Analysis of the Input Current Distortion and Guidelines for Designing High Power Factor Quasi-Resonant Flyback LED Drivers

energies

Article Analysis of the Input Current Distortion and Guidelines for Designing High Power Factor Quasi-Resonant Flyback LED Drivers

Claudio Adragna 1, Giovanni Gritti 1, Angelo Raciti 2, Santi Agatino Rizzo 3,* and Giovanni Susinni 3 1 Industrial & Power Conversion Division Application Laboratory, STMicroelectronics s.r.l, 20864 Agrate Brianza (MB), Italy; [email protected] (C.A.); [email protected] (G.G.) 2 Istituto per la microelettronica e microsistemi, Consiglio Nazionale Delle Ricerche, 95121 Catania, Italy; [email protected] 3 Department of Electrical, Electronic and Computer Engineering, University of Catania, 95125 Catania, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-095-738-2308

Received: 7 April 2020; Accepted: 3 June 2020; Published: 10 June 2020

Abstract: Nowadays, LED lamps have become a widespread solution in different lighting systems due to their high brightness, efficiency, long lifespan, high reliability and environmental friendliness. The choice of a proper LED driver circuit plays an important role, especially in terms of power quality. In fact, the driver controls its own input current in addition to the LED output current, thus it must guarantee a high power factor. Among the various LED drivers available on the market, the quasi-resonant (QR) flyback topology shows interesting benefits. This paper aims at investigating and analyzing the different issues related to the input current distortion in a QR flyback LED driver. Several effects, such as the distortion caused by the ringing current, crossover distortion due to transformer leakage inductance and crossover distortion due to the input storage capacitor have been experimentally reported. These effects, not previously studied for a high power factor (Hi-PF) QR flyback, have been analyzed in depth. Finally, some practical design guidelines for a Hi-PF QR flyback driver for LED applications are provided.

Keywords: converter control; solid-state lighting; power factor correction; Total Harmonic Distortion; primary sensing regulation; ﬂyback; LED lamp

1. Introduction Nowadays, almost a fifth of global electricity consumption is reserved for the lighting system [1]. As an example, in the USA, more than 200 billion kWh each year are expected for lighting systems, and the forecast confirms that the demand is going to increase in the next decade [2]. Traditional lighting equipment, such as incandescent lamps, florescent lamps and tungsten lamps, are environmental unfriendly, due to low efficiency or, in some cases, due to toxic substances that may contribute to the increase in ambient pollution levels [3]. In order to deal with these issues, many governments have already forbidden the trade of incandescent lamps and, on the other hand, encouraged the employment of light-emitting diodes (LEDs). They have been widespread both in private and public lighting systems thanks to their high luminous efficacy, long lifespan, high reliability and environmental friendliness [4]. Generally speaking, a LED lamp can be thought of as a combination of LED semiconductor materials and a driver circuit, thus the choice of the LED driver circuit plays a fundamental role from a power quality point of view. In fact, the driver controls both the LED output current and its own input current. Consequently, it must guarantee a high power factor (PF), and an input current with low total

Energies 2020, 13, 2989; doi:10.3390/en13112989 www.mdpi.com/journal/energies Energies 2020, 13, 2989 2 of 24 harmonic distortion (THD) [5,6]. Of course, it is worth remembering that LED lamps must comply with the national and international standards and regulations concerning harmonic currents, such as the standard IEEE-Std-519 and the IEC 61000-3-2. LED drivers can be roughly grouped into “passive” drivers, since they use only passive components, and “switching” drivers, where there is at least a controllable power switch (e.g., a MOSFET) [7,8]. To further improve the quality of the switching LED drivers, a linear regulator could be added in series to the LEDs array. The linear regulator almost provides a dc output voltage and therefore, it eliminates the low frequency current ripple. Unfortunately, the aforementioned solution increases both the cost and worsens the efficiency. The main drawback of passive LED drivers is the low PF that often cannot comply with the standard limit. Instead, switching LED drivers can realize a better output current regulation and higher power density, and they are more capable of satisfying the standard limits. Generally, there is a wide variety of LED driver topologies depending on the different power ranges. Furthermore, the topology of the driver can also be selected according to other requirements, including cost, galvanic isolation and efficiency. The most common switching LED driver topologies for low and medium power applications are mainly based on single-stage (SS) or two-stage (TS) LED architectures. A SS LED driver consists of a dc-dc converter with constant output current regulation that also acts as a power factor correction (PFC). The SS LED drivers can be classified on the bandwidth of the feedback control systems. In the case of a narrow bandwidth (e.g., boost, buck-boost and flyback) the storage capacitor must be located at the output of the converter, otherwise it is placed between the two semi-stages (such as quadratic topologies) [9–11]. Although, the TS drivers consist of two power stages, where the first one acts as a PFC and the second stage performs output current regulation. Focusing on the SS drivers, one of the main drawbacks is the bulky storage capacitor that also affects both the reliability and the size of the overall system. A PFC topology is usually adopted in the dc-dc stage to obtain high PF, and such a dc-dc stage is also responsible for setting the desired output current. In many of these applications, the power switches and control units are shared and embedded together to ensure a high reliability, efficiency and fast dynamics [12–14]. The reduction in the cost and size of the electronic ballast are additional benefits. The flyback converter is the most used SS LED driver because it obtains high step-down ability and it provides a good tradeoff among the power quality, the cost, the capacitor size and the efficiency. In integrated lighting applications, a SS discontinuous current conduction mode (DCM) flyback PFC converter is commonly used to drive the LED lamps in order to achieve a high PF. The task is performed by means of a simple circuit configuration used to regulate the lamp current. A flyback LED driver operating in DCM where the PFC proprieties can be easily obtained has been frequently adopted. In this context, the quasi-resonant (QR) flyback LED drivers can effectively reduce the transformer size and weight thanks to the high switching frequency [15], which is not fixed since it increases as the load decreases. Furthermore, the switching losses can be strongly reduced by adopting zero voltage switching (ZVS). This work aims at investigating and analyzing the different issues due to the input current distortion in a high power factor (Hi-PF) QR flyback LED driver. The THD performance of the whole converter is the result of the correlation of several causes such as the ringing current, crossover distortion due to transformer leakage inductance and crossover distortion due to the input storage capacitor. The input current distortion caused by the ringing current has been widely studied for the boost converter, while—as far as the authors know—it is the first time the problem has been studied for the QR flyback. Moreover, the crossover distortion due to the input storage capacitor in the case of a Hi-PF flyback LED driver has never been treated in literature. Finally, an accurate analysis is performed in this paper by considering the linear approximation of the input voltage and the related comparison with a sinusoidal waveform. Energies 2020, 4, x FOR PEER REVIEW 3 of 24 design guidelines for a Hi-QR flyback driver for lighting applications will be discussed, along with a few experimental results.

2. Brief Overview of LED Driver Topologies In this section, various topologies of passive and switching LED drivers are described and classified. The passive drivers are simpler and more reliable and they operate at the line frequency. No active control is developed, so a significant current ripple can occur. The switching drivers can strongly reduce the current distortion and improve the PF with the additional advantage of being suitable for high frequency operations, thus enabling reduced size. On the other hand, switching LED drivers are less reliable than passive ones. Energies 2020, 13, 2989 3 of 24 2.1. Passive LED Drivers PassiveThe paper LED is organizeddrivers are as characterized follows: In Section by the2 exclus, a classificationive use of ofpassive several components LED drivers (e.g., is introduced, resistors, capacitors,and their main magnetic characteristics components). are discussed. The insertion The of main an impedance issues related between to input the current ac line distortionand the LED in a lampHi-PF load QR flybackto limit converter the current are is discussed mandatory. in Section One 3of. Inthe Section main 4drawbacks, some practical of these design topologies guidelines is forthe lowa Hi-QR PF and flyback high driverTHD, forwhich lighting are sometimes applications not will enough be discussed, to meet along the standards with a few [16]. experimental results. Passive LED drivers can be classified into lossy and lossless impedance drivers. The strength 2. Brief Overview of LED Driver Topologies and simplicity of the lossy impedance driver is due to the use of a resistor RL or a linear regulator on the dcIn side, this as section, depicted various in Figure topologies 1. In many of passive practical and applications, switching LEDit is widespread drivers are to described use a bulky and step-downclassified. transformer The passive from drivers high are to simpler low voltage. and more It has reliabletwo-fold and benefits—it they operate reduces at the the line voltage frequency. drop onNo the active resistor control with is an developed, increase in so the a significant overall sy currentstem efficiency ripple can and, occur. at the The same switching time, guarantees drivers can galvanicstrongly isolation. reduce the A currentlarge electrolytic distortion capacitor and improve CS is typically the PF with used the in additionalorder to avoid advantage flickering. of being suitableA passive for high driver frequency with operations, a lossless thusimpedance enabling (an reduced inductor, size. capacitor On the other or their hand, combinations), switching LED usuallydrivers placed are less in reliable the ac thanside to passive limit the ones. LED output current, is shown in the example in Figure 2. An inductor Lin is usually adopted with the aim of replacing the less efficient and low-frequency transformer.2.1. Passive LED The DriversLin usually behaves as an additional input filter (by means of Cin) which smooths the inputPassive currentLED and driversleads to are advantages characterized suchby as thereducing exclusive the useinput of current passive distortion components [17–19]. (e.g., resistors,Instead ofcapacitors, using a bulky magnetic electrolytic components). capacitor The (E-CAP) insertion that of ensures an impedance a constant between output the current, ac line a and non-E-CAP the LED islamp often load used to limiton the the dc current side, still is mandatory. achieving Onea higher of the PF main with drawbacks respect to ofthe these lossy topologies counterpart. is the This low solutionPF and highcan ensure THD,which a longer are lifetime sometimes and notsmaller enough size to of meet the overall the standards system, [although16]. there is a small outputPassive current LED ripple drivers in canthe beLED classified load. Notwithstanding, into lossy and lossless lossless impedance passive drivers. drivers Thecan strengthreach high and efficiency, above 90% [18]. simplicity of the lossy impedance driver is due to the use of a resistor RL or a linear regulator on the dc side,Passive as depicted drivers in Figure can be1 .easily In many employed practical in applications, outdoor applications it is widespread and they to useare very a bulky cost step-down effective, especiallytransformer for fromlow-power high to applications. low voltage. The It has main two-fold drawbacks benefits—it of passive reduces LED drivers the voltage are their drop lack on theof aresistor proper withoutput an increasecurrent control in the overalland their system having effi anciency input and, current at the THD same that time, does guarantees not always galvanic meet the standards target. isolation. A large electrolytic capacitor CS is typically used in order to avoid flickering.

Figure 1. Typical application of a passive “lossy” LED driver. Figure 1. Typical application of a passive “lossy” LED driver. A passive driver with a lossless impedance (an inductor, capacitor or their combinations), usually placed in the ac side to limit the LED output current, is shown in the example in Figure2. An inductor L is usually adopted with the aim of replacing the less efficient and low-frequency transformer. in The Lin usually behaves as an additional input filter (by means of Cin) which smooths the input current and leads to advantages such as reducing the input current distortion [17–19]. Instead of using a bulky electrolytic capacitor (E-CAP) that ensures a constant output current, a non-E-CAP is often used on the dc side, still achieving a higher PF with respect to the lossy counterpart. This solution can ensure a longer lifetime and smaller size of the overall system, although there is a small output current ripple in the LED load. Notwithstanding, lossless passive drivers can reach high efficiency, above 90% [18]. EnergiesEnergies 20202020,, 4,13 x, 2989FOR PEER REVIEW 4 4of of 24 24

Energies 2020, 4, x FOR PEER REVIEW 4 of 24

Figure 2. Typical application of a passive “lossless” LED driver. Figure 2. Typical application of a passive “lossless” LED driver.

Passive drivers can be easily employed in outdoor applications and they are very cost effective, 2.2.especially Switching for LED low-power DriverFigure applications. 2. Typical application The main of drawbacks a passive “lossless” of passive LED LED driver. drivers are their lack of a properSwitching output currentLED drivers control bring and various their having advantages an input arising current from THD the thatuse doesof the not switching always meetdevices. the Moreover,2.2.standards Switching target.switching LED Driver LED drivers can embed into a single electronic ballast with various functionalities,Switching LEDsuch driversas circuit bring fault various protection advantages and active arising and from high the PFC use [20].of the They switching are especially devices. 2.2. Switching LED Driver employedMoreover, inswitching indoor applications, LED drivers since can the embed whole ciintorcuit a controlsingle iselectronic compact, ballastreliable withand effectivevarious withfunctionalities, aSwitching low and controllablesuch LED driversas circuit ripple bring fault output various protection current. advantages and Another active arising strengthand from high thepoint PFC use of[20]. of switching the They switching are LED especially devices.drivers isemployedMoreover, their higher switchingin indoor efficiency LEDapplications, in drivers comparison can since embed to the the intowhole passive a single circuit ones. electronic control Switching ballastis compact, LED with driver variousreliable topologies functionalities,and effective can be classifiedwithsuch a as low circuit into and two faultcontrollable main protection categories, ripple and outputSS active and andTS,current. as high shown Another PFC in [20 Figure]. strength They 3. are The point especially SS ofdrivers switching employed consist LED of in adrivers indoorsingle powerisapplications, their stage higher that since efficiency acts the as whole both in comparison circuita dc-dc control regulator to the is compact, passiveand PFC, reliableones. shown Switching and in eFigureffective LED 3a, with driverwith a lowa topologiesstorage and controllable capacitor can be Cclassifiedripples. On outputthe into other two current. hand, main Another thecategories, TS drivers strength SS and are point TS, used as of shownfo switchingr high-power in Figure LED 3. driversapplications. The SS is drivers their The higher consist aforementioned effi ofciency a single in driverspowercomparison stage comprise that to the actstwo passive aspower both ones. aconversion dc-dc Switching regulator stages LED and that driverPFC, can shownperform topologies in different Figure can be3a, func classifiedwithtions. a storage In into general, twocapacitor main the firstCcategories,s. On stage the acts SSother andas ahand, TS,PFC as andthe shown theTS driverssecond in Figure oneare3. Theusedas a SSdc fo-dc driversr high-power regulator consist and ofapplications. afilter single [21], power asThe depicted stageaforementioned that in Figure acts as 3b.driversboth a dc-dccomprise regulator two power and PFC, conversion shown instages Figure that3a, can with perform a storage different capacitor funcCtions.s. On theIn general, other hand, the firstthe TSstage drivers acts areas a used PFC forand high-power the second applications.one as a dc-dc The regulator aforementioned and filter drivers [21], as comprise depicted two in Figure power 3b.conversion stages that can perform different functions. In general, the first stage acts as a PFC and the second one as a dc-dc regulator and filter [21], as depicted in Figure3b.

(a) (b)

Figure 3. (a) SS switching LED driver topology. (b) TS switching LED driver topology. (a) (b)

In the FigurefollowingFigure 3. 3. (a( a)section, )SS SS switching switching there LED LEDis a driver driverbrief topology.focus topology. on (SS (bb)) TSLED TS switching switching drivers LED LEDwith driver driver the maintopology. topology. advantages of different topologies adopted in the literature. In the following section, there is a brief focus on SS LED drivers with the main advantages of SSIn driversthe following usually section, have a therelow co ismponent a brief focus count on and SS among LED drivers them a with power the switch main thatadvantages controls of a different topologies adopted in the literature. powerdifferent conversion topologies stage. adopted However, in the literature.it is often difficult for a SS driver to simultaneously ensure good SS drivers usually have a low component count and among them a power switch that controls performanceSS drivers in usuallymany respects, have a low such co mponentas high efficiency, count and high among PF, themconstant a power current switch output that and controls so on. a a power conversion stage. However, it is often difficult for a SS driver to simultaneously ensure SSpower drivers conversion are suitable stage. for However, low and itmedium is often power difficult class for applicationsa SS driver to (below simultaneously 70 W) where ensure size good and good performance in many respects, such as high efficiency, high PF, constant current output and costperformance are usually in many more respects, critical suchthan asPF high and efficiency, efficiency. high The PF, storage constant capacitor current isoutput usually and placed so on. so on. SS drivers are suitable for low and medium power class applications (below 70 W) where downstreamSS drivers are from suitable the fordc-dc low converter, and medium that poweris on theclass high applications frequency (below side, 70as W)shown where by sizethe andred size and cost are usually more critical than PF and efficiency. The storage capacitor is usually placed capacitorcost are usuallyin Figure more 4, to obtaincritical a thanhigh PF.PF Notwithstanding,and efficiency. The another storage approach capacitor can isbe usuallyfound in placed some downstream from the dc-dc converter, that is on the high frequency side, as shown by the red capacitor embeddeddownstream LED from drivers the dc-dcwhere converter,the capacitor that is placedis on the on thehigh low frequency frequency side, side, as as shownshown byby thethe blue red in Figure4, to obtain a high PF. Notwithstanding, another approach can be found in some embedded capacitor in FigureFigure 4,4. to More obtain specifically, a high PF. suchNotwithstanding, an approach anotherhas sometimes approach been can usedbe found in very in some low LED drivers where the capacitor is placed on the low frequency side, as shown by the blue capacitor in powerembedded applications LED drivers (below where 5 W) the [22]. capacitor Indeed, is placednowadays, on the in low this frequency power range, side, passiveas shown LED by thedrivers blue arecapacitor adopted in toFigure guarantee 4. More an inexpensivespecifically, cost.such Figurean approach 5a shows has the sometimes waveforms been of theused current in very drawn low power applications (below 5 W) [22]. Indeed, nowadays, in this power range, passive LED drivers are adopted to guarantee an inexpensive cost. Figure 5a shows the waveforms of the current drawn

Energies 2020, 13, 2989 5 of 24

Energies 2020, 4, x FOR PEER REVIEW 5 of 24 Figure4. More speciﬁcally, such an approach has sometimes been used in very low power applications Energies(belowby LED 2020 5lamps, 4, W) x FOR [when22 ].PEER Indeed, the REVIEW capacitor nowadays, is placed in this on powerthe low range, and high passive frequency LED drivers side. The are adoptedwaveforms5 of 24 to guaranteeconfirm that an inexpensivethe dc-dc converter cost. Figure can5 aguarantee shows the PFC waveforms only when of the the current storage drawn capacitor by LED is lampsplaced by LED lamps when the capacitor is placed on the low and high frequency side. The waveforms whendownstream the capacitor from the is placeddc-dc converter on the low itself. and Althou high frequencygh the input side. current The waveformsdrawn by the conﬁrm LED lamp that has the confirm that the dc-dc converter can guarantee PFC only when the storage capacitor is placed dc-dca better converter THD when can the guarantee capacitor PFC is placed only when at the the high storage frequency capacitor side, is a placedhigher downstreamoutput current from ripple the downstream from the dc-dc converter itself. Although the input current drawn by the LED lamp has dc-dcoccurs converter in comparison itself. to Although the previous the input solution, current as drawnshown byin Figure the LED 5b. lamp However, has a better the required THD when storage the a better THD when the capacitor is placed at the high frequency side, a higher output current ripple capacitorcapacitance is placedis not always at the highable frequencyto handle both side, the a higher low and output the high current frequency ripple occurs ripple in [23]. comparison to occurs in comparison to the previous solution, as shown in Figure 5b. However, the required storage the previous solution, as shown in Figure5b. However, the required storage capacitance is not always capacitance is not always able to handle both the low and the high frequency ripple [23]. able to handle both the low and the high frequency ripple [23].

Figure 4. Switching-mode single-stage driver: depending on the position of the capacitor Cs, the single Figuredc-dc converter 4. Switching-mode can provide single-stage both PFC driver:and output depending current on regulation. the position of the capacitor Cs, the single Figure 4. Switching-mode single-stage driver: depending on the position of the capacitor Cs, the single dc-dc converter can provide both PFC and output current regulation. dc-dc converter can provide both PFC and output current regulation.

(a) (b) (a) Figure 5. Simulation of the current Iac drawn by the LED lamp ( a), and the output(b) LED load current ILED ((bb).). The The blue blue represents represents the the case case when when the the stor storageage capacitor capacitor is is directly directly connected on the low Figure 5. Simulation of the current Iac drawn by the LED lamp (a), and the output LED load current frequency side and thethe redred onon thethe highhigh frequencyfrequency side.side. ILED (b). The blue represents the case when the storage capacitor is directly connected on the low frequency side and the red on the high frequency side. The SS LED drivers can be roughl roughlyy classified classified on the bandwidth of of the the feedback feedback control control systems. systems. In the the case case of of a narrow a narrow bandwidth bandwidth (e.g., (e.g., boost, boost, buck-boost buck-boost and flyback) and flyback) the storage the storagecapacitor capacitor must be The SS LED drivers can be roughly classified on the bandwidth of the feedback control systems. mustlocated be at located the output at the of the output converter. of the Instead, converter. in the Instead, case of ina wide the case bandwidth of a wide system bandwidth (e.g., quadratic system In the case of a narrow bandwidth (e.g., boost, buck-boost and flyback) the storage capacitor must be (e.g.,or single-stage quadratic single-switch or single-stage inpu single-switcht current shaper input topologies) current shaper the storage topologies) capacitor the is storage placed capacitor between located at the output of the converter. Instead, in the case of a wide bandwidth system (e.g., quadratic isthe placed two semi-stages. between the In the two following, semi-stages. the narrow In the bandwidth following, thesystems narrow have bandwidth been analyzed. systems In detail, have or single-stage single-switch input current shaper topologies) the storage capacitor is placed between beena huge analyzed. number of In conventional detail, a huge LED number drivers ofare conventional widely discussed LED in drivers literature, are such widely as buck discussed [24–26], in the two semi-stages. In the following, the narrow bandwidth systems have been analyzed. In detail, literature,buck-boost such [27], asSEPIC buck [28,29], [24–26 ],flyback buck-boost [30,31], [27 hal], SEPICf-bridge [28 [32–34],29], flyback and push-pull [30,31], half-bridge converters [[35,36].32–34] a huge number of conventional LED drivers are widely discussed in literature, such as buck [24–26], andFurthermore, push-pull PFC converters can be achieved [35,36]. Furthermore,with valley-fill PFC circuits can [37]. be achieved Other approaches, with valley-fill such circuitsas coupled- [37]. buck-boost [27], SEPIC [28,29], flyback [30,31], half-bridge [32–34] and push-pull converters [35,36]. Otherinductor approaches, modified converters such as coupled-inductor [38–41] and valley-fill modified modified converters converters [38– 41[42],] and show valley-fill a simple modified solution Furthermore, PFC can be achieved with valley-fill circuits [37]. Other approaches, such as coupled- convertersto the step-down [42], show ratio a requirement simple solution without to the compromising step-down ratio the requirement efficiency and without system compromising complexity. the inductor modified converters [38–41] and valley-fill modified converters [42], show a simple solution efficiencyThe flyback and system converter complexity. has been widely adopted in LED driver lamps because of its simple to the step-down ratio requirement without compromising the efficiency and system complexity. structureThe flybackand high converter PF. One has of been the widelystrengths adopted is its inefficiency, LED driver which lamps can because be improved of its simple by using structure the The flyback converter has been widely adopted in LED driver lamps because of its simple andleakage high energy PF. One or of using the strengths soft-switching is its effi techniquesciency, which [43]. can In bespite improved of the byflyback using LED the leakagedriver having energy structure and high PF. One of the strengths is its efficiency, which can be improved by using the various advantages, in many practical applications the QR mode operation has become one of the leakage energy or using soft-switching techniques [43]. In spite of the flyback LED driver having most familiar methods in LED driver applications. It is worth noting the decrease in switching losses various advantages, in many practical applications the QR mode operation has become one of the with respect to a flyback converter operated with a fixed frequency. Moreover, the QR driver has an most familiar methods in LED driver applications. It is worth noting the decrease in switching losses with respect to a flyback converter operated with a fixed frequency. Moreover, the QR driver has an

Energies 2020, 13, 2989 6 of 24 or using soft-switching techniques [43]. In spite of the flyback LED driver having various advantages, in many practical applications the QR mode operation has become one of the most familiar methods inEnergies LED 2020 driver, 4, x FOR applications. PEER REVIEW It is worth noting the decrease in switching losses with respect6 toof 24 a flyback converter operated with a fixed frequency. Moreover, the QR driver has an enhanced transient responseenhanced in transient DCM operation response [ 44in, 45DCM] and operation it may have [44,45 a smaller] and it EMImay filterhave [a46 smaller]. In fact, EMI in applicationsfilter [46]. In operatedfact, in applications from the mains, operated the switching from the frequencymains, the is switching modulated frequency at twice the is modulated mains frequency at twice due the to themains voltage frequency ripple due appearing to the acrossvoltage the ripple input appearin capacitance.g across The switchingthe input frequencycapacitance. span The depends switching on thefrequency amplitude span of depends this input on voltage the ampl ripple.itude This of this causes input the voltage spectrum ripple. to be This spread causes over the some spectrum frequency to bands,be spread rather over than some being frequency concentrated bands, on rather single th frequencyan being concentrated values. on single frequency values. InIn addition,addition, thethe QRQR driverdriver hashas aa higherhigher safetysafety degreedegree underunder shortshort circuitcircuit conditions,conditions, sincesince thethe switchswitch isis notnot enabledenabled untiluntil thethe primaryprimary windingswindings areare fullyfullydemagnetized. demagnetized. Therefore,Therefore, transformertransformer saturationsaturation isis notnot possible.possible. On the other hand, the QR flybackflyback LED driver maymay havehave aa highhigh rippleripple outputoutput currentcurrent andand highhigh conductionconduction losseslosses inin comparisoncomparison toto thethe fixedfixed frequencyfrequency driver.driver. ItIt is important to to point point out out that that the the difference difference between between a DCM a DCM flyback flyback and and a QR a QR flyback flyback is in is the in theturn-on turn-on mechanism. mechanism. In a In DCM a DCM flyback flyback converter, converter, the the gate gate driver driver provides provides a a constantconstant switchingswitching frequency,frequency, while while in in a QRa QR flyback flyback a variable a variable frequency frequency is used, is used, where where the off timethe off depends time depends on the resonant on the valleyresonant detection valley ofdetection the drain-to-source of the drain-to-sou voltage ringingrce voltage that follows ringing transformer that follows demagnetization. transformer Figuredemagnetization.6 depicts the Figure drain-to-source 6 depicts the voltage drain-to-s waveformsource voltage in the casewaveforms of a DCM in the flyback, case shownof a DCM in Figureflyback,6a, shown and a in QR Figure flyback, 6a, shownand a QR in Figureflyback,6b, shown thus highlighting in Figure 6b, the thus benefit highlighting in terms the of reducedbenefit in switchingterms of reduced losses. switching losses.

(a) (b)

Figure 6. Drain to source voltage waveforms for a DCM flyback (a), and a QR flyback (b). Figure 6. Drain to source voltage waveforms for a DCM flyback (a), and a QR flyback (b). A simplified schematic of a QR flyback LED driver (not for Hi-PF applications) is shown in A simplified schematic of a QR flyback LED driver (not for Hi-PF applications) is shown in Figure7. The primary current Ip starts to flow into Lp when the power switch is turned on. The voltage Figure 7. The primary current Ip starts to flow into Lp when the power switch is turned on. The voltage at the secondary is such that the diode is reverse biased, hence the capacitance Cout supplies the at the secondary is such that the diode is reverse biased, hence the capacitance Cout supplies the LED LED string. Once the switch is turned off, the current Ip goes to zero and the voltage across both string. Once the switch is turned off, the current Ip goes to zero and the voltage across both windings windings reverses, so that the output diode is forward-biased. Then, the current starts flowing in reverses, so that the output diode is forward-biased. Then, the current starts flowing in the secondary the secondary winding and Cout can be charged by the energy stored in the transformer. It is worth winding and Cout can be charged by the energy stored in the transformer. It is worth noting that, while noting that, while the current flows on the secondary side, the drain–source voltage is equal to the the current flows on the secondary side, the drain–source voltage is equal to the rectified input Vin(t) rectified input Vin(t) plus the reflected output voltage VR at the primary windings. The transformer plus the reflected output voltage VR at the primary windings. The transformer takes a time TFW to takes a time TFW to demagnetize, and as soon as the energy transfer is completed, the drain–source demagnetize, and as soon as the energy transfer is completed, the drain–source voltage starts ringing. voltage starts ringing. The main reason for this energy exchange phenomenon is due to a resonant tank The main reason for this energy exchange phenomenon is due to a resonant tank between the between the inductance Lpp and the capacitance Cpp. The inductance Lpp is the sum between the leakage inductance Lpp and the capacitance Cpp. The inductance Lpp is the sum between the leakage inductance inductance of the copper paths and the primary winding Lp. The latter it is the prevalent contribution, of the copper paths and the primary winding Lp. The latter it is the prevalent contribution, hence Lpp hence Lpp can be simply approximated as Lp. The capacitance Cpp can be expressed as the sum of the can be simply approximated as Lp. The capacitance Cpp can be expressed as the sum of the parasitic parasitic capacitances on the primary circuit and the capacitances of the secondary circuit referred to the capacitances on the primary circuit and the capacitances of the secondary circuit referred to the primary one. The latter are due to the parasitic capacitances of the secondary circuit and to the output primary one. The latter are due to the parasitic capacitances of the secondary circuit and to the output capacitance Cout. The primary circuit capacitance includes various contributions: the output parasitic capacitance Cout. The primary circuit capacitance includes various contributions: the output parasitic capacitance Coss of the MOSFET; the junction capacitance of the diode; the package capacitance; capacitance Coss of the MOSFET; the junction capacitance of the diode; the package capacitance; the intra-winding capacitance of the transformer; plus other stray contributors together and so on [47– 51]. Coss is a strongly nonlinear capacitance and, especially in the latest MOSFET generations, it increases dramatically (100 times or more) when the drain–source voltage, vDS, falls below few tens volt; i.e., Cpp is a function of vDS: Cpp(vDS). In the following, this capacitance will be considered constant

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Energies 2020, 4, x FOR PEER REVIEW 7 of 24 the intra-winding capacitance of the transformer; plus other stray contributors together and so or,on [at47 –least,51]. Cnotoss issignificantly a strongly nonlinear impacting capacitance the overall and, CDS especially. Hence, the in the resonant latest MOSFET frequency generations, fr can be representedit increases dramaticallyas: (100 times or more) when the drain–source voltage, vDS, falls below few tens volt; i.e., C is a function of v : C (v ). In the following, this capacitance will be considered pp DS=≈pp DS 11 fr constant or, at least, not signiﬁcantly impactingπ the overall CπDS. Hence, the resonant frequency fr can(1) 2 LCpp pp() v DS 2 LCpDS be represented as: In applications, the inductance Lp is measured1 with an impedance1 meter or by measuring the fr = q p (1) di/dt when a voltage square wave is applied, while ≈C2DSπ is LesteemedpC by Equation (1) from the 2π LppCpp(vDS) DS measurement of the drain–source voltage ringing.

Figure 7. QR resonant ﬂyback valley-switching. Figure 7. QR resonant flyback valley-switching.

In applications, the inductance Lp is measured with an impedance meter or by measuring the di/dt Bearing in mind that, in a conventional QR topology a feedback voltage loop is used, and this when a voltage square wave is applied, while CDS is esteemed by Equation (1) from the measurement loop controls the average value of the secondary current Is, it follows that the switching frequency is of the drain–source voltage ringing. continuously adjusted depending on the output current load. In this way the switching device turns Bearing in mind that, in a conventional QR topology a feedback voltage loop is used, and this on when necessary, provided that the inner controller is able to detect a valley in the ringing drain– loop controls the average value of the secondary current Is, it follows that the switching frequency is source voltage. The switching period T can be expressed as: continuously adjusted depending on the output current load. In this way the switching device turns on =+ + when necessary, provided that the innerTT controlleron T FW is able T ring to detect a valley in the ringing drain–source   voltage. The switching period T can be expressed=+−1 as:() (2) TkTring12 1 r ( 2 T = Ton + TFW + Tring where Ton is the on time of the power switch, T1FW is the time interval where the current flows on the(2) Tring = 2 [1 + 2(k 1)]Tr secondary side and Tring is the time interval during which− the drain–source voltage rings. Ton is reducedwhere Ton asis the the load on timereduces, of the hence power at very switch, lightTFW loadis thethe timefrequency interval is high. where Therefore, the current the flows maximum on the on switchingsecondary sidefrequency and Tring imposesis the time a minimum interval during T . In which this thecase, drain–source a further load voltage reduction rings. T oninvolvesis reduced an incrementas the load of reduces, Tring. In hence detail, at T veryring strictly light load depends the frequency on the load is high. level Therefore, and it is imposed the maximum by keeping switching off thefrequency switch imposes for some a minimumvalley pointsTon. Inin thisthe case,drain–so a furtherurce loadvoltage. reduction In other involves terms, an it increment depends ofonT thering. numberIn detail, ofT ringthe strictlyvalley points depends k “skipped” on the load by level the andcontroller. it is imposed More specifically, by keeping ok ffincreasesthe switch as forthe someload levelvalley decreases. points in theThe drain–source term k is equal voltage. to 1 until In other Ton is terms, greater it depends than its onminimum the number value, of thei.e. the valley switch points is turnedk “skipped” on at bythe the fist controller. valley on Morethe drain–source specifically, vok increasesltage waveform as the load (one-half level decreases.of the resonant The termperiod).k is Thus,equal the to 1 QR until modeTon isoperation greater than of the its converter minimum represents value, i.e. the the condition switch is turnedk = 1. on at the fist valley on the drain–sourceThe reason voltagebehind waveformthe turn-on (one-half during of a the valley resonant point period). of the Thus, drain–source the QR mode voltage operation is the of achievementthe converter of represents lower capacitive the condition switchingk = 1. losses: The reason behind the turn-on during a valley1 point of the drain–source voltage is the achievement ECV= 2 (3) of lower capacitive switching losses: sDSDS2 1 This mechanism is able to minimize inE a =significC antV way2 the switching losses. It is worth noting(3) s 2 DS DS that the bill of material of a QR driver contains only a few simple components, which implies an inexpensive solution [51,52].

Energies 2020, 13, 2989 8 of 24

EnergiesThis 2020 mechanism, 4, x FOR PEER is REVIEW able to minimize in a significant way the switching losses. It is worth noting8 of 24 that the bill of material of a QR driver contains only a few simple components, which implies an inexpensiveFinally, solutionFigure 8 [compares51,52]. the performance between a QR flyback, a buck-boost and a hybrid solutionFinally, [36–49] Figure in terms8 compares of some the key performance factors. The betweenhigh step-down a QR flyback, ability a(HSDA) buck-boost [36–38], and efficiency a hybrid (EFF)solution [38–41], [36–49 compact] in terms capacitor of some volume key factors.(CAP) [42], The cost high (COST) step-down [43–46] ability and low (HSDA) power [ 36range–38], applicationsefficiency (EFF) (LP) [ 38[47,48]–41], compacthave been capacitor taken into volume account (CAP) for [ 42the], costthree (COST) converters. [43–46 The] and comparison low power highlightsrange applications the superior (LP) performance [47,48] have beenof the taken QR flyb intoack account converter, for the which three makes converters. it the preferred The comparison choice forhighlights LED driver the superiorapplications. performance of the QR flyback converter, which makes it the preferred choice for LED driver applications.

Figure 8. Performance of three main SS LED driver topologies: high step-down ability (HSDA), Figure 8. Performance of three main SS LED driver topologies: high step-down ability (HSDA), eﬃciency (EFF), reduced capacitor (CAP), cost (COST) and low power applications (LP). efficiency (EFF), reduced capacitor (CAP), cost (COST) and low power applications (LP).

3.3. Analysis Analysis of of Input Input Current Current Distortion Distortion due due to to Power Power Processing Processing and and Power Power Circuit Circuit

3.1.3.1. Generic Generic Control Control Method Method Obtaining Obtaining High High Power Power Factor Factor AlthoughAlthough the the QR QR flyback flyback converter converter topology topology is is extr extremelyemely popular popular since since it it is is a a very very cost-effective cost-effective solutionsolution with with high high performance, performance, the the converter converter featur featureses inherent inherent distortion distortion of of the the input input current current [48]. [48]. Normally,Normally, this distortion is not not a a concern concern for for compliance compliance with with the the IEC61000-3-2, IEC61000-3-2, however, however, some some input input currentcurrent THDTHD targetstargets (e.g., (e.g.,< 10%<10% at at full full power) power) are becomingare becoming market market requirements requirements that are that very are di ffiverycult difficultto achieve, to achieve, especially especia whenlly working when working with lighting with lighting equipment equipment over 25 Wover [53 25]. W [53]. AsAs shown in FigureFigure7 ,7, the the storage storage capacitor capacitor of of the the QR QR flyback flyback is placed is placed on theon lowthe frequencylow frequency side sideof the of dc-dcthe dc-dc converter. converter. With With the aim the to aim reduce to reduce the input the input current current THD, theTHD, capacitor the capacitor should should be placed be placeddownstream downstream from the from converter, the converter, according according to the waveform to the waveform simulations simulations in Figure5a (redin Figure trace) 5a related (red trace)to the related capacitor to positionthe capacitor of Figure position4. In of this Figure perspective, 4. In this the perspective, converter canthe beconverter considered can asbe aconsidered Hi-PF QR asflyback, a Hi-PF which QR flyback, has a rectified which has voltage a rectified in input voltage and thein input storage and capacitor the storage downstream capacitor downstream of the power ofstage. the power The aforementioned stage. The aforementioned solution strongly solution reduces strongly the harmonics reduces the distortions harmonics of distortions the input current of the inputdrawn current by the drawn LED lamp. by the LED lamp. TheThe control control gear gear in in a a Hi-PF Hi-PF QR QR flyback flyback converter converter has has a a twofold twofold task: task: firstly, firstly, it it is is responsible responsible for for regulatingregulating the outputoutput voltagevoltage or or current current and, and, simultaneously, simultaneously, it has it has to maintain to maintain a low a THDlow THD of the of input the inputcurrent. current. For the For sake the ofsake completeness, of completeness, the characteristics the characteristics and proprietiesand proprieties of the of controlthe control method method that thathave have to be to considered be considered to achieve to achieve a Hi-PF a Hi-PF QR flybackQR flyback converter converter will bewill briefly be briefly discussed. discussed. TheThe control control method, method, shown shown in in Figure Figure 99,, isis responsibleresponsible forfor thethe turnturn onon andand turnturn ooffff of the switch. Henceforth,Henceforth, the the quantities quantities depending depending on on the the inst instantaneousantaneous line line voltage voltage will will be be considered considered as as a a functionfunction of of the the term term θθ = 22ππfflinelinet.t . TheThe target target of of the the control control method method is is to to obtain obtain an an input input current current very very similar similar to to a a sinusoid sinusoid in-phase in-phase withwith the inputinput voltagevoltage to to achieve achieve high-PF. high-PF. Regardless Regardless of theof the specific specific control control method, method, the previous the previous target targetcan be can partially be partially converted converted into obtaining into obtaining a primary a primary current, current, as shown as shown by Ip inby Figure Ip in Figure7, whose 7, whose peak peakenvelope, envelope, detailed detailed by the by the green green elements elements in Figurein Figu9re, leads9, leads to to a sinusoida sinusoid in-phase in-phase with with the the inputinput voltage.voltage. To To obtain obtain such such an an envelope, envelope, the the control control method method must must turn turn off off thethe switch switch when when the the current current on the primary side reaches one of the (green) peaks in Figure 9. To obtain a similar result for the secondary current, shown by the red elements in Figure 9, a similar control must be adopted for the switch turn on. The turn on must occur when the transformer is fully demagnetized, thanks to a zero- current detector. In other terms, the combination of these switching rules, which result in a variable

Energies 2020, 13, 2989 9 of 24 on the primary side reaches one of the (green) peaks in Figure9. To obtain a similar result for the secondary current, shown by the red elements in Figure9, a similar control must be adopted for theEnergies switch 2020 turn, 4, x FOR on. PEER The REVIEW turn on must occur when the transformer is fully demagnetized, thanks9 of to24 a zero-current detector. In other terms, the combination of these switching rules, which result in a variableswitching switching frequency, frequency, shown shownin Figure in 9, Figure ensures9, ensures that the that peaks the peaksof the primary of the primary current, current, in a half in period a half of the rectified voltage Vin(θ), can be enveloped by a rectified sinusoid. The primary current Ip(θ) in a period of the rectified voltage Vin(θ), can be enveloped by a rectified sinusoid. The primary current switching cycle is triangular shaped and flows only during the switch on-time, as sketched by the Ip(θ) in a switching cycle is triangular shaped and flows only during the switch on-time, as sketched green triangles shown in Figure 9. During the off time, the secondary current Is(θ) flows and it is by the green triangles shown in Figure9. During the o ff time, the secondary current Is(θ) flows and itrepresented is represented by the by red the triangles. red triangles. Therefore, Therefore, the switching the switching frequency frequency SW is SWvariable, is variable, where wherethe TON the(θ) and TOFF(θ) are modulated to achieve a sinusoidal current envelope. TON(θ) and TOFF(θ) are modulated to achieve a sinusoidal current envelope.

Figure 9. Primary and secondary currents in a QR-controlled Hi-PF flyback converter. Figure 9. Primary and secondary currents in a QR-controlled Hi-PF flyback converter. However, the primary current Ip(θ) flows only during the on-time of the power switch, and this meansHowever, that the averagethe primary value current of the Ip primary(θ) flows current only during deviates the on-time significantly of the from power an switch, ideal sinusoid. and this 2 Tomeans ensure that good the average performance value andof the low primary THD inputcurrent current deviates for significantly a Hi-PF driver from LED, an theideal quantity sinusoid.D ToT (ensureD is the good duty performance ratio) must beand constant low THD along input each current line half-cycle, for a Hi-PF thus driver a unity LED, PF the is obtainedquantity [D542T]. (D is the dutyIn this ratio) paper, must it isbe assumed constant thatalong the each generic line half-cycle, control method thus a implementsunity PF is obtained the previous [54]. features. For theIn sakethis paper, of simplicity it is assumed to obtain that a quantitativethe generic control expression method of the implements input current the previousIin, the following features. assumptionsFor the sake haveof simplicity been considered: to obtain a quantitative expression of the input current Iin, the following 1.assumptionsThe line voltagehave been is sinusoidal, considered: and the input bridge rectifier is ideal, thus the voltage at the bridge 1. outputThe line terminal voltage is asinusoidal, rectified sinusoid. and the input bridge rectifier is ideal, thus the voltage at the bridge 2. Theoutput voltage terminal drop is acrossa rectified the powersinusoid. switch in the on-state is negligible and there is negligible 2. energyThe voltage accumulation drop across on the the dc power side ofswitch the bridge. in the on-state is negligible and there is negligible 3. Theenergy transformer accumulation windings on the are dc perfectly side of the coupled bridge. (i.e., no leakage inductance).

4.3. TheThe transformer turn-oﬀ transient windings of the are power perfectly switch coupled has (i.e., negligible no leakage duration inductance). so that TFW immediately FW 4. followsThe turn-off TON. transient of the power switch has negligible duration so that T immediately 5. Thefollows converter TON. is operated so the power switch is turned on in each cycle after the secondary 5. currentThe converter becomes is zero,operated therefore so the in power either QR-modeswitch is turned (i.e., on on the in ﬁrst each valley cycle of after the ringingthe secondary in the drain–sourcecurrent becomes voltage) zero, ortherefore DCM. in either QR-mode (i.e., on the first valley of the ringing in the 6. Thedrain–source output voltage voltage) is constantor DCM. along a line half-cycle. 6. The output voltage is constant along a line half-cycle. 7. During the time interval elapsing from the instant when the transformer demagnetizes to the 7. During the time interval elapsing from the instant when the transformer demagnetizes to the instant when the power switch is turned on, the transformer current is zero; consequently, instant when the power switch is turned on, the transformer current is zero; consequently, the the initial current during the on-time is zero too. This time interval is equal to Tr/2 in the case of initial current during the on-time is zero too. This time interval is equal to Tr/2 in the case of the the converter being used in QR mode. converter being used in QR mode.

It is worth generalizing the relation to D2T when a variable switching frequency is used, in particular:

Energies 2020, 13, 2989 10 of 24

Energies 2020, 4, x FOR PEER REVIEW 10 of 24 It is worth generalizing the relation to D2T when a variable switching frequency is used, in particular: 2 2 " TTON()#θθ2 ON ()2 2 T===ON(θ) ()TON(θ) D2TDT= ()T(θ T) =θconstant()= constant (4)(4) T(θTT) θθT(θ) wherewhere thethe dependencesdependences ofofT TONON and T on θ point out that they are a function of the instantaneous lineline voltagevoltageV Vacac. Assuming Assuming that the rectifiedrectified voltagevoltage VVin is sinusoidal inin 00 ≤ θ ≤ ππ,, according according to assumption in ≤ ≤ 1,1, itit cancan bebe writtenwritten as:as: VinVV(θ)()=θsinθV= in,pk sin(θ ()) (5) in in, pk (5) where Vin,pk is the amplitude of the rectified voltage. where Vin,pk is the amplitude of the rectified voltage. By considering the current flowing in the primary windings during the time interval TON, By considering the current flowing in the primary windings during the time interval TON, and andbearing bearing in mind in mind the theinductance inductance current–voltage current–voltage differential differential relation, relation, the the peak peak value value of primaryprimary current Ipkp(θ) can be expressed as: current Ipkp(θ) can be expressed as: 1 IVT()θsinθθ=1 ()() () Ipkp(pkpθ) = Vin,pkin,sin pk(θ) TON ON(θ) (6)(6) LpLp

TheThe genericgeneric controlcontrol methodmethod mustmust ensureensure thatthat thethe heightheight ofof thethe trianglestriangles depicteddepicted inin FigureFigure9 9 variesvaries alongalong aa lineline cyclecycle asas expressed expressed by by Equation equation (6). 6. To To reach reach this target, the controlcontrol methodmethod mustmust properly vary the width of TON(θ) as well as TOFF(θ). The input current Iin(θ) is the average value of properly vary the width of TON(θ) as well as TOFF(θ). The input current Iin(θ) is the average value of eacheach triangletriangle overover aa switchingswitching cycle,cycle, hence,hence, takingtaking EquationEquation (6)(6) intointo account:account: 2 11TTON()θθ ON () II()θθ==() () V sinθ() 2 (7) in1 pkp TON(θ()) 1 in, pk TON(θ()) Iin(θ) = I2θ2pkp(θ) TL= Vp in,pk sin(θ) T θ (7) 2 T(θ) 2Lp T(θ) It is worth noting that, if the ratio Ton(θ)2/T(θ) is maintained constant by the controller, the input 2 currentIt is Iin worth(θ) can noting be assumed that, if thesinusoidal. ratio Ton Figure(θ) /T( θ10) issumma maintainedrizes the constant key current by the waveform controller, in the the input Hi- currentPF driverIin (LED.θ) can be assumed sinusoidal. Figure 10 summarizes the key current waveform in the Hi-PF driver LED.

Figure 10. Current waveforms of the circuit in Figure7: line cycle time scale of the primary current Ip, Figure 10. Current waveforms of the circuit in Figure 7: line cycle time scale of the primary current Ip, secondary current Is, input rectified current Iin and current in the ac side Iac. secondary current Is, input rectified current Iin and current in the ac side Iac. Despite the fact that a suitable control method leads to a Hi-PF QR flyback driver according to theDespite previous the features, fact that therea suitable are di controlfferent method inherent leads causes to a of Hi-PF distortion QR flyback in the inputdriver current according to beto faced.the previous The distortion features, there due to are these different causes inherent is reported causes in of the distortion next section in the where input current the THD to hasbe faced. been experimentallyThe distortion evaluated.due to these After causes that, is in thereport followinged in the subsections, next section these where causes, the due THD to thehas power been processingexperimentally mechanism evaluated. of the After QR flyback that, in converter the following that are subsections, not ascribable these to thecauses, specific due control to the method power (althoughprocessing the mechanism control method of the may QR mitigateflyback them),converter are that described. are not ascribable to the specific control method (although the control method may mitigate them), are described.

3.2. Experimental Verification of the Input Current Distortion in a Hi-PF QR Flyback LED Driver A prototype of a Hi-PF QR flyback LED driver was set up to highlight the distortion occurring, even when a control method implementing the previous features is adopted. The main parameters of the converter are summarized in Table 1.

Energies 2020, 13, 2989 11 of 24

3.2. Experimental Veriﬁcation of the Input Current Distortion in a Hi-PF QR Flyback LED Driver A prototype of a Hi-PF QR ﬂyback LED driver was set up to highlight the distortion occurring, even when a control method implementing the previous features is adopted. The main parameters of the converter are summarized in Table1.

Table 1. Main characteristics of the Hi-PF QR ﬂyback converter.

Parameter Value

Input voltage range [Vac] 90–265 V Line frequency range [fl] 47–63 Hz Rated output voltage [Vout] 48 V Regulated dc output current [Iout] 700 mA Expected full-load eﬃciency [η] 86% Transformer primary inductance [Lp] 500 µH Reﬂected voltage [VR] 120 V Drain–Source capacitance [CDS] 150 pF

The prototype was built and its performance evaluated on the bench. The PF was greater than 0.98 over the input voltage range at full load. At 50% load, at low line it was nearly equal to that at 100% load, but at high line it dropped to about 0.97 at 230 Vac. Figure 11 depicts the experimental waveforms of the input current Iac(θ) (blue trace), the output voltage (red trace), the drain current of the MOSFET (green trace) and the current sense voltage reference for the controller (purple trace). The measurements were carried out at full load both at 110 Vac (60 Hz) and 230 Vac (50 Hz). From the experimental evidence, the shape of the measured Iac(θ) was very close to an ideal sinusoid waveform. It is worth noting that the harmonic contribution the prototype boards met the European norm EN61000-3-2 Class-C and Japanese norm JEITA_MITI Class-C, both of which are relevant to lighting equipment, at full load and nominal input voltage mains, as depicted in Figure 11. A small distortion in the input current is apparent in Figure 11 (blue trace) by looking at the zero crossing of the waveform. Therefore, as previously mentioned, notwithstanding the enhanced performance of a Hi-PF QR flyback LED driver, there are still different inherent causes of distortion in the input current that are not ascribable to the specific control method. The causes are the ringing current, the crossover distortion due to transformer leakage inductance and crossover distortion due to the input storage capacitor. Generally speaking, being the converter in a nonlinear system, the overall THD of the input current cannot be the sum of each individual distortion contribution. Due to the number and the complexity of the distortion causes and, above all, due to the complexity of their mutual multi-interactions, the only way to have a sensible estimate of the overall result in terms of input current THD is to resort to simulations. It is worth noting that the distortion of the input current Iac(θ) caused by the ringing effect has already been studied theoretically for boost PFC [55–57], while in the case of a Hi-PF QR flyback it has not yet been treated. Furthermore, the crossover distortion due to the input storage capacitor in the case of a Hi-PF flyback LED driver has never been studied. Finally, the crossover distortion due to transformer leakage inductance has also been theoretically analyzed in depth. Energies 2020, 13, 2989 12 of 24 Energies 2020, 4, x FOR PEER REVIEW 12 of 24

Measured value JEITA-MITI Class-C limits Measured value EN61000-3-2 Class-C limits

1 1

0.1 0.1

0.01 0.01 Har m onicCur rent[A] Harmonic Current [A] Current Harmonic 0.001 0.001

0.0001 0.0001 123579111315171921232527293133353739 1 2 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Harmonic Order [n] Harmonic Order [n]

(a) (b)

Figure 11. Steady-state waveforms at full-load: (a) V = 110 V at 60 Hz. (b) V = 230 V at 50 Hz. Figure 11. Steady-state waveforms at full-load: (a) Vin =in 110 V at 60 Hz. (b) Vin = 230in V at 50 Hz. They haveThey havebeen beenoverlapped overlapped by the by theharmonic harmonic contributi contributionsons of ofthe the input input current current (green (green bars) bars) with with the Japanese norm JEITA_MITI Class-C ( a), and European norm EN61000-3-2 Class-C ( (b)) (red bars) for lighting equipment. 3.3. Distortion Caused by the Ringing Current 3.3. Distortion Caused by the Ringing Current In a flyback, the drain-to-source voltage rings as soon as the secondary winding is fully In a flyback, the drain-to-source voltage rings as soon as the secondary winding is fully demagnetized. The energy is exchanged between the total capacitance CDS of the drain node demagnetized. The energy is exchanged between the total capacitance CDS of the drain node and the and the primary inductance of the flyback transformer Lp. For the sake of simplicity, assumption 3 primary inductance of the flyback transformer Lp. For the sake of simplicity, assumption 3 has been has been assumed, and consequentially it has been carried out an equivalent circuit of the QR flyback assumed, and consequentially it has been carried out an equivalent circuit of the QR flyback during during the time interval Tneg, which is the duration of the negative portion of the primary current, Ip. the time interval Tneg, which is the duration of the negative portion of the primary current, Ip. The The duration of the positive portion, Tpos, of the primary current is equal to TON when the current in duration of the positive portion, Tpos, of the primary current is equal to TON when the current in the the turn-on instant of the power switch is zero. The simplified circuit model and the key waveforms turn-on instant of the power switch is zero. The simplified circuit model and the key waveforms are are depicted in Figure 12. depicted in Figure 12. In Figure 12a, the voltage across Lp is equal to the reflected output voltage VR, which is almost In Figure 12a, the voltage across Lp is equal to the reflected output voltage VR, which is almost constant during TFW, according to the previous assumption 6. After that, the voltage across Lp and CDS constant during TFW, according to the previous assumption 6. After that, the voltage across Lp and CDS and the current trough them start to oscillate. More specifically, the analytical expressions of VDS(t) and the current trough them start to oscillate. More specifically, the analytical expressions of VDS(t) and Ip(t) in the time interval Tneg, can be written as: and Ip(t) in the time interval Tneg, can be written as:  t  Vin(t) + VR cos 2tπ 0 < t Tz V (t) = Vt()+<≤ Vcos 2π Tr ≤ 0 tT (8) DS =  in R z Vt()  0 Tr Tz < t Tneg (8) DS  ≤  <≤ 0 TtTzneg   t  YLVR sin 2π 0 < t Tz ( ) =  − Tr ≤ Ip t  Vin (9)  Ip(Tz) + t Tz < t Tneg Lp ≤

Energies 2020, 4, x FOR PEER REVIEW 13 of 24

 t −<≤YVsin 2π 0 t T LRT z =  r Itp ()  (9) V Energies 2020, 13, 2989 I ()Tt+<≤in TtT 13 of 24  pz z neg  Lp s C Y = CDSDS YL =L (10)(10) LpLp wherewhereY LYLis is the the characteristic characteristic admittance admittance of of the theC DSCDS-L-Lp ptank tank circuit circuit and andT Tz zis is the the time time interval interval needed needed forfor the theV VDSDSto to fallfall zerozerowhen whenV Vinin < VVRR; ;that that is: is:  TπTz =− VVTRinzcos 2z () (11) VR cos 2π T= Vin(Tz) (11) Tr r −

WhenWhen QR-mode QR-mode is adopted,is adopted, the devicethe device can becan turned be turned on, but on, according but according to assumption to assumption 5, the current 5, the current must reach zero. Tzz is the time interval needed for the primary current Ip to ramp linearly must reach zero. Tzz is the time interval needed for the primary current Ip to ramp linearly until zero until zero from the current value Ip(Tz). In DCM operation, considering that Ip oscillates around zero, from the current value Ip(Tz). In DCM operation, considering that Ip oscillates around zero, and that ringingand that is damped,ringing is afterdamped, a few after ringing a few cycles, ringing assumption cycles, assumption 7 can be considered 7 can be considered exactly true; exactly with true; QR operation,with QR operation, the negative the current negative just current after demagnetization just after demagnetization is not compensated is not compensated by subsequent by subsequent positive positive contributions as in DCM. Considering zero the average value of Ip, as per assumption 7, is contributions as in DCM. Considering zero the average value of Ip, as per assumption 7, is already a already a better approximation as compared to totally neglecting TR (i.e., assuming the operation is better approximation as compared to totally neglecting TR (i.e., assuming the operation is exactly at theexactly boundary at the between boundary DCM between and CCM). DCM However, and CCM). in thisHowever, context, in assumption this context, 7 isassumption just a simpliﬁcation, 7 is just a whosesimplification, impact on whose the shape impact of on the the input shape current, of the quantitativelyinput current, quantitatively expressed by itsexpressed THD, needs by itsto THD, be assessed.needs to be Being assessed. the ringing Being the current ringing at thecurrent turn-on at th instante turn-on of instant the power of the switch power equal switch to equal the initial to the currentinitial current during theduring on-time the on-time interval, interval, this current this maycurrent or maymay notor may be zero, not be which zero, has which an impact has an onimpact the shapeon the of shape the input of the current input and,current consequently, and, consequently, on its THD on too.its THD too. When Vin > VR, in QR-mode, the turn on occurs at the first valley of the ringing in the drain voltage When Vin > VR, in QR-mode, the turn on occurs at the ﬁrst valley of the ringing in the drain that follows the transformer demagnetization. Therefore, in this case, Tneg is equal to Tr/2. voltage that follows the transformer demagnetization. Therefore, in this case, Tneg is equal to Tr/2.

(a) (b)

FigureFigure 12. 12. (a)(a) Simplified Simpliﬁed equivalent equivalent circuit circuit during during TnegTneg , (b,() keyb) keywaveforms waveforms in QR-mode in QR-mode and DCM and DCMoperations. operations.

ByBy neglecting neglecting the the input input capacitor capacitorC Cs sfrom from the the QR QR ﬂyback flyback schematic schematic in in Figure Figure7, 7, the the following following considerationsconsiderations are are valid: valid: duringduring TTonon aa charge charge QQpospos isis provided provided from from the the input input source source and and stored stored in inthe thetransformer; transformer; during during TFWT, FWthe, energy the energy is mostly is mostly delivered delivered to the to output; the output; finally, ﬁnally, in Tneg in, aT negativeneg, a negative charge charge Qneg is returned to the input source. The average input current during a switching cycle can therefore be expressed as: D E Qpos Qneg I = − (12) p T Energies 2020, 13, 2989 14 of 24

As shown in Figure9, the positive charge Qpos is clearly given by:

1 Q = I T (13) pos 2 pkp on

To evaluate the absolute value of the negative charge Qneg, it is necessary to consider Equations (8)–(10), that describe the VDS(t) and Ip(t) variations in the time interval Tneg (when QR operation is only considered) and distinguish two cases.

1. Vin > VR. In the time interval (0, Tneg) VDS(t) is always greater than zero and the current Ip(t) is sinusoidal; Tneg equals half the ringing period. The average value of Ip(t) during Tneg is 2/π times the negative peak value |Ivyp| = YLVR, therefore:

2 Tr Qneg = Tneg Y V = Y V = 2V C (14) π L R π L R R DS

2. V V . The current Ip(t) is sinusoidal in the subinterval (0, Tz). Tz can be expressed as: in ≤ R Tr 1 1 Vin Tz = 1 cos− (15) 2 − π VR

and the current Ip(t) evaluated when t = Tz is: s 2 Vin Ip(Tz) = YLVR 1 (16) − − VR

As depicted in Figure 12b, in the time interval Tzz, Ip(t) ramps up linearly to zero. Hence, Tzz can be expressed as: L p Tzz = Ip(Tz) (17) Vin

Since Tneg is the sum of Tz and Tzz, it can be written:  s   2  Tr  VR Vin 1 Vin  Tneg = Tz + Tzz = 1 + 1 cos−  (18) 2  Vin − VR − VR  which is always greater than Tr/2, except when Vin = VR. Finally, Qneg is given by the sum of the two contributions, Qneg1 during the subinterval (0, Tz) and Qneg2 during the subinterval (Tz, Tneg). After some mathematical steps, Qneg can be evaluated as:

Z Tz Z Tneg 2 t Vin 1 (Vin + VR) Qneg = Qneg1 + Qneg2 = YLVR cos 2π dt + Ip(Tz) + tdt = CDS (19) 0 Tr Tz Lp 2 Vin

Considering Equation (12), the overall input current Iin(θ) can be found by adding the contributions obtained in Equations (7) and (13), which also takes into account the ringing oscillations along each line half-cycle. Hence it can be written as:

 T (θ)  1 I (θ) on 2 V C V > V  2 pkp T(θ) T(θ) R DS in R Iin(θ) = − 2 (20)  1 Ton(θ) 1 (Vin+VR)  Ipkp(θ) CDS Vin VR 2 T(θ) − 2T(θ) Vin ≤ It is evident that the positive term in Equation (20) does not introduce any distortion, provided that the generic control method provides Ipkp(θ), as in Equation (6), and satisfies Equation (4). The contribution of the ringing current, related exclusively to the negative terms, takes into account Energies 2020, 13, 2989 15 of 24 the ringing contributions derived in Equations (14) and (19). They represent a twofold effect: they downwards offset the input current waveform, which produces crossover distortion and, considered that the offset is a function of the instantaneous line voltage, they distort its shape as well. It is worth noting that the distortion contribution is not ascribable to the control method; that is, it is not due to the control but is inherent in the power processing mechanism of any Hi-PF QR flyback converter. On the other hand, the control method may compensate or mitigate it. First of all, the contribution of the ringing current can be reduced by lowering the switching frequency (i.e., a longer T(θ), obtained with a larger Lp value) and using a low reflected voltage VR. However, a larger Lp value implies a bigger flyback transformer, and a lower VR increases the primary rms current and, consequently, the conduction losses. A trade-off is therefore required. A smaller CDS also helps to reduce the ringing contribution, however, it is important to underline that a tradeoff between switching losses and EMI should be found. In fact, the lower the CDS, the faster the VDS transient at turn-off, and this faster transient may adversely affect both the efficiency of the power switch, as well as the EMI. LED driver applications are typically specified to accommodate a certain range of output voltages Vout to power different types and lengths of LED string. Thus the contribution of the ringing current is expected to be maximum at the upper end of the Vout range and minimum at the lower end of the Vout range. In fact, lowering Vout will simultaneously reduce VR and increase T(θ).

3.4. Crossover Distortion Due to the Input Capacitor The storage capacitor Cs placed downstream of the input bridge, as depicted in Figure7, is part of the EMI filter, which is always needed in an SMPS connected to the power line to restrict the conducted emission within the limits envisaged by the relevant EMI regulations. The capacitor has a twofold effect: it contributes to the voltage–current phase-shift and worsens the THD by maintaining a residual voltage on the dc side of the input bridge rectifier. The latter causes a non-conduction zone as the line voltage approaches zero, therefore, assumption 2 is no longer valid for the following analysis. This phenomenon also occurs for systems without a PFC and is an additional source of crossover itself, which interacts with the other distortion mechanisms. A quantitative analysis of this crossover distortion can be carried out independently from the topology employed in a PFC. In these terms, any PFC and, consequently, the Hi-PF QR flyback converter can be modeled with an equivalent resistor (Req) that can be expressed as:

2 VPK Req = (21) 2Pin where VPK is the peak of the rectiﬁed line voltage Vin(θ) and Pin is the input power of the converter. Generally speaking, the dead zone in Iac(θ) increases at large CS values, high line voltage and low load (large Req). In quantitative terms, the dead zone starts, near the zero-crossing, when the rate of fall in the line voltage Vac(θ) exceeds the rate of the voltage Vin (θ) across CS, limited by the time constant Req CS, so that from that instant on it is Vac(θ) < Vin(θ), as shown in Figure 13. Focusing on the Vac(θ) and Vin(θ) near the zero-crossing zone, shown in Figure 13a, and neglecting the voltage drop across the input bridge rectiﬁer, the phase angle α at which the slope of the two voltages are equal can be found with the following relation:

dvac(θ) dvin(θ) 1 = 2π fLVPK cos α = VPK sin α (22) dθ dθ ⇒ ReqCs

Solving for α: tan α = 2π fLReqCs (23) EnergiesEnergies 20202020,, 4,13 x, 2989FOR PEER REVIEW 1616 of of 24 24

0.4 π − α π π + β Rectified π + β sinusoid 0.1 Rectified 0.3 Exponential Exponential (θ) (θ) in

in sinusoid V Expon. linear V Vin(θ) 0.09 θ approximation Vin1 ( ) V (θ) 0.2 Vin(θ) in1 Sine linear Expon. linear approximation |Vac(θ)| Line s egment 0.08 approximation 0.1 Normalized Normalized Normalized Normalized Rectified |Vac(θ)| sinusoid π + βa 0.07 0 2.8 3 3.2 3.4 3.2 3.22 3.24 θ [rad] θ [rad]

(a) (b)

Figure 13. Detail of Vac(θ) (red trace) and Vin(θ) (blue trace) near voltage zero-crossing (a), and zoomed Figure 13. Detail of Vac(θ) (red trace) and Vin(θ) (blue trace) near voltage zero-crossing (a), and zoomed view of the yellow area, intersection of Vac(θ) and Vin(θ)(b). Values are normalized to VPK. view of the yellow area, intersection of Vac(θ) and Vin(θ) (b). Values are normalized to VPK.

From π-α on, Vin(θ) follows an exponential decay until the phase θ is equal to π + β, where it is From π-α on, Vin(θ) follows an exponential decay until the phase θ is equal to π + β, where it is again |Vac(θ)| V (θ), which marks the end of the dead zone. In this interval the expression of V (θ) ≥≥ in in againcan be |V assumedac(θ)| V as:in(θ), which marks the end of the dead zone. In this interval the expression of Vin(θ) θ (π α) can be assumed as: − − Vin(θ) = sin αe− tan α (24) θ−−()πα − Beyond the angle π + β, Vac(θ) and V (θ)= are againα tan bothα sinusoidal and overlapping until they(24) Veinin()θsin have a phase shift equal to α away from the next zero-crossing. The duration of the dead zone is clearly α + βBeyond. the angle π + β, Vac(θ) and Vin(θ) are again both sinusoidal and overlapping until they have Thea phase phase shift angle equalβ can to beα away found from at the the intersection next zero-crossing. of an exponential The duration curve with of the a sinusoidal dead zone one, is clearlywhich resultsα + β. in a transcendental equation with no closed-form solution. Since β is small, it is possible The phase angle β can be found at the intersection of an exponential curve with a sinusoidal one, to find an approximate solution βa, substituting the last part of the exponential function (from θ = π to whichθ = π +resultsβ) with in a its transcendental expansion in Taylorequation series with to no the closed-form first order (solution.e θ 1 Sinceθ). In β a is similar small, way it is possible it can be − ≈ − todone find for an the approximate rectified sinusoid solution (sin βa,θ substitutingθ). Therefore, the last the valuepart ofΛ theof theexponential input voltage, function evaluated (from θ when = π ≈ −θ tothe θ phase= π + β angle) with is itsπ, expansion can be computed in Taylor as series follows: to the first order (e ≈ 1−θ). In a similar way it can be done for the rectified sinusoid (sin θ ≈ θ). Therefore, the value Λ of the input voltage, evaluated when the phase angle is π, can be computed as follows: α sin α Λ = Vin(π) = sin αe− tan α (25) α ≈ e − sinα Λ=πα =tanα ≈ Vein () sin (25) As shown in Figure 13b, at the intersection of the two straighte lines, the phase angle is π + βa.

 a As shown in Figure 13b, at the intersection of the twoθ straightπ lines, the phase angle is π + β .  V (θ) = Λ 1 −  in1 − tan α   (θ) θ θ −π  VVin2 ()θ1==Λπ − (26)  in1 −α  Λ tanα tan βa =  ()=−Λ+tanπ α Vin2 θθ (26) The truncation of the exponential Equation (24) to the ﬁrst order introduces an underestimation, Λ tanα as depicted in Figure 13b, hence Equation (26)β provides= an approximated value βa < β. On the other a hand, if Equation (24) is approximated to the secondΛ+ order,tanα which provides a better approximation of EquationThe truncation (24), an overestimation of the exponential will resultEquation and (24) the approximatedto the first order value introducesβa > β isan less underestimation, than 4% larger asthan depicted the value in Figure provided 13b, by hence Equation Equati (26),on (26) which provides proves an that approximated the accuracy value of the β ﬁrsta < β. assumption On the other is hand,valid anyway.if Equation (24) is approximated to the second order, which provides a better approximation of equationThe previous 24, an overes analysistimation shows thatwill theresult storage and the capacitor approximatedCS can be value assumed βa > β as is a less source than of 4% crossover larger thandistortion, the value even provided if Req is aby real Equation resistor. (26), The which key point proves is the that inability the accuracy of Vin(θ of) to the keep first pace assumption with Vac (isθ) validon the anyway. falling edge of the sinusoid, due to the maximum discharge rate of CS through Req. However, its netThe impact previous is the analysis result of shows the interaction that the withstorage the othercapacitor sources CS ofcan distortion, be assumed previously as a source analyzed. of crossoverIn fact, distortion, it has been even assumed if Req is that a real a resistor resistor.Req Thethat key represents point is the the whole inability converter, of Vin( whichθ) to keep implicitly pace withmeans Vac that(θ) on the the ratio falling of Vin edge(θ) to ofIin the(θ) sinusoid, is constant due in (0,to πthe); i.e., maximumIin(θ) is puredischarge sinusoidal rate of in C (0,S throughπ). Actually, Req. However,the distortion its net of impact the current is the shape, result causedof the interaction by all the previouslywith the other considered sources of sources, distortion, tends previously to reduce analyzed.

Energies 2020, 4, x FOR PEER REVIEW 17 of 24

In fact, it has been assumed that a resistor Req that represents the whole converter, which Energiesimplicitly2020 ,means13, 2989 that the ratio of Vin (θ) to Iin(θ) is constant in (0, π); i.e., Iin(θ) is pure sinusoidal17 in of (0, 24 π). Actually, the distortion of the current shape, caused by all the previously considered sources, tends to reduce Iin(θ) with respect to the undistorted case. Instead, considering the same model θ Iapproach,in( ) with respectif it can tobe the assumed undistorted that the case. Req Instead,changes consideringalong the sinusoid, the same then model Req = approach, Req(θ) = V ifin( itθ)/ canIin(θ be). θ θ θ assumedFigure 14 thatshows the RReqeq(θchanges) obtained along by dividing the sinusoid, a sinusoidal then Req input= Req (voltage) = Vin and( )/ Ithein( current). Figure drawn 14 shows that Rtakeseq(θ) into obtained account by dividingthe distortion a sinusoidal caused inputby the voltage ringing and current the currentin Equation drawn (20). that takes into account the distortion caused by the ringing current in Equation (20).

× 5

] 110 Ω [ Pin = 40 W eq R

4 110×

V = 115 V V = 230 V 3 ac ac 110×

Equivalent resistance 100 0 1 2 3 θ [rad]

Figure 14. Variation of Req(θ) along the sinusoid for the prototype converter caused by the Figure 14. Variation of Req(θ) along the sinusoid for the prototype converter caused by the ringing ringing current. current.

Req(θ) is almost ﬂat for a wide range of the sinusoid except when the current is approaching the Req(θ) is almost flat for a wide range of the sinusoid except when the current is approaching the zero crossings. The increase in Req(θ) produces an early dead zone with respect to the case with a ﬁxed zero crossings. The increase in Req(θ) produces an early dead zone with respect to the case with a Req. The corresponding value of α would be the solution of the equation: fixed Req. The corresponding value of α would be the solution of the equation: απ== ( () α) tantanα 2 2π fLfRLeqRCeq α Cs s (27)(27)

TheThe solutionsolution ofof EquationEquation (27) mustmust be defineddefined in the intervalinterval (π//2,2, ππ),), wherewhere tantan αα hashas aa finitefinite valuevalue andand consequentiallyconsequentially alsoalso RReqeq(α). As As a result, the deaddead zonezone causedcaused byby thethe inputinput capacitorcapacitor CCss startsstarts beforebefore RReqeq(θ)) diverges,diverges, andand IIacac(θ) mustmust crosscross zerozero beforebefore IIinin(θ).). In other words, asas longlong asas thethe bridgebridge rectifierrectifier conducts,conducts, IIacac(θ)) isis thethe sumsum ofof IIinin(θ)) andand thethe currentcurrent throughthrough CCSS,, whichwhich isis essentiallyessentially aa sinusoidsinusoid leadingleading byby 9090 degrees.degrees. ThisThis causescauses IIacac(θ)) toto leadlead IIinin(θ),), whereaswhereas withoutwithout CCSS theythey wouldwould bebe essentiallyessentially coincident.coincident. WithinWithin thethe deaddead zone,zone, the the relation relationR Reqeq((θθ))= = VVinin(θ)//Iin((θθ)) is is no no longer longer the one shown in the diagramdiagram ofof FigureFigure 1414 becausebecause ofof bothboth thethe voltagevoltageV Vinin((θθ)) andand thethe IIinin((θ)) areare notnot sinusoidal.sinusoidal. Besides,Besides, thethe voltagevoltage retainedretained byby CCss,, whichwhich makesmakes VVin((θθ)) > |V |Vac((θθ)|)|,, andand TTONON(θ) and TT((θ)) willwill bebe muchmuch smallersmaller thanthan thosethose predictedpredicted duringduring thethe controlcontrol methodmethod design,design, whichwhich waswas carriedcarried outout assumingassuming aa sinusoidalsinusoidal inputinput voltage,voltage, asas perper EquationEquation (5).(5). The resultingresulting switchingswitching frequencyfrequency maymay bebe higher.higher. As long asas somesome energyenergy is delivered to to the the output, output, CSC (whichS (which is the is the only only source source of en ofergy, energy, since since no current no current comes comes from fromthe reverse-biased the reverse-biased bridge bridge rectifier), rectifier), keeps keepson discharging on discharging and as and long as as long Iin(θ as) >I in0.( Ifθ) the> 0. peak If the current peak currentbecomes becomes lower than lower the than critical the criticalpeak primary peak primary current currentfor no energy for no energytransfer, transfer, the energy the energyis no longer is no longertransferred transferred to the tooutput the output and andbounces bounces back back and and forth forth between between CSC andS and LpL, pexcept, except for for the the energyenergy dissipateddissipated duringduring thisthis process.process. AsAs aa consequence,consequence,C CSS isis dischargeddischarged atat aa lowerlower rate.rate. Instead,Instead, ifif thethe peakpeak currentcurrent doesdoes notnot gogo belowbelow thatthat criticalcritical value,value, thethe dischargedischarge raterate ofofC CSS hashas nono change.change. ≥ WhenWhen| V|Vacac(θ(θ)|)| V V(θin()θ and) and the the dead dead zone zone ends ends at θ = atβ ,θ it = is β possible, it is possible to observe to anobserve abrupt an variation abrupt ≥ in invariationIac(θ). This in Iac is(θ caused). This is by caused the bridge by the rectifier bridge rectifier that is forward-biased, that is forward-biased, as Iac(θ as) has Iac(θ to) has transition to transition from zerofrom to zeroIac( βto), I whichac(β), which is already is already greater greater than zerothan becausezero because of the of leading the leading phase phase of Iac( θof). Iac(θ). ItIt isis worthworth mentioningmentioning that,that, nearnear thethe zerozero crossingcrossing interval,interval, thethe inputinput voltagevoltage isis farfar fromfrom constantconstant inin aa switchingswitching cyclecycle andand thethe switchingswitching frequencyfrequency maymay comecome closeclose toto thethe frequencyfrequency relatedrelated toto thethe S p in resonanceresonance betweenbetweenC S Cand andLp. So,L . the So,V inthe(θ) hasV ( sinusoidalθ) has sinusoidal behavior, ratherbehavior than, constant.rather than Additionally, constant. theAdditionally, resonance the of the resonance inductors of andthe capacitorsinductors and in the capacitors EMI filter in can the also EMI be filter stimulated, can also so be that stimulated, different

Energies 2020, 4, x FOR PEER REVIEW 18 of 24 so that different resonance phenomena may coexist. This means that the modeling approach used throughout the present discussion is not always valid in describing the practical behavior.

3.5. Crossover Distortion Due to Transformer’s Leakage Inductance

EnergiesIn practical2020, 13, 2989 cases, the real transformer’s windings are not perfectly coupled. This phenomenon18 of 24 causes a crossover distortion that affects the input current shape and degrades the THD. Therefore, in the following subsection, the analysis of the distortion no longer takes into account assumptions 3 andresonance 4. phenomena may coexist. This means that the modeling approach used throughout the presentIn a real discussion transformer, is not a always portion valid of the in energy describing stored the in practical the primary behavior. winding cannot be transferred to the secondary winding because of imperfect magnetic coupling. In other words, this can be 3.5. Crossover Distortion Due to Transformer’s Leakage Inductance modeled by considering the primary inductance Lp split into two distinct elements: the magnetizing inductanceIn practical LM perfectly cases, the coupled real transformer’s to the secondary windings winding are not perfectly and the coupled. leakage This inductance phenomenon Llk (uncoupled).causes a crossover In these distortion terms, it is that useful affects to define the input the currentcoupling shape coefficient and degrades σ such that the L THD.M = σ L Therefore,p and Llk =in (1− the σ) followingLp. Typical subsection, values of σ the range analysis from of0.95 the to distortion 0.99. no longer takes into account assumptions 3 andThe 4. energy stored in the leakage inductance is not transferred to the output and this energy wouldIn overcharge a real transformer, CDS well a over portion Vin( ofθ) the + V energyR, in most stored cases in theexceeding primary the winding voltage cannot rating be of transferred the power to switch.the secondary A typical winding solution because is to use of imperfecta clamp circuit magnetic which coupling. limits V InDS other(t) at words,a well-defined this can and be modeled properly by selectedconsidering value the VCL primary (> VR) above inductance Vin(θ).L Figurep split into15 shows two distinct the equivalent elements: circuit the magnetizing of the Hi-PF inductance QR flybackLM converterperfectly during coupled the to theoff time secondary and the winding current and waveforms the leakage underlining inductance theLlk effect(uncoupled). of Llk. In these terms, it isWhen useful the to define power the switch coupling is turned coeffi cientoff, theσ such primary that Lcurrent= σ L firstlyp and Lcharges= (1 theσ) L capacitancep. Typical values CDS, M lk − thusof σ therange VDS from(t) evolves 0.95 to until 0.99. the voltage of the magnetizing inductance LM, VM, equals –VR. During this timeThe interval energy T storedps, the inprimary the leakage current inductance Ip is flowing is not through transferred LM and to L thelk, so output there is and an thisinductive energy voltagewould divider. overcharge Therefore:CDS well over Vin(θ) + VR, in most cases exceeding the voltage rating of the power switch. A typical solution is to use a clampLL circuit which limits VDS(t) at a well-defined and properly VVVVt=−MM =()θ − () selected value V (> V ) above V MpinDS(θ). Figure 15 shows the equivalent circuit of the Hi-PF QR flyback(28) CL R in LLpp converter during the off time and the current waveforms underlining the effect of Llk.

(b) (a)

Figure 15. (a) Simpliﬁed equivalent circuit during oﬀ time. (b) Current in the real primary windings Figure 15. (a) Simplified equivalent circuit during off time. (b) Current in the real primary windings (blue trace) and in the secondary windings (black trace). (blue trace) and in the secondary windings (black trace).

When the power switch is turned off, the primary current firstly charges the capacitance CDS, In the instant when the voltage VM equals –VR, the voltage Vp across the primary winding can be thus the VDS(t) evolves until the voltage of the magnetizing inductance LM, VM, equals –VR. During this expressed as: time interval Tps, the primary current Ip is flowing through LM and Llk, so there is an inductive voltage divider. Therefore: L p 1 VVV== LM pRRLM σ (29) VM = Vp =LM [Vin(θ) VDS(t)] (28) − Lp Lp − Immediately after the time interval Tps, the current starts flowing through D1 and the energy In the instant when the voltage VM equals –VR, the voltage Vp across the primary winding can be starts being transferred to the output. VDS(t) keeps on ramping up until it reaches Vin (θ) + VCL in a time expressed as: TLK. After that, VDS(t) is clamped, Llk starts being demagnetized with a rate equal to (VR−VCL)/Llk and Lp 1 the current through D1 reduces at the sameV prate.= TheV portionR = V Rof the primary current Ip(t) of the leakage(29) LM σ inductance varies from the peak value, Ipkp(θ), until zero in a time interval equal to TLK(θ). Hence: Immediately after the time interval Tps, the current starts flowing through D1 and the energy starts being transferred to the output. VDS(t) keeps on ramping up until it reaches Vin (θ) + VCL in a time T . After that, V (t) is clamped, L starts being demagnetized with a rate equal to (V V ) L and LK DS lk R− CL / lk Energies 2020, 13, 2989 19 of 24

the current through D1 reduces at the same rate. The portion of the primary current Ip(t) of the leakage inductance varies from the peak value, Ipkp(θ), until zero in a time interval equal to TLK(θ). Hence:

VCL VR Ip(t) = Ipkp(θ) − t (30) − (1 σ)Lp −

As soon as Ip(t) is equal to zero, it can be written in the analytical expression of TLK:

(1 σ)Lp TLK(θ) = − Ipkp(θ) (31) V VR CL −

Bearing in mind the current that ﬂows in LM can be written as:

VR IM(t) = Ipkp(θ) t (32) − σLp

Meanwhile, the secondary current reaches its peak value Ipks(θ): " # VR 1 σ Ipks(θ) = nIM(TLK) = n 1 − Ipkp(θ) (33) − V VR σ CL − After that, the magnetizing inductance on the secondary side starts being demagnetized at a rate VR/LM until it zeroes in a time TDEM, as shown in Figure 15. The presence of Llk splits the time interval TFW in a ﬁrst subinterval TLK needed to demagnetize the leakage inductance (and magnetize the secondary winding) and a second subinterval TDEM needed to demagnetize the secondary winding. It is possible to prove that the resulting TFW can be expressed as:

LM Lp TFW(θ) = Ipkp(θ) = σ Ipkp(θ) (34) VR VR

(i.e., the one calculated neglecting the leakage inductance multiplied by the coupling coeﬃcient σ). The presence of Llk clearly compromises the input-to-output transfer energy, thus the dead zone where there is no energy transfer is expected to occur over a wider phase angle range of the line voltage. In fact, the “no energy transfer” condition concerns only the portion of the primary voltage Vp across the magnetizing inductance LM. Combining Equations (8), (9) and (28), it is possible to ﬁnd that the condition for no energy transfer, solved for Ipkp(θ), yields: s 2 VR 2 I (θ) YL V (θ) (35) pkp ≤ σ − in

The critical peak primary current for no energy transfer, expressed by Equation (35), is larger than the ideal case (σ = 1), thus conﬁrming the forecast of a wider dead zone related to this phenomenon. It is noteworthy to compare the critical value of Ipkp(θ) in Equation (35) to that determined by the ringing current after demagnetization and the relative positions of the dead zones they generate. Bearing in mind that the on time can be written as:

Lp TON = Ipkp (36) Vin

The region around zero crossings where there is no input-to-output energy transfer (Iin(θ) = 0), can be written by considering the combination of the input current in Equation (20) in the case of Vin < VR, and Equation (36): 2 (Vin + VR) CDS Vin Ipkp = (37) Vin Lp Ipkp Energies 2020, 13, 2989 20 of 24

Combining Equations (10)–(37), the ringing current after demagnetization related to the dead zone can be expressed as: I (θ) YL[V (θ) + VR] (38) pkp ≤ in Figure 16 shows the ratio of the critical value of Ipkp(θ) for no input-to-output energy transfer, Energies 2020, 4, x FOR PEER REVIEW 20 of 24 as in Equation (35), to that caused by the ringing current, as in Equation (38). The ratio is expressed as K a function of the parameter v: V K V= PK =v PK (39) Kv VR (39) VR where VPK is the peak value of the rectified line voltage Vin(θ). The dashed blue line represents the where VPK is the peak value of the rectiﬁed line voltage Vin(θ). The dashed blue line represents the idealideal case. case. As As is evidentis evident from from Figure Figure 16 16,, the the dead dead zonezone becomesbecomes even even larger larger near near the the zero zero crossings. crossings. Therefore, the lack of input-to-output energy transfer may override the dead zone caused by the Therefore, the lack of input-to-output energy transfer may override the dead zone caused by the ringing current needed for the demagnetization of the secondary winding in a time TDEM. ringing current needed for the demagnetization of the secondary winding in a time TDEM.

Figure 16. Ratio of the critical value of I (θ) for no input-to-output energy transfer in dead zone, Figure 16. Ratio of the critical value of pkpIpkp(θ) for no input-to-output energy transfer in dead zone, causedcaused by leakageby leakage inductance, inductan asce, inas Equation in Equation (35), (35), to that to that caused caus byed ringingby ringing current, current, as in as Equation in equation (38), as a38, function as a function of Kv forof Kvσ =for0.95. σ = 0.95. 4. Conclusions 4. Conclusions The standard limitations regarding the input current distortion in LED driver applications may The standard limitations regarding the input current distortion in LED driver applications may be stringent, especially in large lighting systems. Therefore, the choice of a proper LED driver be stringent, especially in large lighting systems. Therefore, the choice of a proper LED driver aiming aimingat current at current harmonics harmonics reduction reduction is crucial. is crucial. At the sa Atme the time, same the time,solution the must solution be economical, must be economical, compact compactand reliable. and reliable. TheThe QR QR flyback flyback shows shows the highestthe highest figure figure of merit of (FOM)merit (FOM) among among the diff theerent different LED driver LED solutions driver analyzedsolutions in theanalyzed literature. in the With literature. this aim, With this this work aim, this has work investigated has investigated and has analyzedand has analyzed the diff erentthe inherentdifferent causes inherent of distortion causes of in distortion the input currentin the in andput thecurrent power and processing the power mechanism processing ofmechanism the proposed of QRthe flyback proposed converter QR flyback that are converter not ascribable that are tonot the ascribable control method.to the control method. SeveralSeveral eff effects,ects, such such as as the the distortion distortion caused by by the the ringing ringing current, current, crossover crossover distortion distortion due dueto to transformertransformer leakage leakage inductance inductance and and crossover crossover distortion distortion due to due the to input the inputstorage storage capacitor, capacitor, have havebeen been analyzed analyzed in indepth. depth. It is It important is important to topoint point out out that that the the converter converter is isa nonlinear a nonlinear system system and, and, consequentially,consequentially, the the overall overall THD THD of theof the input input current current cannot cannot be evaluatedbe evaluated as aas simple a simple sum sum of the of THDthe associatedTHD associated to each individual to each individual contribution, contribution where the, where aforementioned the aforementioned issues are issues mutually are interactingmutually or partlyinteracting overlapping, or partly suchoverlapping, as those such creating as those a dead creating zone a in dead the linezone current. in the line current. SomeSome useful useful design design hints hints have beenhave extractedbeen extracte and discussedd and discussed in order in to provideorder to some provide suggestions some forsuggestions an optimized for design an optimized of a QR design flyback of LED a QR driver flyback converter. LED driver In detail, converter. the following In detail, list the summarizes following list summarizes all the possible precautions to bear in mind. all the possible precautions to bear in mind. 1. The impact of the ringing current after transformer demagnetization can be mitigated by 1. The impact of the ringing current after transformer demagnetization can be mitigated by lowering lowering the switching frequency, using a low reflected voltage VR or choosing a power MOSFET the switching frequency, using a low reflected voltage VR or choosing a power MOSFET with a with a RDS(on) with an optimized RDS(on)/Coss FOM. These criteria also help to reduce the phenomenon of the lack of input-to-output energy transfer near the zero crossings of the line voltage. 2. The leakage inductance of the transformer should be kept as low as practically possible. This choice essentially optimizes the converter efficiency but does not impact the reduction in the

Energies 2020, 13, 2989 21 of 24

RDS(on) with an optimized RDS(on)/Coss FOM. These criteria also help to reduce the phenomenon of the lack of input-to-output energy transfer near the zero crossings of the line voltage. 2. The leakage inductance of the transformer should be kept as low as practically possible. This choice essentially optimizes the converter eﬃciency but does not impact the reduction in the dead zones near the zero crossings of the line voltage caused by other phenomena (essentially, the input capacitor CS). 3. The input storage capacitor CS should be minimized to reduce the dead zone near the line voltage zero crossings and the current leap occurring in the proximity of the dead zone. However, particular attention should be paid to the following points.

a. The diodes of the input bridge rectifier are usually slow-recovery ones, so the primary current at the switching frequency may require an enhanced filter on the ac side of the bridge and may cause the diodes of the bridge to overheat. b. Close to the zero crossings, the switching frequency can be very low. If the ringing frequency related to CS and Lp is comparable with the switching one, it may generate current spikes that would degrade the current THD. 4. Class-X capacitors are generally used along with inductors for EMI filtering, necessary for the certification of the final product. Class-X capacitors can degrade the PF, although they do not contribute to the THD. From this perspective, on the one hand, the design of the filter must make the device compliant with the standards. On the other hand, there is a degree of freedom that can be exploited to minimize PF lowering at high line and light load. The filters should be designed with the largest inductance and the smallest capacitance practically possible.

Author Contributions: All authors contributed equally to this work. Writing-Review & Editing, C.A.; Writing-Review & Editing, G.G.; Writing-Review & Editing, A.R.; Writing-Review & Editing, S.A.R.; Writing-Review & Editing, G.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

Vac(t, θ) [V] Line voltage input. Vin(t, θ) [V] Rectified line voltage. VPK [V] Peak of the rectified line voltage Vin (θ). Iac(t, θ) [A] Line input current. Ip(t, θ) [A] Current on the primary windings. Is(t, θ) [A] Current on the secondary windings. Iout [A] Output constant current. Vout [V] Output constant voltage. LP [Ω] Inductance of the primary windings. Ls [Ω] Inductance of the secondary windings. Ton [s] On time of the power switch TFW [s] Time interval where the current flows on the secondary side. Tneg [s] Time interval where the drain–source voltage rings T [s] Time interval of the switching period. CDS [F] Drain to source capacitance of the MOSFET. VDS(t) [V] Drain to source voltage of the MOSFET. Ipkp(θ) [A] Current amplitude on the primary windings. VR [V] Reflected voltage. Tr [s] Time period of drain voltage ringing. Tz [s] Time interval needed for VDS to fall to zero. Tzz [s] Time interval needed for primary current to ramp linearly until zero. YL [Ω] Characteristic admittance of the CDS-Lp tank circuit. Qpos [C] Charge accumulates from the input source. Energies 2020, 13, 2989 22 of 24

Qneg [C] Charge returned to the input source. LM [Ω] Magnetizing inductance. TLK [s] Time interval needed to demagnetize the leakage inductance. σ Coupling coeﬃcient of Lm. Ipks(θ) [A] Current amplitude on the secondary windings. Req [Ω] Equivalent resistor of the converter. Pin [W] Input power.

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