Analysis and Design of Function Decoupling High Voltage Gain DC/DC Converter

Abstract

Traditional boost converters have difficulty realizing high efficiency and high voltage gain conversion due to 1) extremely large duty cycles, 2) high voltage and current stresses on devices, and 3) low conversion efficiency. Therefore, a function decoupling high voltage gain DC/DC converter composed of a DC transformer (DCX) and an auxiliary converter is proposed. The role of DCX is to realize fixed gain conversion with high efficiency, whereas the role of the auxiliary converter is to regulate the output voltage. In this study, different forms of combined high voltage gain converters are compared and analyzed, and a structure is selected for the function decoupling high voltage gain converter. Then, topologies and control strategies for the DCX and auxiliary converter are discussed. On the basis of the discussion, an optimal design method for circuit parameters is proposed, and design procedures for the DCX are described in detail. Finally, a 400 W experimental prototype based on the proposed optimal design method is built to verify the accuracy of the theoretical analysis. The measured maximum conversion efficiency at rated power is 95.56%.

I. INTRODUCTION

Environmental problems and shortage of fossil energy have resulted in increased demand for renewable energy, such as solar and wind energy [1], [2]. Distributed photovoltaic (PV) systems have been widely used due to their superior reliability and efficiency. However, the input voltage (380-400 V) of a grid-connected inverter is much higher than the output voltage of a PV array (40-60 V). Therefore, a step-up DC/DC converter is required to realize voltage matching.

The most fundamental step-up converter is the boost converter. Theoretically, a boost converter can have infinite voltage gain when its duty cycle is close to unity; however, when parasitic parameters are considered, boost converter operation is suitable only under conditions with voltage gains less than four, and it cannot satisfy the requirements of PV system. To achieve high efficiency and high gain conversion, substantial research on this topic has been published. Nonisolated high step-up converters can be divided into three types: I) switched-capacitor or switched-inductor high step-up converters [3], [4], in which high voltage ratio is realized by combining a single or several switched-capacitor or switched-inductor units with a basic DC/DC converter; II) coupled-inductor high step-up converters [5], in which voltage gain depends on the duty cycle and turns ratio of the coupled inductor; III) combined high step-up converters [6], in which voltage gain is improved by combing several basic DC/DC converters. For type I, the voltage gain is related to the number of switched-capacitor or switched-inductor units; therefore, in relatively high voltage gain applications, the complexity of the circuit is increased; in addition, the voltage gain can only be an integer, which limits its application. For type II, the voltage stress on devices and losses are increased due to resonance between the leakage inductance of the coupled inductor and the parasitic capacitance of switches or diodes; passive and active methods should be adopted to suppress the voltage spike caused by resonance. Type III maximizes the advantages of basic DC/DC converters; at the same time, the system structure can be simplified by sharing components. In addition, in some studies that focused on the combination of different techniques, such as [7] and [8], coupled inductor and switched capacitor techniques are adopted together to enhance performance, which is also a good solution. Overall, type III is a competent solution, and the proposed converter belongs to this type.

Combined high voltage gain converters mainly have two different forms: cascaded and input-parallel output-series (IPOS) forms. Voltage gain is the square of a basic converter for the cascaded form; it can achieve high voltage gain with low duty cycle [9], [10]; however, back-end converters have high voltage stress. For the IPOS form, voltage gain is the sum of two basic converters; it can not only improve system voltage gain but also reduce voltage stress; boost-flyback [11], boost-SEPIC [12], boost-zeta [13], and flyback-SEPIC [14] all belong to this type.

In this study, an IPOS function decoupling high gain converter composed of a DC transformer (DCX) and an auxiliary converter is proposed. The rest of this paper is organized as follows. In Section II, analysis and comparison between cascaded and IPOS forms are presented according to system voltage gain and conversion efficiency, and the structure of the proposed function decoupling high gain converter is presented. The topology selection and control strategy for each unit are discussed in Section III. In Section IV, the proposed optimal design method is introduced, and the detailed design procedures are presented. Experimental results are analyzed in Section V. Finally, conclusions are drawn in Section VI.

II. ANALYSIS OF TWO FORMS OF COMBINED HIGH VOLTAGE GAIN CONVERTER

In this section, analysis and comparison between two forms of combined high gain converters are presented. Figs. 1 and 2 show the structures of the cascaded and IPOS forms, respectively.

Fig. 1. Structure of the cascaded form.

Fig. 2. Structure of the IPOS form.

The voltage gains of the two converter units are M1 and M2, and their efficiencies are η1 and η2. From Fig. 1, the voltage gain and efficiency for the cascaded form can be obtained.

\(M_{\mathrm{casc}}=M_{1} M_{2}\)       (1)

\(\eta_{\mathrm{casc}}=\eta_{1} \eta_{2}\)       (2)

Similarly, from Fig. 2, the voltage gain and efficiency for the IPOS form can be derived as

\(M_{\text {ipos }}=M_{1}+M_{2},\)       (3)

\(\eta_{\text {ipos }}=\frac{P_{o1}+P_{o2}}{P_{\text {in1 } }+P_{\text {in2 }}}=\eta_{1} \frac{P_{\text {inl }}}{P_{\text {in }}}+\eta_{2} \frac{P_{\text {in2 }}}{P_{\text {in }}}.\)       (4)

For the sake of analysis, the following assumptions are made: 1) M₁∈[0, 5], M₂∈[0, 5]; 2) η₁∈[0.8, 1), η₂∈[0.8, 1). On the basis of the assumptions and Equations (1) and (3), Fig. 3, which shows the voltage gain comparison between the two forms, can be obtained. When M₁ and M₂ are lower than 1, M_ipos is higher than M_casc; when M₁ and M₂ are greater than 1, M_casc is higher than M_ipos.

Fig. 3. Comparison of voltage gain between cascaded and IPOS forms.

A comparison of efficiency between the two forms based on Equations (2) and (4) is shown in Fig. 4. The blue, green, and yellow curves illustrate when P_in1/P_in equals 0.1, 0.3, and 0.5, respectively. Under these conditions, the efficiency of the IPOS form is higher than that of the cascaded form. In addition, for the IPOS form, the higher the power that is handled by the high-efficiency power conversion unit, the higher the overall system efficiency.

Fig. 4. Comparison of efficiency between cascaded and IPOS forms.

According to the above analysis, even though the IPOS form cannot achieve high voltage gain conversion, its voltage gain characteristic can satisfy PV system requirements, and it has better system efficiency than does the cascaded form. Therefore, the IPOS form is more advantageous than the cascaded form in PV system application.

DCX can achieve high efficiency and constant voltage gain conversion [15], [16]; thus, it is selected to handle as much power as possible. Voltage regulation is realized by another converter unit, which is called auxiliary converter. The structure of the proposed function decoupling high gain converter is shown in Fig. 5.

Fig. 5. Proposed function decoupling high gain converter structure.

Although the proposed converter has more components, it has the following advantages compared with a single LLC resonant converter.

1) It has high power handling ability, given the two units handling the total power.

2) Constant duty cycle and frequency operation can be implemented for the LLC resonant converter, whereas a variable switching frequency range is required for a single LLC resonant converter, which will complicate the magnetic components and electromagnetic interference (EMI) design.

3) Although the output voltage of an LLC converter can nearly remain constant at the resonant frequency operation point, this principle will be challenged during wide-range operation.

III. TOPOLOGY SELECTION AND CONTROL STRATEGY

In this section, based on the functions of the DCX and auxiliary converter, the topology for each unit and corresponding control strategy are determined.

A. DCX

The role of DCX is to handle most of the power; high conversion efficiency is required to improve the overall system efficiency. Given that the outputs of DCX and auxiliary converter are in series connection, higher power handling ability means higher output voltage. Thus, a high gain conversion for DCX is also required. LLC resonant converters have been widely used in many applications due to their soft-switching capability in wide load ranges. However, frequency control strategy is often adopted to regulate output voltage, thereby complicating the optimal design of resonant components; in addition, system efficiency is reduced because of high reactive circulation current when the system operation point is far from the resonant operation point [17]. Fig. 6 shows an asymmetric half-bridge LLC resonant converter. DCX is not required to regulate output voltage; therefore, constant duty cycle and constant frequency operation can be implemented. On the one hand, by optimal design, zero-voltage switching (ZVS) operation for primary switches and zero-current switching (ZCS) operation for secondary diodes can be guaranteed; on the other hand, with the selection of a reasonable transformer turns ratio, high voltage gain conversion can be achieved, and voltage doublers composed of D_m1, D_m2, and C_mo1, C_mo2 can further improve the system voltage gain. In addition, the leakage inductor of the transformer can be used as resonant inductor L_r, and L_m is the magnetic inductor of the transformer; thus, only resonant capacitor C_r is added into the circuit. This topology has high power density due to its high-frequency operation capability.

Fig. 6. Asymmetric half-bridge LLC resonant converter.

For LLC resonant converters, three subintervals exist during half of a switching period. During the positive of a half switching period, the P subinterval is defined when the voltage across the magnetic inductor is clamped by V_mo/2N; the N subinterval exists when the voltage across the magnetic inductor is clamped by -V_mo/2N; the O subinterval happens when no power flows to the secondary side of the transformer. Based on the sequences of these subintervals, an LLC resonant converter mainly has the following operation modes: PO, PON, PN, NP, NOP, and OPO. Take PO operation mode as an example. During half of the switching period, the LLC resonant converter operates at the P subinterval first, and then at the O subinterval.

Fig. 7 shows the main waveforms when an LLC resonant converter operates at PO mode. Before the drive signal of S_m1 comes, the switch current i_Sm1 is negative; thus, its parallel diode will be conducted, and ZVS operation for Sm1 is achieved. Similarly, S_m2 can achieve ZVS operation. According to the waveform of transformer secondary current i_Sec, the secondary diodes D_m1 and D_m2 can achieve ZCS operation. In other operation modes, such as PON, ZVS operation cannot be guaranteed for primary switches and leads to reduced efficiency; in OPO mode, three subintervals are involved, which complicates the mathematical derivation of this mode; in P mode, parasitic parameters and a slight variation of resonant components exist; thus, this operation cannot be guaranteed. Therefore, the design consideration for DCX is to ensure that it can operate in PO mode during the entire operation.

Fig. 7. Main waveforms during operation in PO mode.

When the operation point of an LLC resonant converter is close to the resonant frequency operation point, its voltage gain nearly remains constant with different input voltages and loads. Therefore, constant frequency and constant duty cycle operation are adopted for the converter to simplify the optimal design and the control circuit.

B. Auxiliary Converter

The output voltage of DCX varies with input voltage; thus, the role of the auxiliary converter is to regulate the output voltage. When the input voltage reaches the maximum value V_in-max, the output voltage of DCX also reaches the maximum value V_mo-max, whereas the output voltage of the auxiliary converter reaches its minimum value V_ao-min. For the sake of system efficiency, V_ao-min is generally less than V_in-max; thus, the auxiliary converter needs to realize step-down conversion. When the input voltage reaches the minimum value V_in-min, the output voltage of DCX also reaches the minimum value V_mo-min, whereas the output voltage of the auxiliary converter reaches its maximum value V_ao-max. For wide input voltage applications, V_ao-max is generally higher than V_in-min; thus, step-up conversion for the auxiliary converter is required.

Based on the above analysis, a noninverted buck-boost converter, as shown in Fig. 8, is selected for the auxiliary converter. In addition, it has the following advantages:

1) Compared with other step-up and step-down converters, such as buck-boost, cuk, and SEPIC, it has lower voltage stress;

2) Two switches S_a1 and S_a2 are present in the topology, and they can work simultaneously or separately according to the requirements, thus improving the control freedom of the system; compared with the buck-boost converter, it has a wider output voltage range and lower inductor current ripple [18];

3) Compared with cuk, SEPIC, and zeta, the selected converter requires fewer passive components; thus, high power density conversion can be achieved [19];

4) The input and output of buck-boost and cuk are inverted, which complicates the corresponding power supply and drive circuit.

Fig. 8. Noninverted buck-boost converter.

According to Fig. 8, when S_a2 is off, the output voltage can be regulated by adjusting the duty cycle of S_a1, and the converter is working in buck mode; when S_a1 is on, the output voltage is regulated by controlling S_a2, and the converter is working in boost mode. However, when the output voltage is close to the input voltage in both operation modes, extreme duty cycle will appear; specifically, in buck mode, the duty cycle of S_a1 should be close to 1, and in boost mode, the duty cycle of S_a2 should be zero. Therefore, when the converter voltage gain M_a∈[0.8,1], the duty cycle of S_a2 is set as d_Boost=0.2, whereas the duty cycle of S_a1 d_Buck∈[0.64,0.8]; when the converter voltage gain M_a∈[1,1.25], the duty cycle of S_a1 is set as d_Buck=0.8, and the duty cycle of S_a2 d_Boost∈[0.2,0.36]. Based on the above analysis, the relationship between the auxiliary converter voltage gain M_a and S_a1, S_a2 duty cycle d_Buck and d_Boost can be obtained as shown in Fig. 9. With the adoption of this control strategy, the extreme duty cycle situation can be avoided, and the output voltage can be regulated.

Fig. 9. Relationship between M_a, d_Buck and d_Boost.

The proposed converter topology is shown in Fig. 10.

Fig. 10. Topology of the proposed converter.

IV. OPTIMAL DESIGN METHOD

In this section, the design considerations are presented. The design of DCX is the key point. Given the simplicity and the similarity of the design of the auxiliary converter to the traditional design, only the results are provided.

Fundamental harmonic analysis (FHA) is usually adopted to design an LLC resonant converter [20]. However, errors occur in this methodology [21], [22] due to the following assumptions.

1) The LLC resonant converter is always working in continuous conduction mode.

2) The input voltage of the resonant tank and the output voltage of the transformer are continuous square-wave voltages that can be replaced by their fundamental harmonic components.

3) The equivalent AC resistance can be expressed as Rac=8n2RL/π2. With increased load, the error will further increase.

Therefore, in this study, an operation mode-based optimal design method is proposed. The precision of this method is highly improved compared with that of the FHA method.

The unknown parameters of DCX include transformer turns ratio N, resonant components L_r and C_r, and L_m. First, the transformer turns ratio N is determined according to the voltage gain of the DCX. Then, the resonant components are selected based on the proposed optimal design method. Finally, simulation cases are introduced to verify the theoretical analysis, and the design procedures are summarized.

The known parameters are V_i=40–60 V, V_o=400 V, P_o=100–400 W, f_s=100 kHZ.

A. Selection of Transformer Turns Ratio N

The voltage gain of the DCX is independent of the input voltage and load. When the input voltage reaches the maximum value V_in-max, the auxiliary converter operates in buck mode, and d_buck reaches the minimum value d_buck-min. To avoid extreme duty cycle, the following assumption is made:

\(d_{\text {buck-min }} \geq 0.2.\)       (5)

According to Fig. 5, the following equation can be derived:

\(V_{\text {in-max }} d_{\text {buck-min }}+V_{\text {in-max }} M_{\mathrm{m}}=V_{\mathrm{o}}.\)       (6)

According to Equations (5) and (6), an inequality of M_m can be expressed as

\(M_{m}\le 6.5.\)       (7)

Similarly, when the input voltage equals the minimum value V_in-min, the auxiliary converter works in boost mode, and d_boost reaches the maximum value d_boost-max. To avoid extreme duty cycle, the following assumption is made:

\(d_{\text {boost-max }} \leq 0.8.\)       (8)

According to Fig. 5, the following equation can be obtained:

\(\frac{V_{\text {in-min }}}{1-d_{\text {boost-max }}}+V_{\text {in-min }} M_{\mathrm{m}}=V_{\mathrm{o}}.\)       (9)

With the combination of Equations (8) and (9), another inequality of M_m can be obtained.

\(M_{\mathrm{m}} \geq 5\)       (10)

From Equations (7) and (10), Mm can be selected as

\(M_{\mathrm{m}} = 6.5.\)       (11)

The operating frequency is close to the resonant frequency; thus, the transformer turns ratio N is nearly the same as the voltage gain M_m.

\(N \approx M_{\mathrm{m}}\)       (12)

In practice, the voltage gain is slightly higher than 1; finally, N=6.25 is selected.

B. Selection of Resonant Components

The design consideration for the resonant components is to ensure that the DCX is in PO mode during its entire operation, thus keeping its voltage gain constant.

First, parameters K, ω_r, and ω_m are defined as follows:

\(\left\{\begin{array}{c} K=\frac{L_{\mathrm{m}}}{L_{\mathrm{r}}} \\ \omega_{\mathrm{r}}=\frac{1}{\sqrt{L_{\mathrm{r}} C_{\mathrm{r}}}} \\ \omega_{\mathrm{m}}=\frac{1}{\sqrt{\left(L_{\mathrm{r}}+L_{\mathrm{m}}\right) C_{\mathrm{r}}}}. \end{array}\right.\)       (13)

The equivalent circuits when the DCX operates in the P and O subintervals during the positive half of the switching period are shown in Figs. 11(a) and (b), respectively.

Fig. 11. Equivalent circuits of an LLC resonant converter. (a) In P subinterval. (b) In O subinterval.

As shown in Fig. 11(a), when an LLC resonant converter operates in the P subinterval, the circuit can be described as

\(L_{\mathrm{r}} C_{\mathrm{r}} \frac{d^{2}}{d t^{2}} v_{\mathrm{Cr}}+v_{\mathrm{Cr}}=V_{\mathrm{i}}-\frac{V_{\mathrm{mo}}}{2 N}.\)       (14)

The general solution for Equation (14) is

\(v_{\mathrm{Cr}}(t)=k_{1} \cos \left(\omega_{\mathrm{r}} t\right)+k_{2} \sin \left(\omega_{\mathrm{r}} t\right)+V_{\mathrm{i}}-\frac{V_{\mathrm{mo}}}{2 N},\)       (15)

where k₁ and k₂ depend on the initial condition of the circuit.

According to Equation (15), the general solution for i_Lr can be derived as

\(i_{\mathrm{Lr}}(t)=C_{\mathrm{r}} \omega_{\mathrm{r}}\left[-k_{1} \sin \left(\omega_{\mathrm{r}} t\right)+k_{2} \cos \left(\omega_{\mathrm{r}} t\right)\right].\)       (16)

In the following section, the expressions of k₁ and k₂ are derived on the basis of the initial circuit condition.

According to Figs. 6 and 7, during half of the switching period, the voltage across the resonant capacitor increases from V_Cr,tf to V_Cr,tc. Therefore, the total charge of the resonant tank during this period can be expressed as

\(Q_{\mathrm{mi}}=C_{\mathrm{r}}\left(V_{\mathrm{Cr}, \mathrm{tc}}-V_{\mathrm{Cr}, \mathrm{tf}}\right).\)       (17)

Fig. 7 shows that V_Cr,tf and V_Cr,tc are symmetric with V_i/2; thus,

\(\frac{V_{i}}{2}-V_{\mathrm{Cr}, \mathrm{tf}}=V_{\mathrm{Cr}, \mathrm{tc}}-\frac{V_{\mathrm{i}}}{2}.\)       (18)

With the combination of Equations (17) and (18), the total charge can be expressed as

\(Q_{\mathrm{mi}}=C_{\mathrm{r}}\left(V_{\mathrm{i}}-2 V_{\mathrm{Cr}, \mathrm{tf}}\right).\)       (19)

The system efficiency is assumed to be 100%; thus, during one switching period, the input energy E_mi equals the output energy E_mo.

\(E_{\mathrm{mi}}=E_{\mathrm{mo}}\)       (20)

On the basis of the operation of the DCX, the following expressions for E_mi and E_mo can be obtained:

\(E_{\mathrm{mi}}=V_{\mathrm{i}} Q_{\mathrm{mi}},\)       (21)

\(E_{\mathrm{mo}}=\frac{V_{\mathrm{mo}}^{2}}{R_{\mathrm{mL}} f_{\mathrm{s}}}.\)       (22)

From Equations (19)–(22), the resonant capacitor voltage at t_f can be expressed as

\(V_{\mathrm{Cr}, \mathrm{tf}}=\frac{C_{\mathrm{r}} R_{\mathrm{mL}} V_{\mathrm{i}}^{2} f_{\mathrm{s}}-V_{\mathrm{mo}}^{2}}{2 R_{\mathrm{mL}} f_{\mathrm{s}} C_{\mathrm{r}} V_{\mathrm{i}}}.\)       (23)

t_f is considered the initial point of the P subinterval; thus, ω_rt_f=0 can be obtained. With substitution of this condition into Equation (15) and combination with Equation (23), the final expression for k₁ can be calculated as

\(k_{1}=\frac{C_{\mathrm{r}} R_{\mathrm{mL}} V_{\mathrm{mo}} V_{\mathrm{i}} f_{\mathrm{s}}-N C_{\mathrm{r}} R_{\mathrm{mL}} V_{\mathrm{i}}^{2} f_{\mathrm{s}}-N V_{\mathrm{mo}}^{2}}{2 N R_{\mathrm{mL}} f_{\mathrm{s}} C_{\mathrm{r}} V_{\mathrm{i}}}.\)       (24)

According to Fig. 7, at t_f and t_x, the resonant inductor current i_Lr equals the magnetic inductor current i_m, that is,

\(\left\{\begin{array}{l} i_{\mathrm{Lr}, \mathrm{tf}}=i_{\mathrm{Lm}, \mathrm{tf}} \\ i_{\mathrm{Lr}, \mathrm{tx}}=i_{\mathrm{Lm}, \mathrm{tx}} \end{array}\right..\)       (25)

The relationship between i_Lm,tx and i_Lm,tf can be described as

\(i_{\mathrm{Lm}, \mathrm{tx}}=i_{\mathrm{Lm}, \mathrm{tf}}+\frac{V_{\mathrm{mo}} \theta}{2 N \omega_{\mathrm{r}} L_{\mathrm{m}}}.\)       (26)

With the substitution of ω_rt_f=0 and ω_rt_x=θ into Equation (16) and their combination with Equations (25) and (26), respectively, the expression of k₂ can be solved as

\(\begin{array}{l} k_{2}=\frac{V_{\mathrm{mo}} \theta}{2 N K(\cos \theta-1)}+ \\ \frac{\left(C_{\mathrm{r}} R_{\mathrm{mL}} V_{\mathrm{mo}} V_{\mathrm{i}} f_{\mathrm{s}}-N C_{\mathrm{r}} R_{\mathrm{mL}} V_{\mathrm{i}}^{2} f_{\mathrm{s}}-N V_{\mathrm{mo}}^{2}\right) \sin \theta}{2 N R_{\mathrm{mL}} f_{\mathrm{s}} C_{\mathrm{r}} V_{\mathrm{i}}(\cos \theta-1)}. \end{array}\)       (27)

The derivation for the P subinterval is thus completed. (The discussion of the O subinterval is similar to that of the P subinterval.)

From Fig. 11(b), the circuit can be described as

\(\left(L_{\mathrm{r}}+L_{\mathrm{m}}\right) C_{\mathrm{r}} \frac{d^{2}}{d t^{2}} v_{\mathrm{Cr}}+v_{\mathrm{Cr}}=V_{\mathrm{i}}.\)       (28)

The general solution for V_cr can be obtained as

\(v_{\mathrm{Cr}}(t)=k_{3} \cos \left(\omega_{\mathrm{m}} t\right)+k_{4} \sin \left(\omega_{\mathrm{m}} t\right)+V_{\mathrm{i}}.\)       (29)

From Equation (29), the general solution for i_Lr is derived as

\(i_{\mathrm{Lr}}(t)=C_{\mathrm{r}} \omega_{\mathrm{m}}\left[-k_{3} \sin \left(\omega_{\mathrm{m}} t\right)+k_{4} \cos \left(\omega_{\mathrm{m}} t\right)\right].\)       (30)

Similarly, k₃ and k₄ are determined on the basis of the initial circuit condition.

The resonant capacitor voltage V_cr at t_x calculated by two subintervals should be the same due to the continuity of V_cr. In the P subinterval, with the substitution of ω_rt_x=θ into Equation (15), the resonant capacitor voltage at t_x can be expressed as

\(V_{\mathrm{Cr}, \mathrm{tx}}=k_{1} \cos \theta+k_{2} \sin \theta+V_{\mathrm{i}}-\frac{V_{\mathrm{mo}}}{2 N}.\)       (31)

In the O subinterval, t_x is the initial point; with the substitution of ω_mt_x=0 into Equation (29), the resonant capacitor voltage at t_x can be derived as

\(V_{\mathrm{Cr}, \mathrm{tx}}=k_{3}+V_{\mathrm{i}}.\)       (32)

From Equations (31) and (32), the following equation can be obtained:

\(k_{3}=k_{1} \cos (\theta)+k_{2} \sin (\theta)-\frac{V_{\mathrm{mo}}}{2 N}.\)       (33)

Similarly, the resonant inductor current i_Lr at t_x calculated by two subintervals should be the same due to the continuity of i_Lr. In the P subinterval, with the substitution of ω_rt_x=θ into Equation (16), the expression of the resonant inductor current at t_x can be derived as

\(I_{\mathrm{Lr}, \mathrm{tx}}=C_{\mathrm{r}} \omega_{\mathrm{r}}\left(-k_{1} \sin \theta+k_{2} \cos \theta\right).\)       (34)

In the O subinterval, with the substitution of ω_rt_x =0 into Equation (30), the expression of the resonant inductor current at t_x can be calculated as

\(I_{\mathrm{Lr}, \mathrm{tx}}=k_{4} C_{\mathrm{r}} \omega_{\mathrm{m}}.\)       (35)

From Equations (13), (34), and (35), the following equation can be obtained:

\(k_{4}=\sqrt{(K+1)}\left(-k_{1} \sin \theta+k_{2} \cos \theta\right).\)       (36)

From Equations (15), (16), (24), (27), (29), (30), (33), and (36), for quantitative calculation in PO mode, the relationship between C_r and θ needs to be determined.

With the substitution of ω_rt_f=0 into Equation (15), the resonant capacitor voltage at t_f can be expressed as

\(V_{\mathrm{Cr}, \mathrm{tf}}=k_{1}+V_{\mathrm{i}}-\frac{V_{\mathrm{mo}}}{2 N}.\)       (37)

With the substitution of ω_mt_c=φ into Equation (29), the resonant capacitor voltage at t_c can be calculated as

\(V_{\mathrm{Cr}, \mathrm{tc}}=k_{3} \cos \varphi+k_{4} \sin \varphi+V_{\mathrm{i}}.\)       (38)

From Equations (18), (37), and (38), the following equation is obtained:

\(k_{3} \cos \varphi+k_{4} \sin \varphi=-k_{1}+V_{\mathrm{i}}-\frac{V_{\mathrm{mo}}}{2 N}.\)       (39)

Fig. 7 shows that the resonant inductor current value at t_f equals the value at t_c with the opposite sign; thus,

\(I_{\mathrm{Lr}, \mathrm{tf}}+I_{\mathrm{Lr}, \mathrm{tc}}=0.\)       (40)

In the P subinterval, with the substitution of t=0 into Equation (16), the resonant inductor current value at t_f can be calculated as

\(I_{\mathrm{Lr}, \mathrm{tf}}=C_{\mathrm{r}} \omega_{\mathrm{r}}\left(-k_{1}+k_{2}\right).\)       (41)

In the O subinterval, with the substitution of ω_mt_c=φ into Equation (30), the resonant inductor current value at t_c can be calculated as

\(I_{\mathrm{Lr}, \mathrm{tc}}=C_{\mathrm{r}} \omega_{\mathrm{m}}\left(-k_{3} \sin \varphi+k_{4} \cos \varphi\right).\)       (42)

From Equations (13), (40), (41), and (42), the following equation is derived:

\(-k_{3} \sin \varphi+k_{4} \cos \varphi=\sqrt{K+1}\left(k_{1}-k_{2}\right).\)       (43)

With the combination of Equations (39) and (43), the following equation can be derived:

\(k_{3}^{2}+k_{4}^{2}=(K+1)\left(k_{1}-k_{2}\right)^{2}+\left(k_{1}-V_{\mathrm{i}}+\frac{V_{\mathrm{mo}}}{2 N}\right)^{2}.\)       (44)

From Equations (24), (27), (33), (36), and (40), the relationship between C_r and θ can be derived as

\(\begin{aligned} &C_{\mathrm{r}}\left(\begin{array}{l} 4 R_{\mathrm{mL}} M_{\mathrm{m}} f_{\mathrm{s}} / N-4 R_{\mathrm{mL}} M_{\mathrm{m}}^{2} f_{\mathrm{s}} \\ +R_{\mathrm{mL}} f_{\mathrm{s}} \theta^{2}+4 R_{\mathrm{mL}} f_{\mathrm{s}} \cos \theta \\ +R_{\mathrm{mL}} M_{\mathrm{m}} f_{\mathrm{s}} \theta^{2} / N- \\ 4 R_{\mathrm{mL}} M_{\mathrm{m}} \cos \theta / N+ \\ R_{\mathrm{mL}} f_{\mathrm{s}} \theta^{2} \cos \theta+ \\ R_{\mathrm{mL}} M_{\mathrm{m}} f_{\mathrm{s}} \theta^{2} \cos \theta / N- \\ 4 R_{\mathrm{mL}} M_{\mathrm{m}} f_{\mathrm{s}} \theta \sin \theta / N \end{array}\right)\\ &-\left(\begin{array}{l} 4 M_{\mathrm{m}}^{2}-4 N M_{\mathrm{m}} \\ +M_{\mathrm{m}}^{2} \theta^{2}-4 M_{\mathrm{m}}^{2} \cos \theta \\ +4 N M_{\mathrm{m}} \cos \theta+ \\ M_{\mathrm{m}}^{2} \theta^{2} \cos \theta- \\ 4 M_{\mathrm{m}}^{2} \theta \sin \theta \end{array}\right)=0 \end{aligned}\)       (45)

When the DCX is operating in PO mode, with a given resonant capacitor C_r, the corresponding θ can be calculated according to Equation (45). Then, given K, the values of k₁, k₂, k₃, and k₄ can be derived according to Equations (24), (27), (33), and (36), and the value of φ can be obtained on the basis of Equation (39) or (43). Finally, the quantitative calculation of the circuit can be performed.

From Fig. 7, the following expression can be obtained:

\(\frac{\theta}{\omega_{\mathrm{r}}}+\frac{\varphi}{\omega_{\mathrm{m}}}=\frac{1}{2 f_{\mathrm{s}}}.\)       (46)

Equation (46) can be changed into the form of Equation (47).

\(\frac{\theta}{\omega_{\mathrm{r}}}+\frac{\varphi}{\omega_{\mathrm{r}}} \sqrt{K+1}=\frac{1}{2 f_{\mathrm{s}}}.\)       (47)

The values of θ and φ can be determined by the given C_r and K. Then, with the substitution of θ and φ into Equation (47), the value of ω_r can be obtained. Finally, according to Equation (13), the values of the resonant inductor L_r and magnetic inductor L_m can be calculated. The boundary condition and values of C_r and K need to be discussed to finalize the design.

According to [22], the adjunct operation modes of PO mode are the PON and OPO modes. The boundary condition between the PO and PON modes is as follows. When the DCX is operating in PO mode, at t_c, the voltage across the magnetic inductor V_Lm,tc equals −V_mo/2N. Therefore, for operation in PO mode, the following inequality should be satisfied:

\(V_{\mathrm{Lm}, \mathrm{tc}} \geq-\frac{V_{\mathrm{mo}}}{2 N}.\)       (48)

When the DCX is operating in the O subinterval, according to the voltage divider law, the voltage across the magnetic inductor V_Lm,tc can be expressed as

\(V_{\mathrm{Lm}, \mathrm{tc}}=\frac{K}{K+1}\left(V_{\mathrm{i}}-V_{\mathrm{Cr}, \mathrm{tc}}\right).\)       (49)

From Equations (18) and (49), the voltage across the magnetic inductor V_Lm,tc can be expressed as

\(V_{\mathrm{Lm}, \mathrm{tc}}=\frac{K}{K+1} V_{\mathrm{Cr}, \mathrm{tf}}.\)       (50)

With the combination of Equations (23), (48), and (50), the inequality of the load can be derived as

\(R_{\mathrm{mL}} \leq \frac{M_{\mathrm{m}}}{C_{\mathrm{r}} f_{\mathrm{s}}\left(\frac{1}{M_{\mathrm{m}}}+\frac{1}{N K}-\frac{1}{N}\right).}\)       (51)

Similarly, the boundary condition between the PO and OPO modes can be described as follows. When the DCX is operating in PO mode, at t_f, the voltage across the magnetic inductor V_Lm,tf is V_mo/2N. Therefore, for operation in PO mode, the following inequality should be satisfied:

\(\left.L_{\mathrm{m}} \frac{d i_{\mathrm{Lm}}}{d t}\right|_{t=t_{\mathrm{f}}}=\left.L_{\mathrm{m}} \frac{d i_{\mathrm{Lr}}}{d t}\right|_{t=t_{\mathrm{f}}} \geq \frac{V_{\mathrm{mo}}}{2 N}.\)       (52)

With the combination of Equations (13), (16), (24), and (52), the inequality of the load can be derived as

\(R_{\mathrm{mL}} \geq \frac{M_{\mathrm{m}}}{C_{\mathrm{r}} f_{\mathrm{s}}\left(\frac{1}{M_{\mathrm{m}}}+\frac{1}{N K}+\frac{1}{N}\right).}\)       (53)

C. Parameter Design and Simulation Verification

On the basis of the above analysis, the proposed optimal design method can be summarized as follows.

1) According to the known parameters, the value of M_m, N and R_mL are determined.

2) According to Inequalities (51) and (53), the figures, which show the boundary condition of PO mode, are obtained. On the basis of the figures, the reasonable range of C_r and K can be found.

3) The range of C_r and K is further constrained by optimizing efficiency; one is the soft-switching capability, which is related to switching losses; the other is the RMS current of the resonant tank, which is related to the conduction losses.

4) Reasonable values of C_r and K based on 2) and 3) are selected.

5) Then, the value of θ can be calculated according to Equation (45).

6) The values for k₁, k₂, k₃, and k₄ are obtained on the basis of Equations (24), (27), (33), and (36).

7) The value of φ can be determined according to Equation (39) or (43).

8) The value of ω_r is calculated according to Equation (47).

9) The values of the resonant inductor L_r and magnetic inductor L_m can be determined according to Equation (13).

On the basis of the above analysis, M_m=6.5 and N=6.25 are selected. When the input voltage equals the maximum value V_in-max, the output voltage of the DCX reaches the maximum value V_mo-max. If the output current equals the minimum value I_o-min, then the equivalent resistance of the DCX reaches its maximum value R_mL-max, and it can be expressed as

\(R_{\mathrm{mL}-\max }=\frac{V_{\mathrm{mo}-\max }}{I_{\mathrm{o}-\min }}=\frac{M_{\mathrm{m}} V_{\mathrm{i}-\max } V_{\mathrm{o}}}{P_{\mathrm{o}-\min }}=1560 \Omega.\)       (54)

Similarly, the minimum value of R_mL can be derived as

\(R_{\mathrm{mL}-\min }=\frac{V_{\mathrm{mo}-\min }}{I_{\mathrm{o}-\max }}=\frac{M_{\mathrm{m}} V_{\mathrm{i}-\min } V_{\mathrm{o}}}{P_{\mathrm{o}-\max }}=260 \Omega.\)       (55)

According to Inequalities (51) and (53), by using MATLAB, a three-dimensional graph regarding C_r, K, and R_mL is obtained as shown in Fig. 12(a). When the DCX is operating between two surfaces, PO mode operation is guaranteed for the DCX. For ensuring that the DCX is always operating in PO mode during a wide load range, a plane R_mL=1560 Ω is made, and the intersection line is the red line on the upper surface; similarly, a plane R_mL=260 Ω is made, and the intersection line is the blue line on the lower surface. Fig. 11(b) is the overhead view of Fig. 11(a), and the intersection point A shown in Fig. 11(b) can be calculated according to Inequalities (51) and (53); the result is (0.6771uF, 2.8889).

Fig. 12. Boundary conditions among different operating modes.

To verify the accuracy of the theoretical analysis, point B (0.75uF, 8.21) at the boundary between PON and PO modes and point C (0.8uF, 3.484) at the boundary between OPO and PO modes are selected. At point B, R_mL=260 Ω, and L_r=2.67uH and L_m=21.91uH can be calculated according to the design procedures. At point C, RmL=1560 Ω; similarly, Lr=2.85uH and Lm=9.94uH can be calculated. In Fig. 13, the simulation waveforms at point B are shown, and the DCX is operating in PO mode; at t_c, the voltage across the magnetic inductor V_Lm,tc is −V_mo/2N, which means that the DCX is operating at the boundary between the PO and PON modes. Fig. 14 shows the simulation waveforms at point C. Similarly, at t_f, the voltage across the magnetic inductor V_Lm,tf is V_mo/2N, which means that the DCX is operating at the boundary between the PO and OPO modes. Therefore, the simulation results agree with the theoretical analysis findings.

Fig. 13. Simulation waveforms at point B.

Fig. 14. Simulation waveforms at point C.

To ensure that switches S_a1 and S_a2 can achieve ZVS operation, the charge stored in the drain-to-source capacitance C_ds of the switch should be extracted completely during the dead time t_dead. The resonant current during the dead time is nearly the same as in the P subinterval, and the dead time is extremely shorter than the switch period; thus, the resonant current can be linearized. Therefore, the condition for ZVS operation can be derived as

\(Q=\left[\left|i_{\mathrm{Lr}}(0)\right|+\left|i_{\mathrm{Lr}}\left(t_{\mathrm{dead}}\right)\right|\right] t_{\mathrm{dead}} / 2 \geq 2 C_{\mathrm{ds}} V_{\mathrm{i}-\mathrm{max}}.\)       (36)

In steady state, the output power is confirmed, and the primary side current is constant; thus, if the converter can achieve ZVS operation when the input voltage is at its maximum value, then ZVS operation for the whole operation range can be guaranteed. In this research, the dead time is selected to be 80 ns, and C_ds is 1.92nF; therefore, we can calculate the value on the right side of Inequality (56), which is 230 nF. Then, i_Lr(0) and i_Lr(tdead) can be calculated according to Equation (16), and ω_r, k₁, and k₂ can be determined on the basis of the proposed design method. Finally, a three-dimensional graph regarding C_r, K, and Q can be obtained as shown in Fig. 15(a). Fig. 15(b) is obtained by combining the intersection line in Fig. 15(a) with that in Fig. 12(b).

Fig. 15. Boundary condition for soft switching.

The RMS current of the resonant tank is related to the conduction losses; thus, a limitation on I_Lr is required to improve efficiency further. In the following section, FHA is adopted to approximate the limitation value of I_Lr.

According to Equation (31), the fundamental voltage RMS value of the half-bridge input voltage can be expressed as

\(V_{\mathrm{HB}}=\frac{\sqrt{2}}{\pi} V_{i}.\)       (57)

Given an assumption that the fundamental power factor λ≥0.95, without consideration of the losses, the expression of λ can be derived as

\(\lambda=\frac{V_{\mathrm{mo}} I_{\mathrm{o}}}{V_{\mathrm{HB}} I_{\mathrm{Lr}}} \geq 0.95.\)       (58)

According to Equation (58), the limitation for I_Lr can be expressed as I_Lr≤15.2 A. On the basis of the design procedures, the RMS value of I_Lr can be obtained. Fig. 16(a) shows a threedimensional graph of C_r, K, and RMS current of I_Lr. The shaded area of Fig. 16(b) shows the final range of C_r and K.

Fig. 16. Limitation of the resonant current.

Theoretically, all points in the shaded area can be selected. However, in practice, the following conditions are considered:

1) the commercially available capacitor values and their corresponding voltage stresses;

2) a high K, which usually reduces the resonant tank RMS current while increasing in size.

Therefore, a reasonable value based on these considerations must be selected. In the experiment, C_r=0.97uF and K=5 are selected, and the corresponding resonant components are calculated as

\(\left\{\begin{array}{l} C_{\mathrm{r}}=0.97 \mu \mathrm{F} \\ L_{\mathrm{r}}=2.27 \mu \mathrm{H} \\ L_{\mathrm{m}}=11.2 \mu \mathrm{H} \end{array}\right..\)       (59)

Based on the input and output of the auxiliary converter, L_a=1mH and C_a=10µF are selected for the auxiliary converter.

V. EXPERIMENTAL RESULTS

A 400 W experimental prototype based on the calculated parameters is built to verify the theoretical analysis. Fig. 17(a) shows the setup of the system, and Fig. 17(b) shows the experimental prototype of the proposed converter. Table I shows the circuit parameters and component part numbers for the proposed converter.

Fig. 17. Photographs of experimental setup. (a) System setup. (b) Experimental prototype of the proposed converter.

TABLE I CIRCUIT PARAMETERS AND COMPONENT PART NUMBERS

Fig. 18(a) shows the experimental waveforms when the input voltage is 40 V. Under this condition, S_a1 of the auxiliary converter is always on, the output voltage is regulated by controlling the duty cycle of S_a2, the auxiliary converter is working in boost mode, the output voltage V_ao is 138 V, and the voltage gain M_a is 3.45. The output voltage of the DCX is 262 V, and its voltage gain M_m is 6.55.

Fig. 18. Experimental waveforms with different input voltages. (a) V_i=40 V. (b) V_i=51 V. (c) V_i=54 V. (d) V_i=60 V.

Fig. 18(b) shows the experimental waveforms when the input voltage is 51 V. Under this condition, S_a1 of the auxiliary converter is operating at a constant duty cycle of 0.8, the output voltage is regulated by controlling the duty cycle of Sa2, the auxiliary converter is working in buck-boost mode, the output voltage V_ao is 62 V, and the voltage gain M_a is 1.21. The output voltage of the DCX is 338 V, and its voltage gain M_m is 6.62.

Fig. 18(c) shows the experimental waveforms when the input voltage is 54 V. Under this condition, S_a2 of the auxiliary converter is operating at a constant duty cycle of 0.2, the output voltage is regulated by controlling the duty cycle of S_a1, the auxiliary converter is working in buck-boost mode, the output voltage V_ao is 44 V, and the voltage gain M_a is 0.81. The output voltage of the DCX is 359 V, and its voltage gain M_m is 6.64.

Fig. 18(d) shows the experimental waveforms when input voltage is 60 V. Under this condition, S_a2 of the auxiliary converter is always off, the output voltage is regulated by controlling the duty cycle of S_a1, the auxiliary converter is working in buck mode, the output voltage V_ao is 12 V, and the voltage gain M_a is 0.2. The output voltage of the DCX is 388 V, and its voltage gain M_m is 6.46.

During the whole input voltage range, the auxiliary converter has boost, buck-boost, and buck operation modes, and the voltage gain of the DCX nearly remains constant at 6.5, which agrees with the theoretical analysis. Notably, the resonant inductor current i_Lr, which is also the input current of the DCX, almost remains constant. This phenomenon can be explained by power conservation; the output power and output voltage are constant in different conditions, and the output current is constant. Meanwhile, a constant gain conversion is achieved in the DCX; thus, the resonant inductor current depends on the output current.

Fig. 19 shows the ZVS operation of S_m2 under the most difficult condition for achieving ZVS, that is, when the input voltage reaches its maximum value (V_i=60 V), and the output power reaches its minimum value (P_o=100 W). Before its drive signal Q_sm2 comes, the resonant current i_Lr is positive. Thus, the drain-to-source current of S_m2 is negative, which means its parallel diode is conducting. Therefore, the voltage across S_m2 becomes zero, and ZVS operation is achieved. Similarly, S_m1 can achieve ZVS operation.

Fig. 19. ZVS operation waveforms of S_m2 when V_i=60 V and P_o=100 W.

Fig. 20 shows the theoretical and experimental results of the ratio of the DCX transmitted power to total power with different input voltages. These results agree with the theoretical analysis findings. Therefore, with increased input voltage, the power transmitted by the DCX increases.

Fig. 20. Theoretical and experimental results of the ratio of the DCX transmitted power to total power with different input voltages.

Fig. 21 shows the dynamic response of the proposed converter when the output power changes from full power to half power. The DCX is operating at the resonant frequency operation point; thus, this system is not sensitive to load changes. Therefore, it requires a short time to regulate the output voltage.

Fig. 21. Dynamic response of the proposed converter when load changes from full power to half power.

Fig. 22 shows the efficiency curves of the system under different input voltages, including auxiliary converter efficiency η_a, DCX efficiency η_dcx, and overall efficiency η. η_dcx is higher than η_a, and with an increase in the input voltage, the difference between η_dcx and η becomes smaller. In addition, the system efficiency reaches its highest point at 95.56% when input voltage is 52 V.

Fig. 22. Efficiency curve with different input voltages.

Fig. 23 shows the efficiency comparison results among different topologies. The proposed converter has the highest efficiency of nearly 97.2% at 100 W, and with the increase in output power, the efficiency decreases, and the efficiency at rated power (400 W) is approximately 95.56%. The system efficiency is above 95% when the output power is higher than 100 W. The main reason behind this trend is that the converters’ losses are proportional to the operating current. Therefore, with the increase in output power, the operating current increases, and the losses increase, leading to low efficiency.

Fig. 23. Efficiency comparison among different topologies.

Fig. 24 shows the power loss distribution of the proposed converter with rated output power (400 W) and input voltage V_i=52 V. Clearly, switch loss and magnetic loss are dominant, and both are proportional with the system operating current. This finding can explain why the proposed converter has low efficiency at high output power.

Fig. 24. Power loss distribution when V_i=52 V and P_o=400 W.

VI. CONCLUSION

In this study, an input-parallel output-series function decoupling high gain DC/DC converter that consists of a DCX and an auxiliary converter is proposed. The topology selection and control strategy for the DCX and auxiliary converter are discussed. On the basis of the discussion, an optimal design method, which is also the main contribution of this study, is proposed. On the basis of the proposed method, quantitative calculation for the DCX can be achieved, which can guarantee the DCX operating in PO mode; thus, ZVS and ZCS operation for primary switches and secondary diodes can be realized. Meanwhile, by combining with other limitations, the system efficiency can be further optimized. Finally, the quantitative design for the system can be achieved. A simulation and an experiment are implemented to verify the accuracy of the theoretical analysis. The main contribution of this study is summarized as follows:

1) a function decoupling high voltage gain DC/DC converter composed of an LLC resonant converter (acting as a DCX to transfer most of the system’s power) and a noninverted buck-boost converter (acting as an auxiliary converter to regulate system output);

2) with the adoption of the function decoupling idea, a constant duty cycle and switching frequency for the LLC resonant converter can be implemented, thereby simplifying the magnetic components and EMI design;

3) instead of a two-mode operation for the noninverted buck-boost converter, a four-mode operation strategy is adopted to avoid extreme duty cycles;

4) an optimal design strategy is proposed. Initially, resonant capacitor C_r and inductor ratio K are selected as input parameters; with the addition of an operation mode boundary limitation (ZVS operation and RMS current limitations), an accurate parameter range is determined.