Unified Classification of Operation-State Regions for Switching Converters with Ramp Compensation

Unified Classification of Operation-StateRegions for Switching Converters withRamp CompensationBocheng Bao,Guohua Zhou,Student Member,IEEE,Jianping Xu,Member,IEEE,and Zhong Liu,Member,IEEEAbstract—With the variation of circuit parameters,the operation-state regions of current-mode-controlled switching dc–dc converters can shift among stable period region,robust chaos region in continuous conduction mode,and intermittent chaos re-gion in discontinuous conduction mode.This paper presents a uni-fied approach to the operation-states analysis of switching dc–dc converters with ramp compensation.The piecewise map model of the converters isfirst derived.With the bifurcation analyses,two boundary classification equations of the orbit state shifting are then obtained.Finally,the operation-state regions are well classified.To verify the theoretical analysis results,2-D bifurcation diagrams are simulated and experimentations with current-mode-controlled buck converter are conducted.It is revealed that regular and ir-regular(chaotic or intermittent)operation states can be generated depending on circuit parameters or control of ramp-compensation current.Index Terms—Bifurcation,chaos,ramp compensation,switch-ing dc–dc converter.I.I NTRODUCTIOND UE to strong nonlinearity of switching dc–dc convert-ers,it is well known that nonlinear phenomena,such as bifurcations and chaos,occur naturally in switching dc–dc con-verters[1]–[9].In particular,the operation states of current-mode-controlled switching dc–dc converters may shift from continuous conduction mode(CCM)to discontinuous conduc-tion mode(DCM)with the variation of converter circuit pa-rameters,such as input voltage or reference current.It has been found that weak chaos and strong intermittency occur during CCM–DCM transition in current-mode-controlled con-verters[10]–[12].Therefore,three operation-state regions clas-sified by stable period region,robust chaos region in CCM,and intermittent chaos region in DCM exist in the current-mode-controlled switching dc–dc converters.Manuscript received August12,2009;revised October15,2009,December 2,2009,January30,2010,and October11,2010;accepted December2,2010. Date of current version August5,2011.This work was supported by the Na-tional Natural Science Foundation of China under Grant60971090and Grant 50677056,and by the Natural Science Foundations,Jiangsu,China under Grant BK2009105.Recommended for publication by Associate Editor F.L.Luo. B.Bao is with the School of Information Science and Engineering,Changzhou University,Changzhou21300,China(e-mail:mervinbao@).G.Zhou and J.Xu are with the School of Electrical Engineering,Southwest Jiaotong University,Chengdu610031,China(e-mail:ghzhou-swjtu@; jpxu-swjtu@).Z.Liu is with the Department of Electronic Engineering,Nanjing University of Science and Technology,Nanjing210094,China(e-mail: eezliu@).Digital Object Identifier10.1109/TPEL.2010.2098421Chaos in switching dc–dc converters is generally undesirable. However,in some cases,the robust chaotic behavior may be uti-lized to improve the performance of the power converters,for example,to improve electromagnetic compatibility[13],[14] or reduce electromagnetic interference[15],[16],while the in-termittency chaotic behavior should be avoided.Therefore,it is highly expected that the boundary between the robust chaos region and intermittent chaos region can be located in switching dc–dc converters.Some methods have been developed to locate the boundary of the period-one zone of switching dc–dc convert-ers[17]–[19].However,full exploration on the operation-state regions of switching dc–dc converters,such as stable periodic orbit,robust chaotic orbit,and intermittent chaotic orbit,has not been conducted.Once two boundaries of these three oper-ating regions are located,it is easy to adjust the parameters in such a way that the dynamics can be stabilized or the robust chaos can be utilized.Furthermore,it is well known that ramp compensation is one of the most effective ways to eliminate subharmonic oscillation in current-mode-controlled switching dc–dc converters[20]–[23].In this paper,with the bifurcation analysis,we will de-velop a unified approach to locate the operation-state regions of switching dc–dc converters with ramp compensation.Three ba-sic current-mode-controlled switching dc–dc converters,buck, boost,and buck–boost,operating in CCM and DCM with ramp compensation are studied.Based on the unified discrete map model,two boundary classification equations,which divide three smooth operating-state regions over the parameter space of switching dc–dc converters,can be easily derived and2-D bifur-cation diagrams can be obtained.The effects of compensating ramp on the dynamical behaviors of switching dc–dc converters are also illustrated.II.U NIFIED M AP M ODEL OF S WITCHINGDC–DC C ONVERTERSA.Switching DC–DC Converters With Ramp Compensation Buck,boost,and buck–boost converters are three basic switching dc–dc converters.Fig.1shows the schematics of current-mode-controlled buck converter by utilizing ramp-compensation current in which the compensated reference current i ref is given by the following:i ref=I ref−m c(t mod T)(1)0885-8993/$26.00©2011IEEEFig.1.Current-mode-controlled buck converter with ramp compensation. where I ref is the reference current,m c is the compensation ramp slope,T is the clock period,and(t mod T)is the t modulo T.The switch S turns ON at the beginning of each switching cycle and turns OFF when the inductor current increases to the compensated reference current i ref.In CCM,the inductor current is always nonzero,while in DCM,the inductor current decreases to zero during the switch-OFF period and remains at zero until the end of switching cycle.In this paper,the dynamics of switching dc–dc converters are analyzed through a discrete iterative map model that covers both CCM and DCM.Because the switching frequency of switching dc–dc converters is usually much higher than the natural fre-quency of the converters,the dynamics of the outer voltage loop is much slower and can be ignored,and the output side can be represented by a constant voltage source[12],[24],[25].Under this assumption,switching dc–dc converters become1-D and the inductor current waveform becomes piecewise linear.Let the slopes of inductor current during switch-ON state and switch-OFF state be m1and m2,respectively,and the inductor current at the beginning of the n th and the(n+1)th switch-ing cycle be i n and i n+1,respectively.Then,for three basic current-mode-controlled switching dc–dc converters,there ex-ists following relationships[10]:m1=E−V oLm2=V oL,(2)for the buck converterm1=ELm2=V o−EL,(3)for the boost converterm1=ELm2=V oL,(4)for the buck–boost converter.In the following,two normalized parameters m1and m2are used for the analysis of operation states of switching dc–dc converters.For the switching dc–dc converters operating in DCM,there are two inductor-current borders in the discrete state space. Thefirst inductor-current border I b1is defined as the value of inductor current at the beginning of switching cycle,which reaches i ref just at the end of the switching cycle.The second inductor-current border I b2is defined as the value of inductor current at the beginning of switching cycle,which decreases to zero just at the end of the switchingcycle.Fig.2.Illustration of inductor-current waveform of switching dc–dc convert-ers with two inductor-current borders.(a)i n=I b1,i n+1=I ref−m c T.(b)i n =I b2,i n+1=0.Fig.2shows the inductor-current waveform of current-mode-controlled switching dc–dc converters at these two borders when the ramp-compensation current is added.Fig.2(a)shows the evolution of inductor current,if i n=I b1.In this case,the in-ductor current reaches i ref at the end of the n th switching cycle, and the switch S remains ON throughout this switching cycle. Fig.2(b)shows the evolution of inductor current,if i n=I b2, where the current decreases to zero at the end of the n th switch-ing cycle.B.Unified Discrete Iterative Map ModelBased on the definitions of two inductor-current borders dis-cussed earlier,we can easily obtain these two inductor-current borders I b1and I b2as follows:I b1=I ref−(m1+m c)T(5)I b2=(m1+m2)I ref−m2(m1+m c)Tm2−m c.(6)With these two borders,there are three types of orbits between consecutive clock instants[25].1)i n≤I b1.The switch remains ON throughout the switchingcycle,and the map is easily given by the following:i n+1=i n+m1T.(7)2)I b1<i n<I b2.The inductor current increases to i ref andthen decreases until the end of the switching cycle,and the map is given by the following:i n+1=−m2−m cm1+m c(i n−I b2).(8)3)i n≥I b2.The inductor current decrease to zero duringthe n th switching cycle,i.e.,the converter enters DCM.Thus,at the end of the n th switching cycle,we have the following:i n+1=0.(9) By introducing a new state variable x=i/T and scaling two borderline currents and a reference current as B1=I b1/T,B2= I b2/T,andη=I ref/T,respectively,the discrete iterative map model of current-mode-controlled switching dc–dc converters with ramp compensation can then be written in the form asfollows: x n+1=⎧⎪⎪⎪⎨⎪⎪⎪⎩x n+m1,x n≤B1−m2−m cm1+m c(x n−B2),B1<x n<B20,x n≥B2(10)whereB1=η−(m1+m c)andB2=(m1+m2)η−m2(m1+m c)m2−m c.Based on the aforementioned three piecewise linear equa-tions containing two borders,the unified smooth discrete map model of current-mode-controlled switching dc–dc converters is obtained,from which the dynamics of the switching dc–dc converters depend only on four parameters m1,m2,m c,and η.Different from the piecewise linear stroboscopic map model presented in[12],the unified discrete map model proposed in this paper includes ramp compensation for the switching dc–dc converters,without loss of the information of major circuit parameters.C.Dynamical Behavior AnalysisWith the model(10),the bifurcation to chaotic behavior of current-mode-controlled switching dc–dc converters with cir-cuit parameters variation can be effectively exhibited.When ramp compensation is not implemented,if we take m1as bifur-cation parameter,then from(10),we can obtain its bifurcation diagram as shown in Fig.3(a),and its corresponding Lyapunov exponent as shown in Fig.3(b).In Fig.3(a),two borders B1and B2are simultaneously plotted by using dashed line and dash-dot line,respectively.The numerical simulations are performed by using MATLAB software platform with the following parame-ters:m c=0,m2=12,andη=9.From Fig.3(a),it is found that the current-mode-controlled switching dc–dc converter has complex dynamical behaviors.A reverse period-doubling route to chaos exists with the increas-ing of m1.Itsfirst period-doubling bifurcation occurs when m1=12,where the Lyapunov exponent just increases to zero from negative.Moreover,the period-doubling bifurcation and the border-collision bifurcation occur at the same parameter value.Some special dynamical phenomena as stated in the fol-lowing occur in the converter:1)after thefirst period-doubling bifurcation occurs,the converter operates in DCM,i.e.,the in-ductor current decreases to zero;2)the unstable periodic orbit with period-two collides with the second borderline B2,result-ing in border-collision bifurcation;and3)there exists some su-perstable periodic orbits in DCM,the occurrence of superstable periodic orbits has been analyzed in detail in[12].The converter shows weak chaos and strong intermittency,which means that the chaotic behavior becomes weaken in DCM.Where the be-havior is chaotic in the DCM,the bifurcation diagrams show high density of points in the neighborhood of unstable periodic orbits,which implies that there is intermittent periodic behavior within chaos[11].Fig.3.Dynamics of the current-mode-controlled switching dc–dc converter without ramp compensation.(a)Bifurcation diagram with the increasing of m1.(b)Corresponding Lyapunov exponent.III.C LASSIFICATION OF O PERATION-S TATE R EGIONS FORS WITCHING DC–DC C ONVERTERSA.Two Boundary Classification EquationsIt is clear that during stable period-one operation,x n≤B1 does not exist.Thus,the stability of the switching converter is only affected when B1<x n<B2.From(10),the eigenvalueλof the characteristic equation for the inner compensated loop is given byλ=−m2−m cm1+m c.(11) To ensure stable operation,λmust fall between−1and1[23]. Thefirst period doubling occurs whenλ=−1.Hence,by putting λ=−1,thefirst boundary classification equation L1for the operation-mode shifting from stable period-one to subharmon-ics and chaos will be as follows:L1:2m c+m1−m2=0(12) or thefirst critical ramp-compensation slope M c1can be ob-tained as follows:M c1=m2−m12.(13) The aforementioned result is consistent with the criterion of ramp compensation to eliminate the subharmonic oscillation or chaotic oscillation for the converter in[20].It should be noted that M c1increases linearly with the decreasing of m1and the increasing of m2.From Fig.2,it is clear that the maximum inductor current at the end of the n th switching cycle is as follows:i n +1,max =I ref −m c T(14)i.e.,the scaled maximum inductor current is as follows:x n +1,max =η−m c .(15)When the converter orbit,i.e.,the inductor current,reaches the second borderline B 2,the border-collision bifurcation and operation-state shift occur.Under this condition,there exists B 2=x n +1,max ,i.e.,η−m c =(m 1+m 2)η−m 2(m 1+m c )m 2−m c.(16)Therefore,the second boundary classification equation L 2for the operation-region shifting from CCM robust chaos to DCM weak chaos with strong intermittency will be as follows:L 2:m 2c −ηm c +m 1(m 2−η)=0(17)or the second critical ramp-compensation slope M c 2can beobtained as follows:M c 2=0.5η−0.5η2+4m 1(η−m 2).(18)It is remarkable that the second critical ramp-compensationslope M c 2is nonlinear with m 1,m 2,and η.In other words,M c 2depends on all of circuit parameters of the current-mode-controlled switching dc–dc converter.When ramp-compensation current is not implemented,i.e.,m c =0,(17)can be rewritten as follows:L 2:m 2=η.(19)For current-mode-controlled buck converter,the critical induc-tance for the occurrence of transition from CCM to DCM can be obtained as follows:L critical =T V oI ref .(20)If the inductance is less than L critical ,the buck converter will operate in DCM,otherwise in CCM.B.Control of Operation States by Ramp Compensation Fig.4(a)shows the bifurcation diagram of the switching dc–dc converters under the variation of ramp-compensation slope m c ,with m 1=5,m 2=12,and η=9.It can be observed that the converter has a reverse period-doubling route to chaos with the increasing of m c .Its first period-doubling bifurcation occurs when m c =3.5,which well coincides with M c 1=3.5calculated from (13).After the first period-doubling bifurcation occurs,the border-collision bifurcation occurs,since the fixed point of the second iterative map of the converter collides with the borderline B 1separating two smooth regions.The converter enters robust chaos region in CCM from stable period-one region.In the neighborhood of m c =2.21,which well coincides with M c 2=2.2087calculated from (18),the chaotic orbit collides with the second borderline B 2separating three smooth regions;the converter enters intermittent chaos region in DCM from robust chaos region inCCM.Fig.4.Dynamics of the current-mode-controlled switching dc–dc converter with ramp compensation.(a)Bifurcation diagram with the increasing of m c .(b)Bifurcation diagram with the increasing of m 1after introducing m c =2.Fig.4(b)illustrates the dynamics of the current-mode-controlled switching dc–dc converter with ramp paring Fig.4(b)with Fig.3(a),it can be found that partial region of weak chaos with strong intermittency in DCM shifts into the robust chaos region in CCM and the region of stable period-one becomes wider by introducing ramp-compensation slope m c =2.The first period-doubling bifurcation and the operation-mode shift from CCM to DCM occur at m 1=8and m 1=4.67,respectively,which well coincide with the values of parameter m 1determined by (12)and (17).Fig.5shows the discrete iterative waveform of the state vari-able.During 0≤n <100,the converter operates in DCM with ramp-compensation slope m c =0.5,i.e.,the inductor current decreases to zero in some switching cycles,and shows high-periodic behavior and intermittent locking into the periodic or-bits.At n =100,by using ramp-compensation current with m c =2.5,the converter enters into CCM and shows chaotic behavior.At n =150,with m c =4,the converter comes out of chaos soon,which means that ramp-compensation current can be used to control chaotic behavior of switching dc–dc converter,and thus,to realize stabilization control of current-mode-controlled switching dc–dc converter.In the aforementioned discussion,the dynamics of the current-mode-controlled switching dc–dc converters are stud-ied under the assumption that the output voltage is constant.To illustrate the effect of output voltage variation on the dy-namics,a current-mode-controlled buck–boost converter with ramp compensation is considered by establishing a 2-D discreteFig.5.Waveform of state variable with different m c.Fig.6.Bifurcation diagrams for the current-mode-controlled buck–boost con-verter with ramp compensation when the output voltage is not constant.(a)Inductor current versus m c .(b)Capacitor voltage versus m c .iterative map model of the state variables of inductor current and capacitor voltage.The bifurcation diagrams are plotted in Fig.6(a)and (b),respectively,with circuit parameters as:E =5V ,L =1H,C =1F,R =8Ω,I ref =9A,and T =1s.As the ramp-compensation slope m c increases,the buck–boost converter undergoes three different dynamics similar to those of current-mode-controlled switching dc–dc converters under con-stant output voltage assumption,with minor difference between Fig.6(a)and Fig.4(a).However,when the output voltage of current-mode-controlled switching dc–dc converters is not con-stant,the dynamical model is much more complicated,making the derivation of mathematical expressions much more difficult and the unified analysis approach impossible.IV .E STIMATION OF O PERATION -S TATE R EGIONS FORS WITCHING DC–DC C ONVERTERS In the aforementioned discussion,some parameters of current-mode-controlled switching dc–dc converters areas-Fig.7.Estimation of operation-state regions for switching dc–dc converters.(a)Parameter space map of m c and η,where darker regions imply chaos.(b)Corresponding division of operation-state regions.sumed to vary continuously,while the other parameters are assumed constant.With wider parameter variation,the switch-ing dc–dc converters may undergo bifurcation routes covering three smooth operation regions containing two boundaries.It is,therefore,necessary to study the bifurcation patterns over the parameter space [17]–[19].Consider the circuit parameters with ranges m c =0–5and η=6–16,and letting m 1=5and m 2=12,we can obtain the parameter space map,i.e.,the 2-D dynamical behavior distri-bution diagram,as shown in Fig.7(a).The higher periodicities are depicted with deeper gray levels,the darker shade areas im-ply chaos,while the white and shallower gray areas mean low period.The three smooth operation regions of current-mode-controlled switching dc–dc converters over the parameter space can be divided by the aforementioned boundary classifica-tion equations L 1and L 2.Fig.7(b)shows the regions ofFig.8.Estimation of operation-state regions for switching dc–dc converters.(a)Parameter space map of m c and m 1,where darker regions imply chaos.(b)Corresponding division of operation-state regions.the orbit-operating states corresponding to Fig.7(a).From (12)and (17),the two boundaries are L 1:m c =3.5and L 2:η=(m 2c +60) (m c +5),respectively.The boundary L 1is called as the first period-doubling bifurcation borderline,and the boundary L 2is called as the operation-mode shifting border-line.The regions of stable period-one,robust chaos in CCM,and weak chaos and intermittency in DCM are shown in Fig.7(b).From the parameter space map,the different operation re-gions of the switching dc–dc converters can be demonstrated clearly.Fig.8(a)shows the simulation results of 2-D bifurcation di-agram illustrating the dependence of the dynamical behavior of the switching dc–dc converters on two control parameters m c and m 1,where the circuit parameters variation ranges are m c =0–5and m 1=2–10,with m 2=12and η=9.Fig.8(b)shows the regions of the orbit-operating states corresponding to Fig.8(a),with two boundaries L 1:m 1=12−2m c ,and L 2:m 1=3m c −m 2c /3.The regions of stable period-one,ro-bust chaos in CCM,and weak chaos and intermittency in DCM divided by boundaries L 1and L 2are shown in Fig.8(b).It is clearly visible that the 1-D bifurcation diagrams of x n versus m c and m 1in Fig.4(a)and (b)can be obtained along the paths from point A to point B in Fig.7(a)and from point C to point D in Fig.8(a),respectively.It should also be noted that the boundary classification equation L 2will lose its physical significance when switching dc–dc converters are located in stable operation-state region.V .E XPERIMENTAL V ERIFICATION BY U SINGC URRENT -M ODE -C ONTROLLED B UCKC ONVERTER AS AN E XAMPLETo verify the analysis results given earlier and to illustrate the effect of ramp compensation on the operation-state con-trol of switching dc–dc converters,the experimental studies of current-mode-controlled buck converter are conducted as an ex-ample [26].By substituting (2)and η=I ref /T into (12)and (17),two boundary classification equations with practical circuit pa-rameters can be rewritten as follows:L 1:2m c L +E −2V o =0(21)L 2:m c L 2(m c T −I ref )+(E −V o )(V o T −LI ref )=0.(22)It is well known that m c is usually selected as 0.5m 2or m 2to restrain the instability due to input-voltage variation or the instability due to inductor-current perturbation for all duty ratios [27].However,if m c is selected as 0.5m 2or m 2,the current-mode-controlled switching dc–dc converters are stable and no CCM–DCM chaotic behavior can be observed.To verify that ramp compensation can make the switching dc–dc converters shift from DCM to CCM and control the converters to operate in stable period-one region,the value of m c less than 0.5m 2is used in the following experiments.Considering four sets of experimental circuit parameters with-out and with utilizing ramp-compensation current,the corre-sponding constitutive relations and operation states can be ob-tained,as summarized in Table I,where σ1and σ2represent the left-hand sides of (21)and (22),respectively.In experimen-tal verifications,the other circuit parameters are set as I ref =1A,T =100μs (switching frequency f s =10kHz),and out-put filtering capacitor C =3300μF.Remember that the stable period-two and period-four operation in DCM also deems as the subharmonic oscillation of DCM.Fig.9shows the inductor-current waveforms of the current-mode-controlled buck converter without utilizing ramp-compensation current,i.e .,m c =0,where Fig.9(a)and (d)indicate that the converter operates in DCM with weak chaotic oscillation and Fig.9(b)and (c)in DCM with subharmonic os-cillation.According to (21)and (22),we can know that these four sets of circuit parameters satisfy the constitutive relations:σ1<0,and σ2>0.Fig.10shows the inductor-current waveforms of the current-mode-controlled buck converter with ramp-compensation cur-rent.In Fig.10(a)and (d),the circuit parameters satisfy σ1<0,and σ2<0;the converter operates in CCM with chaotic oscilla-tion.In Fig.10(b)and (c),the circuit parameters satisfy σ1>0,and the converter operates with periodic oscillation.These ex-perimental results demonstrate that ramp-compensation current can be used to shift DCM to CCM and to realize stabilization control of switching dc–dc converters.When we perform the theoretical analysis,it is assumed that the switching frequency of switching dc–dc converters is much higher than the natural frequency of the converters so that the output voltage can be regarded as constant,and the inductor current is thus piecewise linear.However,Fig.9(a)and 9(c),TABLE IE XPERIMENTAL R ESULTS W ITH D IFFERENT C IRCUIT PARAMETERSFig.9.Inductor-current waveforms of buck converter without ramp compen-sation.(a)E =3.7V ,V o =3V ,L =240μH.(b)E =5.6V ,V o =3V ,L =240μH.(c)E =9V ,V o =5V ,L =240μH.(d)E =6.4V ,V o =5V ,L =330μH.and Fig.10(a)and (c)show that inductor-current waveforms are curvature in some extents.The observed curvature inductor-current waveforms are due to nonconstant output voltage in practical application.VI.C ONCLUSIONScaling of variable and parameters has been proved to be use-ful to analyze the dynamical behaviors and locate the bound-aries of the operation regions of switching dc–dc converters.Specifically,it allows the number of essential parameters to be reduced from five to four without loss the information of circuit parameters.When current-mode-controlled switching dc–dc converters shift from CCM to DCM,the state trajectory of switching dc–dc converters is no longer strong chaotic as in the case of CCM.If the inductor current decreases to zero during at least one clock period,the orbit becomes intermittently periodic,even when the converters are 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