two-dimensional bifurcation diagrams. background pattern of ...
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 periodic. In this case, the global Poincar´e 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 different linear configurations. This must be implemented with an appropriate control of the switches. In a noise perturbation free environment, given the desired output voltage, the switching frequency can be selected and the switches can be turned ON and OFF according to a fixed pattern; this is referred to as the open loop system. In contrast, in industrial applications, noise and perturbations are always present, and also the parameters of the circuits may be affected by external disturbances. Thus the use of an appropriate control to counteract the modifications on the output voltage in the system
The basic DC–DC converters Buck, Boost and Buck–Boost are a family of circuits which allow the conversion of energy from one level to another without taking into account, theoretically, losses in the components. They are used extensively in power supplies for electronic circuits and in the control of the flow of energy between DC to DC systems, and in any industrial application where there is a need of stabilizing an output voltage to a desired value. Also, they are widely used in small spacecrafts such as satellites where DC power is generated by solar arrays.
427
428 L. Benadero et al.
is recommended; this is referred to as the closed loop system. The most popular control strategy used in the literature is Pulse Width Modulation (PWM) where electronic control of the basic power electronic converter circuit is achieved by controlling the duty cycle d of the controlled switch S (the duty cycle is the ratio of the ON phase of the switch to the period of the periodic ON–OFF operation). We will refer to the ON phase when the switch S is closed and diode D is open; the OFF phase refers to when the switch S is open and diode D closed; and mode OFF’ (or discontinuous mode) takes place when both switch and diode are open (see Fig. 2). There are many ways that fixed frequency PWM control can be implemented. Nevertheless, the basic ingredients of almost all existing PWM controllers that are used for voltage control are: 1. an output voltage error amplifier 2. a T -periodic sawtooth signal generator (driving signal) 3. a comparator that compares the error amplifier output with the sawtooth waveforms. The most interesting dynamics of these systems, from a classical design point of view, is the T -periodic orbit (periodic evolution with the same period as the driving signal). Nonlinear phenomena in the PWM voltage controlled DC–DC basic power electronic regulators have been studied in the past years. Various kinds of bifurcational behaviors are found for different converters with different control schemes. Flip bifurcations and period doubling route to chaos are found in the Buck converter [Deane & Hamill, 1990; Fossas & Olivar, 1996; Tse, 1994a], Neimark–Sacker bifurcation and quasiperiodicity route to chaos are found to occur in the PWM Boost and Buck–Boost converters [El Aroudi et al., 2000] and border collision bifurcations are found to occur in the Buck and the Boost converter with different control strategies [Banerjee et al., 2000; Yuan et al., 1998]. Up to now there are very few works that try to characterize the bifurcational phenomena in the parameter space [Chakrabarty et al., 1996; Banerjee & Chakrabarty, 1998; El Aroudi et al., 2000; Olivar, 1997; Toribio et al., 2000]. The aim of this paper is to investigate in the parameter space the mechanisms of losing the stability of the T -periodic orbit, and the transition between the different bifurcations in these systems.
2. Continuous Time Model of the Basic Switching Regulators 2.1. State equations The basic DC–DC switching converters are shown in Fig. 2. The differential equations, modeling each one of the three configurations that use every converter, can be derived by using the standard Kirchoff’s laws. Let us define matrices A 1 , A2 , A3 , B1 and B2 as follows: 1 1 1 − − 0 RC C , , A2 = RC A1 = 1 1 1 − − 0 − L RS L
1 − RC A3 = 0 B2 =
' (
0
0
0
0
(1)
, B1 = VIN ,
L
0
where R is the output load resistance, L is the inductance which is supposed to have an Equivalent Series Resistance ESR RS , C is the capacitance, and VIN is the input voltage. During each phase (ON, OFF and OFF’), and until a switching condition is fulfilled, the dynamics of the system is described by: X˙ = AX + B (2) X = (vC , iL )T is the vector of the state variables and the overdot stands for derivation with respect to time t(X˙ = dX/dt). Table 1 shows the A’s and B’s matrices for the three basic converters Buck, Boost and Buck–Boost during each phase.
2.2. Analytical solutions for each configuration Since the previous differential equations are piecewise linear (PWL), a closed form solution is Table 1. The A’s and B’s matrix for the basic converters during phases ON, OFF and OFF’. Converter
AON
AOFF
AOFF!
BON
BOFF
BOFF!
Buck
A1
A1
A3
B1
B2
B2
Boost
A2
A1
A3
B1
B1
B2
Buck–Boost
A2
A1
A3
B1
B2
B2
Two-Dimensional Bifurcation Diagrams 429
available for each configuration. Let us write: kC =
1 , 2RC ω0 =
kL =
)
1 LC
RS , 2L
k = kc + kL ,
*
RS 1+ R
+
(3) −
k2
Therefore, the solution for each configuration can be written as: • RC circuit and L in series with VIN (matrices A2 and B1 ) (this configuration corresponds to the ON phase for the Boost and the Buck–Boost): vC (t) = vC0 e−2kC (t−t0 ) iL (t) =
*
VIN VIN + iL0 − RS RS
+
e−2kL (t−t0 )
(4)
• RLC oscillator in the free regime (matrices A 1 and B2 ) (this configuration corresponds to the OF F phase for the Buck and the Buck–Boost): vC (t) = e−k(t−t0 ) [vC0 cos ω0 (t − t0 ) +
*
+
kvC0 iL0 − sin ω0 (t − t0 )] Cω0 ω0
iL (t) = exp−k(t−t0 ) [iL0 cos ω0 (t − t0 ) + * vC0 iL0 − sin ω0 (t − t0 )] + ω0 Lω0
(5)
• RLC oscillator forced with VIN (matrices A1 and B1 ) (this configuration corresponds to the ON phase for the Boost, and the OF F phase for the Buck–Boost): vC (t) = VC∞ + e−k(t−t0 ) [(vC0 − VC∞ ) cos ω0 (t − t0 ) + −
+
− IL∞ ) cos ω0 (t − t0 ) + −
*
(iL0 − IL∞ ) Cω0
k(vC0 − VC∞ ) sin ω0 (t − t0 )] ω0
iL (t) = IL∞ + exp−k(t−t0 ) [(iL0
+
*
(iL0 − IL∞ ) ω0
(vC0 − VC∞ ) sin ω0 (t − t0 )] Lω0
(6) • The capacitor connected to the load when the converter works in the so-called Discontinuous Conduction Mode (DCM), characterized by iL (t) = 0) (matrices A3 and B2 ): vC (t) = vC0 e−2kC (t−t0 ) iL (t) = 0
where t0 is the initial time at which the system switches from one configuration to another, and i L0 and vC0 are the states of the system at the switching instant t0 . The values for VC∞ and IL∞ are VC∞ =
VIN , RS 1+ R
IL∞ =
VC∞ R
The above solutions are only valid when we have * + 1 RS 1+ − k2 > 0 (8) LC R in such a way that ω0 is real and positive. From the design point of view this is the most important case, since it gives oscillatory solutions.
2.3. The switching conditions The PWM control of a switched converter is achieved by the comparison of the control voltage vcon which is a linear combination of the capacitor voltage vC and the inductor current iL in the form vcon = A(vC + Zr iL − VREF )
(9)
with a driving signal, generalized as a triangular function [Fig. 1(a)]
vtriang (t) =
VU −VL t VL + pT
VU −VL VU − (t−pT )
(1−p)T
if 0 < t < pT if pT < t < T
(10) where A is the gain of the error amplifier, Z r is the impedance used to convert the inductor current to a voltage, VREF is the reference voltage, VL and VU are the lower and upper values of the driving triangular signal, T and p are the periode and symmetry factor of this signal. Let us define the function Vcomp as: Vcomp (t) = vcon (t) − vtriang (t)
= A(vC + Zr iL − VREF ) − vtriang (t) (11)
The switching condition is therefore:
(7) 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) Fig. 1. (a) Triangular signal vtriang (t) used as driving signal in control. (b) Normalized function h(τ ).
Since the expressions of the trajectories in each configuration include exponential and trigonometric functions, this equation is transcendental. Thus, a closed form expression for the solution is not possible [Hamill et al., 1992; El Aroudi et al., 1999; Fossas & Olivar, 1996]. Hence, one must resort to numerical methods to compute the switching instants. It should be noted that when the converter enters in discontinuous conduction mode (that is, changes from iL (t) "= 0 to iL (t) = 0), a new switching condition (which is iL = 0) appears.
2.4. Numerically computed orbits Numerical methods usually play a major role when the system is nonlinear and parameters must be varied in certain ranges. Although there exist some very useful packages for the study of the behavior of dynamical systems, to our knowledge, none is specially suited for piecewise linearities. In these systems, one may take profit of the analytically-computed solutions, but the switching instants must be numerically computed. The PSPICE package [Rashid, 1990] is the more real-
istic waveform simulator for DC to DC converters because it is designed for electrical and electronic circuits. However, since the simulations are very time-consuming, one cannot rely on this package for extensive computations. Other packages, like LOCBIF [Khibnik et al., 1993], INSITE [Parker & Chua, 1987], DSTOOL [Guckenheimer et al., 1991] and AUTO [Deodel & Wang, 1995] are well suited only for smooth systems. They compute equilibrium points, eigenvalues, characteristic multipliers, Lyapunov exponents and invariant manifolds assuming that the vector field is smooth enough. But the switching action in power electronic converters makes these systems very different from those characterized by a smooth vector field. The vector field for DC–DC switching converters is discontinuous and PWL in the form: X˙ =
0
f1 (X, t)
if
g(X, t) < 0
f2 (X, t)
if
g(X, t) > 0
(13)
In DCM, the previous model is still valid and the functions f1 or f2 are in turn PWL. Moreover, these numerical packages do not allow to handle two-dimensional bifurcation diagrams easily. Finally, as it was mentioned before, the numerical method used for integrating the system of differential equations is not optimized for PWL systems, where closed form solutions in each sub adjacent
Two-Dimensional Bifurcation Diagrams 431
interval are available. For these reasons, some specific programs written in C code are used in this paper, using the closed form solution in each linear configuration. The equation for the switching condition is solved by a specific root-finding method (Newton, secant, bisection, . . . etc.). In our computations, the bisection method is used. Therefore the analytical solutions combined with the bisection method to locate the switching instants will reduce considerably the time needed for a given simulation. In this paper, the global dynamics is obtained in this way. The simulations are therefore very fast compared to simulations computed with standard packages. This method was already used by the authors in [Deane & Hamill, 1990] and [Fossas & Olivar, 1996] to compute trajectories of the PWM voltage controlled DC–DC Buck converter.
3. Dimensionless Formulation 3.1. Background The large number of parameters associated with the PWM DC–DC converters is a major handicap to the characterization of all the possible dynamics. To deal with this problem, some authors have fixed the values of the parameters according to specific examples [Hamill et al., 1992; Tse, 1994b] and then they varied the parameters near these values. Instead, we will define dimensionless parameters achieving a significant reduction in the number of independent parameters of the circuit. We begin with linear transformation of the state variables in such a way that the resulting new variables are dimensionless. We choose Ts , Vs , Is scale parameters 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 (related 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 Among all possible choices of the scale parameters,
the following will be chosen: √ Ts = T0 = 2π LC, Vs = VIN ,
VIN Is = 1 L/C
Then, we define the normalized variables and parameters of the power circuit as follows: vC (t) v(t) = , VIN
1
L/C t iL (t), τ = , VIN T0 1 L/C R , QS = Q= 1 R L/C S (14) where Q is the quality factor associated to the load, and QS is the quality factor associated to the equivalent series resistance ESR of the inductance. Finally, τ is the dimensionless time. i(t) =
3.3. Scaling the switching condition Introducing the normalization in the control condition, we get fS (v, i, τ ) = v + Zi − VR − VD h(τ ) = 0
(15)
where VR is the normalized reference voltage, V D is the normalized width of the external control signal and Z is the normalized impedance. h(τ ) becomes a triangular function with amplitude one, null average value, period TN and symmetry factor p [see Fig. 1(b)]. VR =
Fig. 3. (v, i).
VU − VL VREF VU + VL + , VD = , VIN 2AVIN AVIN (16) Zr T 1 Z= , TN = T0 L/C
The switching band in the normalized phase plane
432 L. Benadero et al.
It should be noted that the evolution of the driving signal voltage may be represented by a switching band in the (v, i) phase plane (Fig. 3). Within this band the switchings from one configuration to another can occur. With the proposed normalization, the resulting state variables and parameters are dimensionless. Their number is reduced from 10 to 5. The normalization means that different circuit designs with equal normalized parameters have equivalent dynamics. After scaling variables and parameters, the dimensionless differential equations of each configuration can be written Y˙ = CY + D
(17)
where the overdot denotes for the derivation with respect to the dimensionless time τ . Y = (v, i) T is the vector of the normalized state variables and the dimensionless matrices are C1 , C2 , C3 , D1 and D2 as follows:
1 − Q
C1 = 2π
−1
−
C3 = 2π
1 Q
0
1 − Q
1 , C2 = 2π 1 − 0 QS
0
0
, D1 =
'
0 2π
(
0 , 1 − QS
, D2 =
' (
3.4. Normalized analytical solutions of each configuration Note that Eq. (8) is equivalent to 1+
1 − QQS
* *
1 2
1 1 + Q QS
++2
>0
Assuming that the condition above is fulfilled, the closed form solutions for each configuration are: • The LRC parallel oscillator with the source V IN in series with the inductor.
v(τ ) = V∞ + exp−α(τ −τ0 ) [(v0 − V∞ ) cos ωN (τ − τ0 ) +
*
+
i0 − I∞ − β(v0 − V∞ ) sin ωN (τ − τ0 )] δ
i(τ ) = I∞ + exp−α(τ −τ0 ) [(i0 − I∞ ) cos ωN (τ − τ0 ) *
+
v0 − V∞ − β(i0 − I∞ ) − sin ωN (τ − τ0 )] δ
0 0
(18)
Matrices Ci and vectors Di correspond to normalized ones Ai and Bi respectively (see Table 1). Therefore, the converter depends on four essential dimensionless parameters Q, TN , VD , VR and an additional (parasitic) parameter Q S (the essential and parasitic classification is due to circuit considerations, but this will not affect the analysis). T N is the period of the driving signal normalized to the natural period of the LC-circuit, and finally, in the event of Rs = 0 (not considered here), Q is the quality factor of the power circuit. Thus, a DC– DC converter can be characterized by the normalized parameter set C = {Q, QS , TN , VR , VD }. The set C of dimensionless parameters of one specific PWM DC–DC regulator stands for a whole family of regulators whose dimensionless parameters are those in C, and thus they display the same dynamics (obviously, this is true only if the converter is the same and its model remains valid for the entire family).
(19)
(20) Note that the equilibrium point (QQ S /(Q + QS ), QS /(Q + QS )) of this configuration is a stable spiral sink. • The LRC parallel oscillator in the free regime. v(τ ) = exp−α(τ −τ0 ) [v0 cos ωN (τ − τ0 ) *
+
i0 − βv0 + sin ωN (τ − τ0 )] δ i(τ ) = exp−α(τ −τ0 ) [i0 cos ωN (τ − τ0 ) −
*
(21)
+
v0 − βi0 sin ωN (τ − τ0 )] δ
The equilibrium point (0, 0) of this configuration is a stable spiral sink. • The inductor connected to the source, and the RC circuit: v(τ ) = v0 exp
− 2π (τ −τ0 ) Q
i(τ ) = QS + (i0 − QS ) exp
2π (τ −τ0 ) −Q S
(22)
Two-Dimensional Bifurcation Diagrams 433
4. The Poincar´ e Map In the case of periodically driven systems, the usual Poincar´e section considered is a plane Σ (with equation 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 configurations that use the Boost converter.
The equilibrium point (0, QS ) of this configuration is a stable sink. • The RC circuit (this corresponds when the converter works in DCM): v(τ ) = v0 exp i(τ ) = 0
(τ −τ0 ) − 2π Q
(23)
The equilibrium point (i = 0) of this configuration is a stable one-dimensional1 sink. In the expressions of the solutions, we have used V∞ =1QQS /(1 + QQS ), I∞ = V∞ /Q, ωN = 2πδ, δ = (1 − β 2 ), β = 1/2(1/Q − 1/QS ) and α = π(1/Q + 1/QS ). Also, v0 , i0 are the state variables at the switching instant τ0 . As we have seen, the dynamics in each configuration is well determined. But the parameters of the circuit are selected in such a way that when we make the control work, the converter switches from one configuration to another. Therefore, since the system is highly nonlinear due to the switching action, a great variety of dynamics are possible (limit cycles, subharmonics, quasiperiodicity, chaos). In Fig. 4, the switching lines for a PWM voltage mode controlled Boost regulator and the trajectories near the equilibrium points of each configuration are schematically plotted. 1
Assume we have a converter with all parameters 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 beginning and those at the end of each cycle of the driving signal and, on the other hand, forcing the continuity of the state variables at the asynchronous switching instants [di Bernardo et al., 1997]. By imposing these conditions, a system of two nonlinear equations with two unknowns veq and ieq is obtained. By imposing the asynchronous switching condition, another equation is added, being the switching time τS another unknown. These three nonlinear equations with three unknowns can be introduced to a standard mathematics program (for example Maple V) to find the fixed points (v eq , ieq ) for a set of parameter values.
5. Mechanisms of Losing the Stability of the One-Periodic Orbit The basic elements which will be used in this section to study the stability of the one-periodic orbit are the Poincar´e map and the characteristic multipliers of its fixed points. The stability of the one-periodic orbit is analyzed by means of the response to a small perturbation near this fixed point. To obtain the stability character of a fixed point (veq , ieq ) of this map is enough to compute the image of a perturbed state near the fixed point in the phase plane (vn , in ). This transforms the problem of analyzing the stability of the one-periodic orbit to the study of the eigenvalues of the linearized
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 associated to the linearization near a fixed point are obtained by computing the transformation of the state space points in the neighbor of the fixed points. From the eigenvalues m1 and m2 of DP (the characteristic multipliers of the orbit), or from the Lyapunov exponents (λ = log |m|/TN ) the following bifurcations will be detected: • Flip or period doubling bifurcation of the stable one-periodic orbit. It is characterized by the fact that one of the characteristic multipliers is equal to −1 and the other one has absolute value less than 1. • Hopf or Neimark–Saker bifurcation. It is characterized by the fact that both characteristic multipliers cross the unit circle, being complex conjugates. • Border-Collision bifurcations. They are characterized by a sudden change in the system behavior accompanied by a jump in the values of the characteristic multipliers. These anomalous bifurcations are well explained in [Yuan et al., 1998; Banerjee et al., 2000; Nusse et al., 1994] and [Nusse & Yorke, 1992].
6. Two-Dimensional Bifurcation Diagrams In order to obtain the bifurcational structure when two parameters are varied, we will plot the twodimensional bifurcation diagrams corresponding to these parameters. The two-dimensional bifurcation diagram is color-coded depending on the periodicity of the attractor corresponding to the point in the parameter space. Due to hysteresis phenomena and in general to possible coexisting attractors, only one of them can be identified at each point of the parameter space. In this paper the diagrams are produced as follows: first we give the system with extreme values of both varying parameters and the initial condition xini = (vini , iini ); then, one-dimensional bifurcation diagrams are successively computed fixing one parameter (primary) and varying the other one (secondary). The final state of the system is taken as initial condition to the next value of the primary parameter. The initial condition when the secondary parameter is varied is the last state calculated using the preceding value of the secondary parameter, both with the same extreme value of the primary parameter. Arrows in the diagrams
will specify the sense of variations of the parameters; the larger arrow corresponds to the parameter chosen as primary (in most of the diagrams this is TN ). Also, in order to understand the bifurcation patterns better, some simulations have been made using an ideal switch instead of a diode.
6.1. A point in the parameter space To obtain a good comparison of all the twodimensional bifurcation diagrams, we choose a central point in the parameter space and then, we vary the parameters around this point. In this zone we will explore all the possible dynamics of the three converters. We fix the values of this parameter point to Q = 4, QS = 6 and Z = 0. Moreover we take VR = 0.5, VD = 0.3 for the Buck, and VR = 1.4, VD = 1.8 for the Boost and the Buck–Boost. We cannot ideally take the same VR in the three converters since its value tightly depends on the voltage gain. Thus it is convenient to take V R less than one in the case of the Buck, which reduces the input voltage; and more than one in the case of the Boost, which increases the input voltage. In the case of the Buck–Boost it can be any value since this converter can be designed to increase or reduce the input voltage. The range of variation of the normalized period is large enough to consider not only low periods (TN ( 1), which is the usual range in circuits, but also high periods near TN = 0.7 are checked. It will be shown that some bifurcations which pose limits on the TN -periodic dynamics are met near TN = 1. With regards to parameter p, we will use only its extreme values p = 0 and p = 1.
6.2. Choosing the parameters to vary For the two-dimensional bifurcation diagrams, we have to choose two parameters from the seven (T N , Q, QS , Z, VD , VR , p) which correspond to a given circuit. The bifurcation patterns will be obtained from different pairs of parameters. We classify all our possible parameters according to three different classes. One class contains the parameter associated to time in the control loop TN , and it will be one of the parameters varied in each of our bifurcation diagrams; another class will be made with the parameters associated to the power circuit, these are Q and QS , which will be kept fixed in each of the 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 specifically, they define the regulation band (Z, V D , VR and p). One of them will be taken to be varied when we compute the two-dimensional bifurcation 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 case parameters VD and Z will be varied. Parameter Z allows to detect more easily all the possible dynamics.
7. Bifurcation Diagrams for the Boost Converter in the (TN , Z)-Parameter Space. Regions in the Parameter Space 7.1. Basic regions according to the dynamical behavior Using the numerical simulator which has been 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 PWM control, and without diode, this is, with both ideal and complementary switches. The parameters are fixed according to the central point defined before (Q = 4, QS = 6, VR = 1.4, VD = 1.8, p = 0) and we vary Z and TN (see Fig. 5). The resulting bifurcation diagram is simpler than the one obtained using a diode, and we can distinguish four big different regions, which are described in the following: First, we have a big two-periodic region of lobar type, which will be called 2T region, on the righthand side of the figure. Into this zone the orbits display the chattering phenomenon and too much time is required to decide the periodicity. The same phenomenon is detected on other similar lobar areas, which yield sliding modes like in Fig. 6, with Z = 0.82 and TN = 0.55. Second, we have a region whose stationary behavior is the fixed point of the ON topology, in the lower part of the figure. This region will be called the FP region. On the right part of its upper border we found that the impedance is constant at Z = 0.08, and the left part is made of a curve which starts at a point with TN = 0.18 approximately. Third, between the region FP and the twoperiodic lobar region, we found chaotic and quasiperiodic orbits with some little periodic islands. The chaotic and quasiperiodic orbits are found after Hopf bifurcation following the torus breaking route to chaos. This zone will be called QP. Fourth, there is a large one-periodic dynamics (blue) which is in the remainder of the figure. This is the usual region of operation of the regulators and we will denote it by TN -periodic or simply 1T.
Its linearized dynamics corresponds to a focus, with negative Lyapunov exponents. It is clear that the border of the region 1T shows a discontinuity at a point of the parameter space which corresponds approximately to T N = 0.50 and Z = 0.19. This type of behavior, which is common in a certain sense, is associated to a variation pattern in the characteristic multipliers when the parameters are moved, and this implies two different types of borders.
7.2. Curve of Hopf bifurcations At the border between the zones 1T and QP a Hopf bifurcation occurs, which corresponds to a null Lyapunov exponent. The bifurcation occurs smoothly when two complex characteristic multipliers cross the unit circle. If we decrease the impedance Z the characteristic multipliers begin to grow and finally they cross the unit circle at the border. We will call this border the Hopf border. For TN near 0 this border begins at Z = −0.05, which can be analytically predicted when averaging techniques are used. This is tightly related with the critical value of the impedance Z associated to the equilibrium [Benadero et al., 1999] which is related to the orientation of the vector field. The value of the critical impedance fixes, with the exception of a certain correction which depends on the values of Q and QS , the impedance Z which makes the transition to instability. For different values of T N , the bifurcation value for Z is changed and the period one always appears in the upper region of the 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 The border between the region 1T and the 2T lobe corresponds to a flip bifurcation. At this point, both Lyapunov exponents are real and the larger one becomes null. We will call this border the flip border. Figure 7(a) shows both two exponents when parameter TN is varied, and three important facts are observed: first, if we decrease Z then the average value of the Lyapunov exponents increases; second, if we increase TN the average value also increases when p = 0 (this effect is inverted when p = 1, see Sec. 9.2); and third, a bifurcation from focus to a node is produced. From this bifurcation point and onwards, one exponent grows until it becomes null (this exponent is associated with a characteristic multiplier equal to −1) and a flip bifurcation 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 corresponds to the point Z = 0.19 and TN = 0.50, where the two curves coalesce. Figure 7(b) corresponds to the cuspid of the lobe 2T; for instance, with the arbitrary value Z = 0.82, we have bifurcations at TN = 0.47 and
TN = 0.63. As Z is increased, the larger Lyapunov exponent does not pass through zero, and thus no bifurcation is produced. The limit case is obtained approximately for Z = 1.01 and TN = 0.51.
7.4. Curve of grazing bifurcations Finally, the border between the region FP and QP corresponds to a grazing bifurcation, and thus will be called the grazing border. A grazing bifurcation is a subtype of the so-called border collision bifurcations and can be only observed in nonsmooth systems. This can be observed in Fig. 8 which corresponds to an orbit of the regulator with Z = 0.08. Observe that the left line border of the switching band passes through the fixed point of the ON state.
8. Two-Dimensional Bifurcation Diagrams for the Boost Converter with a Diode It should be expected that the two-dimensional bifurcation diagram for a Boost converter with a
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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 without it, since we add one more topology to the circuit. As we shall see, this is really so. The new attractors which are obtained visit periodically or intermittently, the discontinuous conduction mode, and can be periodic, quasiperiodic or chaotic. We will use CCM for dynamics of continuous conduction mode and DCM have been denoted for discontinuous conduction mode dynamics. As a general rule, DCM stabilizes quasiperiodic and chaotic behavior and contributes to reduce the multistability phenomenon. There exists a border between orbits of DCM and CCM, although it is not shown in the figures when both orbits at each side of the border are stable. If one of the orbits on one side is unstable, this effect can be observed in the diagram. It is worth to note that Hopf bifurcations do not appear for TN -periodic orbits in the DCM. Figure 9 shows that, with the parameters used, the addition of a diode in the circuit does not modify the curves of Hopf and Flip bifurcations (left side) in the CCM, and the cuspid of the 2T-lobe has also been retained. But some important facts have really changed:
First, a region of DCM dynamics of TN -periodic type has eroded the 2T-lobe from its right part (this is, for high values of TN and Z). To check this point, we plot in Fig. 10 three different orbits for different values of TN and Z fixed to 1.01 (the value corresponding to the cuspid of the 2T-lobe). The orbit which corresponds to TN = 0.37 is TN -periodic of CCM type; the orbit for TN = 0.65 is TN -periodic of DCM type, and the orbit for TN = 0.51 is placed just within the border between TN -periodic CCM type and TN -periodic DCM type. Like in the CCM type, the central zone of the 2T-lobe (light gray) corresponds to 2TN -periodic orbits with sliding. Second, the region QP shows more complex dynamics than without diode. It can be observed that each kTN -periodic Arnold tongue has an accumulation of 2nkTN -periodic regions (n = 1, 2, 3, . . .) in the lower part which corresponds to a period doubling route to chaos. Figure 11 shows a onedimensional bifurcation diagram for Z = 0 varying TN . It can be observed as a torus-breaking route to chaos. The rotation number has been computed as the quotient of the number of cycles in the state 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.
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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 in both central figures. The bottom figures are Poincar´e 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. The same figure includes the Poincar´e sections for some orbits corresponding to several values for T N . Some of them correspond to phase-locking orbits; for instance, we have period 8TN for TN = 0.15 and period 15TN for TN = 0.16 (its period is the addition of the periods of the neighboring 8T N and 7TN periodic orbits). Finally, for TN near 0.153 we obtain a quasiperiodic attractor [El Aroudi et al., 1999]. Third, the grazing border is found for lower values for Z. This means that the QP region is bigger than before. Near this border there is also multistability. To show that, we plot a new bifurcation diagram of the same zone sweeping the parameter ranges in the opposite order (see Fig. 12), and we obtain slightly different patterns, specially in the overlapping of the Arnold tongues. Moreover depending on the initial conditions, if Z < 0.08 a fixed point attractor is also possible.
9. Effect of the Other Parameters in the Two-Dimensional Bifurcation Diagram for the Boost Converter 9.1. Effect of Q and QS In Fig. 13, we plot the values of the intensity current at the switching instant against the bifurcation parameter TN . The quality factors Q in (a), and QS in (b), are taken as secondary parameter. High
values for Q and QS are related to DCM. This is due to the increase in the relative amplitude of the oscillations. On the other hand, the evolution of the Lyapunov exponents shows that the stability of the region 1T-CCM depends on Q and QS in different ways (see Fig. 14). Concretely, stability grows with Q and decreases when QS is made higher. If we increase QS then the losses in the inductor decrease and thus the system is made less stable. The reason why the converter gets better stability when Q is increased is not so easy to explain. On one side, Q has a similar role like QS . But on the other side, Q moves the orbit in the phase space in a sense that the critical impedance gets lower, and the stability is favored [Benadero et al., 2002]. This latter effect has predominance over the decrease of the losses in the inductor. Figure 15(a) shows a two-dimensional bifurcation diagram with Q = 8 instead of Q = 4. As it 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 approximately from Z = −0.05 (for Q = 4) to Z = −0.12 (for Q = 8). Also, since DCM is favored, the region 1T-DCM moves to the left lower end. This makes dissappear the region 2T-CCM and makes appear a little 1T-DCM region inside the QP. Inside this zone of the parameter space we have 2T-DCM behavior beside the right part of the 1T-DCM region. Figure 15(b) shows a two-dimensional bifurcation diagram with QS = 3 instead of QS = 6. In
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(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.
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(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 also moves to a lower position (from Z = −0.05 for QS = 6 to Z = −0.18 for QS = 3). Also the QP region is stretched.
9.2. Effect of the parameters of the control (regulation band) VR, VD, p We show in Fig. 16(a) a two-dimensional bifurcation diagram for VR = 2.0 instead of VR = 1.4. This moves the curve of Hopf bifurcations higher
and thus the values for Z are always positive. This result, which agrees with the fact that when V R is increased the gain voltage rises up, can also be explained from the point of view of the critical impedance, which increases with VR . We show in Fig. 16(b) a two-dimensional bifurcation diagram for VD = 0.8 instead of VD = 1.8, which has the same effect as before, moving the curve of Hopf bifurcations to a higher position. Qualitatively, to make VD lower means to stretch the regulation band; thus in some sense, the
Two-Dimensional Bifurcation Diagrams 445
Fig. 17. Lyapunov exponents for a Boost regulator with parameters Q = 4, QS = 6, VR = 1.4, VD = 1.8. Both cases p = 0 and p = 1 are represented.
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 bigger when TN grows (this is, the border moves down); on the contrary, if p = 0 the movement is in the opposite direction. This can be observed in Figs. 9 and 18. The patterns which were discussed before do not change qualitatively if we vary parameter V D or VR instead of Z; the different dynamics are visited 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 converter 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.
Fig. 18. 2D bifurcation diagram for a Boost regulator. Varying parameters are period TN and impedance Z and fixed are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 1.
11. Two-Dimensional Bifurcation Diagrams for a PWM Buck Converter
possibility of TN -periodicity is reduced and we have a loss in the stability. The bifurcation diagram in the Boost converters also depends on the symmetry factor p in the comparator signal. For p = 0, when TN grows, the limit cycle is moved to the region of higher currents, and this destabilizes the system. The inverse effect is observed for p = 1. Lyapunov exponents are represented in Fig. 17 showing different signs for slope depending on p. Figure 18 shows that for p = 1 (ramp with positive slope) there is quasiperiodic behavior. For TN = 0 the bifurcation value for Z does not change; but it does when TN is varied. In real applications, which correspond to T N near zero, the stability character of the regulator does not change when one or the other ramp is used.
Figure 21 shows a two-dimensional bifurcation diagram for a PWM controlled Buck converter with diode. In this plot, the values of the central vector of parameters have been slightly modified. Concretely, we take VR = 0.5 and VD = 0.3. This figure shows some similar patterns with regards to the other bifurcation diagrams presented before but there are also some differences which will be commented on. Like in the other bifurcation diagrams, the region 1T is beside the region QP; the system changes from 1T to QP behavior through a Hopf bifurcation. And also, a little region of 1T-DCM behavior is inside the QP region. The main differences which can be distinguished are the following: First, for the values of the parameters used in the simulations, the bifurcation of the T N -periodic
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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.
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orbit occurs for negative values of the impedance Z. This is due to the fact that the Boost and Buck– Boost regulators have a positive critical impedance but the Buck has null critical impedance. Thus the system loses stability when it is fedback positive (Z negative), and show quasiperiodic dynamics.
Second, the Buck does not have an ON fix point simultaneously with other dynamics. The position of its fixed points in the topologies do not allow that, in the skipping cycle regime, the dynamics could be attracted to this type of behavior. The 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 values for impedances and high periods) is replaced by a period adding route (we will denote this region by ADD). In the phase space, the orbits present the skipping cycle phenomenon. We observed similarities between this ADD region and the QP region (see the zone on the right-hand side of Fig. 16(b)). Third, the effect of the parameters Q and Q S acts in the same direction: a higher loss of stability corresponds to bigger quality factors Q and Q S . These effects can be observed in Fig. 22. It is worth noting that increasing Q favors DCM.
12. Two-Dimensional Bifurcation Diagrams for a PWM Voltage-Controlled Buck Converter
12.2. With a diode
Most practical applications of a Buck converter use a voltage-controlled PWM loop (Z = 0). This means that a Hopf bifurcation will not appear. On the other hand, since the stability character almost do not depend on VR we will plot the bifurcation diagrams in this section with parameter V D instead of Z.
12.1. Without a diode Figure 23(a) shows a two-dimensional bifurcation diagram for a voltage-controlled PWM Buck converter without a diode. It can be observed that the lobe has changed with regards to the diagrams in other sections. Its middle zone contains chaotic dy-
(a)
namics instead of 2TN -periodic orbits with chattering. Moreover between the 2TN -periodic region and the chaotic zone there exist periodic narrow bands of periods 2TN , 4TN , 8TN , and so on, which correspond to a period doubling route to chaos. On the other hand, for TN near to 1, these can be observed a region with chaotic dynamics bordered by a curve of grazing type bifurcations. Figure 23(c) shows the Lyapunov exponents against TN with different values for VD . We can see flip bifurcations when λ = 0, which correspond with Fig. 23(a). VD = 1.34 is the value for the cuspid of the 2T lobe.
(b)
Figure 23(b) corresponds to a voltage-controlled PWM converter with a diode. A region with 1T-DCM dynamics is clearly visible. The twodimensional bifurcation diagrams with and without diode are similar for TN < 0.15.
12.3. Two-dimensional bifurcation diagrams with VIN as bifurcation parameter In 1990, chaos via period doubling and other nonlinear phenomena are shown to occur in PWM voltage-controlled Buck converter both numerically and experimentally when input voltage V IN was taken as a bifurcation parameter [Deane & Hamill, 1990]. Sweeping the parameter VIN means varying simultaneously the dimensionless parameters V R
(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. (b) Bifurcation diagram showing period doubling route to chaos, where the intensity current for Poincar´e 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.
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and VD keeping a linear relation between them. A two-dimensional bifurcation diagram for a Buck without a diode taking the values of the parameters like in [Deane & Hamill, 1990] (Q = 1, Q S = 7, VR = VD /20, Z = 0, TN = 0.066 and p = 0) has been computed with VD in the range [0, 0.05] [Fig. 24(a)]. It is worth to note that the bifurcation diagram for TN = 0.066 [Fig. 24(b)] and the attractor observed for VD = 0.015 [Fig. 24(c)] perfectly agree with those obtained in the same work of Deane and Hamill.
13. Conclusions In this work two-dimensional bifurcation diagrams have been extensively studied for a wide class of basic DC–DC converters, which are used in practical applications. The similarities and differences among the Buck, Boost and Buck–Boost regulators, with a diode, and without, are explained from a dynamical systems and engineering points of view. The three subclasses show a general bifurcation pattern which includes smooth flip and Hopf bifurcation, quasiperiodicity, nonsmooth bifurcations and chaotic dynamics. The effect due to the existence of a change from continuous to discontinuous conduction mode has also been discussed. These two-dimensional bifurcations have been precisely computed with ad hoc code taking advantage of the piecewise linearity topologies. Analytical closed-form solutions are exploited and only a numerical method is used to find the switching instants between the two different topologies. A large part of this work presents only a macroscopical view of what happens to these systems when some meaningful parameters are varied. The authors are working on the details of some interesting practical regions to provide also a microscopic view which can be useful for design.
Acknowledgments The authors would like to acknowledge David Carri´ o for his invaluable help in building up a great part of the numerical simulator. This work was supported by the Spanish CYCIT under Grant DPI2000-1509-C03-02 and TIC2000-1019-C02-01.
References
Banerjee, S. & Chakrabarty, K. [1998] “Nonlinear modelling and bifurcations in the boost converter,” IEEE Trans. Power Electron. 13, 252–260.
Banerjee, S., Ranjan, P. & Grebogi, C. [2000] “Bifurcations in two-dimensional piecewise smooth maps-theory and applications in switching circuits,” IEEE Trans. Circuits Syst. I 47, 633–647. Benadero, L., El Aroudi, A., Toribio, E., Olivar, G. & Mart´ınez-Salamero, L. [1999] “Characteristic curves to analyze limit cycles behavior of DC–DC converters,” Electron. Lett. 687–789. Benadero, L., El Aroudi, A., Olivar, G., Toribio, E. & Moreno, V. [2002] “Bifurcations analysis in PWM regulated DC–DC converters using averaged models,” EPE-PEMC’02, Dubrovnik & Cavtat (Croatia), SSIN-04. Chakrabarty, K., Podar, G. & Baberjee, S. [1996] “Bifurcation behavior of the buck converter,” IEEE Trans. Circuits Syst. I 11, 439–447. Deane, J. H. B. & Hamill, D. C. [1990] “Analysis, simulation and experimental study of chaos in the buck converter,” IEEE Power Electronic Specialist Conf. PESC’90, pp. 491–498. Deodel, E. J. & Wang, X. J. [1995] AUTO94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Technical Report CRPC-95-2, California Institute of Technology. di Bernardo, M., Fossas, E., Olivar, G. & Vasca, F. [1997] “Secondary bifurcations and high periodic orbits in voltage controlled buck converter,” Int. J. Bifurcation and Chaos 7, 2755–2771. El Aroudi, A., Olivar, G., Benadero, L. & Toribio, E. [1999] “Hopf bifurcation and chaos from torus breakdown in a PWM voltage-controlled DC–DC Boost converter,” IEEE Trans. Circuits Syst. I 11, 1374–1382. El Aroudi, A., Benadero, L., Toribio, E. & Machiche, S. [2000] “Quasiperiodicty and chaos in the DC–DC Buck–Boost converter,” Int. J. Bifurcation and Chaos 10, 359–371. Fossas, E. & Olivar, G. [1996] “Study of chaos in the buck converter,” IEEE Trans. Circuits Syst. I 43, 13–25. Guckenheimer, J., Myers, M., Wicklin, R. & Worfolk, P. [1991] “DStool: Dynamical system toolkit with interactive interface,” Graphic Center of Applied Mathematics, Cornell Univ. www.cam.cornell.edu/guckenheimer/dstool Hamill, D. C. & Jefferies, D. J. [1988] “Subharmonics and chaos in a controlled switched–mode power converter,” IEEE Trans. Circuits Syst. I 35 1059–1060. Hamill, D. C., Deane, J. H. B. & Jefferies, D. J. [1992] “Modeling of chaotic DC–DC converters by iterated nonlinear mappings,” IEEE Trans. Power Electron. 7 25–36. Khibnik, A. I., Kuznestov, Y. A., Levitin, V. V. & Nikolaev, E. V. [1993] “Continuation thecniques and interactive software for bifurcation analisis of ODEs and iterated maps,” Physica D62, 360–371.
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Nusse, H., Ott, E. & Yorke, J. A. [1994] “Border–collision bifurcations: An explanation for observed bifurcation phenomena,” Phys. Rev. E49, 1073–1076. Nusse, H. & Yorke, J. A. [1992] “Border–collision bifurcations including ‘period two to period three’ for piecewise smooth systems,” Physica D57, 39–57. Olivar, G. [1997] Chaos in the Buck Converter, Ph.D. Thesis, Servei de Publicacions de la UPC, Barcelona. Parker, T. & Chua, L. O. [1987] “INSITE–a software toolkit for the analysis of nonlinear dynamical systems,” Proc. IEEE 75, 1081–1088. Rashid, M. H. [1990] Spice for Circuits and Electronics Using PSpice (Prentice-Hall).
Toribio, E., El Aroudi, A., Olivar, G. & Benadero, L. [2000] “Numerical and experimental study of the region of period-one operation of a PWM boost converter,” IEEE Trans. Power Electron. 15, 1163–1171. Tse, C. K. [1994a] “Chaos from a buck switching regulator operating in discontinuous mode,” Int. J. Circuits Th. Appl. 22, 263–278. Tse, C. K. [1994b] “Flip bifurcation and chaos in threestate boost switching regulators,” IEEE Trans. Circuits Syst. I 41, 16–23. Yuan, G., Banerjee, S., Ott, E. & Yorke, J. A. [1998] “Border collision bifurcations in the buck converter,” IEEE Trans. Circuits Syst. I 45, 707–715.
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 periodic. In this case, the global Poincar´e 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 different linear configurations. This must be implemented with an appropriate control of the switches. In a noise perturbation free environment, given the desired output voltage, the switching frequency can be selected and the switches can be turned ON and OFF according to a fixed pattern; this is referred to as the open loop system. In contrast, in industrial applications, noise and perturbations are always present, and also the parameters of the circuits may be affected by external disturbances. Thus the use of an appropriate control to counteract the modifications on the output voltage in the system
The basic DC–DC converters Buck, Boost and Buck–Boost are a family of circuits which allow the conversion of energy from one level to another without taking into account, theoretically, losses in the components. They are used extensively in power supplies for electronic circuits and in the control of the flow of energy between DC to DC systems, and in any industrial application where there is a need of stabilizing an output voltage to a desired value. Also, they are widely used in small spacecrafts such as satellites where DC power is generated by solar arrays.
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is recommended; this is referred to as the closed loop system. The most popular control strategy used in the literature is Pulse Width Modulation (PWM) where electronic control of the basic power electronic converter circuit is achieved by controlling the duty cycle d of the controlled switch S (the duty cycle is the ratio of the ON phase of the switch to the period of the periodic ON–OFF operation). We will refer to the ON phase when the switch S is closed and diode D is open; the OFF phase refers to when the switch S is open and diode D closed; and mode OFF’ (or discontinuous mode) takes place when both switch and diode are open (see Fig. 2). There are many ways that fixed frequency PWM control can be implemented. Nevertheless, the basic ingredients of almost all existing PWM controllers that are used for voltage control are: 1. an output voltage error amplifier 2. a T -periodic sawtooth signal generator (driving signal) 3. a comparator that compares the error amplifier output with the sawtooth waveforms. The most interesting dynamics of these systems, from a classical design point of view, is the T -periodic orbit (periodic evolution with the same period as the driving signal). Nonlinear phenomena in the PWM voltage controlled DC–DC basic power electronic regulators have been studied in the past years. Various kinds of bifurcational behaviors are found for different converters with different control schemes. Flip bifurcations and period doubling route to chaos are found in the Buck converter [Deane & Hamill, 1990; Fossas & Olivar, 1996; Tse, 1994a], Neimark–Sacker bifurcation and quasiperiodicity route to chaos are found to occur in the PWM Boost and Buck–Boost converters [El Aroudi et al., 2000] and border collision bifurcations are found to occur in the Buck and the Boost converter with different control strategies [Banerjee et al., 2000; Yuan et al., 1998]. Up to now there are very few works that try to characterize the bifurcational phenomena in the parameter space [Chakrabarty et al., 1996; Banerjee & Chakrabarty, 1998; El Aroudi et al., 2000; Olivar, 1997; Toribio et al., 2000]. The aim of this paper is to investigate in the parameter space the mechanisms of losing the stability of the T -periodic orbit, and the transition between the different bifurcations in these systems.
2. Continuous Time Model of the Basic Switching Regulators 2.1. State equations The basic DC–DC switching converters are shown in Fig. 2. The differential equations, modeling each one of the three configurations that use every converter, can be derived by using the standard Kirchoff’s laws. Let us define matrices A 1 , A2 , A3 , B1 and B2 as follows: 1 1 1 − − 0 RC C , , A2 = RC A1 = 1 1 1 − − 0 − L RS L
1 − RC A3 = 0 B2 =
' (
0
0
0
0
(1)
, B1 = VIN ,
L
0
where R is the output load resistance, L is the inductance which is supposed to have an Equivalent Series Resistance ESR RS , C is the capacitance, and VIN is the input voltage. During each phase (ON, OFF and OFF’), and until a switching condition is fulfilled, the dynamics of the system is described by: X˙ = AX + B (2) X = (vC , iL )T is the vector of the state variables and the overdot stands for derivation with respect to time t(X˙ = dX/dt). Table 1 shows the A’s and B’s matrices for the three basic converters Buck, Boost and Buck–Boost during each phase.
2.2. Analytical solutions for each configuration Since the previous differential equations are piecewise linear (PWL), a closed form solution is Table 1. The A’s and B’s matrix for the basic converters during phases ON, OFF and OFF’. Converter
AON
AOFF
AOFF!
BON
BOFF
BOFF!
Buck
A1
A1
A3
B1
B2
B2
Boost
A2
A1
A3
B1
B1
B2
Buck–Boost
A2
A1
A3
B1
B2
B2
Two-Dimensional Bifurcation Diagrams 429
available for each configuration. Let us write: kC =
1 , 2RC ω0 =
kL =
)
1 LC
RS , 2L
k = kc + kL ,
*
RS 1+ R
+
(3) −
k2
Therefore, the solution for each configuration can be written as: • RC circuit and L in series with VIN (matrices A2 and B1 ) (this configuration corresponds to the ON phase for the Boost and the Buck–Boost): vC (t) = vC0 e−2kC (t−t0 ) iL (t) =
*
VIN VIN + iL0 − RS RS
+
e−2kL (t−t0 )
(4)
• RLC oscillator in the free regime (matrices A 1 and B2 ) (this configuration corresponds to the OF F phase for the Buck and the Buck–Boost): vC (t) = e−k(t−t0 ) [vC0 cos ω0 (t − t0 ) +
*
+
kvC0 iL0 − sin ω0 (t − t0 )] Cω0 ω0
iL (t) = exp−k(t−t0 ) [iL0 cos ω0 (t − t0 ) + * vC0 iL0 − sin ω0 (t − t0 )] + ω0 Lω0
(5)
• RLC oscillator forced with VIN (matrices A1 and B1 ) (this configuration corresponds to the ON phase for the Boost, and the OF F phase for the Buck–Boost): vC (t) = VC∞ + e−k(t−t0 ) [(vC0 − VC∞ ) cos ω0 (t − t0 ) + −
+
− IL∞ ) cos ω0 (t − t0 ) + −
*
(iL0 − IL∞ ) Cω0
k(vC0 − VC∞ ) sin ω0 (t − t0 )] ω0
iL (t) = IL∞ + exp−k(t−t0 ) [(iL0
+
*
(iL0 − IL∞ ) ω0
(vC0 − VC∞ ) sin ω0 (t − t0 )] Lω0
(6) • The capacitor connected to the load when the converter works in the so-called Discontinuous Conduction Mode (DCM), characterized by iL (t) = 0) (matrices A3 and B2 ): vC (t) = vC0 e−2kC (t−t0 ) iL (t) = 0
where t0 is the initial time at which the system switches from one configuration to another, and i L0 and vC0 are the states of the system at the switching instant t0 . The values for VC∞ and IL∞ are VC∞ =
VIN , RS 1+ R
IL∞ =
VC∞ R
The above solutions are only valid when we have * + 1 RS 1+ − k2 > 0 (8) LC R in such a way that ω0 is real and positive. From the design point of view this is the most important case, since it gives oscillatory solutions.
2.3. The switching conditions The PWM control of a switched converter is achieved by the comparison of the control voltage vcon which is a linear combination of the capacitor voltage vC and the inductor current iL in the form vcon = A(vC + Zr iL − VREF )
(9)
with a driving signal, generalized as a triangular function [Fig. 1(a)]
vtriang (t) =
VU −VL t VL + pT
VU −VL VU − (t−pT )
(1−p)T
if 0 < t < pT if pT < t < T
(10) where A is the gain of the error amplifier, Z r is the impedance used to convert the inductor current to a voltage, VREF is the reference voltage, VL and VU are the lower and upper values of the driving triangular signal, T and p are the periode and symmetry factor of this signal. Let us define the function Vcomp as: Vcomp (t) = vcon (t) − vtriang (t)
= A(vC + Zr iL − VREF ) − vtriang (t) (11)
The switching condition is therefore:
(7) 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) Fig. 1. (a) Triangular signal vtriang (t) used as driving signal in control. (b) Normalized function h(τ ).
Since the expressions of the trajectories in each configuration include exponential and trigonometric functions, this equation is transcendental. Thus, a closed form expression for the solution is not possible [Hamill et al., 1992; El Aroudi et al., 1999; Fossas & Olivar, 1996]. Hence, one must resort to numerical methods to compute the switching instants. It should be noted that when the converter enters in discontinuous conduction mode (that is, changes from iL (t) "= 0 to iL (t) = 0), a new switching condition (which is iL = 0) appears.
2.4. Numerically computed orbits Numerical methods usually play a major role when the system is nonlinear and parameters must be varied in certain ranges. Although there exist some very useful packages for the study of the behavior of dynamical systems, to our knowledge, none is specially suited for piecewise linearities. In these systems, one may take profit of the analytically-computed solutions, but the switching instants must be numerically computed. The PSPICE package [Rashid, 1990] is the more real-
istic waveform simulator for DC to DC converters because it is designed for electrical and electronic circuits. However, since the simulations are very time-consuming, one cannot rely on this package for extensive computations. Other packages, like LOCBIF [Khibnik et al., 1993], INSITE [Parker & Chua, 1987], DSTOOL [Guckenheimer et al., 1991] and AUTO [Deodel & Wang, 1995] are well suited only for smooth systems. They compute equilibrium points, eigenvalues, characteristic multipliers, Lyapunov exponents and invariant manifolds assuming that the vector field is smooth enough. But the switching action in power electronic converters makes these systems very different from those characterized by a smooth vector field. The vector field for DC–DC switching converters is discontinuous and PWL in the form: X˙ =
0
f1 (X, t)
if
g(X, t) < 0
f2 (X, t)
if
g(X, t) > 0
(13)
In DCM, the previous model is still valid and the functions f1 or f2 are in turn PWL. Moreover, these numerical packages do not allow to handle two-dimensional bifurcation diagrams easily. Finally, as it was mentioned before, the numerical method used for integrating the system of differential equations is not optimized for PWL systems, where closed form solutions in each sub adjacent
Two-Dimensional Bifurcation Diagrams 431
interval are available. For these reasons, some specific programs written in C code are used in this paper, using the closed form solution in each linear configuration. The equation for the switching condition is solved by a specific root-finding method (Newton, secant, bisection, . . . etc.). In our computations, the bisection method is used. Therefore the analytical solutions combined with the bisection method to locate the switching instants will reduce considerably the time needed for a given simulation. In this paper, the global dynamics is obtained in this way. The simulations are therefore very fast compared to simulations computed with standard packages. This method was already used by the authors in [Deane & Hamill, 1990] and [Fossas & Olivar, 1996] to compute trajectories of the PWM voltage controlled DC–DC Buck converter.
3. Dimensionless Formulation 3.1. Background The large number of parameters associated with the PWM DC–DC converters is a major handicap to the characterization of all the possible dynamics. To deal with this problem, some authors have fixed the values of the parameters according to specific examples [Hamill et al., 1992; Tse, 1994b] and then they varied the parameters near these values. Instead, we will define dimensionless parameters achieving a significant reduction in the number of independent parameters of the circuit. We begin with linear transformation of the state variables in such a way that the resulting new variables are dimensionless. We choose Ts , Vs , Is scale parameters 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 (related 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 Among all possible choices of the scale parameters,
the following will be chosen: √ Ts = T0 = 2π LC, Vs = VIN ,
VIN Is = 1 L/C
Then, we define the normalized variables and parameters of the power circuit as follows: vC (t) v(t) = , VIN
1
L/C t iL (t), τ = , VIN T0 1 L/C R , QS = Q= 1 R L/C S (14) where Q is the quality factor associated to the load, and QS is the quality factor associated to the equivalent series resistance ESR of the inductance. Finally, τ is the dimensionless time. i(t) =
3.3. Scaling the switching condition Introducing the normalization in the control condition, we get fS (v, i, τ ) = v + Zi − VR − VD h(τ ) = 0
(15)
where VR is the normalized reference voltage, V D is the normalized width of the external control signal and Z is the normalized impedance. h(τ ) becomes a triangular function with amplitude one, null average value, period TN and symmetry factor p [see Fig. 1(b)]. VR =
Fig. 3. (v, i).
VU − VL VREF VU + VL + , VD = , VIN 2AVIN AVIN (16) Zr T 1 Z= , TN = T0 L/C
The switching band in the normalized phase plane
432 L. Benadero et al.
It should be noted that the evolution of the driving signal voltage may be represented by a switching band in the (v, i) phase plane (Fig. 3). Within this band the switchings from one configuration to another can occur. With the proposed normalization, the resulting state variables and parameters are dimensionless. Their number is reduced from 10 to 5. The normalization means that different circuit designs with equal normalized parameters have equivalent dynamics. After scaling variables and parameters, the dimensionless differential equations of each configuration can be written Y˙ = CY + D
(17)
where the overdot denotes for the derivation with respect to the dimensionless time τ . Y = (v, i) T is the vector of the normalized state variables and the dimensionless matrices are C1 , C2 , C3 , D1 and D2 as follows:
1 − Q
C1 = 2π
−1
−
C3 = 2π
1 Q
0
1 − Q
1 , C2 = 2π 1 − 0 QS
0
0
, D1 =
'
0 2π
(
0 , 1 − QS
, D2 =
' (
3.4. Normalized analytical solutions of each configuration Note that Eq. (8) is equivalent to 1+
1 − QQS
* *
1 2
1 1 + Q QS
++2
>0
Assuming that the condition above is fulfilled, the closed form solutions for each configuration are: • The LRC parallel oscillator with the source V IN in series with the inductor.
v(τ ) = V∞ + exp−α(τ −τ0 ) [(v0 − V∞ ) cos ωN (τ − τ0 ) +
*
+
i0 − I∞ − β(v0 − V∞ ) sin ωN (τ − τ0 )] δ
i(τ ) = I∞ + exp−α(τ −τ0 ) [(i0 − I∞ ) cos ωN (τ − τ0 ) *
+
v0 − V∞ − β(i0 − I∞ ) − sin ωN (τ − τ0 )] δ
0 0
(18)
Matrices Ci and vectors Di correspond to normalized ones Ai and Bi respectively (see Table 1). Therefore, the converter depends on four essential dimensionless parameters Q, TN , VD , VR and an additional (parasitic) parameter Q S (the essential and parasitic classification is due to circuit considerations, but this will not affect the analysis). T N is the period of the driving signal normalized to the natural period of the LC-circuit, and finally, in the event of Rs = 0 (not considered here), Q is the quality factor of the power circuit. Thus, a DC– DC converter can be characterized by the normalized parameter set C = {Q, QS , TN , VR , VD }. The set C of dimensionless parameters of one specific PWM DC–DC regulator stands for a whole family of regulators whose dimensionless parameters are those in C, and thus they display the same dynamics (obviously, this is true only if the converter is the same and its model remains valid for the entire family).
(19)
(20) Note that the equilibrium point (QQ S /(Q + QS ), QS /(Q + QS )) of this configuration is a stable spiral sink. • The LRC parallel oscillator in the free regime. v(τ ) = exp−α(τ −τ0 ) [v0 cos ωN (τ − τ0 ) *
+
i0 − βv0 + sin ωN (τ − τ0 )] δ i(τ ) = exp−α(τ −τ0 ) [i0 cos ωN (τ − τ0 ) −
*
(21)
+
v0 − βi0 sin ωN (τ − τ0 )] δ
The equilibrium point (0, 0) of this configuration is a stable spiral sink. • The inductor connected to the source, and the RC circuit: v(τ ) = v0 exp
− 2π (τ −τ0 ) Q
i(τ ) = QS + (i0 − QS ) exp
2π (τ −τ0 ) −Q S
(22)
Two-Dimensional Bifurcation Diagrams 433
4. The Poincar´ e Map In the case of periodically driven systems, the usual Poincar´e section considered is a plane Σ (with equation 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 configurations that use the Boost converter.
The equilibrium point (0, QS ) of this configuration is a stable sink. • The RC circuit (this corresponds when the converter works in DCM): v(τ ) = v0 exp i(τ ) = 0
(τ −τ0 ) − 2π Q
(23)
The equilibrium point (i = 0) of this configuration is a stable one-dimensional1 sink. In the expressions of the solutions, we have used V∞ =1QQS /(1 + QQS ), I∞ = V∞ /Q, ωN = 2πδ, δ = (1 − β 2 ), β = 1/2(1/Q − 1/QS ) and α = π(1/Q + 1/QS ). Also, v0 , i0 are the state variables at the switching instant τ0 . As we have seen, the dynamics in each configuration is well determined. But the parameters of the circuit are selected in such a way that when we make the control work, the converter switches from one configuration to another. Therefore, since the system is highly nonlinear due to the switching action, a great variety of dynamics are possible (limit cycles, subharmonics, quasiperiodicity, chaos). In Fig. 4, the switching lines for a PWM voltage mode controlled Boost regulator and the trajectories near the equilibrium points of each configuration are schematically plotted. 1
Assume we have a converter with all parameters 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 beginning and those at the end of each cycle of the driving signal and, on the other hand, forcing the continuity of the state variables at the asynchronous switching instants [di Bernardo et al., 1997]. By imposing these conditions, a system of two nonlinear equations with two unknowns veq and ieq is obtained. By imposing the asynchronous switching condition, another equation is added, being the switching time τS another unknown. These three nonlinear equations with three unknowns can be introduced to a standard mathematics program (for example Maple V) to find the fixed points (v eq , ieq ) for a set of parameter values.
5. Mechanisms of Losing the Stability of the One-Periodic Orbit The basic elements which will be used in this section to study the stability of the one-periodic orbit are the Poincar´e map and the characteristic multipliers of its fixed points. The stability of the one-periodic orbit is analyzed by means of the response to a small perturbation near this fixed point. To obtain the stability character of a fixed point (veq , ieq ) of this map is enough to compute the image of a perturbed state near the fixed point in the phase plane (vn , in ). This transforms the problem of analyzing the stability of the one-periodic orbit to the study of the eigenvalues of the linearized
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 associated to the linearization near a fixed point are obtained by computing the transformation of the state space points in the neighbor of the fixed points. From the eigenvalues m1 and m2 of DP (the characteristic multipliers of the orbit), or from the Lyapunov exponents (λ = log |m|/TN ) the following bifurcations will be detected: • Flip or period doubling bifurcation of the stable one-periodic orbit. It is characterized by the fact that one of the characteristic multipliers is equal to −1 and the other one has absolute value less than 1. • Hopf or Neimark–Saker bifurcation. It is characterized by the fact that both characteristic multipliers cross the unit circle, being complex conjugates. • Border-Collision bifurcations. They are characterized by a sudden change in the system behavior accompanied by a jump in the values of the characteristic multipliers. These anomalous bifurcations are well explained in [Yuan et al., 1998; Banerjee et al., 2000; Nusse et al., 1994] and [Nusse & Yorke, 1992].
6. Two-Dimensional Bifurcation Diagrams In order to obtain the bifurcational structure when two parameters are varied, we will plot the twodimensional bifurcation diagrams corresponding to these parameters. The two-dimensional bifurcation diagram is color-coded depending on the periodicity of the attractor corresponding to the point in the parameter space. Due to hysteresis phenomena and in general to possible coexisting attractors, only one of them can be identified at each point of the parameter space. In this paper the diagrams are produced as follows: first we give the system with extreme values of both varying parameters and the initial condition xini = (vini , iini ); then, one-dimensional bifurcation diagrams are successively computed fixing one parameter (primary) and varying the other one (secondary). The final state of the system is taken as initial condition to the next value of the primary parameter. The initial condition when the secondary parameter is varied is the last state calculated using the preceding value of the secondary parameter, both with the same extreme value of the primary parameter. Arrows in the diagrams
will specify the sense of variations of the parameters; the larger arrow corresponds to the parameter chosen as primary (in most of the diagrams this is TN ). Also, in order to understand the bifurcation patterns better, some simulations have been made using an ideal switch instead of a diode.
6.1. A point in the parameter space To obtain a good comparison of all the twodimensional bifurcation diagrams, we choose a central point in the parameter space and then, we vary the parameters around this point. In this zone we will explore all the possible dynamics of the three converters. We fix the values of this parameter point to Q = 4, QS = 6 and Z = 0. Moreover we take VR = 0.5, VD = 0.3 for the Buck, and VR = 1.4, VD = 1.8 for the Boost and the Buck–Boost. We cannot ideally take the same VR in the three converters since its value tightly depends on the voltage gain. Thus it is convenient to take V R less than one in the case of the Buck, which reduces the input voltage; and more than one in the case of the Boost, which increases the input voltage. In the case of the Buck–Boost it can be any value since this converter can be designed to increase or reduce the input voltage. The range of variation of the normalized period is large enough to consider not only low periods (TN ( 1), which is the usual range in circuits, but also high periods near TN = 0.7 are checked. It will be shown that some bifurcations which pose limits on the TN -periodic dynamics are met near TN = 1. With regards to parameter p, we will use only its extreme values p = 0 and p = 1.
6.2. Choosing the parameters to vary For the two-dimensional bifurcation diagrams, we have to choose two parameters from the seven (T N , Q, QS , Z, VD , VR , p) which correspond to a given circuit. The bifurcation patterns will be obtained from different pairs of parameters. We classify all our possible parameters according to three different classes. One class contains the parameter associated to time in the control loop TN , and it will be one of the parameters varied in each of our bifurcation diagrams; another class will be made with the parameters associated to the power circuit, these are Q and QS , which will be kept fixed in each of the 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 specifically, they define the regulation band (Z, V D , VR and p). One of them will be taken to be varied when we compute the two-dimensional bifurcation 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 case parameters VD and Z will be varied. Parameter Z allows to detect more easily all the possible dynamics.
7. Bifurcation Diagrams for the Boost Converter in the (TN , Z)-Parameter Space. Regions in the Parameter Space 7.1. Basic regions according to the dynamical behavior Using the numerical simulator which has been 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 PWM control, and without diode, this is, with both ideal and complementary switches. The parameters are fixed according to the central point defined before (Q = 4, QS = 6, VR = 1.4, VD = 1.8, p = 0) and we vary Z and TN (see Fig. 5). The resulting bifurcation diagram is simpler than the one obtained using a diode, and we can distinguish four big different regions, which are described in the following: First, we have a big two-periodic region of lobar type, which will be called 2T region, on the righthand side of the figure. Into this zone the orbits display the chattering phenomenon and too much time is required to decide the periodicity. The same phenomenon is detected on other similar lobar areas, which yield sliding modes like in Fig. 6, with Z = 0.82 and TN = 0.55. Second, we have a region whose stationary behavior is the fixed point of the ON topology, in the lower part of the figure. This region will be called the FP region. On the right part of its upper border we found that the impedance is constant at Z = 0.08, and the left part is made of a curve which starts at a point with TN = 0.18 approximately. Third, between the region FP and the twoperiodic lobar region, we found chaotic and quasiperiodic orbits with some little periodic islands. The chaotic and quasiperiodic orbits are found after Hopf bifurcation following the torus breaking route to chaos. This zone will be called QP. Fourth, there is a large one-periodic dynamics (blue) which is in the remainder of the figure. This is the usual region of operation of the regulators and we will denote it by TN -periodic or simply 1T.
Its linearized dynamics corresponds to a focus, with negative Lyapunov exponents. It is clear that the border of the region 1T shows a discontinuity at a point of the parameter space which corresponds approximately to T N = 0.50 and Z = 0.19. This type of behavior, which is common in a certain sense, is associated to a variation pattern in the characteristic multipliers when the parameters are moved, and this implies two different types of borders.
7.2. Curve of Hopf bifurcations At the border between the zones 1T and QP a Hopf bifurcation occurs, which corresponds to a null Lyapunov exponent. The bifurcation occurs smoothly when two complex characteristic multipliers cross the unit circle. If we decrease the impedance Z the characteristic multipliers begin to grow and finally they cross the unit circle at the border. We will call this border the Hopf border. For TN near 0 this border begins at Z = −0.05, which can be analytically predicted when averaging techniques are used. This is tightly related with the critical value of the impedance Z associated to the equilibrium [Benadero et al., 1999] which is related to the orientation of the vector field. The value of the critical impedance fixes, with the exception of a certain correction which depends on the values of Q and QS , the impedance Z which makes the transition to instability. For different values of T N , the bifurcation value for Z is changed and the period one always appears in the upper region of the 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 The border between the region 1T and the 2T lobe corresponds to a flip bifurcation. At this point, both Lyapunov exponents are real and the larger one becomes null. We will call this border the flip border. Figure 7(a) shows both two exponents when parameter TN is varied, and three important facts are observed: first, if we decrease Z then the average value of the Lyapunov exponents increases; second, if we increase TN the average value also increases when p = 0 (this effect is inverted when p = 1, see Sec. 9.2); and third, a bifurcation from focus to a node is produced. From this bifurcation point and onwards, one exponent grows until it becomes null (this exponent is associated with a characteristic multiplier equal to −1) and a flip bifurcation 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 corresponds to the point Z = 0.19 and TN = 0.50, where the two curves coalesce. Figure 7(b) corresponds to the cuspid of the lobe 2T; for instance, with the arbitrary value Z = 0.82, we have bifurcations at TN = 0.47 and
TN = 0.63. As Z is increased, the larger Lyapunov exponent does not pass through zero, and thus no bifurcation is produced. The limit case is obtained approximately for Z = 1.01 and TN = 0.51.
7.4. Curve of grazing bifurcations Finally, the border between the region FP and QP corresponds to a grazing bifurcation, and thus will be called the grazing border. A grazing bifurcation is a subtype of the so-called border collision bifurcations and can be only observed in nonsmooth systems. This can be observed in Fig. 8 which corresponds to an orbit of the regulator with Z = 0.08. Observe that the left line border of the switching band passes through the fixed point of the ON state.
8. Two-Dimensional Bifurcation Diagrams for the Boost Converter with a Diode It should be expected that the two-dimensional bifurcation 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 without it, since we add one more topology to the circuit. As we shall see, this is really so. The new attractors which are obtained visit periodically or intermittently, the discontinuous conduction mode, and can be periodic, quasiperiodic or chaotic. We will use CCM for dynamics of continuous conduction mode and DCM have been denoted for discontinuous conduction mode dynamics. As a general rule, DCM stabilizes quasiperiodic and chaotic behavior and contributes to reduce the multistability phenomenon. There exists a border between orbits of DCM and CCM, although it is not shown in the figures when both orbits at each side of the border are stable. If one of the orbits on one side is unstable, this effect can be observed in the diagram. It is worth to note that Hopf bifurcations do not appear for TN -periodic orbits in the DCM. Figure 9 shows that, with the parameters used, the addition of a diode in the circuit does not modify the curves of Hopf and Flip bifurcations (left side) in the CCM, and the cuspid of the 2T-lobe has also been retained. But some important facts have really changed:
First, a region of DCM dynamics of TN -periodic type has eroded the 2T-lobe from its right part (this is, for high values of TN and Z). To check this point, we plot in Fig. 10 three different orbits for different values of TN and Z fixed to 1.01 (the value corresponding to the cuspid of the 2T-lobe). The orbit which corresponds to TN = 0.37 is TN -periodic of CCM type; the orbit for TN = 0.65 is TN -periodic of DCM type, and the orbit for TN = 0.51 is placed just within the border between TN -periodic CCM type and TN -periodic DCM type. Like in the CCM type, the central zone of the 2T-lobe (light gray) corresponds to 2TN -periodic orbits with sliding. Second, the region QP shows more complex dynamics than without diode. It can be observed that each kTN -periodic Arnold tongue has an accumulation of 2nkTN -periodic regions (n = 1, 2, 3, . . .) in the lower part which corresponds to a period doubling route to chaos. Figure 11 shows a onedimensional bifurcation diagram for Z = 0 varying TN . It can be observed as a torus-breaking route to chaos. The rotation number has been computed as the quotient of the number of cycles in the state 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 in both central figures. The bottom figures are Poincar´e 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. The same figure includes the Poincar´e sections for some orbits corresponding to several values for T N . Some of them correspond to phase-locking orbits; for instance, we have period 8TN for TN = 0.15 and period 15TN for TN = 0.16 (its period is the addition of the periods of the neighboring 8T N and 7TN periodic orbits). Finally, for TN near 0.153 we obtain a quasiperiodic attractor [El Aroudi et al., 1999]. Third, the grazing border is found for lower values for Z. This means that the QP region is bigger than before. Near this border there is also multistability. To show that, we plot a new bifurcation diagram of the same zone sweeping the parameter ranges in the opposite order (see Fig. 12), and we obtain slightly different patterns, specially in the overlapping of the Arnold tongues. Moreover depending on the initial conditions, if Z < 0.08 a fixed point attractor is also possible.
9. Effect of the Other Parameters in the Two-Dimensional Bifurcation Diagram for the Boost Converter 9.1. Effect of Q and QS In Fig. 13, we plot the values of the intensity current at the switching instant against the bifurcation parameter TN . The quality factors Q in (a), and QS in (b), are taken as secondary parameter. High
values for Q and QS are related to DCM. This is due to the increase in the relative amplitude of the oscillations. On the other hand, the evolution of the Lyapunov exponents shows that the stability of the region 1T-CCM depends on Q and QS in different ways (see Fig. 14). Concretely, stability grows with Q and decreases when QS is made higher. If we increase QS then the losses in the inductor decrease and thus the system is made less stable. The reason why the converter gets better stability when Q is increased is not so easy to explain. On one side, Q has a similar role like QS . But on the other side, Q moves the orbit in the phase space in a sense that the critical impedance gets lower, and the stability is favored [Benadero et al., 2002]. This latter effect has predominance over the decrease of the losses in the inductor. Figure 15(a) shows a two-dimensional bifurcation diagram with Q = 8 instead of Q = 4. As it 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 approximately from Z = −0.05 (for Q = 4) to Z = −0.12 (for Q = 8). Also, since DCM is favored, the region 1T-DCM moves to the left lower end. This makes dissappear the region 2T-CCM and makes appear a little 1T-DCM region inside the QP. Inside this zone of the parameter space we have 2T-DCM behavior beside the right part of the 1T-DCM region. Figure 15(b) shows a two-dimensional bifurcation diagram with QS = 3 instead of QS = 6. In
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(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.
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(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 also moves to a lower position (from Z = −0.05 for QS = 6 to Z = −0.18 for QS = 3). Also the QP region is stretched.
9.2. Effect of the parameters of the control (regulation band) VR, VD, p We show in Fig. 16(a) a two-dimensional bifurcation diagram for VR = 2.0 instead of VR = 1.4. This moves the curve of Hopf bifurcations higher
and thus the values for Z are always positive. This result, which agrees with the fact that when V R is increased the gain voltage rises up, can also be explained from the point of view of the critical impedance, which increases with VR . We show in Fig. 16(b) a two-dimensional bifurcation diagram for VD = 0.8 instead of VD = 1.8, which has the same effect as before, moving the curve of Hopf bifurcations to a higher position. Qualitatively, to make VD lower means to stretch the regulation band; thus in some sense, the
Two-Dimensional Bifurcation Diagrams 445
Fig. 17. Lyapunov exponents for a Boost regulator with parameters Q = 4, QS = 6, VR = 1.4, VD = 1.8. Both cases p = 0 and p = 1 are represented.
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 bigger when TN grows (this is, the border moves down); on the contrary, if p = 0 the movement is in the opposite direction. This can be observed in Figs. 9 and 18. The patterns which were discussed before do not change qualitatively if we vary parameter V D or VR instead of Z; the different dynamics are visited 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 converter 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.
Fig. 18. 2D bifurcation diagram for a Boost regulator. Varying parameters are period TN and impedance Z and fixed are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 1.
11. Two-Dimensional Bifurcation Diagrams for a PWM Buck Converter
possibility of TN -periodicity is reduced and we have a loss in the stability. The bifurcation diagram in the Boost converters also depends on the symmetry factor p in the comparator signal. For p = 0, when TN grows, the limit cycle is moved to the region of higher currents, and this destabilizes the system. The inverse effect is observed for p = 1. Lyapunov exponents are represented in Fig. 17 showing different signs for slope depending on p. Figure 18 shows that for p = 1 (ramp with positive slope) there is quasiperiodic behavior. For TN = 0 the bifurcation value for Z does not change; but it does when TN is varied. In real applications, which correspond to T N near zero, the stability character of the regulator does not change when one or the other ramp is used.
Figure 21 shows a two-dimensional bifurcation diagram for a PWM controlled Buck converter with diode. In this plot, the values of the central vector of parameters have been slightly modified. Concretely, we take VR = 0.5 and VD = 0.3. This figure shows some similar patterns with regards to the other bifurcation diagrams presented before but there are also some differences which will be commented on. Like in the other bifurcation diagrams, the region 1T is beside the region QP; the system changes from 1T to QP behavior through a Hopf bifurcation. And also, a little region of 1T-DCM behavior is inside the QP region. The main differences which can be distinguished are the following: First, for the values of the parameters used in the simulations, the bifurcation of the T N -periodic
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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.
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orbit occurs for negative values of the impedance Z. This is due to the fact that the Boost and Buck– Boost regulators have a positive critical impedance but the Buck has null critical impedance. Thus the system loses stability when it is fedback positive (Z negative), and show quasiperiodic dynamics.
Second, the Buck does not have an ON fix point simultaneously with other dynamics. The position of its fixed points in the topologies do not allow that, in the skipping cycle regime, the dynamics could be attracted to this type of behavior. The 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 values for impedances and high periods) is replaced by a period adding route (we will denote this region by ADD). In the phase space, the orbits present the skipping cycle phenomenon. We observed similarities between this ADD region and the QP region (see the zone on the right-hand side of Fig. 16(b)). Third, the effect of the parameters Q and Q S acts in the same direction: a higher loss of stability corresponds to bigger quality factors Q and Q S . These effects can be observed in Fig. 22. It is worth noting that increasing Q favors DCM.
12. Two-Dimensional Bifurcation Diagrams for a PWM Voltage-Controlled Buck Converter
12.2. With a diode
Most practical applications of a Buck converter use a voltage-controlled PWM loop (Z = 0). This means that a Hopf bifurcation will not appear. On the other hand, since the stability character almost do not depend on VR we will plot the bifurcation diagrams in this section with parameter V D instead of Z.
12.1. Without a diode Figure 23(a) shows a two-dimensional bifurcation diagram for a voltage-controlled PWM Buck converter without a diode. It can be observed that the lobe has changed with regards to the diagrams in other sections. Its middle zone contains chaotic dy-
(a)
namics instead of 2TN -periodic orbits with chattering. Moreover between the 2TN -periodic region and the chaotic zone there exist periodic narrow bands of periods 2TN , 4TN , 8TN , and so on, which correspond to a period doubling route to chaos. On the other hand, for TN near to 1, these can be observed a region with chaotic dynamics bordered by a curve of grazing type bifurcations. Figure 23(c) shows the Lyapunov exponents against TN with different values for VD . We can see flip bifurcations when λ = 0, which correspond with Fig. 23(a). VD = 1.34 is the value for the cuspid of the 2T lobe.
(b)
Figure 23(b) corresponds to a voltage-controlled PWM converter with a diode. A region with 1T-DCM dynamics is clearly visible. The twodimensional bifurcation diagrams with and without diode are similar for TN < 0.15.
12.3. Two-dimensional bifurcation diagrams with VIN as bifurcation parameter In 1990, chaos via period doubling and other nonlinear phenomena are shown to occur in PWM voltage-controlled Buck converter both numerically and experimentally when input voltage V IN was taken as a bifurcation parameter [Deane & Hamill, 1990]. Sweeping the parameter VIN means varying simultaneously the dimensionless parameters V R
(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. (b) Bifurcation diagram showing period doubling route to chaos, where the intensity current for Poincar´e 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.
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and VD keeping a linear relation between them. A two-dimensional bifurcation diagram for a Buck without a diode taking the values of the parameters like in [Deane & Hamill, 1990] (Q = 1, Q S = 7, VR = VD /20, Z = 0, TN = 0.066 and p = 0) has been computed with VD in the range [0, 0.05] [Fig. 24(a)]. It is worth to note that the bifurcation diagram for TN = 0.066 [Fig. 24(b)] and the attractor observed for VD = 0.015 [Fig. 24(c)] perfectly agree with those obtained in the same work of Deane and Hamill.
13. Conclusions In this work two-dimensional bifurcation diagrams have been extensively studied for a wide class of basic DC–DC converters, which are used in practical applications. The similarities and differences among the Buck, Boost and Buck–Boost regulators, with a diode, and without, are explained from a dynamical systems and engineering points of view. The three subclasses show a general bifurcation pattern which includes smooth flip and Hopf bifurcation, quasiperiodicity, nonsmooth bifurcations and chaotic dynamics. The effect due to the existence of a change from continuous to discontinuous conduction mode has also been discussed. These two-dimensional bifurcations have been precisely computed with ad hoc code taking advantage of the piecewise linearity topologies. Analytical closed-form solutions are exploited and only a numerical method is used to find the switching instants between the two different topologies. A large part of this work presents only a macroscopical view of what happens to these systems when some meaningful parameters are varied. The authors are working on the details of some interesting practical regions to provide also a microscopic view which can be useful for design.
Acknowledgments The authors would like to acknowledge David Carri´ o for his invaluable help in building up a great part of the numerical simulator. This work was supported by the Spanish CYCIT under Grant DPI2000-1509-C03-02 and TIC2000-1019-C02-01.
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