Performance Comparison of PI and P Compensation in DSP-Based Average-Current-Controlled Three-Phase Six-Switch Boost PFC Rectifier

Transcription

1 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics erformace Comariso of ad Comesatio i DS-Based Average-Curret-Cotrolled Three-hase Six-Switch Boost FC Rectifier Abstract I this aer, it is show that i the three-hase sixswitch boost FC rectifier with average-curret cotrol with mismatched iut-voltage ad iut-curret sesig gais as well as offset errors i iut-voltage ad iut-curret sesig, the curret cotroller with roortioal () comesatio exhibits lower total harmoic distortio (THD) ad higher ower factor (F) comared with that of roortioal ad itegral () comesatio. It is also show that comesatio with iutvoltage feedforward is effective i imrovig outut-voltage trasiet resose with resect to iut-voltage chages oly if duty-cycle feedforward is also imlemeted. Fially, it is show that zero-sequece-sigal (ZSS) ijectio, i additio to eablig the outut-voltage regulatio i a wider iut-voltage rage, also imroves the THD of the iut currets. Idex Terms Three-hase six-switch boost FC rectifier, average-curret cotrol, comesatio, comesatio, voltage feedforward (VFF), duty-cycle feedforward (DFF), zerosequece sigal (ZSS) ijectio, mismatched sesig gais, offset errors, total harmoic distortio (THD), ower factor (F). I. INTRODUCTION Today, active three-hase FC rectifiers eed to meet very challegig erformace requiremets. I the majority of alicatios, the iut curret of active three-hase FC rectifiers is required to have a total harmoic distortio (THD) less tha % ad a ower factor (F) greater tha.99 []. Oe of the most cost-effective toologies that ca meet these requiremets is the three-hase six-switch boost FC rectifier [], which is usually imlemeted without eutral-oit coectio. May cotrol methods that ca achieve a high quality of iut currets i the three-hase six-switch boost FC rectifier are available [3], [4]. Geerally, aroaches usig direct cotrol of iut curret result i better quality of the iut currets comared to those usig direct ower cotrol []. Today, the cotrol circuit is usually imlemeted with digital techology. Oe direct curret cotrol method, well suited for digital imlemetatio, is average-curret cotrol [6], [7]. Fudametally, the average-curret cotrol of the threehase six-switch boost FC rectifier sulied from a threewire ower system, i.e., without eutral-oit coectio, ca be imlemeted with three ideedet curret cotrollers with a commo triagular carrier [], [8]. It was show i [] that although the three-hase currets i such a system are ot ideedet because without the eutral-oit coectio the sum of the hase currets is zero, the whole rectifier system is stable if the equivalet sigle-hase cotrol loos are stable. Laszlo Huber, Misha Kumar, ad Mila M. Jovaović By addig aroriate zero-sequece-sigals to the outut of the curret cotrollers, advaced cotrol methods ca be achieved which are equivalet to differet cotiuous ad discotiuous sace vector modulatio methods [9]-[]. I the average-curret cotrol, the outut-voltage cotroller is usually imlemeted with comesatio to make the outut voltage follow the outut-voltage referece with miimum error, whereas, the curret cotroller ca be imlemeted with [3] or comesatio []. The average-curret cotrol i most imlemetatios also icludes voltage feedforward (VFF), zero-sequece-sigal (ZSS) ijectio, ad duty-cycle feedforward (DFF). With VFF, it ca be achieved that the outut voltage is ractically isesitive to the lie-voltage variatios [6]. Geerally, i boost FC rectifiers, ZSS ijectio is emloyed to exted the iut-voltage regulatio rage ad/or to reduce the outut voltage. With ZSS ijectio, the maximum iut-voltage amlitude for a give outut voltage ca be icreased by u to % []. Fially, DFF is mostly emloyed whe curret cotrollers with comesatio are used to reduce the hase shift betwee a hase voltage ad hase curret ad, cosequetly, imrove the F [4]. It should be oted that if the outut-filter caacitace is very small, the low-frequecy rile i the outut voltage ca be cosiderable, resultig i icreased distortios i the iut currets. To reduce these distortios, the triagular carrier ca be modulated with the outut voltage waveform []. However, i alicatios where the outut-filter caacitace is relatively large such as i alicatios that require hold-u time, which are rimarily cosidered i this aer, the outut-voltage rile is geerally very small ad its effect o the iut-curret distortios is egligible. Although desig cosideratios for the curret cotrollers i three-hase rectifiers are almost idetical to those for siglehase rectifiers, the erformace of the three-hase three-wire rectifiers uder ubalaced coditios, caused by ubalaced iut voltages, mismatched sesig gais, ad offset errors i iut-voltage ad iut-curret sesig, ca be sigificatly differet from that of sigle-hase rectifiers due to the coulig of the iut hase currets. I this aer, it is show that i the three-hase six-switch boost FC rectifier with average-curret cotrol ad mismatched iut-voltage ad iut-curret sesig gais as well as offset errors i iut-voltage ad iut-curret sesig, the curret cotroller with comesatio exhibits better erformace, i.e., lower THD ad higher F, comared with that of the comesatio. It is also show that the DFF, i additio to reducig the hase shift betwee the resective iut voltages ad currets i the -comesatio imlemetatio, sigificatly imroves the effectiveess of VFF i the -comesatio imlemetatio. Fially, it is IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

2 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics v a v b v c.v Offset LINE VOLTAGE SENSING K vs ANTIALIASING FILTER.V Offset HASE CURRENT SENSING K cs show that ZSS ijectio imroves the THD of iut-currets i both curret cotrollers with ad comesatio. The oeratio with ad comesatio is illustrated with Matlab/Simulik simulatio waveforms ad exerimetally verified o a 3-kW rototye. II. OWER STAGE AND CONTROL CIRCUIT The simlified circuit diagram of the three-hase six-switch boost FC rectifier used i this study is show i Fig.. The evaluatio circuit was desiged with the followig basic secificatios: iut hase-voltage rage ± % Vrms lie-frequecy rage 4-6 Hz omial outut voltage 4 V maximum outut ower 3 kw The switches are imlemeted with a IGBT six-ack module [6]. The switchig frequecy is selected as f sw = khz, which is the maximum recommeded f sw for the selected IGBT module. The values of boost iductors ad outut filter caacitors are L a = L b = L c = mh ad C = C = 4 µf, resectively. The block diagram of the average-curret cotrol that is imlemeted with digital sigal rocessor (DS) TMS3F88 from TI [7] is also show i Fig.. For average-curret cotrol, the iut hase-to-hase voltages, hase currets, ad the outut voltage are sesed. The iut hase-to-hase voltages ad the outut voltage are sesed by usig differetial amlifiers, whereas, the hase currets are sesed by usig Hall sesors ad differetial amlifiers. As show i Fig., the sesig gai of the iut hase-to-hase voltages, hase currets, ad outut voltage are deoted as K vs, K cs, ad K d, resectively. The sesed voltages ad currets are coverted to digital sigals through the -bit aalog-digital coverter (ADC) of the DS. The iut-voltage rage of the ADC is -3 V, i.e., the full-scale rage = 3 V. Sice oly ositive voltages ca be alied to the i a i b i c LOW-ASS FILTER L a L b L c S a S b S c S a v a v ZSS ADC ADC S b DUTY-CYCLE FEEDFORWARD + ZSS INJECTION v a + v ZSS v ab v oref CURRENT CONTROLLER v ca i a D FFa 3 D ZSS i aerr S c hase A hase B hase C C C S a D a D CCa VOLTAGE FEEDFORWARD i aref K mab B C v a AVG C v a v b DWM A v c S a + v o - v EA OUTUT VOLTAGE SENSING K d ANTIALIASING FILTER ADC VOLTAGE CONTROLLER v oref iut of the ADC, the biolar hase-to-hase voltages ad the bidirectioal hase currets are scaled to ±/ ad level shifted by /. The outut sigals of the DS are the digital WM (DWM) gate sigals for the bottom switches S x, xϵ{a,b,c}. The gate sigals for the uer switches are the comlemetary sigals of the bottom switches. The DWM is imlemeted with a triagular carrier, i.e., u-dow couter. As the clock frequecy of the DS is f sysclock = MHz, the eak value of the triagular carrier (i.e., the maximum couter value) is C k = f sysclock /(f sw ) =. All the sesed sigals are samled at the eak of the triagular carrier. The samlig frequecy fs = fsw. As show i Fig., the average-curret cotrol is imlemeted with VFF rereseted by the K m AB/C block [6]. The voltage cotroller is imlemeted with a adative comesatio so that the voltage-loo badwidth is Hz i steady-state oeratio to meet the strict hase-curret THD requiremets ad Hz durig load trasiets to reduce the outut voltage overshoot ad udershoot. The roer comesator is determied by moitorig the outut-voltage error. Wheever the absolute error exceeds. V, the cotroller chages from the low-badwidth to the highbadwidth comesator. To avoid cotroller-outut boucig, the chage from the high-badwidth to the low-badwidth comesator is imlemeted with a.-v hysteresis, i.e., the chage occurs whe the absolute error falls below.6 V. The arameters of the adative digital comesator (z-domai arameters) are K V,LBW = 3., K iv,lbw = ad K V,HBW = 3.9, K iv,hbw = for the -Hz ad -Hz badwidth, resectively. The curret cotroller with ad comesatio is desiged so that the curret-loo badwidth is -khz ad.-khz, resectively, which both result i 4 o hase margi. The comesatio also icludes a coditioal ati-widu imlemetatio [8]. The z-domai arameters of the ad comesator are K I, = 64, K ii, = 4 ad K I, = 3337, resectively. The desig of the v o v ZSS ZERO-SEQUENCE SIGNAL (ZSS) GENERATOR DS otatio for digital values Fig. Simlified circuit diagram of ower stage ad block diagram of cotrol circuit. i o IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

3 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics V a Vb Vc THDc=.% THDa=.% THDb=.% - - THDc=8.7% THDc=4.68% ϕ~6.º w/ VFF t[sec] w/o DFF, ZSS Fig. Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (Vrms, kw) with comesatio (with VFF, without ZSS ijectio ad DFF), without offset errors i sesig : hase voltages [V]; iductor currets [A] at matched sesig gais, mismatched curret-sesig gais (K csa =.9K csb, K csb =K csc ), mismatched voltage-sesig gais (K vsab =.9K vsbc, K vsbc =K vsca ). curret ad voltage cotrollers is outlied i Aedix A. It is also show i Fig. that the cotrol imlemetatio icludes ZSS ijectio ad DFF. Ijected ZSS value v ZSS is obtaied as the egative average of the ositive ad egative eveloes of the iut hase voltages (symmetrical ZSS) [], i.e., as ZSS max( v a,v b,v c ) mi( v a,v b,v c v ) ev v ev v.() The value of DFF sigal D FFx for each hase is obtaied directly from the corresodig hase voltage. It should be oted that without DFF, the ZSS sigal ca also be obtaied from the outut sigals of the curret cotrollers D CCx. However, whe DFF is emloyed, the duty cycle of the switches i steady-state oeratio is domiated by the DFF sigals ad, therefore, the ZSS ijectio obtaied from the outut sigals of the curret cotrollers is ot effective. As show i Fig., the duty cycle of the bottom switches is obtaied as vx vzss vx vzss Dx DFFx DZSS DCCx D CCx voref voref, x { a, b, c}, () where outut of curret cotroller DCCx is much smaller tha the DFF+ZSS term D FFx D ZSS. Fially, it should be oted that i order to meet the deadtime requiremets for the IGBT module [6] ad takig ito THDa=4.4% THDa=3.64% Vrms, kw THDb=4.84% THDb=3.46% V a Vb Vc w/ VFF t[sec] w/o DFF, ZSS Fig. 3 Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (Vrms, kw) with comesatio (with VFF, without ZSS ijectio ad DFF), without offset errors i sesig: hase voltages [V]; iductor currets [A] at matched sesig gais, mismatched curret-sesig gais (K csa =.9K csb, K csb =K csc ), mismatched voltage-sesig gais (K vsab =.9K vsbc, K vsbc =K vsca ). accout the roagatio delay times of the otocoulers i the iterface circuit betwee the DS ad IGBT module [9], the duty-cycle rage is limited from D mi,lim =.7 to D max,lim =.93. III. ERFORMANCE COMARISON OF CURRENT CONTROLLERS WITH AND COMENSATION A. Steady-State Oeratio First, steady-state erformace comariso of curret cotrollers with ad comesatio imlemeted oly with VFF, i.e., without ZSS ijectio ad DFF, is doe. The comariso is erformed for balaced iut voltages for both matched ad mismatched sesig gais, ad without offset errors i iut-voltage ad iut-curret sesig. As a examle, simulated waveforms of hase voltages ad iductor currets at omial hase voltage of Vrms ad -kw load with ad comesatio are reseted i Figs. ad 3, resectively. The iductor-curret waveforms i Figs. ad 3 are obtaied with matched sesig gais (K vsab = K vsbc = K vsca ad K csa = K csb = K csc ), i Figs. ad 3 with mismatched curret-sesig gais (K csa =.9K csb, K csb =K csc ), ad i Figs. ad 3 with mismatched voltage-sesig gais (K vsab =.9K vsbc, K vsbc =K vsca ). With matched sesig gais, the waveform of the iductor currets with both ad comesatio is almost erfectly siusoidal with THDs slightly greater tha %. However, as show i Fig., the iductor currets with comesatio are sigificatly hase shifted with resect to the hase voltages (ϕ ~6. o ), which results i a reduced F. To exlai THDc=.8% THDa=.8% THDb=.8% Vrms, kw THDc=.% THDa=.8% THDb=.8% THDc=.% THDa=.96% THDb=.6% IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

4 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics the origi of the hase shift, Fig. 4 shows the iut ad outut waveforms of the curret cotroller. To achieve a siusoidal curret i hase x, where xє{a, b, c}, cotroller outut D CCx must be siusoidal (ad roortioal to the egative value of the hase voltage), as show i Fig. 4. Sice the itegrator itroduces a hase shift of 9 o, to obtai a siusoidal sigal at the outut of the cotroller, the iut of the cotroller i xerr = i xref - i x must be a cosie sigal, as illustrated i Fig. 4. This cosie sigal at the iut of the cotroller ca oly be geerated as a differece of two haseshifted sie sigals, i.e., if there is a hase shift betwee siusoidal sigal i xref that is roortioal to the hase voltage ad siusoidal sigal i x that is roortioal to the hase curret, as show i Fig. 4. It should be oted that the hase shift is roortioal to the amlitude of the required iut sigal i xerr, which for a give -cotroller gai at lie frequecy is roortioal to the amlitude of the outut sigal D CCx. Sice whe DFF is emloyed the outut of the curret cotroller D CCx is very much reduced (because it oly eeds to correct for relatively small imerfectios of DFF), the hase shift with DFF is also dramatically reduced. With mismatched curret-sesig gais, the quality of the iductor currets with comesatio, Fig. 3, is almost the same as that with erfectly matched sesig gais, Fig. 3. However, with mismatched curret-sesig gais, the quality of the iductor currets with comesatio, Fig., is oticeably deteriorated comared to that with matched sesig gais, Fig.. This behavior ca be exlaied by cosiderig the outut sigal of the curret cotrollers ad the duty-cycle limits. I steady-state oeratio, with matched sesig gais, without ZSS ijectio ad DFF, the outut sigal of the i x i x i xerr D CCx D CURRENT CCx CONTROLLER i xerr i xref i xref ~ v x xϵ{a,b,c} HASE SHIFT Fig. 4 Illustratio of oeratio of cotroller with comesatio ad siusoidal referece sigal: block diagram, iut sigals, error sigal, outut sigal. t t t curret cotrollers is siusoidal with amlitude DCCx,max Vm C k Vo, x { a, b, c}, (3) where, V m is the amlitude of the iut hase voltages ad V o is the outut voltage. It should be oted i (3) that DCCx,max is scaled to eak value C k of the triagular carrier. At the omial hase voltage of Vrms, D CCx,max 6. (4) 4 Cosiderig the duty-cycle limits, the liear-oeratio rage is determied as Dmax,Lim Dmi,Lim DCCx,Lim 7. () From (4) ad (), it ca be cocluded that i this desig the margi to comesate errors, such as dc offsets, mismatched sesig gais, ad offset errors i iut-voltage ad iutcurret sesig, is very arrow, i.e., D 7 6 CCx. 3%. (6) 6 This arrow margi for comesatio of errors has a detrimetal effect o the curret cotrollers with comesatio. The simulatios show that the outut sigal of the curret cotrollers with comesatio cotais a egative dc offset such that eve with matched sesig gais the miimum duty cycle is very close to the miimum limit. This dc offset is geerated by the itegrator ad it is due to the o-zero steady-state error of the cotroller. I fact, whe the referece sigals are siusoidal, the cotroller caot achieve zero steady-state error due to the fiite gai at the lie frequecy []. With mismatched curret-sesig gais, the miimum duty cycle i hases b ad c, which have larger curret-sesig gai tha hase a, becomes saturated, resultig i distorted hase currets as show i Fig.. The effect of the duty cycle limits o the liear oeratio rage of the curret cotroller with comesatio is illustrated i Fig.. With mismatched voltage sesig gais, for both ad comesatio, the quality of iductor currets is similar to that with mismatched curret sesig gais, as show i Figs., ad 3,, resectively. Referrig to Fig., this similarity ca be exlaied by recogizig that the mismatchig of the curret sesig gais roduces differeces at the egative iut of the curret cotrollers, i.e., differeces i i x, whereas the mismatchig of the voltage sesig gais roduces differeces at the ositive iut, i.e., differeces i i xref. Sice the iut of the curret cotrollers is i xerr =i xref -i x, the effect of the mismatchig of the curret ad voltage gais at the iut of the cotroller is the same. However, as show i Figs., for the imlemetatio with comesatio, the THD of iductor currets with mismatched voltage sesig gais is slightly lower tha that with the same mismatchig of the curret sesig gais. This is due to the fact that a % mismatch i hase-to-hase voltage sesig IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

5 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics gais results i oly % mismatch i calculated hase-toeutral voltages which are used i determiig i xref. Geerally, for a give iut voltage, the duty-cycle saturatio margi ca be icreased by selectig a higher outut voltage ad/or extedig the available duty-cycle rage. Both aroaches icrease the differece betwee the required maximum (miimum) duty cycle ad the duty-cycle limits. However, both aroaches also suffer from major drawbacks that make them imractical i ay alicatios. Besides icreasig the size ad cost of the outut filter, icreasig the outut voltage of the three-hase frot ed beyod 4-4 V revets the use of dowstream dc-dc coverters for sigle-hase alicatios that are tyically desiged for 34-4-V rage. Wideig the duty-cycle limits requires lowerig the switchig frequecy ad/or selectig faster switches ad faster iterface circuits, if available. Lowerig the switchig frequecy below khz is ot desirable because of icreased size of magetic comoets ad ossible audio-oise issues. It should be oted that istead of cotroller, a roortioal lus resoat (R) cotroller ca be emloyed, which results i zero steady-state error due to the ifiite gai at resoat frequecy, whe the resoat frequecy is set to the lie frequecy [4], []-[4]. It should be also oted that zero steady-state error ca also be achieved with cotroller if the cotrol is imlemeted istead of i the statioary (a,b,c) referece frame i the rotatig (d,q) referece frame, where the siusoidal sigals are trasformed to dc sigals [3], [4], []-[]. The ower factor of the iductor currets with comesatio ca be vastly imroved by imlemetig DFF. As show i Fig. 6, with DFF, the hase shift of the hase currets with resect to the hase voltages is ϕ ~. o, which is a sigificat reductio comared to ϕ ~6. o without DFF i Fig.. As already exlaied, this hase-shift reductio is the result of the fact that with DFF the duty cycle of the switches i steady-state oeratio is domiated by the DFF sigals, which are roortioal to the hase voltages, while the effect of the outut sigals of the curret cotrollers o the duty cycles is ractically egligible. However, it should be also oted that with DFF, the distortio of the iductor currets with comesatio ad with mismatched curretsesig gais is aroximately the same as that without DFF, as it ca be see i Figs. ad 6. D x offset 6 7 D x 3 7 D x Fig. Illustratio of effect of duty cycle limits o liear oeratio rage of curret cotroller with comesatio: without ZSS, with ZSS ijectio D x 7 The THD of the iductor currets with comesatio ad with mismatched sesig gais ca be imroved by imlemetig ZSS ijectio, as illustrated i Fig. 7. I fact, with symmetrical ZSS ijectio, which is imlemeted i this aer, the effective amlitude of the iut hase voltages is reduced to 86.6% [si(6 o )], as show Fig.. As a result, the maximum duty cycle is roortioally reduced,.866 D CCx, max 98, (7) 4 ad, therefore, more margi is available for comesatio of errors, i.e., 7 98 D CCx 7.%. (8) 98 This icreased margi is sufficiet to kee the duty cycle of the switches away from saturatio at omial hase voltage of Vrms, as illustrated i Fig. 7. However, at maximum hase voltage of 38 Vrms, eve with ZSS ijectio, the iductor currets are distorted with mismatched sesig gais, as show i Fig. 8. It should be oted i Fig. 7 that with ZSS ijectio, the THD of the iductor currets with comesatio ad matched sesig gais is also imroved comared to that without ZSS ijectio show i Fig 6. Secifically, the THD of the iductor currets is reduced from ~.% to ~.6%. ZSS ijectio also imroves the THD of iductor currets with comesatio, as show i Fig. 9. Secifically, at - Vrms hase voltage ad matched sesig gais, the THD of the iductor currets with comesatio ad with ZSS ijectio is reduced to ~.7% from ~.% without ZSS ijectio. This is the result of the reduced eak-to-eak rile i the iductor currets at the eak values of the hase voltages, as it ca be observed by comarig Figs. 9 ad 9. I fact, it ca be see i Fig. 9 that the eak-to-eak rile i the iductor curret waveforms icreases ad decreases four times i a lie cycle, havig maximum values at the ositive eak, egative eak, ad zero-crossigs of the iut hase voltages. This meas that the rile curret of iductors cotais low-frequecy ( Hz ad 4 Hz) harmoics, which cotribute to the THD of the iductor currets. Sice with ZSS ijectio the eak-to-eak rile i the iductor currets decreases at the eaks of the iut hase voltages comared to that without ZSS ijectio, the lowfrequecy rile comoets of the iductor currets decrease, ad, cosequetly, the THD decreases. A aalysis of the eak-to-eak rile i the iductor currets with ad without ZSS ijectio is give i the Aedix B. Fially, steady-state erformace comariso of curret cotrollers with ad comesatio imlemeted with VFF, DFF, ad with/without ZSS ijectio is doe with offset errors i the iut-curret sesig (with matched sesig gais). As a examle, with a equal offset error of.% of (ΔI offset = -(Q)) i all three iut-curret sesig circuits (same.-v offset alied i all three hases, as show i Fig. ), simulated waveforms of hase voltages ad iductor currets at omial hase voltage of Vrms ad - kw load with ad comesatio are reseted i Figs IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

6 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics V a Vb Vc THDc=.9% THDa=.% THDb=.9% - - THDc=7.% THDa=4.% THDb=4.3% w/ VFF, DFF t[sec] w/o ZSS Fig. 6 Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (Vrms, kw) with comesatio (with VFF ad DFF, without ZSS ijectio), without offset errors i sesig: hase voltages [V]; iductor currets [A] at matched sesig gais, mismatched curret-sesig gais (K csa =.9K csb, K csb =K csc ). ad, resectively. The iductor-curret waveforms i Figs. ad are obtaied without ZSS ijectio ad i Figs. ad with ZSS ijectio. As show i Fig., with comesatio, the iductor currets with offset errors i the curret sesig are oly slightly distorted comared to those without offset errors i Fig. 9. However, as show i Fig., with comesatio, the quality of iductor currets with offset errors i the curret sesig is sigificatly deteriorated (THD a,b,c 3%) comared to those without offset errors i Fig. 6. ZSS ijectio, as show i Figs. ad, slightly imroves the quality of iductor currets for both ad comesatio. However, eve with V a Vb Vc ϕ~.º w/ VFF, DFF, ZSS t[sec] Fig. 8 Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (38Vrms, kw) with comesatio (with VFF, DFF, ad ZSS ijectio), without offset errors i sesig: hase voltages [V]; iductor currets [A] at matched sesig gais, mismatched curret-sesig gais (K csa =.9K csb, K csb =K csc ). Vrms, kw 38Vrms, kw THDc=.87% THDa=.87% THDb=.88% THDc=6.47% THDa=3.64% THDb=3.64% V a Vb Vc w/ VFF, DFF, ZSS t[sec] Fig. 7 Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (Vrms, kw) with comesatio (with VFF, DFF, ad ZSS ijectio), without offset errors i sesig: hase voltages [V]; iductor currets [A] at matched sesig gais, mismatched curret-sesig gais (K csa =.9K csb, K csb =K csc ). ZSS ijectio, the iductor currets with comesatio are cosiderably distorted (THD a,b,c 9%). This distortio i the iductor currets with comesatio is caused by the itegratio error. It should be oted that idetical offset errors i iut-voltage sesig do ot affect the quality of the iut currets because a offset error i the hase-to-hase voltage sesig is cacelled i the calculated hase voltages because they are give by differece of hase-to-hase voltages. The THDs of iductor currets with ad comesatio i steady-state oeratio at omial hase voltage of Vrms ad -kw load at differet oeratig coditios are summarized i Tables I ad II, resectively. It ca be V a Vb Vc THDc=.6% THDc=.79% THDc=.4% THDc=.7% w/ VFF, DFF Fig. 9 Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (Vrms, kw) with comesatio (with VFF, DFF) with matched sesig gais ad o offset errors i sesig: hase voltages [V]; iductor currets [A] without ZSS ijectio, with ZSS ijectio. THDa=.6% THDa=.7% THDa=.3% THDa=.7% t[sec] Vrms, kw THDb=.6% THDb=.8% Vrms, kw w/o ZSS THDb=.% w/ ZSS THDb=.7% IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

7 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics Vrms, kw Vrms, kw V a Vb Vc - V a Vb Vc THDc=.7% THDa=.74% w/ VFF,DFF,w/o ZSS THDb=.74% - - THDc=.3% THDa=.3% w/ VFF,DFF,w/o ZSS THDb=.3% - THDc=8.9% THDa=8.9% THDb=8.9% - THDc=.68% THDa=.68% THDb=.69% w/ VFF, DFF, ZSS t [sec] Fig. Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (Vrms, kw) with comesatio (with VFF, DFF), with matched sesig gais ad with offset errors i curret sesig (ΔI offset = -(Q)) : hase voltages [V]; iductor currets [A] without ZSS ijectio, with ZSS ijectio. cocluded from Tables I ad II that with mismatched sesig gais, o offset errors, ad without ZSS ijectio, the THD of iductor currets with comesatio is sigificatly higher tha that with comesatio; whereas, with ZSS ijectio, the THD of iductor currets with comesatio is similar to that with comesatio. However, i the whole iut voltage rage, comesatio does ot erform as well as comesatio. I fact, at the maximum hase voltage of 38 Vrms, eve with ZSS ijectio, with mismatched sesig gais the iductor currets with comesatio are distorted, as show i Fig. 8. Fially, it ca be cocluded from Tables I ad II that with offset errors ad matched sesig gais, the THD of iductor currets with comesatio, eve with ZSS ijectio, is much higher tha that with comesatio. B. Oeratio with hase-voltage Trasiets First, erformace comariso of curret cotrollers with ad comesatio durig hase-voltage trasiets is erformed with balaced iut voltages, matched sesig gais, o offset errors i iut-curret sesig, ad for a cotrol imlemeted with VFF ad ZSS ijectio ad without DFF. I this evaluatio, the hase voltage is steed from Vrms to 38 Vrms ad back to Vrms at -kw load. It should be oted that ZSS ijectio was ecessary to achieve outut-voltage regulatio i the etire iut-voltage rage from Vrms to 38 Vrms. Durig the hase-voltage trasiets, the voltage-loo badwidth was Hz. Key simulated waveforms with ad comesatio are reseted i Figs. ad 3, resectively. It ca be see i Fig. that the outut voltage with comesatio ad VFF is almost isesitive to iut voltage trasiets. The outut voltage overshoot ad udershoot whe the amlitude of the iut hase voltages ste chages betwee 44 V ad 9 V is aroud 3-4 V. However, as show i Fig. 3, desite the VFF, w/ VFF,DFF,ZSS t [sec] Fig. Simulated waveforms of hase voltages ad iductor currets i steady-state oeratio (Vrms, kw) with comesatio (with VFF, DFF), with matched sesig gais ad with offset errors i curret sesig (ΔI offset = -(Q)) : hase voltages [V]; iductor currets [A] without ZSS ijectio, with ZSS ijectio. the outut voltage with comesatio exhibits a sigificat overshoot ad udershoot (~V) whe the amlitude of the iut hase voltages stes u from 44 V to 9 V ad stes dow from 9 V to 44 V, resectively. As ca be see from the waveforms i Figs. ad (e), i the imlemetatio with comesatio the sesed iductor currets follow the referece currets, whereas i the imlemetatio with comesatio the sesed iductor currets are very differet from the referece currets, as show i waveforms i Figs. 3 ad (e). This behavior of the comesatio ca be exlaied by aalyzig the outut sigal of the curret cotrollers. For simlicity, the ZSS ijectio is ot icluded i the followig cosideratios. I steady-state oeratio, without DFF ad ZSS ijectio, the curret cotrollers geerate the desired duty cycles. Usig the otatios i Fig., the followig relatioshi ca be writte at the eak value of the hase voltage, Kvs Vm V EA Kcs Vm K Km I m C k K Vo vs Vm K FF, (9) where, the first ad secod term i the bracket rereset the referece curret ad sesed curret at the iut of the curret cotroller, resectively, K is the gai of the comesatio, I m is the amlitude of the iut hase currets, ad K FF is the coversio factor from rms to average value, i.e., Vm KFF. 9 Vm. () The amlitude of the hase currets I m ca be exressed through the iut-outut ower balace as IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

8 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics TABLE I THD OF INDUCTOR CURRENTS WITH COMENSATION (Vrms, kw) hase curret sesig hase-hase voltage sesig w/o DFF, w/o ZSS w/ DFF, w/o ZSS w/ DFF, w/ ZSS K csa Kcs K csb Kcs K csc K cs ΔIoffset K vsab Kvs Kvsbc Kvs K ΔV offset vsca Kvs THD a THD b THD c THD a THD b THD c THD a THD b THD c Sesig gais ormalized to their matched values TABLE II THD OF INDUCTOR CURRENTS WITH COMENSATION (Vrms, kw) hase curret sesig hase-hase voltage sesig w/o DFF, w/o ZSS w/ DFF, w/o ZSS w/ DFF, w/ ZSS K csa Kcs K csb Kcs K csc K cs ΔIoffset K vsab Kvs Kvsbc Kvs K ΔV offset vsca Kvs THD a THD b THD c THD a THD b THD c THD a THD b THD c Sesig gais ormalized to their matched values Im 3 o Vm, () where, o is the outut ower ad efficiecy of % is assumed. After substitutig () ito (9), it is obtaied that V a Vb Vc V o 4 V oref V oref 3.6V V o 39 VEA 9 ref ref ref - Da Db Dc (f) Fig. Key simulated waveforms at hase-voltage trasiets Vrms 38Vrms Vrms, -kw load, with comesatio (with VFF ad ZSS ijectio, without DFF), with matched sesig gais ad without offset errors i sesig: hase voltages [V], iductor currets [A], outut voltage ad referece outut voltage [V], outut of voltage cotroller [digital value i Q format], (e) referece hase currets [A], (f) duty cycles [digital values scaled to C k =]. (e) t[sec].6v I(a,b,c) follow I(a,b,c)ref Kvs K FF K C cs k V EA o Vm Km 3 K V, () o where, it is show that the voltage cotroller outut V EA varies roortioally to the square of the eak value of the hase voltages Vm. I fact, this variatio is sigificat, as show i Fig. 3. The voltage cotroller outut V EA chages aroximately betwee ad -8 (i Q format) whe the amlitude of the iut hase voltages ste chages betwee 44 V to 9 V. Due to the slow voltage loo, V EA slowly chages, resultig i sigificat overshoot ad udershoot i the outut voltage. The trasiet erformace of the comesator with VFF ca be vastly imroved by simultaeously imlemetig DFF. The effect of the DFF o hase-voltage trasiets with comesatio is reseted i Fig. 4. It ca be see i Fig. 4 that the outut voltage with comesatio whe VFF is used alog with DFF becomes ractically isesitive to iutvoltage trasiets. It should be oted that with DFF, the sesed iductor currets with comesatio roerly follow the referece currets, as see comarig waveforms i Figs. 4 ad (e). This behavior of the comesatio ca be exlaied by cosiderig how the switch duty cycles are geerated. As already exlaied, i steady-state oeratio with DFF, the switch duty cycles are domiated by the DFF sigals, whereas, the effect of the outut sigals of the curret cotrollers o the duty cycles ca be eglected, i.e., the outut sigals of the curret cotrollers ca be cosidered as zero. By equatig the left-had side of (9), which reresets the outut IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

9 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics V a Vb Vc V o 4 V oref 38 - VEA - ref ref - ref - Da Db Dc V o 8V V oref Fig. 3 Key simulated waveforms at hase-voltage trasiets Vrms 38Vrms Vrms, -kw load, with comesatio (with VFF ad ZSS ijectio, without DFF), with matched sesig gais ad without offset errors i sesig: hase voltages [V], iductor currets [A], outut voltage ad referece outut voltage [V], outut of voltage cotroller [digital value i Q format], (e) referece hase currets [A], (f) duty cycles [digital values scaled to C k =]. of the curret cotrollers, with zero, ad by usig (), it is obtaied that VEA Kvs KcsKFF o 3Km. (3) From (3), it ca be see that the voltage-cotroller outut is ideedet of the eak value of the hase voltages. I fact, whe the amlitude of the iut hase voltages ste chages betwee 44 V to 9 V, the voltage cotroller outut VEA chages oly by aroximately ± aroud 4 (i Q format) to comesate the small chages i the outut voltage due to the fact that the iductor currets caot istatly chage. I the case of the comesatio, whe VFF is used with DFF, the outut voltage overshoot ad udershoot is ractically egligible (less tha V) whe the amlitude of the iut hase voltages ste chages betwee 44 V to 9 V. IV. EXERIMENTAL RESULTS Exerimetal waveforms of hase voltage v a ad hase currets i a, i b, ad i c i steady-state oeratio, at omial hase voltage of Vrms ad -kw load, at balaced iut voltages are reseted i Figs. -. It should be oted that the iut-voltage ad iut-curret sesig gais are ot comletely matched ad that the offset i iut-voltage ad iut-curret sesig is ot exactly.-v due to comoet toleraces. The waveforms i Figs. ad 6 are obtaied with ad comesatio, resectively, for oeratio with VFF ad ZSS ijectio, ad without DFF. The waveforms i Figs. 7 ad 8 are obtaied with ad comesatio, resectively, for oeratio with VFF ad DFF, ad without ZSS ijectio. Fially, the waveforms i Figs. 9 ad are (e) (f) t[sec] 9. V I(a,b,c) do ot follow I(a,b,c)ref V a Vb Vc V o 4 V oref VEA 4 ref ref ref - Da Db Dc.4V V o V oref (e).3.3 (f).4.4. t[sec].3v I(a,b,c) follow I(a,b,c)ref Fig. 4 Key simulated waveforms at hase-voltage trasiets Vrms 38Vrms Vrms, -kw load, with comesatio (with VFF, ZSS ijectio, ad DFF), with matched sesig gais ad without offset errors i sesig: hase voltages [V], iductor currets [A], outut voltage ad referece outut voltage [V], outut of voltage cotroller [digital value i Q format], (e) referece hase currets [A], (f) duty cycles [digital values scaled to C k =]. obtaied with ad comesatio, resectively, for oeratio with VFF, DFF, ad ZSS ijectio. Geerally, these results are i good agreemet with the simulated waveforms ad verify that the curret cotroller with comesatio is more sesitive to mismatches i iut-voltage ad iutcurret sesig gais ad offset errors i iut-curret sesig tha the curret cotroller with comesatio. It should be oted by comarig the measured hase currets with comesatio i Figs. 7 ad 9 with the corresodig simulated hase currets i Figs. ad that the distortios i the measured hase currets are maily caused by the offset errors i iut-curret sesig. Fially, comarig the measured currets ad Fs with comesatio i Figs., 7 ad 9, it ca be see that DFF imroves the F with comesatio from to.998. The THD measuremets with comesatio i Figs. ad 9 comared to Fig. 7, ad THD measuremets with comesatio i Figs. 6 ad comared to Fig. 8 show that ZSS ijectio imroves the quality of the iut currets i both imlemetatios. V. CONCLUSION I this aer, it is show that i the three-hase six-switch boost FC rectifier with average-curret cotrol ad mismatched iut-voltage ad iut-curret sesig gais as well as offset errors i iut-voltage ad iut-curret sesig, the curret cotroller with comesatio exhibits better erformace, i.e., lower THD ad higher F, comared with that of the comesatio. The sesitivity of the curret cotroller with comesatio to both mismatches i the sesig gais ad offset errors i the iut-curret sesig is the result of the relatively small margi i the liear oeratio rage (due to the limits of the switch duty cycles), which is IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

10 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics w/ VFF,ZSS w/ VFF,ZSS w/o DFF w/o DFF Va Va V/div V/div ϕ~6.º A/div A/div THDa=.48% Fa=.99 THDb=4.86% 4msec/div Fb=.96 THDc=4.83% Fc=.964 Fig. Steady-state exerimetal waveforms (Vrms, kw) with comesatio (with VFF ad ZSS ijectio, without DFF): hase-voltage Va, hase currets,,. w/ VFF,DFF THDa=.78% Fa=.9988 THDc=.76% Fc=.9983 THDb=3.% Fb= msec/div Fig. 6 Steady-state exerimetal waveforms (Vrms, kw) with comesatio (with VFF ad ZSS ijectio, without DFF): hase-voltage Va, hase currets,,. w/ VFF,DFF w/o ZSS w/o ZSS Va Va V/div V/div ϕ~3.º A/div A/div THDa=.7% Fa=.998 THDc=6.49% Fc=.9977 THDb=6.3% Fb=.9977 THDa=4.4% Fa=.998 4msec/div Fig. 7 Steady-state exerimetal waveforms (Vrms, kw) with comesatio (with VFF ad DFF, without ZSS ijectio): hase-voltage Va, hase currets,,. THDb=4.4% Fb=.997 4msec/div Fig. 8 Steady-state exerimetal waveforms (Vrms, kw) with comesatio (with VFF ad DFF, without ZSS ijectio): hase-voltage Va, hase currets,,. w/ VFF,DFF,ZSS THDc=4.6% Fc=.9978 w/ VFF,DFF,ZSS Va Va V/div V/div ϕ~3.º A/div A/div THDa=4.98% Fa=.998 THDc=4.73% Fc=.9983 THDb=4.9% Fb=.998 4msec/div Fig. 9 Steady-state exerimetal waveforms (Vrms, kw) with comesatio (with VFF, DFF ad, ZSS ijectio): hase-voltage Va, hase currets,,. THDa=.8% Fa=.9987 THDc=.7% Fc=.9976 THDb=3.8% 4msec/div Fb=.998 Fig. Steady-state exerimetal waveforms (Vrms, kw) with comesatio (with VFF, DFF ad, ZSS ijectio): hase-voltage Va, hase currets,, IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

11 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics available for comesatio of various arameter mismatches ad offset errors. It is also show that the DFF, i additio to reducig the hase shift betwee the resective iut voltages ad currets i the imlemetatio with comesatio, sigificatly imroves the effectiveess of VFF i imrovig the oututvoltage trasiet resose with resect to iut-voltage chages i the -comesatio imlemetatio. This behavior of the curret cotroller with comesatio is the result of the variatio of the voltage-cotroller outut V EA with the chages of the iut-voltage amlitude. Without DFF, EA V varies with the square of the iut-voltage amlitude, whereas, with DFF, V EA is ideedet of the iut-voltage amlitude. Fially, it is show that ZSS ijectio imroves the THD of iut-currets i curret cotrollers with both ad comesatio. This is the result of the reduced eak-to-eak rile i the iductor currets aroud the eak values of the hase voltages. I additio, ZSS ijectio icreases the margi i the liear oeratio rage for comesatio of various arameter mismatches ad offset errors. AENDICES AENDIX A - CONTROLLER DESIGN I the desig of the voltage ad curret cotrollers, the digital redesig aroach is used, i.e., the cotrollers are desiged i cotiuous-time domai (s-domai) ad the traslated to discrete-time domai (z-domai) which ca be hadled by the DS. Sice to achieve low distortios of the iut currets, the voltage-loo (outer-loo) badwidth must be well below the lie frequecy, whereas the curret-loo (ier-loo) badwidth should be i the khz rage, the cotrol loos are well searated so that they ca be ideedetly desiged. The faster curret loo is desiged first assumig that the voltage loo is oe ad the the slow voltage loo is desiged by cosiderig the boost ower stage with the curret loo closed to be a ew lat. A. Curret-Cotroller Desig For the three-hase-rectifier-cotrol imlemetatio with three ideedet curret cotrollers, the desig of the hasecurret cotrollers reduces to the sigle-hase desig. The equivalet cotiuous-time small-sigal block diagram of the hase-curret loo is show i Fig. A, which is obtaied directly from the block diagram i Fig. assumig that the ^ I ERR CURRENT CONTROLLER G CC (s) DELAY e -st DLY T I DWM C K DUTY-CYCLE-TO- OUTUT-CURRENT TRANSFER FUNCTION ^ d G id (s) ^ I L voltage loo is oe, i.e., v EA, ad that for several switchig cycles the iut hase voltage ca be cosidered costat, i.e., v ˆ x, where xϵ{a,b,c}. I Fig. A, G id (s) is the duty-cycle-to-iductor-curret trasfer fuctio, G LF (s) is the curret-sesig low-ass filter trasfer fuctio, G CC (s) is the hase-curret cotroller trasfer fuctio, /C k is the gai of the digital WM, ad / is the gai of the ADC coverter. The curret loo i Fig. A also icludes delay time T DLY, which cosists of the trasort delay (delay from the istat the iductor currets are samled at the eak of the triagular carrier to the istat the duty cycle is udated at the begiig of the ext switchig cycle) T DLY_Trasort = T sw /, where T sw is the switchig eriod, ad the WM delay of the triagular carrier T DLY_WM = T sw / [6]. Therefore, the total delay i the curret loo is T DLY = T sw. Sice the crossover frequecy of the curret loo is relatively high (tyically greater tha - khz), trasfer fuctio G id (s) at frequecies aroud the crossover frequecy of the curret loo ca be aroximated as [7] ˆ Vo id Gid ( s), (A) sl s where ωid = Vo/L. The, the lat trasfer fuctio is obtaied as st G DLY L, I ( s) Gid ( s) Kcs GLF( s) e Ck Kcs st e DLY C s s k LF id, (A) where, ωlf = πflf is the corer frequecy of the low-ass filter. To facilitate the exlaatio of the curret cotroller desig, Bode lots of trasfer fuctios G id (s), G L,I (s), ad G CC (s), as well as loo-gai trasfer fuctio T I (s) are show i Figs. A ad A for ad comesatio, resectively. To achieve a 4 o hase margi, the curret-loo badwidth f CI with ad comesatio is selected as. khz ad khz, resectively. For the comesatio, gai of the curret-cotroller K I, is K I, GCC, ( fci, ) 3337 GL, I ( fci, ), (A3) whereas, for the comesatio, roortioal gai K I, is GCC, ( fci, ) K I, 64, (A4) f ZCC, (, ) f G ZCC L I fci fci, fci, ^ I ^ I ref = ADC G L,I (s) G LF (s) LOW-ASS FILTER K cs CURRENT SENSING GAIN LANT Fig. A Equivalet cotiuous-time small-sigal block diagram of curret loo. ad itegral gai K ii, is K ii, f ZCCK I, 49768, (A) where the zero of comesatio is selected f ZCC = 3 Hz < f CI, to achieve 4 o hase margi IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

12 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics Traslatio from s-domai to z-domai is erformed by biliear (Tusti s) aroximatio z s, (A6) Ts z where, T s = /f s is the samlig eriod. Fially, the curret cotroller arameters i z-domai are obtaied as K Iz, K I, 3337 for comesatio, ad G id [db] G L,I [db] - - High frequecy aroximatio f CI, f id f LF. k 63.7 k 9.7 k f [Hz] -7.dB (A7) ad K Iz, K I, 64 (A8) Ts K iiz, KiI, 4 (A9) for comesatio. B. Voltage-Cotroller Desig The equivalet cotiuous-time small-sigal block diagram of the voltage loo is show i Fig. A3. Because the badwidth of the voltage loo is very low, i.e., it is below the lie frequecy, the effect of time delays i the voltage loo is egligible. The low-frequecy small-sigal cotrol-tooutut-voltage trasfer fuctio G vc (s) i Fig. A3 is derived i [8]. With resistive load R Load, which was used i the reseted exerimetal erformace evaluatio, G vc (s) is give by [8] G CC, [db] [db] T I, G id [db] G L,I [db] G CC, [db] T I, [db] f ZCC 3 f CI, f CI, G CC, +7.dB (s) K I, - f LF f [Hz] - f [Hz] f CI, f id f LF k 63.7 k 9.7 k f [Hz] -68.dB G f CI, CC, +68.dB f ICC High frequecy aroximatio (s) K I, - K s ii, f [Hz] where, vˆ o gcrload Gvc( s) vˆ EA s VC, (A) VC C orload RLoad C C, (A) ad g C is the trascoductace, defied i steady state as, gc VEA Io. (A) To rovide a adequate margi for outut-voltage lowfrequecy rile ad trasiets, the scaled value of V EA is selected as V EAmax =.8 V at I omax = omax / V oom = 3/ 4 = 7. A. Therefore, trascoductace g C is obtaied as Io A g C max (A3) VEAmax V The lat trasfer fuctio for the voltage loo is give as gckd RLoad GL V ( s) s s AAF CV,, (A4) where, ωaaf = πfaaf is the corer frequecy of the ati-aliasig filter. The desig of the voltage cotroller with comesatio is illustrated with Bode lots i Fig. A4. For low distortios of the iut currets, the voltage-loo badwidth f CV is selected as Hz. CONTROL-TO-OUTUT-VOLTAGE TRANSFER FUNCTION ^ v EA G vc (s) ^ v o K d G L,V (s) AAF G AAF (s) ADC LANT f LF f ZCC f CI, f [Hz] - - Fig. A Curret cotroller desig with: comesatio, comesatio. T V G CV (s) VOLTAGE CONTROLLER v ^ ref = Fig. A3 Equivalet cotiuous-time small-sigal block diagram of voltage loo IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

13 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics G vc [db] G L,V [db] G CV [db] T V [db] f f CV f AAF db f [Hz] - - KiV - GCV(s) KV s +.db 6 f ZCV f CV f ICV f [Hz] - - f CV f ZCV f AAF f [Hz] - - AENDIX B INDUCTOR-CURRENT RILE Simulated waveforms of iductor curret i a aroud the ositive eak of iut hase voltage v a i steady-state oeratio at omial hase voltage of Vrms ad -kw load with comesatio (with VFF ad DFF), for matched sesig gais ad o offset errors i sesig, without ad with ZSS ijectio are show i Figs. B ad B, resectively. To better uderstad the rile waveform of iductor curret i a, the gate ulses of the bottom switches are also icluded i Figs. B ad B, whereas, the sloe of iductor curret i a i differet switchig states is summarized i Table BI. The equivalet circuit for calculatio of the sloe of iductor currets is give i Fig. B3. As show i Figs. B ad B, for both cases (without ad with ZSS ijectio), the eak-to-eak rile of iductor curret i a i a switchig eriod T sw aroud the ositive eak of iut hase voltage v a is obtaied durig switchig state, where iductor curret icreases with sloe (v a +v ZSS )/L (see Table BI). Without ZSS ijectio, the duratio of switchig state, T, is determied as The voltage-cotroller arameters are obtaied as ad K V GCV ( fcv ) 3. f ZCV f, ( ) ZCV (A) GL V fcv fcv fcv K iv fzcv K V 3, (A6) where the zero is selected as f ZCV = 6 Hz < f CV, which results i a hase margi of 77 o >> 4 o. Fially, the voltage cotroller arameters i z-domai by usig the biliear (Tusti s) aroximatio are obtaied as ad Fig. A4 Voltage cotroller desig with comesatio. K Vz K V 3. Ts 3 ivz KiV 3.3 (A7) K. (A8) V m T Tsw Vo. (B) Multilyig T with the curret sloe dia va Vm (B) dt L L at the ositive eak of the hase voltage, the eak-to-eak rile of iductor curret without ZSS ijectio is obtaied as V m V m ia L f sw Vo, (B3) where, /f sw = T sw. With symmetrical ZSS ijectio [], used i this aer, the duratio of switchig state, T, is determied as 3 V T m Tsw. (B4) 4 Vo TABLE BI SLOE OF INDUCTOR CURRENTS ar br cr VaR VbR VcR V dix L Vx -VxR -V dt x=a x=b x=c V a V b V c V o -V o /3 V a +V o /3 V b +V o /3 V c -V o /3 V o -V o /3 V a +V o /3 V b -V o /3 V c +V o /3 V o V o -V o /3 V a +V o /3 V b -V o /3 V c -V o /3 V o -V o /3 V a -V o /3 V b +V o /3 V c +V o /3 V o V o -V o /3 V a -V o /3 V b +V o /3 V c -V o /3 V o V o -V o /3 V a -V o /3 V b -V o /3 V c +V o /3 V o V o V o -V o V a V b V c With ZSS ijectio,v x is relaced with V x +V ZSS IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.

14 This article has bee acceted for ublicatio i a future issue of this joural, but has ot bee fully edited. Cotet may chage rior to fial ublicatio. Citatio iformatio: DOI.9/TEL , IEEE Trasactios o ower Electroics Va Vb Vc - - Vrms, kw x L i x xr Sa Sb Sc 8 6 VCAR Tsw.834 (f) w/ VFF, DFF t [sec] w/o ZSS Fig. B Key simulated waveforms obtaied durig steady-state oeratio (Vrms, kw) with comesatio (with VFF, DFF, ad without ZSS ijectio), with matched sesig gais ad without offset errors i sesig: hase voltages [V] iductor curret of hase a [A]; state of bottom switch of hase a, b, (e) c, (f) triagular carrier [eak of carrier=c k =]. Va Vb - Vc - Sa Sb Sc 8 6 VCAR Tsw (f) w/ VFF, DFF t [sec] w/ ZSS Fig. B Key simulated waveforms obtaied durig steady-state oeratio (Vrms, kw) with comesatio (with VFF, DFF, ad ZSS ijectio), with matched sesig gais ad without offset errors i sesig: hase voltages [V] iductor curret of hase a [A]; state of bottom switch of hase a, b, (e) c, (f) triagular carrier [eak of carrier=c k =]. Multilyig T with the curret sloe dia va vzss 3 Vm (B) dt L 4 L at the ositive eak of the hase voltage, the eak-to-eak rile of iductor curret with ZSS ijectio is obtaied as (e) (e) 3V m 3 V m ia L f. (B6) 8 sw Vo Vrms, kw v x V xr v Fig. B3 Equivalet circuit for calculatio of sloe of iductor currets, xϵ{a,b,c}. From Eqs. (B)-(B6), it ca be see that with ZSS ijectio both the sloe of iductor curret i a ad the duratio of switchig state are decreased comared to those without ZSS ijectio, which results i a decreased eak-to-eak rile of the iductor curret with ZSS ijectio comared to that without ZSS ijectio. Secifically, i this examle, the sloe of the iductor curret is decreased.33 times ad the duratio of switchig state is decreased.9 times, resultig i a total reductio of the eak-to-eak rile of the iductor curret aroximately two times, which ca also be see i Figs. B ad B. REFERENCES [] J.W. Kolar ad T. Friedli, The essece of three-hase FC rectifier systems art I, IEEE Tras. ower Electro., vol. 8, o, , Ja. 3. [] T. Friedli, M. Hartma, ad J.W. Kolar, The essece of three-hase FC rectifier systems art II, IEEE Tras. ower Electro., vol. 9, o,. 43-6, Ja. 4. [3] M.. Kazmierkowski ad L. Malesai, Curret cotrol techiques for three-hase voltage-source WM coverters: A survey, IEEE Tras. Id. Electro., vol. 4, o, , Oct [4] M. Maliowski ad M.. Kazmierkowski, Cotrol of three-hase WM rectifiers, Cotrol i ower Electroics Selected roblems, Academic ress, Sa Diego, CA,. [] M. Hartma, H. Ertl, ad J.W. Kolar, Curret cotrol of three-hase rectifier systems usig three ideedet curret cotrollers, IEEE Tras. ower Electro., vol. 8, o 8, , Aug. 3. [6].C. Todd, UC384 cotrolled ower factor correctio circuit desig, Uitrode Alicatio Note, U-34, [7] M. Fu ad Q. Che, A DS based cotroller for ower factor correctio (FC) i a rectifier circuit, roc. Al. ower Electro. Cof., , Mar.. [8] Y. Jiag, H. Mao, F.C. Lee, ad D. Borojević, Simle higherformace three-hase boost rectifiers, Rec. IEEE ower Electro. Sec. Cof.,. 8-63, Ju [9] A.M. Hava, R.J. Kerkma, ad T.A. Lio, A high-erformace geeralized discotiuous WM algorithm, IEEE Tras. Id. Al., vol. 34, o,. 9-7, Se./Oct [] A.M. Hava, R.J. Kerkma, ad T.A. Lio, Simle aalytic ad grahical methods for carrier-based WM-VSI drives, IEEE Tras. ower Electro., vol. 4, o,. 49-6, Ja [] K. Zhou ad D. Wag, Relatioshi betwee sace-vector modulatio ad three-hase carrier-based WM: a comrehesive aalysis, IEEE Tras. Id. Electro., vol. 49, o, , Feb.. [] X. We ad X. Yi, The uified WM imlemetatio method for three-hase iverters, roc. It. Electric Mach. & Drives Cof.,. 4-46, 7. [3] B. Tamyurek, A. Ceyha, E. Birdae, ad F. Keles, A simle DS based cotrol system desig for a three-hase high ower factor boost rectifier, roc. Al. ower Electro. Cof.,. 46-4, Feb IEEE. ersoal use is ermitted, but reublicatio/redistributio requires IEEE ermissio. See htt:// for more iformatio.