https://doi.org/10.6113/JPE.2019.19.1.24

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718

Analysis and Modeling of AC–AC Switched Capacitor Converters

Hui Cai^*, Liting Bao^*, Qian Guo^†, Ying Wang^*, and Weimin Chen^*

^†,*College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou, China

Abstract

A new modeling method for AC–AC switched capacitor converters (SCCs) is introduced in this study. The proposed analytical method aims to accurately describe the input–output characteristics of AC–AC SCCs and establish a mathematical model for static voltage conversion ratio and equivalent resistance, which are key performance metrics for SCCs. A quantitative analysis of converter regulation capability is addressed on the basis of the modeling method. In this analysis, the effects of the control parameters and individual components on SCCs are illustrated extensively. Component stresses, such as the peak value and transient variation of the voltage/current of the converter, are also presented. The effectiveness of the proposed method is verified by comparing it with the existing modeling method and applying it to an AC–AC SCC with a conversion ratio of three. Two 1 kW prototypes are built in a laboratory, and their experimental results exhibit good agreement with the theoretical analysis.

Key words: AC–AC converters, Modeling, Switched capacitor, Transient analysis

Manuscript received Oct. 6, 2017; accepted Oct. 15, 2018

Recommended for publication by Associate Editor Sangshin Kwak.

^†Corresponding Author: guoqmail@cjlu.edu.cn Tel: +86-571-86914550, China Jiliang University

^*College of Mechanical and Electrical Eng., China Jiliang Univ., China

Ⅰ. INTRODUCTION

Switched capacitor converters (SCCs) have been an important research topic for many years. They present the advantages of high power density, small size, and low weight due to the absence of inductors or transformers [1]. Different circuit models have been presented for DC–DC SCCs [2], [3]. Applications that benefit from the switched capacitor (SC) principle include power supplies for mobile electronic systems, electric vehicles, and battery equalizer circuits [4]-[8]. Reference [9] extended the SC principle to AC–AC static conversion for the first time. This literature presented a brief analysis and the experimental results of a step-down/step-up converter. On the basis of [9] and by using bidirectional switches, a new SC- based bidirectional AC–AC static power converter with novel characteristics, such as the stress voltages on the components being equal to half of the high-side voltage and the use of a single SC leg, was introduced in [10]. SCCs have also been used in three-phase systems for high-power applications [11]-[13].

An SCC is characterized by a conversion ratio and an equivalent resistance [14], as shown in Fig. 1, which are determined by a selection of switching frequency, duty cycle, and other component parameters. An understanding of the basic mechanisms is crucial to achieve optimized performance. Previous studies have provided several modeling methods for DC–DC SCCs. A unified state-space averaging approach for switching mode converters, which combined the state-space and averaging methods, was presented in [15], but the derived mathematical model did not consider switching frequency. References [16] and [17] offered a behavioral average circuit model, which is valid for SCCs within all operational ranges, but is limited by the derivation assumptions that subcircuits can be described or approximated by a first- order resistor–capacitor circuit. Another modeling approach addressed in [18] analyzes an individual SC cell to derive an equivalent resistance formula for various operation modes but cannot be used to provide generic symbolic results for a complete converter. The charge-balance transient-calculation modeling method introduced in [19] and elaborated in [20] provides an accurate prediction of circuit behavior under varying duty cycle and frequency conditions; this method has been applied to obtain a steady-state model and peak current formulas. Nevertheless, modeling methods for SCCs that are implemented for static AC–AC conversion are rarely found in the literature.

Fig. 1. Ideal model of an SCC.

Motivated by References [19], [20], the primary objective of the current study is to find an appropriate modeling method for AC–AC SCCs. The input–output characteristics of AC–AC SCCs can be accurately described using the proposed method. A mathematical model is also built for calculating static voltage conversion ratio and equivalent resistance. The major characteristics and transient behavior of the converter are analyzed in detail. The impacts of the electric elements on the converter are also illustrated. Component stresses, such as the voltage variation and peak voltage value of the capacitors, are addressed in detail to enhance the understanding of SCCs. The experimental results confirm the correctness of the proposed modeling method.

Ⅱ. MODELING OF AN AC–AC SCC WITH A CONVERSION RATIO OF TWO

In this section, a typical step-up AC–AC SCC, as shown in Fig. 2(a), is analyzed using the conventional and proposed methods. The analyzed converter has a single SC cell, and complex converters that consist of multistage SC cells can be generated. The converter has four groups of bidirectional switches, and each group consists of two metal-oxide- semiconductor field-effect transistors (MOSFETs) that share a common source.

Fig. 2. Topology of the analyzed AC–AC SCC and operational high-frequency waveforms of capacitors: (a) Step-up converter; (b) Low-frequency input–output voltages and voltages across the capacitors; (c) Voltages; (d) Currents.

(a)

(b)

(c)

(d)

The expected low-frequency waveforms of the input–output voltages and the voltages across the capacitors are presented in Fig. 2(b), with the input voltage as the peak value. The gate drive signals and the operational high-frequency waveforms of the capacitors are shown in Figs. 2(c) and 2(d).

Three operational modes, namely, partial charge (PC), no charge, and complete charge (CC), exist for the capacitors depending on the relationship between the duration and the time constant [17]. The converter analyzed in this section operates in PC mode due to the trade-off between performance and cost.

To simplify the analysis, all MOSFETs are assumed to have the same on resistance R_dson and all capacitors have the same capacitance C and equivalent series resistance (ESR).

A. Conventional Modeling Method

In accordance with the modeling method presented in [18] and its application example in [10], the equivalent resistance of the circuits shown in Fig. 2(a) is defined in (1) for different duty cycle values.

 (1)

where f_s is the switching frequency, R_on is the total on resistance of a bidirectional switch that is twice that of a MOSFET, and D is the duty cycle.

When D is fixed at 0.5, the limit value of the equivalent resistance is

                                (2)

B. Proposed Modeling Method

A new modeling technique for conversion ratio and equivalent resistance based on the state equations derived from Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law can be used for the circuit in Fig. 2(a). This modeling technique for AC–AC SCCs can also be used to model its DC–DC counterpart due to the similar subcircuits and operating principle.

Fig. 3. Subcircuits of an SCC: (a) Stage 1; (b) Stage 2.

(a)

(b)

The converter has two operational stages. The subcircuits are presented in Fig. 3. Stage 1, as shown in Fig. 3(a), starts when switches S₁ and S₃ are turned on and capacitor C₁ is charged by the input voltage source throughout the stage. Accordingly,

           (3)

where r is the ESR of the capacitor.

When the boundary condition is considered,

                                    (4)

The switching frequency is relatively higher than the grid frequency; therefore, the input voltage is assumed to be constant during a switching period, and the following equation can be derived:

  (5)

The maximum voltage of C₁ can be derived as

                             (6)

Given the subcircuits in Fig. 3(b), the charge stored in C₁ is delivered to C₂ during Stage 2 [DT_s, T_s]. The following equation based on charge balance can be derived:

                     (7)

The voltage across the load is nearly imperceptible; thus, the load current can be assumed to be a constant current source.

                                        (8)

The analyzed converter is symmetrical; hence, the output voltage can be expressed as

                             (9)

From (3)–(9), the conversion ratio of the projected converter can be derived as

    (10)

When R_L→∞, the ideal conversion ratio of the converter can be obtained as

                                (11)

From the idealized model of SCC in Fig. 1, the equivalent resistance can be derived as

      (12)

When D is fixed at 0.5 and f_s approaches infinity, the minimum limit of equivalent resistance can be derived as

     (13)

The deduced limit value of equivalent resistance presented in (13) coincides with that in (1), which is 16 times of a single MOSFET’s on resistance.

The modeling method is also valid for an SCC with a step-down configuration.

C. Comparison Study of the Two Methods

The two modeling methods include frequency and duty cycle regulation and involve the impacts of capacitors, on resistance, duty cycle, and switching frequency. A comparison of equivalent resistance between the two modeling methods under varying duty cycle and switching frequency conditions is presented in Fig. 4 and the experimental curve. The component parameters selected for the theoretical analysis and experiments are provided in Table I.

Fig. 4. Behavior comparison of the two modeling methods and the experimental results: (a) Conversion ratio vs. duty cycle; (b) Equivalent resistance vs. duty cycle.

(a)

(b)

TABLE I COMPONENT PARAMETERS FOR THE THEORETICAL ANALYSIS AND EXPERIMENTS

Fig. 4	Vin_rms (V)	f (Hz)	fs (kHz)	Ron (Ω)	C (μF)	D	RL (Ω)
(a)	110	50	50	0.07	10	0.05–0.5	100
(b)	110	50	50	0.07	10	0.05–0.5	100

The curves in Fig. 4 show that compared with the conventional modeling method, the proposed method tends to exhibit better agreement with the experimental results. The primary reason for the imprecision of the conventional method is that the voltage ripples in C₂ and C₃ are assumed to be constant in the modeling process, and the ESR of the capacitors is disregarded. Discrepancies in the proposed method occur when the duty cycle is between 0.05 and 0.10 because the actual value of the duty cycle deviates from the setting value as the impacts of finite dead time, rise time, and fall time become significant when D tends to approach zero (or is extremely small). Nevertheless, the proposed method is suitable for the modeling of AC–AC SCCs.

Ⅲ. APPLICATION OF THE PROPOSED MODELING METHOD

A. AC–AC SCC with a Conversion Ratio of Three

In this section, a step-up AC–AC SCC with a conversion ratio of three is analyzed using the proposed method to demonstrate the effectiveness of this method. The converter topology is presented in Fig. 5.

Fig. 5. Topology of an AC–AC SCC with a conversion ratio of three.

The analyzed converter consists of five capacitors and six bidirectional switches and can achieve step-up and step-down configurations with minimal change in the points where the input source and the load are connected. The gate drive signals are the same as those in Section II.A. The converter has two major operation stages as shown in Figs. 6(a) and 6(b).

Fig. 6. Operation stages of an SCC and high-frequency waveforms of capacitors: (a) Stage 1; (b) Stage 2; (c) Voltages; (d) Currents.

(a)	(b)
(c)	(d)

Stage 1 starts when switches S₁, S₃, and S₅ are turned on. Capacitor C₁ is charged by C₄, and C₅ is charged by the input voltage source through C₂. C₃ discharges and delivers energy to the load throughout the stage.

Stage 2 starts when S₂, S₄, and S₆ are turned on and S₁, S₃, and S₅ are turned off. Capacitor C₃ is charged by C₅, and C₄ is charged by the input voltage source through C₂. C₁ discharges and delivers energy to the load throughout the stage.

The voltage across capacitor C₂ is always equal to the input source during the two stages.

The expected operation waveforms of voltages and currents are presented in Figs. 6(c) and 6(d), respectively. The converter is assumed to work in PC mode.

The current waveforms of capacitors C₁ and C₃ are nearly identical to those in Fig. 2(d) given that they all receive energy in one working stage (PC mode) and deliver energy to the load in the other. However, the waveforms in CC mode differ and are further illustrated in Section III.D. The current waveforms of capacitors C₄ and C₅ vary from those of C₁ because they work in PC mode throughout the charge–discharge cycle.

B. Modeling of the Converter

The subcircuits of the converter can be obtained by analyzing the operational stages and expected waveforms of the components, as shown in Figs. 7(a) and 7(b).

Fig. 7. Subcircuits of an SCC: (a) Stage 1; (b) Stage 2.

(a)

(b)

To simplify the analysis, all MOSFETs are assumed to have the same on resistance R_dson and all capacitors have the same capacitance C. The ESR of the capacitor is disregarded.

On the basis of the subcircuits in Fig. 7(a), when t∈ [0, DT_s], we yield

        (14)

From KCL,

                (15)

For the charging loop of C₅ in Fig. 7(b), the following equation can be derived:

   (16)

Considering the law of charge–balance and assuming the load current to be constant due to the high frequency, the following equations can be derived:

              (17)

                    (18)

Given the symmetrical structure of the converter, the output voltage can be expressed as follows:

                  (19)

From (14)-(19) and considering the boundary conditions, the expression of the conversion ratio is obtained as follows:

   (20)

The equivalent resistance can be derived as

   (21)

The currents that flow through capacitors C₄ and C₅ are numerically identical due to the similar charge–discharge process. Therefore, when decoupling the subcircuits, the on resistance of S₃ can be equalized to 2R_on, which leads to R=3R_on.

From (20), the ideal conversion ratio can be derived as the load approaches infinity,

   (22)

The minimum limit of equivalent resistance can be derived as follows (D=0.5):

                  (23)

The obtained R_eqmin is represented in the same form as that of the converter in Fig. 2(a).

C. Analysis of Control and Component Parameters

The model derived in Section III.B can be used to provide a solid design principle for the converter in terms of modulation and selection of component parameters. Capacitance, resistance, duty cycle, switching frequency, and load will exert impacts on the conversion ratio. The selection of component parameters is provided in Table II.

TABLE II CONTROL AND COMPONENT PARAMETERS FOR THEORETICAL ANALYSIS

Fig. 8	f (Hz)	fs (kHz)	Ron (Ω)	C (μF)	D	RL (Ω)
(a)	50	10–100	0.07	1,10,100	0.05–0.5	100
(b)	50	10,50,100	0.07	10	0.5	100
(c)	50	10,50,100	0.07	10	0.5	100

The effects of the converter parameters are illustrated in Fig. 8.

Fig. 8. Effects of control and converter components: (a) Conversion ratio vs. duty cycle; (b) Conversion ratio vs. switching frequency; (c) Conversion ratio vs. loop resistance.

(a)

(b)

(c)

The duty cycle can be used to regulate converter output. As shown in Fig. 8(a), the conversion ratio increases with an increment in duty cycle in the “low duty-cycle” region and reaches a limit at approximately 0.3 for the 50 kHz and 100 kHz curves. The conversion ratio reaches an earlier limit for the 10 kHz curve. The duty cycle is strictly limited within 0.5 to avoid shooting through.

Switching frequency can also be used to regulate the conversion ratio. The modulation capability is noticeable within the low-frequency region. Although a high switching frequency leads to a high conversion ratio, switching losses and parasitic elements (e.g., ESR and ESL) prevent it from increasing indefinitely. As shown in Fig. 8(b), the conversion ratio remains nearly unchanged when the frequency reaches 60 kHz for the given parameters. The conversion ratio is slightly higher with a switching frequency of 100 kHz than that of 50 kHz, as shown in Figs. 8(a) and 8(c).

Frequency considerably influences capacitance modulation capability when it is relatively small (e.g., when C=1 μF). Increasing capacitance is another method for raising the conversion ratio. A high charging efficiency is obtained when △V (=V_cmax−V_cmin) is small, and a large capacitance leads to a small △V, and consequently, high efficiency [21].

Loop resistance is the total resistance in the charging and discharging circuits, and it consists of the ESR of capacitors and on resistances. A large loop resistance leads to a small charging current in the path. Loop resistance is supposed to be small for improving efficiency and decreasing conduction losses, as shown in Fig. 8(c).

D. Component Stress Analysis

Using the proposed method, the peak stress of components, such as extreme voltage and current stresses, can be estimated for the analyzed SCC in Fig. 5. The ESR of the capacitor is disregarded to simplify the derivation. The expressions for V_c1(t) during the charging period can be derived as follows:

      (24)

where and .

The SCC operates in different modes due to the varying component parameters of the converter.

The expected operational high-frequency waveforms of the voltage across C₁ in different operation modes can be obtained from the theoretical analysis in Figs. 9(a) and 9(b).

Fig. 9. Expected high-frequency waveforms of capacitor C₁: (a) PC mode; (b) CC mode.

Fig. 9(a) indicates that for the SCC operating in PC mode, C₁ charges throughout Stage 1 [0, DTs] and discharges throughout Stage 2 [DT_s, T_s] in a switching period. Therefore, the maximum value is reached at the end of Stage 1, whereas the minimum value is reached at the end of Stage 2.

Fig. 9(b) indicates that for the SCC operating in CC mode, unlike the converter operating in PC mode, the maximum value is reached at △t during interval [0, DTs]. The capacitor C₁ starts to discharge in Stage 1. The minimum value is reached at the end of Stage 2.

Nevertheless, the high-frequency waveforms of capacitors C₄ and C₅ in CC mode remain the same as those in Figs. 6(c) and 6(d).

For the converter operating in PC mode, the maximum voltage of capacitor C₁ can be derived on the basis of the conversion ratio derivation process described in Section III.B as follows:

             (25)

where .

The minimum voltage of capacitor C₁ can be derived as

      (26)

The input voltage source delivers energy to C₄ through Stage 1, and the peak current occurs at the start of the charging process.

The minimum voltage of C₄ can be derived as

  (27)

Therefore, I_c4max can be derived as follows:

   (28)

where and .

The derived peak currents and voltages can be useful when designing converter topology and selecting control and component parameters. The component stress of other switches and capacitors can be derived by following a similar procedure illustrated in this section.

Ⅳ. EXPERIMENTAL VERIFICATION

A series of experiments is performed with two prototypes in a laboratory to demonstrate the correctness of the proposed modeling method. Prototype 1 is an SCC with a conversion ratio of two, whereas Prototype 2 has a conversion ratio of three. The major specifications and component parameters used in the prototypes are presented in Table III.

TABLE III CONTROL AND COMPONENT PARAMETERS FOR EXPERIMENT VERIFICATION

Description	Value
Output power (Po)	1 kW
Input voltage (Vin)	110 V (50 Hz)
Switching frequency (fs)	50 kHz
Duty cycle (D)	0.5
Capacitance (C)	20 μF
Load (RL)	50 Ω
MOSFETs	FQA62N25C (Rdson=0.035 Ω) IRFP460 (Rdson=0.27 Ω)

A. Experimental Results of Prototype 1

The input and output voltages are presented in Fig. 10(a). The measured output voltage is 216.88 Vrms. The experimental conversion ratio is 1.972, which is close to the ideal value of 2 V_in. The slight voltage drop results from the switching and conduction losses caused by the resistance and other parasitic elements.

Fig. 10. Experimental results of Prototype 1: (a) Waveforms of input and output voltages; (b) Waveform of current through capacitor C₁; (c) Waveform of voltage across switch S₁; (d) Conversion ratio vs. switching frequency.

(a)

(b)

(c)

(d)

The current through capacitor C₁ is presented in Fig. 10(b), and the waveform agrees with the theoretical analysis in Fig. 2(d). Several current spikes occur due to the distortion of the grid voltage.

The voltage across MOSFET S₁ is given in Fig. 10(c). As expected, the voltage stress of the switch is equal to the input voltage and half of the output voltage.

The theoretical curves of the conversion ratio and the switching modulation using the proposed modeling method are compared with the experimental results. The duty cycle regulation curves are presented in Fig. 4(a). The frequency regulation curves are provided in Fig. 10(d). The experimental and theoretical curves exhibit good agreement when switching frequency is between 30 kHz and 60 kHz. The peak ratio for the experimental curve is reached when f_s=60 kHz.

B. Experimental Results of Prototype 2

The input and output voltages are shown in Fig. 11(a). The input voltage is 75 V_rms, and the selection of control and other component parameters is presented in Table III. The measured output voltage is 215.65 Vrms, and the conversion ratio is 2.875.

Fig. 11. Experimental results of Prototype 2: (a) Waveforms of input and output voltages; (b) Waveform of current through capacitor C₄; (c) Waveform of current through C₁ (PC mode); (d) Waveform of current through C₁ (CC mode).

(a)

(b)

(c)

(d)

The current through capacitor C₄ is presented in Fig. 11(b), and the waveform agrees with that in Fig. 6(d). The currents through C₁ in PC and CC modes are shown in Figs. 11(c) and 11(d), respectively. The experimental current waveforms agree with those of the theoretical analysis in Figs. 9(a) and 9(b).

The duty cycle regulation curves are shown in Fig. 12(a). The experimental curve matches with the theoretical curve when the duty cycle is above 0.2. The difference in the two curves is mostly attributed to the finite dead time leading to the deviation of the actual duty cycle from the setting values.

Fig. 12. Duty cycle and frequency regulation curves: (a) Conversion ratio vs. duty cycle; (b) Conversion ratio vs. switching frequency.

(a)

(b)

The frequency regulation curves are shown in Fig. 12(b). The curves are in good agreement when switching frequency is lower than 70 kHz. The deviations result from the effect of parasitic elements (e.g., ESL and ESR) and the increasing switching losses.

Ⅴ. CONCLUSION

A new modeling method for AC–AC SCCs is introduced in this study. The method provides input–output characteristics, including low-frequency and high-frequency behavior and the impacts of component parameters. The feasibility of this method is validated by comparing it with an existing modeling method for a typical SCC with a conversion ratio of two and through experimental verification.

Using the proposed method, another AC–AC SCC with a conversion ratio of three is analyzed extensively. The operational characteristics of the converter are described and analyzed. The impacts of control and component parameters on the converter are also presented. Component stress analysis for the converter operating in PC and CC modes is performed comprehensively. The effectiveness of the theoretical analysis is confirmed by the experimental results.

We will work on the development of a generalized modeling method for AC–AC SCCs with multiple ratios in the future.

ACKNOWLEDGMENT

This project is supported by the National Natural Science Foundation of China (51407173) and the Zhejiang Provincial Natural Science Foundation (LGG18E070004, LQ17E070003).

REFERENCES

[1] Y.-C. Wong, O.-C. Mak, and A. Ioinovici, “Development of boost converter based on switched-capacitor circuits,” in Proc. of IEEE TENCON, pp. 522-525, 1993.

[2] S. Xiong, S.-C. Wong, S.-C Tan, and C. K. Tse, “A family of exponential step-down switched-capacitor converters and their applications in two-stage converters,” IEEE Trans. Power Electron., Vol. 29, No. 4, pp. 1870-1880, Apr. 2014.

[3] B. Wu, S. Li, K. M. Smedley, and S. Singer, “A family of two-switch boosting switched-capacitor converters,” IEEE Trans. Power Electron., Vol. 30, No. 10, pp. 5413-5424, Oct. 2015.

[4] D. Kilani, M. Alhawari, B. Mohammad, H. Saleh, and M. Ismail, “An efficient switched-capacitor DC-DC buck converter for self-powered wearable electronics,” IEEE Trans. Circuits Syst. I, Reg. Papers, Vol. 63, No. 10, pp. 1557-1566, Oct. 2016.

[5] T. Tong, S. K. Lee, X. Zhang, D. Brooks, and G.-Y. Wei, “A fully integrated reconfigurable switched-capacitor DC-DC converter with four stacked output channels for voltage stacking applications,” IEEE J. Solid-State Circuits, Vol. 51, No. 9, pp. 2142-2152, Sep. 2016.

[6] Z. Amjadi and S. S. Williamson, “A novel control technique for a switched-capacitor-converter-based hybrid electric vehicle energy storage system,” IEEE Trans. Ind. Electron., Vol. 57, No. 3, pp. 926-934, Mar. 2010.

[7] Y. Ye, K. and W. E. Cheng, “Modeling and analysis of series–parallel switched-capacitor voltage equalizer for battery/supercapacitor strings,” IEEE J. Emerg. Sel. Top. Power Electron., Vol. 3, No. 4, pp. 977-983, Dec. 2015.

[8] S. Xiong and S.-C Tan, “Cascaded high-voltage-gain bidirectional switched-capacitor DC-DC converters for distributed energy resources applications,” IEEE Trans. Power Electron., Vol. 32, No. 2, pp. 1220-1231, Feb. 2017.

[9] T. B. Lazzarin, R. L. Andersen, G. B. Martins, and I. Barbi, “A 600-W switched-capacitor AC-AC converter for 220 V/110 V and 110 V/220 V applications,” IEEE Trans. Power Electron., Vol. 27, No. 12, pp. 4821-4826, Dec. 2012.

[10] R. L. Andersen, T. B. Lazzarin, and I. Barbi, “A 1-kW step-up/step-down switched-capacitor AC-AC converter,” IEEE Trans. Power Electron., Vol. 28, No. 7, pp. 3329-3340, Jul. 2013.

[11] T. B. Lazzarin, M. D. Vecchia, and I. Barbi, “Experimental validation of a proposal for a 3.5 kVA three-phase magnetic-less solid-state autotransformer (SSAT) based on the switched-capacitor principle,” in Proc. of IEEE Int. Conf. Ind. Technol., pp. 993-998, 2015.

[12] M. D. Vecchia, T. B. Lazzarin, and I. Barbi, “A three-phase AC-AC converter in open-delta connection based on switched capacitor principle,” IEEE Trans. Ind. Electron., Vol. 62, No. 10, pp. 6035-6041, Oct. 2015.

[13] T. B. Lazzarin, R. L. Andersen, and I. Barbi, “A switched- capacitor three-phase AC-AC converter,” IEEE Trans. Ind. Electron., Vol. 62, No. 2, pp. 735-745, Feb. 2015.

[14] S. Li, B. Wu, K. M. Smedley, and S. Singer, “Analysis and design of a 1-kW 3X interleaved switched-capacitor DC-DC converter,” in Proc. of IEEE Energy Convers. Congr. Expo., pp. 1692-1698, 2014.

[15] R. Middlebrook and S. Cuk, “A general unified approach to modelling switching-converter power stages,” in Proc. of IEEE Power Electron. Spec. Conf., pp. 18-34, 1976.

[16] S. Ben-Yaakov and M. Evzelman, “Generic and unified model of switched capacitor converters,” in Proc. of IEEE Energy Convers. Congr. Expo., pp. 3501-3508, 2009.

[17] S. Ben-Yaakov, “Behavioral average modeling and equivalent circuit simulation of switched capacitors converters,” IEEE Trans. Power Electron., Vol. 27, No. 2, pp. 632-636, Feb. 2012.

[18] J. W. Kimball, P. T. Krein, and K. R. Cahill, “Modeling of capacitor impedance in switching converters,” IEEE Power Electron. Lett., Vol. 3, No. 4, pp. 136-140, Dec. 2005.

[19] G. Zhu and A. Ioinovici, “Switched-capacitor power supplies: DC voltage ratio, efficiency,ripple, regulation,” in Proc. of IEEE Int. Symp. Circuits Syst., pp. 553-556, 1996.

[20] B. Wu, S. Li, K. M. Smedley, and S. Singer, “Analysis of high-power switched-capacitor converter regulation based on charge-balance transient-calculation method,” IEEE Trans. Power Electron., Vol. 31, No. 5, pp. 3482–3494, May 2016.

[21] C. K. Cheung, S.-C. Tan, C. K. Tse, and A. Ioinovici, “On energy efficiency of switched-capacitor converters,” IEEE Trans. Power Electron., Vol. 28, No. 2, pp. 862-876, Feb. 2013.

Hui Cai was born in Zhejiang, China in 1980. He obtained his B.S.E.E. and Ph.D. from Zhejiang University, Hangzhou, China in 2002 and 2008, respectively. He is currently an Associate Professor in the College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou, China. His research interests include power converter systems, AC drives, and new energy power generation.

Liting Bao was born in Hangzhou, China. He obtained his B.S. in Electronics Engineering from Xi’an Jiaotong University, Xi’an, China in 2014 and his M.S. in Control Theory and Control Engineering from China Jiliang University, Hangzhou, China in 2017. He is currently working toward a Ph.D. in Electrical Engineering in Zhejiang University, Hangzhou, China. His research interests include distributed generation, microgrids, and soft open points.

Qian Guo was born in Henan, China. She obtained her B.S. and Ph.D. in Electrical Engineering from Zhejiang University, Hangzhou, China in 2010 and 2016, respectively. She is currently a Lecturer in the College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou, China. Her research interests include control of inverters in microgrids and distributed generation systems.

Ying Wang was born in Henan, China. She obtained her Ph.D. from Zhejiang University, Hangzhou, China in 2017. She is currently a Lecturer in the College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou, China. Her research interests include protection and control in microgrids and distributed generation systems.

Weimin Chen was born in Jiangxi, China. He obtained his M.S. and Ph.D. in Electrical Engineering from Shanghai University, Shanghai, China in 2006 and 2011, respectively. He is currently an Associate Professor in the College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou, China. His research interests include control of inverters and photovoltaic generation.