Switched Capacitor DC-DC Converters: A Survey on the Main Topologies, Design Characteristics, and Applications

Abstract

This work presents a review of the main topologies of switched capacitors (SCs) used in DC-DC power conversion. Initially, the basic configurations are analyzed, that is, voltage doubler, series-parallel, Dickson, Fibonacci, and ladder. Some aspects regarding the choice of semiconductors and capacitors used in the circuits are addressed, as well their impact on the converter behavior. The operation of the structures in terms of full charge, partial charge, and no charge conditions is investigated. It is worth mentioning that these aspects directly influence the converter design and performance in terms of efficiency. Since voltage regulation is an inherent difficulty with SC converters, some control methods are presented for this purpose. Finally, some practical applications and the possibility of designing DC-DC converters for higher power levels are analyzed.

1. Introduction

DC-DC converters are widely used in residential, commercial, and industrial applications, such as renewable energy conversion systems, electric traction devices, and, mainly, power supplies. Currently, it is estimated that more than 70% of electricity is processed by electronic devices [1]. In this context, increasing the power processing capacity and power density of converters associated with reduced manufacturing costs is of major concern.

The evolution of low-power electronics is mainly related to the increase in the purity of materials and advanced techniques used in the manufacture of integrated circuits (ICs) [2]. This also has a direct impact on power electronics, which seeks to increase the power levels and maximum operating frequency associated with reduced dimensions of power converters [3]. Recently, new semiconductors have become commercially available, which are based on silicon carbide (SiC) and gallium nitride (GaN). These elements have promising characteristics that allow the gradual replacement of silicon-based (Si) devices owing to their higher efficiency and operating frequency. Besides, there is the possibility of combining both manufacturing technologies in a single component [4,5,6].

In turn, energy storage devices such as capacitors and inductors, which are typically used as filters, also contribute directly to the improved performance of power converters. This is owing to the use of dielectric materials with optimized characteristics in capacitors [7,8]. Besides, new ferrosilicon alloys are capable of achieving higher magnetic permeability with reduced hysteresis and eddy current losses in the inductor cores [9,10,11].

In the constant search for higher power density in power converters, topologies using switched capacitors (SCs) have recently aroused the interest of industry and academia. These structures have been adopted in low power electronic applications, especially in systems with limited physical dimensions and involving high energy density. Their characteristics allow monolithic integration [12,13], minimized levels of electromagnetic interference (EMI) [14,15,16], as well as reduced weight and volume.

However, despite the aforementioned advantages, these circuits may present low efficiency [17]. This aspect is particularly influenced by the intrinsic characteristics of the switches and capacitors used in the circuit, and the number of components must also be carefully considered [18]. The regulation of the load voltage is another challenge because, in certain operation conditions, the duty cycle does not have a linear relationship with the output voltage, which implies an increase in the complexity of the control systems [19].

SC circuits can also be combined with traditional structures based on inductors for obtaining families of hybrid converters, resulting in improved load voltage regulation and extended conversion ratio when compared with topologies composed only of capacitors and semiconductors [20,21]. An example of a pseudo SC bandpass filter can also be found in [22]. Another hybrid approach lies in resonant SC converters, which allow increasing the power processing capacity and power density as demonstrated in [23]. Considering that the combination of SCs and inductors leads to a wide variety of topologies, this analysis is beyond the scope of this work.

Considering that SC-based structures have inherent advantages and disadvantages, the existence of adequate guides that detail different design aspects for practical applications is of great interest to professionals and researchers. In this context, this work describes a synthesis of basic concepts and design guidelines associated with the conception of SC-based dc-dc converter topologies. Key issues such as classic configurations, semiconductors, efficiency, regulation, and operation modes are analyzed. The rest of the work is organized as follows. Section 2 addresses voltage multiplier circuits, as some SC cells are derived from these structures. Next, the SC DC-DC converters are presented. In turn, Section 3 is dedicated to analyzing the main design parameters, which includes the influence of components and operation modes in the converter efficiency, regulation, and control. Section 4 presents the main applications of these circuits, as well as future perspectives of their application. Section 5 summarizes the main aspects investigated in the study.

2. SC Structures

Although voltage multiplier circuits are simple structures widely known and described in analog electronics textbooks, they are still used in applications where high DC voltages must be obtained from an AC voltage source. Voltage multipliers are typically used in X-ray machines, scanning electron microscopes, and particle accelerators, among other devices, because they are simple and low-cost circuits [24,25,26].

In the literature, there are some SC structures dedicated to specific applications, which are often derived from basic converters [27,28,29,30,31,32,33,34]. A description of SC topologies and voltage multipliers used in the conception of non-isolated dc-dc converters for high step-up applications is presented in [35]. However, this study is specifically focused on the thorough analysis of existing techniques for extending the conversion range of non-isolated dc-dc converters. Some relevant issues for the design of SC-based converters are not addressed in [35], e.g., the proper choice of components, the charging mode of capacitors, evaluation of efficiency associated with relevant practical aspects, and control techniques aiming to achieve the output voltage regulation.

It is worth mentioning that there is a wide variety of circuits that employ inductors and SCs in the form of hybrid topologies, especially for wide conversion range applications [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65]. Owing to the existence of a great diversity of combinations with distinct characteristics, this work is dedicated to the analysis of classical topologies based on the use of only semiconductors and capacitors, resulting in the so-called “pure SC” converters.

2.1. Greinacher Voltage Doubler (1914)

The simplest structure of a step-up converter that uses switches and capacitors is the Greinacher voltage doubler, which is shown in Figure 1 [66]. This circuit is powered by an AC source and operates according to two stages. In the first one, the capacitor is charged to the peak value of the source voltage. In the second one, the previously charged capacitor is placed in series with the source and supplies a load. This combination generates a voltage that is ideally equal to twice the input voltage [67].

Being a simple structure, this was one of the first topologies capable of stepping up the voltage across a load powered by an AC source using only passive semiconductors and capacitors. This circuit or cell can be connected in a modular way in series or parallel with phase opposition, thus allowing the achievement of high voltages [68,69].

2.2. SC Voltage Doubler

Although the voltage step-up circuit shown in Figure 1 may be useful in some applications, its use is only feasible when an AC voltage source is available. However, many electronic circuits are powered by DC sources. The SC voltage doubler (SCVD) circuit shown in Figure 2 is a suitable alternative in this case [70,71].

This structure employs active switches to control the charge and discharge of the capacitors. The circuit operates in the same way as the voltage multiplier, i.e., there are two stages. In the first one, the charge of capacitor C₁ occurs. In the second one, capacitor C₁ is connected in series with the source and the load. It is possible to employ the connection of several cells in series to obtain a given gain $2^n$, where n is the number of associated stages. Thus, it is possible to obtain high voltage gains, which allow supplying loads with high output voltages.

2.3. Cockcroft Walton Voltage Multiplier or Ladder Structure (1932)

To supply a particle accelerator with a voltage of 1 MV, Cockcroft and Walton used a circuit powered by an AC source, substantially increasing the output voltage [72]. This circuit became known as Cockcroft and Walton voltage multiplier (CWVM), or simply voltage multiplier, which is exemplified in Figure 3 in terms of several stages. This is one of the best-known structures in electronics to obtain high gains owing to the simplicity of the circuit, low cost, and simple implementation. Considering the components as ideal, the output voltage V_o can be obtained by n⋅V_i, where n is the number of associated cells and V_i is the input voltage.

Recent works point to the possibility of using this structure in several modern applications involving high-gain DC-DC converters associated with photovoltaic modules [73,74] and even high power factor rectifiers [75,76,77].

In the CWVM, the source provides some charge Q to capacitor C₁ and, in each half cycle of the AC input voltage, the charge Q is transferred from capacitor to capacitor (C₂, C₃, …), until it reaches the last element C_n. This process involves shifting the charge through each diode similarly to a ladder, which justifies the name given to the circuit.

An inherent problem with this topology is that its output impedance increases by an n³ factor as the number of stages increases. This characteristic limits the output current to low values, as in the case of high currents; the output voltage will decrease rapidly owing to losses [78].

2.4. Series-Parallel Converter (1971)

A proposed alternative to overcome the main limitations of the CWVM in terms of voltage regulation is the series-parallel converter (SPC) shown in Figure 4 [79]. This converter is very versatile since each cell can be associated in series or parallel, and the output voltage can be modified at any time.

As the name suggests, the energy transfer from the input to the output occurs when connecting some of the capacitors in parallel with the source during the charge stage. In a second moment, the capacitors are associated in series and supply the load. This arrangement has some remarkable advantages because it does not present the same problems as the CWVM, especially concerning the high output impedance. However, there are other drawbacks, such as the simultaneous charging of many capacitors, which can demand a high current from the source in a short time. Also, if many capacitors are in charging process, the voltage ripple at the output may be high, directly impacting the converter efficiency. Depending on the length of the connection tracks on the printed circuit boards (PCBs), the arrangement of the elements, and the number of capacitors, there may still be problems due to parasitic capacitances, drastically reducing the efficiency of the circuit in the event of high voltage gains [80].

2.5. Series-Parallel Multiphase Converter (1973)

Another alternative to overcome the problems presented by the CWVM and the SPC is the series-parallel multiphase converter (SPMC) [81], which is shown in Figure 5. In this way, it is possible to mitigate the transients of the input current and the effect of parasitic capacitances. While part of the capacitors is connected to the source during the charge cycle, the other components that are fully charged are connected to the load. Meanwhile, part of the other series-connected capacitors remains in an intermediate charge state. Therefore, there are at least two or three phase-shifted signals controlling the connection of the capacitors with the source and the load.

2.6. Dickson Converter (1976)

Dickson implemented the first integrated multiphase voltage multiplier in which the circuit reached an output voltage of 40 V from a 15 V source [82]. Besides the innovation inherent in the introduction of the topology itself, it should be noted that this was the first structure built experimentally in a totally encapsulated form. Figure 6 shows the Dickson converter.

In the literature, there are several publications dedicated to the study of this topology aiming at improving aspects such as efficiency, dynamic response, and regulation [83,84,85]. A limitation of this structure lies in the fact that it does not present good performance in high-power applications owing to the difficulty of obtaining adequate regulation and the low efficiency when it is desired to obtain high voltage gains.

2.7. Fibonacci Converter (1991)

When the first SC converters were designed, the power levels were initially limited to milliwatts or a few units of watts. A natural next step would be to reduce the number of components of these structures, aiming at increasing the gain, power levels, and efficiency. In this sense, the Fibonacci converter (FC) was proposed as a multiphase topology in which the capacitors are associated in series and parallel. In this case, the cells are combined with the source and with each other to achieve the desirable voltage gain [86]. The FC is presented in Figure 7 and receives this name because of its gain characteristic, since each cell/stage coupled to the converter leads to an increase in the output voltage following a proportion defined by the Fibonacci sequence (2, 3, 5, 8, 13, …).

The FC is very similar to the SPC, differing mainly in the way the capacitors are connected. At this point, it should be noted that among the topologies presented so far that use the combination of capacitors in series and parallel, the highest gain by the association of stages is obtained with the FC [15,87].

2.8. Ladder Converter (1992)

The CWVM can produce high voltages from a low-voltage source, but there are limitations to this structure, such as the high output impedance that prevents high gains from being achieved. In general, it can be stated that as the gain increases, the output impedance also does by a cubic factor [79], which can compromise the output voltage regulation for loads requiring high currents.

The basic ladder cells are shown in Figure 8, which are derived from a study regarding the proposal of a three-level neutral point clamped (NPC) inverter topology [88,89]. The inverter described in [88,89] works with three voltage levels: V_i, V_i/2, and 0 V, and it is called capacitor-clamped multilevel inverter or flying capacitor multilevel inverter (FCMLI). This is a circuit similar to the NPC inverter, but the diodes are replaced by capacitors.

These cells can be arranged in the form of boost and buck configurations as shown in Figure 8a,b, respectively. A converter based on this topology can consist of one or more cells according to the desired conversion ratio. When the converter works in voltage step-up mode, the source is associated in series with n capacitors charged with a voltage V_i. The flying capacitor at a given moment is connected in parallel with the source to be charged. Still, later it is connected in parallel with the other capacitor to charge it with V_i. In this way, the voltage source in series with n charged capacitors can produce an output voltage V_o according to Equation (1).

$$V_o=V_i+n{V}_i$$

(1)

when used as a step-down topology, the converter operates similarly to a capacitive voltage divider formed by n capacitors in series, in which the load is connected in parallel with one of the capacitors. In this circuit, there are also one or more flying capacitors C_fly, which at a given moment are connected in parallel with the source and later on associated in parallel with the elements that form the capacitive divider. This arrangement produces an output voltage that varies according to the number of capacitors used in the converter as defined by Equation (2):

$$V_o=\frac{V_i}{n_{cap}}$$

(2)

where n_cap is the number of series capacitors in the circuit. Figure 9a shows a boost-type ladder converter with a gain equal to two and Figure 9b presents a buck converter with a gain equal to 0.5. This topology is also efficient in AC-AC conversion, with good regulation and high power factor, and it is a feasible option to replace ferromagnetic core transformers [90].

3. Design Parameters

In this section, the main design parameters of SC converters are addressed, such as operation modes, main components, efficiency, and control techniques.

3.1. Components

3.1.1. Switches

In general, the efficiency and regulation of SC converters are dependent on the intrinsic resistances of the switches and the dielectrics that constitute the capacitors. In low-power and medium-power DC-DC converters, metal-oxide-semiconductor field-effect transistors (MOSFETs) are widely used because they can operate at high frequencies. For higher power levels, insulated gate bipolar transistors (IGBTs) are better recommend because they support higher currents and voltages, but switching losses increase significantly with the switching frequency. Some studies have shown that SiC MOSFETs have a performance similar to Si-based IGBTs [91,92,93]. Thus, the analysis presented in this work considers only MOSFETs.

Table 1. Comparison of the characteristics of different MOSFETs.

Component	Type	Vds(max) (V)	Id(max) (A)	Rds(on) (mΩ)	tr (ns)	tf (ns)	Coss (pF)	Ciss (pF)
IRFP240	Si	200	20	180	51	36	400	1300
IRF530	Si	100	14	160	34	24	250	670
IRFP460	Si	500	20	270	51	36	430	1300
IRF840	Si	500	8	850	21	20	200	1225
C2M0080120D	SiC	1200	36	80	22	14	92	1130
SCT3080KL	SiC	1200	31	80	22	24	75	785
IMW120R045M1XKSA1	SiC	1200	52	59	24	13	115	1900
NTHL080N120SC1	SiC	1200	44	80	20	10	80	1112

shows some characteristics of different commercial Si and SiC MOSFETs for comparison purposes. In these semiconductors, conduction losses must be carefully analyzed, as they depend on the drain-source on-resistance R_ds(on). This parameter is provided in the manufacturer’s datasheet and expression 3 can be used to estimate the conduction losses [94,95]:

$$P_{cond\left(\mathrm{MOSFET}\right)}=R_{ds\left(on\right)}{I}_{d\left(rms\right)}{}^2$$

(3)

where I_d(rms) is the root-mean-square (RMS) drain current.

The switching losses are essentially related to the time interval required by a given semiconductor to change from the ON state to the OFF state, or vice-versa. In other words, the shorter the rise time t_r and fall time t_f, the faster the dynamic response of a MOSFET. Table 1 shows the aforementioned parameters, as well as the parasitic capacitances that affect the switching losses. It is worth mentioning that C_iss and C_oss do vary significantly with the drain-source voltage V_ds, but they are only slightly affected by the temperature. Capacitance C_iss is an important parameter, since its charge and discharge are directly influenced by drive circuit connected to the gate. Besides, capacitance C_oss is charged by the voltage source of the circuit and discharged through the body diode when the MOSFET is ON and OFF, respectively.

In turn, the switching losses are calculated according to the energy dissipated in the switch during the turn-on and turn-off intervals. These losses depend on the switching frequency f_s, the drain-source voltage V_ds, the drain current I_d, and the time interval required for turn-on and turn-off of the transistor given by t_on and t_off, respectively. Thus, both parts are considered in expression (4) [96]:

$$P_{sw\left(\mathrm{MOSFET}\right)}=\left(\frac{f_s{V}_{ds}{I}_d}{6}\right)\left(t_{on}+t_{off}\right)$$

(4)

Substituting the values assumed by distinct parameters defined in Table 1 in (3) and (4), it is observed that SiC MOSFETs present improved performance with respect to the conduction and switching losses when compared with their Si counterparts. Thus, SiC MOSFETs can be regarded as an adequate choice for applications that high conversion efficiency.

The transistors shown in Table 1 have good performance at high frequencies, but as the frequency increases, the switching losses can significantly compromise the efficiency of the converter. It is observed that the drain current also influences this behavior. As an example, Figure 10 represents the losses in a MOSFET model IRFP460 operating at 25 °C, subjected to a drain-source voltage of 100 V, while the switching frequency ranges from 0 to 500 kHz and, at the same time, the drain current varies from 0 to 10 A. This plot demonstrates the profile of losses in that particular component, but a similar analysis can be developed for any other commercial device, considering that its characteristics are provided by the manufacturer. Thus, one can consider these aspects in detail concerning the design and estimate of the converter efficiency.

3.1.2. Diodes

Some SC topologies like Dickson, ladder, and SMPC use diodes besides active switches, which can influence the conversion efficiency significantly. Therefore, it is important to know the characteristics of such semiconductors, especially considering the commercial availability of Si and SiC components.

Table 2. Comparison of the characteristics of different diodes.

Component	Type	VRRM (V)	IF(avg.)	VF (V)	Trr (ns)	Qrr/QC (nC)
MUR860	Si	600	8	1.20	60	195
RHRP8120	Si	1200	8	3.2	70	165
HFA15TB60S	Si	600	15	1.2	50	84
60EPU04	Si	400	60	1.05	85	375
IDT08S60C	SiC	600	8	1.5	-	20
C4D10120A	SiC	1200	33	1.5	-	52
SCS210KGHR	SiC	1200	10	1.4	-	34
IDH15S120	SiC	1200	15	1.65	-	54

summarizes the main parameters associated with some diode models, which include the maximum value of the repetitive peak reverse voltage V_RRM, the average forward current I_F(avg.), the forward voltage drop V_F, the reverse recovery time t_rr, and the reverse recovery charge Q_rr as informed by the manufacturer at 25 °C. It is worth mentioning that the total capacitive charge Q_C of SiC diodes is equivalent to Q_rr in Si ones. Only ultrafast Si diodes are listed because SC converters often operate at high switching frequencies. Besides, only SiC Schottky diodes are described since this technology presents improved characteristics when compared with other solutions.

SiC diodes are often much faster than Si ones and do not present significant problems due to the reverse recovery phenomenon, thus contributing to the reduction of switching losses. In either case, such semiconductors can be represented by the simple equivalent model shown in Figure 11, in which V_to represents the voltage drop across the semiconductor junction and r_t is the intrinsic resistance. The values of V_to and r_t are not often provided by manufacturers, even though they can be estimated from characteristic curves available in datasheets.

The aforementioned model can be used to estimate the losses in a diode. The conduction losses are owing to the current flowing through the component and can be calculated as in Equation (5).

$$P_{cond\left(D\right)}=V_{to}{I}_{F\left(avg.\right)}+r_t{I}_{F\left(rms\right)}{}^2$$

(5)

where I_F(avg.) and I_F(rms) represent the average and RMS values of the forward current.

In turn, the switching losses occur during the turn-on and turn-off of the diode when both the current and voltage are not zero, being estimated from Equations (6) and (7), respectively.

$$P_{sw\left(on\right)\left(D\right)}=0.5\left(V_F{}_{(\max )}-V_F\right)I_{F\left(avg.\right)}{t}_{rr}{f}_s$$

(6)

$$P_{sw\left(off\right)\left(D\right)}=Q_{rr}{V}_{F(\max )}{f}_s$$

(7)

where V_F(max) is the maximum value assumed by the forward voltage drop. From Equation (6), it is observed that the longer t_rr, the higher the switching losses, and the lower the efficiency as a consequence. Besides, the efficiency can be further increased if diodes with low Q_rr or Q_C are chosen in practical designs.

3.1.3. Capacitors

Besides semiconductors, capacitors are essential devices in SC converters, as they are the only energy storage elements used in the circuit when hybrid topologies are not adopted, which also have inductors as in the case of [97]. The equivalent model of a capacitor basically consists of the series association of a capacitance C, the equivalent series resistance (ESR), and the equivalent series inductance (ESL) as shown in Figure 12.

Capacitance C is related to the type of dielectric used in the construction of the component. It has a direct impact on the energy density and, consequently, on the dimensions of the converter. Parameter ESR cannot be accurately estimated from the simple measurement performed with an ohmmeter because it is strongly influenced by the dielectric, electrodes, and the ohmic losses in the terminals of the component.

The losses in dielectrics correspond to the delayed bias or relaxation, varying according to the dielectric medium, temperature, and frequency [98]. The losses in electrodes in low frequencies are essentially ohmic and depend on the dimensions and type of material. At high frequencies, the losses increase significantly owing to the skin and proximity effects.

Overall, it is reasonable to state that the losses in capacitors are influenced by a wide variety of aspects. In SC-based converters, this element can be satisfactorily represented by C and ESR, resulting in the representation shown in Figure 13. Parameter δ is called dissipation factor (DF) or loss angle, whose value is often provided in datasheets [99,100] In this way, it is possible to estimate the ESR using Equation (8) [101].

$$ESR=\frac{\tan \left(\delta \right)}{2\pi f_{s}C}$$

(8)

The ESL can be hardly measured because it depends on the pad layout, capacitor height, and power plane spreading inductance [102]. It can be neglected at low frequencies, but its influence should be considered as the operating frequency gets close to the resonance frequency of the capacitor. The ESL limits the maximum switching frequency at which the converter can operate since, at very high frequencies, the inductive reactance is much higher than the capacitive reactance. Figure 14 presents the behavior of the capacitor impedance as a function of the frequency and ESR [103].

Table 3. Comparison among different types of capacitors.

Cap.	Manufacturer	Type	Series	Vdc	Vol. (mm3)	E (mJ)	Dens. (μJ/mm3)	Max(tan (δ)) @1 kHz
1	Hitano	Electrolytic	ECR	47	215.98	1.25	5.78	0.10 *
2	Nichicon	Electrolytic	UMA	50	62.83	1.25	19.89	0.10 *
3	TDK	Polyester	B32560	63	244.8	1.98	8.10	0.008
4	Panasonic	Polyester	ECQE	100	1125.6	5	4.44	0.01
5	TDK	Polypropylene	B32672L	250	3663	31.25	8.53	0.0008
6	Hitano	Polypropylene	MKT	100	2592	5	1.92	0.01
7	TDK	Ceramic (ML)	FA24	50	67.5	1.25	18.51	0.03
8	Hitano	Ceramic (ML)	R25	50	132	1.25	9.46	0.1

* Value measured at 120 Hz as informed by the manufacturer.

presents a comparison among eight commercially available capacitor models that use different types of dielectrics, which have a capacitance of 1 μF. Also, the maximum operating voltage varies according to the type of material used. An analysis is then performed considering important aspects such as volume, energy density, and the maximum loss angle measured at 1 kHz, so that the designer can be assisted in choosing the most adequate type of capacitor. The energy density has an impact on the size of the converter and, therefore, it is important to know how these parameters vary from one capacitor to the other according to the type of dielectric.

The power dissipated in a capacitor depends on the root mean square (RMS) current I_C(rms) and the ESR according to Equation (9).

$$P_{cap}=ESR\cdot I_{C\left(rms\right)}{}^2 \left(\mathrm{W}\right)$$

(9)

As the ESR depends on the DF and the frequency, it is possible to combine Equations (8) and (9) to approximately determine the losses in a capacitor according to Equation (10).

$$P_{cap}=\left[\frac{\tan \left(\delta \right)}{2\pi f_{s}C} \right]I_{C\left(rms\right)}{}^2$$

(10)

As an example, and for the eventual efficiency analysis of the converters, capacitor 8 was chosen in Table 3. By varying the RMS current from 0 to 10 A and the source frequency from 0 to 500 kHz, it is possible to plot the component losses as shown in Figure 15 with the aid of Equation (10).

Of course, each type of capacitor has its own electrical and constructive characteristics. The objective of this work is not to present a detailed analysis of these components, but only to address some important aspects that allow the choice of the most suitable element for a given application. In general, it is reasonable to state that the most popular capacitor used in SC converters is of film type. This component is widely used mainly in applications in which it is necessary to operate at high voltages and frequencies. The film dielectric has low DF, reduced ESR, and good response at high frequencies [104]. The dielectric consists of a thin layer of plastic, the most common materials being polyester and polypropylene. Capacitors made of metallized film have vaporized dielectrics with a thin layer of aluminum, as this additional layer provides the dielectric with the self-regeneration ability [105,106,107]. Given the possible occurrence of a voltage higher than the nominal value supported by the capacitor, which in turn would damage the dielectric, this characteristic allows rapid heating to occur owing to the short circuit of the plates. When subjected to high temperatures, this layer of aluminum around the hole turns into aluminum oxide, which has an insulating character and thus eliminates the short circuit.

Polyester capacitors can be found commercially from some units of nanofarads to values on the order of 220 µF, with voltages greater than or equal to 1000 V. This material often has limited use since, among the film-based capacitors, this is the type most sensitive to the increase of temperature [108] and frequency [104]. Its use is recommended mainly when it is desired to obtain converters with reduced dimensions since polyester capacitors have the smallest size for a given capacitance value in comparison to its remaining counterparts.

When compared to other components, polypropylene-based capacitors demonstrate excellent characteristics, such as the fact that the capacitance varies little with temperature, frequency, and voltage variations. Polypropylene absorbs less moisture than polyester, which makes this material suitable for a wide range of applications in harsh environments. Commercial capacitance values can reach up to 270 µF.

Ceramic capacitors are the passive devices most used in low power electronics today. In this case, the capacitance does not vary significantly with the frequency and temperature, while the components present low loss factor [109]. The dielectric materials used can be porcelain, mica, and other silicates. Also, the components can be produced on a single disc or in multiple layers, i.e., in the form of multilayer ceramic capacitors (MLCCs). A negative aspect lies in the fact that fractures in the dielectric can occur owing to voltage spikes and rapid changes in temperature, which can cause failures and even a short circuit in the capacitor [110]. These components are often manufactured with low capacitances, which are typically rated on the order of picofarads.

Within the most common commercial capacitors, the electrolytic capacitor has the most different construction characteristics. This element is formed by a sheet of porous paper embedded in an electrolyte and, to improve the contact of the plate with the welding terminal, the paper is wrapped in aluminum foil, so that this set forms the negative terminal of the capacitor (cathode). The other terminal of the capacitor is connected internally to a second aluminum foil that constitutes the positive terminal (anode). This sheet receives an electrochemical treatment called anodic oxidation, which causes the appearance of a thin layer of an insulating material called aluminum oxide (Al₂O₃) deposited on that same sheet, thus forming the dielectric.

This Al₂O₃ oxide layer has the advantage of a high dielectric constant and high rigidity with a very thin thickness on the order of 0.7 µm. With this characteristic, it is possible to reach high capacitances in a very small volume, which can vary from 1 µF to 330 mF. Thus, this is one of the most important characteristics of this type of capacitor. On the other hand, the maximum voltage stresses supported by this component are low in relation to the other elements previously presented. If this type of component is subjected to transients that exceed the rated voltage, even if very briefly, the dielectric is compromised, causing a change in the capacitance value and reducing the component’s useful life. In general, this type of capacitor has defined polarity and, if it is connected to a circuit in an inverted way, electrochemical reactions can occur inside it, causing leakage and even explosion. When high capacitances are required, their use is often essential, but these capacitors have a shorter useful life than film and ceramic components. As they are more susceptible to failure, their use on a large scale is avoided.

3.2. Operation Modes

As mentioned, SC converters have a simpler structure than conventional power electronic converters that employ inductors, but the control system becomes more complex in this case. Theoretically, it is possible to achieve high voltage gains with some structures. However, it should be noted that the higher the gain, the more components will be employed and, consequently, the lower the efficiency. Figure 16 presents the behavior of the theoretical voltage gain as a function of the number of cells for the voltage doubler, SPC, SPMC, Dickson, and FC topologies.

To analyze the behavior of the output voltage, the circuit represented in Figure 17 is adopted. In this case, the converter can be modeled as an ideal transformer with turns ratio m:n, where R_o is the equivalent output resistance of the SC converter, which is connected in series with the load R_L [86,111]. In this model, the resistance R_o controls the power transferred to the load and the output voltage regulation. The variation of R_o in practice is mainly associated with the switching frequency and the internal resistance of the converter elements.

As for the switching frequency, two asymptotic limits can be established: slow switching limit (SSL) and fast switching limit (FSL) [112,113]. When the converter operates in SSL, the switching frequency is low enough for the capacitors to fully charge and discharge. In this scenario, the currents have short-term peaks. Besides, all resistances of the switches and capacitors are neglected, as these impedances do not affect the full charge of the capacitors. In this operation mode, the duty cycle has little influence on the behavior of the converter. In the FSL condition, the currents that flow through the capacitors can be considered constant, and the series resistances of the components cannot be neglected, as they can affect the full charge of the capacitors.

Other analytical methods are based on the charge and discharge of capacitors [114]. The circuit shown in Figure 18 can be used for this study, where an active switch S controls the energy transferred from a DC voltage source V_i to a capacitor C_o. This representation is simpler than the model corresponding to Figure 17 and leads to similar results, and is adopted in this work in the study of topologies. It addresses the behavior of the circuit according to three different modes shown in Figure 19, where the instantaneous voltage v_Co(t) and instantaneous current i_Co(t) are represented for the full charge, partial charge, and no charge conditions.

In this case, the equivalent model of the converter is considered as a first-order R_iC_o circuit with time constant τ, where R_i = R_sw + ESR, R_sw being the resistance of the switch, while C_o represents the capacitor in the charge or discharge regime. The switch is turned on to charge the capacitors during a time interval T_on over a switching period T_s. Thus, it is reasonable to state that SC circuits are often designed based on the time constant rather than the switching frequency. Besides, such converters are capable of operating in full charge condition even at high frequencies according to the specifications of capacitors and switches used in the design.

The three modes of operation are defined [115,116]:

(1)

If T_on >> R_iC_o, that is, when the time interval during which the switch remains on is much greater than the time constant of the circuit, the capacitors charge and discharge completely. Thus, the current becomes zero in each cycle, and then it can be assumed that the system operates under full charge condition according to Figure 19a.

(2)

If T_on ≈ R_iC_o, the capacitor current does not become zero and, therefore, the capacitor is not fully discharged [117]. The converter then operates on a partial charge condition. The representation of the current in the capacitor is shown in Figure 19b.

(3)

If the switching frequency is very high, that is, if the value of T_on is small in relation to T_s, then T_on << R_iC_o. Thus, the current in the capacitor is almost zero, and the voltage across it can also be considered constant. In this mode, the converter operates at no charge condition according to Figure 19c.

In general, the equivalent resistance of the circuit represented in Figure 18 can be defined by Equation (11) [118]:

$$R_{eq}=\frac{1}{2f_s{C}_o}\frac{\left(1+e^{-\beta }\right)}{\left(1-e^{-\beta }\right)}$$

(11)

where β is given by Equation (12).

$$\beta =\frac{T_{on}}{R_i{C}_o}$$

(12)

The time constant associated with the circuit is given by Equation (13).

$$\tau =R_i{C}_o$$

(13)

The duty cycle is defined as the ratio between the on-time of the switch T_on and the switching period T_s as in Equation (14):

$$D=\frac{T_{on}}{T_s}$$

(14)

where T_s is defined according to Equation (15).

$$T_s=\frac{1}{f_s}$$

(15)

Defining D = 0.5, Expression (11) can be written as Expression (16).

$$R_{eq}=\frac{1}{2f_s{C}_o}\frac{\left[1+e^{\left(-\frac{0.5}{\tau f_s}\right)}\right]}{\left[1-e^{\left(-\frac{0.5}{\tau f_s}\right)}\right]}$$

(16)

Expression (16) is plotted in Figure 20a, where the behavior of the converter equivalent resistance is represented as a function of product τ⋅f_s. Proper tradeoffs must be made between such parameters to achieve good overall performance. Increasing the switching frequency may lead to higher efficiency as observed in Expression (16). However, it is worth mentioning that the switching losses and the existence of parasitic inductances and capacitances may affect the behavior of R_eq in practice, which tends to increase after a given point in the curve shown in Figure 20b [119]. Therefore, the efficiency will be reduced as a consequence. This condition depends on the characteristics of switches and capacitors used in the design, and such components must be wisely chosen in practical applications.

Performing an analysis in terms of β and T_on, the output resistance of the converter can be estimated as follows:

(1)

If β > 1, the converter operates in full charge mode, where T_on > 5τ, while the output resistance depends on the inverse of the product of f_s and C_o. In other words, the resistances of the switches and capacitors represented by R_i has little influence on the converter behavior.

(2)

If β = 1, the capacitors operate in partial charge mode and the output resistance of the converter depends on R_i, C_o, and f_s.

(3)

If β < 1, the converter operates in no charge mode. In this case, it is possible to employ very short switching time intervals compared to the circuit time constant. Thus, the output resistance of the converter is very dependent on R_i and D.

Table 4. Operation modes of SC converters.

Operation Mode	Time Constant	β	Variation of Req
Full charge	Ton >> RiCo	β > 1	$\frac{1}{2f_s{C}_o}$
Partial charge	Ton ≈ RiCo	β ≈ 1	$\beta ,\frac{1}{2f_s{C}_o}$
No charge	Ton << RiCo	β < 1	$R_i\frac{T_s}{T_{on}}=\frac{R_i}{D}$

presents a comparison among the operation modes of SC converters.

3.3. Voltage Balance

SCs rely on the principle that the energy stored in a given element is transferred to another, while the capacitors can be associated in series or in parallel depending on the configuration. Of course, small differences will exist in the rated capacitances even if components with the very same specifications are used. In most SC converters, the energy transfer occurs in equivalent circuits represented by the parallel association of capacitors. A given capacitor C_i previously charged with a voltage V_Ci transfers energy to another capacitor C_o, which has no charge or a smaller amount of charge according to the circuit shown in Figure 21. It is worth mentioning that a given series resistance R_i also exists in practice owing to capacitors C_i and C_o, as well as to the active switches.

The series resistance is neglected a priori in the forthcoming analysis. Thus, the total amount of charge in the circuit Q_t is given by Equation (17):

$$Q_t=Q_{Ci}+Q_C{}_o$$

(17)

where Q_Ci and Q_Co are the charges stored in capacitors C_i and C_o, respectively.

Since charge balance is supposed to occur, the initial charge Q_i must be equal to the final charge Q_f according to Equation (18).

$$Q_f=Q_i$$

(18)

Considering this condition, Expression (19) is valid:

$$V_{Ci}{C}_i+V_{Co}{C}_o=V_f{C}_i+V_f{C}_o$$

(19)

where V_Co is the initial voltage across capacitor C_o; and V_f is the final value assumed by the equilibrium voltage.

Thus, it is possible to obtain V_f from Expression (19) as in Expression (20).

$$V_f=\frac{V_{Ci}{C}_i+V_{Co}{C}_o}{C_i+C_o}$$

(20)

If C_o is initially discharged, then V_Co = 0 and Expression (19) becomes (21).

$$V_{Ci}{C}_i=V_f{C}_i+V_f{C}_o$$

(21)

Thus, Expression (22) results.

$$V_f=V_{Ci}\left(\frac{C_i}{C_i+C_o}\right)$$

(22)

The behavior of the voltages across the capacitors does depend on the charge mode as shown in Figure 19. On the other hand, the rated value of the equilibrium voltage achieved when one capacitor transfers charge to another depends on the capacitances, which may vary in practical conditions owing to distinct factors, e.g., aging, temperature, among others. If V_Ci remains constant while considering distinct combinations of C_o and C_i, it is observed that the equilibrium voltage changes when the capacitances are not the same. Another important issue is the fact that the value of R_i does not affect the value assumed by V_f, but only the time constant of the circuit. This aspect also becomes evident in Section 3.4, where it is demonstrated that R_i does not affect the conversion efficiency during the charge cycle.

3.4. Efficiency

There are many works dedicated to analyzing the variables that influence the efficiency of SC converters. Some studies point out the constructive characteristics of capacitors [120], the intrinsic resistance of switches [121], or even a combination of both as the main factors responsible for affecting this behavior [122]. Other aspects are also brought up, such as the voltage conversion ratio and the charge regime, that is, full charge or no charge [123].

As previously mentioned, in each cycle when in full charge mode, the capacitor discharges fully. For a circuit in which the capacitor is initially discharged, the maximum theoretical efficiency achievable is 50% according to Equation (23). As the initial voltage on the capacitor increases, the efficiency also does. The same behavior occurs when operating with partial charge, and the efficiency does not depend on the resistance of the switches and capacitors according to Equation (24). The charging process is affected only by the initial and final values of the capacitor voltage. Therefore, the greater the voltage variation, the lower the efficiency during the charge cycle. The increase of the resistances in series with the capacitors during the charge cycle only modifies the time constant, but it does not affect the efficiency itself. An increase in such series resistances during the charge cycle decreases the peak current and, at the same time, increases the charge time. Thus, the efficiencies under full charge and partial charge conditions are given by η_FC and η_PC in Equations (23) and (24), respectively.

$${\eta }_{FC}=\frac{1}{2}\left(1+\frac{V_{initial}}{V_i}\right)$$

(23)

$${\eta }_{PC}=\frac{1}{2}\left(\frac{V_{initial}+V_{final}}{V_i}\right)$$

(24)

where V_initial and V_final are the initial and final values of the capacitor voltage, respectively.

During the discharge cycles, the efficiency can be estimated in at least three different ways: capacitor discharging in a resistance; capacitor discharging into another capacitor; and capacitor discharging into a parallel RC circuit.

For the first case, the efficiency is given by (25):

$$n_R=\frac{R_L}{R_L+R_i}$$

(25)

where R_L is the resistance on which the capacitor is discharged. It is observed that the lower the series resistance, the higher the circuit efficiency. Thus, it is interesting to know the parameters presented in Table 1 and Table 3.

However, for the capacitor discharge in another capacitor, the efficiency can be estimated in two ways that depend on the operation mode in which the circuit works, that is, full charge or no charge. As an example, Figure 21 shows the circuit used in the analysis, whose waveforms in Figure 22 represent the moment after which switch S is turned on.

As previously mentioned, capacitor C_i is initially charged with voltage V_Ci, whereas capacitor C_o is completely discharged. Thus, a charge balancing occurs so that at the end of the process, the voltage on both capacitors becomes nearly balanced and equal to V_Ci/2 if the capacitances are equal to each other. Therefore, the voltages across C_o and C_i are increased and decreased, respectively, as the efficiency can be calculated according to Equation (26).

$$n_{FC}=\frac{V_f+V_{Co}}{V_f+V_{Ci}}$$

(26)

Considering a capacitor discharging into another capacitor, but in partial charge mode, a similar analysis can be performed. Now, the capacitor C_i initially charged with V_Ci is discharged into another capacitor until reaching V_Ci(min). In this way, the capacitor C_o charges from a minimum voltage V_Co to V_Co(max), and then the cycle is interrupted. The behavior of the capacitor voltages in this case is shown in Figure 23. The efficiency of this type of discharge cycle can be given by Equation (27):

$$n_{PC}=\frac{V_{Co\left(\max \right)}+V_{Co}}{V_{Ci}+V_{Ci\left(\min \right)}}$$

(27)

where V_Co(max) is the maximum voltages on capacitor C_o; and V_Ci(min) is the minimum voltage on capacitor C_i.

A third possibility is the discharge of a capacitor in a parallel RC circuit, as shown in Figure 24. In this case, there is a capacitor C_i charged with an initial voltage V_Ci, to which a series resistance R_i is connected, corresponding to the sum of the resistances of the switch and the capacitors. The load R_L is associated in parallel with a capacitor C_o, which is initially discharged. Thus, the discharge occurs in two events. At first, the charge of capacitor C_i is divided between C_o and R_L. When the charge between the two capacitors is balanced, both start supplying R_L until they are completely discharged, as represented in Figure 25. At each stage, there is a different expression for the efficiency.

In the first stage, the efficiency can be given by Equation (28).

$${\eta }_1=\left(\frac{V_C{}_{o(\max )}+V_{Co}}{V_f+V_{Ci}}\right)$$

(28)

In the second stage, the efficiency can be estimated by Equation (29).

$${\eta }_2=\frac{R_L}{R_L+R_i}$$

(29)

In this analysis, the switching losses in the semiconductors must also be considered, although they have not been incorporated in the previous expressions. This is justified because adequately quantifying them depends on the intrinsic characteristics of each type of switch. In the case of MOSFETs, the increase in the switching frequency has direct impact on the switching losses as shown in Equation (4) [18,113]. On the other hand, when the switching frequency is increased, the voltage ripple is reduced. With a smaller voltage variation, the efficiency increases according to Expressions (23), (24), (26)–(28). Thus, it is up to the designer to carry out a detailed analysis regarding the best operation point for the circuit.

Analyzing the behavior of capacitors operating in no charge, partial charge, and full charge modes, it is reasonable to state that this methodology becomes valid for any converter topology because the operating principle of SC circuits is essentially based on the charge transfer from one element to another.

3.5. Regulation and Control

In this section, the most common control techniques applicable to SC converters are discussed since there is a significant voltage decrease when the load is connected to the output of such structures. The simplest way to increase the voltage gain in the case of reducing the output voltage is to associate more cells in series. However, the higher the component count, the greater the series resistance in the circuit and, consequently, the lower the efficiency. A problem that can occur is the accuracy of the regulation capacity, as the connection of a second stage can excessively increase the output voltage.

Another solution that has less impact on efficiency is the achievement of voltage regulation using pulse width modulation (PWM) [124,125,126]. This technique is widely used to control the output voltage in the conventional buck, boost, and buck-boost converters. In SC circuits, this operation mode is divided into two stages per cycle. In the first stage, the capacitor is charged, and the duty cycle can assume values between 0 < D < 0.5. In the second stage, the capacitor is connected to the load, and its discharge process begins. This situation is repeated during each cycle. It should be noted that when the duty cycle approaches zero, the voltage ripple increases and, consequently, the input current peaks increase [127].

Another strategy presented in the literature is the interleaved connection of cells [128]. However, this arrangement may or may not work with overlapping phases [128,129]. The main advantage lies in the fact that this method reduces the output voltage ripple, providing not only voltage regulation, but also an improvement in efficiency. However, as mentioned, it should be considered that a greater number of components causes an increase in losses. An advantage of this technique is the possibility of increasing the power levels considering that the current is shared among the interleaved cells.

The pulse frequency modulation (PFM) control technique is also well known in electronics. The advantage of this strategy lies in the simplicity of producing a variable frequency signal from monitoring the output voltage or output current. This signal increases the switching frequency according to the load current [130,131]. However, this technique does not guarantee reduced voltage ripple under heavy load conditions [132]. Another important point is that, when MOSFETs are used, increasing the switching frequency can increase switching losses according to (4).

A prominent control technique that takes advantage of the MOSFET transconductance is called quasi-switched capacitor (QSC). This strategy was formerly introduced in [133] as a new SC cell. Subsequent work used this concept in a somewhat obscure way while adopting the terms “QSC cell” and “QSC control”. However, it can be regarded as technique for controlling the gate-source voltage in a MOSFET. When operating in the active region, that is, in the triode region, the drain current is proportional to the gate-source voltage [134]. If the MOSFET always operates in that region, the drain current can be controlled. The main difference in relation to the other techniques mentioned above is that the input current waveform does not change during the charge stage, while in other cases, there are short-term current peaks. In this way, this allows to reduce the current stresses on the switches and prevents the occurrence of conducted EMI. This control approach does not affect the converter efficiency, because, according to Expressions (23) and (24), the efficiency during the charge stage is dependent only on the difference between the voltage on the capacitor in charging process and the source voltage.

3.6. Comparison among SC Topologies

The existing SC topologies are very similar to each other in terms of constructive aspects, although they may differ with respect to the voltage gain, number of components, and stresses. The component count associated with the capacitors and switches required by a given basic cell is shown in

Table 5. Comparison among SC topologies.

Parameters	SCVD	SPC	SPMC	Dickson	FC	Ladder *
Capacitors	N	N	3N	2N	N	2N
Switches	4N	3N	3N	2N	3N	2N
Maximum voltage on the capacitors for N-th cell	$2^{(N-1)}{V}_i$	$V_i$	$V_i$	$N{V}_i$	$F\left(N-1\right)V_i$	$V_i$
Maximum voltage on the switches for N-th cell	$2^{(N-1)}{V}_i$	$N{V}_i$	$N{V}_i$	$V_i$	$F\left(N-1\right)V_i$	$V_i$

* Boost mode.

for a generic number of cells N = 1, 2, 3, …. Besides, F(N − 1) is the (N − 1)-th Fibonacci number, where F(0) = F(1) = 1. It is also worth mentioning that Table 5 summarizes the main characteristics of basic topologies used in the conception of SC dc-dc converters, which can assist the designer in choosing which configuration is more adequate for a given application.

It is observed that the voltage stresses on the capacitors increase as more cells are added to extend the conversion ratio in the SVCD and FC configurations. The maximum voltage stress on the active switches is constant and equal to the input voltage V_i in the Dickson and ladder converters, which makes them proper choices for applications involving high output voltages. The current stresses cannot be estimated in a such a simple manner because they do not depend only on the characteristics of the load connected to the converter, but also on the charge modes of the capacitors, which were discussed in Section 3.2.

SC circuits operating in full charge mode present high short-term current peaks, which occur when the capacitors are charged. The maximum value of the charging current must be taken into account carefully because it may damage the switch. Besides, the pulsating nature of the input current may lead to the increase of conducted electromagnetic emissions. Operation at high frequencies and/or the use of high capacitances to ensure the full charge or no charge condition are possible solutions that allow for mitigating such undesirable problems. The QSC is a control approach that can be used for this purpose as well as explained in Section 3.5. It allows obtaining a nearly continuous input current, thus causing the current stresses on the switches to be reduced as a consequence. Interleaved converters are also less susceptible to the undesirable effects caused by the voltage ripple on the capacitors as discussed in Section 3.5.

4. Applications

Conventional SC converters are widely used in practice, mainly at low power levels and when high power density is desired [35]. Such characteristics make them quite attractive for specific applications involving embedded electronic systems, biomedical equipment, energy harvesting, and general-purpose microelectronics. In this context, it is worth mentioning that some modern portable electronic devices have very low power consumption. For instance, smartphones consume approximately 1 W, while cardiac pacemakers consume 50 μW [135].

Several studies have discussed the relevance of using the available energy in the environment where a device or equipment is placed. Various forms of energy can be harvested and converted into electrical energy to power the aforementioned devices. Photovoltaic solar energy is a typical example [136,137,138,139,140], but other more unusual sources can be considered, such as pyroelectric [141,142,143] and blood sugar [144,145,146].

Regardless of the source, the applicability of SC converters in these cases has also drawn the attention of researchers in the case of other low-power applications [147,148,149,150]. Its use in the supply of sensors or very-low-power circuits is also possible, precisely in conditions in which space is limited [151,152,153,154,155].

Hybrid vehicles, which combine fossil fuels and energy storage devices such as supercapacitors and batteries, also constitute a favorable scenario for the use of SC converters [156,157]. In this case, the topologies can be employed to increase the voltage across the batteries and also to control the bidirectional power flow [47,158] aiming at the energy management and charge of the batteries [159,160,161,162,163]. Owing to the small size, one can even consider the integration of the whole set to the battery system itself as a consolidated solution.

SCVDs are often reported in the literature for applications rated at some milliwatts, which include electrically-erasable programmable read-only memories (EEPROMs), very large-scale integration (VLSI) systems VLSI [164,165], energy harvesting [152,154,166,167], and liquid crystal display (LCD) technology [168,169].

This technology can be used to supply electronic circuits such as operational amplifiers and analog-to-digital (A/D) converters, since this is the only type of SC configuration available in the form of commercial ICs. Table 6 shows some voltage doubler ICs available on the market, which are rated at low output currents and low input voltages.

SPC structures can be configured more simply with an array of encapsulated capacitors aiming at a wide range of options not only in series to increase the gain, but also in parallel to decrease the current ripple [170]. Existing applications are reported in a smaller number of publications owing to the control complexity. Among them, some circuits are proposed for energy harvesting in [150,151,153,171] and biomedical systems [172]. Since the SP topology is very versatile with respect to the voltage gain, the output voltage may remain constant even when there are large variations in the input voltage.

The Dickson converter was one of the first structures developed for practical applications involving memories [173,174,175,176]. The evolution of existing circuits is mainly concerned with the increase of energy density and efficiency. Some devices can even be integrated with the same memory chip [120,177].

The FC is similar to the SPC, but it requires a larger number of switches and capacitors. An emergency power supply for a computer system based on this structure was proposed in [178] as an emerging application, but LCD drivers can also be found in [179,180].

The ladder topology is also adequate for some specific applications, even though it is not very often used in DC-DC power conversion when compared with its remaining counterparts [181,182]. Among the topologies presented, this is the one with the highest efficiency and versatility. In experimental tests, the efficiency can be greater than 90% with low voltage ripple even at power levels on the order of 1 kW [183]. Thus, this is the only promising topology developed so far for high-power applications, while the others are only feasible at low power levels [170].

Table 6. Parameters of SCVD ICs.

IC		Vi (V)	Io (mA)	ηmax (%)
MAX660 [180]		1.8 to 5.5	100	88
LM2660 [184]		2.5 to 5.5	100	88
LM2685 [185]		2.85 to 6.5	50	80
MAX1680 [186]		2 to 5.5	5	90
MAX860 [187]		2 to 5.5	50	87
LT1054 [188]		3.5 to 15	100	NA
LM2665 [189]		2.5 to 5.5	40	90
LM2767 [190]		1.8 to 5.5	15	96
ADP3610 [191]		3 to 3.6	320	90
TCM828 [192]		1.5 to 5.5	25	95
GS7660 [193]		3 to 6	200	98

Table 6. Parameters of SCVD ICs.

IC		Vi (V)	Io (mA)	ηmax (%)
MAX660 [180]		1.8 to 5.5	100	88
LM2660 [184]		2.5 to 5.5	100	88
LM2685 [185]		2.85 to 6.5	50	80
MAX1680 [186]		2 to 5.5	5	90
MAX860 [187]		2 to 5.5	50	87
LT1054 [188]		3.5 to 15	100	NA
LM2665 [189]		2.5 to 5.5	40	90
LM2767 [190]		1.8 to 5.5	15	96
ADP3610 [191]		3 to 3.6	320	90
TCM828 [192]		1.5 to 5.5	25	95
GS7660 [193]		3 to 6	200	98

5. Conclusions

SC converters have drawn significant attention from academia and industry recently owing to their prominent characteristics associated with high energy density and low EMI levels in DC-DC, DC-AC, AC-DC, and AC-AC conversion. Considering that many hybrid SC topologies composed of capacitors, semiconductors, and inductors have been reported in the literature, this work has been specifically concerned with a general review of important concepts associated with “pure SC” DC-DC converters. The existing configurations were analyzed in detail, while a brief explanation on voltage multipliers was also included, considering that some structures are derived from such popular circuits used in low-power ac-dc power conversion.

It should be noted that pure SC converters employ only semiconductors and capacitors. Thus, it is reasonable to state that a successful design relies on the careful choice of such components, whose characteristics and impact on overall performance could be addressed in detail. Among controlled switches, MOSFETs are most used in practice owing to their ability to operate at high frequencies. In this context, the characteristics of required parts have an important role aimed at the increase of the converter efficiency. The capacitors, which are also a fundamental part of the circuit design, were analyzed in terms of the main characteristics of different types of technologies found on the market, such as film, ceramic, and electrolytic. It is worth mentioning that capacitors have direct impact on the converters size, while both the ESL and ESR limit the maximum operating frequency of the circuit.

During the design stage, it is very important to know inherent characteristics in operation modes of SC converters. Essentially, there are three distinct modes, each of which has a different impact on the circuit performance in terms of efficiency and regulation. From the previous knowledge of the intrinsic resistances of both capacitors and switches, it is possible to define a proper time constant aimed at achieving an adequate operating frequency for the converter. This aspect is also directly associated with the behavior of the output resistance of the circuit, which is inversely proportional to the conversion efficiency.

The control and voltage regulation techniques for SC DC-DC converters are still reported in a smaller number in the literature compared to those dedicated to the classic isolated and non-isolated DC-DC converters. Thus, this work was dedicated to the detailed analysis of the existing strategies for regulating the output voltage and obtaining reduced high-frequency ripple, since these aspects directly influence the efficiency significantly.

The classic control approach is based on PWM, but other techniques may also lead to good overall performance, e.g., PFM, especially because the duty cycle has little influence on the output voltage depending on the operation mode. The use of interleaved structures also provides good output voltage regulation and high efficiency due to the ripple minimization. Other more complex solutions, e.g., QSC, can also be used, with special concern to the reduction of the current peaks due to the operation of MOSFETs in the triode region.

Typical low-power applications of SC converters include general-purpose ICs, memories, and also energy harvesting for biomedical equipment owing to inherent high energy density. Considering the low efficiency typically achieved by SC structures in high-power applications, few publications are dedicated to the conception of power electronic converters based on capacitors and semiconductors only, i.e., without using inductors and transformers. However, more recently, the attention of many researchers has turned to the development of prominent structures for higher power levels. In this scenario, the efficiency analysis of SC converters considering the operation modes associated with the charging and discharging of capacitors is of major importance. It is also noteworthy that the ladder structure is an attractive solution for the development of power converters rated on the order of a few units of kilowatts.

Despite the apparently reduced complexity of the topologies, the design procedure must consider many variables. The characteristics of semiconductors and capacitors can drastically reduce the circuit efficiency if adequate components are not chosen. Another important aspect is the fact that the control methods are relatively complex, and do not act so responsively in the output voltage.

Even in the face of all the difficulties encountered in the development of these circuits, several modern applications highlight the great application potential of SC converters. Considering the constant search for miniaturization of electronic devices and the growing use of system on chip (SOC) type ICs, the topologies in question are among the main solutions that can be fully integrated into a single chip, since inductors do not allow that feature.