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fs=1/T, and the duty cycleD=ton/T, where tonis the intervalwhen the switch is on. The DCM waveform of Buck-Boostconverter is shown in Fig.3.

Figure 2. The equivalent circuit with parasitic components of Buck-Boost

converter

Figure 3. DCM waveform of Buck-Boost converter

From Fig.3, the relationships among the dc componentsof the current in Buck-Boost converter are obtained:

22 2 DD

II

D

I

D

I LPKDS

+

=== (1)

WhereIs,IDare the dc component of the current throughthe switch and the diode, respectively.IL,IPKare the averagecurrent and the peak current through the inductor,respectively.

The component values of the averaged model can bederived by using the principle of energy conservation.

The rms value of the switch current is:

PK

T

SSrms IDdtiT

I3

10

2 == (2)

Thus, the power loss inRONis:

2

2

2

2

)(3

4LononSrmsRon IR

DD

DRIP

+

== (3)

The rms value of the diode current can be written as:

PK

T

DDrms ID

dtiT

I3

1 2

0

2== (4)

The power loss inRFand VFare:

2

22

22

)(3

4

LFFDrmsRF IRDD

D

RIP +== (5)

LFDF IVDD

DIV

+

=

2

2 (6)

The rms value of the inductor current is:

PK

T

LLrms IDD

dtiT

I3

1 2

0

2 +== (7)

The power loss inRLis:

2

2

2 1

3

4LLLLrmsRL IR

DDRIP

+

== (8)

In this way, the EAR of RON, RFin the inductor branch

are onR

DD

D2

2 )(3

4

+

, FR

DD

D2

2

2

)(3

4

+

, respectively, in

terms of the reflection rule and the energy conservationprinciple. The equivalent averaged voltage of VF in the

switch branch is FVDD

D

2

2

+

. Thus, the ideal PWM

switches are replaced by dependent current and voltagesources, the large signal model of Buck-Boost converter asshown in Fig.4.

Figure 4. Large signal model of Buck-Boost converter

Where, FFE VDD

DV

2

2

+

=

2

2

2

2

2

2 3

4

)(3

4

)(3

4

DD

RR

DD

DR

DD

DR LFonE

+

+

+

+

+

=

From Fig.4, the relationship between current and voltageare obtained of the switches:

LCS idi = (9)

apCcp vdv = (10)

D

ac

S

cp

Ci

v

L

Td

i

v

L

Td

d

dd ===

22

22

2

(11)

The iS, dC, iL, vcp, vapcan be expressed as the sum ofthe dc and ac components.

B. DC model of Buck-Boost converterThe DC model of Buck-Boost converter is obtained, as

shown in Fig. 5

Figure 5. DC model of Buck-Boost converter

From Fig.5, the dc voltage transfer function can beobtained:

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FELC

i

C

CO VRI

DV

D

DV

=

1

1

1 (12)

Where

O

FEi

O

Si

DO

V

V

R

R

D

DDD

D

V

V

IV

IV

+

++

===2

2

2

2

1

1

ac

D

cp

S

V

I

DT

L

V

I

DT

LD ==

222 ,

2

2

DD

DDC

+

=

If the parasitic components are not considered, RonRFRL0,VF0thus:

ii

C

CO V

D

DV

D

DV

21=

= (13)

Eq.(13) agrees with [11][12].

C. Small-signal model of Buck-Boost converterThe voltage, current, and duty cycle can be expressed as

the sum of the dc and ac components.

DDD iIi+= (14)

cpcpcp vVv += (15)

dDd += (16)

SSS iIi+= (17)

acacac vVv += (18)

Substitution of (14), (15), (16), (17), (18) into (9), (10),(11), Neglecting the high order terms gives:

dkvgd

L

TDVv

L

TDi iaci

acacS

2

2

+=+= (19)

cpooacfcp

cp

DDac

cp

SD vgdkvgv

V

Id

D

Iv

V

Ii

2

2 +=+= (20)

The circuit models of Buck-Boost converter for small-signal operation is obtained based on (19), (20), as shown inFig.6.

Figure 6. Small-signal model of Buck-Boost converter

From Fig.6, the open loop transfer functions of Buck-Boost converter in DCM can be obtained:

22

211

02

))((

)(

)()(

nn

ZZ

v

oVd

ss

ssG

sd

svsG

i

++

++==

=

(21)

Where,

CiofioC

oifiioC

RRgggggRR

gkgkgkRRG

++++

=

))((

)(1 ,

C

ZCR

11 = ,

L

kR oEZ

=2 ,

]))([()(1

CiofioC

ioEoEfio

nRRgggggRRLC

ggRRRgRggg++++

+++++= ,

32

43121

2

+++= ,

)(1 CRRC += , LRE += 12 ,

)1(3 iEOC gRgCRR += , )(1 fio ggg ++= ,

ioEoE ggRRRgR +++= 12 1 ,

][ 113 Cio RRgCgL += In fact, a driver delay is added necessarily to keep from

reducing EMI in close-loop system of PWM converterswitch. Therefore, there will be a MOSFET delay in Buck-Boost converter which is caused by the duty-cyclemodulator, MOSFET driver, and the MOSFET itself. If the

delay time of MOSFET is Td, the functiondsTe

can be

represented as the MOSFET delay, where Td is theMOSFET delay time. A first-order Pade function can be

utilized to approximate dsT

e as s rational function:

d

d

d

dsT

Ts

Ts

sT

sTe d

/2

/2

2/1

2/1

+

=

+

(22)

Form (22), the transfer function of the duty cycle tooutput voltage with MOSFET delay considered in DCM can

be obtained:

0

1 21 2 2

( )( )

( )

2 /( )( )

2 2 /

d

i

sToVdT Vd

v

dZ Z

n n d

v sG s G e

d s

s Ts sG

s s s T

=

= =

+ +=

+ + +

(23)

If the parasitic components and MOSFET delay are notconsidered, by substituting VF=0, RL=Ron=RC=RF=0, Td=0into (21) and (23), the results are agree with [11][12], whichshows that the state-space averaging method and equivalentcircuit modeling method are a special exception of thisconstitutive relation. The modeling method presented in this

paper features in more accurate and reasonable, and issuitable for nonideal PWM converter.

III. SIMULATIONAs the figure 2 shown, a Buck-Boost converter with the

following parameters is assumed: VI=32V, VO=48V,L=50H,Ron=0.05, RL=0.01, RF=0.01, VF=0.5V, C=100F,RC=0.04, R=96, fs=100kHz, Td=1s. The GVd(s) andGVdT(s) frequency characteristics can be obtained by using

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MATLAB, shown as in Fig.7 and Fig.8. Form Fig.7, it canbe seen that parasitic components have influence on thetransfer functions, time domain and frequency domaincharacteristics by comparing the Bode plots with and withoutconsidering the parasitic components.

-60

-40

-20

0

20

40

Magnitude(dB)

101

102

103

104

105

106

107

-90

0

90

180

Phase(deg)

Bode Diagram

Frequency (rad/sec)

w ith parasitic components

w ithout parasitic component

Figure 7. Bode Plot of GVd(s). Solid line is for considering the parasitic

components, dash line is for without considering the parasitic components

-40

-20

0

20

40

Magnitude(dB)

101

102

103

104

105

106

107

0

45

90

135

180

Phase(deg)

Bode Diagram

Frequency (rad/sec)

Td=0

Td=1us

Figure 8. Bode plot of GVdT(s).. Solid line is for Td=0, and dashed line is for

Td=1s

The GVdT(s) frequency characteristics of Buck-Boostconverter are shown in Fig.8. Two curves of the gaincharacteristics with Td=0 and Td=1s are well coincident.Butthe MOSFET delay has effect on high-frequencyperformance and loop stability. Based on the classic controltheory, the amplitude frequency characteristics of the delaysection is constant, not to be independent of angularfrequency, but the phase of the delay section is proportionalto the angular frequency. The simulation results obtainedagree with the theoretical analysis well, confirming the

validity of this modeling method presented in this paper.Compared with [11][12], this model is better intuitive andaccurate.

IV. CONCLUSIONA new method is described for modeling nonideal Buck-

Boost converter operation in DCM, considering the parasiticcomponents of Buck-Boost converter, based on the principleof energy conservation and equivalent circuit modelingmethod. Dynamic large signal, dc and small signal circuit

models of the nonideal Buck-Boost converter operating inDCM are derived. The dc component relationship in theoutput and input voltage can be calculated, and open looptransfer functions are educed, the duty cycle to outputvoltage transfer function with considering MOSFET delayis studied. The simulation results show that: (1) Theparasitic components have influence on the transfer

functions, time domain and frequency domaincharacteristics. (2) The MOSFET delay only has effect onhigh-frequency performance and loop stability. Therefore,the influence of MOSFET delay can not be ignored in high-frequency domain. The simulation results obtained are inexcellent agreement with the theoretical analysis. Thismethod has better precision compared with [11][12]. Themodel obtained is desirable and widely used. Themethodology used can be applied to other convertertopologies.

REFERENCES

[1] Middlebrook R D, Cuk S. A general unified approach to modelingswitching converter power stages. IEEE Power ElectronicsSpecialists Conference Record, 1976, pp.18-34

[2] Robert W. Erickson, Dragan Maksimovi, Fundamentals of PowerElectronics, Kluwer Academic Publishers, 2001.

[3] Vorperian V. Simplified analysis of PWM converter using the PWMswitch, Part I: Continuous conduction mode. IEEE Trans. Aerospaceand Electronic Systems, 1990, Vol. 26, No.3, pp.490~496

[4] Vorperian V, McLyman W T. Analysis of a PWM-resonantconverter. IEEE Transactions on Aerospace and Electronic Systems,1997, Vo1.33, No. 1, pp.163~170

[5] Jovanovic M M, Huber L. Small-signal modeling of non-idealmagnetic amplifier PWM switch. IEEE Transactions on PowerElectronics, September 1999, Vo1.14, No.5,882~889

[6] B.Bryant and M.K.Kazimierczuk, Small-signal duty cycle toinductor current transfer function for boost PWM dc-dc converter incontinuous conduction mode, in Proc. IEEE ISCAS, Vancouver,

C,Canada, May 23-26,2004,pp. 856-859

[7] B.Bryant, M.K.Kazimierczuk, Voltage-loop power-stage transferfunction with MOSFET delay for boost PWM converter operating inCCM, IEEE Transactions on industrial electronics,.Feb.2007, Vol54,

No.1, pp.347-353

[8] Francesco Alonge, Filippo DIppolito, Francesco Maria Raimondi,and Salvatore Tumminaro. Nonlinear modeling of DC/DCconverters using the hammiesteins aproach. IEEE Transactions onPower Electronics, July 2007, Vo1.22, No. 4, pp.1210-1221

[9] Czarkowski D, Kazimierczuk M K. Static- and dynamic- circuitmodels of PWM buck-derived DC-DC converters ,IEE Proceeding C,Circuits, Devices and Systems, 1992, Vol.6, pp.669-679

[10] Czarkowski D, Kazimierczuk M K. A new and systematic method ofmodeling PWM DC-DC converters, IEEE International Conferenceon Systems Engineering, 1992, pp. 628-631

[11] Cai Xuanshan, Gong Shaowen. High frequency power electronicsDC-DC conversion. Beijing: Science Press. 1993

[12] Zhang Weiping. Modeling and controlling of switching converter.Beijing:China power press.2005

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