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A Unified Hybrid Model with Two Level Networksfor Peak Current Mode Controlled Buck-Boost

Converters Operating in DCM and CCM

Jiaxin Chen 1,2, Jianguo Zhu 2, Youguang Guo 2,1College of Electromechanical Engineering, Donghua University, Shanghai 200051, China

2Faculty of Engineering, University of Technology, Sydney, NSW 2007, Australia

Abstract -This paper presents a unified hybrid model with two levelnetworks for peak current mode controlled buck-boost convertersoperating in continuous conduction mode (CCM) anddiscontinuous conduction mode (DCM). It is a network model withtwo levels in which both a detailed switching model and a smallsignal average model are included correspondingly. Thenonlinearity of some electronic components and the parasiticaleffects of circuit elements are considered. The nonlinearinductance is calculated by finite element analysis (FEA). Thedynamical performance of the buck-boost converter can beobtained from the detailed switching model in the first levelnetwork, and then the frequency domain performance can beobtained from the average model in the second level network. Asan automatic mechanism of exchanging data between two modelsis proposed, all the performances can be obtained automatically.By implementing the proposed model in Simulink surrounding,both the large signal transient performance and the frequencydomain behavior of converters are obtained efficiently.

I. I NTRODUCTION

In order to obtain the stability and dynamic performance ofPWM switching DC to DC converter, the detailed switchingmodel is often enrolled to obtain its large-signal (transient)dynamical performance, and the small signal average model isoften used for analyzing the frequency domain performance[1-2]. It is well known that the small signal average model isdeduced from the method of linearizing the large signal averagemodel around a desired operating point, which is also deducedfrom the method of averaging the switching model. There aremany different operating states which have to be obtained forcalculating their corresponding average models, and this willcause heavy work. For that, this paper introduces a unified hybridmodel with two level networks for buck-boost converters

operating in discontinuous conduction mode (DCM) andcontinuous conduction mode (CCM), in which the detailedswitching model, the small signal average model and theirautomatic mechanism of exchanging data are included. Thenonlinear inductance is calculated by the method of finiteelement analysis (FEA), and both the nonlinear factors and the

parasitical effects of circuit elements are readily considered. Byimplementing the proposed model in Matlab/Simulinksurrounding, both the transient performance and the frequencydomain behavior of converters are obtained efficiently.

II. TYPICAL CIRCUIT OF BUCK -BOOST CONVERTER WITHPEAK CURRENT MODEL CONTROLLED

Fig. 1 shows the typical circuit of the buck-boost converterwith peak current model controlled which is realized based onUC3842. To effectively analyze the flyback converter, thefollowing assumptions and simplifications are made: (1) the

power MOSFET in the ON state is modeled by a zero resistanceand in the OFF state by an infinite resistance, R INF , and theoutput capacitance and inductance of the leading wires areignored; (2) The diode in the ON state is modeled by a constantvoltage source, V F , with a constant forward resistance, R F , and inthe OFF state by an infinite resistance. The diode junctioncapacitance is negligible; (3) All the leakage inductances andstray capacitances of the transformer are neglected. (4) Passivecomponents are linear, time invariant, and frequencyindependent. Then the main equivalent circuit is obtained andshown in Fig. 2.

Fig. 1. Buck-boost converter with peak current model controlled

Fig. 2. Main equivalent circuit of buck-boost converter

Proceeding of International Conference on Electrical Machines and Systems 2007, Oct. 8~11, Seoul, Korea

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III. NONLINEAR I NDUCTANCE OF TRANSFORMERCONSIDERING THE MAGNETIC SATURATION

In Fig. 2, the inductance, L, of the primary winding oftransformer is a differential inductance, which is calculated byusing the nonlinear finite element method (FEM), including twosteps [3]: (1) For a given winding current , i, conduct a non-linearfield analysis to find the flux linkage of windings, m; (2) When

the flux densities in two consecutive time steps are obtained, thedifferential inductances can be calculated by

( ) ( ) ( 1)( )

( ) ( ) ( 1)m m m

md k B k B k

L i Andi k i k i k = =

(1)

where A is the cross-sectional area, n1 is the number of turns ofthe primary winding, and subscripts ( k ) and ( k -1) refer to the k thand ( k -1)th steps, respectively.

Fig. 3(a) shows the 2D magnetic force lines of an E25transformer with air gap. Both the differential inductance andapparent inductance of transformer are shown in Fig. 3(b). Themajor data of the E25 transformer with PC40 ferrite core include:number of turns of primary winding, n1=58, number of turns ofsecondary winding, n2=12, and length of air gap, g =0.16 mm.Fig. 2 is the actual equivalent circuit of a flyback converter withinput voltage of 102 V, 15V/1.2A rated output, and PWMfrequency of 100 KHz.

(a)

(b)

Fig. 3. Calculation of nonlinear inductance: (a) magnetic force lines, and (b)apparent and differential inductances vs. primary current.

IV. DETAILED SWITCHING MODEL OF BUCK -BOOSTCONVERTER WITH PEAK CURRENT MODEL CONTROLLED

According to Figs. 1 and 2, the current flowing through theinductance may be in continuous conduction mode (CCM) or indiscontinuous conduction mode (DCM). The typical currentwaveform flowing through the inductance in DCM is shown inFig. 4, where d 1, d 2 and d 3 are duty ratios of PWM duty cycle,

respectively. When d 3=0, CCM can be seen as a specialcondition of DCM. Fig. 5 shows the current in CCM.

Fig. 4. Typical inductor current waveform in DCM

Fig. 5. Typical inductor current waveform in CCM

In order to describe the converter easily, considering that acomplete converter is constituted by two parts: one is the maintopology, and the other is the feed back circuits, and the dutyratios d 1, d 2 and d 3 can be seen as variables, then the generalcontrol scheme of PWM switching DC to DC converter isobtained and shown in Fig. 6. In other words, not only thecurrent, i L(t ) flowing through the inductance and the voltage, V c(t ), across the capacitance, but also the duty ratios d 1, d 2 and d 3 are considered as the system state variables. The input voltage,V i, and the constant voltage source of diode, V F , are the inputvariables of system, and the output voltage, V o, is the outputvariable of system.

Fig. 6. General control scheme of PWM switching DC to DC converter

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According to the KVL and KCL circuit laws, the detailedswitching dynamical model of buck-boost converter can beobtained and shown in Table I with a set of state spacedifferential equations. By omitting the control delay in feedingloop, the states of the converter, s1, s2 and s3, corresponding tothe duty cycles, d 1, d 2 and d 3 respectively, are also decided bythe corresponding equations shown in the table.

TABLE IDETAILED SWITCHING MODEL OF THE BUCK -BOOST CONVERTER

t or S Model

T d t 10

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disturbance ones, respectively. By using the method oflinearizing the large signal average model around a desired localoperating point, the small signal average model is deduced asthe follows.

3

1

3

1

[( ) ( ) ]

( )

k k k k

k k k

D D d

D d

=

=

= + + +

= +

k k k k

k k

x A x B u A X B U

y C x C X

(10)

and

=

=

=

=

0

1

3

1

3

1

k k

k k

d

D (11)

By using the Laplace transform, one can obtain

3 31

1 1

3

1

( ) ( ) [ ( ) ]

( ) [ ( ) ( )]

k k k k k

k k k

s s D D d

s D s d s

= =

=

= + +

= +

k k k k

k k

x I A B u A X B U

y C x C X

(12)

By using (12), some small signal transfer functions can beobtained. For example, G vdk can be derived as

3 3 31

1 1 1

( )( )( ) ( ) ( )

k k k k k

s s D

d s

= = =

= = + k vd k k k k

yG C X I A A X B U (13)

Through the transfer functions, some frequency domain behaviors of the converter can be obtained.

VI. MECHANISM FOR EXCHANGING DATA BETWEEN THEDETAILED SWITCHING MODEL AND THE AVERAGE MODEL

In (12), all the system matrices, Ak , Bk and C k , are constantswhich can be preset before running the system model. Other

parameters D k , ,X U can also be preset. However, the relationamong D k , ,X U is not unique, so it is preferred to set themfrom the detailed switching model in each PWM cycleautomatically. There are many methods for executing the smallsignal model. The manual method, in which a switching is used

to control the execution of the small signal model, is adopted inthis paper.

VII. MODEL IMPLEMENTATION In Fig. 6, the complete simulation model is composed of two

parts: one is the block of the converter topology, and the other isthe block of the peak current mode control. According to Table I,the simulation model for the block of the converter topology isobtained and shown in the converter block of Fig. 7(a), wherethe nonlinear differential inductance is calculated in advanceand stored into a look-up table. The simulation model for the

block of the closed-loop current mode control is shown in thecurrent mode control block of Fig. 7(a), in which the systemswitching states, s1, s2 and s3, are calculated. By connectingthese two blocks, the complete simulation model of buck-boostconverter is obtained and shown in Fig. 7(a).

By inputting the parameters of parts into the proposed modelshown in Fig. 7(a) and running it in MATLAB/Simulinksurrounding, all the system output and system state variables can

be obtained and shown in Fig. 7(b)-(e), which are in agreementwith the rated specification of the converter.

1Vc

s

iL

Vc

Vi

VF

Cp

SW

iLA

VCA

ViA

VFA

D1

D2

D3

bode

Tsr

Tsf

1

6

1.314

0.047e-6s+1

Scope3

Scope2

Scope1

Scope

Manual Switch2

L(i)

12/58

0.0812/58

0

iL

iL~

Cp

PA

Vf

Vr

Vo~

VH

VL

Tsr_2

Tsf_2

Uai

s

Tsr_1

Tsf_1

Dui

Current-Mode

L

s

Rdon

Rc

R

Rqon

Ri

m

b

VF

Vs

C

g

iL

iLc

Vo

Vc

Converter

6.5e-6

102

3.383

0

0

0

1

292

0

1

1

0

0

1

12.08

10.234

1.168Cp

VC

iL

iL

iL~

Vo

Io

Vc

Vo

S1

S1

S2

S3

(a)

(b)

(c)

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(d)

(e)Fig. 7. Implementation of buck-boost converters with peak current mode control:(a) complete simulation model, (b) output voltage, (c) output current, (d) voltage

across the capacitor, and (e) current flowing through the inductor.

According to (10)-(13), the small signal simulation model can be easily built and shown in Fig. 8. The average value of Dk , X and U in each PWM cycle is calculated by the function ofumocav and transferred to the function of bplot. These twofunctions are programmed according to (10)-(13). Fig. 9 showsthe Bode plot of G vd1 when the converter operates with the ratedcondition.

7

D3

6

D2

5

D1

4

VFA

3

ViA

2

VCA

1

iLA

MATLABFunction

umocav

MATLABFunction

bplot

STOP

Stop Simulation

Memory

0

Clock

7

SW

6

Cp

5

VF

4

Vi

3

Vc

2

iL

1

s

iLa

Vca

U

VF

D1

D2

D3

Fig. 8. Small signal average model of buck-boost converters with peak current

mode control

0

10

20

30

40

50

60

M a g n

i t u d e

( d B )

103

104

105

106

-135

-90

-45

0

P h a s e

( d e g

)

Bode Diagram

Frequency (rad/sec) Fig. 9. Bode plot of G Vd1

VIII. CONCLUSION In this paper, a unified hybrid model is proposed for peak

current mode controlled buck-boost converters operating in bothcontinuous conduction mode and discontinuous conductionmode. The model consists of two level networks including adetailed switching model and a small signal average model,respectively. Nonlinearity such as the nonlinear inductance isconsidered by using magnetic field finite element analysis.

The hybrid model has been implemented in Matlab/Simulink

for simulating the transient performance and frequency domaincharacteristics of a buck-boost converter in different conductionmodes. Theoretical analysis and simulation results show thatcomparing with the traditional method, the proposed model ismore feasible and more efficient.

R EFERENCES [1] A. Davoudi, D. J. Jatskevich, and T. Rybel, Numerical state-space

average-value modeling of PWM DC-DC converters operating in DCMand CCM, IEEE Trans. Power Electronics , Vol. 21, No. 4, pp.1003-1012, July 2006.

[2] J. Sun, D. M. Mitchell, F. M. Greuel, T. P. Krein, and R. M. Bass,Averaged modeling of PWM converters operating in discontinuousconduction mode , IEEE Trans. Power Electronics , Vol. 16, No. 4, pp.482-492, July 2001.

[3] J. X. Chen, Y. G. Guo, and J. G. Zhu, A generalized dynamic model forflyback switching converter based on nonlinear finite element analysis, in

Proc. Int. Technology and Innovation Conf., Advanced ManufacturingTechnology , Hangzhou, China, Nov. 2006, paper no. 01147.

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