Mathematical modeling of buck–boost dc–dc converter and investigation of converter elements on transient and steady state responses

Electrical Power and Energy Systems 44 (2013) 949–963

Contents lists available at SciVerse ScienceDirect

Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Mathematical modeling of buck–boost dc–dc converter and investigationof converter elements on transient and steady state responses

Hamed Mashinchi Mahery, Ebrahim Babaei ⇑Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 51664, Iran

a r t i c l e i n f o

Article history:Received 31 December 2011Received in revised form 15 August 2012Accepted 19 August 2012Available online 2 October 2012

Keywords:Buck–boost dc–dc converterZ-transformLaplace transformCCMModeling

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.08.035

⇑ Corresponding author. Tel./fax: +98 411 3300829E-mail addresses: h_mashinchi.mahery@iau-ahar.

[email protected] (E. Babaei).

a b s t r a c t

In this paper, a new method is proposed for mathematical modeling of buck–boost dc–dc converter incontinuous conduction mode (CCM). In proposed method, using the Laplace transform the relations ofinductor current and output voltage are obtained. In the next step, in each switching interval using theZ-transform the initial values of inductor current and output voltage are calculated. Then, the transientand steady states responses of these quantities are calculated. In addition, the effect of the values of con-verter components on each of these responses is investigated. Finally, the simulation results in PSCAD/EMTDC software as well as the experimental results are used to reconfirm the validity of the theoreticalanalysis.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Mathematical modeling of dc–dc converters is one of the basicsubjects in analysis of their operation. In order to have a suitableefficiency and desired operation, the values of converter compo-nents should be designed properly. Modeling is one of the mainsteps in design and control of a system.

Nowadays, by development in power electronic fields, there is agood attention to dc–dc converters [1–4]. In order to achieve aproper design and control, it is necessary to have an exact modelof converter. By modeling the dc–dc converters, the operation ofconverter in different operational modes can be investigated inboth transient and steady states. High accuracy and low responsetime are major features of a good modeling [5–12].

The significant point in mathematical modeling of dc–dcconverters is the existence of power switch in the structure ofthese converters. This causes nonlinearity model for converters.As a result, it is needed to solve nonlinear equations. Differentmathematical modeling methods such as impedance method[13], small signal analysis method [14,15], state space method[16], and state space average value method [17,18] have been pre-sented in literature. One of the disadvantages of them is to usenumerical solution or simplification in extracting the models.Because of approximation in these models, the results of themare not enough accurate. The other considerable point in analyzing

ll rights reserved.

.ac.ir (H. Mashinchi Mahery),

of each system is the response time of system. In conventionalmethods, it is usually needed to use mathematical operators suchas matrix inversion or to solve the complicated algebraic equa-tions. As a result, in systems with high rank matrix, the responsetime is increased.

In [19,20], a new method has been presented for modeling theboost and buck dc–dc converters in CCM and discontinuous con-duction mode (DCM) operations, respectively. In this paper, it isaimed to improve the analysis accomplished in [19,20] on abuck–boost converter in CCM operation. It should be noted thatthe proposed method also is applicable for DCM operation. Consid-ering which the operational conditions are different in each ofoperational modes, so the results for each will not be the same.For this reason and also for avoiding an overlong paper, in thispaper, the results for CCM operation of buck–boost dc–dcconverter are presented.

The proposed method is based on the Laplace and Z transforms.The Laplace transform is used to obtain the inductor current andoutput voltage equations, and Z-transform is used to determinethe initial values of them. It is important to mention that if Laplacetransform is used to determine the initial values, the transfer func-tion of inductor current and output voltage consist of s and esT. Inthis case, the time function which is obtained using by inverse ofLaplace transfer will be an infinite series. For solving this problem,the Z-transform has been used. The final value theorem of Z-trans-form is used for analyzing the converter in steady state.

One of main disadvantage of small signal and state spaceaverage value models is high variation of the system parametersaround the dc parameter. In this case by neglecting the dc

http://dx.doi.org/10.1016/j.ijepes.2012.08.035

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.ijepes.2012.08.035

http://www.sciencedirect.com/science/journal/01420615

http://www.elsevier.com/locate/ijepes

950 H. Mashinchi Mahery, E. Babaei / Electrical Power and Energy Systems 44 (2013) 949–963

component, the results will not have enough accuracy [13,16]. Theproposed method in this paper can be applied to analyse the dc–dcconverters by high variations of parameters.

In [19,21], the presented methods are based on Laplace and Ztransforms. In these methods, only the transient response of con-verter has been investigated. Using the proposed method of thispaper, both of the transient and steady state responses of convertercan be analyzed. In addition, the electrical parameters of convertersuch as output voltage and inductor current ripples can be investi-gated. Also, the proposed method in this paper can be used to ana-lyze dc–dc converters with large variations of parameters. In thispaper, the effect of converter components on transient and steadystates responses is investigated, too. Finally, the validity of pre-sented theoretical subjects is proved by experimental and simula-tion results in PSCADnEMTDC software.

2. The proposed mathematical model

The equivalent circuit of buck–boost dc–dc converter is shownin Fig. 1. In this figure, the diode D and switch S are consideredideal and RL is the equivalent resistance of inductor.

2.1. Converter analysis in CCM

The CCM is the mode which the inductor current is always con-tinuous and existence in all time intervals. Considering Fig. 1 andapplying Kirchhoff current and voltage laws, we have:

LdiL

dtþ RLiL ¼ f1ðtÞVi � f2ðtÞvo ð1Þ

Cdvo

dtþ vo

R¼ f2ðtÞiL ð2Þ

Eqs. (1) and (2) are the general equations of the output voltageand inductor current for buck–boost dc–dc converter. By applyingthe values of functions f1(t) and f2(t) in (1) and (2), the equations ofoutput voltage and inductor current into each of switching inter-vals can be obtained. The functions f1(t) and f2(t) are defined todetermine the converter equations during on (t1) and off (t2) statesof switch S as follows:

f1ðtÞ ¼X1n¼0

uðt � nTÞ � uðt � t1 � nTÞ ð3Þ

f2ðtÞ ¼X1n¼0

uðt � t1 � nTÞ � uðt � T � nTÞ ð4Þ

+

−

ovL

D

C R

oiciLi

S

iV+

−

cVLR

Fig. 1. Buck–boost dc–dc converter.

1t

T

2t

Tt +1t Tn+1

)(1 tf

t1

0

(a)Fig. 2. Waveforms of functio

where n denotes the number of switching time intervals and T is theswitching period.

Fig. 2 shows the waveforms of functions f1(t) and f2(t). It is ob-served that for time interval (0, t1), the values of functions f1(t) andf2(t) are equal with 1 and 0, respectively. By applying these valuesin (1) and (2), the equations corresponding to time interval that theswitch S is on are obtained. Also for time interval (t1, T), the valuesof functions f1(t) and f2 (t) are 0 and 1, respectively. By applyingthese values in (1) and (2), the equations of output voltage andinductor current corresponding to time interval that the switch Sis off are obtained.

Considering (3) and (4), it is observed that the functions f1(t)and f2(t) are defined as sum of the two step functions for n inter-vals. So, for analyzing the converter in a specified time, the valuesof functions f1(t) and f2(t) should be determined. With these condi-tions, the analysis of converter will be difficult. For independentanalysis of converter in each switching time interval based on Z-transform, the following variable exchange can be used:

t ¼ ðnþmÞT for n ¼ 0;1;2; . . . 0 6 m < 1 ð5Þ

where m is an unit time variable which its value is same for allswitching intervals and always varies between zero and one.

By applying (5) in (3) and (4), the functions f1(m) and f2 (m) canbe expressed as follows, respectively:

f1ðmÞ ¼1 0 6 m < D

0 D 6 m < 1

�ð6Þ

f2ðmÞ ¼0 0 6 m < D

1 D 6 m < 1

�ð7Þ

Prove of (6) and (7) has been given in Appendix A. Considering(6) and (7), it is observed that the interval [0,1] is divided into twointervals [0,D] and [D,1]. In interval [0,D], the switch S is on and inthe interval [D,1], the switch S is off. Considering (1), (2), (5), (6),(7), we have:

diL;ndm

dvo;ndm

" #¼ � RLT

L 00 � T

RC

" #iL;n

vo;n

� �þ

TL

0

� �Vi for 0 6 m < D

iL;nð0Þ ¼ iL0;n ¼ 0; vo;nð0Þ ¼ vo0;n

ð8Þ

diL;ndm

dvo;ndm

" #¼ � RLT

L � TL

TC � T

RC

" #iL;n

vo;n

� �þ

00

� �Vi for D 6 m < 1

iL;nðDÞ ¼ iL1;n; vo;nðDÞ ¼ vo1;n

ð9Þ

where (iL0,n, vo0,n) and (iL1,n, vo1,n) are the initial values of inductorcurrent and output voltage in intervals [0,D]and [D,1], respectively.

As it is observed, the initial values of output voltage and induc-tor current are functions of n, so the value of each of these param-eters for each switching time interval will be different.

1t

T

2t

Tt +1t Tn+1

t1

0

)(2 tf

(b)n; (a) f1(t) and (b) f2(t).

H. Mashinchi Mahery, E. Babaei / Electrical Power and Energy Systems 44 (2013) 949–963 951

2.2. Solving output voltage and inductor current equations usingLaplace transform

Considering (8) and (9), it is observed that the obtained equa-tions are differential equations. One of the methods to solve theseequations is to use Laplace transform. By applying the Laplacetransform in (8) and (9), the followings are obtained:

sþ RLTL

� �IL;nðsÞ ¼ iL0;n þ

TLs

Vi for 0 6 m < D ð10Þ

sþ TRC

� �Vo;nðsÞ ¼ vo0;n for 0 6 m < D ð11Þ

sþ TRL

L

� �IL;nðsÞ ¼ iL1;n �

TL

Vo;nðsÞ for D 6 m < 1 ð12Þ

sþ TRC

� �Vo;nðsÞ ¼ vo1;n þ

TC

IL;nðsÞ for D 6 m < 1 ð13Þ

By arranging (10)–(13) in matrix form, the equations of induc-tor current and output voltage in Laplace domain will be asfollows:

IL;nðsÞVo;nðsÞ

� �¼

1sþða�cÞT iL0;n þ T

LVis

� �v0;n

sþðaþcÞT

24

35 for 0 6 m < D ð14Þ

IL;nðsÞVo;nðsÞ

� �¼ 1

s2 þ 2aTsþ T2ða2 þx2Þ�

sþ Tðaþ cÞ � TL

TC sþ Tða� cÞ

" #iL1;n

vo1;n

� �

for D 6 m < 1 ð15Þ

In (14) and (15), a, x, and c are defined as follows:

a ¼ 12

RL

Lþ 1

RC

� �ð16Þ

x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

LCRL

Rþ 1

� �� a2

sð17Þ

c ¼ 12

1RC� RL

L

� �ð18Þ

By applying the inverse of Laplace transform in (14) and (15),the inductor current and output voltage can be expressed in m do-main as follows:

iL;nðmÞ ¼iL0;ne�ða�cÞmT þ Vi

RL½1� e�ða�cÞmT � for 0 6 m < D

e�aðmT�t1Þ iL1;n cos xðmT � t1Þ þ cx sin xðmT � t1Þ

�� vo1;n

xL sin xðmT � t1Þ

for D 6 m < 1

8>><>>:

ð19Þ

vo;nðmÞ ¼

vo0;ne�ðaþcÞmT for 0 6 m < D

e�aðmT�t1Þ iL1;nxC sin xðmT � t1Þn

þvo1;n cos xðmT � t1Þ � cx sin xðmT � t1Þ

� for D 6 m < 1

8>><>>:

ð20Þ

where iL0,n, iL1,n, vo0,n and vo1,n are unknown parameters.In order to determine the values of inductor current (iL(t)) and

output voltage (vo(t)) in each instant of time, these parametersshould be determined. The values of iL1,n and vo1,n can be expressedversus iL0,n and vo0,n. Considering the continuity characteristic ofiL(t) and vo(t), the followings are valid:

limt!ðt1þnTÞþ

iLðtÞ ¼ limt!ðt1þnTÞþ

iLðtÞ ð21Þ

Eq. (21) can be expressed versus m as follows:

limm!ðDÞ�

iL;nðmÞ ¼ limm!ðDÞþ

iL;nðmÞ ð22Þ

By applying (19) in (22), the value of iL1,n versus iL0,n is obtainedas:

iL1;n ¼ iL0;ne�ða�cÞt1 þ Vi

RL½1� e�ða�cÞt1 � ð23Þ

The following is valid about the output voltage:

limt!ðt1þnTÞ�

voðtÞ ¼ limt!ðt1þnTÞþ

voðtÞ ð24Þ

Eq. (24) can be rewritten as follows:

limm!ðDÞ�

vo;nðmÞ ¼ limm!ðDÞþ

vo;nðmÞ ð25Þ

Applying (20) in (25), the value of vo1,n will be equal with:

vo1;n ¼ vo0;ne�ða�cÞt1 ð26Þ

Considering (19), (20), (23), and (26), the relations of inductorcurrent and output voltage as follows:

iL;nðmÞ ¼

iL0;ne�ða�cÞmT þ ViRL½1� e�ða�cÞmT � for 0 6 m < D

iL0;n � ViRL

� �ect1�amT þ Vi

RLe�aðmT�t1Þ

h icos xðmT � t1Þ þ c

x sin xðmT � t1Þ �� 1

xL vo0;ne�ct1�amT sin xðmT � t1Þ for D 6 m < 1

8>>>>><>>>>>:

ð27Þ

vo;nðmÞ ¼

vo0;ne�ðaþcÞmT for 0 6 m < D1

xC iL0;n � ViRL

� �� ect1�amT þ Vi

RLe�aðmT�t1Þ

h isin xðmT � t1Þ

þvo0;ne�ct1�amT cos xðmT � t1Þ � cx sin xðmT � t1Þ

�for D 6 m < 1

8>>>><>>>>:

ð28Þ

2.3. Determining the initial values of inductor current and outputvoltage using Z-transform

In (27) and (28), the only unknown parameters are iL0,n and vo0,n.Considering which iL0,n and vo0,n are functions of n and n is a dis-continuous variable, so to determine the values of these parame-ters, Z-transform can be used.

Considering the continuous characteristic of inductor currentand output voltage, in t = nT the following relation is always valid:

limt!ðnTÞ�

iLðtÞ ¼ limt!ðnTÞþ

iLðtÞ ð29Þ

Eq. (29) can be rewritten as follows:

limm!1�

iL;nðmÞ ¼ limm!0þ

iL;nþ1ðmÞ ð30Þ

The following is valid about the output voltage:

limt!ðnTÞ�

voðtÞ ¼ limt!ðnTÞþ

voðtÞ ð31Þ

The above equation can be rewritten as follows:

limm!1�

vo;nðmÞ ¼ limm!0þ

vo;nþ1ðmÞ ð32Þ

Considering t2 = T � t1, for the initial values of inductor currentand output voltage in time interval n + 1 the following equationsare obtained:

iL0;nþ1 ¼ iL0;n �Vi

RL

� �ect1�aT þ Vi

RLe�at2

� �cos xt2 þ

cx

sin xt2

� �

� vo0;n

xLe�ct1�aT sin xt2

ð33Þ


vo0;nþ1 ¼1

xCiL0;n �

Vi

RL

� �ect1�aT þ Vi

RLe�at2

� �sin xt2

þ vo0;ne�ct1�aT cos xt2 �cx

sin xt2

� �ð34Þ

To solve (33) and (34), the Z-transform can be used. The follow-ing relations are always valid about a discontinuous function:

ZfiL0;ng ¼ IL0ðzÞ ð35ÞZfv0;ng ¼ V0ðzÞ ð36Þ

In addition, we have:

x½nþ 1�$Z zXðzÞ � zxð0Þ ð37Þ

Considering (37), the following equations are valid:

ZfiL0;nþ1g ¼ zIL0ðzÞ � ziL0;0 ð38ÞZfv0;nþ1g ¼ zV0ðzÞ � zv0;0 ð39Þ

where iL0,0 and vo0,0 are the initial values of inductor current andoutput voltage in t = 0, respectively.

By applying Z-transform in (33) and (34), the equations of IL0 (z)and Vo0(z) are obtained as follows:

½X� �IL0ðzÞV0ðzÞ

� �¼

zz�1 b1 þ ziL0;0

zz�1 b2 þ zv0;0

" #ð40Þ

The values of X, b1, and b2 have been given in Appendix B.Considering (40), the values of IL0(z) and Vo0(z) are given by:

IL0ðzÞVo0ðzÞ

� �¼ X�1 �

zIL0;0 þ b1z

z�1

zVo0;0 þ b2z

z�1

" #ð41Þ

The value of X�1 in (41) is equal with:

X�1 ¼ 1jXj�

z�e�aT�ct1 cos xt2� cx sin xt2

� �� e�aT�ct1 sin xt2

xL

e�aTþct1 sin xt2xC z�e�aTþct1 cos xt2þ c

x sin xt2� �

" #

ð42Þ

The value of jXj in (42) is equal with:

jXj ¼ z2 � 2ze�aT cos uT þ e�2aT ð43Þ

which

cos uT ¼ cos xt2 cosh ct1 þcx

sin xt2 sinh ct1 ð44Þ

Applying X�1 in (41), the values of IL0(z) and Vo0 (z) can be cal-culated by:

IL0ðzÞ¼z

z2�2ze�aT cosuTþe�2aT

� z�e�aT�ct1 cosxt2�cx

sinxt2

� �h iiL0;0

�

�vo0;0e�aT�ct1 sinxt2

xLþ z�e�aT�ct1 cosxt2�

cx

sinxt2

� �h ib1

�

�e�aT�ct1 sinxt2

xLb2

�1

z�1

�ð45Þ

Vo0ðzÞ¼z

z2�2ze�aT cosuTþe�2aT

iL0;0e�aTþct1 sinxt2

xC

�

þ z�e�aTþct1 cosxt2þcx

sinxt2

� �h ivo0;0

þ e�aTþct1 sinxt2

xCb1þ z�e�aTþct1 cosxt2þ

cx

sinxt2

� �h ib2

� �

� 1z�1

�ð46Þ

Eqs. (45) and (46) show the initial values of inductor currentand output voltage in Z domain. To obtain their initial values insteady state, the final value theorem of Z-transform can be used.The final value theorem of Z-transform for IL0(z) and Vo0(z) is ex-pressed as follows:

iL0;ss ¼ ‘imZ!1ðz� 1ÞIL0ðzÞ ð47Þ

vo0;ss ¼ ‘imZ!1ðz� 1ÞVo0ðzÞ ð48Þ

In (47) and (48), iL0,ss and v0,ss denote the initial values of iL,n(m)and vn(m) in steady state, respectively.

Considering (45)–(48), the initial values of inductor current andoutput voltage in steady state can be calculated as follows:

iL0;ss ¼1� e�aT�ct1 cos xt2 � c

x sin xt2� � �

b1 � e�aT�ct1 sin xt2xL b2

1� 2e�aT cos uT þ e�2aTð49Þ

vo0;ss ¼e�aTþct1 sin xt2

xC b1 þ 1� e�aTþct1 cos xt2 þ cx sin xt2

� � �b2

1� 2e�aT cos uT þ e�2aTð50Þ

By applying the inverse of Z-transform for (45) and (46), the ini-tial values of inductor current and output voltage for each switch-ing time interval can be obtained in discontinuous time domain.For obtaining the inverse of Z-transform, Eqs. (45) and (46) shouldbe expanded to partial fractions. After this, Eqs. (45) and (46) canbe expressed as follows:

IL0ðzÞ ¼z½z� e�aT cosðuTÞ þ ke�aT �iL0;0

z2 � 2ze�aT cosðuTÞ þ e�2aT

� ze�aT�ct1 sin xt2


vo0;0

xLþ ziL0;0

z� 1

�z iL0ssz� 2iL0sse�aT cosðuTÞ þ iL0ss � b1 �

z2 � 2ze�aT cosðuTÞ þ e�2aTð51Þ

Vo0ðzÞ ¼ze�aT�ct1 sin xt2


iL0;0

xC

þ z½z� e�aT cosðuTÞ � ke�aT �vo0;0

z2 � 2ze�aT cosðuTÞ þ e�2aTþ zvo0ss

z� 1

� z½vo0ssz� 2vo0sse�aT cosðuTÞ þ vo0ss � b2�z2 � 2ze�aT cosðuTÞ þ e�2aT

ð52Þ

where k is given by:

k ¼ cosðuTÞ � e�ct1 cos xt2 �cx

sin xt2

� �¼ c

xcosh ct1 sin xt2 þ sinh ct1 cos xt2 ð53Þ

By applying the inverse of Z-transform for (51) and (52), the ini-tial values of inductor current and output voltage in discontinuoustime domain can be calculated as follows:

iL0;n ¼ iL0ss þ ðiL0;0 � iL0ssÞe�anT cosðunTÞ

þ kiL0;0 �e�ct1 sinxt2

xLv0;0 þ

iL0sse�aT cosðuTÞ � iL0ss þ b1

e�aT

� �

� e�anT sinðunTÞsinðuTÞ ð54Þ

vo0;n ¼ vo0ss þ ðvo0;0 � vo0ssÞe�anT cosðunTÞ

þ e�ct1 sin xt2

xLvo0;0 � kvo0;0 þ

vo0sse�aT cosðuTÞ� vo0ss þ b2

e�aT

� �

� e�anT sinðunTÞsinðuTÞ ð55Þ


Eqs. (54) and (55) show the initial values of inductor currentand output voltage in each switching interval versus n. By applying(54) and (55) in (27) and (28), the values of inductor current andoutput voltage can be obtained in each instant of the time.

3. Theoretical analyses

In this section, the time response of buck–boost dc–dc con-verter is investigated based on presented mathematical model inSection 2. The time response of dc–dc converter consists of tran-sient and steady states responses. The transient response of con-verter is that part of the response which the inductor currentand the output voltage have not reached to their steady values.This part of time response starts from zero instant and will con-tinue to achieve to steady state.

Fig. 3 shows the time response of inductor current and outputvoltage of buck–boost dc–dc converter in CCM. These curves areplotted for Vi = 17 V, L = 8 mH, C = 0.2 mF, R = 20 X, RL = 0.5 X, andf = 1 kHz. Considering Fig. 3, it is observed that the time responsesof iL(t) and vo(t) consist of two transient and steady parts. The steadystate response of the converter is formed from the transient stateswhich are repeated in each switching interval. After that the induc-tor current and output voltage passed their transient states andreached to their final values, the inductor and capacitor chargeand discharge between the minimum and maximum values. Thiscauses to create the transient state in each switching interval.

3.1. Analysis of transient response

The time constant of the transient response in a system speci-fies the damping mode of the transient response. In each system,the roots of denominator of transfer function (the poles of system)are the time constant of system. Considering buck–boost dc–dcconverter, the time constants of iL(t) and vo(t) are determined bythe relations of these parameters in discontinuous time domain.The following relation is always valid between the roots of discon-tinuous and continuous time domains:

k1;2 ¼1T

lnðq1;2Þ ð56Þ

where k1,2 and q1,2 are the roots of characteristic equation in contin-uous and discontinuous time domains, respectively.

By calculating the dominator roots of (45) and (46), and apply-ing them in (56), the time constants of transient response are ob-tained. The dominator roots of (45) and (46) are given by:

q1;2 ¼ e�aT�juT ð57Þ

0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

4

5

][A

i L

Time [sec]

(a)Fig. 3. Time step response; (a) inducto

Considering (56) and (57), the time constants of iL(t) and vo (t)are calculated by:

k1;2 ¼ �a� ju ð58Þ

Considering (58), it is observed that the time constant of con-verter is a function of duty cycle, inductance, capacitance, loadresistance, and switching frequency. Variation of each of theseparameters changes the damping time of the transient responseof the system.

Considering (16), it is observed that the value of a in (58) is realand positive. But according to (45) and considering the values ofinductance, capacitance, load resistance, and the inductor equiva-lent resistance, u can be real or imaginary. If u is an imaginary inte-ger, the transient response of converter is severely damped. If u be areal integer, the transient response of the converter is weaklydamped with time constant e�aT. Considering the effect of induc-tance and capacitance on the value of u, the effect of these parame-ters on transient response can be investigated. Fig. 4 shows thevariation of real part of characteristic equation roots for differentvalues of duty cycle versus inductance and capacitance. As it is ob-served, for high values of L and C, the real part of roots will have high-er values. Considering which the real part of roots shows thedamping constant of transient response, so the time responses ofiL(t) and vo(t) have slow transient responses for higher values ofinductance and capacitance. As shown in Fig. 4, because the real partof roots is equal with�a and the value of a is independent of duty cy-cle, so for different values of duty cycle, the value of a is constant.

3.1.1. Analysis of overshoot and setting timeThe values of settling time and overshoot of the system step

response are important parameters in investigating dynamic re-sponse of a system. In this section, the effect of each of buck–boostdc/dc converter elements on the values of settling time and over-shoot are investigated by the plotted curves in Figs. 5 and 6. Thesefigures are obtained by using the related relations to settling timeand overshoot and the poles of transfer function of output voltageand inductor current which are expressed in (58). Figs. 5 and 6show the variations of overshoot and settling time versus induc-tance and capacitance for different values of duty cycle, respec-tively. As shown in Fig. 5a, the value of overshoot has a reverserelation with the value of inductance. In other words, by increasingthe value of inductance, the value of overshoot decreases. Fig. 5bshows that by increasing the value of capacitance the value ofovershoot increases. Considering Fig. 5b, it is observed that for aspecified value of capacitance, by increasing the duty cycle, the va-lue of overshoot decreases. Considering Fig. 6, it is observed that byincreasing the values of inductance and capacitance, the settlingtime of step response of output voltage and inductor current

0 0.01 0.02 0.03 0.04 0.05 0.060

5

10

15

20

25

30

][V

v o

Time [sec]

(b)r current and (b) output voltage.

][mFC

][

)R

e(H

zλ

0 0.2 0.4 0.6 0.8 1-5000

-4000

-3000

-2000

-1000

0

9.0

5.0

1.0

===

D

D

D

][HL

0 0.02 0.04 0.06 0.08-800

-600

-400

-200

0

9.0

5.0

1.0

===

D

D

D]

[)

Re(

Hz

λ

(a) (b)Fig. 4. Variation of real part of roots of iL(t) and vo(t) versus; (a) inductance and (b) capacitance.

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

3.0=D

6.0=D

7.0=D

over

shoo

t [%

]

over

shoo

t [%

]

2 4 6 8 10 12 14 160

10

20

30

40

50

3.0=D

8.0=D

6.0=D

]mFC [mHL ][

(a) (b)Fig. 5. Variations of overshoot of iL(t) and vo(t) versus; (a) inductance and (b) capacitance.

0 0.2 0.4 0.6 0.8 1

0.02

0.04

0.06

0.08

[sec

]st

2 4 6 8 10 12 14 160.015

0.02

0.025

0.03

[sec

]st

mFC ][][mHL

(a) (b)Fig. 6. Variation of setting time of iL(t) and vo(t) versus; (a) inductance and (b) capacitance.

0.2 0.4 0.6 0.8 1

25.6

sec]

m[st

D

Fig. 7. Variation of setting time of step response of iL(t) and vo(t) versus duty cycle.


increases. This means that the increment of the values of capaci-tance and inductance slows the transient response of converter.Fig. 6 has been plotted for different values of duty cycle, i.e.D = 0.3, D = 0.6, and D = 0.7. As shown in this figure, it isconcluded that the value of settling time is independent of theduty cycle. Fig. 7 shows the variations of settling time versus dutycycle for L = 8 mH and C = 0.2 mF. As it is observed, fordifferent values of duty cycle, the value of settling time remainsconstant.


3.2. Analysis of steady state response

The analysis of steady state response of buck–boost dc–dc con-verter is performed in CCM which the minimum value of inductorcurrent is more than the load current. Fig. 8a shows the equivalentcircuit of converter for time interval t1 that switch S is on and thediode D is off. Fig. 8b shows the equivalent circuit of converter fortime interval t2 that switch S is off and the diode is on. The wave-forms of inductor voltage, inductor current, capacitor current, andoutput voltage are shown in Fig. 9. In this figure, iL1,ss is the maxi-mum value of inductor current and iL0,ss is the minimum value ofinductor current. Also vo1,ss and vo0,ss show the minimum and max-imum values of output voltage, respectively. In time interval t1, theinductor current increases from its minimum value to its maxi-mum value, linearly. In this case, because the diode is off, thecapacitor current is equal with the load current but with oppositedirection. In this time interval, the capacitor energy discharges toload and as a result, the value of capacitor voltage decreases fromits maximum value to its minimum value. In time interval t2, theinductor provides the energy of capacitor and load. So, by discharg-ing the inductor energy, the value of its current is decreased fromits maximum value to its minimum value.

3.2.1. Calculation of average value of output voltageThe average value of output voltage can be calculated by:

Vo ¼1T

Z t0þT

t0voðtÞdt ð59Þ

where t0 is an arbitrary instant of time.Considering (5) and (59), Eq. (59) can be rewritten as follows:

Vo ¼Z 1

0vo;ssðmÞdm ð60Þ

+

−

ovL

D

C R

oiciLi

S

iV+

−

cVLR

+

−

ovL

D

C R

oiciLi

S

iV+

−

cVLR

(a) (b)Fig. 8. Equivalent circuit of converter in time interval; (a) t1 if iL > Io and (b) t2 ifiL > Io.

ossL Ii −,1

oI− 1t 3tt

ci

Lv

iV

cV−

1t 2t

T

t

Li

oI

oV

oVssov ,0

ssov ,1

t

T2

ssLi ,1

t1t 3t

ossL Ii −,0

ssLi ,0

Fig. 9. Waveforms converter in CCM.

where vo,ss(m) is the value of output voltage in steady state.By calculating the integral in a switching time interval, the aver-

age value of output voltage will be equal with:

Vo ¼vo0;ss

ðaþ cÞT ð1� e�ðaþcÞDTÞ þ iL1;ss

xCða0a1 þxa3Þ

þ vo1;ss a0 a2 �cx

a1

� �� a3ðc� aÞ

h ið61Þ

The values of a0 to a3 have been given in Appendix B.Fig. 10 shows the variations of average value of output voltage

in steady state versus duty cycle. This figure has been plottedfor lossless and losses condition and for Vi = 17 V, R = 40 X,C = 0.25 mF, L = 7 mH, and f = 1 kHz. Considering this figure, it isobserved that by increasing the duty cycle, the average value ofoutput voltage increases. In the loss condition, for higher valuesof duty cycle, by increasing the value of duty cycle the average va-lue of output voltage decreases.

3.2.2. Calculation of output voltage ripple in steady stateOne of the effective parameters in operation of dc–dc converters

is the value of output voltage ripple. Because dc–dc convertershave so much application in dc devices, so they should have min-imum value of output ripple. As a result, it can be concluded thatstudying of output voltage ripple is one of important subjects indc–dc converters. In this section, by using the proposed mathemat-ical model, the output voltage ripple can be calculated.

Considering Fig. 9, the value of output voltage ripple is equalwith:

DVo ¼ vo1;ss � vo0;ss ð62Þ

Considering (26) and (50), the value of vo1,ss is expressed asfollows:

vo1;ss¼e�aTþct1 sin xt2

xC b1þ z�e�aTþct1 cos xt2þ cx sin xt2

� � �b2

1�2e�aT cos uTþe�2aTe�ða�cÞt1

ð63Þ

Applying (50) and (63) in (62), the value of output voltage ripplein steady state is obtained as follows:

DVo;ss ¼e�aTþct1 sin xt2

xC b1 þ z� e�aTþct1 cos xt2 þ cx sin xt2

� � �b2

1� 2e�aT cos uT þ e�2aT

� ðe�ða�cÞt1 � 1Þ ð64Þ

In continuous, the effect of each of converter parameters onoutput voltage ripple is investigated. In a buck–boost dc–dc con-verter, the operational modes can be classified based on parame-ters such as duty cycle, input voltage, and load resistance [22,23].The boundary between CCM and DCM is obtained as follows [22]:

][V

V o

D0 0.2 0.4 0.6 0.8 1

0

20

40

60

80

loss condition

lossless condition

Fig. 10. Variation curves of average value of output voltage in steady state versusthe duty cycle of converter.


LC ¼RT8½4ð1� Dþ rÞ2 � 4rD� ð65Þ

where LC is the critical inductance between CCM and DCM.The buck–boost dc–dc converter operates in DCM for L < LC and

operates in CCM for L > LC [22,23]. In (65), r is equal with:

r ¼ RL

Rð1� DÞ ð66Þ

For theoretical analysis, the converter parameters are consid-ered as follows:

C ¼ 0:15� 2 mF; Vi ¼ 15� 20 V; R ¼ 20� 100 X;

f ¼ 1 kHz; RL ¼ 0:5 X

For theoretical analysis, the following values are selected con-sidering the above ranges:

Vi ¼ 17 V; R ¼ 40 X; D ¼ 0:6; C ¼ 0:25 lF

Considering the selected values by using (65), the value of thecritical inductance between CCM and DCM is obtained LC = 3.2 mH.For operating the converter in CCM, L = 7 mH is considered.

Fig. 11a shows the variations of output voltage ripple versusload resistance and duty cycle for specified values of input voltageand inductance. Considering this figure, it is observed that the va-lue of output voltage ripple has a reverse relation with load resis-tance. As a result, by increasing the value of load resistance, thevalue of output voltage ripple decreases. In addition, it is clear thatby increasing the value of duty cycle, the value of output voltageripple increases. Fig. 11b shows the variations of output voltageripple versus inductance and duty cycle for specified values of in-put voltage and load resistance. Considering this figure, it is ob-served that, the value of inductance does not affect on the valueof output voltage ripple. Also Fig. 11b shows that by increasingthe duty cycle, the value of output voltage ripple increases.

Fig. 11c shows the variations of output voltage ripple versus thecapacitance and duty cycle for specified values of Vi, R, and L. Con-sidering this figure, it is observed that by increasing the value ofcapacitance, the value of output voltage ripple decreases. In addi-tion, by increasing the value of duty cycle, the value of output volt-age ripple increases. It is important to note that for higher values ofcapacitance, the effect of duty cycle on the value of output voltageripple is less.

In modeling the converter, the value of equivalent series resis-tance (ESR) of the output capacitor has been neglected. The valueof ESR has negligible effect on the different parameters of convertersuch as output voltage ripple. This has been proven with more de-tails in [22].

3.2.3. Calculation of inductor current ripple in steady stateOne of the other effective parameters on the operation of dc–dc

converters is the inductor current ripple. Considering Fig. 9, it isobserved that the value of the inductor current ripple in steadystate is obtained by:

00.5

10

5

10

[%]

,

, sso

sso

VVΔ

D

2040608010000.5

10

10

20

[%]

,

, sso

sso

VVΔ

D ][ΩR

(a) (Fig. 11. Variations of output voltage ripple versus; (a) duty cycle and load resi

DIL;ss ¼ iL1;ss � iL0;ss ð67Þ

The value of iL1,ss in (67) is obtained by applying (49) in (23) asfollows:

iL1;ss ¼1� e�aT�ct1 cos xt2 � c




( )

� e�ða�cÞt1 þ Vi


By applying (49) and (68) in (67), the value of inductor currentripple is given by:

DIL;ss ¼1� e�aT�ct1 cos xt2 � c




( )

� e�ða�cÞt1 � 1� �

þ Vi


Fig. 12a shows the variations of inductor current ripple versusthe load resistance and duty cycle for specific values of input volt-age and inductance. Considering this figure, it is clear that byincreasing the value of load resistance, the value of inductor cur-rent ripple increases. Also the value of inductor current ripple in-creases by increasing the value of duty cycle, but this incrementwill continue until a specific value of duty cycle and after it byincreasing the value of duty cycle, the value of inductor current rip-ple decreases. Fig. 12b shows the variations of inductor current rip-ple versus inductor current and duty cycle for specific values ofload resistance, input voltage, and capacitance. Considering thisfigure, it is observed that by increasing the value of inductance,the value of inductor current ripple decreases. Fig. 12c shows thevariations of inductor current ripple versus the values of capaci-tance and duty cycle for specific values of load resistance and inputvoltage. Considering this figure, it is clear that the value of inductorcurrent ripple is independent of the value of capacitance. As shownin Fig. 12c, by increasing the duty cycle, the value of inductor cur-rent ripple increases. This increment will continue until a specificvalue of duty cycle and after a specific value by increasing the dutycycle, the value of inductor current ripple decreases.

4. Calculation of power loss and efficiency of the buck–boostdc–dc converter in CCM

In this section, the power loss and efficiency analysis of thebuck–boost dc–dc converter is presented considering non-idealcomponents for the converter. Fig. 13a and b shows the equivalentcircuit of the converter in on and off-state of the switch, respec-tively. In the figure, RS, RD, RL, and RC are the resistance of switchin its on-state, resistance of the diode, resistance of the inductorand capacitor, respectively. For the loss calculations, ripple of theinductor current is neglected [24].

According to Fig. 13, the switch current for its on and off-statecan be expressed as follows:

00.41.2

200.5

10

10

20

[%]

,

, sso

sso

VVΔ

D ][mFC

0515

25 ][mHL

b) (c)stance; (b) duty cycle and inductance and (c) duty cycle and capacitance.

00.41.2

200.5

10

1

2][

,A

Iss

LΔ

D

0515

2500.5

10

1

2][

,A

Iss

LΔ

D

2040608010000.5

10

1

2][

,A

Iss

LΔ

D

(a) (b) (c)][mFC][mHL][ΩR

Fig. 12. Variations of inductor current ripple versus; (a) duty cycle and load resistance; (b) duty cycle and inductance; and (c) duty cycle and capacitance.

+

−

ov

L CR

oiciLi

iVLR CRDR DVS

Di

Li +

−

ov

L

D

CR

oici

Si

iVLR CRSR SV

(a) (b)Fig. 13. Buck–boost dc–dc converter equivalent circuit in; (a) on-state of the switch and (b) off-state of the switch.


iS ¼IL for 0 6 t < DT

0 for DT 6 t < 1

�ð70Þ

In (70), IL is the average value of the inductor current which isobtained by substituting (49) and (50) in (27) and averaging (27).

The loss of the power switch can be written as follows:

PS ¼ RSi2S þ VSiS ð71Þ

Substituting (70) in (71), the power loss of the switch is ob-tained as:

PS ¼ RSI2L þ VSIL ð72Þ

Considering Fig. 13, the diode current can be writes as:

iD ¼0 for 0 6 t < DT

IL for DT 6 t < 1

�ð73Þ

The power loss of the diode is as follows:

PD ¼ RDi2D þ VDiD ð74Þ

Using (73) and (74), the power loss of the diode is obtained asthe following equation:

PD ¼ RDI2L þ VDIL ð75Þ

The power loss of the inductor resistance can be written asfollows:

PRL ¼ RLI2L ð76Þ

The capacitor current can be written as:

iC ¼�Io for 0 6 t < DT

IL � Io for DT 6 t < 1

�ð77Þ

In (77), Io is the average output current which is obtained fromthe following equation:

Io ¼Vo

Rð78Þ

where Vo is the average output voltage which can be achieved using(61).

The power loss of the equivalent resistance of the capacitor issum of the corresponding losses in the both on and off-state ofthe switch. This power loss can be written as follows:

PRC ¼ RCI2o þ RCðIL � IoÞ2 ð79Þ

The total power loss of the buck–boost dc–dc converter in CCMcan be expressed as:

PLoss ¼ PS þ PRD þ PRC þ PRL ð80Þ

As the power loss is calculated, the efficiency of the convertercan be obtained using the following equation:

g ¼ Po

Po þ PLossð81Þ

where Po is the output power of the converter.Fig. 14 shows the variation of the efficiency of the converter

versus the duty cycle for different value of the converter compo-nents. The curves have been plotted for Vi = 17 V, L = 7 mH,C = 0.25 mF, RL = 0.1 X, RC = 6 mX, RD = 20 mX, RS = 0.11 X,VD = 0.7 V, VS = 1 V and f = 1 kHz.

Fig. 14a shows the variation of the converters efficiency versusduty cycle for different values of the load resistance. The figureindicates that the efficiency is minimum for the lowest and high-est values of the duty cycle. Also, for a specific value of duty cycle,the efficiency of the converter improves as the load resistance in-creases. The variation of the converter efficiency versus dutycycle for different values of the inductor resistance is indicatedin Fig. 14b. As the figure shows, the converter efficiency decreasesas the inductor resistance increases. Moreover, the effect of theinductor resistance on the efficiency is not considerable for lowervalues of the duty cycle. Fig. 14c shows the variations of theconverter efficiency versus the duty cycle for various values ofthe capacitor equivalent resistance. For a specific value of RC,the value of efficiency decreases for lowest and highestvalues of the duty cycle. Also, for a specific value of duty cyclethe value of efficiency is inversely related with the equivalentresistance of the capacitor. Fig. 14d and e shows the variationsof the efficiency versus duty cycle for different values of theswitch resistance and diode resistance, respectively. As the fig-ures indicate, for a specific value of duty cycle, the converter effi-ciency decreases as the value of the switch and diode resistancesincrease.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Ω=Ω=Ω=

2.0

09.0

10

D

D

D

R

R

mR

D

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Ω=Ω=

Ω=

2.0

11.0

7

S

S

S

R

R

mR

D

][p

uη

][p

uη

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Ω=Ω=Ω=

2.0

1.0

5

C

C

C

R

R

mR

D

0 0.2 0.4 0.6 0.8 1

][2.0

][1.0

][05.0

Ω=Ω=

Ω=

L

L

L

R

R

R

][p

uη

][p

uη

][p

uη

D0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Ω=Ω=Ω=

100

60

15

R

R

R

D

(a) (b)

(c) (d)

(e)Fig. 14. Variations of the converter efficiency versus duty cycle for different values of (a) load resistance, (b) equivalent resistance of the inductor, (c) equivalent resistance ofthe capacitor, (d) power switch resistance, and (e) diode resistance.

0 0.01 0.02 0.03 0.040

10

20

30

][V

v o

Time [sec]

Average Model

Proposed Model

][A

i L

Time [sec]0 0.01 0.02 0.03 0.04

0

1

2

3

4

5

Average Model

Proposed Model

(a) (b)Fig. 15. Step response of; (a) inductor current and (b) output voltage.



5. Comparison of the proposed method with conventionalmethod

In this section, the proposed method is compared with one ofthe common modeling methods. One of the modeling methods of

021.0

22.0

23.0

24.0

25.0

21.5

23.5

22.5

24.5

0.001 0.002 0.003 0.004

0.0

1.0

2.0

3.0

Time [sec]

[ ]

LiA

[ ]

ovV

021.0

22.0

23.0

24.0

25.0

21.5

23.5

22.5

24.5

0.001 0.002 0.003 0.004

0.0

1.0

2.0

3.0

Time [sec]

[ ]

LiA

[ ]

ovV

Time [sec]0

21.0

22.0

23.0

24.0

25.0

21.5

23.5

22.5

24.5

0.001 0.002 0.003 0.004

0.0

1.0

2.0

3.0

[ ]

LiA

[ ]

ovV

(

(

Fig. 16. Waveforms of inductor current and output voltage; left column: experimental rL = 6 mH; (d) L = 15 mH; (e) C = 0.4 mF; and (f) C = 1.2 mF.

dc/dc converters is state space average method [17]. In this meth-od, the state variables of dc–dc converter are expanded to Fourierseries. For linearizing the equations, a switching function is used.The switching function is defined as a combination of cosine andsinusoidal functions which are obtained from Fourier expansion.

0.1000 0.1010 0.1020 0.1030 0.1040

0.0

1.0

2.0

3.0 IL[A]

21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

Vo[V]

Time [sec]

0.1000 0.1010 0.1020 0.1030 0.1040

0.0

1.0

2.0

3.0 IL[A]

21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

Vo[V]

Time [sec]

0.1000 0.1010 0.1020 0.1030 0.1040

0.0

1.0

2.0

3.0 IL[A]

21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

Vo[V]

Time [sec]a)

b)

(c)esults; right column: simulation results using PSCAD; (a) R = 30 X; (b) R = 70 X; (c)

021.0

22.0

23.0

24.0

25.0

21.5

23.5

22.5

24.5

0.001 0.002 0.003 0.004

0.0

1.0

2.0

3.0

Time [sec]

021.0

22.0

23.0

24.0

25.0

21.5

23.5

22.5

24.5

0.001 0.002 0.003 0.004

0.0

1.0

2.0

3.0

Time [sec]

0.1000 0.1010 0.1020 0.1030 0.1040

0.0

1.0

2.0

3.0 IL[A]

21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

Vo[V]

Time [sec]

0.1000 0.1010 0.1020 0.1030 0.1040

0.0

1.0

2.0

3.0 IL[A]

21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

Vo[V]

Time [sec]

0.1000 0.1010 0.1020 0.1030 0.1040

0.0

1.0

2.0

3.0 IL[A]

21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

Vo[V]

Time [sec]0

21.0

22.0

23.0

24.0

25.0

21.5

23.5

22.5

24.5

0.001 0.002 0.003 0.004

0.0

1.0

2.0

3.0

Time [sec]

[]

LiA

[]

LiA

[]

ovV

[]

LiA

[]

ovV

[]

ovV

(d)

(e)

(f)Fig. 16. (continued)


In this method, for modeling the converter, two approximationsare used. In the first order approximation, the state variables andswitching function contain three terms of Fourier expansion andother terms are ignored. By applying the related Fourier relations

to the state variables and switching function of converter in (1)and (2), the differential equations of order 6 are obtained that com-plicated mathematical methods such as numerical solution meth-ods should be used to solve them, In the proposed method in


this paper, the obtained differential equations for the output volt-age and inductor current are order two which solving these equa-tions is easier by using the Laplace transform.

One of the other advantages of the proposed method in compar-ison with the presented method in [17] is the ability to analyze thesteady state parameters of converter such as output voltage andinductor current ripples. Although the proposed method has com-plex equations, it has the ability to analyze the converter in eachtwo transient and steady states.

Fig. 15 shows the step responses of output voltage and inductorcurrent of converter. These curves are plotted for Vi = 17 V,L = 8 mH, C = 0.2 mF, R = 20 X, RL = 0.5 X, and f = 1 kHz. In this fig-ure, the step response of proposed method is compared with theobtained results of presented method in [17]. Considering this fig-ure, it is observe that by the proposed method have better accuracythan the average state space method.

6. Experimental and simulation results

To prove the validity of presented theoretical subjects, theexperimental and simulation results are obtained for Vi = 17 V,D = 0.6, L = 7 mH, C = 0.25 mF, R = 40 X, RL = 0.5 X, and f = 1 kHz.The transistor (the switch S) and the diode D used in the prototypeare MJ13005 and BUR460, respectively. In order to generate thegate commands of the power switch, the 89C52 type microcontrol-ler by ATMEL Company has been used. The dc supply existing inthe laboratory has been used as the dc voltage source. TektronixTDS 2024B four channel digital storage oscilloscope has been usedfor measurements in laboratory.

1

1

2

2

3

0.000 0.010 0.020 0.030 0.040 0.050 0.060

0.0

1.0

2.0

3.0

4.0

5.0 IL[A]

Time [sec]

(a)Fig. 17. Step response based on simulation; (a

0 0.01 0.02 0.03 0.04 0.05 0.060

1

2

3

4

5

][A

i L

Time [sec]

alExperiment

lTheoretica

(a)Fig. 18. Experimental and theoretical results of step res

Fig. 16a and b shows the simulation and experimental results ofinductor current and output voltage for load resistance R = 30 Xand R = 70 X in the steady state, respectively. Considering the fig-ures, it is observed that by increasing the value of load resistance,the value of inductor current ripple increases and the value of out-put voltage ripple decreases. According to Fig. 16a, the average va-lue of output voltage is equal with 22.867 V By applying the abovevalues in (61), the value of Vo is equal with 22.983 V. Also inFig. 16a, the initial values of output voltage and inductor currentin each switching interval are equal with:

iL1;ss ¼ 2:593 A; iL0;ss ¼ 1:219 A; vo1;ss ¼ 21:953 V; vo0;ss ¼ 23:781 V

Considering (49) and (50), the initial values of inductor currentand output voltage in steady state are equal with 1.222 A and23.830 V, respectively. Also, considering (23) and (26), the initialvalues of inductor current and output voltage during off time inter-val of switch are equal with 2.597 A and 21.998 V, respectively. Asit is observed, the results of the theoretical analysis prove thevalidity of the simulation and experimental results and the valueof error is about 0.5%.

For L = 6 mH and L = 15 mH, the waveforms of inductor currentand output voltage are shown in Fig. 16c and d, respectively.According to the figure, it is observed that by increasing the valueof inductance, the value of the output voltage ripple does not varybut the value of inductor current ripple decreases by increasing thevalue of inductance. In Fig. 16c, the values of output voltage andinductor current ripples are 1.404 V and 1.625 A, respectively. Con-sidering (64) and (69), the values of output voltage and inductorcurrent ripples are equal with:

0.000 0.010 0.020 0.030 0.040 0.050 0.060

0.0

5.0

0.0

5.0

0.0

5.0

0.0 Vo[V]

Time [sec]

(b)) inductor current and (b) output voltage.

0 0.01 0.02 0.03 0.04 0.05 0.060

10

20

30

][V

v o

Time [sec]

alExperiment

lTheoretica

(b)ponse; (a) inductor current and (b) output voltage.


DVo ¼ 1:406V ; DIL ¼ 1:626A

As it is observed, the experimental and simulation results provethe validity of results of theoretical analysis. Fig. 16e and f showsthe waveforms of inductor current and output voltage forC = 0.4 mF and C = 1.2 mF, respectively. According to figure, it is ob-served that, by increasing the capacitance, the value of output volt-age ripple decreases, but the value of inductor current ripple doesnot vary.

Fig. 17a and b shows the step response of inductor current andoutput voltage based on simulation. In order to simulation, the se-lected values for plotting Fig. 3 have been used. Fig. 18 shows thestep response of output voltage and inductor current in two exper-imental and theoretical cases. In this case, the selected values forplotting Fig. 3 are used. As shown in Figs. 3, 17, and 18, there isgood agreement between simulation, theoretical, and experimen-tal results.

7. Conclusion

In this paper, a mathematical model was proposed for modelingthe buck–boost dc–dc converter in CCM. The proposed method isbased on Laplace and Z transforms. The Laplace transform is usedto determine the equations of inductor current and output voltage.The Z-transform is applied as a tool to obtain the initial values ofinductor current and output voltage. Also it was shown, the Z-transform can be used for analyzing the responses of transientand steady states. By using the proposed method, the effect of con-verter elements was investigated on time response of converter intransient and steady states. It was shown that the buck–boost dc–dc converter in CCM has slow transient response for higher valuesof inductance and capacitance. In addition, the effect of converterelements on output voltage and inductor current ripples wasinvestigated. It was shown that the value of output voltage rippleis independent of the value of inductance, but by increasing thevalues of load resistance and capacitance, the value of output volt-age ripple decreases. In addition the inductor current ripple has adirect relation with the load resistance. In CCM, the inductor cur-rent ripple is independent of the value of capacitance and has anindirect relation with the value of inductance. The proposed mod-eling method can be used as a tool for analyzing the operation ofconverter by high variations of parameters. The theoretical resultswere proved by experimental and simulation results.

Appendix A

Considering (3)–(5), the functions f1(t) and f2(t) can be ex-pressed as follows, respectively:

f1ðtÞ ¼X1n¼0

½uðnT þmT � nTÞ � uðnT þmT � t1 � nTÞ�

¼X1n¼0

½uðmTÞ � uðmT � t1Þ� ¼ nf1ðmÞ ðA1Þ

f2ðtÞ ¼X1n¼0

½uðnT þmT � t1 � nTÞ � uðnT þmT � T � nTÞ�

¼X1n¼0

½uðmT � t1Þ � uðmT � TÞ� ¼ nf2ðmÞ ðA2Þ

In (A1) and (A2), the values of f1(m) and f2(m) are equal with:

f1ðmÞ ¼ uðmTÞ � uðmT � t1Þ ðA3Þf2ðmÞ ¼ uðmT � t1Þ � uðmT � TÞ ðA4Þ

The values of the functions u(mT), u(mT � t1), and u(mT � T) areexpressed as follows, respectively:

uðmTÞ ¼1 mT P 0 for m P 00 mT < 0 for m < 0

�ðA5Þ

uðmT � t1Þ ¼1 mT � t1 P 0 for m P t1

T

0 mT � t1 < 0 for m < t1T

(ðA6Þ

uðmT � TÞ ¼1 mT � T P 0 for m P 10 mT � T < 0 for m < 1

�ðA7Þ

In (A6), t1T is the duty cycle of converter (D) in CCM and it is de-

fined as follows:

D ¼ t1

TðA8Þ

By applying (A5)–(A7) in (A3) and (A4), we have:

f1ðmÞ ¼1 0 6 m < D

0 D 6 m < 1

�ðA9Þ

f2ðmÞ ¼0 0 6 m < D

1 D 6 m < 1

�ðA10Þ

Appendix B

X ¼z� e�aTþct1 cos xt2 þ c

x sin xt2� � e�aT�ct1 sin xt2

xL

� e�aTþct1 sin xt2xC z� e�aT�ct1 cos xt2 � c

x sin xt2� �

" #

ðB1Þ

b1 ¼Vi

RLðe�at2 � ect1�aTÞ cos xt2 þ

cx

sin xt2

� �h iðB2Þ

b2 ¼Vi

RL

1xCðe�at2 � ect1�aTÞ sin xt2

� �ðB3Þ

a0 ¼e�at2

ða2 þx2ÞT2 ðB4Þ

a1 ¼ �aT sin xt2 �xT cos xt2 ðB5Þa2 ¼ �aT cos xt2 þxT sin xt2 ðB6Þ

a3 ¼1

ða2 þx2ÞT ðB7Þ

References

[1] Algazar M, AL-monier H, Abd EL-halim H, El Kotb Salem ME. Maximum powerpoint tracking using fuzzy logic control. Int J Electr Power Energy Syst2012;39:21–8.

[2] Mazouz N, Midoun A. Control of a DC/DC converter by fuzzy controller for asolar pumping system. Int J Electr Power Energy Syst 2011;33:1623–30.

[3] Alonso J, Vina J, Vaquero DG, Martı́nez G, Osorio R. Analysis and design of theintegrated double buck–boost converter as a high-power-factor driver forpower-led lamps. IEEE Trans Indust Electron 2012;59:1689–97.

[4] Ramasamy M, Thangavel S. Photovoltaic based dynamic voltage restorer withpower saver capability using PI controller. Int J Electr Power Energy Syst2012;39:51–9.

[5] Ben-Yaakov S, Adar D. Average models as tools for studying the dynamics ofswitch mode dc–dc converters. In: Proceeding power electronics specialistsconference (PESC), 20–25 June, 1994. p. 1369–76.

[6] Maksimovic D, Zane R. Small-signal discrete-time modeling of digitallycontrolled PWM converters. IEEE Trans Power Electron 2007;22:2552–6.

[7] Wu T, Chen Y. A systematic and unified approach to modeling PWM dc–dcconverters based on the graft scheme. IEEE Trans Indust Electron1998;45:88–98.

[8] Hongbo M, Quanyuan F. Hybrid modeling and control for buck–boostswitching converters. In: Proceeding international conference oncommunications, circuits and systems (ICCCAS), 23–25 July, 2009. p. 678–82.

[9] Hajizadeh A, Aliakbar Golkar MA. Control of hybrid fuel cell/energy storagedistributed generation system against voltage sag. Int J Electr Power EnergySyst 2010;32:488–97.


[10] Yildiz HA, Goren-Sumer L. Lagrangian modeling of dc–dc buck–boost and flyback converters. In: Proceeding 20th European conference on circuit theoryand design (ECCTD), 23–27 August, 2009. p. 245–8.

[11] Gong RX, Xie LL, Wang K, Ning CD. A novel modeling method of non-idealbuck–boost converter in DCM. In: Proceeding international conference oninformation and computing (ICIC), 4–6 June, 2010. p. 182–5.

[12] Gatto G, Marongiu I, Perfetto A, Serpi A. Modeling and predictive control of abuck–boost dc–dc converter. In: Proceeding international symposium onpower electronics, electrical drives, automation and motion (SPEEDAM), 14–16 June, 2010. p. 1430–5.

[13] Luo FL, Ye H. Small signal analysis of energy factor and mathematical modelingfor power dc–dc converters. IEEE Trans Power Electron 2007;22:69–79.

[14] Wang Q, Shi L, Chang C. Small-signal transfer functions for a single-switchbuck–boost converter in continuous conduction mode. In: Proceeding 10thIEEE international conference on solid-state and integrated circuit technology(ICSICT), 20–23 October, 2008. p. 2016–9.

[15] Czarkowski D, Kazimierczuk MK. Energy-conservation approach to modelingPWM dc–dc converters. IEEE Trans Aerospace Electron Syst 1993;29:1059–63.

[16] Rim CT, Joung B, Cho GH. Practical switch based state-space modeling of dc–dcconverters with all parasitics. IEEE Trans Power Electron 1991;6:611–7.

[17] Davoudi A, Jatskevich J, De Rybel T. Numerical state space average valuemodeling of PWM dc–dc converters operating in DCM and CCM. IEEE TransPower Electron 2006;21:1003–12.

[18] Cuk S, Middlebrook RD. A general unified approach to modeling switchingconverter power stage. In: Proceeding PESC; 1976, p. 18–34.

[19] Tricoli P. Analytical closed-form solution for transient analysis of boost dc–dcconverters. COMPEL: Int J Comput Math Electr Electron Eng 2011;30:706–25.

[20] Babaei E, Mashinchi Mahery H. Mathematical modeling and analysis oftransient and steady states of buck dc–dc converter in DCM. COMPEL: Int JComput Math Electr Electron Eng; in press.

[21] Biolek D, Biolkova V, Dobes J. Modeling of switched dc–dc converters by mixeds–z description. In: Proceeding IEEE ISCAS; 2006. p. 831–4.

[22] Babaei E, Mahmoodieh MES, Mashinchi Mahery H. Operational modes andoutput voltage ripple analysis and design considerations of buck–boost dc–dcconverters. IEEE Trans Indust Electron 2012;59:381–91.

[23] Babaei E, Mahmoodieh MES, Sabahi M. Investigating buck dc–dc converteroperation in different operational modes and obtaining the minimum outputvoltage ripple considering filter size. J Power Electron 2011;11:793–800.

[24] Kazimierczuk MK. Pulse-width modulated dc–dc power converters. 1sted. UK: Wiley; 2008.

Documents

Mathematical modeling of buck–boost dc–dc converter and investigation of converter elements on transient and steady state responses