{s1}^{\prime }=\frac{i_L\left(t\right)-i_L(t-\frac{T_{sw}}{3})}{3}$ goes to zero which gives the following Equation

$$\begin{array}{c}2\sin \left(\frac{w_r{T}_{sw}}{6}\right)(\cos \left(w_r(\frac{T_{sw}}{6}+t_1-t_2)\right)+2\cos \left(\frac{w_r{T}_{sw}}{6}\right))G-3(2\cos \left(\frac{w_r{T}_{sw}}{6}\right)\sin \left(w_r(\frac{T_{sw}}{6}-t_1)\right)\hfill \\ \hfill -\sin \left(w_r(\frac{T_{sw}}{3}+t_1-D{T}_{sw})\right))=0\end{array}$$

(17)

where $G=\frac{V_o^{\prime }}{V_{in}}$ is the gain of the converter. In a similar manner, the zero crossing of the secondary current $i_{s3}^{\prime }$ at $t=t_2$ gives (18).

$$\begin{array}{c}2\sin \left(\frac{w_r{T}_{sw}}{6}\right)(\cos \left(w_r(\frac{T_{sw}}{6}+t_1-t_2)\right)+2\cos \left(\frac{w_r{T}_{sw}}{6}\right))G-3(\sin \left(w_r(\frac{T_{sw}}{3}-t_2)\right)\hfill \\ \hfill -2\cos \left(\frac{w_r{T}_{sw}}{6}\right)\sin \left(w_r(\frac{T_{sw}}{6}+t_2-D{T}_{sw})\right))=0\end{array}$$

(18)

The rectified current $i_o^{\prime }\left(t\right)$ can be described as below.

$$\begin{array}{cc}\hfill i_o^{\prime }\left(t\right)& =\left\{\begin{array}{cc}{i}_{s1}^{\prime }\left(t\right)\hfill & \mathrm{for}{t}_2<t<t_1+\frac{T_{sw}}{3}\hfill \\ -i_{s3}^{\prime }\left(t\right)\hfill & \mathrm{for}{t}_1+\frac{T_{sw}}{3}<t<t_2+\frac{T_{sw}}{3}\hfill \end{array}\right.\hfill \end{array}$$

(19)

The average of the output current $I_o^{\prime }$ is given by

$$I_o^{\prime }=\frac{3}{T_{sw}}\left\{{\int }_{t_2}^{t_1+\frac{T_{sw}}{3}}{i}_{s1}^{\prime }\left(t\right)dt-{\int }_{t_1+\frac{T_{sw}}{3}}^{t_2+\frac{T_{sw}}{3}}{i}_{s3}^{\prime }\left(t\right)dt\right\}$$

(20)

Equation (21) gives the gain of the converter ($G=\frac{V_o^{\prime }}{V_{in}}=\frac{I_o^{\prime }{R}_L^{\prime }}{V_{in}}$). The quality factor is $Q=\frac{Z_o}{R_{ac}}$ and the effective load resistance is $R_{ac}=\frac{6}{{\pi }^2}{\left(\frac{n}{\sqrt{3}}\right)}^2{R}_L$ [17,21].

$$\begin{array}{c}G=\frac{1}{\frac{2}{{\pi }^2}{w}_r{T}_{sw}Q\sin \left(\frac{w_r{T}_{sw}}{2}\right)}\left\{\sin \left(\frac{w_r{T}_{sw}}{6}\right)(\cos \left(w_r(\frac{T_{sw}}{3}-t_2)\right)+\cos \left(w_r(\frac{T_{sw}}{3}+t_1-D{T}_{sw})\right)\right.\hfill \\ \hfill \left.+2\cos \left(\frac{w_r{T}_{sw}}{6}\right)(\cos \left(w_r(\frac{T_{sw}}{6}-t_1)\right)+\cos \left(w_r(t_2+\frac{T_{sw}}{6}-D{T}_{sw})\right)))-2\sin \left(\frac{w_r{T}_{sw}}{2}\right)\right\}\end{array}$$

(21)

Solving Equations (17), (18) and (21), the unknown variables $t_1$, $t_2$ and G can be found. The theoretical and simulated waveforms from PSIM are plotted in Figure 6a–c with the converter operating at $f_{sw}=214$ kHz, $D=0.4$, and $V_{in}=160$ V with a 1 kW, 400 V output. A close correspondence can be observed between the theoretical and the simulated waveforms proving the accuracy of the analysis presented.

Figure 7 shows the waveforms for the remaining CCM modes. The analysis of these modes can be carried out in the same manner and the results are presented in

Table 1. CCM mode definitions.

Mode	Region of Operation	Initial Conditions	Steady-State Equations to Solve for ${\mathit{V}}_{\mathit{o}},{\mathit{t}}_1$ and ${\mathit{t}}_2$
CCM1	$D>0.33$ $0<t_1<T_{sw}(D-\frac{1}{3})$ $T_{sw}(D-\frac{1}{3})<t_2<\frac{T_{sw}}{3}$	$X\left(0\right)={\alpha }_1{V}_o^{\prime }+{\beta }_1{V}_{in}$	(17), (18), and (21)
CCM2	$D>0.33$ $t_1<0$ $0<t_2<T_{sw}(D-\frac{1}{3})$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_1{V}_{in}$	(A7)–(A9)
CCM3	$D>0.33$ $t_1>T_{sw}(D-\frac{1}{3})$ $T_{sw}(D-\frac{1}{3})<t_2<\frac{T_{sw}}{3}$	$X\left(0\right)={\alpha }_1{V}_o^{\prime }+{\beta }_1{V}_{in}$	(17), (18), and (21)
CCM4	$D>0.33$ $t_1<0$ $t_2>T_{sw}(D-\frac{1}{3})$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_1{V}_{in}$	(A7), (A10), and (A11)
CCM5	$D<0.33$ $0<t_1<D{T}_{sw}$ $t_2<D{T}_{sw}$	$X\left(0\right)={\alpha }_1{V}_o^{\prime }+{\beta }_2{V}_{in}$	(17), (18), and (21)
CCM6	$D<0.33$ $T_{sw}(D-\frac{1}{3})<t_1<0$ $0<t_2<D{T}_{sw}$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_2{V}_{in}$	(A7), (A10), and (A11)
CCM7	$D<0.33$ $t_1<T_{sw}(D-\frac{1}{3})$ $0<t_2<D{T}_{sw}$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_2{V}_{in}$	(A7), (A10), and (A11)

. The variables used to describe the initial conditions are defined in Appendix A and the relevant steady-state equations are presented in Appendix A.

3.3. Discontinuous Conduction Modes

The converter enters into discontinuous conduction modes when the secondary currents $i_{s1}$, $i_{s2}$ and $i_{s3}$ become discontinuous. During this discontinuity in the secondary current, two diodes of the rectifier bridge are in conduction as opposed to three diodes seen in the CCM modes discussed previously. As a result, $v_3$ and $v_4$ no longer remain as six-stepped waveforms. This section describes the converter operation in four possible DCM modes.

The converter operates in DCM1 mode with the conditions $D>0.33$, $t_1<0$, and $t_2>T_{sw}(D-1/3)$. Figure 8 shows the waveforms in this mode. As seen from the figure, new voltage levels appear in $v_3$ and $v_4$ waveforms during the discontinuous intervals. These voltage levels are described using the variable $x=\frac{V_d}{3}$, with $V_d$ being the voltage across the top diode of the rectifier leg that is out of conduction as shown in Figure 9. Similar to the CCM modes, the converter has 12 different stages within a switching period and the initial values of the state vector can be found as

$$X\left(0\right)=\left[\begin{array}{c}{i}_L\left(0\right)\\ v_c\left(0\right)\end{array}\right]={\alpha }_2{V}_o^{\prime }+{\beta }_1{V}_{in}+{\gamma }_{1}x$$

(22)

where ${\alpha }_2$ and ${\gamma }_1$ are defined in Appendix A. From Figure 8, it can be seen that the secondary current $i_{s1}$ is discontinuous during the interval $t_1+T_{sw}$ to $T_{sw}$. By equating $i_{s1}$ to zero at the limits of the interval, Equations (A12) and (A13) can be obtained. Equation (A14) is obtained from the zero crossing of $i_{s3}$ while $V_o^{\prime }$ is found as (A15). Solving these four equations, the unknown parameters $t_1$, $t_2$, x and $V_o^{\prime }$ of the converter can be found. The waveforms obtained from the theoretical analysis and from simulation using PSIM software are plotted in Figure 10a–c considering the converter operation at $f_{sw}=118.4$ kHz, $D=0.415$, and $V_{in}=80$ V with a 0.5 kW, 400 V output.

Figure 11 shows the waveforms for the remaining DCM modes. The DCM2 mode has two discontinuous intervals within each switching period as shown in Figure 11a. In this case, the voltages $x_1$ and $x_2$ are used to describe the waveforms where $x_1=\frac{V_{d1}}{3}$ and $x_2=\frac{V_{d2}}{3}$ with $V_{d1}$ being the voltage appearing across the top diode of the inverter leg for the discontinuous interval that occurs after the negative half cycle, whereas $V_{d2}$ refers the diode voltage for the discontinuous interval that occurs after the positive half cycle of the secondary current. The rest of the analysis follows the same procedure as presented for DCM1 mode. The initial conditions, operating regions, and relevant steady-state equations for all the DCM modes have been summarised in

Table 2. DCM mode definitions.

Mode	Region of Operation	Initial Conditions	Steady-State Equations to Solve for Unknown Variables
DCM1	$D>0.33$ $t_1<0$ $t_2>T_{sw}(D-1/3)$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_1{V}_{in}+{\gamma }_{1}x$	(A12)–(A15)
DCM2	$D>0.33$ $t_1<0$ $t_2<T_{sw}(D-1/3)$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_1{V}_{in}+{\gamma }_1{x}_1+{\gamma }_2{x}_2$	(A16)–(A20)
DCM3	$D<0.33$ $t_1<T_{sw}(D-1/3)$ $t_2>0$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_2{V}_{in}+{\gamma }_{1}x$	(A12)–(A15)
DCM4	$D<0.33$ $T_{sw}(D-1/3)<t_1<0$ $t_2>0$	$X\left(0\right)={\alpha }_2{V}_o^{\prime }+{\beta }_2{V}_{in}+{\gamma }_{1}x$	(A12)–(A15)

.

3.4. Mode Boundaries and ZVS Region

Figure 12 shows the mode boundaries when the converter is operated at full load condition with $V_{in}=160$ V. The boundaries are obtained by solving for the mode constraints with the use of the derived steady-state equations. Zero voltage switching (ZVS) at the switch turn-on can be achieved when there exists a mechanism for the drain to source voltage ($v_{ds}$) of the switch to discharge before the switch is gated. Consequently, the switches $S_1$, $S_3$, and $S_5$ can be turned on under ZVS if $i_L\left(0\right)$ is negative while $i_L\left(D{T}_{sw}\right)$ should be positive for the switches $S_2$, $S_4$, and $S_6$ to achieve ZVS. In practice, to obtain complete ZVS in the upper switches, the condition $i_L\left(0\right)<I_{ZVS}$ should be satisfied, where $I_{ZVS}$ is the minimum current necessary for $v_{ds}$ to discharge within the specified dead time. The upper switches will lose ZVS first since they are operated at a lower duty cycle ($D\le 0.5$). Hence, it is sufficient to ensure $i_L\left(0\right)<I_{ZVS}$ to guarantee the ZVS operation of all the switches. The zero-voltage switching region of the converter under full load condition at a line voltage of 160 V is shown in Figure 12.

4. Operating Modes of the Converter

4.1. Design Specifications of the Converter

Table 3. Design specifications.

Parameter	Value
Output power ($P_o$)	1 kW
Output voltage ($V_o$)	400 V
Input voltage range	80 V–160 V
Resonant frequency ($f_r$)	100 kHz
Switching frequency range ($f_{sw}$)	100–250 kHz
Turns ratio ($n=N_p/N_s$)	$1/3$
Rated Quality factor ($Q_{rated}$)	1

shows the specifications of a three-phase series resonant converter with a rated power of 1kW. The MPP voltage of a single PV panel generally ranges from 20 V–40 V. The operating range of input voltage for the converter is chosen as 80 V–160 V so that four PV panels may be used. The converter is designed to have a low quality factor with $Q=1$ at rated load conditions.

4.2. Three-Phase Operating Mode

This is the primary operating mode of the converter where the switches $S_1$ to $S_6$ are operated with variable frequency and duty cycle. Due to the two degrees of freedom, there are multiple ($f_{sw}$,D) operating points that can maintain regulation for a given load condition and gain. Figure 13a shows the possible operating points for 160 V input, full load condition with output voltage of 400 V. Figure 13b shows the normalized resonant current rms ($I_{base}=P_o/V_{inmin}$) for the different operating points. It can be observed that as the duty cycle deviates from $0.5$, $i_{Lrms}$ increases. Hence, at higher load conditions, where conduction losses are dominant, operating with a fixed 50% duty cycle using variable frequency modulation provides higher efficiency. The modulation scheme can be switched to the proposed hybrid modulation at light load conditions to maintain wide range regulation.

In the three-phase mode, the converter frequency may vary from 100–250 kHz while the duty cycle D is limited to the range 0.5–0.2. A further reduction in the duty cycle may cause the primary currents to become discontinuous resulting in the loss of ZVS in the upper switches $S_1,S_3$ and $S_5$. At light loads, the converter has a very low quality factor and regulation may not be attained within the set limits of $f_{sw}$ and D. Under such conditions, the converter is transitioned to a different operating mode.

4.3. Single-Phase Operating Mode

In the single-phase mode, only the switches $S_1$ to $S_4$ are gated. $S_1$ and $S_4$ are operated with a duty cycle D whereas switches $S_2$ and $S_3$ are operated with a duty cycle $D^{\prime }=(1-D)$. The converter operates with a switching frequency range of 100–250 kHz with the duty cycle in the range $50\%$ to $13\%$. The possible circuit conditions in this mode are shown in Figure 14. As seen from the figure, in the single-phase mode, the resonant inductors and capacitors of two phases are connected in series. The equivalent parameters of the resonant tank are given below.

$$\begin{array}{cc}\hfill L_{eq}=2L_s{C}_{eq}=\frac{C_s}{2}& \hfill \end{array}$$

This results in an increased quality factor with the same resonant frequency $f_r$. Furthermore, since the three-phase transformer no longer operates with a $\Delta -Y$ connection, the additional $\sqrt{3}$ gain associated with the configuration is lost, and instead a gain factor of 1.5 is present. Due to these factors, the single-phase mode is capable of maintaining regulation along with soft-switching at lighter loads compared to the three-phase mode. However, for the current design with a turns ratio of 1:3, this mode can be utilized only for $V_{in}>90$ V owing to the decreased gain of the transformer configuration.

4.4. Proposed Operating Modes

Figure 15 shows the region of operation for the proposed modes of the converter based on the load condition and line voltage. The converter is predominantly operated in the three-phase mode under variable frequency modulation or hybrid modulation based on the load condition. This mode can maintain soft-switching and regulation up to 10% load at 80 V input and up to 35% load at 160 V input.

When the regulation and soft-switching cannot be maintained by the three-phase mode, the converter is switched to single-phase mode. This mode can only be utilized when the input voltage $V_{in}$ lies in the range of 90 V–160 V. Under extreme light load conditions (<10%), where the single-phase mode fails to maintain regulation within the operating limits, burst mode operation is used.

5. Experimental Results

An experimental 1 kW prototype with the component specifications provided in

Table 4. Component specifications.

Component	Value
Primary switches ($S_1-S_6$)	IPB600N25N3
Secondary diodes ($D_1-D_6$)	STPSC406B
Series resonant inductance ($L_s$)	$5.4\mathsf{\mu }$H
Resonant capacitance ($C_s$)	$0.44\mathsf{\mu }$H
Transformer leakage inductance ($L_{lk}$)	300 nH
Transformer magnetizing inductance ($L_m$)	$180\mathsf{\mu }$H

was implemented to assess the performance of the converter. The experimental setup is shown in Figure 16. Figure 17 shows the waveforms at full load condition with $V_{in}=80$ V when the converter is operated in three-phase mode. The converter operation is in CCM2 mode with $f_{sw}=108$ kHz and $D=0.5$. To achieve ZVS at the turn-on of the switches $S_1$ and $S_2$, the current $i_{pr1}$ should be negative at the turn-off of $S_2$ and positive at the turn-off of $S_1$ as seen in Figure 17a. The primary currents $i_{pr1}$, $i_{pr2}$ and $i_{pr3}$ are also identical and 120° phase-shifted as seen in Figure 17c, and hence it can be inferred that zero voltage switching is attained at the turn-on of all the switches.

Figure 18 shows the converter waveforms at 35% load and 160 V input operating in DCM3 mode with $f_{sw}=240$ kHz and $D=0.21$ and Figure 19 shows the obtained waveforms with 80 V input and 10% load in DCM1 mode ($f_{sw}=238$ kHz, $D=0.35$) with ZVS being achieved in both the cases.

Figure 20a shows the converter operation in single-phase mode with $V_{in}=90$ V and 10% load with $f_{sw}=121$ kHz and $D=0.5$. It is observed that the switches $S_1$ to $S_4$ are turned on at zero voltage conditions. Figure 20b shows the converter waveforms at 10% load with 160 V input at $0.14$ duty cycle and 250 kHz switching frequency.

Figure 21a shows the experimental efficiency for different load conditions at $V_{in}=80$ V and $V_{in}=160$ V. It can be seen that a peak efficiency of $96.7$% is obtained at 70% load with 80 V input. Moreover, the converter almost exhibits a flat efficiency curve with less than 1% drop in efficiency when the load is varied from full load to light load. Figure 21b shows the theoretical power loss breakdown at two different operating conditions. Since the waveforms are non-sinusoidal, the improved Generalized Steinmetz Equation (iGSE) was used to model the core losses. At full load conditions, the conduction loss in the switches contributes mainly to the power loss. The efficiency of the converter can be further improved by utilizing switches with a lower drain-to-source resistance.

Table 5. Comparison of a high Q converter with conventional modulation and a low Q converter with the proposed modulation.

Parameters	80 V Full Load		80 V Half Load		160 V Full Load		160 V Half Load
	High Q	Low Q	High Q	Low Q	High Q	Low Q	High Q	Low Q
Switching frequency ($f_{sw}$)	$103.2$ kHz	$109.6$ kHz	$106.3$ kHz	$118.4$ kHz	125 kHz	$221.5$ kHz	$153.9$ kHz	250 kHz
Duty cycle (D)	$0.5$	$0.5$	$0.5$	$0.5$	$0.5$	$0.5$	$0.5$	$0.24$
Resonant inductor current ($i_{Lrms}$)	$9.64$ A	$9.79$ A	$4.81$ A	$5.01$ A	$9.59$ A	$9.61$ A	$4.80$ A	$5.23$ A
Resonant capacitor voltage ($V_{cp-p}$)	$381.97$ V	$92.19$ V	$185.26$ V	$44.01$ V	$312.23$ V	$44.09$ V	$126.99$ V	$21.06$ V
Conduction loss	$32.87$ W	$31.23$ W	$9.80$ W	$9.73$ W	$33.42$ W	$30.29$ W	$10.02$ W	$10.31$ W
Core loss	$5.58$ W	$2.95$ W	$2.79$ W	$2.46$ W	$6.39$ W	$3.07$ W	$2.51$ W	$1.49$ W
Turn-off loss	$0.12$ W	$0.18$ W	$0.07$ W	$0.13$ W	$0.89$ W	$1.67$ W	$0.56$ W	1 W
Efficiency	$96.24\%$	$96.63\%$	$97.41\%$	$97.51\%$	$96.03\%$	$96.57\%$	$97.36\%$	$97.41\%$
Experimental Efficiency	$95.41\%$	$95.97\%$	$96.32\%$	$96.49\%$	$94.46\%$	$95.2\%$	$95.6\%$	$95.64\%$

presents a theoretical comparison between two converters—a low Q converter operated with the proposed modulation and a high Q converter ($Q=4,L_s=22.934u$ and $C_s=0.1104u$) with conventional variable frequency modulation and resonant frequency of 100 kHz. The theoretical inductor rms currents and capacitor peak-to-peak voltages listed in the table were obtained using the analysis method detailed in Section 2. Compared to the low Q converter, resonant inductors of the higher cross-sectional area has to be utilized in the high Q case due to the increased inductance to avoid core saturation. While an RM10 core (effective volume $V_e=3360$ mm${}^3$) was sufficient for the low Q design while an RM12 core ($V_e=8340$ mm${}^3$) with more than twice the effective volume has to be utilized for the high Q design. While the high Q design enables converter operation over a much narrow frequency range, this comes at the cost of high voltage stresses across the resonant capacitor and inductor. Though the resonant RMS currents are comparatively lower in the high Q design due to operation close to the resonant frequency, the inductor ESR also increases due to the increased number of turns contributing to the higher conduction loss. Hence, it can be seen that the proposed modulation enables the operation of a low Q converter over a wide operating region with high efficiency, more compact magnetic design, and reduced voltage stress across the resonant components.

6. Conclusions

In this article, a low Q three-phase series resonant converter was proposed to reduce the voltage stress across the resonant components. To reduce the operating frequency range, a hybrid control technique employing variable switching frequency along with asymmetric pulse width was introduced. A time-domain analysis method was presented to model the converter under the proposed modulation with high accuracy. It was further proposed to operate the converter in different modes based on the line voltage and load condition to ensure soft-switching in the entire operating region. For experimental validation, a 1kW prototype was developed achieving a peak efficiency of $96.7\%$.It was observed that the converter presented a drop in efficiency of less than 1% with load variation. Under three-phase operation, the converter was able to achieve ZVS for upto $10\%$ load at an input of 80 V and upto $35\%$ load at an input of 160 V. The soft-switching region was further extended for the entire line voltage range by operating in single-phase mode. A comparison was made between a high Q converter with conventional VF modulation and a low Q converter with the proposed modulation. It was shown that the proposed modulation was able to achieve higher efficiency along with a compact magnetic design and reduced voltage stress across resonant components.