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TWO DIMENSIONAL BIFURCATION DIAGRAMS BACKGROUND

VIEWS: 11 PAGES: 25

									International Journal of Bifurcation and Chaos, Vol. 13, No. 2 (2003) 427–451
 c World Scientific Publishing Company




              TWO-DIMENSIONAL BIFURCATION DIAGRAMS.
               BACKGROUND PATTERN OF FUNDAMENTAL
               DC DC CONVERTERS WITH PWM CONTROL

                                                                         ´
               L. BENADERO, A. EL AROUDI, G. OLIVAR, E. TORIBIO and E. G OMEZ
                               Departament Fisica Aplicada, UPC,
                            Jordi Girona 1-3, Campus Nord-modul B4,
                                      08034 Barcelona, Spain

                               Received May 14, 2001; Revised February 22, 2002


           One of the usual ways to build up mathematical models corresponding to a wide class of DC–DC
           converters is by means of piecewise linear differential equations. These models belong to a class
           of dynamical systems called Variable Structure Systems (VSS). From a classical design point of
           view, it is of interest to know the dynamical behavior of the system when some parameters are
           varied. Usually, Pulse Width Modulation (PWM) is adopted to control a DC–DC converter.
           When this kind of control is used, the resulting mathematical model is nonautonomous and
                                                        e
           periodic. In this case, the global Poincar´ map (stroboscopic map) gives all the information
           about the system.
                The classical design in these electronic circuits is based on a stable periodic orbit which has
           some desired characteristics. In this paper, the main bifurcations which may undergo this orbit,
           when the parameters of the circuit change, are described. Moreover, it will be shown that in
           the three basic power electronic converters Buck, Boost and Buck–Boost, very similar scenarios
           are obtained. Also, some kinds of secondary bifurcations which are of interest for the global
           dynamical behavior are presented. From a dynamical systems point of view, VSS analyzed in
           this work present some kinds of bifurcations which are typical in nonsmooth systems and it is
           impossible to find them in smooth systems.

           Keywords: Nonlinear phenomena; bifurcation diagrams; multistability; DC–DC converters;
           PWM control.



1. Introduction                                                       The operation of power electronic converter
                                                                 circuits is mainly based on the switching between
The basic DC–DC converters Buck, Boost and                       different linear configurations. This must be imple-
Buck–Boost are a family of circuits which allow the              mented with an appropriate control of the switches.
conversion of energy from one level to another with-             In a noise perturbation free environment, given the
out taking into account, theoretically, losses in the            desired output voltage, the switching frequency can
components. They are used extensively in power                   be selected and the switches can be turned ON and
supplies for electronic circuits and in the control of           OFF according to a fixed pattern; this is referred
the flow of energy between DC to DC systems, and                  to as the open loop system. In contrast, in indus-
in any industrial application where there is a need              trial applications, noise and perturbations are al-
of stabilizing an output voltage to a desired value.             ways present, and also the parameters of the circuits
Also, they are widely used in small spacecrafts                  may be affected by external disturbances. Thus
such as satellites where DC power is generated by                the use of an appropriate control to counteract the
solar arrays.                                                    modifications on the output voltage in the system


                                                           427
428 L. Benadero et al.

is recommended; this is referred to as the closed        2. Continuous Time Model of the
loop system. The most popular control strategy              Basic Switching Regulators
used in the literature is Pulse Width Modulation
                                                         2.1. State equations
(PWM) where electronic control of the basic power
electronic converter circuit is achieved by control-     The basic DC–DC switching converters are shown
ling the duty cycle d of the controlled switch S (the    in Fig. 2. The differential equations, modeling
duty cycle is the ratio of the ON phase of the switch    each one of the three configurations that use ev-
to the period of the periodic ON–OFF operation).         ery converter, can be derived by using the standard
We will refer to the ON phase when the switch S is       Kirchoff’s laws. Let us define matrices A 1 , A2 , A3 ,
closed and diode D is open; the OFF phase refers         B1 and B2 as follows:
                                                               
to when the switch S is open and diode D closed;                     1      1           
                                                                                                1
                                                                                                         
and mode OFF’ (or discontinuous mode) takes place                −
                                                                RC                        −          0 
                                                                            C           
                                                                                 , A2 =  RC
when both switch and diode are open (see Fig. 2).         A1 =                                          ,
                                                                    1       1                       1
There are many ways that fixed frequency PWM                       −       −                   0     −
                                                                    L       RS                         L
control can be implemented. Nevertheless, the basic                                                
ingredients of almost all existing PWM controllers                  1                            0
                                                               − RC           0                                (1)
that are used for voltage control are:                   A3 =                      , B1 =  VIN  ,
                                                                  0            0                 L
1. an output voltage error amplifier
2. a T -periodic sawtooth signal generator (driving                  0
   signal)                                               B2 =
3. a comparator that compares the error amplifier                     0
   output with the sawtooth waveforms.                   where R is the output load resistance, L is the in-
                                                         ductance which is supposed to have an Equivalent
     The most interesting dynamics of these sys-         Series Resistance ESR RS , C is the capacitance, and
tems, from a classical design point of view, is the      VIN is the input voltage. During each phase (ON,
T -periodic orbit (periodic evolution with the same      OFF and OFF’), and until a switching condition is
period as the driving signal).                           fulfilled, the dynamics of the system is described by:
     Nonlinear phenomena in the PWM voltage con-                             ˙
                                                                            X = AX + B                     (2)
trolled DC–DC basic power electronic regulators
have been studied in the past years. Various kinds       X = (vC , iL )T is the vector of the state variables
of bifurcational behaviors are found for different        and the overdot stands for derivation with respect
converters with different control schemes. Flip bi-                  ˙
                                                         to time t(X = dX/dt). Table 1 shows the A’s and
furcations and period doubling route to chaos are        B’s matrices for the three basic converters Buck,
found in the Buck converter [Deane & Hamill, 1990;       Boost and Buck–Boost during each phase.
Fossas & Olivar, 1996; Tse, 1994a], Neimark–Sacker
bifurcation and quasiperiodicity route to chaos are      2.2. Analytical solutions for each
found to occur in the PWM Boost and Buck–Boost                configuration
converters [El Aroudi et al., 2000] and border colli-
sion bifurcations are found to occur in the Buck and     Since the previous differential equations are piece-
the Boost converter with different control strategies     wise linear (PWL), a closed form solution is
[Banerjee et al., 2000; Yuan et al., 1998]. Up to now
there are very few works that try to characterize        Table 1. The A’s and B’s matrix for the basic converters
the bifurcational phenomena in the parameter space       during phases ON, OFF and OFF’.
[Chakrabarty et al., 1996; Banerjee & Chakrabarty,
                                                         Converter       AON       AOFF   AOFF   BON      BOFF   BOFF
1998; El Aroudi et al., 2000; Olivar, 1997; Toribio
et al., 2000]. The aim of this paper is to investigate   Buck            A1         A1     A3        B1    B2     B2
in the parameter space the mechanisms of losing
                                                         Boost           A2         A1     A3        B1    B1     B2
the stability of the T -periodic orbit, and the tran-
sition between the different bifurcations in these        Buck–Boost      A2         A1     A3        B1    B2     B2
systems.
                                                                                Two-Dimensional Bifurcation Diagrams 429

available for each configuration. Let us write:                 where t0 is the initial time at which the system
             1                 RS                              switches from one configuration to another, and i L0
     kC =       ,      kL =       ,   k = kc + kL ,            and vC0 are the states of the system at the switching
            2RC                2L
                                                         (3)   instant t0 . The values for VC∞ and IL∞ are
                       1       RS
            ω0 =            1+         −   k2
                      LC       R                                                      VIN               VC∞
                                                                          VC∞ =            ,    IL∞ =
Therefore, the solution for each configuration can                                       RS               R
                                                                                     1+
be written as:                                                                           R
• RC circuit and L in series with VIN (matrices                    The above solutions are only valid when we
  A2 and B1 ) (this configuration corresponds to the            have
  ON phase for the Boost and the Buck–Boost):                               1       RS
                                                                                1+       − k2 > 0          (8)
      vC (t) = vC0 e−2kC (t−t0 )                                           LC        R
                 VIN         VIN                         (4)
      iL (t) =       + iL0 −             e−2kL (t−t0 )         in such a way that ω0 is real and positive. From
                 RS          RS                                the design point of view this is the most important
• RLC oscillator in the free regime (matrices A 1              case, since it gives oscillatory solutions.
  and B2 ) (this configuration corresponds to the
  OF F phase for the Buck and the Buck–Boost):
                                                               2.3. The switching conditions
    vC (t) = e−k(t−t0 ) [vC0 cos ω0 (t − t0 )
                                                               The PWM control of a switched converter is
                    iL0   kvC0
              +         −      sin ω0 (t − t0 )]               achieved by the comparison of the control voltage
                    Cω0    ω0                                  vcon which is a linear combination of the capacitor
                                                         (5)
     iL (t) = exp−k(t−t0 ) [iL0 cos ω0 (t − t0 )               voltage vC and the inductor current iL in the form
                  iL0      vC0
               +      −           sin ω0 (t − t0 )]                            vcon = A(vC + Zr iL − VREF )         (9)
                  ω0       Lω0
• RLC oscillator forced with VIN (matrices A1 and
  B1 ) (this configuration corresponds to the ON                with a driving signal, generalized as a triangular
  phase for the Boost, and the OF F phase for the              function [Fig. 1(a)]
  Buck–Boost):                                                                 
                                                                               V + VU −VL t            if 0 < t < pT
   vC (t) = VC∞ + e−k(t−t0 ) [(vC0                                              L
                                                                               
                                                                                     pT
                                               (iL0 − IL∞ )    vtriang (t) =
            − VC∞ ) cos ω0 (t − t0 ) +                                         
                                                   Cω0                         V − VU −VL (t−pT )
                                                                               
                                                                                U                      if pT < t < T
                                                                                     (1−p)T
                 k(vC0 − VC∞ )                                                                                  (10)
             −                 sin ω0 (t − t0 )]
                      ω0                                       where A is the gain of the error amplifier, Z r is the
   iL (t) = IL∞ + exp−k(t−t0 ) [(iL0                           impedance used to convert the inductor current to
                                                               a voltage, VREF is the reference voltage, VL and VU
                                           (iL0 − IL∞ )        are the lower and upper values of the driving trian-
            − IL∞ ) cos ω0 (t − t0 ) +
                                                ω0             gular signal, T and p are the periode and symmetry
                 (vC0 − VC∞ )                                  factor of this signal.
             −                sin ω0 (t − t0 )]                    Let us define the function Vcomp as:
                     Lω0
                                           (6)
• The capacitor connected to the load when the                  Vcomp (t) = vcon (t) − vtriang (t)
  converter works in the so-called Discontinu-
  ous Conduction Mode (DCM), characterized by                              = A(vC + Zr iL − VREF ) − vtriang (t) (11)
  iL (t) = 0) (matrices A3 and B2 ):
                                                                    The switching condition is therefore:
                  vC (t) = vC0 e−2kC (t−t0 )
                                                         (7)
                  iL (t) = 0                                                          Vcomp (t) = 0                (12)
430 L. Benadero et al.




                              (a)




                                                                   Fig. 2. The three basic power electronic converters from up
                                                                   to down, Buck, Boost and Buck–Boost.


                              (b)
                                                                   istic waveform simulator for DC to DC converters
Fig. 1. (a) Triangular signal vtriang (t) used as driving signal   because it is designed for electrical and electronic
in control. (b) Normalized function h(τ ).
                                                                   circuits. However, since the simulations are very
                                                                   time-consuming, one cannot rely on this package
     Since the expressions of the trajectories in each             for extensive computations. Other packages, like
configuration include exponential and trigonometric                 LOCBIF [Khibnik et al., 1993], INSITE [Parker &
functions, this equation is transcendental. Thus, a                Chua, 1987], DSTOOL [Guckenheimer et al., 1991]
closed form expression for the solution is not pos-                and AUTO [Deodel & Wang, 1995] are well suited
sible [Hamill et al., 1992; El Aroudi et al., 1999;                only for smooth systems. They compute equilib-
Fossas & Olivar, 1996]. Hence, one must resort                     rium points, eigenvalues, characteristic multipliers,
to numerical methods to compute the switching                      Lyapunov exponents and invariant manifolds as-
instants.                                                          suming that the vector field is smooth enough. But
     It should be noted that when the converter                    the switching action in power electronic convert-
enters in discontinuous conduction mode (that is,                  ers makes these systems very different from those
changes from iL (t) = 0 to iL (t) = 0), a new switch-              characterized by a smooth vector field. The vector
ing condition (which is iL = 0) appears.                           field for DC–DC switching converters is discontinu-
                                                                   ous and PWL in the form:

2.4. Numerically computed orbits                                                     f1 (X, t)   if   g(X, t) < 0
                                                                             ˙
                                                                             X=                                          (13)
Numerical methods usually play a major role when                                     f2 (X, t)   if   g(X, t) > 0
the system is nonlinear and parameters must be
varied in certain ranges.      Although there ex-                       In DCM, the previous model is still valid and
ist some very useful packages for the study of                     the functions f1 or f2 are in turn PWL. Moreover,
the behavior of dynamical systems, to our knowl-                   these numerical packages do not allow to handle
edge, none is specially suited for piecewise lineari-              two-dimensional bifurcation diagrams easily. Fi-
ties. In these systems, one may take profit of the                  nally, as it was mentioned before, the numerical
analytically-computed solutions, but the switch-                   method used for integrating the system of differ-
ing instants must be numerically computed. The                     ential equations is not optimized for PWL systems,
PSPICE package [Rashid, 1990] is the more real-                    where closed form solutions in each sub adjacent
                                                                           Two-Dimensional Bifurcation Diagrams 431

interval are available. For these reasons, some spe-       the following will be chosen:
cific programs written in C code are used in this                          √                                VIN
paper, using the closed form solution in each linear         Ts = T0 = 2π LC, Vs = VIN ,            Is =
                                                                                                            L/C
configuration. The equation for the switching con-
dition is solved by a specific root-finding method               Then, we define the normalized variables and
(Newton, secant, bisection, . . . etc.). In our compu-     parameters of the power circuit as follows:
tations, the bisection method is used.                                 vC (t)           L/C               t
     Therefore the analytical solutions combined              v(t) =          ,   i(t) =     iL (t), τ =     ,
                                                                        VIN            VIN               T0
with the bisection method to locate the switching
instants will reduce considerably the time needed                              R                 L/C
                                                                       Q=           , QS =
for a given simulation. In this paper, the global dy-                         L/C                RS
namics is obtained in this way. The simulations are                                                         (14)
therefore very fast compared to simulations com-           where Q is the quality factor associated to the
puted with standard packages. This method was              load, and QS is the quality factor associated to the
already used by the authors in [Deane & Hamill,            equivalent series resistance ESR of the inductance.
1990] and [Fossas & Olivar, 1996] to compute tra-          Finally, τ is the dimensionless time.
jectories of the PWM voltage controlled DC–DC
Buck converter.                                            3.3. Scaling the switching condition
                                                           Introducing the normalization in the control
3. Dimensionless Formulation                               condition, we get
                                                                fS (v, i, τ ) = v + Zi − VR − VD h(τ ) = 0      (15)
3.1. Background
                                                           where VR is the normalized reference voltage, V D is
The large number of parameters associated with             the normalized width of the external control signal
the PWM DC–DC converters is a major handicap               and Z is the normalized impedance. h(τ ) becomes
to the characterization of all the possible dynam-         a triangular function with amplitude one, null av-
ics. To deal with this problem, some authors have          erage value, period TN and symmetry factor p [see
fixed the values of the parameters according to spe-        Fig. 1(b)].
cific examples [Hamill et al., 1992; Tse, 1994b] and                  VREF VU + VL         VU − VL
then they varied the parameters near these values.           VR =        +        , VD =          ,
                                                                     VIN   2AVIN           AVIN
Instead, we will define dimensionless parameters                                                     (16)
achieving a significant reduction in the number of                           Zr         T
                                                                       Z=       , TN =
independent parameters of the circuit. We begin                             L/C        T0
with linear transformation of the state variables in
such a way that the resulting new variables are di-
mensionless. We choose Ts , Vs , Is scale parame-
ters with physical dimensions of time, voltage and
current respectively in such a way that the new
variables t/Ts , vC /Vs , iL /Is are dimensionless.
     The normalization of variables and parameters
allows to carry out an easy analysis, it facilitates the
understanding of the differences between converters
and especially shows how the quality factors Q (re-
lated to the resistive load R) and QS (related to the
resistor RS in series with the inductor) play their
role in an outstanding way (the quality factors will
be defined below).


3.2. Scaling the power stage
                                                           Fig. 3.   The switching band in the normalized phase plane
Among all possible choices of the scale parameters,        (v, i).
432 L. Benadero et al.

     It should be noted that the evolution of the       3.4. Normalized analytical solutions
driving signal voltage may be represented by a               of each configuration
switching band in the (v, i) phase plane (Fig. 3).
Within this band the switchings from one con-           Note that Eq. (8) is equivalent to
figuration to another can occur. With the pro-
posed normalization, the resulting state variables                    1          1     1   1        2
                                                                1+       −               +              >0      (19)
and parameters are dimensionless. Their number                       QQS         2     Q QS
is reduced from 10 to 5. The normalization means
that different circuit designs with equal normalized     Assuming that the condition above is fulfilled,
parameters have equivalent dynamics.                    the closed form solutions for each configuration
     After scaling variables and parameters, the        are:
dimensionless differential equations of each config-
uration can be written                                  • The LRC parallel oscillator with the source V IN
                      ˙                                   in series with the inductor.
                      Y = CY + D                (17)

where the overdot denotes for the derivation with
respect to the dimensionless time τ . Y = (v, i) T is   v(τ ) = V∞ + exp−α(τ −τ0 ) [(v0 − V∞ ) cos ωN (τ − τ0 )
the vector of the normalized state variables and the
dimensionless matrices are C1 , C2 , C3 , D1 and D2                  i0 − I∞ − β(v0 − V∞ )
                                                               +                           sin ωN (τ − τ0 )]
as follows:                                                                    δ
                                                 
            1                           1
         − Q       1               − Q     0        i(τ ) = I∞ + exp−α(τ −τ0 ) [(i0 − I∞ ) cos ωN (τ − τ0 )
                                              
C1 = 2π               , C2 = 2π               ,
                    1                       1                    v0 − V∞ − β(i0 − I∞ )
             −1   −                 0       −                  −                           sin ωN (τ − τ0 )]
                    QS                        QS                              δ
                     
              1
         −       0            0               0                                                               (20)
C3 = 2π      Q       , D1 =        , D2 =
             0    0             2π              0
                                                          Note that the equilibrium point (QQ S /(Q + QS ),
                                                (18)      QS /(Q + QS )) of this configuration is a stable
                                                          spiral sink.
     Matrices Ci and vectors Di correspond to nor-      • The LRC parallel oscillator in the free regime.
malized ones Ai and Bi respectively (see Table 1).
     Therefore, the converter depends on four essen-           v(τ ) = exp−α(τ −τ0 ) [v0 cos ωN (τ − τ0 )
tial dimensionless parameters Q, TN , VD , VR and an
additional (parasitic) parameter Q S (the essential                         i0 − βv0
                                                                        +            sin ωN (τ − τ0 )]
and parasitic classification is due to circuit consid-                           δ
erations, but this will not affect the analysis). T N                                                            (21)
is the period of the driving signal normalized to the           i(τ ) = exp−α(τ −τ0 ) [i0 cos ωN (τ − τ0 )
natural period of the LC-circuit, and finally, in the                        v0 − βi0
event of Rs = 0 (not considered here), Q is the                         −            sin ωN (τ − τ0 )]
                                                                               δ
quality factor of the power circuit. Thus, a DC–
DC converter can be characterized by the normal-          The equilibrium point (0, 0) of this configuration
ized parameter set C = {Q, QS , TN , VR , VD }. The       is a stable spiral sink.
set C of dimensionless parameters of one specific        • The inductor connected to the source, and the
PWM DC–DC regulator stands for a whole family             RC circuit:
of regulators whose dimensionless parameters are
those in C, and thus they display the same dynam-                                − 2π (τ −τ0 )
ics (obviously, this is true only if the converter is           v(τ ) = v0 exp     Q


the same and its model remains valid for the entire                                                2π
                                                                                                                (22)
                                                                                                 − Q (τ −τ0 )
family).                                                        i(τ ) = QS + (i0 − QS ) exp         S
                                                                                Two-Dimensional Bifurcation Diagrams 433

                                                                               e
                                                                 4. The Poincar´ Map
                                                                 In the case of periodically driven systems, the usual
                                                                          e
                                                                 Poincar´ section considered is a plane Σ (with equa-
                                                                 tion τ = 0) in the cylindrical space (v, i, τ ) ∈
                                                                 R+ × R+ × S 1 (we identify (v, i, τ = 0) with
                                                                 (v, i, τ = TN )). At every period of the dimensionless
                                                                 driving signal the trajectory intersects Σ. A map
                                                                 which lies on two successive points in the Poincar´  e
                                                                 section can be defined.

                                                                               P :Σ→Σ
                                                                        (v(τ0 ), i(τ0 )) → (v(τ0 + TN ), i(τ0 + TN ))


                                                                 4.1. Obtaining the one-periodic orbit

Fig. 4. Schematic representation for the three configura-         Assume we have a converter with all parameters
tions that use the Boost converter.                              fixed. The fixed points of the stroboscopic map are
                                                                 obtained equaling, on one hand, the values of the
                                                                 capacitor voltage and inductor current at the be-
                                                                 ginning and those at the end of each cycle of the
  The equilibrium point (0, QS ) of this configura-               driving signal and, on the other hand, forcing the
  tion is a stable sink.                                         continuity of the state variables at the asynchronous
• The RC circuit (this corresponds when the con-                 switching instants [di Bernardo et al., 1997]. By
  verter works in DCM):                                          imposing these conditions, a system of two non-
                                                                 linear equations with two unknowns veq and ieq is
                    v(τ ) = v0 exp
                                     − 2π (τ −τ0 )
                                       Q                         obtained. By imposing the asynchronous switch-
                                                        (23)     ing condition, another equation is added, being the
                    i(τ ) = 0                                    switching time τS another unknown. These three
                                                                 nonlinear equations with three unknowns can be in-
     The equilibrium point (i = 0) of this configura-             troduced to a standard mathematics program (for
     tion is a stable one-dimensional1 sink.                     example Maple V) to find the fixed points (v eq , ieq )
                                                                 for a set of parameter values.
     In the expressions of the solutions, we have used
V∞ = QQS /(1 + QQS ), I∞ = V∞ /Q, ωN = 2πδ,
δ = (1 − β 2 ), β = 1/2(1/Q − 1/QS ) and α =                     5. Mechanisms of Losing the Stability
π(1/Q + 1/QS ). Also, v0 , i0 are the state variables               of the One-Periodic Orbit
at the switching instant τ0 .
     As we have seen, the dynamics in each config-                The basic elements which will be used in this section
uration is well determined. But the parameters of                to study the stability of the one-periodic orbit are
the circuit are selected in such a way that when                                 e
                                                                 the Poincar´ map and the characteristic multipliers
we make the control work, the converter switches                 of its fixed points. The stability of the one-periodic
from one configuration to another. Therefore, since               orbit is analyzed by means of the response to a small
the system is highly nonlinear due to the switch-                perturbation near this fixed point.
ing action, a great variety of dynamics are pos-                       To obtain the stability character of a fixed point
sible (limit cycles, subharmonics, quasiperiodicity,             (veq , ieq ) of this map is enough to compute the im-
chaos). In Fig. 4, the switching lines for a PWM                 age of a perturbed state near the fixed point in the
voltage mode controlled Boost regulator and the                  phase plane (vn , in ). This transforms the problem
trajectories near the equilibrium points of each con-            of analyzing the stability of the one-periodic orbit
figuration are schematically plotted.                             to the study of the eigenvalues of the linearized
1
    When the converter works in the DCM in some cycles, the system order changes from two to one in these cycles.
434 L. Benadero et al.

map DP . The four coefficients of the matrix DP            will specify the sense of variations of the parame-
associated to the linearization near a fixed point        ters; the larger arrow corresponds to the parameter
are obtained by computing the transformation of          chosen as primary (in most of the diagrams this is
the state space points in the neighbor of the fixed       TN ). Also, in order to understand the bifurcation
points.                                                  patterns better, some simulations have been made
    From the eigenvalues m1 and m2 of DP (the            using an ideal switch instead of a diode.
characteristic multipliers of the orbit), or from the
Lyapunov exponents (λ = log |m|/TN ) the following       6.1. A point in the parameter space
bifurcations will be detected:
                                                         To obtain a good comparison of all the two-
• Flip or period doubling bifurcation of the stable      dimensional bifurcation diagrams, we choose a
  one-periodic orbit. It is characterized by the fact    central point in the parameter space and then, we
  that one of the characteristic multipliers is equal    vary the parameters around this point. In this zone
  to −1 and the other one has absolute value less        we will explore all the possible dynamics of the three
  than 1.
                                                         converters.
• Hopf or Neimark–Saker bifurcation. It is char-              We fix the values of this parameter point to
  acterized by the fact that both characteristic         Q = 4, QS = 6 and Z = 0. Moreover we take
  multipliers cross the unit circle, being complex       VR = 0.5, VD = 0.3 for the Buck, and VR = 1.4,
  conjugates.                                            VD = 1.8 for the Boost and the Buck–Boost. We
• Border-Collision bifurcations. They are charac-        cannot ideally take the same VR in the three con-
  terized by a sudden change in the system be-           verters since its value tightly depends on the volt-
  havior accompanied by a jump in the values of          age gain. Thus it is convenient to take V R less than
  the characteristic multipliers. These anomalous        one in the case of the Buck, which reduces the input
  bifurcations are well explained in [Yuan et al.,       voltage; and more than one in the case of the Boost,
  1998; Banerjee et al., 2000; Nusse et al., 1994]       which increases the input voltage. In the case of
  and [Nusse & Yorke, 1992].                             the Buck–Boost it can be any value since this con-
                                                         verter can be designed to increase or reduce the
6. Two-Dimensional Bifurcation                           input voltage.
   Diagrams                                                   The range of variation of the normalized pe-
                                                         riod is large enough to consider not only low periods
In order to obtain the bifurcational structure when      (TN      1), which is the usual range in circuits, but
two parameters are varied, we will plot the two-         also high periods near TN = 0.7 are checked. It will
dimensional bifurcation diagrams corresponding to        be shown that some bifurcations which pose limits
these parameters. The two-dimensional bifurcation        on the TN -periodic dynamics are met near TN = 1.
diagram is color-coded depending on the periodicity      With regards to parameter p, we will use only its
of the attractor corresponding to the point in the       extreme values p = 0 and p = 1.
parameter space. Due to hysteresis phenomena and
in general to possible coexisting attractors, only one   6.2. Choosing the parameters to vary
of them can be identified at each point of the param-
eter space. In this paper the diagrams are produced      For the two-dimensional bifurcation diagrams, we
as follows: first we give the system with extreme         have to choose two parameters from the seven (T N ,
values of both varying parameters and the initial        Q, QS , Z, VD , VR , p) which correspond to a given
condition xini = (vini , iini ); then, one-dimensional   circuit. The bifurcation patterns will be obtained
bifurcation diagrams are successively computed fix-       from different pairs of parameters. We classify all
ing one parameter (primary) and varying the other        our possible parameters according to three different
one (secondary). The final state of the system is         classes. One class contains the parameter associ-
taken as initial condition to the next value of the      ated to time in the control loop TN , and it will be
primary parameter. The initial condition when the        one of the parameters varied in each of our bifurca-
secondary parameter is varied is the last state cal-     tion diagrams; another class will be made with the
culated using the preceding value of the secondary       parameters associated to the power circuit, these
parameter, both with the same extreme value of           are Q and QS , which will be kept fixed in each of the
the primary parameter. Arrows in the diagrams            bifurcation diagrams. Finally a third class includes
                                                                                 Two-Dimensional Bifurcation Diagrams 435




Fig. 5. 2D bifurcation diagram for a Boost regulator in continuous conduction mode. Varying parameters are period T N
and impedance Z and the fixed ones are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. The color for each point in the
diagram shows the period of the resulting orbit. Initial conditions are those corresponding to the orbit of the preceding point
as indicated by the arrows. The same upper color codes are used in all diagrams.



all the parameters which define the control. More                 7. Bifurcation Diagrams for the
specifically, they define the regulation band (Z, V D ,               Boost Converter in the
VR and p). One of them will be taken to be varied                   (TN , Z)-Parameter Space. Regions
when we compute the two-dimensional bifurcation                     in the Parameter Space
diagrams. It will be shown that VR has a meaningful
effect in the Boost and the Buck–Boost converters,
but has little or no effect in the Buck. Thus, in this
                                                                 7.1. Basic regions according to the
case parameters VD and Z will be varied. Param-
                                                                      dynamical behavior
eter Z allows to detect more easily all the possible             Using the numerical simulator which has been
dynamics.                                                        described before, we obtained two-dimensional
436 L. Benadero et al.




Fig. 6. Time (left) and phase plane (right) representation for a 2TN -periodic orbit showing chattering. This corresponds to
a Boost regulator in continuous conduction mode. Parameters are TN = 0.55, Z = 0.82 and like in Fig. 5, (Q = 4, QS = 6,
VR = 1.4, VD = 1.8 and p = 0).




bifurcation diagrams for the Boost converter with               Its linearized dynamics corresponds to a focus, with
PWM control, and without diode, this is, with both              negative Lyapunov exponents.
ideal and complementary switches. The parameters                      It is clear that the border of the region 1T shows
are fixed according to the central point defined be-              a discontinuity at a point of the parameter space
fore (Q = 4, QS = 6, VR = 1.4, VD = 1.8, p = 0)                 which corresponds approximately to T N = 0.50 and
and we vary Z and TN (see Fig. 5). The resulting bi-            Z = 0.19. This type of behavior, which is com-
furcation diagram is simpler than the one obtained              mon in a certain sense, is associated to a variation
using a diode, and we can distinguish four big dif-             pattern in the characteristic multipliers when the
ferent regions, which are described in the following:           parameters are moved, and this implies two differ-
     First, we have a big two-periodic region of lobar          ent types of borders.
type, which will be called 2T region, on the right-
hand side of the figure. Into this zone the orbits
display the chattering phenomenon and too much                  7.2. Curve of Hopf bifurcations
time is required to decide the periodicity. The same
phenomenon is detected on other similar lobar ar-               At the border between the zones 1T and QP a Hopf
eas, which yield sliding modes like in Fig. 6, with             bifurcation occurs, which corresponds to a null Lya-
Z = 0.82 and TN = 0.55.                                         punov exponent. The bifurcation occurs smoothly
     Second, we have a region whose stationary be-              when two complex characteristic multipliers cross
havior is the fixed point of the ON topology, in                 the unit circle. If we decrease the impedance Z the
the lower part of the figure. This region will be                characteristic multipliers begin to grow and finally
called the FP region. On the right part of its upper            they cross the unit circle at the border. We will call
border we found that the impedance is constant at               this border the Hopf border.
Z = 0.08, and the left part is made of a curve which                 For TN near 0 this border begins at Z = −0.05,
starts at a point with TN = 0.18 approximately.                 which can be analytically predicted when averaging
     Third, between the region FP and the two-                  techniques are used. This is tightly related with the
periodic lobar region, we found chaotic and                     critical value of the impedance Z associated to the
quasiperiodic orbits with some little periodic is-              equilibrium [Benadero et al., 1999] which is related
lands. The chaotic and quasiperiodic orbits are                 to the orientation of the vector field. The value of
found after Hopf bifurcation following the torus                the critical impedance fixes, with the exception of
breaking route to chaos. This zone will be called               a certain correction which depends on the values
QP.                                                             of Q and QS , the impedance Z which makes the
     Fourth, there is a large one-periodic dynamics             transition to instability. For different values of T N ,
(blue) which is in the remainder of the figure. This             the bifurcation value for Z is changed and the pe-
is the usual region of operation of the regulators              riod one always appears in the upper region of the
and we will denote it by TN -periodic or simply 1T.             figure.
                                                                            Two-Dimensional Bifurcation Diagrams 437




                                                         (a)




                                                         (b)
Fig. 7. Computed Lyapunov exponents versus TN for a Boost regulator. Fixed parameters are as in Fig. 5, Q = 4, QS = 6,
VR = 1.4, VD = 1.8 and p = 0.




7.3. Curve of flip bifurcations                                 TN = 0.63. As Z is increased, the larger Lyapunov
                                                               exponent does not pass through zero, and thus no
The border between the region 1T and the 2T lobe
                                                               bifurcation is produced. The limit case is obtained
corresponds to a flip bifurcation. At this point, both
                                                               approximately for Z = 1.01 and TN = 0.51.
Lyapunov exponents are real and the larger one be-
comes null. We will call this border the flip border.
Figure 7(a) shows both two exponents when pa-                  7.4. Curve of grazing bifurcations
rameter TN is varied, and three important facts are
observed: first, if we decrease Z then the average              Finally, the border between the region FP and QP
value of the Lyapunov exponents increases; second,             corresponds to a grazing bifurcation, and thus will
if we increase TN the average value also increases             be called the grazing border. A grazing bifurca-
when p = 0 (this effect is inverted when p = 1,                 tion is a subtype of the so-called border collision
see Sec. 9.2); and third, a bifurcation from focus             bifurcations and can be only observed in nonsmooth
to a node is produced. From this bifurcation point             systems. This can be observed in Fig. 8 which corre-
and onwards, one exponent grows until it becomes               sponds to an orbit of the regulator with Z = 0.08.
null (this exponent is associated with a character-            Observe that the left line border of the switching
istic multiplier equal to −1) and a flip bifurcation            band passes through the fixed point of the ON state.
occurs. Note that, depending on Z, we are favoring
one bifurcation or the other, and thus two different
curves of bifurcations appear. The limit case corre-           8. Two-Dimensional Bifurcation
sponds to the point Z = 0.19 and TN = 0.50, where                 Diagrams for the Boost
the two curves coalesce.                                          Converter with a Diode
     Figure 7(b) corresponds to the cuspid of the
lobe 2T; for instance, with the arbitrary value                It should be expected that the two-dimensional bi-
Z = 0.82, we have bifurcations at TN = 0.47 and                furcation diagram for a Boost converter with a
438 L. Benadero et al.




Fig. 8. Switching band and attractor associated with the critical value Z = 0.08 for boundary between FP and QP, T N = 0.21
and the other parameters are the same set Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0.




diode should display more complex dynamics than                      First, a region of DCM dynamics of TN -periodic
without it, since we add one more topology to the              type has eroded the 2T-lobe from its right part (this
circuit. As we shall see, this is really so. The new           is, for high values of TN and Z). To check this point,
attractors which are obtained visit periodically or            we plot in Fig. 10 three different orbits for different
intermittently, the discontinuous conduction mode,             values of TN and Z fixed to 1.01 (the value corre-
and can be periodic, quasiperiodic or chaotic. We              sponding to the cuspid of the 2T-lobe). The orbit
will use CCM for dynamics of continuous conduc-                which corresponds to TN = 0.37 is TN -periodic of
tion mode and DCM have been denoted for discon-                CCM type; the orbit for TN = 0.65 is TN -periodic
tinuous conduction mode dynamics. As a general                 of DCM type, and the orbit for TN = 0.51 is placed
rule, DCM stabilizes quasiperiodic and chaotic be-             just within the border between TN -periodic CCM
havior and contributes to reduce the multistability            type and TN -periodic DCM type. Like in the CCM
phenomenon. There exists a border between orbits               type, the central zone of the 2T-lobe (light gray)
of DCM and CCM, although it is not shown in the                corresponds to 2TN -periodic orbits with sliding.
figures when both orbits at each side of the border                   Second, the region QP shows more complex dy-
are stable. If one of the orbits on one side is unsta-         namics than without diode. It can be observed that
ble, this effect can be observed in the diagram. It is          each kTN -periodic Arnold tongue has an accumu-
worth to note that Hopf bifurcations do not appear             lation of 2nkTN -periodic regions (n = 1, 2, 3, . . .)
for TN -periodic orbits in the DCM.                            in the lower part which corresponds to a period
     Figure 9 shows that, with the parameters used,            doubling route to chaos. Figure 11 shows a one-
the addition of a diode in the circuit does not mod-           dimensional bifurcation diagram for Z = 0 varying
ify the curves of Hopf and Flip bifurcations (left             TN . It can be observed as a torus-breaking route
side) in the CCM, and the cuspid of the 2T-lobe                to chaos. The rotation number has been computed
has also been retained. But some important facts               as the quotient of the number of cycles in the state
have really changed:                                           space to the number of cycles of the driving sig-
                                                                              Two-Dimensional Bifurcation Diagrams 439




Fig. 9. 2D bifurcation diagram for a Boost regulator with diode. As in Fig. 5, varying parameters are period T N and
impedance Z and the fixed ones are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0.




Fig. 10. Intensity current versus TN for the TN -periodic orbit of a Boost regulator in both switching instants. Parameters
are Z = 1.01 and the set Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0. Some orbits are also represented for three values of
TN in order to show the discontinuous mode bifurcation.
440 L. Benadero et al.




Fig. 11. Upper figures correspond to the bifurcation diagram for a Boost regulator with diode and period T N , which present
a Hopf bifurcation and the associated rotation number, showing clearly the devil staircase pattern. Partial zoom is showed
                                                         e
in both central figures. The bottom figures are Poincar´ map orbits for three periods showing phase locking (8T N ) (left),
quasiperiodicity (center), and phase locking (30/2TN ) (right). In the last one only the attractor is represented; the others
show also the divergence from the unstable 1T orbit. Parameters are Z = 0 and the set Q = 4, Q S = 6, VR = 1.4, VD = 1.8
and p = 0.
                                                                             Two-Dimensional Bifurcation Diagrams 441




Fig. 12. This figure is the same as Fig. 5, but changing the sweep as specified in the text; the sweeping direction is also
indicated by the arrows.




nal and a typical devil staircase has been obtained.          values for Q and QS are related to DCM. This is
The same figure includes the Poincar´ sections for
                                        e                     due to the increase in the relative amplitude of the
some orbits corresponding to several values for T N .         oscillations. On the other hand, the evolution of
Some of them correspond to phase-locking orbits;              the Lyapunov exponents shows that the stability of
for instance, we have period 8TN for TN = 0.15 and            the region 1T-CCM depends on Q and QS in differ-
period 15TN for TN = 0.16 (its period is the ad-              ent ways (see Fig. 14). Concretely, stability grows
dition of the periods of the neighboring 8T N and             with Q and decreases when QS is made higher. If
7TN periodic orbits). Finally, for TN near 0.153 we           we increase QS then the losses in the inductor de-
obtain a quasiperiodic attractor [El Aroudi et al.,           crease and thus the system is made less stable. The
1999].                                                        reason why the converter gets better stability when
     Third, the grazing border is found for lower val-        Q is increased is not so easy to explain. On one side,
ues for Z. This means that the QP region is bigger            Q has a similar role like QS . But on the other side,
than before. Near this border there is also multi-            Q moves the orbit in the phase space in a sense that
stability. To show that, we plot a new bifurcation            the critical impedance gets lower, and the stability
diagram of the same zone sweeping the parameter               is favored [Benadero et al., 2002]. This latter effect
ranges in the opposite order (see Fig. 12), and we            has predominance over the decrease of the losses in
obtain slightly different patterns, specially in the           the inductor.
overlapping of the Arnold tongues. Moreover de-                    Figure 15(a) shows a two-dimensional bifurca-
pending on the initial conditions, if Z < 0.08 a fixed         tion diagram with Q = 8 instead of Q = 4. As it
point attractor is also possible.                             has been explained, this makes the stability grow
                                                              and favors DCM. The effect in the diagram is that
                                                              the curve of Hopf bifurcations near TN = 0 moves
9. Effect of the Other Parameters in                           approximately from Z = −0.05 (for Q = 4) to
   the Two-Dimensional Bifurcation                            Z = −0.12 (for Q = 8). Also, since DCM is fa-
   Diagram for the Boost Converter                            vored, the region 1T-DCM moves to the left lower
                                                              end. This makes dissappear the region 2T-CCM
9.1. Effect of Q and QS                                        and makes appear a little 1T-DCM region inside
                                                              the QP. Inside this zone of the parameter space we
In Fig. 13, we plot the values of the intensity cur-          have 2T-DCM behavior beside the right part of the
rent at the switching instant against the bifurcation         1T-DCM region.
parameter TN . The quality factors Q in (a), and                   Figure 15(b) shows a two-dimensional bifurca-
QS in (b), are taken as secondary parameter. High             tion diagram with QS = 3 instead of QS = 6. In
442 L. Benadero et al.




                                                           (a)




                                                           (b)
Fig. 13. Intensity current at the switching instants for a Boost regulator versus period T N . (a) QS is fixed and Q takes
different values; (b) Q is fixed and QS is varied. The remaining parameters are Z = 0, VR = 1.4, VD = 1.8 and p = 0.




                                                           (a)




                                                           (b)
Fig. 14. Computed Lyapunov exponents for a Boost regulator versus period TN . (a) QS is fixed and Q takes different values;
(b) Q is fixed and QS is varied. The remaining parameters are Z = 0, VR = 1.4, VD = 1.8 and p = 0.
                                                                             Two-Dimensional Bifurcation Diagrams 443




                                                           (a)




                                                           (b)
Fig. 15. 2D bifurcation diagram for a Boost regulator. Varying parameters are period T N and impedance Z. Fixed parameters
are VR = 1.4, VD = 1.8, p = 0. (a) Q = 8, QS = 6; (b) Q = 4, QS = 3.
444 L. Benadero et al.




                                                           (a)




                                                           (b)
Fig. 16. 2D bifurcation diagram for a Boost regulator. Varying parameters are period T N and impedance Z. Fixed parameters
are Q = 4, QS = 6, p = 0. (a) VR = 2, VD = 1.8; (b) VR = 1.4, VD = 0.8.




this case the curve of Hopf bifurcations near T N = 0            and thus the values for Z are always positive. This
also moves to a lower position (from Z = −0.05 for               result, which agrees with the fact that when V R
QS = 6 to Z = −0.18 for QS = 3). Also the QP                     is increased the gain voltage rises up, can also be
region is stretched.                                             explained from the point of view of the critical
                                                                 impedance, which increases with VR .
9.2. Effect of the parameters of the                                   We show in Fig. 16(b) a two-dimensional bi-
     control (regulation band) VR, VD, p                         furcation diagram for VD = 0.8 instead of VD =
                                                                 1.8, which has the same effect as before, moving
We show in Fig. 16(a) a two-dimensional bifurca-                 the curve of Hopf bifurcations to a higher posi-
tion diagram for VR = 2.0 instead of VR = 1.4.                   tion. Qualitatively, to make VD lower means to
This moves the curve of Hopf bifurcations higher                 stretch the regulation band; thus in some sense, the
                                                                      Two-Dimensional Bifurcation Diagrams 445

                                                         But when higher periods are needed, the stability
                                                         gets better when the ramp has positive tangent. It
                                                         is worth to note that in the Boost and Buck–Boost
                                                         converters with p = 1 the stability region gets big-
                                                         ger when TN grows (this is, the border moves down);
                                                         on the contrary, if p = 0 the movement is in the op-
                                                         posite direction. This can be observed in Figs. 9
Fig. 17. Lyapunov exponents for a Boost regulator with   and 18.
parameters Q = 4, QS = 6, VR = 1.4, VD = 1.8. Both            The patterns which were discussed before do
cases p = 0 and p = 1 are represented.                   not change qualitatively if we vary parameter V D
                                                         or VR instead of Z; the different dynamics are vis-
                                                         ited in a similar order. In Fig. 19 a two-dimensional
                                                         diagram is shown using VD and VR as bifurcation
                                                         parameters.


                                                         10. Two-Dimensional Bifurcation
                                                             Diagrams for the PWM
                                                             Buck–Boost Converter
                                                         Figure 20 shows a two-dimensional bifurcation
                                                         diagram for a PWM controlled Buck–Boost con-
                                                         verter with a diode with the same main parameters
                                                         like the Boost. Both regulators display a similar
                                                         pattern. The presence of a 1T-DCM little region
                                                         inside the QP zone occurs like in Sec. 9.1. The
                                                         DCM dynamics is more favored in the Buck–Boost
                                                         converter than in the Boost since the Buck–Boost
                                                         works with lower currents.


                                                         11. Two-Dimensional Bifurcation
Fig. 18. 2D bifurcation diagram for a Boost regulator.       Diagrams for a PWM Buck
Varying parameters are period TN and impedance Z and
fixed are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 1.
                                                             Converter
                                                         Figure 21 shows a two-dimensional bifurcation di-
                                                         agram for a PWM controlled Buck converter with
possibility of TN -periodicity is reduced and we have    diode. In this plot, the values of the central vector
a loss in the stability.                                 of parameters have been slightly modified. Con-
     The bifurcation diagram in the Boost convert-       cretely, we take VR = 0.5 and VD = 0.3. This
ers also depends on the symmetry factor p in the         figure shows some similar patterns with regards to
comparator signal. For p = 0, when TN grows, the         the other bifurcation diagrams presented before but
limit cycle is moved to the region of higher cur-        there are also some differences which will be com-
rents, and this destabilizes the system. The inverse     mented on.
effect is observed for p = 1. Lyapunov exponents              Like in the other bifurcation diagrams, the
are represented in Fig. 17 showing different signs        region 1T is beside the region QP; the system
for slope depending on p. Figure 18 shows that for       changes from 1T to QP behavior through a Hopf
p = 1 (ramp with positive slope) there is quasiperi-     bifurcation. And also, a little region of 1T-DCM
odic behavior. For TN = 0 the bifurcation value for      behavior is inside the QP region.
Z does not change; but it does when TN is varied.            The main differences which can be distin-
In real applications, which correspond to T N near       guished are the following:
zero, the stability character of the regulator does          First, for the values of the parameters used in
not change when one or the other ramp is used.           the simulations, the bifurcation of the T N -periodic
446 L. Benadero et al.




Fig. 19. 2D bifurcation diagram for a Boost regulator. Varying parameters are normalized voltages for reference V R and the
width of the modulating signal VD . Fixed parameters are TN = 0.2, Z = 0, Q = 4, QS = 6 and p = 0.




Fig. 20. 2D bifurcation diagram for a Buck–Boost regulator with diode. Varying parameters are period T N and impedance
Z and the fixed ones are Q = 4, QS = 6, VR = 1.4, VD = 1.8 and p = 0.
                                                                             Two-Dimensional Bifurcation Diagrams 447




Fig. 21. 2D bifurcation diagram for a Buck regulator with diode. Varying parameters are period T N and impedance Z and
the fixed ones are Q = 4, QS = 6, VR = 0.5, VD = 0.3 and p = 0.




Fig. 22. 2D bifurcation diagram for a Buck regulator. Varying parameters are period T N and impedance Z. Fixed parameters
are VR = 0.5, VD = 0.3, p = 0. (a) Q = 8, QS = 6; (b) Q = 4, QS = 3.
448 L. Benadero et al.

orbit occurs for negative values of the impedance Z.                   Second, the Buck does not have an ON fix point
This is due to the fact that the Boost and Buck–                  simultaneously with other dynamics. The position
Boost regulators have a positive critical impedance               of its fixed points in the topologies do not allow
but the Buck has null critical impedance. Thus the                that, in the skipping cycle regime, the dynamics
system loses stability when it is fedback positive (Z             could be attracted to this type of behavior. The
negative), and show quasiperiodic dynamics.                       zone of the parameter space where other regulators




                                                            (a)




                                                            (b)




                                                            (c)
Fig. 23. (a and b) 2D bifurcation diagrams for a Buck regulator. Varying parameters are period T N and the width of the
modulating signal VD . Fixed parameters are Q = 4, QS = 3, Z = 0, VR = 0.5 and p = 0, (a) for continuous conduction mode,
and (b) with diode. (c) Lyapunov exponents versus TN is represented for three values of VD , where VD = 1.34 corresponds to
the maximum value of VD for 2TN -periodic orbits.
                                                                              Two-Dimensional Bifurcation Diagrams 449

showed ON fixed point behavior (high negative val-              namics instead of 2TN -periodic orbits with chatter-
ues for impedances and high periods) is replaced by            ing. Moreover between the 2TN -periodic region and
a period adding route (we will denote this region              the chaotic zone there exist periodic narrow bands
by ADD). In the phase space, the orbits present                of periods 2TN , 4TN , 8TN , and so on, which corre-
the skipping cycle phenomenon. We observed simi-               spond to a period doubling route to chaos. On the
larities between this ADD region and the QP region             other hand, for TN near to 1, these can be observed
(see the zone on the right-hand side of Fig. 16(b)).           a region with chaotic dynamics bordered by a curve
     Third, the effect of the parameters Q and Q S              of grazing type bifurcations.
acts in the same direction: a higher loss of stabil-                Figure 23(c) shows the Lyapunov exponents
ity corresponds to bigger quality factors Q and Q S .          against TN with different values for VD . We can
These effects can be observed in Fig. 22. It is worth           see flip bifurcations when λ = 0, which correspond
noting that increasing Q favors DCM.                           with Fig. 23(a). VD = 1.34 is the value for the
                                                               cuspid of the 2T lobe.

12. Two-Dimensional Bifurcation
    Diagrams for a PWM                                         12.2. With a diode
    Voltage-Controlled Buck                                    Figure 23(b) corresponds to a voltage-controlled
    Converter                                                  PWM converter with a diode. A region with
Most practical applications of a Buck converter use            1T-DCM dynamics is clearly visible. The two-
a voltage-controlled PWM loop (Z = 0). This                    dimensional bifurcation diagrams with and without
means that a Hopf bifurcation will not appear. On              diode are similar for TN < 0.15.
the other hand, since the stability character almost
do not depend on VR we will plot the bifurcation               12.3. Two-dimensional bifurcation
diagrams in this section with parameter V D instead                  diagrams with VIN as
of Z.                                                                bifurcation parameter
                                                               In 1990, chaos via period doubling and other
12.1. Without a diode
                                                               nonlinear phenomena are shown to occur in PWM
Figure 23(a) shows a two-dimensional bifurcation               voltage-controlled Buck converter both numerically
diagram for a voltage-controlled PWM Buck con-                 and experimentally when input voltage V IN was
verter without a diode. It can be observed that the            taken as a bifurcation parameter [Deane & Hamill,
lobe has changed with regards to the diagrams in               1990]. Sweeping the parameter VIN means vary-
other sections. Its middle zone contains chaotic dy-           ing simultaneously the dimensionless parameters V R




                (a)                                 (b)                                           (c)
Fig. 24. (a) 2D bifurcation diagram for a Buck regulator. Varying parameters are period T N and the width of the modulating
signal VD . The voltage reference VR = 20VD is also varying. Fixed parameters are Q = 1.1, QS = 7, Z = 0 and p = 1.
                                                                                                       e
(b) Bifurcation diagram showing period doubling route to chaos, where the intensity current for Poincar´ map is represented
versus VD (VR = 20VD ) and TN = 0.066. (c) Corresponds to the chaotic attractor for VD = 0.015 (VR = 0.3) and TN = 0.066.
450 L. Benadero et al.

and VD keeping a linear relation between them.          Banerjee, S., Ranjan, P. & Grebogi, C. [2000]
A two-dimensional bifurcation diagram for a Buck           “Bifurcations in two-dimensional piecewise smooth
without a diode taking the values of the parameters        maps-theory and applications in switching circuits,”
like in [Deane & Hamill, 1990] (Q = 1, Q S = 7,            IEEE Trans. Circuits Syst. I 47, 633–647.
VR = VD /20, Z = 0, TN = 0.066 and p = 0)               Benadero, L., El Aroudi, A., Toribio, E., Olivar, G. &
has been computed with VD in the range [0, 0.05]           Mart´ ınez-Salamero, L. [1999] “Characteristic curves
[Fig. 24(a)]. It is worth to note that the bifurca-        to analyze limit cycles behavior of DC–DC convert-
tion diagram for TN = 0.066 [Fig. 24(b)] and the           ers,” Electron. Lett. 687–789.
                                                        Benadero, L., El Aroudi, A., Olivar, G., Toribio, E. &
attractor observed for VD = 0.015 [Fig. 24(c)] per-
                                                           Moreno, V. [2002] “Bifurcations analysis in PWM
fectly agree with those obtained in the same work
                                                           regulated DC–DC converters using averaged mod-
of Deane and Hamill.
                                                           els,” EPE-PEMC’02, Dubrovnik & Cavtat (Croatia),
                                                           SSIN-04.
13. Conclusions                                         Chakrabarty, K., Podar, G. & Baberjee, S. [1996]
                                                           “Bifurcation behavior of the buck converter,” IEEE
In this work two-dimensional bifurcation diagrams          Trans. Circuits Syst. I 11, 439–447.
have been extensively studied for a wide class of       Deane, J. H. B. & Hamill, D. C. [1990] “Analysis, simu-
basic DC–DC converters, which are used in prac-            lation and experimental study of chaos in the buck
tical applications. The similarities and differences        converter,” IEEE Power Electronic Specialist Conf.
among the Buck, Boost and Buck–Boost regulators,           PESC’90, pp. 491–498.
with a diode, and without, are explained from a         Deodel, E. J. & Wang, X. J. [1995] AUTO94: Soft-
dynamical systems and engineering points of view.          ware for Continuation and Bifurcation Problems in
The three subclasses show a general bifurcation            Ordinary Differential Equations, Technical Report
pattern which includes smooth flip and Hopf bi-             CRPC-95-2, California Institute of Technology.
furcation, quasiperiodicity, nonsmooth bifurcations     di Bernardo, M., Fossas, E., Olivar, G. & Vasca, F. [1997]
and chaotic dynamics. The effect due to the exis-           “Secondary bifurcations and high periodic orbits in
tence of a change from continuous to discontinuous         voltage controlled buck converter,” Int. J. Bifurcation
conduction mode has also been discussed.                   and Chaos 7, 2755–2771.
                                                        El Aroudi, A., Olivar, G., Benadero, L. & Toribio,
     These two-dimensional bifurcations have been
                                                           E. [1999] “Hopf bifurcation and chaos from torus
precisely computed with ad hoc code taking advan-
                                                           breakdown in a PWM voltage-controlled DC–DC
tage of the piecewise linearity topologies. Analyt-
                                                           Boost converter,” IEEE Trans. Circuits Syst. I 11,
ical closed-form solutions are exploited and only          1374–1382.
a numerical method is used to find the switching         El Aroudi, A., Benadero, L., Toribio, E. & Machiche,
instants between the two different topologies. A            S. [2000] “Quasiperiodicty and chaos in the DC–DC
large part of this work presents only a macroscop-         Buck–Boost converter,” Int. J. Bifurcation and Chaos
ical view of what happens to these systems when            10, 359–371.
some meaningful parameters are varied. The au-          Fossas, E. & Olivar, G. [1996] “Study of chaos in the buck
thors are working on the details of some interesting       converter,” IEEE Trans. Circuits Syst. I 43, 13–25.
practical regions to provide also a microscopic view    Guckenheimer, J., Myers, M., Wicklin, R. &
which can be useful for design.                            Worfolk, P. [1991] “DStool: Dynamical sys-
                                                           tem toolkit with interactive interface,” Graphic
                                                           Center of Applied Mathematics, Cornell Univ.
Acknowledgments                                            www.cam.cornell.edu/guckenheimer/dstool
The authors would like to acknowledge David             Hamill, D. C. & Jefferies, D. J. [1988] “Subharmonics
Carri´ for his invaluable help in building up a
     o                                                     and chaos in a controlled switched–mode power con-
great part of the numerical simulator. This work           verter,” IEEE Trans. Circuits Syst. I 35 1059–1060.
was supported by the Spanish CYCIT under Grant          Hamill, D. C., Deane, J. H. B. & Jefferies, D. J. [1992]
                                                           “Modeling of chaotic DC–DC converters by iterated
DPI2000-1509-C03-02 and TIC2000-1019-C02-01.
                                                           nonlinear mappings,” IEEE Trans. Power Electron. 7
                                                           25–36.
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