Author name used in this publication: Chi K. TseRefereed conference paper2007-2008 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Chu, G

Tan, SC

Tse, CKM

Wong, SC

Author name used in this publication: Chi K. TseRefereed conference paper2007-2008 > Academic research: refereed > Refereed conference paperVersion of RecordPublishedVoR allowe

PolyU Institutional Repository

4452General Control for Boost PFC Converter from aSliding Mode ViewpointGrace Chu, Siew-Chong Tan, Chi K. Tse and Siu Chung WongDepartment of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kongemail: grace@eie.polyu.edu.hkAbstract— In this paper, we propose a current controlmethodology for the boost PFC converter based on sliding modecontrol theory. The resulting control is very general in thesense that it provides input current tracking under all practicalconditions by systematically including all necessary controlfunctions that compensate for the possible nonlinearity suchas the zero-crossing distortion in high line frequency condition.By choosing suitable state control variables and a proper slidingsurface, the derivation arrives at a feedforward control which issimilar to a previously proposed method. The implementationtakes the form of a typical pulse-width modulation (PWM)control. It can be designed either as an individual currentcontroller or as feedforward control applied to a standard PFCcontroller.I. INTRODUCTIONIn this paper, a general current control rule is derivedfor the boost PFC converter. The aim is to provide inputcurrent tracking under all practical conditions. Specifically,the proposed general control systematically and automaticallyincludes the necessary compensation functions to eliminatethe phenomena due to the inherent nonlinearity, which maymanifest as zero-crossing distortion under certain line fre-quency condition [1]. The derivation is based on generalsliding mode (SM) control theory. A fixed-frequency versionof sliding mode control [2] is employed and the resultingcontrol takes the form of a typical PWM control, which canbe readily implemented with analog circuits. Interestingly,this SM-based derivation is simpler and more straight forwardthan the conventional large signal approach, as complicatedcurrent loop modeling is not required. The resulting gen-eral control scheme turns out to be consistent with somevoltage feedforward schemes proposed previously, with allthe required feedforward signal paths included systemati-cally [4], [5]. SM operation is ensured by the existencecondition and stability of the current loop can be assessedby frequency response analysis. Experimental measurementsare presented to verify the analytical results.II. THEORETICAL DERIVATIONIn this section, a general current loop control rule for theboost PFC converter is derived. Conventional linear controlis adopted for the outer voltage loop to regulate the outputvoltage and to generate the half-sinusoidal reference current.Continuous conduction mode (CCM) is considered here as itis commonly used in medium to high power applications. Asimplified schematic is shown in Fig. 1A. Deciding the Control State VariablesThe proposed current controller employs the current error,x1, the integral of the current error, x2, and the doubleCurrent loopcontrollerLDCSWPWMsignaliLvoRviirefiLvgsFig. 1. A boost PFC converter with a general current controller for theinner current loop.integral of the current error, x3, as the controlled statevariables. The double-integral term is included to furtheralleviate the steady-state error caused by the finite switchingfrequency [3]. The state variables are described asx1 = iref − iL; x2 =∫x1dt; x3 =∫ ∫x1dtdt, (1)where iL denotes the instantaneous inductor current and i refrepresent the reference input current, as shown in Fig. 1.The switching function is defined as:u =12(1 + sign(S)) (2)where u represents the logic state of the power switch.B. Deciding the Sliding SurfaceThe state variables trajectory is defined as a linear combi-nation of the state variables, i.e.,S = α1x1 + α2x2 + α3x3 (3)where α1, α2 and α3 represent the sliding mode coefficients.The sliding surface is defined by setting S = 0, i.e.,α1x1 + α2x2 + α3x3 = 0. (4)Considering that the boost converter is operating in CCM,the time differentiation of (1) gives the dynamic model ofthe proposed system asx˙1 =ddtiref − 1L(vi − u¯vo); x˙2 = x1; x˙3 = x2, (5)where u¯ = 1 − u represents the inverse logic of u, and v idenotes the rectified input voltage. Under the invariance ofSM control, S = 0 and S˙ = 0. Thus,S˙ = α1x˙1 + α2x˙2 + α3x˙3 = 0. (6)978-1-4244-1668-4/08/$25.00 ©2008 IEEE4453LDCSWPWMsignalirefiLvoRviPWMviSM-derived non-linear current controllervcvovrampPILiLvgs∫? dt1312????dtdFig. 2. A boost PFC converter with the current loop employing the proposedgeneral control rule.C. Synthesizing the Control Signal by Equivalent ControlUnder the equivalent control, the logic signal u¯ is substi-tuted by the continuous signal u¯eq, such that0 < u¯eq < 1. (7)Substituting (5) and (6) into (7) givesα1direfdt− α1L(vi − u¯eqvo) + α2(iref − iL)+ α3∫(iref − iL)dt = 0. (8)Solving (8) for u¯eq givesu¯eq =1vo[−Ldirefdt+ vi − LK1x1 − LK2x2](9)whereK1 =α2α1; K2 =α3α1.Equation (9) is a general control rule for the boost PFCconverter and is illustrated in Fig. 2.III. EXISTENCE CONDITIONFor sliding mode operation to occur, the existence condi-tion has to be satisfied, i.e., limS→0 S · S˙ < 0.For S > 0, i.e., u = 1, u¯ = 0 and S˙ < 0, we haveα1direfdt− α1Lvi + α2x1 + α3x2 < 0. (10)For S < 0, i.e., u = 0, u¯ = 1 and S˙ > 0, we haveα1direfdt− α1L(vi − vo) + α2x1 + α3x2 > 0. (11)The derivative of the reference current can be obtained asdirefdt=2V 2refRVid sin(ωt)dt=2V 2refRViω cos(ωt), (12)where ω is the line frequency in rads−1. Here we assumevo = Vref and the converter has a very high efficiency η withη ≈ 1.Substituting (12) into (10) and (11), and re-arranging, theexistence condition becomesK1x1 + K2x2 < −2V2refRViω cos(ωt) +ViLsin(ωt)K1x1 + K2x2 > −2V2refRViω cos(ωt) +ViLsin(ωt)− VrefL.(13)In the right hand side of (13), x1 is the current error which isbasically the inverted function of the inductor current rippleand x2 is the integral of the current error. The right handsides of (13) define the upper and lower boundaries for theamplified current error and its integral. Thus, satisfactionof the existence condition can be verified numerically byplotting the existence condition boundaries along with thecurrent error and its integral for a rectified line cycle.IV. FREQUENCY RESPONSE ANALYSISUnder the assumption of ideal sliding mode operation,the current loop behavior is confined to the sliding surfaceS = 0 defined in (4) and the dynamics of the system can bedescribed by S˙ = 0 in (6). Re-arranging (6) givesdiLdt=direfdt+ K1(iref − iL)−K2∫(iref − iL)dt. (14)Due to the linear nature, (14) can be readily expressed ins-domain assiL = siref + (iref − iL)(K1 + K21s). (15)The block diagram of (15) is depicted in Fig. 3 (a), which canbe manipulated into a general form as shown in Fig. 3 (b).Upon inspection of Fig. 3 (b), the round-trip (open-loop) loopgain is obtained asG(s) =1s(K1 + K21s)=K2(1 + K1K2 s)s2, (16)which is a second-order system with dc gain of K2 and azero at K2K1 rads−1. In practice, the round-trip loop gain isG′(s) = G(s) ·H(s), (17)irefiLiLs1sKK121 ?s(a)irefiLiLsKKssKK112121???ssKKs1121 ⎟⎠⎞⎜⎝⎛ ??(b)Fig. 3. (a) Block diagram of the current loop under the general controlrule; (b) current loop block diagram in general form.4454PWMviVmVrampG1simoRlG2vcvcavgs(a)PWMviVmVrampG1simoRlG2Vmvcvcavgs(b)Fig. 4. Implementation of feedforward control for (a) a LEM controller;(b) a TEM controller.where H(s) accounts for the non-ideal properties of thepower converter and the practical op-amps, such as the non-ideal integral and the limited dc gain, which will be discussedin detail in the following section.V. IMPLEMENTATIONThe resulting general control rule is readily implementedby discrete components as an individual current controller.Alternatively, it can be incorporated in standard PFC con-trollers, such as UC3854 and UCC3817, as an extra feed-forward control. To formulate the feedforward control, thecurrent variables iL− iref in (9) is replaced by the equivalentvoltages across the sensing resistance RsiL −Rlimo, whereimo is the current generated by the practical multiplier.The instantaneous output voltage vo is approximated by theconstant reference voltage Vref .In doing so, all the integrators in (16) are non-ideal andthus the first two poles are non-zero. Here we denote thefrequencies of the first two poles by fp1 and fp2. Besides,the dc gain is limited by the open-loop gain Ao of practicalop-amps. For feasible implementation, a low-pass filter withpole frequency fp3 is inserted for filtering noise of frequencyhigher than half the converter switching frequency. Therefore,the practical loop gain transfer function becomesG′(s) =Ao(1 + K1K2 s)(1 + s2πfp1)(1 + s2πfp2)(1 + s2πfp3) , (18)where fp1 and Ao can be obtained analytically based on theop-amp’s open-loop characteristics provided in the datasheet;fp2 is approximately calculated by rl+RsL rads−1; and rl andRs are the damping resistance and current sensing resistanceof L respectively. Experimental frequency responses mea-sured from our circuit (Figs. 5, 6 and Table I) give f p1 ≈10 Hz, fp2 ≈ 59 Hz, fp3 ≈ 64 kHz and Ao ≈ 90 dB. Basedon (18), Bode plots can be obtained by drawing asymptotesmanually or using softwares such as Matlab. Therefore, theSM coefficients can be selected by the desired crossoverfrequency and phase margin, as in the conventional Bodeplots design approach used in linear controllers [6].1 mHIRFL460Load0.01 F500 Hz110 Vrms0.25 220 FSTW26NM60LRsCvoviPFC ControlCircuitryDSWRvgsFig. 6. Schematic of the boost PFC converter.The equivalent control signal for leading edge modulation(LEM) is denoted by u¯′eq and is given byu¯′eq =1VrefRs[−LRldimodt + Rsvi − LK1(Rlimo −RsiL)−LK2∫(Rlimo −RsiL)dt]. (19)The control voltage vc is obtained by multiplying (19) withthe amplitude of the ramp function Vm, givingvc =VmVrefvi − VmLRlVrefRsdimodt− LK1 VmVrefRs(Rlimo −RsiL)−LK2 VmVrefRs∫(Rlimo −RsiL)dt. (20)On the other hand, the current amplifier output of a LEMcontroller, e.g. UCC3817, can be expressed asvca = (RsiL −Rlimo)Hca, (21)where Hca is the transfer function of a type 2 amplifier, whichcan be expressed in the s-domain asHca =RfRl+1RlCzs, (22)where Rf , Rl and Cz are the component values shown inFig. 5. Here the high-frequency pole for noise filtering isignored. If we design the component values such thatRfRl=VmLRsVrefK1;1RlCz=VmLRsVrefK2, (23)the general control rule can be implemented as a kind offeedforward control scheme for a LEM controller. Simplify-ing (20) with consideration of (21)–(23) givesvc = G1vi − sG2Rlimo + vca, (24)where G1 = VmVref and G2 =VmLVrefRs. The resulting feedfor-ward control is illustrated in Fig. 4(a) and it is consistentwith that proposed in [5].Alternatively, the general control rule is readily appliedto a trailing edge modulation (TEM) controller. The currentamplifier output of a TEM controller can be represented byvca = (Rlimo −RsiL)(1 + Hca) ≈ (Rlimo −RsiL)Hca.(25)Following a similar procedure as that for LEM, the feed-forward control scheme for a TEM controller can be obtainedasvc = Vm −G1vi + sG2Rlimo + vca, (26)which is illustrated in Fig. 4 (b).4455UCC381721.5 k120 pF620 k100 F2.2 F1.8 k10 k4.7 F150 nF0.01 F560 pF560 pF4 5 3 2 91671121314683.9 kVcc 18VvivOVrefRlRmo510 k111N5819RacRviRs0.25Rvd50 k11.3 k1N58191.8 nF1N5819620 kvo10 k2.2 F100 kVramp10vcao3.9 k15Vref+15V-15VLM311MC34152driver-15VLM31810 k10 k10 k10 k+15V-15VLM318100 k1 k+15V-15VLM31810 k10 k1 k+15V-15VLM31810 k10 k1 k+15V-15VLM31810 k10 k10 k+15V100 nF1 nF1 k1 nF510 k10 k30kvcaovcVramp100 10 k10 k100 vgsRfCzCpvcFig. 5. Schematic of the PFC control circuitry.VI. EXPERIMENTAL VERIFICATIONSTwo boost PFC converters are constructed to verify theproposed current control rule. One uses the general controlrule, which is implemented as feedforward control usinga standard LEM controller UCC3817. Another adopts theconventional ACM control using the same controller IC. Thecomplete schematics of the feedforward control are shownin Figs. 5 and 6. The key circuit parameters are shown inTable I.Comparing Figs. 7 (a) with (b), under line frequency fof 800 Hz, severe zero-crossing distortion is observed underconventional ACM control. Under the general control rule,viiL(a)viiL(b)Fig. 7. Inductor current waveforms (500mA/div) resulting from (a) theconventional ACM control and (b) the proposed general control rule. f =800 Hz.TABLE ICIRCUIT PARAMETERS USED IN THE EXPERIMENTSParameters of PFC stage ValuesRectified line voltage Vi,rms 110 VNominal line frequency f 500 HzReference output voltage Vref 270 VSwitching frequency fsw 100 kHzInductance L 1 mHOutput capacitance C 220 μFNominal resistive load R 1.2 kΩlow harmonic distortion is observed in the input current.Fig. 8 (a) shows the measured power factor PF of thetwo converters as f varies from 50 Hz to 800 Hz. UnderACM control, PF drops to around 0.98 at f = 800 Hz,while PF under the proposed control is maintained above0.994. Likewise, as the input voltage Vi,rms is increased from70 Vrms to 140 Vrms, under ACM control, PF drops by 0.03to around 0.97, while PF of the converter using the proposedcontrol only reduces for less than 0.01, as shown in Fig. 8(b). This verifies the robustness of the general control againstthe variation of line frequency and line voltage.Under the general current control, frequency responsesof the current loop are measured and plotted along withthe theoretical Bode plots in Fig. 9 (a). Here we intend todemonstrate an unstable current loop and the SM coefficientsare chosen such that the phase margin is less than 10◦. Themeasured frequency responses generally agree with the theo-retical Bode plots and an oscillation at a frequency coherentto the crossover frequency is observed in the inductor currentwaveform shown in Fig. 9 (b). The small discrepancies can beattributed to the tolerance of the compensation componentsand the presence of parasitic components in the powerconverter, which have not been taken into account in ourderivation.4456 0.975 0.98 0.985 0.99 0.995 1 0  200  400  600  800  1000Power factor, PFLine frequency, f [Hz]SM-based general controlACM control(a) 0.97 0.975 0.98 0.985 0.99 0.995 1 60  80  100  120  140  160Power factor, PFInput voltage Vi, rms [V]SM-base general controlACM control(b)Fig. 8. PF measured over the variations of (a) line frequency and (b) inputvoltage.VII. CONCLUSIONIn this paper, we have presented the derivation, design andexperimental evaluation of a general current control rule forthe boost PFC converter. The resulting control rule achieveslow input current distortion against the variation of linecondition. It takes a common form of PWM control andcan be either implemented as an individual controller, orfeedforward control applied to a standard PFC controller. SMoperation can be ensured by the compliance of the existencecondition. The current loop gain is deduced from the slidingsurface equation. Stability and performance of the systemcan be assessed by Bode plots, and based on that the SM-60-40-20 0 20 40 60 100  1000  10000  100000-360-315-270-225-180-135-90-45 0Magnitude [dB]Phase [degree]Frequency [Hz]K1 = 1.9 · 104K2 = 4.0 · 109GainPhaseExperimental resultsTheoretical results(a)(b)Fig. 9. (a) Bode plots of the current loop with an unstable set of SMcoefficients; (b) the corresponding inductor current waveform (500 mA/div).K1 = 1.9 · 104, K2 = 4 · 109.coefficients can be designed in conjunction with the existencecondition. Experimental measurements generally agree withthe analytical results. The alleviation of the zero-crossingdistortion and the robustness of the proposed control havebeen verified.REFERENCES[1] J. Sun, “On the zero-crossing distortion in single-phase PFC converter,”IEEE Trans. Power. Electron. vol. 19, no. 3, pp. 685–692, May 2004.[2] S. C. Tan, Y. M. Lai, C. K. Tse, and K. H. Cheung, “A fixed-frequencypulsewith modulation based quasi-sliding-mode controller for buckcontrollers,” IEEE Trans. Power. Electron., vol. 20, no. 6, pp. 1379–1392, Nov 2005.[3] S. C. Tan, Y. M. Lai and C. K. Tse, “Indirect sliding mode controlof power converters via double integral sliding surface,” IEEE Trans.Power. Electron., vol. 23, no. 2, pp.600 611, Mar. 2008.[4] S. Wall and R. Jackson, “Fast controller design for single-phase power-factor-correction systems,” IEEE Trans. Ind. Electron., vol. 44, no. 5,pp. 654–660, 1997.[5] M. Chen and J. Sun, “Feedforward current control of boost single-phase PFC converters,” IEEE Trans. Power. Electron., vol. 21, no. 2,pp. 338–345, Mar 2006.[6] L. H. Dixon, “High power factor switching pre-regulator design opti-mization,” Unitrode Power Supply Design Seminar Manual SEM700,1990.

General control for boost PFC converter from a sliding mode viewpoint

Abstract

Similar works

Full text

Available Versions

PolyU Institutional Repository