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

: 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. 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.


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 character-istics 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.

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 . 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.

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 Energies 2021, 14, 2231 3 of 33 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].
Energies 2021, 14, x FOR PEER REVIEW 3 of 33 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].

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.

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 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].

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].

Greinacher Voltage Doubler (1914)
The simplest structure of a step-up converter that uses switches and capacitors is th Greinacher voltage doubler, which is shown in Figure 1 [66]. This circuit is powered b an AC source and operates according to two stages. In the first one, the capacitor i 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 semiconductor and capacitors. This circuit or cell can be connected in a modular way in series or paralle with phase opposition, thus allowing the achievement of high voltages [68,69].

SC Voltage Doubler
Although the voltage step-up circuit shown in Figure 1 may be useful in som 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 th capacitors. The circuit operates in the same way as the voltage multiplier, i.e., there ar two stages. In the first one, the charge of capacitor C1 occurs. In the second one, capacito C1 is connected in series with the source and the load. It is possible to employ th connection of several cells in series to obtain a given gain 2 n , where n is the number o associated stages. Thus, it is possible to obtain high voltage gains, which allow supplyin loads with high output voltages.

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]. Thi circuit became known as Cockcroft and Walton voltage multiplier (CWVM), or simpl voltage multiplier, which is exemplified in Figure 3 in terms of several stages. This is on of the best-known structures in electronics to obtain high gains owing to the simplicity o 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 C 1 occurs. In the second one, capacitor C 1 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.

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 Cn. 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 n 3 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].

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 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 C 1 and, in each half cycle of the AC input voltage, the charge Q is transferred from capacitor to capacitor (C 2 , C 3 , . . . ), 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 n 3 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].

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. the circuit, low cost, and simple implementation. Considering the components as ideal the output voltage Vo can be obtained by nVi, where n is the number of associated cell and Vi is the input voltage. Recent works point to the possibility of using this structure in several modern appli cations involving high-gain DC-DC converters associated with photovoltaic module [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 hal cycle of the AC input voltage, the charge Q is transferred from capacitor to capacitor (C2 C3, ), until it reaches the last element Cn. 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 n 3 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].

Series-Parallel Converter (1971)
A proposed alternative to overcome the main limitations of the CWVM in terms o voltage regulation is the series-parallel converter (SPC) shown in Figure 4 [79]. This con verter is very versatile since each cell can be associated in series or parallel, and the outpu 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 arrange ment has some remarkable advantages because it does not present the same problems a the CWVM, especially concerning the high output impedance. However, there are othe drawbacks, such as the simultaneous charging of many capacitors, which can demand a 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 Energies 2021, 14, 2231 5 of 33 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].

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.
high current from the source in a short time. Also, if many capacitors are in chargin cess, the voltage ripple at the output may be high, directly impacting the convert ciency. Depending on the length of the connection tracks on the printed circuit b (PCBs), the arrangement of the elements, and the number of capacitors, there may problems due to parasitic capacitances, drastically reducing the efficiency of the cir the event of high voltage gains [80].

Series-Parallel Multiphase Converter (1973)
Another alternative to overcome the problems presented by the CWVM and th is the series-parallel multiphase converter (SPMC) [81], which is shown in Figure 5. way, it is possible to mitigate the transients of the input current and the effect of pa capacitances. While part of the capacitors is connected to the source during the cycle, the other components that are fully charged are connected to the load. Mean part of the other series-connected capacitors remains in an intermediate charge Therefore, there are at least two or three phase-shifted signals controlling the conn of the capacitors with the source and the load.

Dickson Converter (1976)
Dickson implemented the first integrated multiphase voltage multiplier in wh circuit reached an output voltage of 40 V from a 15 V source [82]. Besides the inno inherent in the introduction of the topology itself, it should be noted that this was th structure built experimentally in a totally encapsulated form. Figure 6 shows the D converter.

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.
high current from the source in a short time. Also, if many capacitors are in charg cess, the voltage ripple at the output may be high, directly impacting the conver ciency. Depending on the length of the connection tracks on the printed circuit (PCBs), the arrangement of the elements, and the number of capacitors, there may problems due to parasitic capacitances, drastically reducing the efficiency of the c the event of high voltage gains [80].

Series-Parallel Multiphase Converter (1973)
Another alternative to overcome the problems presented by the CWVM and is the series-parallel multiphase converter (SPMC) [81], which is shown in Figure 5 way, it is possible to mitigate the transients of the input current and the effect of p capacitances. While part of the capacitors is connected to the source during the cycle, the other components that are fully charged are connected to the load. Mea part of the other series-connected capacitors remains in an intermediate charg Therefore, there are at least two or three phase-shifted signals controlling the con of the capacitors with the source and the load.

Dickson Converter (1976)
Dickson implemented the first integrated multiphase voltage multiplier in w circuit reached an output voltage of 40 V from a 15 V source [82]. Besides the inn inherent in the introduction of the topology itself, it should be noted that this was structure built experimentally in a totally encapsulated form. Figure 6 shows the 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.

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, . . . ).
Energies 2021, 14, x FOR PEER REVIEW In the literature, there are several publications dedicated to the study of this to aiming at improving aspects such as efficiency, dynamic response, and regulation A limitation of this structure lies in the fact that it does not present good perform high-power applications owing to the difficulty of obtaining adequate regulation low efficiency when it is desired to obtain high voltage gains.

Fibonacci Converter (1991)
When the first SC converters were designed, the power levels were initially to milliwatts or a few units of watts. A natural next step would be to reduce the n of components of these structures, aiming at increasing the gain, power levels, a ciency. In this sense, the Fibonacci converter (FC) was proposed as a multiphase to in which the capacitors are associated in series and parallel. In this case, the cells a bined with the source and with each other to achieve the desirable voltage gain [8 FC is presented in Figure 7 and receives this name because of its gain characterist each cell/stage coupled to the converter leads to an increase in the output voltage ing 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 a nected. At this point, it should be noted that among the topologies presented so use the combination of capacitors in series and parallel, the highest gain by the asso of stages is obtained with the FC [15,87].

Ladder Converter (1992)
The CWVM can produce high voltages from a low-voltage source, but there itations to this structure, such as the high output impedance that prevents high gai being achieved. In general, it can be stated that as the gain increases, the output ance also does by a cubic factor [79], which can compromise the output voltage reg for loads requiring high currents.
The basic ladder cells are shown in Figure 8, which are derived from a study ing the proposal of a three-level neutral point clamped (NPC) inverter topology The inverter described in [88] and [89] works with three voltage levels: Vi, Vi/2, a and it is called capacitor-clamped multilevel inverter or flying capacitor multil verter (FCMLI). This is a circuit similar to the NPC inverter, but the diodes are r by capacitors. 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].

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 stepup mode, the source is associated in series with n capacitors charged with a voltage Vi. 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 Vi. In this way, the voltage source in series with n charged capacitors can produce an output voltage Vo according to Equation (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 Cfly, 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): where ncap 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]. 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).
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): where n cap is the number of series capacitors in the circuit. Figure 9a shows a boosttype 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].

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

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 lowpower 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 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 Rds(on). This parameter is provided in the manufacturer's datasheet and expression 3 can be used to estimate the conduction losses [94,95]: where Id(rms) is the root-mean-square (RMS) drain current.

Component
Type

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

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 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]: where I d(rms) is the root-mean-square (RMS) drain current.

Component
Type 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 Energies 2021, 14, 2231 9 of 33 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]: 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.
The switching losses are essentially related to the time interval required b semiconductor to change from the ON state to the OFF state, or vice-versa. In oth the shorter the rise time tr and fall time tf, the faster the dynamic response of a M Table 1 shows the aforementioned parameters, as well as the parasitic capacita affect the switching losses. It is worth mentioning that Ciss and Coss do vary sig with the drain-source voltage Vds, but they are only slightly affected by the tem Capacitance Ciss is an important parameter, since its charge and discharge are d fluenced by drive circuit connected to the gate. Besides, capacitance Coss is charg voltage source of the circuit and discharged through the body diode when the is ON and OFF, respectively.
In turn, the switching losses are calculated according to the energy dissipa switch during the turn-on and turn-off intervals. These losses depend on the s frequency fs, the drain-source voltage Vds, the drain current Id, and the time in quired for turn-on and turn-off of the transistor given by ton and toff, respective both parts are considered in expression (4) [96]: Substituting the values assumed by distinct parameters defined in Table 1 (4), it is observed that SiC MOSFETs present improved performance with resp conduction and switching losses when compared with their Si counterparts. T MOSFETs can be regarded as an adequate choice for applications that high co efficiency.
The transistors shown in Table 1 have good performance at high frequenci the frequency increases, the switching losses can significantly compromise the of the converter. It is observed that the drain current also influences this behavi example, Figure 10 represents the losses in a MOSFET model IRFP460 operating subjected to a drain-source voltage of 100 V, while the switching frequency rang to 500 kHz and, at the same time, the drain current varies from 0 to 10 A. This plo strates the profile of losses in that particular component, but a similar analysis c veloped for any other commercial device, considering that its characteristics are by the manufacturer. Thus, one can consider these aspects in detail concerning t and estimate of the converter efficiency.

Diodes
Some SC topologies like Dickson, ladder, and SMPC use diodes besid switches, which can influence the conversion efficiency significantly. Therefore portant to know the characteristics of such semiconductors, especially consid commercial availability of Si and SiC components. Table 2 summarizes the main parameters associated with some diode mode include the maximum value of the repetitive peak reverse voltage VRRM, the ave ward current IF(avg.), the forward voltage drop VF, the reverse recovery time trr 10 20 Loss (W)

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 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. Table 2. Comparison of the characteristics of different diodes.

Component
Type 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.  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 Vto represents the voltage drop across the semiconductor junction and rt is the intrinsic resistance. The values of Vto and rt 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).
where IF(avg.) and IF(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.
where VF(max) is the maximum value assumed by the forward voltage drop. From Equation (6), it is observed that the longer trr, the higher the switching losses, and the lower the 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).
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.
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.

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.
Energies 2021, 14, x FOR PEER REVIEW 11 of 3 efficiency as a consequence. Besides, the efficiency can be further increased if diodes wit low Qrr or QC are chosen in practical designs.

Capacitors
Besides semiconductors, capacitors are essential devices in SC converters, as they ar the only energy storage elements used in the circuit when hybrid topologies are no adopted, which also have inductors as in the case of [97]. The equivalent model of a ca pacitor basically consists of the series association of a capacitance C, the equivalent serie 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 com ponent. It has a direct impact on the energy density and, consequently, on the dimension of the converter. Parameter ESR cannot be accurately estimated from the simple measure ment performed with an ohmmeter because it is strongly influenced by the dielectric, elec trodes, and the ohmic losses in the terminals of the component.

C ESR ESL
The losses in dielectrics correspond to the delayed bias or relaxation, varying accord ing to the dielectric medium, temperature, and frequency [98]. The losses in electrodes i low frequencies are essentially ohmic and depend on the dimensions and type of materia At high frequencies, the losses increase significantly owing to the skin and proximity e fects.
Overall, it is reasonable to state that the losses in capacitors are influenced by a wid variety of aspects. In SC-based converters, this element can be satisfactorily represente by C and ESR, resulting in the representation shown in Figure 13. Parameter δ is calle 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)  The ESL can be hardly measured because it depends on the pad layout, capacito height, and power plane spreading inductance [102]. It can be neglected at low frequen cies, but its influence should be considered as the operating frequency gets close to th resonance frequency of the capacitor. The ESL limits the maximum switching frequenc 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].

Capacitors
Besides semiconductors, capacitors are essential devices in SC converters, as t the only energy storage elements used in the circuit when hybrid topologies adopted, which also have inductors as in the case of [97]. The equivalent model pacitor basically consists of the series association of a capacitance C, the equivalen resistance (ESR), and the equivalent series inductance (ESL) as shown in Figure 12 C ESR ESL Figure 12. Equivalent circuit of a capacitor.
Capacitance C is related to the type of dielectric used in the construction of th ponent. It has a direct impact on the energy density and, consequently, on the dim of the converter. Parameter ESR cannot be accurately estimated from the simple m ment performed with an ohmmeter because it is strongly influenced by the dielectr trodes, and the ohmic losses in the terminals of the component.
The losses in dielectrics correspond to the delayed bias or relaxation, varying ing to the dielectric medium, temperature, and frequency [98]. The losses in electr low frequencies are essentially ohmic and depend on the dimensions and type of m At high frequencies, the losses increase significantly owing to the skin and proxim fects.
Overall, it is reasonable to state that the losses in capacitors are influenced by variety of aspects. In SC-based converters, this element can be satisfactorily repr by C and ESR, resulting in the representation shown in Figure 13. Parameter δ i dissipation factor (DF) or loss angle, whose value is often provided in datasheets In this way, it is possible to estimate the ESR using Equation (8)  The ESL can be hardly measured because it depends on the pad layout, ca height, and power plane spreading inductance [102]. It can be neglected at low f cies, but its influence should be considered as the operating frequency gets clos resonance frequency of the capacitor. The ESL limits the maximum switching fre 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 Energies 2021, 14, 2231 12 of 33 higher than the capacitive reactance. Figure 14 presents the behavior of the capacitor impedance as a function of the frequency and ESR [103]. 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 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.
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).  Table 3 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).
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).
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  Figure 15 with the aid of Equation (10).
rgies 2021, 14, x FOR PEER REVIEW 13 o As an example, and for the eventual efficiency analysis of the converters, capacito was chosen in Table 3. By varying the RMS current from 0 to 10 A and the source quency from 0 to 500 kHz, it is possible to plot the component losses as shown in Fig  15 with the aid of Equation (10). Of course, each type of capacitor has its own electrical and constructive characte tics. The objective of this work is not to present a detailed analysis of these componen but only to address some important aspects that allow the choice of the most suitable e ment for a given application. In general, it is reasonable to state that the most popu capacitor used in SC converters is of film type. This component is widely used mainly applications in which it is necessary to operate at high voltages and frequencies. The f dielectric has low DF, reduced ESR, and good response at high frequencies [104]. The electric consists of a thin layer of plastic, the most common materials being polyester a polypropylene. Capacitors made of metallized film have vaporized dielectrics with a t layer of aluminum, as this additional layer provides the dielectric with the self-regene tion ability [105][106][107]. Given the possible occurrence of a voltage higher than the nomi value supported by the capacitor, which in turn would damage the dielectric, this char teristic allows rapid heating to occur owing to the short circuit of the plates. When s jected to high temperatures, this layer of aluminum around the hole turns into alumin oxide, which has an insulating character and thus eliminates the short circuit.
Polyester capacitors can be found commercially from some units of nanofarads values on the order of 220 μF, with voltages greater than or equal to 1000 V. This mate often has limited use since, among the film-based capacitors, this is the type most sensit to the increase of temperature [108] and frequency [104]. Its use is recommended mai when it is desired to obtain converters with reduced dimensions since polyester capacit have the smallest size for a given capacitance value in comparison to its remaining co terparts.
When compared to other components, polypropylene-based capacitors demonstr excellent characteristics, such as the fact that the capacitance varies little with temperatu frequency, and voltage variations. Polypropylene absorbs less moisture than polyes which makes this material suitable for a wide range of applications in harsh environmen Commercial capacitance values can reach up to 270 μF.
Ceramic capacitors are the passive devices most used in low power electronics tod In this case, the capacitance does not vary significantly with the frequency and tempe ture, while the components present low loss factor [109]. The dielectric materials used be porcelain, mica, and other silicates. Also, the components can be produced on a sin disc or in multiple layers, i.e., in the form of multilayer ceramic capacitors (MLCCs) negative aspect lies in the fact that fractures in the dielectric can occur owing to volt spikes and rapid changes in temperature, which can cause failures and even a short circ 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 (Al 2 O 3 ) deposited on that same sheet, thus forming the dielectric. This Al 2 O 3 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.

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.
ing terminal, the paper is wrapped in aluminum foil, so that this set forms the terminal of the capacitor (cathode). The other terminal of the capacitor is connect nally to a second aluminum foil that constitutes the positive terminal (anode). T receives an electrochemical treatment called anodic oxidation, which causes the ance of a thin layer of an insulating material called aluminum oxide (Al2O3) depo that same sheet, thus forming the dielectric. This Al2O3 oxide layer has the advantage of a high dielectric constant and hi ity with a very thin thickness on the order of 0.7 μm. With this characteristic, it is to reach high capacitances in a very small volume, which can vary from 1 μF to Thus, this is one of the most important characteristics of this type of capacitor other hand, the maximum voltage stresses supported by this component are low tion to the other elements previously presented. If this type of component is sub transients that exceed the rated voltage, even if very briefly, the dielectric is compr causing a change in the capacitance value and reducing the component's usefu general, this type of capacitor has defined polarity and, if it is connected to a circ inverted way, electrochemical reactions can occur inside it, causing leakage and plosion. When high capacitances are required, their use is often essential, but thes itors have a shorter useful life than film and ceramic components. As they are m ceptible to failure, their use on a large scale is avoided.

Operation Modes
As mentioned, SC converters have a simpler structure than conventional pow tronic converters that employ inductors, but the control system becomes more co this case. Theoretically, it is possible to achieve high voltage gains with some str However, it should be noted that the higher the gain, the more components wil ployed and, consequently, the lower the efficiency. Figure 16  To analyze the behavior of the output voltage, the circuit represented in Fig  adopted. In this case, the converter can be modeled as an ideal transformer wi ratio m:n, where Ro is the equivalent output resistance of the SC converter, whic nected in series with the load RL [86,111]. In this model, the resistance Ro con power transferred to the load and the output voltage regulation. The variation practice is mainly associated with the switching frequency and the internal resis the converter elements. 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 establish switching limit (SSL) and fast switching limit (FSL) [112,113]. When the converter in SSL, the switching frequency is low enough for the capacitors to fully charge charge. In this scenario, the currents have short-term peaks. Besides, all resistanc switches and capacitors are neglected, as these impedances do not affect the full c the capacitors. In this operation mode, the duty cycle has little influence on the of the converter. In the FSL condition, the currents that flow through the capacito considered constant, and the series resistances of the components cannot be negl they can affect the full charge of the capacitors.
Other analytical methods are based on the charge and discharge of capacito The circuit shown in Figure 18 can be used for this study, where an active switch S the energy transferred from a DC voltage source Vi to a capacitor Co. This repres is simpler than the model corresponding to Figure 17 and leads to similar resul adopted in this work in the study of topologies. It addresses the behavior of th according to three different modes shown in Figure 19, where the instantaneou vCo(t) and instantaneous current iCo(t) are represented for the full charge, partia and no charge conditions. In this case, the equivalent model of the converter is considered as a first-o circuit with time constant τ, where Ri = Rsw + ESR, Rsw being the resistance of th while Co represents the capacitor in the charge or discharge regime. The switch i on to charge the capacitors during a time interval Ton over a switching period Ts is reasonable to state that SC circuits are often designed based on the time consta than the switching frequency. Besides, such converters are capable of operatin charge condition even at high frequencies according to the specifications of capac switches used in the design.
The three modes of operation are defined [115,116]: (1) If Ton >> RiCo, that is, when the time interval during which the switch rema 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. As for the switching frequency, two asymptotic limits can be establishe switching limit (SSL) and fast switching limit (FSL) [112,113]. When the converter o in SSL, the switching frequency is low enough for the capacitors to fully charge charge. In this scenario, the currents have short-term peaks. Besides, all resistanc switches and capacitors are neglected, as these impedances do not affect the full c the capacitors. In this operation mode, the duty cycle has little influence on the b of the converter. In the FSL condition, the currents that flow through the capacitor considered constant, and the series resistances of the components cannot be negle they can affect the full charge of the capacitors.
Other analytical methods are based on the charge and discharge of capacito The circuit shown in Figure 18 can be used for this study, where an active switch S the energy transferred from a DC voltage source Vi to a capacitor Co. This repres is simpler than the model corresponding to Figure 17 and leads to similar result adopted in this work in the study of topologies. It addresses the behavior of th according to three different modes shown in Figure 19, where the instantaneous vCo(t) and instantaneous current iCo(t) are represented for the full charge, partial and no charge conditions. In this case, the equivalent model of the converter is considered as a first-or circuit with time constant τ, where Ri = Rsw + ESR, Rsw being the resistance of the while Co represents the capacitor in the charge or discharge regime. The switch i on to charge the capacitors during a time interval Ton over a switching period Ts. is reasonable to state that SC circuits are often designed based on the time constan than the switching frequency. Besides, such converters are capable of operatin charge condition even at high frequencies according to the specifications of capaci switches used in the design.
The three modes of operation are defined [115,116]: (1) If Ton >> RiCo, that is, when the time interval during which the switch remai much greater than the time constant of the circuit, the capacitors charge a charge completely. Thus, the current becomes zero in each cycle, and then i assumed that the system operates under full charge condition according to 19a. In this case, the equivalent model of the converter is considered as a first-order R i C 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.
is not fully discharged [117]. The converter then operates on a partial charge condi tion. 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 Ton is small in relation to Ts, then Ton << RiCo. 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]: where β is given by Equation (12).
The time constant associated with the circuit is given by Equation (13).
The duty cycle is defined as the ratio between the on-time of the switch Ton and the switching period Ts as in Equation (14):   The three modes of operation are defined [115,116]: (1) If T on >> R i C 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 i C 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 i C 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]: where β is given by Equation (12).
The time constant associated with the circuit is given by Equation (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): where T s is defined according to Equation (15).
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.
Expression (16) is plotted in Figure 20a, where the behavior of the converter equivalent resistance is represented as a function of product τfs. 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 Req 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 Ton, the output resistance of the converter can be estimated as follows: (1) If β > 1, the converter operates in full charge mode, where Ton > 5τ, while the output resistance depends on the inverse of the product of fs and Co. In other words, the resistances of the switches and capacitors represented by Ri 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 Ri, Co, and fs. (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 Ri and D.  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 R eq
Full charge

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.

Voltage Balance
SCs rely on the principle that the energy stored in a given element is transf another, while the capacitors can be associated in series or in parallel depending configuration. Of course, small differences will exist in the rated capacitances even ponents with the very same specifications are used. In most SC converters, the transfer occurs in equivalent circuits represented by the parallel association of cap A given capacitor Ci previously charged with a voltage VCi transfers energy to capacitor Co, which has no charge or a smaller amount of charge according to th shown in Figure 21. It is worth mentioning that a given series resistance Ri also practice owing to capacitors Ci and Co, as well as to the active switches. The series resistance is neglected a priori in the forthcoming analysis. Thus, amount of charge in the circuit Qt is given by Equation (17): where QCi and QCo are the charges stored in capacitors Ci and Co, respectively. Since charge balance is supposed to occur, the initial charge Qi must be equ final charge Qf according to Equation (18).

Considering this condition, Expression (19) is valid:
where VCo is the initial voltage across capacitor Co; and Vf is the final value assume equilibrium voltage. Thus, it is possible to obtain Vf from Expression (19) as in Expression (20).
If Co is initially discharged, then VCo = 0 and Expression (19) becomes (21). 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): 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).
Considering this condition, Expression (19) is valid: 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).
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.

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.
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): 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 Tables 1 and 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.
Energies 2021, 14, x FOR PEER REVIEW However, for the capacitor discharge in another capacitor, the efficiency can mated in two ways that depend on the operation mode in which the circuit works, full charge or no charge. As an example, Figure 21 shows the circuit used in the a whose waveforms in Figure 22 represent the moment after which switch S is turne As previously mentioned, capacitor Ci is initially charged with voltage VCi, w capacitor Co is completely discharged. Thus, a charge balancing occurs so that at t of the process, the voltage on both capacitors becomes nearly balanced and equal if the capacitances are equal to each other. Therefore, the voltages across Co and increased and decreased, respectively, as the efficiency can be calculated according to tion (26).
Considering a capacitor discharging into another capacitor, but in partial mode, a similar analysis can be performed. Now, the capacitor Ci initially charge VCi is discharged into another capacitor until reaching VCi(min). In this way, the capa charges from a minimum voltage VCo to VCo(max), and then the cycle is interrupted. havior of the capacitor voltages in this case is shown in Figure 23. The efficiency type of discharge cycle can be given by Equation (27): where VCo(max) is the maximum voltages on capacitor Co; and VCi(min) is the minimum on capacitor Ci. A third possibility is the discharge of a capacitor in a parallel RC circuit, as sh 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).
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): where V Co(max) is the maximum voltages on capacitor C o ; and V Ci(min) is the minimum voltage on capacitor C i . mated in two ways that depend on the operation mode in which the circuit works, full charge or no charge. As an example, Figure 21 shows the circuit used in the an whose waveforms in Figure 22 represent the moment after which switch S is turne As previously mentioned, capacitor Ci is initially charged with voltage VCi, w capacitor Co is completely discharged. Thus, a charge balancing occurs so that at t of the process, the voltage on both capacitors becomes nearly balanced and equal if the capacitances are equal to each other. Therefore, the voltages across Co and increased and decreased, respectively, as the efficiency can be calculated according to tion (26).
Considering a capacitor discharging into another capacitor, but in partial mode, a similar analysis can be performed. Now, the capacitor Ci initially charge VCi is discharged into another capacitor until reaching VCi(min). In this way, the capa charges from a minimum voltage VCo to VCo(max), and then the cycle is interrupted. havior of the capacitor voltages in this case is shown in Figure 23. The efficiency type of discharge cycle can be given by Equation (27): where VCo(max) is the maximum voltages on capacitor Co; and VCi(min) is the minimum on capacitor Ci. A third possibility is the discharge of a capacitor in a parallel RC circuit, as sh Figure 24. In this case, there is a capacitor Ci charged with an initial voltage VCi, to a series resistance Ri is connected, corresponding to the sum of the resistances of the 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.
Energies 2021, 14, x FOR PEER REVIEW and the capacitors. The load RL is associated in parallel with a capacitor Co, whic tially discharged. Thus, the discharge occurs in two events. At first, the charge of ca Ci is divided between Co and RL. When the charge between the two capacitors is ba both start supplying RL until they are completely discharged, as represented in Fi At each stage, there is a different expression for the efficiency. In the first stage, the efficiency can be given by Equation (28).
In the second stage, the efficiency can be estimated by Equation (29).
In this analysis, the switching losses in the semiconductors must also be con although they have not been incorporated in the previous expressions. This is because adequately quantifying them depends on the intrinsic characteristics of ea of switch. In the case of MOSFETs, the increase in the switching frequency has di pact on the switching losses as shown in Equation (4) [18,113]. On the other hand the switching frequency is increased, the voltage ripple is reduced. With a smaller variation, the efficiency increases according to Expressions (23), (24), (26)- (28). Th up to the designer to carry out a detailed analysis regarding the best operation p the circuit.
Analyzing the behavior of capacitors operating in no charge, partial charge, charge modes, it is reasonable to state that this methodology becomes valid for a verter topology because the operating principle of SC circuits is essentially based charge transfer from one element to another.

Regulation and Control
In this section, the most common control techniques applicable to SC conver discussed since there is a significant voltage decrease when the load is connecte Energies 2021, 14, x FOR PEER REVIEW and the capacitors. The load RL is associated in parallel with a capacitor Co, whic tially discharged. Thus, the discharge occurs in two events. At first, the charge of ca Ci is divided between Co and RL. When the charge between the two capacitors is ba both start supplying RL until they are completely discharged, as represented in Fig  At each   In the first stage, the efficiency can be given by Equation (28).
In the second stage, the efficiency can be estimated by Equation (29).
In this analysis, the switching losses in the semiconductors must also be cons although they have not been incorporated in the previous expressions. This is j because adequately quantifying them depends on the intrinsic characteristics of ea of switch. In the case of MOSFETs, the increase in the switching frequency has di pact on the switching losses as shown in Equation (4) [18,113]. On the other hand the switching frequency is increased, the voltage ripple is reduced. With a smaller variation, the efficiency increases according to Expressions (23), (24), (26)- (28). Th up to the designer to carry out a detailed analysis regarding the best operation p the circuit.
Analyzing the behavior of capacitors operating in no charge, partial charge, a charge modes, it is reasonable to state that this methodology becomes valid for a verter topology because the operating principle of SC circuits is essentially based charge transfer from one element to another.

Regulation and Control
In this section, the most common control techniques applicable to SC conver In the first stage, the efficiency can be given by Equation (28).
In the second stage, the efficiency can be estimated by Equation (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.

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.

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 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.
Maximum voltage on the switches for N-th cell 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.

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]. 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

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.