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International Journal of Bifurcation and Chaos, Vol. 13, No. 2 (2003) 427–451 c World Scientiﬁc Publishing Company TWO-DIMENSIONAL BIFURCATION DIAGRAMS. BACKGROUND PATTERN OF FUNDAMENTAL DC DC CONVERTERS WITH PWM CONTROL ´ L. BENADERO, A. EL AROUDI, G. OLIVAR, E. TORIBIO and E. G OMEZ Departament Fisica Aplicada, UPC, Jordi Girona 1-3, Campus Nord-modul B4, 08034 Barcelona, Spain Received May 14, 2001; Revised February 22, 2002 One of the usual ways to build up mathematical models corresponding to a wide class of DC–DC converters is by means of piecewise linear diﬀerential equations. These models belong to a class of dynamical systems called Variable Structure Systems (VSS). From a classical design point of view, it is of interest to know the dynamical behavior of the system when some parameters are varied. Usually, Pulse Width Modulation (PWM) is adopted to control a DC–DC converter. When this kind of control is used, the resulting mathematical model is nonautonomous and e periodic. In this case, the global Poincar´ map (stroboscopic map) gives all the information about the system. The classical design in these electronic circuits is based on a stable periodic orbit which has some desired characteristics. In this paper, the main bifurcations which may undergo this orbit, when the parameters of the circuit change, are described. Moreover, it will be shown that in the three basic power electronic converters Buck, Boost and Buck–Boost, very similar scenarios are obtained. Also, some kinds of secondary bifurcations which are of interest for the global dynamical behavior are presented. From a dynamical systems point of view, VSS analyzed in this work present some kinds of bifurcations which are typical in nonsmooth systems and it is impossible to ﬁnd them in smooth systems. Keywords: Nonlinear phenomena; bifurcation diagrams; multistability; DC–DC converters; PWM control. 1. Introduction The operation of power electronic converter circuits is mainly based on the switching between The basic DC–DC converters Buck, Boost and diﬀerent linear conﬁgurations. This must be imple- Buck–Boost are a family of circuits which allow the mented with an appropriate control of the switches. conversion of energy from one level to another with- In a noise perturbation free environment, given the out taking into account, theoretically, losses in the desired output voltage, the switching frequency can components. They are used extensively in power be selected and the switches can be turned ON and supplies for electronic circuits and in the control of OFF according to a ﬁxed pattern; this is referred the ﬂow of energy between DC to DC systems, and to as the open loop system. In contrast, in indus- in any industrial application where there is a need trial applications, noise and perturbations are al- of stabilizing an output voltage to a desired value. ways present, and also the parameters of the circuits Also, they are widely used in small spacecrafts may be aﬀected by external disturbances. Thus such as satellites where DC power is generated by the use of an appropriate control to counteract the solar arrays. modiﬁcations on the output voltage in the system 427 428 L. Benadero et al. is recommended; this is referred to as the closed 2. Continuous Time Model of the loop system. The most popular control strategy Basic Switching Regulators used in the literature is Pulse Width Modulation 2.1. State equations (PWM) where electronic control of the basic power electronic converter circuit is achieved by control- The basic DC–DC switching converters are shown ling the duty cycle d of the controlled switch S (the in Fig. 2. The diﬀerential equations, modeling duty cycle is the ratio of the ON phase of the switch each one of the three conﬁgurations that use ev- to the period of the periodic ON–OFF operation). ery converter, can be derived by using the standard We will refer to the ON phase when the switch S is Kirchoﬀ’s laws. Let us deﬁne matrices A 1 , A2 , A3 , closed and diode D is open; the OFF phase refers B1 and B2 as follows: to when the switch S is open and diode D closed; 1 1 1 and mode OFF’ (or discontinuous mode) takes place − RC − 0 C , A2 = RC when both switch and diode are open (see Fig. 2). A1 = , 1 1 1 There are many ways that ﬁxed frequency PWM − − 0 − L RS L control can be implemented. Nevertheless, the basic ingredients of almost all existing PWM controllers 1 0 − RC 0 (1) that are used for voltage control are: A3 = , B1 = VIN , 0 0 L 1. an output voltage error ampliﬁer 2. a T -periodic sawtooth signal generator (driving 0 signal) B2 = 3. a comparator that compares the error ampliﬁer 0 output with the sawtooth waveforms. where R is the output load resistance, L is the in- ductance which is supposed to have an Equivalent The most interesting dynamics of these sys- Series Resistance ESR RS , C is the capacitance, and tems, from a classical design point of view, is the VIN is the input voltage. During each phase (ON, T -periodic orbit (periodic evolution with the same OFF and OFF’), and until a switching condition is period as the driving signal). fulﬁlled, the dynamics of the system is described by: Nonlinear phenomena in the PWM voltage con- ˙ X = AX + B (2) trolled DC–DC basic power electronic regulators have been studied in the past years. Various kinds X = (vC , iL )T is the vector of the state variables of bifurcational behaviors are found for diﬀerent and the overdot stands for derivation with respect converters with diﬀerent control schemes. Flip bi- ˙ to time t(X = dX/dt). Table 1 shows the A’s and furcations and period doubling route to chaos are B’s matrices for the three basic converters Buck, found in the Buck converter [Deane & Hamill, 1990; Boost and Buck–Boost during each phase. Fossas & Olivar, 1996; Tse, 1994a], Neimark–Sacker bifurcation and quasiperiodicity route to chaos are 2.2. Analytical solutions for each found to occur in the PWM Boost and Buck–Boost conﬁguration converters [El Aroudi et al., 2000] and border colli- sion bifurcations are found to occur in the Buck and Since the previous diﬀerential equations are piece- the Boost converter with diﬀerent control strategies wise linear (PWL), a closed form solution is [Banerjee et al., 2000; Yuan et al., 1998]. Up to now there are very few works that try to characterize Table 1. The A’s and B’s matrix for the basic converters the bifurcational phenomena in the parameter space during phases ON, OFF and OFF’. [Chakrabarty et al., 1996; Banerjee & Chakrabarty, Converter AON AOFF AOFF BON BOFF BOFF 1998; El Aroudi et al., 2000; Olivar, 1997; Toribio et al., 2000]. The aim of this paper is to investigate Buck A1 A1 A3 B1 B2 B2 in the parameter space the mechanisms of losing Boost A2 A1 A3 B1 B1 B2 the stability of the T -periodic orbit, and the tran- sition between the diﬀerent bifurcations in these Buck–Boost A2 A1 A3 B1 B2 B2 systems. Two-Dimensional Bifurcation Diagrams 429 available for each conﬁguration. Let us write: where t0 is the initial time at which the system 1 RS switches from one conﬁguration to another, and i L0 kC = , kL = , k = kc + kL , and vC0 are the states of the system at the switching 2RC 2L (3) instant t0 . The values for VC∞ and IL∞ are 1 RS ω0 = 1+ − k2 LC R VIN VC∞ VC∞ = , IL∞ = Therefore, the solution for each conﬁguration can RS R 1+ be written as: R • RC circuit and L in series with VIN (matrices The above solutions are only valid when we A2 and B1 ) (this conﬁguration corresponds to the have ON phase for the Boost and the Buck–Boost): 1 RS 1+ − k2 > 0 (8) vC (t) = vC0 e−2kC (t−t0 ) LC R VIN VIN (4) iL (t) = + iL0 − e−2kL (t−t0 ) in such a way that ω0 is real and positive. From RS RS the design point of view this is the most important • RLC oscillator in the free regime (matrices A 1 case, since it gives oscillatory solutions. and B2 ) (this conﬁguration corresponds to the OF F phase for the Buck and the Buck–Boost): 2.3. The switching conditions vC (t) = e−k(t−t0 ) [vC0 cos ω0 (t − t0 ) The PWM control of a switched converter is iL0 kvC0 + − sin ω0 (t − t0 )] achieved by the comparison of the control voltage Cω0 ω0 vcon which is a linear combination of the capacitor (5) iL (t) = exp−k(t−t0 ) [iL0 cos ω0 (t − t0 ) voltage vC and the inductor current iL in the form iL0 vC0 + − sin ω0 (t − t0 )] vcon = A(vC + Zr iL − VREF ) (9) ω0 Lω0 • RLC oscillator forced with VIN (matrices A1 and B1 ) (this conﬁguration corresponds to the ON with a driving signal, generalized as a triangular phase for the Boost, and the OF F phase for the function [Fig. 1(a)] Buck–Boost): V + VU −VL t if 0 < t < pT vC (t) = VC∞ + e−k(t−t0 ) [(vC0 L pT (iL0 − IL∞ ) vtriang (t) = − VC∞ ) cos ω0 (t − t0 ) + Cω0 V − VU −VL (t−pT ) U if pT < t < T (1−p)T k(vC0 − VC∞ ) (10) − sin ω0 (t − t0 )] ω0 where A is the gain of the error ampliﬁer, Z r is the iL (t) = IL∞ + exp−k(t−t0 ) [(iL0 impedance used to convert the inductor current to a voltage, VREF is the reference voltage, VL and VU (iL0 − IL∞ ) are the lower and upper values of the driving trian- − IL∞ ) cos ω0 (t − t0 ) + ω0 gular signal, T and p are the periode and symmetry (vC0 − VC∞ ) factor of this signal. − sin ω0 (t − t0 )] Let us deﬁne the function Vcomp as: Lω0 (6) • The capacitor connected to the load when the Vcomp (t) = vcon (t) − vtriang (t) converter works in the so-called Discontinu- ous Conduction Mode (DCM), characterized by = A(vC + Zr iL − VREF ) − vtriang (t) (11) iL (t) = 0) (matrices A3 and B2 ): The switching condition is therefore: vC (t) = vC0 e−2kC (t−t0 ) (7) iL (t) = 0 Vcomp (t) = 0 (12) 430 L. Benadero et al. (a) Fig. 2. The three basic power electronic converters from up to down, Buck, Boost and Buck–Boost. (b) istic waveform simulator for DC to DC converters Fig. 1. (a) Triangular signal vtriang (t) used as driving signal because it is designed for electrical and electronic in control. (b) Normalized function h(τ ). circuits. However, since the simulations are very time-consuming, one cannot rely on this package Since the expressions of the trajectories in each for extensive computations. Other packages, like conﬁguration include exponential and trigonometric LOCBIF [Khibnik et al., 1993], INSITE [Parker & functions, this equation is transcendental. Thus, a Chua, 1987], DSTOOL [Guckenheimer et al., 1991] closed form expression for the solution is not pos- and AUTO [Deodel & Wang, 1995] are well suited sible [Hamill et al., 1992; El Aroudi et al., 1999; only for smooth systems. They compute equilib- Fossas & Olivar, 1996]. Hence, one must resort rium points, eigenvalues, characteristic multipliers, to numerical methods to compute the switching Lyapunov exponents and invariant manifolds as- instants. suming that the vector ﬁeld is smooth enough. But It should be noted that when the converter the switching action in power electronic convert- enters in discontinuous conduction mode (that is, ers makes these systems very diﬀerent from those changes from iL (t) = 0 to iL (t) = 0), a new switch- characterized by a smooth vector ﬁeld. The vector ing condition (which is iL = 0) appears. ﬁeld for DC–DC switching converters is discontinu- ous and PWL in the form: 2.4. Numerically computed orbits f1 (X, t) if g(X, t) < 0 ˙ X= (13) Numerical methods usually play a major role when f2 (X, t) if g(X, t) > 0 the system is nonlinear and parameters must be varied in certain ranges. Although there ex- In DCM, the previous model is still valid and ist some very useful packages for the study of the functions f1 or f2 are in turn PWL. Moreover, the behavior of dynamical systems, to our knowl- these numerical packages do not allow to handle edge, none is specially suited for piecewise lineari- two-dimensional bifurcation diagrams easily. Fi- ties. In these systems, one may take proﬁt of the nally, as it was mentioned before, the numerical analytically-computed solutions, but the switch- method used for integrating the system of diﬀer- ing instants must be numerically computed. The ential equations is not optimized for PWL systems, PSPICE package [Rashid, 1990] is the more real- where closed form solutions in each sub adjacent Two-Dimensional Bifurcation Diagrams 431 interval are available. For these reasons, some spe- the following will be chosen: ciﬁc programs written in C code are used in this √ VIN paper, using the closed form solution in each linear Ts = T0 = 2π LC, Vs = VIN , Is = L/C conﬁguration. The equation for the switching con- dition is solved by a speciﬁc root-ﬁnding method Then, we deﬁne the normalized variables and (Newton, secant, bisection, . . . etc.). In our compu- parameters of the power circuit as follows: tations, the bisection method is used. vC (t) L/C t Therefore the analytical solutions combined v(t) = , i(t) = iL (t), τ = , VIN VIN T0 with the bisection method to locate the switching instants will reduce considerably the time needed R L/C Q= , QS = for a given simulation. In this paper, the global dy- L/C RS namics is obtained in this way. The simulations are (14) therefore very fast compared to simulations com- where Q is the quality factor associated to the puted with standard packages. This method was load, and QS is the quality factor associated to the already used by the authors in [Deane & Hamill, equivalent series resistance ESR of the inductance. 1990] and [Fossas & Olivar, 1996] to compute tra- Finally, τ is the dimensionless time. jectories of the PWM voltage controlled DC–DC Buck converter. 3.3. Scaling the switching condition Introducing the normalization in the control 3. Dimensionless Formulation condition, we get fS (v, i, τ ) = v + Zi − VR − VD h(τ ) = 0 (15) 3.1. Background where VR is the normalized reference voltage, V D is The large number of parameters associated with the normalized width of the external control signal the PWM DC–DC converters is a major handicap and Z is the normalized impedance. h(τ ) becomes to the characterization of all the possible dynam- a triangular function with amplitude one, null av- ics. To deal with this problem, some authors have erage value, period TN and symmetry factor p [see ﬁxed the values of the parameters according to spe- Fig. 1(b)]. ciﬁc examples [Hamill et al., 1992; Tse, 1994b] and VREF VU + VL VU − VL then they varied the parameters near these values. VR = + , VD = , VIN 2AVIN AVIN Instead, we will deﬁne dimensionless parameters (16) achieving a signiﬁcant reduction in the number of Zr T Z= , TN = independent parameters of the circuit. We begin L/C T0 with linear transformation of the state variables in such a way that the resulting new variables are di- mensionless. We choose Ts , Vs , Is scale parame- ters with physical dimensions of time, voltage and current respectively in such a way that the new variables t/Ts , vC /Vs , iL /Is are dimensionless. The normalization of variables and parameters allows to carry out an easy analysis, it facilitates the understanding of the diﬀerences between converters and especially shows how the quality factors Q (re- lated to the resistive load R) and QS (related to the resistor RS in series with the inductor) play their role in an outstanding way (the quality factors will be deﬁned below). 3.2. Scaling the power stage Fig. 3. The switching band in the normalized phase plane Among all possible choices of the scale parameters, (v, i). 432 L. Benadero et al. It should be noted that the evolution of the 3.4. Normalized analytical solutions driving signal voltage may be represented by a of each conﬁguration switching band in the (v, i) phase plane (Fig. 3). Within this band the switchings from one con- Note that Eq. (8) is equivalent to ﬁguration to another can occur. With the pro- posed normalization, the resulting state variables 1 1 1 1 2 1+ − + >0 (19) and parameters are dimensionless. Their number QQS 2 Q QS is reduced from 10 to 5. The normalization means that diﬀerent circuit designs with equal normalized Assuming that the condition above is fulﬁlled, parameters have equivalent dynamics. the closed form solutions for each conﬁguration After scaling variables and parameters, the are: dimensionless diﬀerential equations of each conﬁg- uration can be written • The LRC parallel oscillator with the source V IN ˙ in series with the inductor. Y = CY + D (17) where the overdot denotes for the derivation with respect to the dimensionless time τ . Y = (v, i) T is v(τ ) = V∞ + exp−α(τ −τ0 ) [(v0 − V∞ ) cos ωN (τ − τ0 ) the vector of the normalized state variables and the dimensionless matrices are C1 , C2 , C3 , D1 and D2 i0 − I∞ − β(v0 − V∞ ) + sin ωN (τ − τ0 )] as follows: δ 1 1 − Q 1 − Q 0 i(τ ) = I∞ + exp−α(τ −τ0 ) [(i0 − I∞ ) cos ωN (τ − τ0 ) C1 = 2π , C2 = 2π , 1 1 v0 − V∞ − β(i0 − I∞ ) −1 − 0 − − sin ωN (τ − τ0 )] QS QS δ 1 − 0 0 0 (20) C3 = 2π Q , D1 = , D2 = 0 0 2π 0 Note that the equilibrium point (QQ S /(Q + QS ), (18) QS /(Q + QS )) of this conﬁguration is a stable spiral sink. Matrices Ci and vectors Di correspond to nor- • The LRC parallel oscillator in the free regime. malized ones Ai and Bi respectively (see Table 1). Therefore, the converter depends on four essen- v(τ ) = exp−α(τ −τ0 ) [v0 cos ωN (τ − τ0 ) tial dimensionless parameters Q, TN , VD , VR and an additional (parasitic) parameter Q S (the essential i0 − βv0 + sin ωN (τ − τ0 )] and parasitic classiﬁcation is due to circuit consid- δ erations, but this will not aﬀect the analysis). T N (21) is the period of the driving signal normalized to the i(τ ) = exp−α(τ −τ0 ) [i0 cos ωN (τ − τ0 ) natural period of the LC-circuit, and ﬁnally, in the v0 − βi0 event of Rs = 0 (not considered here), Q is the − sin ωN (τ − τ0 )] δ quality factor of the power circuit. Thus, a DC– DC converter can be characterized by the normal- The equilibrium point (0, 0) of this conﬁguration ized parameter set C = {Q, QS , TN , VR , VD }. The is a stable spiral sink. set C of dimensionless parameters of one speciﬁc • The inductor connected to the source, and the PWM DC–DC regulator stands for a whole family RC circuit: of regulators whose dimensionless parameters are those in C, and thus they display the same dynam- − 2π (τ −τ0 ) ics (obviously, this is true only if the converter is v(τ ) = v0 exp Q the same and its model remains valid for the entire 2π (22) − Q (τ −τ0 ) family). i(τ ) = QS + (i0 − QS ) exp S Two-Dimensional Bifurcation Diagrams 433 e 4. The Poincar´ Map In the case of periodically driven systems, the usual e Poincar´ section considered is a plane Σ (with equa- tion τ = 0) in the cylindrical space (v, i, τ ) ∈ R+ × R+ × S 1 (we identify (v, i, τ = 0) with (v, i, τ = TN )). At every period of the dimensionless driving signal the trajectory intersects Σ. A map which lies on two successive points in the Poincar´ e section can be deﬁned. P :Σ→Σ (v(τ0 ), i(τ0 )) → (v(τ0 + TN ), i(τ0 + TN )) 4.1. Obtaining the one-periodic orbit Fig. 4. Schematic representation for the three conﬁgura- Assume we have a converter with all parameters tions that use the Boost converter. ﬁxed. The ﬁxed points of the stroboscopic map are obtained equaling, on one hand, the values of the capacitor voltage and inductor current at the be- ginning and those at the end of each cycle of the The equilibrium point (0, QS ) of this conﬁgura- driving signal and, on the other hand, forcing the tion is a stable sink. continuity of the state variables at the asynchronous • The RC circuit (this corresponds when the con- switching instants [di Bernardo et al., 1997]. By verter works in DCM): imposing these conditions, a system of two non- linear equations with two unknowns veq and ieq is v(τ ) = v0 exp − 2π (τ −τ0 ) Q obtained. By imposing the asynchronous switch- (23) ing condition, another equation is added, being the i(τ ) = 0 switching time τS another unknown. These three nonlinear equations with three unknowns can be in- The equilibrium point (i = 0) of this conﬁgura- troduced to a standard mathematics program (for tion is a stable one-dimensional1 sink. example Maple V) to ﬁnd the ﬁxed points (v eq , ieq ) for a set of parameter values. In the expressions of the solutions, we have used V∞ = QQS /(1 + QQS ), I∞ = V∞ /Q, ωN = 2πδ, δ = (1 − β 2 ), β = 1/2(1/Q − 1/QS ) and α = 5. Mechanisms of Losing the Stability π(1/Q + 1/QS ). Also, v0 , i0 are the state variables of the One-Periodic Orbit at the switching instant τ0 . As we have seen, the dynamics in each conﬁg- The basic elements which will be used in this section uration is well determined. But the parameters of to study the stability of the one-periodic orbit are the circuit are selected in such a way that when e the Poincar´ map and the characteristic multipliers we make the control work, the converter switches of its ﬁxed points. The stability of the one-periodic from one conﬁguration to another. Therefore, since orbit is analyzed by means of the response to a small the system is highly nonlinear due to the switch- perturbation near this ﬁxed point. ing action, a great variety of dynamics are pos- To obtain the stability character of a ﬁxed point sible (limit cycles, subharmonics, quasiperiodicity, (veq , ieq ) of this map is enough to compute the im- chaos). In Fig. 4, the switching lines for a PWM age of a perturbed state near the ﬁxed point in the voltage mode controlled Boost regulator and the phase plane (vn , in ). This transforms the problem trajectories near the equilibrium points of each con- of analyzing the stability of the one-periodic orbit ﬁguration are schematically plotted. to the study of the eigenvalues of the linearized 1 When the converter works in the DCM in some cycles, the system order changes from two to one in these cycles. 434 L. Benadero et al. map DP . The four coeﬃcients of the matrix DP will specify the sense of variations of the parame- associated to the linearization near a ﬁxed point ters; the larger arrow corresponds to the parameter are obtained by computing the transformation of chosen as primary (in most of the diagrams this is the state space points in the neighbor of the ﬁxed TN ). Also, in order to understand the bifurcation points. patterns better, some simulations have been made From the eigenvalues m1 and m2 of DP (the using an ideal switch instead of a diode. characteristic multipliers of the orbit), or from the Lyapunov exponents (λ = log |m|/TN ) the following 6.1. A point in the parameter space bifurcations will be detected: To obtain a good comparison of all the two- • Flip or period doubling bifurcation of the stable dimensional bifurcation diagrams, we choose a one-periodic orbit. It is characterized by the fact central point in the parameter space and then, we that one of the characteristic multipliers is equal vary the parameters around this point. In this zone to −1 and the other one has absolute value less we will explore all the possible dynamics of the three than 1. converters. • Hopf or Neimark–Saker bifurcation. It is char- We ﬁx the values of this parameter point to acterized by the fact that both characteristic Q = 4, QS = 6 and Z = 0. Moreover we take multipliers cross the unit circle, being complex VR = 0.5, VD = 0.3 for the Buck, and VR = 1.4, conjugates. VD = 1.8 for the Boost and the Buck–Boost. We • Border-Collision bifurcations. They are charac- cannot ideally take the same VR in the three con- terized by a sudden change in the system be- verters since its value tightly depends on the volt- havior accompanied by a jump in the values of age gain. Thus it is convenient to take V R less than the characteristic multipliers. These anomalous one in the case of the Buck, which reduces the input bifurcations are well explained in [Yuan et al., voltage; and more than one in the case of the Boost, 1998; Banerjee et al., 2000; Nusse et al., 1994] which increases the input voltage. In the case of and [Nusse & Yorke, 1992]. the Buck–Boost it can be any value since this con- verter can be designed to increase or reduce the 6. Two-Dimensional Bifurcation input voltage. Diagrams The range of variation of the normalized pe- riod is large enough to consider not only low periods In order to obtain the bifurcational structure when (TN 1), which is the usual range in circuits, but two parameters are varied, we will plot the two- also high periods near TN = 0.7 are checked. It will dimensional bifurcation diagrams corresponding to be shown that some bifurcations which pose limits these parameters. The two-dimensional bifurcation on the TN -periodic dynamics are met near TN = 1. diagram is color-coded depending on the periodicity With regards to parameter p, we will use only its of the attractor corresponding to the point in the extreme values p = 0 and p = 1. parameter space. Due to hysteresis phenomena and in general to possible coexisting attractors, only one 6.2. Choosing the parameters to vary of them can be identiﬁed at each point of the param- eter space. In this paper the diagrams are produced For the two-dimensional bifurcation diagrams, we as follows: ﬁrst we give the system with extreme have to choose two parameters from the seven (T N , values of both varying parameters and the initial Q, QS , Z, VD , VR , p) which correspond to a given condition xini = (vini , iini ); then, one-dimensional circuit. The bifurcation patterns will be obtained bifurcation diagrams are successively computed ﬁx- from diﬀerent pairs of parameters. We classify all ing one parameter (primary) and varying the other our possible parameters according to three diﬀerent one (secondary). The ﬁnal state of the system is classes. One class contains the parameter associ- taken as initial condition to the next value of the ated to time in the control loop TN , and it will be primary parameter. The initial condition when the one of the parameters varied in each of our bifurca- secondary parameter is varied is the last state cal- tion diagrams; another class will be made with the culated using the preceding value of the secondary parameters associated to the power circuit, these parameter, both with the same extreme value of are Q and QS , which will be kept ﬁxed in each of the the primary parameter. Arrows in the diagrams bifurcation diagrams. Finally a third class includes Two-Dimensional Bifurcation Diagrams 435 Fig. 5. 2D bifurcation diagram for a Boost regulator in continuous conduction mode. Varying parameters are period T N and impedance Z and the ﬁxed ones are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. The color for each point in the diagram shows the period of the resulting orbit. Initial conditions are those corresponding to the orbit of the preceding point as indicated by the arrows. The same upper color codes are used in all diagrams. all the parameters which deﬁne the control. More 7. Bifurcation Diagrams for the speciﬁcally, they deﬁne the regulation band (Z, V D , Boost Converter in the VR and p). One of them will be taken to be varied (TN , Z)-Parameter Space. Regions when we compute the two-dimensional bifurcation in the Parameter Space diagrams. It will be shown that VR has a meaningful eﬀect in the Boost and the Buck–Boost converters, but has little or no eﬀect in the Buck. Thus, in this 7.1. Basic regions according to the case parameters VD and Z will be varied. Param- dynamical behavior eter Z allows to detect more easily all the possible Using the numerical simulator which has been dynamics. described before, we obtained two-dimensional 436 L. Benadero et al. Fig. 6. Time (left) and phase plane (right) representation for a 2TN -periodic orbit showing chattering. This corresponds to a Boost regulator in continuous conduction mode. Parameters are TN = 0.55, Z = 0.82 and like in Fig. 5, (Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0). bifurcation diagrams for the Boost converter with Its linearized dynamics corresponds to a focus, with PWM control, and without diode, this is, with both negative Lyapunov exponents. ideal and complementary switches. The parameters It is clear that the border of the region 1T shows are ﬁxed according to the central point deﬁned be- a discontinuity at a point of the parameter space fore (Q = 4, QS = 6, VR = 1.4, VD = 1.8, p = 0) which corresponds approximately to T N = 0.50 and and we vary Z and TN (see Fig. 5). The resulting bi- Z = 0.19. This type of behavior, which is com- furcation diagram is simpler than the one obtained mon in a certain sense, is associated to a variation using a diode, and we can distinguish four big dif- pattern in the characteristic multipliers when the ferent regions, which are described in the following: parameters are moved, and this implies two diﬀer- First, we have a big two-periodic region of lobar ent types of borders. type, which will be called 2T region, on the right- hand side of the ﬁgure. Into this zone the orbits display the chattering phenomenon and too much 7.2. Curve of Hopf bifurcations time is required to decide the periodicity. The same phenomenon is detected on other similar lobar ar- At the border between the zones 1T and QP a Hopf eas, which yield sliding modes like in Fig. 6, with bifurcation occurs, which corresponds to a null Lya- Z = 0.82 and TN = 0.55. punov exponent. The bifurcation occurs smoothly Second, we have a region whose stationary be- when two complex characteristic multipliers cross havior is the ﬁxed point of the ON topology, in the unit circle. If we decrease the impedance Z the the lower part of the ﬁgure. This region will be characteristic multipliers begin to grow and ﬁnally called the FP region. On the right part of its upper they cross the unit circle at the border. We will call border we found that the impedance is constant at this border the Hopf border. Z = 0.08, and the left part is made of a curve which For TN near 0 this border begins at Z = −0.05, starts at a point with TN = 0.18 approximately. which can be analytically predicted when averaging Third, between the region FP and the two- techniques are used. This is tightly related with the periodic lobar region, we found chaotic and critical value of the impedance Z associated to the quasiperiodic orbits with some little periodic is- equilibrium [Benadero et al., 1999] which is related lands. The chaotic and quasiperiodic orbits are to the orientation of the vector ﬁeld. The value of found after Hopf bifurcation following the torus the critical impedance ﬁxes, with the exception of breaking route to chaos. This zone will be called a certain correction which depends on the values QP. of Q and QS , the impedance Z which makes the Fourth, there is a large one-periodic dynamics transition to instability. For diﬀerent values of T N , (blue) which is in the remainder of the ﬁgure. This the bifurcation value for Z is changed and the pe- is the usual region of operation of the regulators riod one always appears in the upper region of the and we will denote it by TN -periodic or simply 1T. ﬁgure. Two-Dimensional Bifurcation Diagrams 437 (a) (b) Fig. 7. Computed Lyapunov exponents versus TN for a Boost regulator. Fixed parameters are as in Fig. 5, Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. 7.3. Curve of ﬂip bifurcations TN = 0.63. As Z is increased, the larger Lyapunov exponent does not pass through zero, and thus no The border between the region 1T and the 2T lobe bifurcation is produced. The limit case is obtained corresponds to a ﬂip bifurcation. At this point, both approximately for Z = 1.01 and TN = 0.51. Lyapunov exponents are real and the larger one be- comes null. We will call this border the ﬂip border. Figure 7(a) shows both two exponents when pa- 7.4. Curve of grazing bifurcations rameter TN is varied, and three important facts are observed: ﬁrst, if we decrease Z then the average Finally, the border between the region FP and QP value of the Lyapunov exponents increases; second, corresponds to a grazing bifurcation, and thus will if we increase TN the average value also increases be called the grazing border. A grazing bifurca- when p = 0 (this eﬀect is inverted when p = 1, tion is a subtype of the so-called border collision see Sec. 9.2); and third, a bifurcation from focus bifurcations and can be only observed in nonsmooth to a node is produced. From this bifurcation point systems. This can be observed in Fig. 8 which corre- and onwards, one exponent grows until it becomes sponds to an orbit of the regulator with Z = 0.08. null (this exponent is associated with a character- Observe that the left line border of the switching istic multiplier equal to −1) and a ﬂip bifurcation band passes through the ﬁxed point of the ON state. occurs. Note that, depending on Z, we are favoring one bifurcation or the other, and thus two diﬀerent curves of bifurcations appear. The limit case corre- 8. Two-Dimensional Bifurcation sponds to the point Z = 0.19 and TN = 0.50, where Diagrams for the Boost the two curves coalesce. Converter with a Diode Figure 7(b) corresponds to the cuspid of the lobe 2T; for instance, with the arbitrary value It should be expected that the two-dimensional bi- Z = 0.82, we have bifurcations at TN = 0.47 and furcation diagram for a Boost converter with a 438 L. Benadero et al. Fig. 8. Switching band and attractor associated with the critical value Z = 0.08 for boundary between FP and QP, T N = 0.21 and the other parameters are the same set Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. diode should display more complex dynamics than First, a region of DCM dynamics of TN -periodic without it, since we add one more topology to the type has eroded the 2T-lobe from its right part (this circuit. As we shall see, this is really so. The new is, for high values of TN and Z). To check this point, attractors which are obtained visit periodically or we plot in Fig. 10 three diﬀerent orbits for diﬀerent intermittently, the discontinuous conduction mode, values of TN and Z ﬁxed to 1.01 (the value corre- and can be periodic, quasiperiodic or chaotic. We sponding to the cuspid of the 2T-lobe). The orbit will use CCM for dynamics of continuous conduc- which corresponds to TN = 0.37 is TN -periodic of tion mode and DCM have been denoted for discon- CCM type; the orbit for TN = 0.65 is TN -periodic tinuous conduction mode dynamics. As a general of DCM type, and the orbit for TN = 0.51 is placed rule, DCM stabilizes quasiperiodic and chaotic be- just within the border between TN -periodic CCM havior and contributes to reduce the multistability type and TN -periodic DCM type. Like in the CCM phenomenon. There exists a border between orbits type, the central zone of the 2T-lobe (light gray) of DCM and CCM, although it is not shown in the corresponds to 2TN -periodic orbits with sliding. ﬁgures when both orbits at each side of the border Second, the region QP shows more complex dy- are stable. If one of the orbits on one side is unsta- namics than without diode. It can be observed that ble, this eﬀect can be observed in the diagram. It is each kTN -periodic Arnold tongue has an accumu- worth to note that Hopf bifurcations do not appear lation of 2nkTN -periodic regions (n = 1, 2, 3, . . .) for TN -periodic orbits in the DCM. in the lower part which corresponds to a period Figure 9 shows that, with the parameters used, doubling route to chaos. Figure 11 shows a one- the addition of a diode in the circuit does not mod- dimensional bifurcation diagram for Z = 0 varying ify the curves of Hopf and Flip bifurcations (left TN . It can be observed as a torus-breaking route side) in the CCM, and the cuspid of the 2T-lobe to chaos. The rotation number has been computed has also been retained. But some important facts as the quotient of the number of cycles in the state have really changed: space to the number of cycles of the driving sig- Two-Dimensional Bifurcation Diagrams 439 Fig. 9. 2D bifurcation diagram for a Boost regulator with diode. As in Fig. 5, varying parameters are period T N and impedance Z and the ﬁxed ones are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. Fig. 10. Intensity current versus TN for the TN -periodic orbit of a Boost regulator in both switching instants. Parameters are Z = 1.01 and the set Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. Some orbits are also represented for three values of TN in order to show the discontinuous mode bifurcation. 440 L. Benadero et al. Fig. 11. Upper ﬁgures correspond to the bifurcation diagram for a Boost regulator with diode and period T N , which present a Hopf bifurcation and the associated rotation number, showing clearly the devil staircase pattern. Partial zoom is showed e in both central ﬁgures. The bottom ﬁgures are Poincar´ map orbits for three periods showing phase locking (8T N ) (left), quasiperiodicity (center), and phase locking (30/2TN ) (right). In the last one only the attractor is represented; the others show also the divergence from the unstable 1T orbit. Parameters are Z = 0 and the set Q = 4, Q S = 6, VR = 1.4, VD = 1.8 and p = 0. Two-Dimensional Bifurcation Diagrams 441 Fig. 12. This ﬁgure is the same as Fig. 5, but changing the sweep as speciﬁed in the text; the sweeping direction is also indicated by the arrows. nal and a typical devil staircase has been obtained. values for Q and QS are related to DCM. This is The same ﬁgure includes the Poincar´ sections for e due to the increase in the relative amplitude of the some orbits corresponding to several values for T N . oscillations. On the other hand, the evolution of Some of them correspond to phase-locking orbits; the Lyapunov exponents shows that the stability of for instance, we have period 8TN for TN = 0.15 and the region 1T-CCM depends on Q and QS in diﬀer- period 15TN for TN = 0.16 (its period is the ad- ent ways (see Fig. 14). Concretely, stability grows dition of the periods of the neighboring 8T N and with Q and decreases when QS is made higher. If 7TN periodic orbits). Finally, for TN near 0.153 we we increase QS then the losses in the inductor de- obtain a quasiperiodic attractor [El Aroudi et al., crease and thus the system is made less stable. The 1999]. reason why the converter gets better stability when Third, the grazing border is found for lower val- Q is increased is not so easy to explain. On one side, ues for Z. This means that the QP region is bigger Q has a similar role like QS . But on the other side, than before. Near this border there is also multi- Q moves the orbit in the phase space in a sense that stability. To show that, we plot a new bifurcation the critical impedance gets lower, and the stability diagram of the same zone sweeping the parameter is favored [Benadero et al., 2002]. This latter eﬀect ranges in the opposite order (see Fig. 12), and we has predominance over the decrease of the losses in obtain slightly diﬀerent patterns, specially in the the inductor. overlapping of the Arnold tongues. Moreover de- Figure 15(a) shows a two-dimensional bifurca- pending on the initial conditions, if Z < 0.08 a ﬁxed tion diagram with Q = 8 instead of Q = 4. As it point attractor is also possible. has been explained, this makes the stability grow and favors DCM. The eﬀect in the diagram is that the curve of Hopf bifurcations near TN = 0 moves 9. Eﬀect of the Other Parameters in approximately from Z = −0.05 (for Q = 4) to the Two-Dimensional Bifurcation Z = −0.12 (for Q = 8). Also, since DCM is fa- Diagram for the Boost Converter vored, the region 1T-DCM moves to the left lower end. This makes dissappear the region 2T-CCM 9.1. Eﬀect of Q and QS and makes appear a little 1T-DCM region inside the QP. Inside this zone of the parameter space we In Fig. 13, we plot the values of the intensity cur- have 2T-DCM behavior beside the right part of the rent at the switching instant against the bifurcation 1T-DCM region. parameter TN . The quality factors Q in (a), and Figure 15(b) shows a two-dimensional bifurca- QS in (b), are taken as secondary parameter. High tion diagram with QS = 3 instead of QS = 6. In 442 L. Benadero et al. (a) (b) Fig. 13. Intensity current at the switching instants for a Boost regulator versus period T N . (a) QS is ﬁxed and Q takes diﬀerent values; (b) Q is ﬁxed and QS is varied. The remaining parameters are Z = 0, VR = 1.4, VD = 1.8 and p = 0. (a) (b) Fig. 14. Computed Lyapunov exponents for a Boost regulator versus period TN . (a) QS is ﬁxed and Q takes diﬀerent values; (b) Q is ﬁxed and QS is varied. The remaining parameters are Z = 0, VR = 1.4, VD = 1.8 and p = 0. Two-Dimensional Bifurcation Diagrams 443 (a) (b) Fig. 15. 2D bifurcation diagram for a Boost regulator. Varying parameters are period T N and impedance Z. Fixed parameters are VR = 1.4, VD = 1.8, p = 0. (a) Q = 8, QS = 6; (b) Q = 4, QS = 3. 444 L. Benadero et al. (a) (b) Fig. 16. 2D bifurcation diagram for a Boost regulator. Varying parameters are period T N and impedance Z. Fixed parameters are Q = 4, QS = 6, p = 0. (a) VR = 2, VD = 1.8; (b) VR = 1.4, VD = 0.8. this case the curve of Hopf bifurcations near T N = 0 and thus the values for Z are always positive. This also moves to a lower position (from Z = −0.05 for result, which agrees with the fact that when V R QS = 6 to Z = −0.18 for QS = 3). Also the QP is increased the gain voltage rises up, can also be region is stretched. explained from the point of view of the critical impedance, which increases with VR . 9.2. Eﬀect of the parameters of the We show in Fig. 16(b) a two-dimensional bi- control (regulation band) VR, VD, p furcation diagram for VD = 0.8 instead of VD = 1.8, which has the same eﬀect as before, moving We show in Fig. 16(a) a two-dimensional bifurca- the curve of Hopf bifurcations to a higher posi- tion diagram for VR = 2.0 instead of VR = 1.4. tion. Qualitatively, to make VD lower means to This moves the curve of Hopf bifurcations higher stretch the regulation band; thus in some sense, the Two-Dimensional Bifurcation Diagrams 445 But when higher periods are needed, the stability gets better when the ramp has positive tangent. It is worth to note that in the Boost and Buck–Boost converters with p = 1 the stability region gets big- ger when TN grows (this is, the border moves down); on the contrary, if p = 0 the movement is in the op- posite direction. This can be observed in Figs. 9 Fig. 17. Lyapunov exponents for a Boost regulator with and 18. parameters Q = 4, QS = 6, VR = 1.4, VD = 1.8. Both The patterns which were discussed before do cases p = 0 and p = 1 are represented. not change qualitatively if we vary parameter V D or VR instead of Z; the diﬀerent dynamics are vis- ited in a similar order. In Fig. 19 a two-dimensional diagram is shown using VD and VR as bifurcation parameters. 10. Two-Dimensional Bifurcation Diagrams for the PWM Buck–Boost Converter Figure 20 shows a two-dimensional bifurcation diagram for a PWM controlled Buck–Boost con- verter with a diode with the same main parameters like the Boost. Both regulators display a similar pattern. The presence of a 1T-DCM little region inside the QP zone occurs like in Sec. 9.1. The DCM dynamics is more favored in the Buck–Boost converter than in the Boost since the Buck–Boost works with lower currents. 11. Two-Dimensional Bifurcation Fig. 18. 2D bifurcation diagram for a Boost regulator. Diagrams for a PWM Buck Varying parameters are period TN and impedance Z and ﬁxed are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 1. Converter Figure 21 shows a two-dimensional bifurcation di- agram for a PWM controlled Buck converter with possibility of TN -periodicity is reduced and we have diode. In this plot, the values of the central vector a loss in the stability. of parameters have been slightly modiﬁed. Con- The bifurcation diagram in the Boost convert- cretely, we take VR = 0.5 and VD = 0.3. This ers also depends on the symmetry factor p in the ﬁgure shows some similar patterns with regards to comparator signal. For p = 0, when TN grows, the the other bifurcation diagrams presented before but limit cycle is moved to the region of higher cur- there are also some diﬀerences which will be com- rents, and this destabilizes the system. The inverse mented on. eﬀect is observed for p = 1. Lyapunov exponents Like in the other bifurcation diagrams, the are represented in Fig. 17 showing diﬀerent signs region 1T is beside the region QP; the system for slope depending on p. Figure 18 shows that for changes from 1T to QP behavior through a Hopf p = 1 (ramp with positive slope) there is quasiperi- bifurcation. And also, a little region of 1T-DCM odic behavior. For TN = 0 the bifurcation value for behavior is inside the QP region. Z does not change; but it does when TN is varied. The main diﬀerences which can be distin- In real applications, which correspond to T N near guished are the following: zero, the stability character of the regulator does First, for the values of the parameters used in not change when one or the other ramp is used. the simulations, the bifurcation of the T N -periodic 446 L. Benadero et al. Fig. 19. 2D bifurcation diagram for a Boost regulator. Varying parameters are normalized voltages for reference V R and the width of the modulating signal VD . Fixed parameters are TN = 0.2, Z = 0, Q = 4, QS = 6 and p = 0. Fig. 20. 2D bifurcation diagram for a Buck–Boost regulator with diode. Varying parameters are period T N and impedance Z and the ﬁxed ones are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. Two-Dimensional Bifurcation Diagrams 447 Fig. 21. 2D bifurcation diagram for a Buck regulator with diode. Varying parameters are period T N and impedance Z and the ﬁxed ones are Q = 4, QS = 6, VR = 0.5, VD = 0.3 and p = 0. Fig. 22. 2D bifurcation diagram for a Buck regulator. Varying parameters are period T N and impedance Z. Fixed parameters are VR = 0.5, VD = 0.3, p = 0. (a) Q = 8, QS = 6; (b) Q = 4, QS = 3. 448 L. Benadero et al. orbit occurs for negative values of the impedance Z. Second, the Buck does not have an ON ﬁx point This is due to the fact that the Boost and Buck– simultaneously with other dynamics. The position Boost regulators have a positive critical impedance of its ﬁxed points in the topologies do not allow but the Buck has null critical impedance. Thus the that, in the skipping cycle regime, the dynamics system loses stability when it is fedback positive (Z could be attracted to this type of behavior. The negative), and show quasiperiodic dynamics. zone of the parameter space where other regulators (a) (b) (c) Fig. 23. (a and b) 2D bifurcation diagrams for a Buck regulator. Varying parameters are period T N and the width of the modulating signal VD . Fixed parameters are Q = 4, QS = 3, Z = 0, VR = 0.5 and p = 0, (a) for continuous conduction mode, and (b) with diode. (c) Lyapunov exponents versus TN is represented for three values of VD , where VD = 1.34 corresponds to the maximum value of VD for 2TN -periodic orbits. Two-Dimensional Bifurcation Diagrams 449 showed ON ﬁxed point behavior (high negative val- namics instead of 2TN -periodic orbits with chatter- ues for impedances and high periods) is replaced by ing. Moreover between the 2TN -periodic region and a period adding route (we will denote this region the chaotic zone there exist periodic narrow bands by ADD). In the phase space, the orbits present of periods 2TN , 4TN , 8TN , and so on, which corre- the skipping cycle phenomenon. We observed simi- spond to a period doubling route to chaos. On the larities between this ADD region and the QP region other hand, for TN near to 1, these can be observed (see the zone on the right-hand side of Fig. 16(b)). a region with chaotic dynamics bordered by a curve Third, the eﬀect of the parameters Q and Q S of grazing type bifurcations. acts in the same direction: a higher loss of stabil- Figure 23(c) shows the Lyapunov exponents ity corresponds to bigger quality factors Q and Q S . against TN with diﬀerent values for VD . We can These eﬀects can be observed in Fig. 22. It is worth see ﬂip bifurcations when λ = 0, which correspond noting that increasing Q favors DCM. with Fig. 23(a). VD = 1.34 is the value for the cuspid of the 2T lobe. 12. Two-Dimensional Bifurcation Diagrams for a PWM 12.2. With a diode Voltage-Controlled Buck Figure 23(b) corresponds to a voltage-controlled Converter PWM converter with a diode. A region with Most practical applications of a Buck converter use 1T-DCM dynamics is clearly visible. The two- a voltage-controlled PWM loop (Z = 0). This dimensional bifurcation diagrams with and without means that a Hopf bifurcation will not appear. On diode are similar for TN < 0.15. the other hand, since the stability character almost do not depend on VR we will plot the bifurcation 12.3. Two-dimensional bifurcation diagrams in this section with parameter V D instead diagrams with VIN as of Z. bifurcation parameter In 1990, chaos via period doubling and other 12.1. Without a diode nonlinear phenomena are shown to occur in PWM Figure 23(a) shows a two-dimensional bifurcation voltage-controlled Buck converter both numerically diagram for a voltage-controlled PWM Buck con- and experimentally when input voltage V IN was verter without a diode. It can be observed that the taken as a bifurcation parameter [Deane & Hamill, lobe has changed with regards to the diagrams in 1990]. Sweeping the parameter VIN means vary- other sections. Its middle zone contains chaotic dy- ing simultaneously the dimensionless parameters V R (a) (b) (c) Fig. 24. (a) 2D bifurcation diagram for a Buck regulator. Varying parameters are period T N and the width of the modulating signal VD . The voltage reference VR = 20VD is also varying. Fixed parameters are Q = 1.1, QS = 7, Z = 0 and p = 1. e (b) Bifurcation diagram showing period doubling route to chaos, where the intensity current for Poincar´ map is represented versus VD (VR = 20VD ) and TN = 0.066. (c) Corresponds to the chaotic attractor for VD = 0.015 (VR = 0.3) and TN = 0.066. 450 L. Benadero et al. and VD keeping a linear relation between them. Banerjee, S., Ranjan, P. & Grebogi, C. [2000] A two-dimensional bifurcation diagram for a Buck “Bifurcations in two-dimensional piecewise smooth without a diode taking the values of the parameters maps-theory and applications in switching circuits,” like in [Deane & Hamill, 1990] (Q = 1, Q S = 7, IEEE Trans. Circuits Syst. I 47, 633–647. VR = VD /20, Z = 0, TN = 0.066 and p = 0) Benadero, L., El Aroudi, A., Toribio, E., Olivar, G. & has been computed with VD in the range [0, 0.05] Mart´ ınez-Salamero, L. [1999] “Characteristic curves [Fig. 24(a)]. It is worth to note that the bifurca- to analyze limit cycles behavior of DC–DC convert- tion diagram for TN = 0.066 [Fig. 24(b)] and the ers,” Electron. Lett. 687–789. Benadero, L., El Aroudi, A., Olivar, G., Toribio, E. & attractor observed for VD = 0.015 [Fig. 24(c)] per- Moreno, V. [2002] “Bifurcations analysis in PWM fectly agree with those obtained in the same work regulated DC–DC converters using averaged mod- of Deane and Hamill. els,” EPE-PEMC’02, Dubrovnik & Cavtat (Croatia), SSIN-04. 13. Conclusions Chakrabarty, K., Podar, G. & Baberjee, S. [1996] “Bifurcation behavior of the buck converter,” IEEE In this work two-dimensional bifurcation diagrams Trans. Circuits Syst. I 11, 439–447. have been extensively studied for a wide class of Deane, J. H. B. & Hamill, D. C. [1990] “Analysis, simu- basic DC–DC converters, which are used in prac- lation and experimental study of chaos in the buck tical applications. The similarities and diﬀerences converter,” IEEE Power Electronic Specialist Conf. among the Buck, Boost and Buck–Boost regulators, PESC’90, pp. 491–498. with a diode, and without, are explained from a Deodel, E. J. & Wang, X. J. [1995] AUTO94: Soft- dynamical systems and engineering points of view. ware for Continuation and Bifurcation Problems in The three subclasses show a general bifurcation Ordinary Diﬀerential Equations, Technical Report pattern which includes smooth ﬂip and Hopf bi- CRPC-95-2, California Institute of Technology. furcation, quasiperiodicity, nonsmooth bifurcations di Bernardo, M., Fossas, E., Olivar, G. & Vasca, F. [1997] and chaotic dynamics. The eﬀect due to the exis- “Secondary bifurcations and high periodic orbits in tence of a change from continuous to discontinuous voltage controlled buck converter,” Int. J. Bifurcation conduction mode has also been discussed. and Chaos 7, 2755–2771. El Aroudi, A., Olivar, G., Benadero, L. & Toribio, These two-dimensional bifurcations have been E. [1999] “Hopf bifurcation and chaos from torus precisely computed with ad hoc code taking advan- breakdown in a PWM voltage-controlled DC–DC tage of the piecewise linearity topologies. Analyt- Boost converter,” IEEE Trans. Circuits Syst. I 11, ical closed-form solutions are exploited and only 1374–1382. a numerical method is used to ﬁnd the switching El Aroudi, A., Benadero, L., Toribio, E. & Machiche, instants between the two diﬀerent topologies. A S. [2000] “Quasiperiodicty and chaos in the DC–DC large part of this work presents only a macroscop- Buck–Boost converter,” Int. J. Bifurcation and Chaos ical view of what happens to these systems when 10, 359–371. some meaningful parameters are varied. The au- Fossas, E. & Olivar, G. [1996] “Study of chaos in the buck thors are working on the details of some interesting converter,” IEEE Trans. Circuits Syst. I 43, 13–25. practical regions to provide also a microscopic view Guckenheimer, J., Myers, M., Wicklin, R. & which can be useful for design. Worfolk, P. [1991] “DStool: Dynamical sys- tem toolkit with interactive interface,” Graphic Center of Applied Mathematics, Cornell Univ. Acknowledgments www.cam.cornell.edu/guckenheimer/dstool The authors would like to acknowledge David Hamill, D. C. & Jeﬀeries, D. J. [1988] “Subharmonics Carri´ for his invaluable help in building up a o and chaos in a controlled switched–mode power con- great part of the numerical simulator. This work verter,” IEEE Trans. Circuits Syst. I 35 1059–1060. was supported by the Spanish CYCIT under Grant Hamill, D. C., Deane, J. H. B. & Jeﬀeries, D. J. [1992] “Modeling of chaotic DC–DC converters by iterated DPI2000-1509-C03-02 and TIC2000-1019-C02-01. nonlinear mappings,” IEEE Trans. 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