Quasi-periodic route to chaos in load resonant DC-DC converters

Quasi-Periodic Route to Chaos in Load ResonantDC-DC Converters

Kuntal Mandal∗, Soumitro Banerjee†, Senior Member, IEEE, Chandan Chakraborty∗, Senior Member, IEEE∗Department of Electrical Engineering, Indian Institute of Technology, Kharagpur-721302, India†Indian Institute of Science Education & Research (Kolkata), Mohanpur Campus 741252, India

Abstract—This paper reports first time, both numericallyand experimentally, the quasi-periodic, subharmonic and chaoticbehavior in a phase shift modulated load resonant converter. Thestudy presented in this paper is restricted to third order series-parallel load resonant converter. From hybrid system point ofview, the system consists of four switching surfaces and ninepossible modes. After undergoing a Hopf bifurcation of a period-one orbit, quasi-periodic behavior is obtained. Two kinds of routelead to the chaotic operation: a quasi-periodic one and a perioddoubling scenario.

I. INTRODUCTION

The elevation of the switching frequency is the moderntrend to reduce the volume of dc-dc converters. As the lossincreases linearly with the switching frequency, in order tokeep the switching losses low and to reduce the problem ofelectromagnetic interference (EMI) a large number of differentresonant dc-dc converter topologies have been proposed in theliterature [19]. Among them, the load resonant dc-dc convert-ers are gaining interest in many potential applications such aspower sources for telecom, medical equipments and computersystems. They are recommended to be applied in the middleand in the higher power ranges. Zero-voltage switching (ZVS)and/or zero-current switching (ZCS) increases the efficiencyof this class of resonant converters and allows high frequencyoperation which minimizes component volumes.

Since first reported by Hamill [5], bifurcation phenomenaand chaotic oscillations in dc-dc basic power electronic con-verters with pulse width modulation (PWM) have been studiedin numerous works both experimentally and numerically [6]-[12]. Several kinds of bifurcational behaviors have been foundwith other control schemes [13]. A review of the main resultsin this field are available in [1]-[4]. The aforementionedstudies mostly focused on the fast-scale instabilities. How-ever dc-dc converters can also exhibit slow-scale instabilitiesi.e. quasi-periodic oscillation, with PWM as well as othercontrol schemes. In practice, these low frequency oscillationscan affect the transfer efficiency and switching stress of theconverters. This was first reported by M. di Bernardo et al. [15]in a PWM boost converter, when the input voltage is perturbedwith a suitable external sinusoidal function. Later Chan andTse observed the same behavior in a current programmed boostconverter [16] by Tse.

The mechanism of loss of stability of the period-one orbitand transition to quasiperiodicity (i.e. torus birth) can occurnot only through the well-known subcritical Neimark-Sacker

+−

vrampVU

VL

t

S2

S1

CsLs

Cp

Lf

CfRL

vCp

iL

v0

Vs

ifD5

D6

D7

D8

vcon

VrefPSMGate Driver

Resonant tank Switch network Rectifier

Ts

Comparator

D1

D2

Leg A Leg B

D2

S2

S1

D1

+ -

+

-

Kp+-

-

+

+

-vsc

Error

vab

G1 G2

G1 G2a

b

Load

Amplifier

LPF

vCs

Fig. 1. Voltage controlled PSM series-parallel power resonant dc-dc converter

bifurcation but also via border collision bifurcation [30], [31].The appearance of a two-dimensional (2-D) torus and itsdestruction is a common route to chaos in nonlinear dynamicalsystems. For smooth dynamical systems, the basic theoremwas given by Afraimovich and Shil’nikov [14] describing thepossible ways for the destruction of torus and the appear-ance of chaotic dynamics. These are: (a) a period-doublingcascade of a mode-locked limit cycle, (b) a saddle-nodebifurcation in the presence of homoclinic structure, and (c)a soft transition due to loss of torus smoothness. Neimark-Sacker bifurcation and quasiperiodicity route to chaos werestudied both numerically and experimentally by El Aroudi andLevya in the PWM DC-DC boost [25], [26] and buck-boostconverters [27]. Zhusubaliyev et al. observed a subcriticaltransition to quasiperiodicity in a three-dimensional DC-DCvoltage converter with two-sided PWM [29] in contrast tothe case of one-sided PWM [28] control. Detailed numericalbifurcation analysis shows that both the converters displaytransitions between resonant and ergodic torus. Moreover, theinner organization of the resonance domains is also quitedifferent from the case of a smooth dynamical system. Someeffort has gone to studying higher-order systems such as theautonomous “free running” cuk converter [17] and parallelconnected boost converter system [18].

Despite extensive investigation toward the basic DC-DCconverters over the past decade, the complex behaviors ofresonant dc-dc converters remain largely unexplored [22].

978-1-4244-5697-0/10/$25.00 ©2010 IEEE 807

G1

G1

G2

G2

s1 s1s2s2D1 D1 D2D2

s1s2 s1 s2

M1 M2 M3 M4 M5 M6

vabvcp

D5 D6 D7 D8D7 D8

vs

- vs

TsTsft

T1 Th

Fig. 2. Typical waveforms of PSM controlled SPRC for operation aboveresonance in CCVM mode

The demand of high bandwidth performance necessitates thedetailed numerical analysis of this class of converter. Thestudy of such systems requires the development of numericalmethods capable of following the large number of modesunder the variation of a parameter. Recently, Nagy and Drangareported the subharmonic, quasiperiodic and chaotic operationin one member of a double channel resonant DC-DC converterfamily [23],[24].

In this paper we will present two different route to chaosin the Phase Shift Modulated (PSM) load resonant converter.Our results are obtained from numerical simulations as wellas experimental prototype of the system.

II. SYSTEM DESCRIPTION

A. Converter Configuration and Operation

Since PSM controlled series-parallel resonant converter(SPRC) shown in fig. 1. has been extensively discussed inthe literature [20],[21] its operation will not be addressedin detail in this paper. Like all other resonant converters ituses a resonant tank circuit which consists of three reactiveelements: series resonant inductor Ls, series resonant capacitorCs, and parallel resonant capacitor Cp. The analysis and designof the studied system operating above resonance (laggingpower factor mode) in continuous capacitor voltage mode(CCVM) is presented here. In CCVM of operation, thereare three basic operating intervals in each half cycle. Thetypical waveforms of different variables and different devicesconducting during different intervals are marked clearly inFig. 2. In discontinuous capacitor voltage mode (DCVM),there is an additional interval in between M1, M2 and M4,M5, in the first half and second half cycle respectively [21].

Phase-shifted gating signals are used to generate a pulse-width-modulated quasi-square wave across a and b. The gatingsignals for the switch pair S2 and S2 of Leg B are phase shiftedwith respect to S1 and S1 of Leg A to obtain the desired zerovoltage interval Tsft in vab.

B. Control of Output Voltage

In order to fully understand the control mechanism [33]PSM block is given in Fig. 3. The control voltage vcon is

Q

Q

D

R

Svramp

vres

vcon

Q

Q

CLKD F/F

RS F/F

oo

+-

+-

+-- - -

Leg A

Leg B

DC

Comparator

vsc

(a)

CLK

vsc

vcon

vresvramp

Leg A

Leg B

(b)

Fig. 3. The conventional PSM (a) scheme block diagram (b) operationwaveforms with a transient

obtained by amplifying the error signal (v0-Vref ) by gainKp. vcon is compared to an externally generated repetitiveramp voltage vramp(t) = VL+(VU -VL)( t

Thmod 1) of time

period Th and upper and lower threshold voltages VU andVL respectively. Whenever the amplified error signal vcon isgreater than the sawtooth waveform, the switch signal vsc

is LOW and the selected controlled switch S2 or S2 turnsOFF. otherwise the switch is ON. S2 and S2 are turned OFFand ON successively, i.e., the switch control signal vsc forS2 is generated in one Th period of the sawtooth wave andin the next Th period vsc controls switch S2. Hence, theperiod of the sawtooth wave Th is half of the switchingperiod:Th = Ts

2 = 1

fs

.

III. MATHEMATICAL MODEL

Assuming ideal components in the circuit (Fig. 1.), the PSMcontrolled SPRC can be described by the following differentialequations with discontinuous right hand sides

diLdt

=1Ls

[−vCs− vCp

+ Vssign(sin(wt))]

dvCs

dt=

iLCs

dvCp

dt=

1Cp

[iL − ifsign(vCp)]

difdt

=1

Lf[abs(vCp

) − v0]

dv0

dt=

1Cf

[if − v0

RL]

In dimensionless form it can be represented as

dx1

dτ= −x1 − x3 + sign(sin(ωτ))

dx2

dτ= x1

808

dx3

dτ= x1 − x4sign(x3)

dx4

dτ= Lrabs(x3) − Lrx5

dx5

dτ= Crx4 − ηCrx5

where, x1 =√

Ls

Cs

1Vs

iL, x2 = vCs

Vs, x3 = vCp

Vs, x4 =√

Ls

Cs

1Vs

if , x5 = v0Vs

, τ = t√LsCs

= πtTh

, ω = 2πfs, ω =√

LsCsω, Lr = Ls

Lf, Cr = Cs

Cf, η = 1

RL

√Ls

Cs.

Circuit symmetry implies an useful relationship among thematrices appearing in different subsystems of Fig. 4: B1 =B2 = B4 = B5 = 0, B3 = WB6, A1 = A5 = A6, A2 = A3 =A4 = WA1W, WW = I. Here, the periodicity matrix

W =

⎡⎢⎢⎢⎢⎣

−1 0 0 0 00 −1 0 0 00 0 −1 0 00 0 0 1 00 0 0 0 1

⎤⎥⎥⎥⎥⎦ , I =

⎡⎢⎢⎢⎢⎣

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

⎤⎥⎥⎥⎥⎦ .

Based on Poincare Map Function (PMF) the stability anal-ysis of the periodic state is done. The PMF relates the coor-dinates of consecutive sampled points in state space separatedby one switching period Ts. The stability of the periodic stateis the same as the stability of the fixed point in the PMF. Byutilizing the periodicity matrix, the state vector at the start ofthe (k + 1)th cycle xk+1 may be expressed in terms of thestate vector at the start of the kth cycle xk, xk+1 = Wf(xk).Function f can easily be determined by the state transition andinput matrices of different modes using sample data modelingtechnique provided that xk, τ1, τsft are known.

The PMF for first half cycle or kth cycle is given by

xk+1 = Wf(xk) = Pk(xk) (1)

The equation and the procedure is the same in second halfcycle or k + 1th cycle as in kth cycle

xk+2 = Wf(xk+1) = Wf(Wf(xk)) (2)

Substituting xn = xk and xk+2 = xn+1 in equation (2), thePMF for full cycle (one switching period) (fig. 4.) is given by

xn+1 = Pn(xn) (3)

In period-one steady state, xk+1 = xn+1 = xk = x0. Puttingthese in equations (2) and (3), x0 = Pk(x0) = Pn(x0). Asthe PMF is the same in each half cycle, the dynamics of thesystem is fully described by Pk(xk). Linearizing (1) at thefixed point x0 now gives

Δxk+1 = Jk(x0)Δxk (4)

where, the Jacobian matrix

Jk(x0) =dPk(x)

dx|x0= W

df(x)dx

|x0 (5)

The relation between full cycle Jacobian matrix and half cycleJacobian matrix is given by Jn(x0) = J2

k(x0) The absolute

M1 M2 M3 M4 M5 M6

vab

τ1 τsft τh

τs

τn

τk+1τk

τh

τk+2

τn+1

τ1 τsft

x3x3=0

x3>0

x3<0

h2

h1

h3

h4

1st half cycle 2nd half cycle

xk xk+1 xk+2

Vs

-Vs

Fig. 4. Time sequence of mode changes in period-one steady state

values of the eigenvalues of Jk or Jn must be smaller than 1for stable operation.

The state equations for the three modes of the kth cycle orfirst-half cycle of the nth cycle in matrix form are given asfollows:

M1 : x = A1x ∨ τ ∈ [τk, τk + τ1)M2 : x = A2x ∨ τ ∈ [τk + τ1, τk + τsft)M3 : x = A3x + B3 ∨ τ ∈ [τk + τsft, τh)

where, x = dxdτ , τ1 = πT1

Th, τsft = πTsft

Th, x =

⎡⎢⎢⎢⎢⎣

x1

x2

x3

x4

x5

⎤⎥⎥⎥⎥⎦, A1 =

⎡⎢⎢⎢⎢⎣

0 −1 −1 0 01 0 0 0 01 0 0 1 00 0 −Lr 0 −Lr

0 0 0 Cr −ηCr

⎤⎥⎥⎥⎥⎦, B3 =

⎡⎢⎢⎢⎢⎣

10000

⎤⎥⎥⎥⎥⎦.

The PMF for the half cycle

xk+1 = Wf(xk, τ1, τsft)

= W[Fxk +∫ τh

τsft

eA3(τh−σ)dσB3] (6)

where, F = φ3(τh − τsft)φ2(τsft − τ1)φ1(τ1), φ1(τ1) =eA1τ1 , φ2(τsft − τ1) = eA2(τsft−τ1), φ3(τh − τsft) =eA3(τh−τsft). For period-one orbit, xk+1 = xk.

The switching instants τ1 and τsft are the solution of twoalgebraic equations,

h1(xk, τ1) = x3(τ1) = Cx(τ1) = 0 (7)

h2(xk, τ1, τsft) =vcon(τsft) − vramp(τsft)

Vs

= Kp[Ex(τsft) − V ∗ref ] − V ∗

L − (V ∗U − V ∗

L )τsft

= 0 (8)

where, x(τ1) = φ1(τ1)xk, x(τsft) = φ2(τsft −τ1)φ1(τ1)xk, C =

[0 0 1 0 0

], E =[

0 0 0 0 1], V ∗

ref = Vref/Vs, V ∗L = VL/Vs, V ∗

U =VU/Vs.

809

Since each mode has linear dynamics, the SPRC is apiecewise-linear system. On the other hand, the whole system,the PSM controlled SPRC becomes a nonlinear system, dueto the dependence of the mode change instants τ1 and τsft onthe state variables x3 and x5 respectively.

To obtain a linearized dynamic model from the half-cyclemodel, the fixed point corresponding to the nominal operatingcondition of the converter must first be found. Given thenominal source voltage and the nominal half switching periodτh, the fixed point x0 and two switching instants τ0

1 and τ0sft

satisfies (6), (7) and (8),

x0 = Wf(x0, τ01 , τ0

sft)h1(x0, τ0

1 ) = 0h2(x0, τ0

1 , τ0sft) = 0

This set of nonlinear equations can be solved numerically byNewton-Raphson’s method. After getting the fixed point(or theperiodic solution in continuous time) we can study the stabilityof the limit cycle by linearizing (6), (7) and (8).

Jk(x0) = W[∂f∂x

− ∂f∂p

(∂h∂p

)−1

(∂h∂x

)T

]|(x0, τ01 , τ0

sft) (9)

where, p =[

τ1 τsft

], h =

[h1 h2

]∂f∂x

= F

∂f∂τ1

= φ3(τh − τsft)φ2(τsft − τ1)[A1 − A2]φ1(τ1)xk

∂f∂τsft

= −φ3(τh − τsft)B3

∂h1

∂x= Cφ1(τ1)

∂h1

∂τ1= CA1φ1(τ1)xk

∂h2

∂x= −KpEφ2(τsft − τ1)φ1(τ1)

∂h2

∂τ1= −KpEφ2(τsft − τ1)[A1 − A2]φ1(τ1)xk

∂h2

∂τsft= −KpEA2x(τsft)[V ∗

U − V ∗L ]/τh

For the stable period-one orbit, the consideration of half-cycle dynamics is sufficient due to circuit symmetry [24].When the period-one orbit loses stability, i.e., beyond the firstinstability, a higher periodic stable orbit may occur; but thesymmetry between the half cycles will break, i.e., relation (5)does not hold. There may be another switching due to thepresence of the DCVM operation in some cycles:

h3 = x4 − abs(x1) = 0. (10)

IV. SIMULATION RESULTS

1) Bifurcation Diagram: In most nonlinear system whenone type of dynamical behavior becomes unstable, anothertype of behavior becomes stable. The transition from one typeof dynamical behavior to other can be nicely represented by

Fig. 5. The bifurcation diagram, with the proportional gain as variableparameter (nominal operating mode is CCVM)

bifurcation diagram. Selecting just a single state variable forobservation, e.g., the sampled output voltage v0, the diagramrecords its change as a function of one system parameter, e.g.,Kp over its range of interest in steady state. The final state fora certain parameter value is taken as the initial condition forthe next parameter value.

−1.5 −1 −0.5 0 0.5 1 1.50.46

0.47

0.48

0.49

0.5

vcon

(V)

i f

(A

)

(a)0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

−5

0

5

time (msec)v

con ,

vram

p (

V)

(b)

Fig. 6. (a) Poincare map and (b) control voltage in quasi-periodic state(Kp = 2.8)

2) Loci of the characteristic multipliers: The eigenvaluesof Jn are used for the stability study and a Matlab programis developed to compute them. According to the stability

−5 0 50.4

0.45

0.5

0.55

vcon

(V)

i f (A

)

Fig. 7. Chaotic attractor for torus destruction (Kp = 4)

810

condition, as long as the characteristic multipliers lie insidethe unit circle, the periodic steady state solution is asymptoti-cally stable. When a complex-conjugate pair of characteristicmultipliers gets outside the unit circle the periodic solutionloses stability by Hopf bifurcation.

The values of parameters used in this study are: Ls =150 μH, Cs = Cp = 0.1 μF, Lf = 3000 μH, Cf = 0.47μF,Vref =12 V, VL = −2.5 V, VU = 2.5 V, Vs = 15 V,fs = 50 kHz, RL = 25 Ω.

At Kp = 2.64, the eigenvalues are −0.1066±0.9944(modulus�1) , 0.4215, 0.1169, 0.03295.

−5 0 50.4

0.45

0.5

0.55

vcon

(V)

i f

(A

)

(a)

−5 0 50.4

0.45

0.5

0.55

vcon

(V)

i f

(A

)

(b)

Fig. 8. Poincare map for (a) period-4 (Kp = 4.15) and (b) period-8 (Kp =4.3)

0.1 0.12 0.14 0.16 0.18 0.2 0.22−5

0

5

time (msec)

vcon , v

ram

p (

V)

(a)

0.1 0.15 0.2 0.25 0.3 0.35−5

0

5

time (msec)

vcon , v

ram

p (

V)

(b)

Fig. 9. Control voltage for (a) period-4 (Kp = 4.15) and (b) period-8(Kp = 4.3)

3) Quasi-periodic route to chaos: When the periodic be-havior undergoes the Hopf bifurcation, quasi-periodic behavior

−5 0 50.4

0.45

0.5

0.55

vcon

(V)

i f (A

)

Fig. 10. Eight-piece chaotic attractor attractor at (Kp = 4.8)

Fig. 11. Experimental Poincare map for quasi-periodic state at Kp = 4.9

is the result. It is readily identified by means of Poincaremap (Fig. 6.(a)). Quasi-periodic behavior appears due tothe interaction of two competing frequencies, The forcingfrequency of the sawtooth signal and the natural frequencyof the low-pass filter. The control voltage vcon, changing withthe natural frequency of the system, hits the ramp wave onceper ramp cycle. The forcing frequency of the ramp wave ispresent in the small ripple of the control voltage (Fig. 6.(a)).Increasing further Kp the system state turns to chaotic one(Fig. 7.) i.e., this is a route to chaos through quasi-periodicstates.

(a) (b)

Fig. 12. Oscilloscope traces of Control voltage for (a) period-4 (Kp = 6.1)and (b) period-8 (Kp = 6.2)

4) Period-doubling route to chaos: The chaotic regionsuddenly changes to a periodic window about at Kp = 4.1(Fig. 7(a).), followed by a finite number of period-doubling.Then eight-piece chaotic attractor is seen at parameter valueKp = 4.8 (Fig. 10.).

V. EXPERIMENTAL RESULTS

In order to verify the simulation results obtained in theprevious section, an experimental setup was built and tested.There is a simple way to display a Poincare map on adigital storage oscilloscope without using extra sample-and-hold circuit. Since the oscilloscope has three different channels(X, Y, Z modulation ports), we can use X-channel and Y-channel for state variables whose sampled values are usedfor drawing the map and impulse waveform generated from

811

the clock is applied to the Z-modulation port. The frequencyof the clock is in synchronism with the falling edge of thesecond ramp of two consecutive cycles. The oscilloscope isset in the X-Y mode, the proportional gain Kp is variedand observation is stored. The oscilloscope traces are shownin Fig. 11 and Fig. 12. Qualitatively, the simulation andexperimental results are in good agreement. The differencein the parameter values at which the bifurcations occur canbe attributed to the nonideality of the components (the ESRof the inductors and capacitors, the forward voltage drops ofthe MOSFETs and diodes, snubbers and delay circuits in theswitch network). It was found that these nonidealities onlyshift the bifurcation points and have negligible influence onthe phenomena reported.

VI. CONCLUSION

This paper has reported for the first time the various dynami-cal behaviors of PSM controlled SPRC. Although attention hasbeen paid to one kind of third-order converter, the approachproposed herein is readily applicable to other resonant convert-ers. In addition to the stability analysis it shows quasi-periodic,subharmonic and chaotic states. Earlier work showed similarnonlinear dynamic behavior in conventional DC-DC convertersbut the number of switching surfaces were less than the systemstudied here. In this system there are four switching surfaces,three are state dependent and remaining one is time dependent.Literature survey shows that dynamical analysis of this typeof system is yet to be done.

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