Modeling of a SEPIC Converter Operating inDiscontinuous Conduction Mode

Vuthchhay Eng and Chanin BunlaksananusornFaculty of Engineering, King Mongkut's Institute of Technology Ladkrabang (KMITL),

Chalongkrung Rd. Ladkrabang, Bangkok 10520vouthchhay_oeung@yahoo.com and kbchanin@kmitl.ac.th

(3)

(2)

(I-c)

(I-a)

(I-b)

accurately predict the converter's characteristics up to one-tenthof the switching frequency.

II. OVERVIEW OF SSA TECHNIQUE IN DCM

A SEPIC converter is shown in Fig. l(a). Due to the switchingaction of the MOSFET, Q, and diode, D, the converter willexhibit three different circuit states in one switching period, T,when operating in DCM. The first state exists when Q is turnedon for a time interval d j T (Fig. 1(b)), the second state when Q isturned off (i.e. D turned on) for a time interval d2T (Fig. l(c)),and the third state when both Q and D are turned off for the restof the time period d3T (Fig. l(d)). Note that dj and d2are the dutycycle of Q and D respectively, and d3 equals l-dj-d2. The generalstate-space equations for these three circuit states are:

{dxldt = A1x+B1u ~ .. IdTlor tIme Interva 1y=C1x+E1u

{dxl dt = A2x+B2u ~ .. I d Tlor tIme Interva 2y=C2x+E2u

{dxldt = A3x+B3u ~ .. IdTlor tIme Interva 3y=C3x+E3u

Since the SEPIC converter is made up of two inductors L j and L2

and two capacitors Cj and C2, the state vector x, thus, comprisesof iLj, iL2, VCj, and VC2. The input voltage vg is typically assignedas the input vector u, and the output voltage Va as the outputvector y. To find the averaged behavior of the converter over oneswitching period, T, equations (I-a) to (I-c) are weighed averageby the duty cycles as:

{dXldt=AX+BUy=Cx+Eu

where A= A1dl+ A2d2+A3d3, B=B1dl+B2d2+B3d3,C=Cldl+C2d2+C3d3' and E=Eldl+E2d2+E3d3.Equation (2) is a nonlinear continuous-time equation. It can belinearized by small-signal perturbation with x=X+x, y=Y+y,u=U+u, dj=Dj+dj, d2=D2+d2, and d3=D3-dj-d2 where the tildesymbol " ,." "represents a small-signal value and the capital lettera DC value. It should be noted that X»x, Y»y, U»u, Dj»dj,D2»d2, and D3»d3• The perturbation yields the steady-stateand linear small-signal state-space equations in (3) and (4)respectively.

{dXldt=AX+BU=OY=CX+EU

Abstract - A SEPIC (Single-Ended Primary Inductor Converter)DC-DC converter is capable of operating in either step-up or step­down mode and widely used in battery-operated equipment. Thereare two possible modes of operation in the SEPIC converter:Continuous Conduction Mode (CCM) and DiscontinuousConduction Mode (DCM). This paper presents modeling of a SEPICconverter operating in DCM using the State-Space Averaging (SSA)technique. The modeling leads to a small-signal linear model of theconverter, from which the transfer functions used for feedbackcontrol design can be determined. It is found that the derived modelis a reduced-order model, which can accurately predict theconverter's characteristics up to one-tenth of the switchingfrequency.

I. INTRODUCTION

A SEPIC (Single Ended Primary Inductor Converter) converteris a fourth-order dc-dc converter capable of delivering an outputvoltage which can be greater or lower than an input voltage.There are two possible modes of operation in the SEPICconverter: Continuous Conduction Mode (CCM) andDiscontinuous Conduction Mode (DCM). Although the convertermay have been designed for the CCM operation, it can plungeinto the DCM operation at light loads. In some cases, theconverter is even intentionally designed to operate in DCMbecause of the faster dynamic response compared with the CCM[1, 2]. Recently, the small-signal dynamic characteristics of theDCM SEPIC converter have been modeled [3]. In this work, theactive switch and passive diode of the SEPIC converter weresubstituted by the PWM-switch model [4]. Transfer functions ofinterest, e.g. duty ratio-to-output or input-to-output transferfunction, were then derived from the resulting equivalent circuit.However, the modeling process in [3] had assumed the converteris being ideal and neglected the Equivalent Series Resistance(ESR) of the capacitors. The ESR affects the value of zeros in thefinal transfer functions - excluding it from the modeling processonly adds to the inaccuracy in the final model.

This papers presents modeling of the DCM SEPIC converterwith State-Space Averaging (SSA) technique [1], taking intoaccount the effect of the capacitors' ESR. The transfer functionsfor feedback control design are derived and compared with theresults from the DCM averaged switch model [2]. It is found thatthe derived model is a reduced-order model, which can

978-1-4244-3388-9/09/$25.00 ©2009 IEEE

(8-b)

(8-c)

(8-a)

(8-d)(8-e)(8-f)

(8-g)

(8-h)

(8-i)(8-j)

R

R

R

(a) SEPIC converter.CI rCI iCI

(b) SEPIC converter during the first state d]T.

iLl 4 CI rCI i CI

+ -

A. State-Space Description

Chosen x = [iLl i L2 VC1 VC2]T, U = v g , and y = vo, the matricesAt, A2, A3, Bt, B2, B3, Ct, C2, C3, Et, E2, and E3 in (1) are

:e:r[t~j:o?:thecgcuitS]in Figs. l(b) to l(d).

.- 0 -I/C 0 0o 0 I 0 -REI(C2rc2R)

[

(RE- rCI)/4 -REI4 -114 -REI(4rC2)]A - -REIL2 -REIL2 0 -REI(L2rc2)

2- llC 0 0 0R/(C2rC2) R/(C2rC2) 0 -REI(C2rc2R)

[

rC/(4 +~) 0 -1/(4+L2) 0 ]A - -rc/(4+~) 0 1/(4+L2) 0

3- l/CI

0 0 0o 0 0 -REI(C2rc2R)

B.=[1/4 0 0 OJT

B2 =[1/4 0 0 OJ T

B3=[1/(4+L2) -1/(4+L2) 0 OJT

c. =[0 0 0 RE lrc2JC2 =[RE RE 0 RElrc2JC3 =[0 0 0 RE lrc2JE.=E2 =E3 =[0]

{~i/dt_=A~+Bii~BdlJl+_Bd2J2 (4)y=Cx+Eu+Edldl+Ed2d2

where A=AIDI+~D2+A3D3' B=BIDI+B2D2+B3D3, C=CIDI+C2D2+C3D3'E=EI~+E2D2+E3D3' Bdl = (AI-A3)X+(BI-B3)U, Bd2=(A2-A3)X+(B2-B3)U,Ed. = (C.-C3)X+(E.-E3)U, and Ed2=(C2-C3)X+(E2-E3)U.

Equations (1) to (4) provide a systematic way to model the DC­DC converters in DCM. When applied to the DCM SEPICconverter, the direct solution of (3) and (4) will not producecorrect results because the chosen state variables, iLl and i L2, areactually not independent from each other. Hence, only one ofthese currents can be said to be a true state variable. If both iLl

and iL2 are to remain as a state variable as in equations (2) to (4),some constraints must be imposed on them. As shown in [1], thesteady-state and linear small-signal state-space equations in (3)and (4) are subject to the following constraints:I =i(~,Vo,4,~,T) (5-a)where I =ILl + IL2 .

{dlldt=O1=(alIOvg)vg+(alIOvo)vo+(allad)dl (5-b)

wehre I =~1+~2 ·To find the steady-state solution, due to the reason above, therelationship between ILl and I L2 must be first established beforethe other unknown, such as VC1 VC2, and Vo , can be found bysolving (3):

{X = -A-IBUY = (-CA-IB + E)U (6)

The constraints in (5) will result in 32 and one inductor current beeliminated from the linear small-signal state-space equations in(4). The disappearance of one inductor current means thesystem's order has been reduced by one. For this reason, theconverter's model derived by the SSA technique in DCM isknown as a reduced-order model [3, 5]. Consequently, the linearsmall-signal state-space can be rewritten as:

{~i/dt~A...i~BmVg +~mdlJl (7)y = Cmx+EmVg +Emd.dl

Finally, by applying the Laplace transform to (7), transferfunctions for feedback control design, e.g. duty ratio-to-output orinput-to-output transfer function, can be derived.

III. MODELING OF SEPIC CONVERTER IN DCM

In Fig. l(a), the resistances rC1 and rC2 are Equivalent SeriesResistances (ESRs) of the capacitors C1 and C2 respectively.Although their values are small, these ESRs cannot be neglectedin the modeling process as they have a direct impact on theaccuracy of the final model. Fig. 2 depicts the current waveformsiLl and i L2 of the converter, when operating in DCM. The currentsiLl and i L2 are increasing during the time interval d1T anddecreasing during the time interval d2T. During the time intervald3T, these currents have a constant value, with the amplitude ofiLl and i L2 being equal but flowing on the opposite direction (Fig.l(d»; that is, iLl = -iL2• The equality in the amplitude of iLl and i L2

is always true during the time interval d3T, and this essentiallymakes iLl and i L2 depend on each other.

(c) SEPIC converter during the second state d2T.

iLl 4 CI rCI i CI

+ -

R

(d) SEPIC converter during the third state d3T.Fig. 1. Operation of the SEPIC converter in DCM.

Fig. 2. Current waveform of iLl and i12.

(13)

(9-f)

(9-e)

(9-a)

From Fig. 2, the average value of ILl and IL2 are expressed as:ILl =DlT(Dl+D2)Vg1(24)+Imin (14-a)IL2=DlT(Dl+D2)Vgl(2L2)-Imin (14-b)Matching the summation of ILl and IL2 in (13) to those in (14), D2can be defined as:

D2=~217LE/(RT) (15)

where LE=(4+~)/(4~) and RE=(R+rC2)/(RrC2).

c. Linear Small-Signal State-Space Equations

Referring to (7), linear small-signal state-space equation is:

{;l~l ~: V~ v~zJ ~A;[~l ~2 vel v~J+BmVg +Bmd1dl (16)vo=Cm[ZLl ZL2 VCl VC2] + Emvg +Emdtdl

where

[

rC1(K23-KtJ2KI2In) 0 -KJ2KI2In-TDIKEI(2~)-K34 D2[KJ/(nLE)-1/4]]A = rC1(KJ2K12ln-K23) 0 KJ2KI2In+TDIKEI(2L)+K34 -D2[KJ/(nLE)-1/4]

m lICI 0 -TDI2/(2~CI) 0rClK12/(nC2) 0 (D2- IRR/ n)TDI/(2~C2)+IKI/(nC,) -[IIR+ID2/(nLE)]/C2

[

(Dl +D2)14+D31(4+~)-KalDl+D2)/(n4)-TDlKEI(24) ]B = -(Dl+D2)/4-D3/(4+~)+KalDl+D2)/(n4)+TDlKEI(24)

m _ TD121(24Cl)

(Dl+D2)Il(n4C2)+(D2- IRR/ n)TD/(24C2)

Cm=[rClREIK12ln 0 RE[IK12ln+(D2-IRRJn)T~/(2~)] l-IRED2/(nLE)]

Em=[REI(~+D2)/(n4)+(D2-IRRrln)T~REI(24)]

[~[114-Ka2/(nLE)]+rCl[Ka2IL2I(n~)-ILl(4+~)]-KTIKEIDl ]

B = -Vg[lI4-Ka2/(nLE)]-rCl[Ka2IL2/(n~)-ILl(4+~)]+KTIKEIDlmdl -2IICI

(VgILE-rClIL21~)II(nC2)+(D2- IRR/ n)II(DlC2)

Emdl =[(VgILE-rcl L21~)REIIn+(D2-IRRJn)REIIDlJ

K12 =Dl/~-D2/4, K23 =D3/(4 +~)-D2/4, K34 =D2/4 +D3/(4 +L2),KE=RED2/4-Ka2~Rln, Ka2=REII4+Vc2/4, n=:::.(REI+Vc2 )ILE, 1]=:::.1,

KT=I/[I+TDlrcl/(2~)]=:::.I, I=:::.DlTVgl(2LE), and RRr=rClDl/~+RED2ILE.

D. Finding Transfer Function

Applying the Laplace transform to (16), various transferfunctions can be derived for the converter. Due to the limitedspace, only two important transfer functions are presented: theduty ratio-to-output voltage and input-to-output transferfunctions, Gd1v(S) and Gvv(s), respectively.

Gdly(S)=vo(s)1dl(s)=C(sI - A)-tBdl +Edl[TD2~/(LEC2)](adlvS3 +bdlvS2+CdlvS+ddlv) (17)

[as+[2+ RE/(RD2)]/(C2R)](S2+bs+c)

Gyy(s) = Vo(s)1 vg(s) =C(sI - A)-tBot +Eot[TD2/(4LiL2ClC2)](aws3 +bvvs

2+cws+dvv) (18)

[as+[2+ RE/(RD2)]/(C2R)](S2 +bs+c)

Coefficients of Gd1v(S) and Gvv(s) are listed in TABLE I.

IV. RESULTS

Fig 4(a) shows a frequency response of Gd1v(S) in (17). Theplot is generated by MATLAB using the SEPIC converter'scircuit parameters in TABLE II (these values cause the converter

(9-b)(9-c)(9-d)

(9-g)(9-h)

(10-a)(10-b)(10-c)

Fig. 3. Current waveform oficl.

Irrnnl: I ~_ .

O~--+--............,.I~......II

:········d~·t····: d2T id

3i

... ~I.. ~I" ~I

Referring back to (3) and (4), the matrices for the steady-stateequations and the linear small-signal state-space equations aredetermined from (8):

retD3 (ret +RE)D2 -RED2 _ K -RED24+L2 4 4 34 4 rC2-ReDz _ retD3 -retDl- RED2 K -ReDz

A= L2 4+L2 L2 34 L2rC2D2+D3 -Dl 0 0

Cl ~

RED2 RED2 0 ~C2rC2 C2rC2 C2rC2R

B=[(Dl+D2)/4+D3/(Ll+L2) -D3/(4+ L2) 0 OJ T

C=[RED2 RED2 0 RElrc2JE=[O]

[

-rClIL/(4+L)+~/4 ]B

dl= rclILl1(4+L2)+(~1-rclIL2 )1L2

-(ILl+IL2)/~o

[

-[(rCl +RE)I4+rc/(4+~)]ILl-(IL2+Vc2Irc2)REI4]B = [rc/(4+L2)-REI4]ILl-(IL2+~2Irc2)REIL2

d2 0RE(ILI+IL2)/(C2rC2)

Edl=[O]

Ed2=[RE(ILl+ IL2)]

B. Steady-state equations

As stated above, the relationship between the steady-stateinductor currents, ILl and IL2, must be determined before othersteady-state values can be found. This is done by averaging thecapacitor current iC1 in Fig. 3 over a switching period, T. In Fig. 3,iC1 can be expressed as:iCl = -iL2 for time interval dlTiCl = iLl for time interval d 2TiCl = iLl = -iL2 for time interval d3T

Average of iC1 over a switching period, T, gives:

Ia =Imin -(IL2+Imin)Dl/(Dl+D2)+(ILl-lmin)D2/(Dl+D2) (11)

In steady-state, I C1 is equal to zero. Thus the relationship betweenILl and IL2 is obtained as:I L2 = I LlD2 / Dl (12)Given the averaged matrices in (9-a) to (9-d) and the relationshipin (12), the steady-state solution of converter can be obtained byusing (5).

[

ILl ] (~ID2)21]1RIL2 =v 1]Dl/(RD2)VCl g 1

VC2 1]Dll D2where 1]=1/[1+REDI I(RD2)+[4 +~(I-Dl)1D2]rClDl/[RD2(4+~)]]

,-..00Q)

S -45Q)cr.J~

~ -9010~1----'----'--''''''''''''''''....i..i.l.L.:02----i..---i....-i.....i....i...i..i..il..L..:03~----i......i.....i....i...i..i....i.L-----i.---..L...L..Ll...i..i...LJl05

Frequency (Hz)(a)

[l] S. Cuk and R. D. Middlebrook, "A general unified approach to modelingswitching DC-to-DC converters in discontinuous conduction mode," inProc. IEEE PESC'77, 1977.

[2] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics,2nd ed., Kluwer Academic Publishers, 2001.

[3] L. G. De Vicuna, F. Guinjoan, J. Majo, and L. Martinez, "Discontinuousconduction mode in the SEPIC converter," Proc. of ElectrotechnicalConference on Integrating Research, Industry and Education in Energy andCommunication Engineering, page. 38-42, April 1989.

[4] E. Niculescu, M. C. Niculescu, and D. M. Purcaru, "Modelling the PWMSEPIC converter in discontinuous conduction mode," Proc. of the 11th

WSEAS International Conference on Circuits, July 2007.[5] Jian Sun, D. M. Mitchell, M. F. Greuel, P. T. Krein, and R. M. Bass,

"Averaged modeling of PWM converters operating in discontinuousconduction mode," IEEE trans. on Power Electronics, July 2001.

v. CONCLUSION

In this paper, modeling of a OCM SEPIC converter with State­Space Averaging (SSA) technique has been presented. Themodeling yielded a steady-state and linear small-signal equationsof the converter in (13) and (16) respectively. From (16), thetransfer functions Gd1 v(s) and Gvv(s), which provide a basis forfeedback control design, were derived. The obtained Gd1v(S) wascompared against the result from the OCM averaged switchmodel [2]. Good consistency between the two results wasobserved only at frequencies below 10kHz, or one tenth of theswitching frequency. Above this frequency, the derived Gd1v(S)became invalid. Oue to this limitation, if the derived models in(17) and (18) were to be used in feedback control design, acrossover frequency must be selected in the frequency regionwhere these models are legitimate.

REFERENCES

Circuit Parameters Values

Vg / Vo 15/5VC j /C2 47 /200JlFrCj/rC2 0.38/0.095il

R 1OilL j /L2 100/30flH

aw = 4 CIC2RE[2L/2D1+D2)-4TrCl]

bw ={24~~(l+REIR)(2D1+D2)+TD12D2C2RE(4+~)

+(2D24 - 4+D14 +D2~)TD12CIC2REr61(4+~)+4~rCl[[D14+2~(2Dl+D2-1)(D1+D2)]2C2RE1(4+~)-(R+RE)TD1

2IR] }

cw=(1+REIR)[TD12D2(~+L2)+[2(Dl+D2)(2Dl+D2-1)~+Dl~]2~CrCl(~+L2)

-(2D2~+Dl~-~+D2~)TD12Cr~/(~+~)]

+4Dl~C2RE[[D14-4-D2~-2D2~D3]TD{Cl/(4~~)+1]

dw=(1+REIR)[4Dl~+[Dl~-~-D2~-2D2L2D3]TD12rCl/~]

TABLE IICONVERTER PARAMETERS

ad1v =C2RE[1-TD1rCI 1(4~)-D1LErCI 1(2D2~R)]

bdlv=(l+REIR)[1-TDlrC/(4~)]+[D2/~-D/~+TDI2rc/(4l3i)]TDIC2R EI(2C1)

D1LEC2RErJI [~b-DzLz [(TDzR +l)D +D]+ DzL,i2Dz-1) (TDzR 1)]2Di~~R ~+~ 2LE 2 I L2 2LE +

-[[TD?~/(4D2~CIR)+1]C2/~+DI(R+RE)/(2D2R2RE)-2C2(DI+D2)/~]LERErCI/~

_ T~ ( RE D2 ~ rp(R+RE) T2Iif 1 TIif4cdlv- 2q l+jf)(Z;--Z:;)+ R [g4C

1+4+~ (l-2D3 4D2~qR)]

~4C2RErQ +~[l [T D L 4D (T n L D )] T~rp]2D2CIR(4+~)2 q(4+~) - L.f+ 2~- 3 L.f~-~ 2 4~4

+ ~rp(~-D2) [C2RE(D ,-_'}nT)-!Q.[Tn2c RR +2'-C(R+R )]]2D2qR(4+~)2 D2 2~ ~LJI 2R ~ 2 E ~ I E

+ ftr~l_ {+(2~CI-T~2C2RE)(D2~+~4+24D2-4) ~:4;R+RE)LJI+~ g~4q 2D2R(4+~)

LF:Dz(2fz-12+A)[~~f (~24C2+D24q+D2~q)+2R+2RE]4D2~R '-1L.f~

_...L..[ qCz4(A-l) +D R ]}44R Di~CI(4+~) 2 E

d =(l+R,:;IR) {1- TD1rCI (T +D T ) I D1rCI [TDI2rCl (T -D,T -D T

dlv CI(~+~) 44~ ~ 2~ 4D2l3iR 2~ ~ l~ 2~

- 2D2~)(2LE-TD2R)+[TDtrCI+2~(DID2 - Dl2- D2)]LEID2]

+ DlrClLE(D2~ - DI~) [T (2D T _ 2D T +TD3,., )-4TD2D R( T +T)] }4l3i4DiR ~ l~ 2~ I CI 2 3 ~ ~

to operate in OCM). Fig. 4(b) shows the same plot from PSPICEsimulation using the OCM averaged switch model [2]. Generallyaccepted as being accurate, this PSPICE result is used to validateGd1v(S) in (17). The two results are closely agreed at frequenciesbelow 10 KHz, i.e. one-tenth of switching frequency, beyondwhich the result from (17) starts to diverse from its PSPICEcounterparts. The discrepancy occurs due to the fact that theformer is a reduced-order model, while the latter is a full-ordermodel [2, 5]. Gd1v(S) in (17) being a reduced-order model isevident in (16), where all elements in the second column of Ammatrix are zero, which literally means iL2 is no longer a statevariable, thus reducing the system's order by one.

TABLE ICOEFFICIENTS OF Gd1v(S) AND Gvv(s).