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

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

Alencar Franco de Souza 1, Fernando Lessa Tofoli 2 and Enio Roberto Ribeiro 1,*

1 Institute of System Engineering and Information Technology, Federal University of Itajubá, Itajubá 37500-903, Brazil; [email protected] 2 Department of Electrical Engineering, Federal University of São João del-Rei, São João del-Rei 36307-352, Brazil; [email protected] * Correspondence: [email protected]

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

Keywords: DC-DC converters; integrated circuits; low-power electronics; semiconductors; switched capacitors Citation: Souza, A.F.d.; Tofoli, F.L.; Ribeiro, E.R. Switched Capacitor DC-DC Converters: A Survey on the Main Topologies, Design 1. Introduction Characteristics, and Applications. DC-DC converters are widely used in residential, commercial, and industrial applica- Energies 2021, 14, 2231. tions, such as renewable energy conversion systems, electric traction devices, and, mainly, https://doi.org/10.3390/en14082231 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 Academic Editor: Ali Mehrizi-Sani 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 Received: 24 February 2021 of materials and advanced techniques used in the manufacture of integrated circuits Accepted: 7 April 2021 Published: 16 April 2021 (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

Publisher’s Note: MDPI stays neutral power converters [3]. Recently, new semiconductors have become commercially available, with regard to jurisdictional claims in which are based on silicon carbide (SiC) and gallium nitride (GaN). These elements have published maps and institutional affil- promising characteristics that allow the gradual replacement of silicon-based (Si) devices iations. owing to their higher efficiency and operating frequency. Besides, there is the possibility of combining both manufacturing technologies in a single component [4–6]. 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 Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. capacitors [7,8]. Besides, new ferrosilicon alloys are capable of achieving higher magnetic This article is an open access article permeability with reduced hysteresis and eddy current losses in the inductor cores [9–11]. distributed under the terms and In the constant search for higher power density in power converters, topologies using conditions of the Creative Commons switched capacitors (SCs) have recently aroused the interest of industry and academia. Attribution (CC BY) license (https:// These structures have been adopted in low power electronic applications, especially in sys- creativecommons.org/licenses/by/ tems with limited physical dimensions and involving high energy density. Their character- 4.0/).

Energies 2021, 14, 2231. https://doi.org/10.3390/en14082231 https://www.mdpi.com/journal/energies Energies 2021, 14, 2231 2 of 33

istics allow monolithic integration [12,13], minimized levels of electromagnetic interference (EMI) [14–16], 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. Section2 addresses voltage multiplier circuits, as some SC cells are derived from these structures. Next, the SC DC-DC converters are presented. In turn, Section3 is dedicated to analyzing the main design parameters, which includes the influence of components and operation modes in the converter efficiency, regulation, and control. Section4 presents the main applications of these circuits, as well as future perspectives of their application. Section5 summarizes the main aspects investigated in the study.

2. SC Structures Although voltage multiplier circuits are simple structures widely known and described in analog electronics textbooks, they are still used in applications where high DC voltages must be obtained from an AC voltage source. Voltage multipliers are typically used in X-ray machines, scanning electron microscopes, and particle accelerators, among other devices, because they are simple and low-cost circuits [24–26]. In the literature, there are some SC structures dedicated to specific applications, which are often derived from basic converters [27–34]. A description of SC topologies and voltage multipliers used in the conception of non-isolated dc-dc converters for high step-up applications is presented in [35]. However, this study is specifically focused on the thorough analysis of existing techniques for extending the conversion range of non-isolated dc-dc converters. Some relevant issues for the design of SC-based converters are not addressed in [35], e.g., the proper choice of components, the charging mode of capacitors, evaluation of efficiency associated with relevant practical aspects, and control techniques aiming to achieve the output voltage regulation. It is worth mentioning that there is a wide variety of circuits that employ inductors and SCs in the form of hybrid topologies, especially for wide conversion range applications [36–65]. Owing to the existence of a great diversity of combinations with distinct characteristics, this work is dedicated to the analysis of classical topologies based on the use of only semiconductors and capacitors, resulting in the so-called “pure SC” converters.

2.1. Greinacher Voltage Doubler (1914) The simplest structure of a step-up converter that uses switches and capacitors is the Greinacher voltage doubler, which is shown in Figure1[ 66]. This circuit is powered by an Energies 2021, 14, x FOR PEER REVIEW 3 of 33

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2.1. Greinacher Voltage Doubler (1914) The simplest structure of a step-up converter that uses switches and capacitors is the 2.1. Greinacher Voltage Doubler (1914) Energies 2021, 14, 2231 Greinacher voltage doubler, which is shown in Figure 1 [66]. This circuit is3 ofpowered 33 by an ACThe soursimplestce and structure operates of a according step-up converter to two stages. that uses In switch the firstes and one, capacitors the capacitor is the is Greinachercharged to thevoltage peak doubler, value of whichthe source is shown voltage. in InFigure the second 1 [66]. one,This the circuit previously is powered charged by ancapacitor AC sour isce placed and operates in series according with the to source two stages. and supplies In the first a load. one, This the combinationcapacitor is AC source and operates according to two stages. In the ﬁrst one, the capacitor is charged to generates a voltage that is ideally equal to twice the input voltage [67]. chargedthe peak valueto the of peak the sourcevalue voltage.of the source In the secondvoltage. one, In thethe previouslysecond one, charged the previously capacitor is charged capacitorplaced in series is placed with the in source series and with supplies the source a load. This and combination supplies a generates load. This a voltage combination C1 generthat isates ideally a voltage equal to that twice is theideally inputD2 equal voltage to [twice67]. the input voltage [67].

C1 D2 + Vi D1 C2 Vo + – Vi D1 C2 Vo

– Figure 1. Greinacher voltage doubler.

Figure 1. Greinacher voltage doubler. FigureBeing 1. Greinacher a simple voltage structure, doubler this. was one of the first topologies capable of stepping up the voltageBeing a simpleacrossstructure, a load powered this was oneby an of theAC ﬁrst source topologies using capableonly passive of stepping semiconductors up theand voltage Beingcapacitors. acrossa simple This a load structure, circuit powered or thiscell by an canw ACas be one source connected of usingthe first only in topologiesa passivemodular semiconductors waycapable in series of stepping and or parallel up thecapacitors.with voltage phase This acrossopposition, circuit a load or cellth poweredus can allowing be connected by tanhe ACachieve in a source modularment using wayof high inonly series voltages passive or parallel [68,69 semiconductors with]. andphase capacitors. opposition, This thus circuit allowing or thecell achievement can be connected of high in voltages a modular [68,69 way]. in series or parallel with2.2. SCphase Voltage opposition, Doubler thus allowing the achievement of high voltages [68,69]. 2.2. SC Voltage Doubler Although the voltage step-up circuit shown in Figure 1 may be useful in some Although the voltage step-up circuit shown in Figure1 may be useful in some appli- 2applications,.2. SC Voltage itsDoubler use is only feasible when an AC voltage source is available. However, cations, its use is only feasible when an AC voltage source is available. However, many electronicmanyAlthough electronic circuits the are circuits voltage powered are step by powered- DCup sources. circuit by DC shown The sources. SC involtage Figure The doubler SC 1 may voltage (SCVD) be dusefuloubler circuit in (SCVD) some applications,showncircuit inshown Figure its in2 isuse Figure a suitableis only 2 is alternativeafeasible suitable when inalternative this an case AC [70 in voltage,71 this]. case source [70,71 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].

S1 S2 S4 Vi S1 S2 + S3 C1 S4Co Vo Vi + – S3 C1 Co Vo FigureFigure 2. 2.SCVD. SCVD. –

This structure employs active switches to control the charge and discharge of the Figurecapacitors.This 2. SCVD. The structure circuit operatesemploys in active the same switc wayhes as theto voltagecontrol multiplier,the charge i.e., and there discharge are two of the capacitors. The circuit operates in the same way as the voltage multiplier, i.e., there are stages. In the ﬁrst one, the charge of capacitor C1 occurs. In the second one, capacitor C1 is connectedtwo Thisstages. instructure seriesIn the with first employs the one, source the active charge and theswitc of load. hescapacitor It to is possiblecontrol C1 occurs. tothe employ charge In the the andsecond connection discharge one, of capacitor of the n capacitors.severalC1 is connected cells The in series circuit in to series obtain operates with a given in the the gain source same2 , where way andn as theis thethe load. numbervoltage It is of multiplier, associatedpossible to stages.i.e., employ there are the n twoThus,connection stages. it is possible Inof theseveral tofirst obtain one,cells high thein series charge voltage to of gains,obtain capacitor which a given C allow1 occurs. gain supplying 2 In, thewhere loads second n with is one,the high number capacitor of output voltages. Cassociated1 is connected stages. in Thus, series it i withs possible the source to obtain and high the voltage load. Itgains, is possible which allow to employ supplying the n connection2.3.loads Cockcroft with ofhigh Walton several output Voltage cells voltages. Multiplier in series or to Ladder obtain Structure a given (1932) gain 2 , where n is the number of associatedTo supply stages. a particle Thus, accelerator it is possible with to a obtain voltage high of 1 voltage MV, Cockcroft gains, andwhich Walton allow used supplying 2.3. Cockcroft Walton Voltage Multiplier or Ladder Structure (1932) loadsa circuit with powered high output by an AC voltages. source, substantially increasing the output voltage [72]. This circuitTo became supply known a particle as Cockcroft accelerator and with Walton a voltage voltage multiplierof 1 MV, Cockcroft (CWVM), orand simply Walton used 2.3.voltagea circuit Cockcroft multiplier, powered Walton which by Voltage an is AC exempliﬁed Multiplier source, substantially in or Figure Ladder3 in Structure terms increasing of (1932) several the stages. output This voltage is one [72]. This ofcircuit theTo best-known becamesupply aknown structuresparticle as accelerator inCockcroft electronics withand to obtain Waltona voltage high voltage of gains 1 MV, owingmultiplier Cockcroft to the (CWVM), simplicity and Walton ofor simply used athevoltage circuit circuit, multiplier,powered low cost, by and which an simple AC is source,exemplifi implementation. substantiallyed in Figu Consideringre increasing 3 in terms the components the of severaloutput as stages.voltage ideal, the This [72] .is This one output voltage V can be obtained by n·V , where n is the number of associated cells and circuitof the bestbecame-known knowno structures as Cockcroft in electronics andi Walton to obtain voltage high multipliergains owing (CWVM), to the simplicity or simply of V is the input voltage. voltagei 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

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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 cells theand circuit, Vi is the low input cost, voltage. and simple implementation. Considering the components as ideal, Energies 2021, 14, 2231 4 of 33 the output voltage Vo can be obtained by nVi, where n is the number of associated cells and Vi is the input voltage. cell C1 Cn cell C1 Cn + Vi D2 Vo D1 Dn +– Vi D2 Vo D1 Dn – C2 Figure 3. CWVM. C2

FigureFigureRecent 3.3. CWVM.CWVM. works point to the possibility of using this structure in several modern applications involving high-gain DC-DC converters associated with photovoltaic modules [73,74Recent] and works workseven pointhigh point topower theto the possibility factor possibility rectifiers of using of using [75 this– structure77 this]. structure in several in modernseveral applica-modern appli- tions involving high-gain DC-DC converters associated with photovoltaic modules [73,74] cationsIn involvingthe CWVM, high the- gainsource DC provides-DC conver someters charge associated Q to capacitor with photovoltaic C1 and, in moduleseach half and even high power factor rectifiers [75–77]. [73,74cycle ]of and the even AC inputhigh powervoltage, factor the charge rectifiers Q is[75 transferred–77]. from capacitor to capacitor (C2, In the CWVM, the source provides some charge Q to capacitor C1 and, in each half C3, ), until it reaches the last element Cn. This process involves shifting the charge cycleIn of the the CWVM, AC input the voltage, source the provides charge Q someis transferred charge Q from to capacitor capacitor C to1 and, capacitor in each half cyclethrough of the each AC diode input similarly voltage, tothe a chargeladder, Qwhich is transferred justifies the from name capacitor given to thecapacitor circuit. ( C2, (C2, C3,...), until it reaches the last element Cn. This process involves shifting the charge Cthrough3, An), eachuntil inherent diode it reaches problem similarly the with to last a ladder, this element topology which C justifiesn. is This that the process its name output involves given impedance to the shifting circuit. increases the ch byarge an throughn3 factorAn inherenteach as the diode number problem similarly of with stages thisto a topology increases.ladder, iswhich thatThis its justifiescharacteristic output the impedance name limits giv increases theen outo tputthe by circuit. an current to nlow3 factor Anvalues, inherent as the as number in problemthe case of stages ofwith high increases. this currents topology This; the characteristic is output that its vol output limitstage will theimpedance outputdecrease current increases rapidly to owingby an nlowto3 factor losses values, as [78] asthe. in number the case ofof highstages currents; increases. the output This characteristic voltage will decrease limits the rapidly output owing current to lowto losses values, [78]. as in the case of high currents; the output voltage will decrease rapidly owing 2.4. Series-Parallel Converter (1971) to2.4. losses Series-Parallel [78]. Converter (1971) AA proposedproposed alternative alternative to to overcome overcome the the main main limitations limitations of the of CWVM the CWVM in terms in terms of 2.4.ofvoltage voltage Series regulation-Parallel regulation Converter is is the the series series-parallel (1971)-parallel converterconverter (SPC) (SPC) shown shown in in Figure Figure4[ 79 4 ].[79] This. This con- converterverterA isproposed very is very versatile versatile alternative since since eachto each overcome cell cell can can bethe be associated associatedmain limitations in series of oror the parallel,parallel, CWVM and and in the the terms output of voltageoutputvoltage voltage regulationcan be can modified be is modifiedthe seriesat any at-parallel anytime. time. 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.cell

cell S1 Sn

S1 Sn + S2 Vi C1 Sn+1 Cn + S2 Vi C1 Sn+1 Cn Co Vo S3 Sn+2 Co Vo- S3 Sn+2

FigureFigure 4. 4.SPC. SPC. -

FigureAsAs 4. the SPC.the name name suggests, suggests, the the energy energy transfer transfer from from the input the toinput the outputto the output occurs when occurs when connectingconnecting some some of of the the capacitors capacitors in parallel in parall withel with the source the source during during the charge the stage.charge In stage. a In a secondsecondAs moment, moment,the name the thesuggests, capacitors capacitors the are energyare associated associated transfer in series in from series and the supply and input supply the to load. the the output This load. arrange- Thisoccurs arrange- when ment has some remarkable advantages because it does not present the same problems as connectingment has some some remarkable of the capacitors advantages in parall becauseel with it the does source not present during the samecharge problems stage. In asa the CWVM, especially concerning the high output impedance. However, there are other the CWVM, especially concerning the high output impedance. However, there are other seconddrawbacks, moment, such asthe the capacitors simultaneous are associated charging of in many series capacitors, and supply which the canload. demand This arrange- mentadrawbacks, high has current some such from remarkable as the the source simultaneous advantages in a short time.charging because Also, itof ifdoes many many not capacitorscapacitors, present arethe which insame charging can problems demand as a theprocess, CWVM, the voltageespecially ripple concerning at the output the mayhigh be output high, directlyimpedance. impacting However, the converter there are other drawbacks, such as the simultaneous charging of many capacitors, which can demand a

Energies 2021, 14, x FOR PEER REVIEW 5 of 33

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 effi- highciency. current Depending from the on source the length in a short of thetime. connection Also, if many tracks capacitors on the printed are in charging circuit boards process,(PCBs), the voltagethe arrangement ripple at theof the output elements, may beand high, the number directly ofimpac capacitors,ting the there converter may still effi- be Energies 2021, 14, 2231 5 of 33 ciency.problems Depending due to parasi on thetic lengthcapacitances, of the connectiondrastically redutrackscing on the the efficiency printed circuitof the circuitboards in (PCBs),the event the ofarrangement high voltage of gains the elements, [80]. and the number of capacitors, there may still be problems due to parasitic capacitances, drastically reducing the efficiency of the circuit in efﬁciency. Depending on the length of the connection tracks on the printed circuit boards 2.5. Series-Parallel Multiphase Converter (1973) the(PCBs), event the of arrangement high voltage of the gains elements, [80]. and the number of capacitors, there may still be problemsAnother due to alternative parasitic capacitances, to overcome drastically the problems reducing the presented efﬁciency ofby the the circuit CWVM in and the SPC 2.5.theis theSeries event series- ofParallel high-parallel voltage Multiphase multiphase gains [80 Converter]. converter (1973) (SPMC) [81], which is shown in Figure 5. In this 2.5.way,Another Series-Parallel it is possible alternative Multiphase to mitigate to Converter overcome the (1973) transients the problems of the inputpresented current by theand CWVM the effect and of the parasitic SPC isc theapacitances. Anotherseries-parallel alternative While multiphase part to overcome of the converter thecapacitors problems (SPMC) presentedis connected [81] by, which the to CWVM the is shown source and the in SPCduring Figure the 5. In charge this way,iscycle, the it series-parallel isthe possible other components multiphaseto mitigate converter thethat transients are (SPMC) fully [charged81of], the which input are is shown currentconnected in Figure and to 5the .the In effect this load. of Meanwhile, parasitic capacitances.way,part it of is possible the otherWhile to mitigate seriespart -o theconnectedf the transients capacitors capacitors of the inputis connected current remains and to in the the an effect source intermediate of parasitic during chargethe charge state. cycle,capacitances.Therefore, the other there While components partare ofat theleast capacitors that two are or is threefully connected chargedphase to the-shifted sourceare connected duringsignals the controlling to charge the cycle,load. the Meanwhile, connection the other components that are fully charged are connected to the load. Meanwhile, part of parttheof the other of capacitors the series-connected other serieswith the capacitors-connected source remains and capacitors the in anload. intermediate remains chargein an state. intermediate Therefore, charge state. Therefore,there are at there least a twore at or least three two phase-shifted or three signalsphase- controllingshifted signals the connection controlling of the the connection ofcapacitors the capacitors with the with source the and source the load. andcell the load. Φ1 Φ2 cell Φ3 Cn+2 Φ1 Cn+3 Φ2 C1 C2 C3 Cn+4 Φ3 D2 D3 Cn+2Dn+2 Dn+4 Do Cn+3 C1 C2 C3 Dn+3 + Vi D1 Cn+4 Co Vo D2 D3 Dn+2 Dn+4 Do – Dn+3 + Vi D1 Φ1 Co Vo Φ2 – Φ1Φ3

Φ2 Figure 5. SPMC. Φ3

FigureFigure2.6. Dickson 5. 5. SPMC. Converter (1976) 2.6. DicksonDickson Converter implemented (1976) the first integrated multiphase voltage multiplier in which the 2.6.circuit DicksonDickson reached Converter implemented an output (1976) the ﬁrstvoltage integrated of 40 multiphase V from a voltage15 V source multiplier [82] in. whichBesides the the innovation circuitinherentDickson reached in the implemented an introduction output voltage the of first 40the V inttopology fromegrated a 15 V itself, sourcemultiphase it [ 82should]. Besides voltag be thenotede multiplier innovation that this in was which the the first circuitinherentstructure reached in thebuilt introduction an experimentally output of voltage the topology in ofa totally40 itself, V from itencapsulated should a 15 be V noted source form. that [82] this Figure. was Besides the 6 shows ﬁrst the innovationthe Dickson structure built experimentally in a totally encapsulated form. Figure6 shows the Dickson inherentconverter. in the introduction of the topology itself, it should be noted that this was the first converter. structure built experimentally in a totally encapsulated form. Figure 6 shows the Dickson converter. cell Φ Φ cell

Φ CA CB CA1 CB1 Φ Do D2 DA1 DB1 + CAD1 CB CA1 CB1 Vi Co Vo Do – D1 D2 DA1 DB1 + Vi Co Vo Φ –

ΦΦ

Φ Figure 6. Dickson converter. Figure 6. Dickson converter.

Figure 6. Dickson converter.

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Energies 2021, 14, 2231 In the literature, there are several publications dedicated to the study6 of 33of this topology aiming at improving aspects such as efficiency, dynamic response, and regulation [83–85]. A limitation of this structure lies in the fact that it does not present good performance in highIn-power the literature, applications there are owing several publicationsto the difficulty dedicated of obtaining to the study adequate of this topology regulation and the lowaiming efficiency at improving when aspects it is suchdesired as efficiency, to obtain dynamic high response,voltage gains. and regulation [83–85]. A limitation of this structure lies in the fact that it does not present good performance in 2.7.high-power Fibonacci applications Converter owing (1991) to the difficulty of obtaining adequate regulation and the low efficiency when it is desired to obtain high voltage gains. When the first SC converters were designed, the power levels were initially limited to2.7. milliwatts Fibonacci Converter or a few (1991) units of watts. A natural next step would be to reduce the number of componentsWhen the first of SC the convertersse structures, were designed, aiming the at power increasing levels were the initially gain, power limited tolevels, and effi- milliwatts or a few units of watts. A natural next step would be to reduce the number of ciency.components In this of these sense, structures, the Fibonacci aiming at converter increasing the(FC) gain, was power proposed levels, and as a efficiency. multiphase topology inIn thiswhich sense, the the capacitors Fibonacci converter are associated (FC) was in proposed series asand a multiphase parallel. topologyIn this case, in which the cells are com- binethe capacitorsd with the are source associated and in with series each and parallel.other to In achieve this case, the the desirable cells are combined voltage gain [86]. The FCwith is the presented source and in with Figure each 7 other and to receives achieve thethis desirable name because voltage gain of its [86 ].gain The characteristic, FC is since presented in Figure7 and receives this name because of its gain characteristic, since each each cell/stage coupled to the converter leads to an increase in the output voltage follow- cell/stage coupled to the converter leads to an increase in the output voltage following a ingproportion a proportion defined bydefined the Fibonacci by the sequence Fibonacci (2, 3, sequence 5, 8, 13, . . .(2, ). 3, 5, 8, 13, ).

cell

S1 Sn+1 + S2 C1 Sn+2 Cn Vi Co Vo

S3 Sn+3 –

FigureFigure 7. 7.FC. FC.

The FC is very similar to the SPC, differing mainly in the way the capacitors are connected.The FC At thisis very point, similar it should to be the noted SPC, that differing among the mainly topologies in presentedthe way sothe far capacitors that are connected.use the combination At this point, of capacitors it should in series be noted and parallel, that among the highest the gain topologies by the association presented so far that useof stages the combination is obtained with of the capacit FC [15ors,87]. in series and parallel, the highest gain by the association of2.8. stages Ladder is Converter obtained (1992) with the FC [15,87]. The CWVM can produce high voltages from a low-voltage source, but there are 2.8.limitations Ladder to Converter this structure, (1992) such as the high output impedance that prevents high gains fromThe being CWVM achieved. can In produce general, it high can be voltages stated that from as the a low gain-voltage increases, source, the output but there are lim- itationsimpedance to alsothis doesstructure, by a cubic such factor as the [79 high], which output can compromise impedance the that output prevents voltage high gains from regulation for loads requiring high currents. beingThe achieved. basic ladder In cells general, are shown it can in Figure be stated8, which that are derivedas the gain from aincreases, study regarding the output imped- ancethe proposal also does of a by three-level a cubic neutral factor point[79], clampedwhich can (NPC) compromise inverter topology the output [88,89]. voltage The regulation forinverter loads described requiring in [ 88high,89] currents. works with three voltage levels: Vi, Vi/2, and 0 V, and it is calledThe capacitor-clamped basic ladder multilevel cells are inverter shown or in ﬂying Figure capacitor 8, which multilevel are derived inverter (FCMLI). from a study regard- This is a circuit similar to the NPC inverter, but the diodes are replaced by capacitors. ing the proposal of a three-level neutral point clamped (NPC) inverter topology [88,89]. The inverter described in [88] and [89] works with three voltage levels: Vi, Vi/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.

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Sn+3 Sn+3 Cn+1 Cn+1 cell cell Sn+4 Sn+4 Cfly(n+1) Cfly(n+1)

+ S1 S1 C1 C1 S2 S2 Cfly1 Vi Cfly1 Vo + S3 S3

Vi Vo S4 S4 – –

(a) (b)

FigureFigure 8. 8. (a(a) )Boost Boost and and ( (bb)) buck buck ladder ladder cells. cells.

TheseThese cells cells can can be be arranged arranged in in the the form form of of boost boost and and buck buck configurations configurations a ass shown shown in in FigureFigure 8a,b,a,b, respectively.respectively. A A converter converter based based on on this this topology topology can can consist consist of one of one or more or more cells cellsaccording according to the to desiredthe desired conversion conversion ratio. ratio. When When the the converter converter works works in in voltage voltage step-up step- upmode, mode, the the source source is associatedis associated in seriesin series with withn capacitors n capacitors charged charged with with a voltage a voltageVi. TheVi. Theflying flying capacitor capacitor at a givenat a given moment moment is connected is connected in parallel in parallel with the with source the to source be charged. to be charged.Still, later Still, it is later connected it is connected in parallel in parallel with the with other the capacitor other capacitor to charge to charge it with itV iwith. In thisVi. Inway, this the way, voltage the voltage source source in series in withseriesn withcharged n charged capacitors capacitors can produce can produce an output an output voltage voltageVo according Vo according to Equation to Equation (1). (1). Vo = Vi + nVi (1) Vo V i  nV i (1) when used as a step-down topology, the converter operates similarly to a capacitive voltage whendivider used formed as a step by n-downcapacitors topology, in series, the inconverter which the operates load is similarly connected to in a parallelcapacitive with volt- one age divider formed by n capacitors in series, in which the load is connected in parallel of the capacitors. In this circuit, there are also one or more flying capacitors Cfly, which at a withgiven one moment of the capacitors. are connected In this in parallel circuit, with there the are source also one and or later more on flying associated capac initors parallel Cfly, whichwith the at a elements given moment that form are theconnected capacitive in parallel divider. with This the arrangement source and produceslater on associated an output involtage parallel that with varies the elements according that to the form number the capacitive of capacitors divider. used This in thearrangement converter asproduces defined anby output Equation voltage (2): that varies according to the number of capacitors used in the converter as defined by Equation (2): V V = i (2) o n Vcap V  i where n is the number of series capacitorso in the circuit. Figure9a shows a boost-(2) cap ncap type ladder converter with a gain equal to two and Figure9b presents a buck converter wherewith a n gaincap is equalthe number to 0.5. of This series topology capacitors is also in efficientthe circuit. in AC-ACFigure 9 conversion,a shows a boost with- goodtype ladderregulation converter and high with power a gain factor,equal andto two it isand a feasible Figure option9b presents to replace a buck ferromagnetic converter with core a gaintransformers equal to 0.5. [90 ].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].

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+ S1 S1

C2 C2 S2 S2 Vi Cfly Co Vo Cfly + S3 S3

Vi Co Vo S4 S4 – –

(a) (b)

FigureFigure 9. ( 9.a)( aBoost) Boost and and (b (b) )buck buck ladder converters.converters.

3. Design Parameters 3. Design Parameters In thisIn this section, section, the the main main design design parameters of of SC SC converters converters are are addressed, addressed, such such as as operationoperation modes, modes, main main components, components, efficiency, efﬁciency, and and control control techniques. techniques.

3.1. 3.1.Components Components 3.1.1. Switches 3.1.1. Switches In general, the efficiency and regulation of SC converters are dependent on the intrinsic resistancesIn general, of thethe switchesefficiency and and the dielectricsregulation that of constituteSC converters the capacitors. are dependent In low-power on the in- trinsicand resistances medium-power of the DC-DC switches converters, and the metal-oxide-semiconductor dielectrics that constitute field-effect the capacitors. transistors In low- power(MOSFETs) and medium are widely-power used DC because-DC converters, they can operate metal at-oxide high- frequencies.semiconductor For fieldhigher-effect transistorspower levels, (MOSFETs) insulated are gate widely bipolar used transistors because (IGBTs) they arecan better operate recommend at high becausefrequencies. they For highersupport power higher levels, currents insulated and voltages, gate bipolar but switching transistors losses (IGBTs) increase are significantly better recommend with the becauseswitching they support frequency. higher Some currents studies have and shown voltages that, but SiC switching MOSFETs have losses a performance increase signifi- cantlysimilar with to the Si-based switching IGBTs frequency. [91–93]. Thus, Some the analysisstudies presentedhave shown in this that work SiC considersMOSFETs only have a MOSFETs. performance similar to Si-based IGBTs [91–93]. Thus, the analysis presented in this work Table1 shows some characteristics of different commercial Si and SiC MOSFETs considersfor comparison only MOSFETs. purposes. In these semiconductors, conduction losses must be carefully Table 1 shows some characteristics of different commercial Si and SiC MOSFETs for analyzed, as they depend on the drain-source on-resistance Rds(on). This parameter is comparisonprovided purposes. in the manufacturer’s In these semiconductors, datasheet and expression conduction 3 can losses be used must to be estimate carefully the analyzed,conduction as they depend losses [94 on,95 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 = 2 losses [94,95]: Pcond(MOSFET) Rds(on) Id(rms) (3)

where Id(rms) is the root-mean-square (RMS) drain current. 2 PRIcondMOSFET  ds on d rms (3) Table 1. Comparison of the characteristics of different MOSFETs. where Id(rms) is the root-mean-square (RMS) drain current. V I R t t C C Component Type ds(max) d(max) ds(on) r f oss iss Table 1. Comparison of the characteristics(V) of different(A) MOSFETs.(mΩ) (ns) (ns) (pF) (pF) IRFP240 Si 200 20 180 51 36 400 1300 Vds(max) Id(max) Rds(on) tr tf Coss Ciss Component Type IRF530 Si 100 14 160 34 24 250 670 IRFP460(V) Si(A) 500 20(mΩ) 270(ns) 51 (ns 36) 430(pF) 1300(pF) IRFP240 Si IRF840200 Si20 500 8180 85051 21 2036 200400 12251300 C2M0080120D SiC 1200 36 80 22 14 92 1130 IRF530 Si SCT3080KL100 SiC14 1200 31160 8034 22 24 75250 785 670 IRFP460 SiIMW120R045M1XKSA1 500 SiC20 1200 52270 5951 24 1336 115430 19001300 IRF840 SiNTHL080N120SC1 500 SiC8 1200 44850 8021 20 1020 80200 11121225 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, x FOR PEER REVIEW 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 tr and fall time tf, the faster the dynamic response of a MOSFET. Energies 2021, 14, 2231 Table 1 shows the aforementioned parameters, as well as the parasitic9 of 33capacitances that affect the switching losses. It is worth mentioning that Ciss and Coss do vary significantly with the drain-source voltage Vds, but they are only slightly affected by the temperature. The switching losses are essentially related to the time interval required by a given Capacitance Ciss is an important parameter, since its charge and discharge are directly in- semiconductor to change from the ON state to the OFF state, or vice-versa. In other words, fluenced by drive circuit connected to the gate. Besides, capacitance Coss is charged by the the shorter the rise time tr and fall time tf, the faster the dynamic response of a MOSFET. Tablevoltage1 shows source the aforementioned of the circuit parameters, and discharged as well asthrough the parasitic the body capacitances diode that when the MOSFET affectis ON the and switching OFF, losses. respectively. It is worth mentioning that Ciss and Coss do vary signiﬁcantly with theIn drain-source turn, the switching voltage Vds ,losses but they are are calculated only slightly according affected by to the the temperature. energy dissipated in the Capacitance C is an important parameter, since its charge and discharge are directly switch duringiss the turn-on and turn-off intervals. These losses depend on the switching inﬂuenced by drive circuit connected to the gate. Besides, capacitance Coss is charged by thefrequency voltage source fs, the of the drain circuit-source and discharged voltage through Vds, the the drain body diode current when Id the, and MOSFET the time interval re- isquired ON and for OFF, turn respectively.-on and turn-off of the transistor given by ton and toff, respectively. Thus, bothIn turn,parts the are switching considered losses in are expression calculated according (4) [96]: to the energy dissipated in the switch during the turn-on and turn-off intervals. These losses depend on the switching

frequency fs, the drain-source voltage Vds, the drainf currents V ds I d Id, and the time interval required for turn-on and turn-off of theP transistorswMOSFET given by t and  tont , trespectively. off  Thus, (4) 6 on  off both parts are considered in expression (4) [96]:

Substituting the values assumed f V byI distinct parameters defined in Table 1 in (3) and P = s ds d t + t (4) (4), it is observed thatsw (SiCMOSFET MOSFETs) 6 presenton improvedo f f performance with respect to the conduction and switching losses when compared with their Si counterparts. Thus, SiC Substituting the values assumed by distinct parameters defined in Table1 in (3) and (4),M itOSFETs is observed can that be SiCregarded MOSFETs as present an adequate improved choice performance for applications with respect tothat the high conversion conductionefficiency. and switching losses when compared with their Si counterparts. Thus, SiC MOSFETsThe can transistors be regarded shown as an adequate in Table choice 1 have for good applications performance that high at conversion high frequencies, but as efficiency.the frequency increases, the switching losses can significantly compromise the efficiency of Thethe transistorsconverter. shown It is observed in Table1 have that good the performancedrain current at high also frequencies, influences but this as behavior. As an the frequency increases, the switching losses can significantly compromise the efficiency ofexample, the converter. Figure It is 10 observed represents that the the drain losses current in a alsoMOSFET influences model this IRFP460 behavior. Asoperating at 25 °C, ansubjected example, Figureto a drain 10 represents-source voltage the losses of in 100 a MOSFET V, while model the switchin IRFP460 operatingg frequency at ranges from 0 25to◦C 500, subjected kHz and, to a drain-sourceat the same voltagetime, the of 100 drain V, while current the switching varies from frequency 0 to 10 ranges A. This plot demon- fromstrates 0 to the 500 kHzprofile and, of at losses the same in time,that particular the drain current component, varies from but 0 a to similar 10 A. This analysis can be de- plot demonstrates the profile of losses in that particular component, but a similar analysis canveloped be developed for any for other any other commercial commercial de device,vice, consideringconsidering that that its characteristics its characteristics are are provided providedby the manufacturer. by the manufacturer. Thus, Thus, one one can can consider consider thesethese aspects aspects in detailin detail concerning concerning the design theand design estimate and estimate of the of converter the converter efficiency. efficiency.

20 ) W (

10 s s o L 500k 9 300k 7 3 5 100k 0 1 fs (Hz) Id(A) FigureFigure 10. 10.Behavior Behavior of switching of switching and conduction and conduction losses in MOSFET losses IRFP460. in MOSFET IRFP460.

3.1.2. Diodes 3.1.2. Diodes Some SC topologies like Dickson, ladder, and SMPC use diodes besides active switches, which canSome influence SC topologies the conversion like efficiency Dickson, significantly. ladder, Therefore, and SMPC it is important use diodes to besides active knowswitches, the characteristics which can of influence such semiconductors, the conversion especially efficiency considering significantly. the commercial Therefore, it is im- availabilityportant toof Si know and SiC the components. characteristics of such semiconductors, especially considering the Table2 summarizes the main parameters associated with some diode models, which commercial availability of Si and SiC components. include the maximum value of the repetitive peak reverse voltage VRRM, the average Table 2 summarizes the main parameters associated with some diode models, which include the maximum value of the repetitive peak reverse voltage VRRM, the average forward current IF(avg.), the forward voltage drop VF, the reverse recovery time trr, and the

Energies 2021, 14, x FOR PEER REVIEW 10 of 33

Energies 2021, 14, 2231 10 of 33 reverse recovery charge Qrr as informed by the manufacturer at 25 C. It is worth mentioning that the total capacitive charge QC of SiC diodes is equivalent to Qrr in Si ones. Only ultrafast Si diodes are listed because SC converters often operate at high switching forward current IF(avg.), the forward voltage drop VF, the reverse recovery time trr, and frequencies. Besides, only SiC Schottky diodes are described since ◦this technology pre- the reverse recovery charge Qrr as informed by the manufacturer at 25 C. It is worth sents improved characteristics when compared with other solutions. mentioning that the total capacitive charge QC of SiC diodes is equivalent to Qrr in Si ones. Only ultrafast Si diodes are listed because SC converters often operate at high switching frequencies.Table 2. Comparison Besides, only of the SiC characteristics Schottky diodes of aredifferent described diodes. since this technology presents improved characteristics when compared with other solutions. VRRM VF Trr Qrr/QC Component Type IF(avg.) Table 2. Comparison(V) of the characteristics of different(V) diodes. (ns) (nC) MUR860 Si 600 8 1.20 60 195 V V T Q Q Component Type RRM I F rr rr/ C RHRP8120 Si 1200 8 (V) F(avg.3.2) (V) 70(ns) (nC) 165 HFA15TB60S Si 600 15 1.2 50 84 MUR860 Si 600 8 1.20 60 195 60EPU04 Si RHRP8120400 Si60 1200 81.05 3.285 70 165 375 IDT08S60C SiC HFA15TB60S600 Si8 600 151.5 1.2 50 84 20 60EPU04 Si 400 60 1.05 85 375 C4D10120A SiC IDT08S60C1200 SiC33 600 81.5 1.5 - 20 52 SCS210KGHR SiC C4D10120A1200 SiC10 1200 331.4 1.5 - 52 34 IDH15S120 SiC SCS210KGHR1200 SiC15 1200 101.65 1.4 - 34 54 IDH15S120 SiC 1200 15 1.65 - 54 SiC diodes are often much faster than Si ones and do not present significant problems due toSiC the diodes reverse are recovery often much phenomenon, faster than Siones thus and contributing do not present to the signiﬁcant reducti problemson of switching 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 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 semicon- model shown in Figure 11, in which Vto represents the voltage drop across the semiconduc- t to t dutorctor junction junction and randt is the r is intrinsic the intrinsic resistance. resistance. The values The of valuesVto and ofrt areV notand often r are provided not often providedby manufacturers, by manufacturers, even though even they though can be they estimated can be from estimated characteristic from curves characteristic available curves availablein datasheets. in datasheets.

A K

Vto rt IF

FigureFigure 11. 11. Diode symbolsymbol and and equivalent equivalent model. model.

TheThe aforementioned model model can can be be used used to estimate to estimate the losses the losses in a diode. in a diode. The conduc- The conduction losses are owing to the current ﬂowing through the component and can be calculated tion losses are owing to the current flowing through the component and can be calculated as in Equation (5). as in Equation (5). 2 Pcond(D) = Vto IF(avg.) + rt IF(rms) (5) P V I  r I 2 where IF(avg.) and IF(rms) represent thecond average D to and F avg RMS. values t F rms of the forward current. (5) In turn, the switching losses occur during the turn-on and turn-off of the diode when whereboth the IF(avg.) current and andIF(rms) voltage represent are notthe zero,average being and estimated RMS values from of Equations the forward (6) and current. (7), respectively.In turn, the switching losses occur during the turn-on and turn-off of the diode when both the current and voltagePsw(on)(D )are= 0.5not Vzero,F(max )bein− VFg estimatedIF(avg.)trr fs from Equations (6)(6) and (7), respectively. Psw(o f f )(D) = QrrVF(max) fs (7)

Psw on D0.5 VF(max)  V F I F avg. t rr f s (6) where VF(max) is the maximum value  assumed  by the forward voltage  drop. From Equation (6), it is observed that the longer t , the higher the switching losses, and the lower the rr P Q V f efﬁciency as a consequence. Besides, thesw efﬁciency off D  canrr F be(max) further s increased if diodes with (7) low Qrr or QC are chosen in practical designs. 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

Energies 2021, 14, x FOR PEER REVIEW 11 of 33

efficiency as a consequence. Besides, the efficiency can be further increased if diodes with low Qrr or QC are chosen in practical designs.

Energies 2021, 14, x FOR PEER REVIEW 11 of 33 3.1.3. Capacitors Besides semiconductors, capacitors are essential devices in SC converters, as they are the only energy storage elements used in the circuit when hybrid topologies are not Energies 2021, 14, 2231 adopted,efficiency which as a consequence. also have inBesides,ductors the as efficiency in the case can ofbe [97]further. The increased equivalent11 if of diodes 33 model with of a ca- pacitolow Qrrr basically or QC are consistschosen in of practical the series designs. association of a capacitance C, the equivalent series resistance (ESR), and the equivalent series inductance (ESL) as shown in Figure 12. 3.1.3. Capacitors 3.1.3. Capacitors BesidesBesides semiconductors, semiconductors capacitors, capacitors are are essential essential devices devices in SC in converters, SC converters, as they as they are arethe the only only energy energy ststorageorage elements elements used used in the in thecircuit circuit when when hybrid hybrid topologies topologies are not are not adopted, which also have inductors as in the case of [97]. The equivalent model of a ca- adopted, which also have inductors as in theESL case of [97]. The equivalent model of a capacitorpacitor basically basicallyC consists consists ofESR of the the series series association association of a capacitance of a capacitance C, the equivalent C, the equivalent series series resistanceresistance (ESR), (ESR), and and the the equivalent equivalent series series inductance inductance (ESL) as(ESL) shown as inshown Figure in 12 Figure. 12. Figure 12. Equivalent circuit of a capacitor.

Capacitance C is related to the type of dielectric used in the construction of the com- ESL ponent. It Chas a direct impactESR on the energy density and, consequently, on the dimensions of the converter. Parameter ESR cannot be accurately estimated from the simple measure- meFigureFigurent pe 12. 12.rformedEquivalent Equivalent with circuit circuit an of a ohmmeterof capacitor. a capacitor. because it is strongly influenced by the dielectric, electrodes,Capacitance and theC ohmicis related losses to the in type the of terminals dielectric used of the in the component. construction of the compo- Capacitance C is related to the type of dielectric used in the construction of the com- nent.The It has losses a direct in impact dielectrics on the energycorrespond density to and, the consequently, delayed bias on or the relaxation, dimensions ofvarying accord- ponent. It has a direct impact on the energy density and, consequently, on the dimensions ingthe converter.to the dielectr Parameteric me ESRdium cannot, temperature, be accurately estimated and frequency from the simple[98]. The measurement losses in electrodes in performedof the converter. with an ohmmeterParameter because ESR cannot it is strongly be accurately influenced estimated by the dielectric, from the electrodes, simple measure- lowandme nt thefrequencies pe ohmicrformed losses arewith in essentially the an terminalsohmmeter ohmic of because thecomponent. and itdepend is strongly on theinfluenced dimensions by the and dielectric, type of elec- material. Attrodes, highThe lossesandfrequencies, the in dielectricsohmic the losses correspond losses in the increase toterminals the delayed significantly of the bias component. or relaxation, owing varyingto the accordingskin and proximity effects.to the The dielectric losses medium, in dielectrics temperature, correspond and frequency to the delayed [98]. The bias losses or relaxation, in electrodes varying in low accord- frequenciesing toOverall, the aredielectr essentiallyit is icreasonable medium ohmic, temperature, andto state depend that on and the dimensionsfrequencylosses in capa [98] and. type citorThe oflossess material.are influenced in electr At odes by in a wide varietyhighlow frequencies,frequencies of aspects. the are losses Inessentially SC increase-based ohmic significantly converters, and depend owing this on toelement thethe skindimensions andcan proximitybe satisfactorilyand type effects. of material. represented Overall, it is reasonable to state that the losses in capacitors are influenced by a wide At high frequencies, the losses increase significantly owing to the skin and proximity ef- byvariety C and of aspects. ESR, resulting In SC-based in converters, the representation this element shown can be satisfactorily in Figure 13 represented. Parameter δ is called dissipationbyfects.C and ESR, factor resulting (DF) in or the loss representation angle, whose shown value in Figure is often 13. Parameter providedδ isin called datasheets [99,100] Indissipation thisOverall, way, factor it it is is (DF) possible reasonable or loss to angle, toestimate state whose that the value the ESR islosses often using in provided capa Equationcitor in datasheetss are (8) influenced [101] [99. ,100 ]by a wide Invariety this way, of itaspects. is possible In SC to estimate-based converters, the ESR using this Equation element (8) can [101 be]. satisfactorily represented by C and ESR, resulting in the representation showntan  in Figure 13. Parameter δ is called tanESR(δ)  dissipation factor (DF) or loss angle, whose= value is often provided in datasheets [99,100] (8) ESR 2 fs C (8) In this way, it is possible to estimate the 2ESRπ fsC using Equation (8) [101]. tan   Complex ESR  (8) 2 fs C

ESR (Ω) Complex Real Capacitive Reactance (Ω) ESR (Ω) Real Capacitive δ Reactance (Ω) – j δ 2πfsC

– j 2πfsC Figure 13. Representation of the loss angle of a capacitor. Figure 13. Representation of the loss angle of a capacitor. The ESL can be hardly measured because it depends on the pad layout, capacitor height,FigureThe and13. ESL Representation power can plane be spreadinghardly of the loss measured inductance angle of [a 102 becausecapacitor.]. It can it be depends neglected aton low the frequencies, pad layout, capacitor heightbut its inﬂuence, and power should plane be considered spreading as the inductance operating frequency [102].gets It can close be to neglected the resonance at low frequen- frequencyThe ofESL the can capacitor. be hardly The ESLmeasured limits thebecause maximum it depends switching on frequency the pad atlayout, which capacitor cies,the converter but its caninfluence operate since,should at verybe cons highidered frequencies, as the the inductiveoperating reactance frequency is much gets close to the resonanceheight, and frequency power plane of thespreading capacitor. inductance The ESL [102] limits. It can the be maximum neglected atswitching low frequen- frequency cies, 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

Energies 2021, 14, x FOR PEER REVIEW 12 of 33 Energies 2021, 14, 2231 12 of 33

at which the converter can operate since, at very high frequencies, the inductive reactance ishigher much than higher the than capacitive the cap reactance.acitive reactance. Figure Figure14 presents 14 presents the behavior the behavior of the of capacitor the ca- pacitorimpedance impedance as a function as a function of the frequency of the frequency and ESR and [103 ESR]. [103].

|Z| ESR

|Z|=1/jωC |Z|=jωL (Ω ) |Z|/ESR

Frequency (Hz) Figure 14. Behavior of the capacitor impedance as a function of the operating frequency and ESR.

Table3 3 presents presents aa comparisoncomparison amongamong eighteight commerciallycommercially availableavailable capacitorcapacitor modelsmodels that useuse differentdifferent types types of of dielectrics, dielectrics, which which have have a capacitancea capacitance of 1ofµ F.1 μ Also,F. Also, the maximumthe maxi- mumoperating operating voltage voltage varies varies according according to the to type the oftype material of material used. used. An analysis An analysis is then is then per- performedformed considering considering important important aspects aspects such such as volume, as volume, energy energy density, density, and theand maximum the maxi- mumloss angle loss angle measured measured at 1 kHz, at 1 sokHz, that so the that designer the desig canner be can assisted be ass inisted choosing in choosing the most the mostadequate adequate type oftype capacitor. of capacitor. The energy The energy density density has an has impact an impact on the on size the of size the of converter the con- verterand, therefore, and, therefore, it is important it is important to know to know how these how parametersthese parameters vary fromvary onefrom capacitor one capac- to itorthe otherto the according other according to the typeto the of type dielectric. of dielectric.

TablTablee 3.3. Comparison among different typestypes ofof capacitors.capacitors. Vol. E Dens. Max(tan (δ)) Cap. Manufacturer Type Series V E Dens. Max(tan (δ)) Cap. Manufacturer Type Series Vdcdc Vol.(mm (mm3) 3) (mJ) (µJ/mm3) @1 kHz (mJ) (μJ/mm3) @1 kHz 1 1Hitano Hitano Electrolytic Electrolytic ECR ECR 4747 215.98215.98 1.25 1.25 5.78 5.78 0.10 0.10 * * 2 Nichicon Electrolytic UMA 50 62.83 1.25 19.89 0.10 * 2 3Nichicon TDK Electrolytic Polyester UMA B32560 6350 244.862.83 1.25 1.98 19.89 8.10 0.10 0.008 * 3 4TDK Panasonic Polyester Polyester B32560 ECQE 10063 1125.6244.8 1.98 5 8.10 4.44 0.008 0.01 4 5Panasonic TDK Polyester Polypropylene ECQEB32672L 250100 36631125.6 31.255 4.44 8.53 0.01 0.0008 5 6TDK Hitano Poly Polypropylenepropylene B32672L MKT 100250 25923663 31.25 5 8.53 1.92 0.0008 0.01 7 TDK Ceramic (ML) FA24 50 67.5 1.25 18.51 0.03 6 Hitano Polypropylene MKT 100 2592 5 1.92 0.01 8 Hitano Ceramic (ML) R25 50 132 1.25 9.46 0.1 7 TDK Ceramic (ML) FA24 50 67.5 1.25 18.51 0.03 * Value measured at 120 Hz as informed by the manufacturer. 8 Hitano Ceramic (ML) R25 50 132 1.25 9.46 0.1 * Value measured at 120 Hz as informed by the manufacturer. The power dissipated in a capacitor depends on the root mean square (RMS) current IC(rms)Thande power the ESR dissipated according in a to capacitor Equation depends (9). on the root mean square (RMS) current

IC(rms) and the ESR according to Equation (9). 2 Pcap = ESR · IC(rms) (W) (9) 2 Pcap  ESR  IC rms (W) (9) As the ESR depends on the DF and the frequency, it is possible to combine Equations (8) andAs (9)the to ESR approximately depends on determinethe DF and the the losses frequency, in a capacitor it is possible according to combine to Equation Equations (10). (8) and (9) to approximately determine the losses in a capacitor according to Equation (10). tan(δ) 2 Pcap = IC(rms) (10) 2πtanfsC   2 PIcap    C rms (10) 2 f C   As an example, and for the eventual efﬁciencys  analysis of the converters, capacitor 8 was chosen in Table3. By varying the RMS current from 0 to 10 A and the source frequency

Energies 2021, 14, x FOR PEER REVIEW 13 of 33

Energies 2021, 14, 2231 As an example, and for the eventual efficiency analysis of the converters,13 of 33 capacitor 8 was chosen in Table 3. By varying the RMS current from 0 to 10 A and the source frequency from 0 to 500 kHz, it is possible to plot the component losses as shown in Figure 15 withfrom the 0 to aid 500 of kHz, Equation it is possible (10). to plot the component losses as shown in Figure 15 with the aid of Equation (10).

3.0 ) W (

1.5 s s o 0 L 9 7 5 300k 100k 3 1 0 500k I (A) c fs (Hz) FigureFigure 15. Behavior 15. Behavior of oflosses losses in in a capacitorcapacitor as as function function of the of RMS the currentRMS current and operation and o frequency.peration frequency. 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, butOf course, only to addresseach type some of important capacitor aspects has its that own allow electrical the choice and of theconstructive most suitable characteristics. Theelement objective for a given of this application. work is In not general, to present it is reasonable a detailed to state analysis that the mostof these popular components, but onlycapacitor to address used in some SC converters important is of aspects film type. that This allow component the choice is widely of the used most mainly suitable ele- mentin for applications a given inapplication. which it is necessary In general, to operate it is reasonable at high voltages to andstate frequencies. that the most The popular film dielectric has low DF, reduced ESR, and good response at high frequencies [104]. The capacitordielectric used consists in SC of converters a thin layer of is plastic, of film the type. most commonThis component materials being is widely polyester used and mainly in applicationspolypropylene. in which Capacitors it is necessary made of metallized to operat filme at have hig vaporizedh voltages dielectrics and frequencies. with a thin The film dielectriclayer ofhas aluminum, low DF, as reduced this additional ESR, layer and provides good response the dielectric at withhigh the frequencies self-regeneration [104]. The di- electricability consists [105–107 of]. a Given thin thelayer possible of plastic, occurrence the ofmost a voltage common higher materials than the nominal being value polyester and polypropylene.supported by Capacitors the capacitor, made which of in turnmetallized would damage film have the dielectric, vaporized this dielectrics characteristic with a thin allows rapid heating to occur owing to the short circuit of the plates. When subjected to layerhigh of aluminum, temperatures, as this this layer additional of aluminum layer around provides the hole the turns dielectric into aluminum with the oxide, self-regeneration abilitywhich has [105 an– insulating107]. Given character the possible and thus eliminatesoccurrence the of short a voltage circuit. higher than the nominal value supportedPolyester capacitorsby the capacitor, can be found which commercially in turn would from some damage units of the nanofarads dielectric, to val- this charac- teristicues allows on the order rapid of 220heatinµF, withg to voltages occur owing greater thanto the or equal short to circuit 1000 V. This of the material plates. often When sub- has limited use since, among the film-based capacitors, this is the type most sensitive to the jectedincrease to high of temperaturetemperatures, [108 ]this and layer frequency of aluminum [104]. Its use around is recommended the hole mainly turns when into it aluminum oxide,is desiredwhich tohas obtain an insulating converters withcharacter reduced and dimensions thus eliminates since polyester the sho capacitorsrt circuit. have thePolyester smallest sizecapacitors for a given can capacitance be found value commercially in comparison from to its remainingsome units counterparts. of nanofarads to values onWhen the order compared of 220 to other μF, components,with voltages polypropylene-based greater than or capacitorsequal to 1000 demonstrate V. This material oftenexcellent has limited characteristics, use since, such among as the the fact thatfilm the-based capacitance capacitors, varies littlethis withis th temperature,e type most sensitive frequency, and voltage variations. Polypropylene absorbs less moisture than polyester, to thewhich increase makes o thisf temperature material suitable [108] for and a wide frequency range of applications [104]. Its inuse harsh is recommended environments. mainly whenCommercial it is desired capacitance to obtain values converters can reach with up to reduced 270 µF. dimensions since polyester capacitors have the Ceramicsmallest capacitors size for are a given the passive capacitance devices most value used in in comparison low power electronics to its remaining today. counterparts.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,When compared mica, and to other other silicates. compone Also,nts, the componentspolypropylene can be-based produced capacitors on a single demonstrate excellentdisc orcharacteristics in multiple layers,, such i.e., as in the the fact form that of multilayer the capacitance ceramic capacitorsvaries little (MLCCs). with temperature, A frequency,negative and aspect voltage lies in thevariations. fact that fractures Polypropylene in the dielectric absorbs can occurless moisture owing to voltage than polyester, whichspikes makes and this rapid materia changesl insuitable temperature, for a whichwide canrange cause of failuresapplications and even in a harsh short circuit environments. in the capacitor [110]. These components are often manufactured with low capacitances, Commercial capacitance values can reach up to 270 μF. which are typically rated on the order of picofarads. CeramicWithin capacitors the most common are the commercialpassive devices capacitors, most the used electrolytic in low capacitorpower electronics has the today. In thismost case, different the capacitance construction characteristics.does not vary This signif elementicantly is formed with the by afrequency sheet of porous and temperature,paper while embedded the components in an electrolyte present and, low to improve loss factor the contact [109] of. The the platedielectric with the materi weldingals used can be porcelain,terminal, the mica, paper and is wrapped other silicates. in aluminum Also, foil, the so that components this set forms can the negativebe produced terminal 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.

Energies 2021, 14, x FOR PEER REVIEW 14 of 33

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 weld- ing 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 Energies 2021, 14, 2231 receives an electrochemical treatment called anodic oxidation, which14 causes of 33 the appearance of a thin layer of an insulating material called aluminum oxide (Al2O3) deposited on that same sheet, thus forming the dielectric. of the capacitorThis Al2O (cathode).3 oxide Thelayer other has terminal the advantage of the capacitor of a high is connected dielectric internally constant to a and high rigid- secondity with aluminum a very thin foil that thickness constitutes on thethe positive order of terminal 0.7 μm. (anode). With Thisthis sheetcharacteristic, receives it is possible anto electrochemicalreach high capacitances treatment called in a anodic very oxidation,small volume, which causeswhich the can appearance vary from ofa 1 μF to 330 mF. thin layer of an insulating material called aluminum oxide (Al2O3) deposited on that same sheet,Thus thus, this forming is one the of dielectric. the most important characteristics of this type of capacitor. On the otherThis hand, Al2O 3theoxide maximum layer has the voltage advantage stresses of a high supported dielectric constant by this and component high rigidity are low in rela- withtion ato very the thin other thickness elements on the previously order of 0.7 presented.µm. With this If characteristic,this type of component it is possible is subjected to totransients reach high that capacitances exceed the in a rated very small voltage, volume, even which if very can briefly, vary from the 1 µ dielectricF to 330 mF. is compromised, Thus,causing this isa onechange of the mostin the important capacitance characteristics value ofand this reducing type of capacitor. the component’s On the other useful life. In hand, the maximum voltage stresses supported by this component are low in relation to the othergeneral, elements this previouslytype of capacitor presented. has If this defined type of polarity component and, is subjected if it is co tonnected transients to a circuit in an thatinverted exceed way, the rated electrochemical voltage, even if reactions very briefly, can the occur dielectric inside is compromised, it, causing causing leakage and even ex- aplosion. change in When the capacitance high capacitances value and reducing are required, the component’s their use usefulis often life. essential, In general, but these capac- thisitors type have of capacitor a shorter has useful defined life polarity than and, film if itand is connected ceramic to components. a circuit in an invertedAs they are more sus- way, electrochemical reactions can occur inside it, causing leakage and even explosion. Whenceptible high to capacitances failure, their are required, use on theira large use isscale often is essential, avoided. but these capacitors have a shorter useful life than film and ceramic components. As they are more susceptible to failure,3.2. Operation their use onModes a large scale is avoided.

3.2. OperationAs mentioned, Modes SC converters have a simpler structure than conventional power elec- tronicAs mentioned,converters SC that converters employ have inductors, a simpler but structure the co thanntrol conventional system becomes power elec- more complex in tronicthis case. converters Theoretically, that employ it inductors, is possible but theto controlachieve system high becomes voltage more gains complex with insome structures. thisHowever, case. Theoretically, it should itbe is possiblenoted that to achieve the higher high voltage the gain, gains the with more some components structures. will be em- However,ployed and, it should consequently, be noted that the the lower higher the the gain,efficien thecy. more Figure components 16 presents will be the em- behavior of the ployed and, consequently, the lower the efﬁciency. Figure 16 presents the behavior of the theoreticaltheoretical voltage voltage gain gain as a function as a function of the number of the ofnumber cells for of the cells voltage for doubler, the voltage SPC, doubler, SPC, SPMC,SPMC, Dickson, Dickson, and and FC topologies. FC topologies.

70 65 Voltage Doubler SPC, SPMC 60 55 Dickson 50 Fibonacci 45 40 35 Vo/Vi 30 25 20 15 10 5 0 0 1 2 3 4 5 6 Number of cells FigureFigure 16. 16.Behavior Behavior of the of voltage the voltage gain of SCgain converters of SC conver as a functionters a ofs a the function number of cells.the number of cells.

To analyze the behavior of the output voltage, the circuit represented in Figure 17 is adopted.To Inanalyze this case, the the behavior converter can of bethe modeled output as voltage, an ideal transformer the circuit with represented turns ratio in Figure 17 is madopted.:n, where RIno is this the equivalentcase, the outputconverter resistance can ofbe the modeled SC converter, as an which ideal is connectedtransformer with turns inratio series m: withn, where the load Ro RisL the[86,111 equivalent]. In this model, output the resistance resistance Rofo controlsthe SC theconverter, power which is con- transferred to the load and the output voltage regulation. The variation of Ro in practice is nected in series with the load RL [86,111]. In this model, the resistance Ro controls the mainly associated with the switching frequency and the internal resistance of the converter elements.power transferred to the load and the output voltage regulation. The variation of Ro in practice is mainly associated with the switching frequency and the internal resistance of the converter elements.

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Energies 2021, 14, 2231 15 of 33 Energies 2021, 14, x FOR PEER REVIEW o 15 of 33 R

Vi RL Ro m:n

FigureVi 17. Simplified model used in the analysisR Lof SC converters.

As for the m:n switching frequency, two asymptotic limits can be established: slow

switching limit (SSL) and fast switching limit (FSL) [112,113]. When the converter operates FigureinFigure SSL, 17. 17.theSimplified Simplified switch modeling model usedfrequency in used the analysis in is the low ofanalysis SC enough converters. of SCfor converters. the capacitors to fully charge and discharge. In this scenario, the currents have short-term peaks. Besides, all resistances of the AsAs for forthe switching the switching frequency, frequency, two asymptotic two limits asymptotic can be established: limits slow can switch- be established: slow ingswitches limit (SSL) and and capacitors fast switching are neglected, limit (FSL) [as112 these,113]. impedances When the converter do not operates affect inthe full charge of switching limit (SSL) and fast switching limit (FSL) [112,113]. When the converter operates SSL,the thecapacitors. switching In frequency this operation is low enough mode, for the the capacitors duty cycle to fully has charge little and influence discharge. on the behavior in SSL, the switching frequency is low enough for the capacitors to fully charge and dis- Inof thisthe scenario, converter. the currentsIn the FSL have condition, short-term peaks. the currents Besides, allthat resistances flow through of the switches the capacitors can be andconsideredcharge. capacitors In thisconstant, are scenario, neglected, and the asthe thesecurrents series impedances resistances have short do notof-term the affect components peaks. the full Besides, charge cannoof all thet resistances be neglected, of theas capacitors. In this operation mode, the duty cycle has little influence on the behavior of theyswitches can affect and c theapacitors full charge are neglected, of the capacitors. as these impedances do not affect the full charge of thethe converter. capacitors. In the In FSL this condition, operation the mode, currents the that duty flow throughcycle has the little capacitors influence can be on the behavior consideredOther constant, analytical and themethod series resistancess are based of theon componentsthe charge cannotand discharge be neglected, of ascapacitors [114]. theyTheof the cancircuit converter. affect shown the full In in charge the Figure FSL of the 18condition, capacitors.can be used the forcurrents this study, that flow where through an active the switchcapacitors S controls can be consideredOther analytical constant, methods and arethe based series on resistances the charge and of dischargethe components of capacitors canno [114t]. be neglected, as the energy transferred from a DC voltage source Vi to a capacitor Co. This representation The circuit shown in Figure 18 can be used for this study, where an active switch S controls isthey simpler can affect than the fullmodel charge corresponding of the capacitors. to Figure 17 and leads to similar results, and is the energyOther transferred analytical from method a DC voltages are sourcebasedV oni to the a capacitor chargeC oand. This discharge representation of capacitors [114]. isadopted simpler thanin this the modelwork correspondingin the study toof Figure topologies. 17 and leads It addresses to similar results,the behavior and is of the circuit The circuit shown in Figure 18 can be used for this study, where an active switch S controls adoptedaccording in this to three work indifferent the study modes of topologies. shown It in addresses Figure the 19 behavior, where ofthe the instantaneous circuit voltage accordingthe energy to three transferred different modesfrom a shown DC voltage in Figure source 19, where Vi theto a instantaneous capacitor C voltageo. This representation vCo(t) and instantaneous current iCo(t) are represented for the full charge, partial charge, visCo (simplert) and instantaneous than the model current correspondingiCo(t) are represented to Figure for the full17 and charge, leads partial to charge,similar results, and is and no charge conditions. andadopted no charge in conditions.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 Ri vCo(t) and instantaneousS current iCo(t) are represented for the full charge, partial charge, and no charge conditions.

iCo + Vi S Ri VCo Co –

iCo + FigureFigureVi 18. 18.RC RC circuit circuit used used in the in analysis the analysis of the charge of theVCo and charge discharge and ofCo capacitorsdischarge in of SC capacito converters.rs in SC converters. In this case, the equivalent model of the converter– is considered as a ﬁrst-order RiCo circuit with time constant τ, where Ri = Rsw + ESR, Rsw being the resistance of the switch, In this case, the equivalent model of the converter is considered as a first-order RiCo while Co represents the capacitor in the charge or discharge regime. The switch is turned Figure 18. RC circuit used in the analysisi of thesw charge andsw discharge of capacitors in SC convert- oncircuit to charge with the time capacitors constant during τ, awhere time interval R = RTon +over ESR, a switching R being period the Tresistances. Thus, it of the switch, ers. iswhile reasonable Co represents to state that the SC capacitor circuits are oftenin the designed charge based or discharge on the time regime. constant The rather switch is turned thanon to the charge switching the frequency.capacitors Besides, during such a time converters interval are T capableon over of a operating switching in fullperiod Ts. Thus, it chargeis reasonableIn condition this ca to evense, state the at high thatequivalent frequencies SC circuits model according are of often the to the designedconverter speciﬁcations basedis considered of capacitorson the time as and a const first-antorder rather RiCo switches used in the design. thancircuit the with switching time constant frequency. τ, where Besides, Ri = suchRsw + convertersESR, Rsw being are capablethe resistance of operating of the switch,in full chargewhile Cconditiono represents even the at capacitorhigh frequencies in the charge according or discharge to the specifications regime. The of switch capacitors is tur andned on to charge the capacitors during a time interval Ton over a switching period Ts. Thus, it switches used in the design. is reasonable to state that SC circuits are often designed based on the time constant rather The three modes of operation are defined [115,116]: than the switching frequency. Besides, such converters are capable of operating in full (1)charge If T conditionon >> RiCo ,even that atis, high when frequencies the time interval according during to the which specifications the switch of capacitorsremains on and is switchesmuch used greater in the than design. the time constant of the circuit, the capacitors charge and dis- chargeThe three completely. modes of Thus, operation the current are def inedbecomes [115 ,116]zero :in each cycle, and then it can be assumed that the system operates under full charge condition according to Figure (1) 19a.If T on >> RiCo, 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.

Energies 2021, 14, x FOR PEER REVIEW 16 of 33

(2) If Ton ≈ RiCo, 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 Ton is small in relation to Ts, then Ton << RiCo. Thus, the current in the capacitor is almost zero, and the voltage Energies 2021, 14, 2231 across it can also be considered constant. In this mode, the converter operates16 of 33 at no charge condition according to Figure 19c.

vCo(t)

Peak Voltage

(a) (b) (c) t (s) Full Partial No Charge Charge Charge

iCo(t)

(a) (b) (c)

Average Current t (s) Full Partial No Charge Charge Charge

FFigureigure 19.19.Capacitor Capacitor charging charging modes: modes: (a) full(a) charge,full charge, (b) partial (b) partial charge, charge, (c) no charge. (c) no charge. The three modes of operation are deﬁned [115,116]: In general, the equivalent resistance of the circuit represented in Figure 18 can be (1) If T >> R C , that is, when the time interval during which the switch remains on is defined onby Equationi o (11) [118]: 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 1 1 e  that the system operates under fullR charge condition according to Figure 19a. eq   (11) T ≈ R C 2 fs C o 1 e  (2) If on i 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. whereThe β is representation given by Equation of the current (12). 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 Ton Ts, then Ton << RiCo. Thus, the current in the = capacitor is almost zero, and the voltage (12) RC across it can also be considered constant. In thisi mode, o the converter operates at no charge condition according to Figure 19c. The time constant associated with the circuit is given by Equation (13). In general, the equivalent resistance of the circuit represented in Figure 18 can be deﬁned by Equation (11) [118]:   RC i o (13) −β The duty cycle is defined as the ratio1 1between+ e the on-time of the switch Ton and the = Req −β (11) switching period Ts as in Equation (14)2 fs:C o 1 − e

where β is given by Equation (12). Ton D  (14) Ton T β = s (12) RiCo whereThe Ts time is defined constant according associated to with Equation the circuit (15) is. given by Equation (13). 1 τ = RiCo (13) Ts  (15) f s The duty cycle is deﬁned as the ratio between the on-time of the switch Ton and the switchingDefining period D T=s 0as.5, in E Equationxpression (14): (11) can be written as Expression (16).

T D = on (14) Ts

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where Ts is deﬁned according to Equation (15).

1 Ts = (15) fs 0.5     1 e fs   Defining D = 0.5, Expression (11) can be written as Expression  (16). 1   Req  (16) 0.5  2 f C (− 0.5 )  s o τ fs   1 + e  fs 1 1 e   Req =   (16) 2 fsCo (− 0.5 )  1 − e τ fs Expression (16) is plotted in Figure 20a, where the behavior of the converter equivalent resistanceExpression is (16) represented is plotted in as Figure a function 20a, where of product the behavior τfs. Proper of the converter tradeoffs equivalent must be made betweenresistance such is represented parameters as to a achieve function good of product overallτ· fperformance.s. Proper tradeoffs Increasing must be the made switching frequencybetween such may parameters lead to higher to achieve efficiency good overall as observed performance. in Expression Increasing (16) the switching. However, it is 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 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 capacitances may affect the behavior of Req in practice, which tends to increase after a given poinpointt in in the the curve shownshown in in Figure Figure 20 b20 [b119 [11].9] Therefore,. Therefore, the efficiencythe efficiency will be will reduced be reduced as as a aconsequence. consequence. This conditioncondition depends depends on on the the characteristics characteristics of switches of switches and capacitors and capacitors usedused in in the the design, design, andand such such components components must must be wiselybe wisely chosen chosen in practical in practical applications. applications.

Req

(b)

(a) τfs

Figure 20. Behavior of the converter equivalent resistance as a function of the time constant and Figureswitching 20. Behavior frequency: of (a )the ideal converter model; ( bequivalent) real model. resistance as a function of the time constant and switching frequency: (a) ideal model; (b) real model. Performing an analysis in terms of β and Ton, the output resistance of the converter can be estimated as follows: Performing an analysis in terms of β and Ton, the output resistance of the converter (1) If β > 1, the converter operates in full charge mode, where T > 5τ, while the output can be estimated as follows: on resistance depends on the inverse of the product of fs and Co. In other words, the (1) Ifresistances β > 1, the ofconverter the switches operates and capacitors in full charge represented mode, by whereRi has littleTon > inﬂuence 5τ, while on the the output resistanceconverter behavior.depends on the inverse of the product of fs and Co. In other words, the (2) If β = 1, the capacitors operate in partial charge mode and the output resistance of the resistances of the switches and capacitors represented by Ri has little influence on the converter depends on R , C , and f . converter behavior. i o s (3) If β < 1, the converter operates in no charge mode. In this case, it is possible to employ (2) Ifvery β = 1 short, the switchingcapacitors time operate intervals in partial compared charge to the mode circuit and time the constant. output resistance Thus, the of the converter depends on Ri, Co, and fs. output resistance of the converter is very dependent on Ri and D. (3) IfTable β < 14, presentsthe converter a comparison operates among in no the charge operation mode. modes In this of case, SC converters. 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. Table 4 presents a comparison among the operation modes of SC converters.

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Table 4. Operation modes of SC converters.

Operation Mode Time Constant β Variation of Req Energies 2021, 14, 2231 18 of 33 1 Full charge Ton >> RiCo β > 1 2 fs C o 1 Partial charge Ton ≈ RiCo β  1  , Table 4. Operation modes of SC converters. 2 fs C o TR Operation Mode Time Constant β Variation of Req R s= i No charge Ton << RiCo β < 1 i TD 1 on Full charge Ton >> R Co β > 1 i 2 fsCo 1 Partial charge Ton ≈ R Co β ≈ 1 β, i 2 fsCo T R 3.3. VoltageNo charge Balance Ton << R Co β < 1 R s = i i i Ton D SCs rely on the principle that the energy stored in a given element is transferred to 3.3.another, Voltage while Balance the capacitors can be associated in series or in parallel depending on the configuration.SCs rely on theOf principlecourse, small that the differences energy stored will in aexist given in element the rated is transferred capacitances to even if com- another,ponents while with the the capacitors very same can be sp associatedecifications in series are orused. in parallel In most depending SC converters, on the the energy conﬁguration. Of course, small differences will exist in the rated capacitances even if componentstransfer occurs with thein veryequivalent same speciﬁcations circuits represented are used. In most by the SC converters,parallel association the energy of capacitors. transferA given occurs capacitor in equivalent Ci previously circuits represented charged by with the parallela voltage association VCi transfers of capacitors. energy to another Acapacitor given capacitor Co, whicCi previouslyh has no charge charged or with a smaller a voltage amountVCi transfers of charge energy toaccording another to the circuit capacitor C , which has no charge or a smaller amount of charge according to the circuit shown in oFigure 21. It is worth mentioning that a given series resistance Ri also exists in shown in Figure 21. It is worth mentioning that a given series resistance Ri also exists in practice owing to capacitors Ci and Co, as well as to the active switches. practice owing to capacitors Ci and Co, as well as to the active switches.

Ri S

+ + Ci VCi Energy VCo Co – –

Figure 21. Circuit representing the energy transfer between two capacitors. Figure 21. Circuit representing the energy transfer between two capacitors. The series resistance is neglected a priori in the forthcoming analysis. Thus, the total amountThe of chargeseries inresistance the circuit isQt neglectedis given by Equationa priori (17):in the forthcoming analysis. Thus, the total amount of charge in the circuit Qt is given by Equation (17): Qt = QCi + QCo (17)

QQQt Ci  C o (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 equal to the where QCi and QCo are the charges stored in capacitors Ci and Co, respectively. ﬁnal charge Qf according to Equation (18). Since charge balance is supposed to occur, the initial charge Qi must be equal to the final charge Qf according to EquationQ f = (18)Qi . (18)

Considering this condition, Expression (19) is valid:QQ f i (18) Considering this condition,VCiCi + V CoExpressionCo = Vf Ci + (19)Vf C ois valid: (19) where V is the initial voltage across capacitor C ; and V is the ﬁnal value assumed by the Co VCVCVCVCCi io Co o f f i  f o (19) equilibrium voltage. whereThus, VCo it is is the possible initial to obtainvoltageVf fromacross Expression capacitor (19) C aso; inand Expression Vf is the (20). final value assumed by the equilibrium voltage. V C + V C = Ci i Co o Vf f (20) Thus, it is possible to obtain V fromCi + C Eo xpression (19) as in Expression (20).

If Co is initially discharged, then VCo = 0 and ExpressionVCVCCi i Co (19) o becomes (21). Vf  (20) CCi o VCiCi = Vf Ci + Vf Co (21)

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

VCVCVCCi i f i  f o (21)

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Thus, Expression (22) results. Ci Vf = VCi (22) Ci + Co

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 VCi remains constant while considering distinct combinations of Co and Ci, 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 Ri does not affect the value assumed by Vf, but only the time constant of the circuit. This aspect also becomes evident in Section 3.4, where it is demonstrated that Ri does not affect the conversion efﬁciency during the charge cycle.

3.4. Efficiency There are many works dedicated to analyzing the variables that influence the efficiency of SC converters. Some studies point out the constructive characteristics of capacitors [120], the intrinsic resistance of switches [121], or even a combination of both as the main factors responsible for affecting this behavior [122]. Other aspects are also brought up, such as the voltage conversion ratio and the charge regime, that is, full charge or no charge [123]. As previously mentioned, in each cycle when in full charge mode, the capacitor discharges fully. For a circuit in which the capacitor is initially discharged, the maximum theoretical efficiency achievable is 50% according to Equation (23). As the initial voltage on the capacitor increases, the efficiency also does. The same behavior occurs when operating with partial charge, and the efficiency does not depend on the resistance of the switches and capacitors according to Equation (24). The charging process is affected only by the initial and final values of the capacitor voltage. Therefore, the greater the voltage variation, the lower the efficiency during the charge cycle. The increase of the resistances in series with the capacitors during the charge cycle only modifies the time constant, but it does not affect the efficiency itself. An increase in such series resistances during the charge cycle decreases the peak current and, at the same time, increases the charge time. Thus, the efficiencies under full charge and partial charge conditions are given by ηFC and ηPC in Equations (23) and (24), respectively. 1 Vinitial ηFC = 1 + (23) 2 Vi 1 Vinitial + Vf inal ηPC = (24) 2 Vi

where Vinitial and Vfinal 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):

RL nR = (25) RL + Ri

where RL 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 Tables1 and3. 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, Energies 2021, 14, x FOR PEER REVIEW 20 of 33

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, Energies 2021, 14, x FOR PEER REVIEW 20 of 33 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, 2231 20 of 33 v(t) However, for the capacitor discharge in another capacitor, the efficiency can be esti- matedVCi 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, 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. whose waveforms in Figure 22 represent the moment after which switch S is turned on. Vf v(t) VCi

VCo t (s) 0 FigureVf 22. Waveforms representing the discharge of one capacitor into another capacitor in full charge mode.

VCoAs previously mentioned, capacitor Ci is inititally (s) charged with voltage VCi, whereas 0 capacitor Co is completely discharged. Thus, a charge balancing occurs so that at the end ofFigureFigure the 22.process,22. WaveformsWaveforms the representingvoltage representing on the both discharge the capacitors dischar of onege capacitorofbecomes one capacitor into nearly another into balanced capacitor another in andcapacitor full equal in to full V Ci/2 charge mode. ifcharge the capacitancesmode. are equal to each other. Therefore, the voltages across Co and Ci are increased and decreased, respectively, as the efficiency can be calculated according to Equa- As previously mentioned, capacitor Ci is initially charged with voltage VCi, whereas tioncapacitor (26)As previously.C o is completely mentioned, discharged. capacitor Thus, a charge Ci is balancing initially occurs charged so that with at the voltage end of VCi, whereas capacitorthe process, C theo is voltagecompletely on both discharged. capacitors becomes Thus, nearlya charge balanced balancing and equal occurs to V Ciso/2 that at the end VV if the capacitances are equal to each other. Therefore,f the voltages Co across Co and C are of the process, the voltage on both capacitorsn  becomes nearly balanced iand equal to VCi/2 increased and decreased, respectively, as the efﬁciencyFC can be calculated according to (26) VVf Ci ifEquation the capacitances (26). are equal to each other. Therefore, the voltages across Co and Ci are increased and decreased, respectively,V asf + theVCo efficiency can be calculated according to Equa- Considering a capacitor dischargingn = into another capacitor, but (26) in partial charge tion (26). FC V + V mode, a similar analysis can be performed.f Ci Now, the capacitor Ci initially charged with Ci Considering a capacitor discharging into another capacitor, butCi in(min partial) charge mode, o V is discharged into another capacitor until reachingVVf Co V . In this way, the capacitor C a similar analysis can be performed. Now, then capacitor C initially charged with V is charges from a minimum voltage VCo to VFCCo(max), andi then the cycle is interrupted.Ci The( be-26) discharged into another capacitor until reaching V VVf. In Ci this way, the capacitor C havior of the capacitor voltages in this case Ciis(min shown) in Figure 23. The efficiencyo of this charges from a minimum voltage VCo to VCo(max), and then the cycle is interrupted. The typebehavior Consideringof discharge of the capacitor cycle a capacitor voltages can be in given discharging this case byis Equation shown into in another Figure(27): 23 . capacitor, The efﬁciency but of this in partial charge mode,type of dischargea similar cycle analysis can be can given be by performed. Equation (27): Now, the capacitor Ci initially charged with VVComax  Co VCi is discharged into another capacitorn until reaching VCi(min). In this way, the capacitor Co VCo(maxPC ) + VCo (27) charges from a minimum voltagenPC = VCo to VCo(max)VVCi , andCimin then the cycle is interrupted.(27) The be- VCi + VCi(min) havior of the capacitor voltages in this case is shown in Figure 23. The efficiency of this where VCo(max) is the maximum voltages on capacitor Co; and VCi(min) is the minimum voltage typewhere ofV dischargeCo(max) is the cycle maximum can be voltages given onby capacitor EquationCo ;(27) and: VCi(min) is the minimum onvoltage capacitor on capacitor Ci. Ci. VVComax  Co nPC  (27) v(t) VV VCi Ci Cimin

where VCo(max) is the maximum voltages on capacitor Co; and VCi(min) is the minimum voltage onVCi capacitor(min) Ci. VCo(max) v(t) VCi VCo t (s) 0

FigureFigureVCi(min) 23.23. WaveformsWaveforms representing representing a capacitor a capacitor discharging discharging into another into another capacitor capacitor in partial in partial charge mode.chargeVCo(max) mode.

A third possibility is the discharge of a capacitor in a parallel RC circuit, as shown A third possibility is the discharge of a capacitor in a parallel RC circuit, as shown in in FigureVCo 24. In this case, there is a capacitor Ci charged witht (s) an initial voltage VCi, to 0 Figurewhich a 24 series. In resistancethis case,R i thereis connected, is a capacitor corresponding Ci charged to the sum with of the an resistances initial voltage of the VCi, to which aFigureswitch series 2 and resistance3. Waveforms the capacitors. Ri isrepresenting con Thenected, load RL a iscorresponding capacitor associated discharging in parallel to the with intosum a another capacitorof the resistancescapacitorCo, which in ofpartial the switch charge mode.

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 Ci charged with an initial voltage VCi, to which

a series resistance Ri is connected, corresponding to the sum of the resistances of the switch

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and the capacitors. The load RL is associated in parallel with a capacitor Co, which is ini- and the capacitors. The load RL is associated in parallel with a capacitor Co, which is ini- tiallytially discharged.discharged. Thus,Thus, thethe dischargedischarge occuroccurss inin twotwo events.events. AtAt first,first, thethe chargecharge ofof capacitorcapacitor Cisi initiallyis divided discharged. between Thus, Co and the dischargeRL. When occurs the charge in two events.between At ﬁrst,the two the chargecapacitors of is balanced, Ci is divided between Co and RL. When the charge between the two capacitors is balanced, capacitor Ci is divided between Co and RL. When the charge between the two capacitors is both start supplying RL until they are completely discharged, as represented in Figure 25. bothbalanced, start both supplying start supplying RL untilRL untilthey theyare arecompletely completely discharged, discharged, as as represented represented in in Figure 25. AtAtFigure eacheach 25 stage,.stage, At each therethere stage, isis there aa differentdifferent is a different expressionexpression expression forfor for thethe the efficiency. efﬁciency.efficiency.

RRii SS + ++ + Ci VCi VCo Co L Ci VCi VCo Co RRL –– – –

FigureFigure 24. 24.24.Circuit CircuitCircuit representing representingrepresenting a capacitor aa capacitorcapacitor discharging dischargingdischarging into a parallel intointo RC aa circuit.parallelparallel RCRC circuit.circuit.

v(t) v(t) VCi VCi

VCi(min) VCi(min) Vf transition between the stages Vf transition between the stages VCo(max) VCo(max)

VCo t (s) VCo t (s) 0 0 FigureFigure 25. 25.25.Waveforms WaveformsWaveforms representing representingrepresenting a capacitor aa capacitorcapacitor discharging dischargingdischarging into a parallel intointo RC aa circuit. paralleparallell RCRC circuit.circuit. In the first stage, the efficiency can be given by Equation (28). InIn thethe firstfirst stage,stage, thethe efficefficiencyiency cancan bebe givengiven byby EquationEquation (28)(28).. ! VCo(max) + VCo η1 = VVCo(max)  Co  (28) V + VVVCo(max)  Co  f 1  Ci  (28) 1     (28) VVVVf Ci   In the second stage, the efficiency can be estimatedf by Equation Ci  (29). In the second stage, the efficiency can be estimated by Equation (29). In the second stage, the efficiency RcanL be estimated by Equation (29). η2 = (29) RL + Ri R R L 2  L (29) In this analysis, the switching losses in the semiconductors2  RR must also be considered, (29) RRL i although they have not been incorporated in the previousL expressions. i This is justified becauseInIn adequatelythisthis analysis,analysis, quantifying thethe switchingswitching them depends losseslosses on inin the thethe intrinsic semiconductorssemiconductors characteristics must ofmust each alsoalso type bebe considered,considered, of switch. In the case of MOSFETs, the increase in the switching frequency has direct impact althoughalthough theythey havehave notnot beenbeen incorporatedincorporated inin thethe previousprevious expressions.expressions. ThisThis isis justifiedjustified becauseon the switching adequately losses quantifying as shown in Equation them depends (4) [18,113 on]. Onthe the intrinsic other hand, characteristics when the of each type becauseswitching adequately frequency is quantifying increased, the them voltage depends ripple is on reduced. the intrinsic With a smaller characteristics voltage of each type ofofvariation, switch.switch. the InIn efficiency thethe casecase increases ofof MOSFETs,MOSFETs, according thethe to increase Expressionsincrease inin (23), thethe (24),switchingswitching (26)–(28). frequencyfrequency Thus, it is hashas directdirect im-im- pactpactup to onon the thethe designer switchingswitching to carry losseslosses out a detailed asas shownshown analysis inin EquationEquation regarding (4) the(4) best[18,113][18,113] operation.. OnOn pointthethe otherother for hand,hand, whenwhen the circuit. thethe switchingswitching frequencyfrequency isis increasincreased,ed, thethe voltagevoltage rippleripple isis reduced.reduced. WithWith aa smallersmaller voltagevoltage variation,Analyzing the theefficiency behavior increases of capacitors according operating to in E noxpressions charge, partial (23), charge,(24), (26) and–(28). Thus, it is variation,full charge modes,the efficiency it is reasonable increases to state according that this methodologyto Expressions becomes (23), valid (24), for (26) any–(28). Thus, it is upupconverter toto thethe topology designerdesigner because toto carrycarry the operatingoutout aa detaileddetailed principle analysisanalysis of SC circuits regardingregarding is essentially thethe bestbest based operationoperation on pointpoint forfor thethe chargecircuit.circuit. transfer from one element to another. Analyzing the behavior of capacitors operating in no charge, partial charge, and full 3.5. RegulationAnalyzing and the Control behavior of capacitors operating in no charge, partial charge, and full chargecharge modes,modes, itit isis reasonareasonableble toto statestate thatthat thisthis methodologymethodology becomesbecomes validvalid forfor anyany con-con- verterIn thistopology section, because the most commonthe operating control techniquesprinciple applicableof SC circuits to SC convertersis essentially are based on the verterdiscussed topology since there because is a significant the operating voltage decreaseprinciple when of theSC loadcircuits is connected is essentially to the based on the chargechargeoutput of transfertransfer such structures. fromfrom oneone The elementelement simplest toto way another.another. to increase the voltage gain in the case of

3.5.3.5. RegulationRegulation andand ControlControl InIn thisthis section,section, thethe mostmost commoncommon controlcontrol techniquestechniques applicableapplicable toto SCSC convertersconverters areare discusseddiscussed sincesince therethere isis aa significantsignificant voltagevoltage decreasedecrease whenwhen thethe loadload isis conneconnectedcted toto thethe outputoutput ofof suchsuch structures.structures. TheThe simplestsimplest wayway toto increaseincrease thethe voltagevoltage gaingain inin thethe casecase ofof reducingreducing thethe outputoutput voltagvoltagee isis toto associateassociate moremore cellscells inin series.series. However,However, thethe higherhigher thethe componentcomponent count,count, thethe greatergreater thethe seriesseries resistanceresistance inin thethe circuitcircuit and,and, consequently,consequently, thethe

Energies 2021, 14, 2231 22 of 33

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) [124–126]. This technique is widely used to control the output voltage in the conventional buck, boost, and buck-boost converters. In SC circuits, this operation mode is divided into two stages per cycle. In the first stage, the capacitor is charged, and the duty cycle can assume values between 0 < D < 0.5. In the second stage, the capacitor is connected to the load, and its discharge process begins. This situation is repeated during each cycle. It should be noted that when the duty cycle approaches zero, the voltage ripple increases and, consequently, the input current peaks increase [127]. Another strategy presented in the literature is the interleaved connection of cells [128]. However, this arrangement may or may not work with overlapping phases [128,129]. The main advantage lies in the fact that this method reduces the output voltage ripple, providing not only voltage regulation, but also an improvement in efficiency. However, as mentioned, it should be considered that a greater number of components causes an increase in losses. An advantage of this technique is the possibility of increasing the power levels considering that the current is shared among the interleaved cells. The pulse frequency modulation (PFM) control technique is also well known in electronics. The advantage of this strategy lies in the simplicity of producing a variable frequency signal from monitoring the output voltage or output current. This signal increases the switching frequency according to the load current [130,131]. However, this technique does not guarantee reduced voltage ripple under heavy load conditions [132]. Another important point is that, when MOSFETs are used, increasing the switching frequency can increase switching losses according to (4). A prominent control technique that takes advantage of the MOSFET transconductance is called quasi-switched capacitor (QSC). This strategy was formerly introduced in [133] as a new SC cell. Subsequent work used this concept in a somewhat obscure way while adopting the terms “QSC cell” and “QSC control”. However, it can be regarded as technique for controlling the gate-source voltage in a MOSFET. When operating in the active region, that is, in the triode region, the drain current is proportional to the gate-source voltage [134]. If the MOSFET always operates in that region, the drain current can be controlled. The main difference in relation to the other techniques mentioned above is that the input current waveform does not change during the charge stage, while in other cases, there are short-term current peaks. In this way, this allows to reduce the current stresses on the switches and prevents the occurrence of conducted EMI. This control approach does not affect the converter efficiency, because, according to Expressions (23) and (24), the efficiency during the charge stage is dependent only on the difference between the voltage on the capacitor in charging process and the source voltage.

3.6. Comparison among SC Topologies The existing SC topologies are very similar to each other in terms of constructive aspects, although they may differ with respect to the voltage gain, number of components, and stresses. The component count associated with the capacitors and switches required by a given basic cell is shown in Table5 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 Table5 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. Energies 2021, 14, 2231 23 of 33

Table 5. Comparison among SC topologies.

Parameters SCVD SPC SPMC Dickson FC Ladder * Capacitors NN 3N 2NN 2N Switches 4N 3N 3N 2N 3N 2N Maximum voltage on the 2(N−1)V V V NV F(N − 1)V V capacitors for N-th cell i i i i i i Maximum voltage on the 2(N−1)V NV NV V F(N − 1)V V switches for N-th cell i i i i i i * Boost mode.

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 Vi in the Dickson and ladder converters, which makes them proper choices for applications involving high output voltages. The current stresses cannot be estimated in a such a simple manner because they do not depend only on the characteristics of the load connected to the converter, but also on the charge modes of the capacitors, which were discussed in Section 3.2. SC circuits operating in full charge mode present high short-term current peaks, which occur when the capacitors are charged. The maximum value of the charging current must be taken into account carefully because it may damage the switch. Besides, the pulsating nature of the input current may lead to the increase of conducted electromagnetic emissions. Operation at high frequencies and/or the use of high capacitances to ensure the full charge or no charge condition are possible solutions that allow for mitigating such undesirable problems. The QSC is a control approach that can be used for this purpose as well as explained in Section 3.5. It allows obtaining a nearly continuous input current, thus causing the current stresses on the switches to be reduced as a consequence. Interleaved converters are also less susceptible to the undesirable effects caused by the voltage ripple on the capacitors as discussed in Section 3.5.

4. Applications Conventional SC converters are widely used in practice, mainly at low power levels and when high power density is desired [35]. Such characteristics make them quite attractive for speciﬁc applications involving embedded electronic systems, biomedical equipment, energy harvesting, and general-purpose microelectronics. In this context, it is worth mentioning that some modern portable electronic devices have very low power consumption. For instance, smartphones consume approximately 1 W, while cardiac pacemakers consume 50 µW[135]. Several studies have discussed the relevance of using the available energy in the environment where a device or equipment is placed. Various forms of energy can be harvested and converted into electrical energy to power the aforementioned devices. Photovoltaic solar energy is a typical example [136–140], but other more unusual sources can be considered, such as pyroelectric [141–143] and blood sugar [144–146]. 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 [147–150]. Its use in the supply of sensors or very-low-power circuits is also possible, precisely in conditions in which space is limited [151–155]. 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 [159–163]. Owing to the small size, one can even consider the integration of the whole set to the battery system itself as a consolidated solution. Energies 2021, 14, 2231 24 of 33

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. Table6 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 [173–176]. The evolution of existing circuits is mainly concerned with the increase of energy density and efficiency. Some devices can even be integrated with the same memory chip [120,177]. The FC is similar to the SPC, but it requires a larger number of switches and capacitors. An emergency power supply for a computer system based on this structure was proposed in [178] as an emerging application, but LCD drivers can also be found in [179,180]. The ladder topology is also adequate for some specific applications, even though it is not very often used in DC-DC power conversion when compared with its remaining counterparts [181,182]. Among the topologies presented, this is the one with the highest efficiency and versatility. In experimental tests, the efficiency can be greater than 90% with low voltage ripple even at power levels on the order of 1 kW [183]. Thus, this is the only promising topology developed so far for high-power applications, while the others are only feasible at low power levels [170].

Table 6. Parameters of SCVD ICs.

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

5. Conclusions SC converters have drawn significant attention from academia and industry recently owing to their prominent characteristics associated with high energy density and low EMI levels in DC-DC, DC-AC, AC-DC, and AC-AC conversion. Considering that many hybrid SC topologies composed of capacitors, semiconductors, and inductors have been reported in the literature, this work has been specifically concerned with a general review of important concepts associated with “pure SC” DC-DC converters. The existing configurations were analyzed in detail, while a brief explanation on voltage multipliers was also included, considering that some structures are derived from such popular circuits used in low-power ac-dc power conversion. Energies 2021, 14, 2231 25 of 33

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. Energies 2021, 14, 2231 26 of 33

Author Contributions: Conceptualization, F.L.T. and E.R.R.; methodology, investigation, writing— original draft preparation, A.F.d.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Federal University of Itajubá. Data Availability Statement: Data available on request from the authors. Acknowledgments: The authors acknowledge CAPES, CNPq, FAPEMIG, and INERGE for the overall support to this work. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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