A CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS


 Lionel Willis
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1 A CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS Isaac Zafrany and Sa BenYaakov Departent of Electrical and Coputer Engineering BenGurion University of the Negev P. O. Box 3, BeerSheva 8 ISRAEL Tel: 977; Fax: ; Eail: ABSTRACT Chaos concepts forulated in a discrete for were applied to exaine instability conditions in a current ode PWM Boost converter under open and closed outerloop conditions. A siple expression for the axiu duty cycle for subharonicfree operation was developed and applied to asses the effects of the outer loop on subharonic oscillations in the converter under study. I. Introduction Switch Mode syste are notorious for their potential to develop instability (by which we ean the onset of parasitic oscillations). A coarse exaination of the nature of these instabilities suggests that they ight have two distinct and possibly unrelated origins. One is associated with 'analog' instability that can be explained in ters of linear feedback theory. The second, a ore devious one, is apparently associated with the sapling or discrete nature of switch ode systes. An exaple of the latter is the onset of subharonic oscillations in current ode (CM) converters. These unstable conditions were recently explained in ters of a Chaos odel [] which sees to fit the nature of switch ode systes. Indeed, it has been shown [3] that subharonic oscillations in CM is a anifestation of a chaotic behavior. This phenoena was originally explained [] by considering the propagation of a disturbance in a CM controlled syste. This fundaental explanation and its extension [] are insufficient, though, to quantize subharonic phenoena encountered in coplex, closed loop systes such as reported in [7]. The objective of this study was to describe and explain by a Chaos odel the behavior of a CM Boost converter under open and closed outer loop situations. The study was otivated by the feeling that a quantitative odel can help to exaine the effect of the outer loop coponents on the onset of subharonic oscillations and to quantize the nature of the oscillation in ters of haronic content under open and closed outer loop conditions. Once developed the odel can be used to exaine other situations in which instability can be expected. II. CHAOS MODEL OF A CURRENT MODE BOOST CONVERTER The CM Boost converter considered in this study (Fig. ) is based on the generic topology as described in []. It is assued that the converter is operating in the continuous conduction ode. The circuit diagra of Fig. serves as a reference for the three cases discussed below:. An open outer loop configuration with a constant control voltage (V c ) (solid line part) and no slope copensation ( c =).. An open outer loop with slope copensation. 3 A closed outer loop with no slope copensation. v g 8V i L i L Fig.. L 9µH R s =.*3. S S Q R i L Rs V s D t C F V out.ω R L 7.KΩ F s =KHz Cf Rf t clock.3µf Vc c R s Vref t R 7.KΩ.KΩ R Circuit diagra of the generic Current Mode controlled boost converter considered in this study. The basic wavefors related to the current controlled prograing (Fig. ) include a current control signal V c /R s (referred to the inductor current), slopes and
2 of the inductor current and a copensation slope c as norally added to ensure stability over the coplete duty cycle range. The turn off instant occurs when the peak inductor current reaches the value of the control current in Fig (a) or the cobined signal of V c plus c (Fig. b). Vc/Rs I(k)= c c I(k) (3) For stability we require: d I(k) d I(k) < () Ι(Κ) Ι(Κ) Ι(Κ) And hence the stability criterion for the case under study can be expressed as: c c < () Don. Doff. Don. Doff. ton(k) toff(k) ton(k) toff(k) In the absence of slope copensation, c = and under steadystate conditions (Fig. (a) solid line): c (a) c Vc/Rs I = Don = Doff () where all paraeters refer to their steadystate values. In this case (no slope copensation, open outer loop), () copresses to: Ι(Κ) Ι(Κ) Ι(Κ) Don. Doff. Don. Doff. ton(k) toff(k) ton(k) toff(k) (b) Fig.. Propagation of perturbation in inductor current, when V c is const (opened outer loop). (a) without c. (b) with c Exaination of the wavefor associated with the propagation of a perturbation over two cycles (Fig., dashedline) reveals that the deviations of inductor current ( I) (at the beginning of each cycle) fro the control signal Vc/Rs are related to other basic paraeters (arked in Fig. b) by the following relationships: I(k) = ( c ) ton(k) () I(k) = toff(k) c ton(k) () where k is the (discrete) cycle index and ton(k) and toff(k) represent the 'on' and 'off' tie in a perturbed cycle k. The discrete tie difference equation of the syste is thus: = Don Doff < (7) which iplies that stability is assured for Don <., as is well known. For nonzero c, we can apply () to deterine the iniu value of c required to ensure stability: or c > ( Doff ) c > ( Don ) (8a) (8b) The relationship (3) can be used to develop the discrete ap of I(k) =f( I(k)). This was accoplished by a MATLAB (MathWorks Inc.) subroutine that was run for a hundred cycles. As evident fro Fig. 3(a), with c = and Don =. the Boost converter Fig. under open outer loop is unstable. In contrast, the single point of Fig. 3(b) iplies stability for the sae converter with a slope copensation.
3 8 7 8 I(k) 3 = I (k) I (k) (k)= I(k) 3 I(k) 7 8 (a) I I (k) [A] Duty Cycle.7.7 (a) I(k) [A] I(k) I(k) (b) Fig. 3. A Discrete ap of I Error!=Constant) conditions with (a) no slope copensation ( c =) (b) with slope copensation ( c =. ). Produced by MATLAB (MathWorks Inc.) for Don=.. Plots represent a sequence of one hundred cycles fro k=9 to k=. Another iportant instruent for exaining and explaining the stability properties of a chaotic syste is the bifurcation diagra [3]. Fig. (a) illustrates the creation of subharonic oscillation (Don >.) as function of duty cycle in open loop converter for a zero c. For nonzero c (Fig. (b) ) the borderline between the stable and chaotic region oves to a higher duty cycle according of (8). III Duty Cycle (b) Fig.. Bifurcation diagra produced by sweeping the duty cycle paraeter (a) over the range. to.7 and c =. (b) over the range. to.9 and c =.. THE EFFECT OF THE OUTER VOLTAGE FEEDBACK LOOP The inductor current wavefor under closed outer voltage loop conditions is shown in Fig. (refer to Fig. for notations). The solid lines represent the steadystate condition whereas the dashed line shows a perturbed wavefors of the inductor current. In Fig. the reference current is denoted Iref corresponding to V c /R s in the open loop case (Fig. (a)). The slopes c and c are an approxiation of the instantaneous rising and falling portions of the control voltage (V c ), scaled by the current feedback network (R s ).
4 Unlikely the case of the open outer loop situation, the voltage control (V c ) in the closed outer loop syste is affected by the output voltage ripple and therefore is not a constant even at steady state. Furtherore, under subharonic oscillation conditions V c could be highly variable. The interception point i top (k) (Fig. ) of the slopes and c can be obtain fro siple geoetrical relationships: I(k) = ( c ) Don = =( c ) Doff = I(k) (9) Consequently, and c c = Don Doff = () I L = I I c () Under steady state conditions, I L and I c are the ripple of inductor current and control voltage(v c ) respectively, scaled by R s. Under perturbed conditions (dashedline): I(k)=( c )ton(k) () I(k) = ( c (k) ) toff(k) (3) Which can be transfored into the difference equation: I(k)= Ιc(k) Iref c c (k) c I(k)( c (k) ) itop(k) c (k) () Ιc(k) Fig.. Propagation of a perturbation in inductor currentwhen the outer loop is closed and c =. A first order approxiation of c and c (k) was obtained by deriving the analytical expression for Vout (Fig. ) and applying Taylor series expansion. It was found that the scaled (by Rs) slopes can be expressed as: c V out R f R R s τ c (k) V g R f ω d toff(k) R R s V g R f DoffR R s τ ω d toff(k) () toff(k) (τ) ω d toff(k) τ toff(k) (τ) ω d toff(k) τ itop(k)r f CR R s toff(k) τ () where τ=r L C (7) ω d = LC (τ) (8) itop(k)=iref ( I(k) I c (k) ) ton(k) (9) V g Iref= Doff V g Don () R L L Applying the above, the borderline Doff between the stable and unstable regions was derived to be Doff ω d τ Doff 3 ω d LCτ (τ) ω d τ R f R R s Cτ LCτ Ι(Κ) Ι(Κ) Doff (τ) ω d R f R R s Cτ R s R LR f Don. ton(k) Doff. toff(k) Doff τ τ R s R LR f τ = () The iportance of this expression is its ability to predict the iniu Doff for subharonicfree operation for the Boost converter. It should be noted that the polynoial equation is only a function of the converter's coponents' values and switching frequency. For exaple, for the noinal values of Fig. [], the
5 liit duty cycle before subharonic develops is Doff=. or Don=.. Changing C to F will ove the liit point significantly to Doff=.8 or Don= I (k) [A] IV Duty Cycle Fig.. Bifurcation diagra for the Boost converter of Fig. under closed outer loop conditions. Fs=KHz. DISCUSSION AND CONCLUSIONS The results of this study clearly show that subharonic oscillation in CM converters can readily be explained by the Chaos odel developed in this investigation. The odel was verified against exact circuit siulation and was found to predict faithfully the behavior of a CM Boost converter under various operating conditions. The Don of., which is often quoted as the borderline for subharonicfree zone is correct for open outer loop conditions. When the outer loop is closed, the borderline ight ove significantly to rather low Don values. This iplies that when slope copensation is not used a Don (ax) of. is no guarantee for stability. When copensation slope is applied, stability is assured only if the slope is adjusted according to the criterion which takes into account the effect of the outer loop. An exaination of the expressions developed in this study reveals that the stability boundary is effected by the ajor power coponents (ain inductor, output capacitor and load) and the voltage feedback network. The ain conclusions are suarized as follows:. A decrease in the values of the switching frequency (Fs), output capacitor (C) and/or load resistor (RL), will lower the Don liit for subharonicfree operation in a current ode Boost converter.. An increase in the values of input inductor (Lin) and/or the high frequency gain of the outer loop (Rf/R), will lower the Don liit for subharonicfree operation in a current ode Boost converter. 3. A preliinary analysis shows that the behavior of a current ode flyback converter is rather siilar to that of a Boost converter.. A cursory exploration suggests that a current ode Buck converter is less sensitive to the to the outer loop as far as subharonic oscillations ar concerned. However, under soe operating conditions the liit Don is appreciably lower than.. REFERENCES [] J. H. B. Deane and D. C. Haill, "Instability, subharonics and chaos in power electronic systes," IEEE Trans. Power Electronics, vol., no.3, pp. 8, July 99. [] D. C. Haill, J. H. B. Deane, and D. J. Jefferies, "Modelling of chaotic dcdc converters by iterated nonlinear apping," IEEE Trans. Power Electronics, vol. 7, Jan. 99. [3] J. H. B. Deane, "Chaos in a currentode controlled boost dcdc converter," IEEE Trans. Circuits Syst., vol.39, no. 8, Aug. 99. [] S. Hsu, A. R. Brown, L. Resnick, and R. D. Middlebrook, "Modeling and analysis of switching dctodc converters in constantfrequency currentprograed ode," in Conf. Rec., IEEE Power Electron. Specialists, 979, pp. 83. [] R. D. Middlebrook, "Modeling currentprograed buck and boost regulators," IEEE Trans. Power Electonics, vol., pp. 3, Jan [] F. D. Tan, and R. D. Middlebrook, "Unified odeling and easureent of currentprograed converters," IEEE PESC Record, pp , 993. [7] W. Tang, E. X. Yang, and F. C. Lee, "Loss coparison and subharonic oscillation issue on flyback power factor correction circuit," IEEE VPEC Record, pp. 3, 993.
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