buck quasi-resonant zvs converter with linear feedback control

BUCK QUASI-RESONANT ZVS CONVERTER WITH LINEAR FEEDBACK CONTROL: ACOMPARISON WITH CURRENT-MODE CONTROLJ. M. Dores CostaINESC-ID, Av. Alves Redol nº 9, 1000 Lisbon, Portugal Escola Náutica Infante D. Henrique,Telephone: +351213100350/ Fax: +351213145843Paço de Arcos, 2780 Oeiras, Portugaljmdc@fidelio.inesc.ptdorescosta@enautica.ptABSTRACTQuasi-resonant (QR) DC-DC converters can becontrolled with either voltage-mode, or current-modecontrol, which leads to similar advantages to those thatoccur in the case of PWM converters. In this paper, alinear feedback control law for QR-ZVS converters iscompared with current-mode control. Instead of using theinstantaneous value of the current, the linear feedbackcontrol uses the average of the inductor current. This isan advantage, since a noiseless and non-distorted sampleof the current becomes more difficult to obtain with highswitching frequency. Optimal control techniques are usedto design a linear quadratic regulator (LQR) for a buckQR-ZVS converter. The method is based on averagedsmall-signal models and it is shown that the dynamicresponses with the LQR can be similar to those obtainedby current-mode control. Experimental results from a 1MHz QR-ZVS buck converter with a LQR are presented.1. INTRODUCTIONTo increase the switching frequency in DC-DCconverters, and to reduce the electromagnetic interferenceproduced due to high di/dt and dv/dt, severalmodifications were made in conventional PWM (Pulse-Width-Modulation) converters. As an example, Fig. 1represents QR converters that are derived from PWMbuck converters by adding reactive components, L 1 andC 1 , which resonant transients make it possible to haveZero Current Switching (ZCS) [1] or Zero VoltageSwitching (ZVS) [2].QR converters with one active switch must be controlledwith variable-frequency in order to maintain ZCS orZVS. This can be done by either voltage-mode control, orcurrent-mode (or multi-loop) control [3], which leads toimproved stability and dynamic performance [4].Current-mode control (CMC) is represented in Fig. 2.Since i L is similar to that of PWM converters, except forshort transient intervals (Fig. 2(b)), CMC acts like thepeak current-mode control in which the active switch, S,is opened when a sample of the inductor-current, A l R s i L ,reaches the error voltage, v E . S is closed when v C1 =0(ZVS), as a consequence of the resonant transient thatinvolves L l and C l .(a)(b)Fig. 1: Buck QR converters; (a) with ZVS; (b) with ZCS.Usually, the design of the voltage regulator is based onsmall-signal models. For QR converters, they can beobtained by replacing the resonant switch by a linearmodel [5, 6], or by using state-space averagingtechniques [7, 8]. The relevant transfer functions arederived and classical control methods are used to designthe compensator [6].More recently, optimal control techniques were applied toPWM and resonant DC-DC converters [9-12]. A LQR for-QR converters was designed in [13] by using the smallsignalmodels developed in [8]. It then was referred thatthe LQR structure has some similarities with the currentmodecontrol model developed in [4]. Thus, it would beinteresting to investigate if averaged models, that use afiltered sample of i L ,, can be adequate for pole placementdesign with optimal control techniques, and if similaradvantages to those of CMC can be achieved. This is the1

For CMC, Fig. 2, the incremental switching frequencycan be given byf ˆ = H vˆ+ H vˆ+ H vˆ+ H iˆ(4)sEEIIOOL Lcontrolled oscillator (VCO), i.e., fˆK ( −βvˆv )with= ,s VCO O + ˆxv = −Kxˆ= −[ k k ]xˆ(6)ˆ x1 2Results QR converters were given in [4]. For the buckQR-ZVS converter, it isHE=2LFsA R K VlssIHIV= K I Fs−(4a)KO2sVIH =OKVO2sVIHL2LFs= K LFs−(4b)K VsIWith CMC in Fig. 2, vˆEis given byVˆ( s)= −βA( s)Vˆ( s)(5)EvOEquation (4) corresponds to feedback and feedforwardsignal paths to be added to (3). The result can berepresented by the block diagram in Fig. 3.Fig. 4: Linear feedback control for QR-ZVS converters.To minimise the steady-state error of the output voltage,v O , an integrator is inserted in the forward path, i.e.,A v (s)=k 3 /s. The output of the error amplifier, v E , is a thirdstate-variable, and the incremental value of the switchingfrequency is given by,fˆsa[ k 1 k 2 k 3 ] xˆa= −KKxˆ= −K(7)VCOVCOwhere x a =[i L v C v E ] t is state-vector of the augmentedsmall-signal model:⎡ A p 0⎤⎡Bp ⎤ ⎡Ep ⎤ 0ˆ.ˆ vˆI fˆ⎡ ⎤x & a = ⎢a + s + vˆref0⎥ x + ⎢0⎥ ⎢0⎥ ⎢1⎥ (8a)⎣−Cβ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦[ C 0] xˆav ˆ =(8b)OFig. 3: Small-signal model of a QR converter with CMC.Besides the feed-forward path, which gain is H I , Fig. 3resembles that of a linear feedback control, although thegains are fixed by the converter circuit and steady-stateDC values.3. LINEAR FEEDBACK CONTROLFig. 3, suggests that the QR converter can be controlledby the linear feedback control law in Fig. 4. Theincremental switching frequency in (3) is madeproportional to an error voltage by using a voltageModel (8) must be controllable and observable. Since î Land vˆO are easily measurable, the ESR of the outputcapacitor is neglected and thus, to simplify the circuit, astate estimator is not considered. The general model of aQR converter with either CMC, or state feedback controlcan be represented by Fig. 5, with the correspondentgains in Table I.With CMC, the line-to-output, T IO (s), and error-to-output,T EO (s), transfer functions can be derived from (3) and (4),and the closed-loop diagram in Fig. 6 is used to designA v (s) by classical control techniques. With state feedbackcontrol, k 1 , k 2 , and k 3 can be obtained by modern poleplacement design.3

−1taK = ρ E P(10)where P is the solution of the Riccati's equation ,t−1ta aa a =A P + PA − ρ PE E P + Q 0(10a)with⎡ Ap0⎤⎡K Ep⎤A a = ⎢ ⎥ E = VCOa⎣-βC0⎢ ⎥⎦⎣ 0 ⎦(10b)Fig. 5: Small-signal model of a QR converter with statevariablefeedback.Table I: Gains for Fig. 5.Control lawCurrent-mode State feedbackg 1 H L -k 1 .K VCO /A l R sg 2 H O /β -k 2 .K VCO /βg 3 H E K VCOg 4 H I 0A v (s)s + zAs( s + p)k − 3s⎡αI0⎤We will use Q = ⎢ ⎥ , where I is a 2x2 identity⎣ 0 q ⎦matrix; q must be large enough for the steady-state errorto be zero, and ρ must be small for a faster response.4. SIMULATION RESULTSTo compare the LQR with CMC, we consider the buckQR-ZVS converter in Fig 1(a) with an half-wave switch,and the following parameters: L 1 =1.3µH, C 1 =4.9nFR=2Ω, L=72µH, C=1000µF. The equivalent lossresistences of L and C are R L =160mΩ and R C =80mΩ,respectively. Nominal voltages are V I =15V, V O =5V;K VCO =170kHz/V, A l R s =1, and β=0.5.The resulting gains for CMC, are listed in Table II.Table II: Results for CMC (eq. (4)).H L H O H E H I1.01 10 7 -2.80 10 4 -1.02 10 7 4.70 10 4Different LQR can be obtained if different weightingcoefficients in (9) are used. With α=0.1, and q=10 8 , someresults for K are listed in Table III.Fig. 6: Closed-loop block diagram.For a LQR, K in (7) is calculated in order to minimise aquadratic cost function,J∞=∫( T2a a E d0x ˆ Qx ˆ + ρvˆ ) t(9)where Q is a symmetric positive-definite matrix, and ρ isa positive scalar that have a strong influence on theclosed-loop poles; vˆE is the error voltage in Fig. 5. Theoptimal solution of (9) is given byTable III: LQR gains (α=0.1, q=10 8 ).LQR gainsρ k 1 k 2 k 30.01 -2.42 -13.73 10 50.1 -0.53 -5.09 3.16 10 41 -0.12 -0.15 10 4With CMC, T EO (s) was calculated in order to obtain 50ºphase margin when the crossover frequency is 2500 rad/s.The result is,4

s + 1506A v ( s)= 20750(11)s(s + 4150)In Fig. 7 we compare loop gains of the buck QR-ZVSconverter: (1) with CMC and (11); (2) with the LQR inTable III for ρ=0.01. Although CMC and LQR havedifferent feedback gains in Fig. 5, the LQR may presentsimilar closed-loop performances to those obtained byCMC, due to the similar dominant poles in closed-loop.This is illustrated by Fig. 8, where theoretical closed-loopresponses of the buck QR-LQR converter, with differentgains, are compared with that obtained by CMC, for thesame sudden change of the reference.R L =160mΩ and R C =80mΩ where measured at 100 kHz.The LQR was implemented for ρ=0.01.Fig. 10 shows some experimental waveforms for R=2.2Ω.Fig. 11 shows perturbations of the output voltage and theinductor current caused by a variation in the referencevoltage.Fig. 9: Experimental buck QR-ZVS converter with LQR.Fig. 7: Loop gains of the buck QR-ZVS converter.Fig. 10: Experimental waveforms (V I =15.5V; V O =5V).Fig. 8: Predicted closed-loop responses of v O .5. EXPERIMENTAL RESULTSA prototype of a 25W, 1 MHz, buck QR-ZVS converterwith LQR is shown in Fig. 9. The parameters are those inthe previous section. The equivalent resistencesThe closed-loop small-signal model with LQR can beobtained by replacing (7) in (8). The resulting equation,with v ˆI = 0 , was solved numerically (with MATLAB)for the experimental values of the reference voltage inFig. 11. The theoretical result is shown in Fig. 12 and itpresents a very good agreement with the experimentalvalue of the output voltage (Fig. 11).5

the weighting factors, the dynamic performance can besimilar to that presented by current-mode control.Simulation and experimental results of a 1 MHz buckQR-ZVS converter confirmed the feasibility of themethod.7. REFERENCESFig. 11: Perturbations of v O and i L for the perturbation ofthe reference voltage.Fig. 12: Theoretical (1) and experimental (2) waveformsof v O for the perturbation of v re f in Fig. 11.6. CONCLUSIONSA linear feedback regulator was proposed to control QR-ZVS converters. It was shown that it has somesimilarities with current-mode control and that it canleads to similar advantages.Optimal control techniques and small-signal models, thatare an extension of those of PWM converters, were usedto design a LQR for QR converters. Results of a buckQR-ZVS converter with a LQR show that, by choosing[1] K. H. Liu, R. Oruganti, F. C. Lee, "Quasi-resonantconverters: topologies and characterisitics", IEEETrans. Power Electronics, vol. 2, pp. 62-71, January1987.[2] K. H. Liu, F. C. Lee, "Zero-voltage switchingtechnique in DC/DC converters", IEEE Trans.Power Electronics, vol. 5, pp. 293-304, July 1990.[3] R. B. Ridley, W. A. Tabisz, F. C. Lee, V. Vorperian,"Multi-Loop Control for Quasi-ResonantConverters", IEEE Trans. on Power Electronics,vol. 6, January 1991, pp. 28-37.[4] J .M. F. Dores Costa, M. Medeiros Silva, "Smallsignalmodels and dynamic performance of quasiresonantconverters with current-mode control",IEEE PESC'94 Rec., pp. 821-829.[5] V. Vorpérien, R. Tymerski, F. C. Lee, "Equivalentcircuit models for resonant and PWM switches",IEEE Trans. Power Electronics, vol. 4, pp. 205-214,April 1989.[6] B. Baha, "The control of quasi-resonant converters",Proc. EPE'93, pp. 304-309.[7] A. F. Witulski, R. W. Erickson, "Extension of statespaceaveraging to resonant switches and beyond",IEEE Trans. Power Electronics, vol. 5, pp. 98-109,January 1990.[8] J .M. F. Dores Costa, M. Medeiros Silva, "Smallsignalmodels of quasi-resonant converters", IEEEISIE'97 Rec., pp. 258-262.[9] J. M. Carrasco, F. Gordillo, L. G. Franquelo, F. R.Rubio, "Control of resonant converters using theLQG/LTR method", IEEE PESC'92 Rec., pp. 814-821.[10] Frank H. F. Leung, Peter K. S. Tam, C. K. Li, "Thecontrol of switching dc-dc converters-A generalLQR problem", IEEE Trans. on IndustrialElectronics, vol. 38, pp. 65-71, February 1991.[11] Frank H. F. Leung, Peter K. S. Tam, C. K. Li, "Animproved LQR-based controller for switching DC-DC converters", IEEE Trans. on IndustrialElectronics, vol. 40, pp. 521-528, October 1993.[12] Cahit Gezgin, Bonnie S. Heck, Richard M. Bass,“Control Structure Optimization of a BoostConverter: An LQR Approach”, IEEE PESC'97Rec., pp. 901-907.6