Study of Nonlinear Behaviour and Chaotic Control in Parallel-Connection Buck Converters

Lili Wang 1,2 , Yufei Zhou 2, Junning Chen 2 1. School of Electronic and Information Engineering,Anhui University of Architechture, Hefei,Anhui,230601,China

2. School of Electronic Science and Technology,Anhui University,Hefei,Anhui,230039,China Lily_wang_502@yahoo.com.cn

Abstract—The article studies on the parallel-connected buck converter under master-slave operation. Firstly, deduces the dynamical equations from the circuit structure, and gains the bifurcation and chaos diagram under the different parameters. Then, open out the reason of bifurcation and chaos by simple theoretical analysis. On basis of it, some chaos control method called phase-shift are putted forward, which are analysed with the mapping method that will map the time variable to some other parameter. The control results are validated by contrasting the power spectrum before and after the control method of phase-shift is adopted

Keywords- bifurcation;parallel-connection buck converters; chaos;breathing;phase-shifting

I. INTRODUCTION

Recently, paralleling converters has become a popular technique in power-supply design for improving power processing capability, reliability and edibility. Nonlinear dynamics and bifurcation behavior are important topics of investigation in power electronics. In this paper, we attempt to probe into some nonlinear phenomena of a system of parallel-connected buck converters controlled under a master–slave current-sharing scheme.

II. MASTER-SLAVE CONTROLLED PARALLEL-CONNECTED BUCK CONVERTERS

The system under study consists of two dc-dc converters which are connected in parallel feeding a common load. The current drawn by the load is shared properly between the two buck converters by the action of a master–slave control scheme, as mentioned briefly in the preceding section. Fig. 1 shows the block diagram of this master–slave configuration. Denoting the two converters as Converter 1 and Converter 2 as shown in Fig. 1, the operation of the system can be described as follows. Both converters are controlled via a simple pulse-width modulation (PWM) scheme, in which a control voltage Vcon is compared with a saw tooth signal to

generate a pulse-width modulated signal that drives the switch, as shown in Fig. 2. The saw tooth signal of the PWM generator is given by

)(11 refvoffsetcon VvKVv −−= 1

where VL and VU are the lower and upper voltage limits of the ramp, and T is the switching period. The PWM output is “high” when the control voltage is greater than Vramp, and is “low” otherwise. For Converter 1, the control voltage is derived from a voltage feedback loop, i.e.,

)()( 1222 miiKVvKVv irefvoffsetcon −−−−= 2

where Voffset is dc offset voltage that gives the steady-state duty cycle ,Vref is reference voltage and Kv1 is voltage feedback gain for Converter 1.

Fig. 1. Block diagram of parallel-connected dc/dc converters

under a master–slave control

Fig. 2. Pulse-width modulation (PWM) showing relationship

between the control voltage and the PWM output

Project supported by the Natural Science Foundation of the Higher Education Institutions of Anhui Province, China (Grant No.

).and the Special Foundation for Young Scientists of Anhui University of Architecture (Grant No. K0245401).

5003978-1-4244-8165-1/11/$26.00 ©2011 IEEE

For Converter 2, an additional current error signal, which is proportional to the weighted difference of the output currents of the two converters, determines the control voltage. Specifically we write the control voltage for Converter 2 as

)mod)(( TtVVVv LULramp −+= 3

where Kv2 is voltage feedback gain of Converter 2, Ki is current feedback gain and m is current weighting factor.

Suppose as rampcon VV > , u=0 and as rampcon VV < ,u=1. Fig. 3 shows two buck converters connected in parallel. When the converters are operating in continuous conduction mode, the equations corresponding to these switch states are generally given by

)(1)(12121 R

vii

Cii

CRCv

v −+=++−=

1

1

1

1

11 L

vEuL

EuLv

i−=+−=

2

2

2

2

22 L

vEuL

EuLv

i−

=+−= 4

This should be taken care of in the simulation and analysis.

III. NONLINERA PHENOMENON IN THE CONTROLLED PARALLEL-CONNECTED BUCK

CONVERTER We now begin our investigation with computer simulations.

Let FCVVsTVE ref μμ 47,24,400,30 ==== , HL 02.01 = ,

VVHL offset 5,04.02 == .Our investigation is divided into two steps. One is the simulation of the same frequencies of the two saw tooth; the other is the simulation of the different frequencies of two saw tooth. A large number of bifurcation diagrams have been obtained. In the following, only representative bifurcation diagrams are shown, which serve to exemplify the main findings concerning the bifurcation behavior of a system of parallel buck converters under a master–slave sharing scheme.

A. Same frequencies of two saw tooth We vary kv1 and kv2 simultaneously, and the

corresponding bifurcation diagram is shown in Fig. 4. The

diagram shows that the converter experiences a typical period-doubling bifurcation and eventually enters chaos.

B. Different frequencies of two saw tooth

Let f1=2500H, f2=2498H, = =6,Ki=5,m=1, the time bifurcation diagram is shown in Fig. 5. The diagram shows that the system changes from period 1 to period-doubling bifurcation ,and to chaos, then change in

opposite directionsto period 1,which is called intermittent chaos or breathing ,and the intermittent period

21/1 ffTin −= is 0.5 seconds.

IV. ANALYSIS OF BIFURCATION From the foregoing simulation study, we have identified

period-doubling bifurcation in a system of parallel buck converters when some parameters are varied such as Kv1 and Kv2. In this and the next sections we analyze these bifurcations in terms of a suitable discrete-time model. We will first derive the model, and examine the Jacobian matrix and the way the system loses stability.

A. Derivation of the Discrete-Time Map The state equations are given in (4) for different switch states. The order in which the system toggles between the switch states depends on d1 and d2 .We will study periodic orbits for which d2 > d1, we have three switch states. These are as follows. 1) For Tdt 10 ≤< both S1 and S2 are turned on.

{ }ETdMxTdNTdx nnnnn )()()( ,11,11,1 += 5 2) For TdtTd 21 ≤< , S1 is turned off and S2 remains on.

{ }ETdMxTddN

Tddx

nnnn

nnn

))1(())(())((

,23,1,23

,1,2

−+−=

− 6

3) For TtTd ≤<2 , both S1 and S2 are off. { }ETdMxTdNx nnnn ))1(())1(( ,24,241 −+−=+ 7

Thus, we can get ),,( ,2,11 nnnn ddxfx =+

{ }ETdMxTdN nnn ))1(())1(( ,24,24 −+−=

nnnnn xTdNTddNTdN )())(())1(( ,11,1,23,24 −−=ETdMTddNTdN nnnn )())(())1(( ,11,1,23,24 −−+

Fig. 3. Two parallel-connected buck converters

Fig. 4. Bifurcation of Kv1 Fig. 5. Bifurcation of time

and Kv2 varies simultaneously

5004

)())(())1(( ,1,24,1,23,24 nnnnn ddMETddMTdN −+−−+ 8 where ξξ iA

i eN =)( [ ] iiii BINAM −= . As AAAAA ==== 4321 and

)()()()()( 4321 ξξξξξ NNNNN ==== , nnnnnn xTdNTddNTdNx )())(())1(( ,1,1,2,21 −−=+

EBAEBBTdN

EBBATdN

EBATNxTNEddM

ETddMTdN

ETdMTddNTdN

n

n

nnn

nnn

nnnn

41

34,2

131

,1

11

,1,24

,1,23,2

,11,1,2,2

))()1((

)())1((

)()()(

))(())1((

)())(())1((

−

−

−

−−−+

−−+

+=−+

−−+

−−+

9

Now ,define ),( ,11 nn dxs and ),,( ,2,12 nnn ddxs as

]mod)([)]([

),(

111

,11

Tt

VVVVvKVvv

dxs

LULrefvoffsetrampcon

nn

−+−−−=−=

])([]))((

)([)(

])([])(

)()[(

])([)]()[(

]mod)([)]()[(

,111

,1

,111

,1,111

,111

,1,111

11

TdVVVEBAITdN

xTdNKVKV

TdVVVETdMK

xTdNKVKV

TdVVVdxKVKVTt

VVVvKVKV

nLULn

nnT

refvoffset

nLULnT

nnT

refvoffset

nLULnT

refvoffset

LULvrefvoffset

−+−−+

++=

−+−+

++=

−+−++=

−+−−++=

−

10

where , )0,0,( 11 vT KK −=

]mod)([)]()([

),,(

122

22,2,12

Tt

VVVmiiKVvKV

vvddxs

LULirefvoffset

rampconnnn

−+−−−−−=

−=

)]()[( 2122 iKimKvKVKV iivrefvoffset −+−++=

]mod)([Tt

VVV LUL −+−

])([)]()[( ,2,222 TdVVVdxKVKV nLULnT

refvoffset −+−++=

])([

])()(

)()([[)(

,2

311

,1,231

,1,2

11

,2,222

TdVVV

EBABEAddNEBATdTdN

EBATdNxTdNKVKV

nLUL

nnnn

nnnT

refvoffset

−+−

−−−−+

+++=−−−

−

11

where ),,( 22 iivT KmKKK −−=

B. Derivation of the Jacobian Matrix

Suppose the equilibrium point is given by 1QX += nx . The Jacobian of the discrete time map evaluated at the equilibrium point can be

qn xxnnnnnn

nnnn

xs

ds

ds

xs

ds

df

xs

ds

df

xf

J

=

−−

−

∂∂

∂∂

∂∂

+∂∂

∂∂

∂∂−

∂∂

∂∂

∂∂−

∂∂=

11

,1

1

,1

221

,2

2

,2

11

,1

1

,112

where

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∂∂

nnn

nnn

nnn

n

if

if

vf

if

if

vf

if

if

vf

xf

,2

3

,1

33

,2

2

,1

22

,2

1

,1

11

,

∂∂

∂∂

∂∂

=∂

∂

n

n

n

n

df

df

df

df

,1

3

,1

2

,1

1

,1

,

∂∂∂∂∂∂

=∂∂

n

n

n

n

isisvs

xs

,2

1

,1

1

1

1 ,

∂∂

∂∂

∂∂

=∂

∂

n

n

n

n

df

df

df

df

,2

3

,2

2

,2

1

,2,

∂∂∂∂∂∂

=∂∂

n

n

n

n

isisvs

xs

,2

2

,1

2

2

2

.

thus we get

)())()(())((

)())((

)())()(()())()1((

)())((

)())()1(()(

13,1,221,22

1,11

,1113,1,22,2234,2

1,11

,1113,1

11

,1

1

,1

221

,2

2

,2

11

,1

1

,1

LUnnT

nnT

LUnnT

nT

nnT

nT

n

LUnnT

nT

n

xxnnnnnnnnnn

VVEBBTddNkEBAxTdNk

VVEBAxTdNk

TdNEkBBTddNkTdNkEBBTdN

VVEBAxTdNk

TdNEkBBTdNTN

xs

ds

ds

xs

ds

df

xs

ds

df

xf

J

qn

−−−−++

−−+

−−−+−−−

−

−−+

−−−−=

∂∂

∂∂

∂∂

+∂∂

∂∂

∂∂

−∂∂

∂∂

∂∂

−∂∂

=

=

−−−

(13)

5005

According the equation 0)(det =− QXJIλ we keep

Ki and m as constant and vary Kv1 and Kv2 simultaneously,

the characteristic multipliers can now be calculated ,as shown in Fig.6. This process of losing stability agrees with Fig.4 exactly.

V. STUDY OF BIFFURCATION CONTROL When the two frequencies of the saw tooth are different

,we define the voltage of the first saw tooth as

1mod)(

1mod)(

1

11

ftVVV

Tt

VVVv

LUL

LULramp

⋅−+=

−+= 14

then , the second one is defined as

[ ]1mod

2)(1mod)(

1mod)()(1mod)(

1mod)(

121

2

22

πθ

LULLUL

LUL

LUL

LULramp

VVVftVVV

ffftVVV

ftVVV

Tt

VVVv

−+=Δ⋅−+=

−+⋅−+=⋅−+=

−+=

15

Now, there is a delay phase θ ,and

tfftf ⋅−=⋅Δ= 2122 ππθ ,so, we can get the bifurcation of

θ as shown as Fig.7. When it change among [ ]π2,0

correspond the intermittent period sTin 5.0= .

From the Fig.7, we can know when 0=θ ,that to say ,there is no phase delay between two saw tooth ,the system is in chaos state, and when 4~2.7=θ the system is steady, that is to say the chaos is controlled. We can prove the conclusion by power spectra as shown as Fig.8. We simulate the system for twice. When 0=θ , the power spectra are continuous while the power spectra are made of some pinnacle in the frequency multiplication when

27.3=θ .

VI. CONCLUSION Despite the popularity of parallel converter systems in

power electronics applications, their bifurcation phenomena are rarely studied. This paper reports some selected bifurcation phenomena in a parallel system of two buck converters which share current under a master–slave control scheme especially describe the processing of intermittent chaos .On basis of it, some chaos control method called phase-shift are putted forward, which are analyzed with the mapping method that will map the time variable to some other parameter. The control results are validated by contrasting the power spectrum before and after the control method of phase-shift is adopted.

REFERENCES

[1] “

”

–

[2] “

”

–

[3] “

”

—

[4]

Fig. 7. Bifurcation of θ

a) chaos b)steady

Fig. 8. power spectra

Fig. 6. Loci of characteristic multipliers as Kv1 and Kv2

varies simultaneously

5006