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

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Mathematical modeling of buck–boost dc–dc converter and investigation of 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 2011 Received in revised form 15 August 2012 Accepted 19 August 2012 Available online 2 October 2012 Keywords: Buck–boost dc–dc converter Z-transform Laplace transform CCM Modeling

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 in continuous conduction mode (CCM). In proposed method, using the Laplace transform the relations of inductor current and output voltage are obtained. In the next step, in each switching interval using the Z-transform the initial values of inductor current and output voltage are calculated. Then, the transient and steady states responses of these quantities are calculated. In addition, the effect of the values of converter 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 theoretical analysis. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Mathematical modeling of dc–dc converters is one of the basic subjects in analysis of their operation. In order to have a suitable efficiency and desired operation, the values of converter components should be designed properly. Modeling is one of the main steps in design and control of a system. Nowadays, by development in power electronic fields, there is a good attention to dc–dc converters [1–4]. In order to achieve a proper design and control, it is necessary to have an exact model of converter. By modeling the dc–dc converters, the operation of converter in different operational modes can be investigated in both transient and steady states. High accuracy and low response time are major features of a good modeling [5–12]. The significant point in mathematical modeling of dc–dc converters is the existence of power switch in the structure of these converters. This causes nonlinearity model for converters. As a result, it is needed to solve nonlinear equations. Different mathematical 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 presented in literature. One of the disadvantages of them is to use numerical solution or simplification in extracting the models. Because of approximation in these models, the results of them are not enough accurate. The other considerable point in analyzing ⇑ Corresponding author. Tel./fax: +98 411 3300829. E-mail addresses: [email protected] (H. Mashinchi Mahery), [email protected] (E. Babaei). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.08.035

of each system is the response time of system. In conventional methods, it is usually needed to use mathematical operators such as matrix inversion or to solve the complicated algebraic equations. As a result, in systems with high rank matrix, the response time is increased. In [19,20], a new method has been presented for modeling the boost and buck dc–dc converters in CCM and discontinuous conduction mode (DCM) operations, respectively. In this paper, it is aimed to improve the analysis accomplished in [19,20] on a buck–boost converter in CCM operation. It should be noted that the proposed method also is applicable for DCM operation. Considering which the operational conditions are different in each of operational modes, so the results for each will not be the same. For this reason and also for avoiding an overlong paper, in this paper, the results for CCM operation of buck–boost dc–dc converter are presented. The proposed method is based on the Laplace and Z transforms. The Laplace transform is used to obtain the inductor current and output voltage equations, and Z-transform is used to determine the initial values of them. It is important to mention that if Laplace transform is used to determine the initial values, the transfer function of inductor current and output voltage consist of s and esT. In this case, the time function which is obtained using by inverse of Laplace transfer will be an infinite series. For solving this problem, the Z-transform has been used. The final value theorem of Z-transform is used for analyzing the converter in steady state. One of main disadvantage of small signal and state space average value models is high variation of the system parameters around the dc parameter. In this case by neglecting the dc

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

component, the results will not have enough accuracy [13,16]. The proposed method in this paper can be applied to analyse the dc–dc converters by high variations of parameters. In [19,21], the presented methods are based on Laplace and Z transforms. In these methods, only the transient response of converter has been investigated. Using the proposed method of this paper, both of the transient and steady state responses of converter can be analyzed. In addition, the electrical parameters of converter such as output voltage and inductor current ripples can be investigated. Also, the proposed method in this paper can be used to analyze dc–dc converters with large variations of parameters. In this paper, the effect of converter components on transient and steady states responses is investigated, too. Finally, the validity of presented theoretical subjects is proved by experimental and simulation results in PSCADnEMTDC software. 2. The proposed mathematical model The equivalent circuit of buck–boost dc–dc converter is shown in Fig. 1. In this figure, the diode D and switch S are considered ideal 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 continuous and existence in all time intervals. Considering Fig. 1 and applying Kirchhoff current and voltage laws, we have:

diL L þ RL iL ¼ f1 ðtÞV i  f2 ðtÞv o dt dv o v o þ ¼ f2 ðtÞiL C dt R

f2 ðtÞ ¼

t ¼ ðn þ mÞT

for n ¼ 0; 1; 2; . . .



f1 ðmÞ ¼

f2 ðmÞ ¼

1 06m
D6m<1

0

06m
uðt  t 1  nTÞ  uðt  T  nTÞ

ð4Þ

n¼0

" di

L;n

#

dm dv o;n dm

" di

L;n

dm dv o;n dm

" ¼

#

" ¼

RL

Vc

L

+

− v C R o +

0

0

T  RC

 þ

v o;n

T L Vi 0

for 0 6 m < D

ic io

#

 TL

iL;n



v o;n v o;n ðDÞ ¼ v o1;n T  RC

þ

  0 Vi 0

for D 6 m < 1

f2 ( t )

f1 ( t ) T

T

t2

t2 1

1 t t1

t1 + T

ð8Þ

v o;n ð0Þ ¼ v o0;n

 RLL T T C

iL;n

ð9Þ

where (iL0,n, vo0,n) and (iL1,n, vo1,n) are the initial values of inductor current and output voltage in intervals [0, D]and [D, 1], respectively. As it is observed, the initial values of output voltage and inductor current are functions of n, so the value of each of these parameters for each switching time interval will be different.

Fig. 1. Buck–boost dc–dc converter.

0

#

 RLL T

iL;n ðDÞ ¼ iL1;n ; D −

ð7Þ

1 D6m<1

iL;n ð0Þ ¼ iL0;n ¼ 0;

iL

ð6Þ

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 two intervals [0, D] and [D, 1]. In interval [0, D], the switch S is on and in the interval [D, 1], the switch S is off. Considering (1), (2), (5), (6), (7), we have:

ð3Þ

n¼0 1 X

Vi

ð5Þ

where m is an unit time variable which its value is same for all switching intervals and always varies between zero and one. By applying (5) in (3) and (4), the functions f1(m) and f2 (m) can be expressed as follows, respectively:

 ð2Þ

1 X uðt  nTÞ  uðt  t 1  nTÞ

S

06m<1

ð1Þ

Eqs. (1) and (2) are the general equations of the output voltage and inductor current for buck–boost dc–dc converter. By applying the values of functions f1(t) and f2(t) in (1) and (2), the equations of output voltage and inductor current into each of switching intervals can be obtained. The functions f1(t) and f2(t) are defined to determine the converter equations during on (t1) and off (t2) states of switch S as follows:

f1 ðtÞ ¼

where n denotes the number of switching time intervals and T is the switching period. Fig. 2 shows the waveforms of functions f1(t) and f2(t). It is observed that for time interval (0, t1), the values of functions f1(t) and f2(t) are equal with 1 and 0, respectively. By applying these values in (1) and (2), the equations corresponding to time interval that the switch S is on are obtained. Also for time interval (t1, T), the values of functions f1(t) and f2 (t) are 0 and 1, respectively. By applying these values in (1) and (2), the equations of output voltage and inductor current corresponding to time interval that the switch S is 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 intervals. So, for analyzing the converter in a specified time, the values of functions f1(t) and f2(t) should be determined. With these conditions, the analysis of converter will be difficult. For independent analysis of converter in each switching time interval based on Ztransform, the following variable exchange can be used:

t1 + nT

0

t t1

(a)

t1 + T

(b) Fig. 2. Waveforms of function; (a) f1(t) and (b) f2(t).

t1 + nT

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

2.2. Solving output voltage and inductor current equations using Laplace transform

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

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

iL1;n ¼ iL0;n eðacÞt1 þ

  RL T T IL;n ðsÞ ¼ iL0;n þ V i for 0 6 m < D sþ L Ls   T V o;n ðsÞ ¼ v o0;n for 0 6 m < D sþ RC   TRL T IL;n ðsÞ ¼ iL1;n  V o;n ðsÞ for D 6 m < 1 sþ L L   T T V o;n ðsÞ ¼ v o1;n þ IL;n ðsÞ for D 6 m < 1 sþ RC C

t!ðt 1 þnTÞ

ð10Þ ð11Þ

IL;n ðsÞ V o;n ðsÞ





2 ¼4

1 sþðacÞT

 iL0;n þ TL v 0;n

Vi s

The following is valid about the output voltage:

lim

v o ðtÞ ¼

lim

t!ðt 1 þnTÞþ

v o ðtÞ

ð24Þ

Eq. (24) can be rewritten as follows:

lim

m!ðDÞ

v o;n ðmÞ ¼

lim

m!ðDÞþ

v o;n ðmÞ

Applying (20) in (25), the value of

ð25Þ

vo1,n will be equal with:

v o1;n ¼ v o0;n e

ð13Þ

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

3 5 for 0 6 m < D

ð23Þ

ð12Þ

By arranging (10)–(13) in matrix form, the equations of inductor current and output voltage in Laplace domain will be as follows:



Vi ½1  eðacÞt1  RL

ð14Þ

ðacÞt1

ð26Þ

8 iL0;n eðacÞmT þ RVLi ½1  eðacÞmT  for 0 6 m < D > > >  i > h > < iL0;n  RVLi ect1 amT þ RVLi eaðmTt1 Þ iL;n ðmÞ ¼  >

> > cos xðmT  t 1 Þ þ xc sin xðmT  t1 Þ > > :  x1L v o0;n ect1 amT sin xðmT  t1 Þ for D 6 m < 1

sþðaþcÞT

ð27Þ "

 IL;n ðsÞ  TL s þ Tða þ cÞ 1  ¼ 2 T 2 2 2 V o;n ðsÞ s þ Tð a  cÞ s þ 2aTs þ T ða þ x Þ C

#



ð15Þ

8 v o0;n eðaþcÞmT for 0 6 m < D > > h  i > > < 1 iL0;n  RVLi  ect1 amT þ RVLi eaðmTt1 Þ sin xðmT  t1 Þ xC v o;n ðmÞ ¼ >

 > þv o0;n ect1 amT cos xðmT  t 1 Þ  xc sin xðmT  t 1 Þ > > : for D 6 m < 1

ð16Þ

ð28Þ

iL1;n

v o1;n

for D 6 m < 1

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

  1 RL 1 þ 2 L RC sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   1 RL 2 x¼ þ1 a LC R   1 1 RL  c¼ 2 RC L

a¼

ð17Þ

ð18Þ

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

8 ðacÞmT > þ RVLi ½1  eðacÞmT  for 0 6 m < D > < iL0;n e

 iL;n ðmÞ ¼ eaðmTt1 Þ iL1;n cos xðmT  t1 Þ þ xc sin xðmT  t 1 Þ >

> v :  xo1;n sin xðmT  t1 Þ for D 6 m < 1 L

2.3. Determining the initial values of inductor current and output voltage 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 discontinuous variable, so to determine the values of these parameters, Z-transform can be used. Considering the continuous characteristic of inductor current and output voltage, in t = nT the following relation is always valid:

lim iL ðtÞ ¼ lim þ iL ðtÞ

t!ðnTÞ

t!ðnTÞ

ð29Þ

Eq. (29) can be rewritten as follows:

ð19Þ 8 v o0;n eðaþcÞmT for 0 6 m < D > > n < v o;n ðmÞ ¼ > eaðmTt1 Þ ixL1;nC sin xðmT  t1 Þ >

 : þv o1;n cos xðmT  t 1 Þ  xc sin xðmT  t 1 Þ

lim iL;n ðmÞ ¼ limþ iL;nþ1 ðmÞ

m!1

for D 6 m < 1

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 parameters should be determined. The values of iL1,n and vo1,n can be expressed versus iL0,n and vo0,n. Considering the continuity characteristic of iL(t) and vo(t), the followings are valid:

lim

iL ðtÞ ¼

lim

t!ðt 1 þnTÞþ

iL ðtÞ

ð21Þ

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

lim iL;n ðmÞ ¼ lim þ iL;n ðmÞ

m!ðDÞ

m!ðDÞ

ð30Þ

The following is valid about the output voltage:

ð20Þ

t!ðt 1 þnTÞþ

m!0

lim

t!ðnTÞ

v o ðtÞ ¼

v o ðtÞ

ð31Þ

The above equation can be rewritten as follows:

lim v o;n ðmÞ ¼ limþ v o;nþ1 ðmÞ

m!1

m!0

ð32Þ

Considering t2 = T  t1, for the initial values of inductor current and output voltage in time interval n + 1 the following equations are obtained:

iL0;nþ1 ¼

    V i ct1 aT V i at2  c e cos xt2 þ sin xt 2 iL0;n  þ e x RL RL 

ð22Þ

lim

t!ðnTÞþ

v o0;n xL

ect1 aT sin xt 2 ð33Þ

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



  1 V i ct1 aT V i at2 e sin xt2 ¼ þ e iL0;n  xC RL RL   c þ v o0;n ect1 aT cos xt2  sin xt 2

v o0;nþ1

x

ð34Þ

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

ZfiL0;n g ¼ IL0 ðzÞ

ð35Þ

Zfv 0;n g ¼ V 0 ðzÞ

ð36Þ

In addition, we have: Z

x½n þ 1 $ zXðzÞ  zxð0Þ

ð37Þ

Considering (37), the following equations are valid:

ZfiL0;nþ1 g ¼ zIL0 ðzÞ  ziL0;0

ð38Þ

Zfv 0;nþ1 g ¼ zV 0 ðzÞ  zv 0;0

ð39Þ

where iL0,0 and vo0,0 are the initial values of inductor current and output 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 

IL0 ðzÞ V 0 ðzÞ

"

 ¼

z b z1 1 z b z1 2

þ ziL0;0

#

þ zv 0;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:



" #  z zIL0;0 þ b1 z1 IL0 ðzÞ ¼ X 1  z zV o0;0 þ b2 z1 V o0 ðzÞ

ð41Þ

The value of X1 in (41) is equal with: X 1 ¼

" #   aTct1 sin xt2 e z  eaTct1 cos xt2  xc sin xt2 1 xL    aTþct1 e sin xt2 jXj z  eaTþct1 cos xt2 þ xc sin xt2 xC ð42Þ

Eqs. (45) and (46) show the initial values of inductor current and output voltage in Z domain. To obtain their initial values in steady 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 expressed as follows:

iL0;ss ¼ ‘imðz  1ÞIL0 ðzÞ

ð47Þ

v o0;ss ¼ ‘im ðz  1ÞV o0 ðzÞ Z!1

ð48Þ

Z!1

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 and output voltage in steady state can be calculated as follows:

  aTct1 sin xt 2 1  eaTct1 cos xt2  xc sin xt 2 b1  e b2 xL ð49Þ 1  2eaT cos uT þ e2aT

   aTþct 1 e sin xt 2 b1 þ 1  eaTþct1 cos xt 2 þ xc sin xt 2 b2 xC ¼ ð50Þ 1  2eaT cos uT þ e2aT

iL0;ss ¼

v o0;ss

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

IL0 ðzÞ ¼

z½z  eaT cosðuTÞ þ ke iL0;0 z2  2zeaT cosðuTÞ þ e2aT zeaTct1 sin xt2 v o0;0 ziL0;0 þ  2zeaT cosðuTÞ þ e2aT xL z1

 z iL0ss z  2iL0ss eaT cosðuTÞ þ iL0ss  b1  z2  2zeaT cosðuTÞ þ e2aT



V o0 ðzÞ ¼

z2

z2

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

jXj ¼ z2  2zeaT cos uT þ e2aT

ð44Þ

Applying X1 in (41), the values of IL0(z) and Vo0 (z) can be calculated by:

z z2 2zeaT cos uT þe2aT h  i c  zeaTct1 cos xt 2  sin xt2 iL0;0 x  i v o0;0 eaTct1 sin xt2 h aTct1  c  cos xt2  sin xt 2 b1 þ z e x xL   eaTct1 sin xt2 1  ð45Þ b2 z 1 xL

IL0 ðzÞ ¼

 iL0;0 eaTþct1 sin xt 2 u xC h  i c aTþct 1 cos xt 2 þ sin xt2 v o0;0 þ ze x  aTþct h  i  1 sin xt e c 2 þ b1 þ zeaTþct1 cos xt 2 þ sin xt 2 b2 x xC  1 ð46Þ  z1

V o0 ðzÞ ¼

z

z2 2zeaT cos

T þe2aT

zeaTct1 sin xt2 iL0;0  2zeaT cosðuTÞ þ e2aT xC aT

ð43Þ

which

c cos uT ¼ cos xt2 cosh ct 1 þ sin xt 2 sinh ct1 x

ð51Þ

þ

z½z  eaT cosðuTÞ  ke v o0;0 zv o0ss þ z2  2zeaT cosðuTÞ þ e2aT z1



z½v o0ss z  2v o0ss eaT cosðuTÞ þ v o0ss  b2  z2  2zeaT cosðuTÞ þ e2aT

ð52Þ

where k is given by:

  c k ¼ cosðuTÞ  ect1 cos xt 2  sin xt 2

x c ¼ cosh ct 1 sin xt2 þ sinh ct1 cos xt 2 x

ð53Þ

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

iL0;n ¼ iL0ss þ ðiL0;0  iL0ss ÞeanT cosðunTÞ   ect1 sin xt 2 i eaT cosðuTÞ  iL0ss þ b1 v 0;0 þ L0ss þ kiL0;0  xL eaT eanT sinðunTÞ  ð54Þ sinðuTÞ

v o0;n ¼ v o0ss þ ðv o0;0  v o0ss ÞeanT cosðunTÞ  ct  e sin xt2 v eaT cosðuTÞ  v o0ss þ b2 þ v o0;0  kv o0;0 þ o0ss aT 1

xL eanT sinðunTÞ  sinðuTÞ

e

ð55Þ

953

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

Eqs. (54) and (55) show the initial values of inductor current and output voltage in each switching interval versus n. By applying (54) and (55) in (27) and (28), the values of inductor current and output voltage can be obtained in each instant of the time. 3. Theoretical analyses In this section, the time response of buck–boost dc–dc converter is investigated based on presented mathematical model in Section 2. The time response of dc–dc converter consists of transient and steady states responses. The transient response of converter is that part of the response which the inductor current and the output voltage have not reached to their steady values. This part of time response starts from zero instant and will continue to achieve to steady state. Fig. 3 shows the time response of inductor current and output voltage of buck–boost dc–dc converter in CCM. 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. Considering Fig. 3, it is observed that the time responses of iL(t) and vo(t) consist of two transient and steady parts. The steady state response of the converter is formed from the transient states which are repeated in each switching interval. After that the inductor current and output voltage passed their transient states and reached to their final values, the inductor and capacitor charge and discharge between the minimum and maximum values. This causes 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 specifies 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–dc converter, the time constants of iL(t) and vo(t) are determined by the relations of these parameters in discontinuous time domain. The following relation is always valid between the roots of discontinuous and continuous time domains:

k1;2 ¼

1 lnðq1;2 Þ T

ð56Þ

where k1,2 and q1,2 are the roots of characteristic equation in continuous and discontinuous time domains, respectively. By calculating the dominator roots of (45) and (46), and applying them in (56), the time constants of transient response are obtained. The dominator roots of (45) and (46) are given by:

q1;2 ¼ eaTjuT

ð57Þ

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

k1;2 ¼ a  ju

ð58Þ

Considering (58), it is observed that the time constant of converter is a function of duty cycle, inductance, capacitance, load resistance, and switching frequency. Variation of each of these parameters changes the damping time of the transient response of the system. Considering (16), it is observed that the value of a in (58) is real and positive. But according to (45) and considering the values of inductance, capacitance, load resistance, and the inductor equivalent resistance, u can be real or imaginary. If u is an imaginary integer, the transient response of converter is severely damped. If u be a real integer, the transient response of the converter is weakly damped with time constant eaT. Considering the effect of inductance and capacitance on the value of u, the effect of these parameters on transient response can be investigated. Fig. 4 shows the variation of real part of characteristic equation roots for different values of duty cycle versus inductance and capacitance. As it is observed, for high values of L and C, the real part of roots will have higher values. Considering which the real part of roots shows the damping constant of transient response, so the time responses of iL(t) and vo(t) have slow transient responses for higher values of inductance and capacitance. As shown in Fig. 4, because the real part of roots is equal with a and the value of a is independent of duty cycle, so for different values of duty cycle, the value of a is constant. 3.1.1. Analysis of overshoot and setting time The values of settling time and overshoot of the system step response are important parameters in investigating dynamic response of a system. In this section, the effect of each of buck–boost dc/dc converter elements on the values of settling time and overshoot are investigated by the plotted curves in Figs. 5 and 6. These figures are obtained by using the related relations to settling time and overshoot and the poles of transfer function of output voltage and inductor current which are expressed in (58). Figs. 5 and 6 show the variations of overshoot and settling time versus inductance and capacitance for different values of duty cycle, respectively. As shown in Fig. 5a, the value of overshoot has a reverse relation with the value of inductance. In other words, by increasing the value of inductance, the value of overshoot decreases. Fig. 5b shows that by increasing the value of capacitance the value of overshoot increases. Considering Fig. 5b, it is observed that for a specified value of capacitance, by increasing the duty cycle, the value of overshoot decreases. Considering Fig. 6, it is observed that by increasing the values of inductance and capacitance, the settling time of step response of output voltage and inductor current 30

5

25 4

vo [V ]

iL [A]

20 3 2

15 10

1

5

0 0

0.01

0.02

0.03

0.04

0.05

0.06

vo (t)

0

0

0.01

0.02

0.03

0.04

Time [sec]

Time [sec]

(a)

(b)

Fig. 3. Time step response; (a) inductor current and (b) output voltage.

0.05

0.06

954

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

0

0

Re(λ ) [ Hz]

Re( λ ) [ Hz ]

D = 0 .1 D = 0 .5 D = 0 .9

-400

-600

-800

D = 0.5 D = 0.9

-2000

-3000 -4000

0

0.02

0.04

0.06

-5000

0.08

0.2

0.4

0.6

C [mF ]

(a)

(b)

0.8

1

vo(t) versus; (a) inductance and (b) capacitance.

50

60

D = 3.0

overshoot [%]

30

D = 6.0

20 10

D = 8.0 2

4

6

8

10

12

D = 3.0

50

40

0

0

L [H ]

Fig. 4. Variation of real part of roots of iL(t) and

overshoot [%]

D = 0.1

-1000

-200

40

D = 6.0

30 20 10

14

0

16

D = 7.0 0

0.2

0.4

0.6

L [mH ]

C [mF ]

(a)

(b)

Fig. 5. Variations of overshoot of iL(t) and

0.8

1

vo(t) versus; (a) inductance and (b) capacitance.

0.03

0.08

0.06

ts [sec]

t s [sec]

0.025

0.04

0.02 0.02

2

4

6

8

10

12

14

16

0

0.2

0.4

0.6

L [mH ]

C [ mF ]

(a)

(b)

Fig. 6. Variation of setting time of iL(t) and

increases. This means that the increment of the values of capacitance 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 is concluded that the value of settling time is independent of the duty cycle. Fig. 7 shows the variations of settling time versus duty cycle for L = 8 mH and C = 0.2 mF. As it is observed, for different values of duty cycle, the value of settling time remains constant.

0.8

1

vo(t) versus; (a) inductance and (b) capacitance.

t s [ m sec]

0.015

25.6

0.2

0.4

0.6

0.8

1

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

955

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

3.2. Analysis of steady state response The analysis of steady state response of buck–boost dc–dc converter is performed in CCM which the minimum value of inductor current is more than the load current. Fig. 8a shows the equivalent circuit of converter for time interval t1 that switch S is on and the diode D is off. Fig. 8b shows the equivalent circuit of converter for time interval t2 that switch S is off and the diode is on. The waveforms of inductor voltage, inductor current, capacitor current, and output voltage are shown in Fig. 9. In this figure, iL1,ss is the maximum value of inductor current and iL0,ss is the minimum value of inductor current. Also vo1,ss and vo0,ss show the minimum and maximum values of output voltage, respectively. In time interval t1, the inductor current increases from its minimum value to its maximum value, linearly. In this case, because the diode is off, the capacitor current is equal with the load current but with opposite direction. In this time interval, the capacitor energy discharges to load and as a result, the value of capacitor voltage decreases from its maximum value to its minimum value. In time interval t2, the inductor provides the energy of capacitor and load. So, by discharging the inductor energy, the value of its current is decreased from its maximum value to its minimum value. 3.2.1. Calculation of average value of output voltage The average value of output voltage can be calculated by:

Vo ¼

1 T

Z

t 0 þT t0

v o ðtÞdt

ð59Þ

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

Vo ¼

Z

1

v o;ss ðmÞdm

ð60Þ

where vo,ss(m) is the value of output voltage in steady state. By calculating the integral in a switching time interval, the average value of output voltage will be equal with:

Vo ¼

v o0;ss iL1;ss ð1  eðaþcÞDT Þ þ ða a þ xa3 Þ ða þ cÞT xC 0 1 h  i c  þ v o1;ss a0 a2  a1  a3 ðc  aÞ

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 plotted for 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 is observed that by increasing the duty cycle, the average value of output voltage increases. In the loss condition, for higher values of duty cycle, by increasing the value of duty cycle the average value of output voltage decreases. 3.2.2. Calculation of output voltage ripple in steady state One of the effective parameters in operation of dc–dc converters is the value of output voltage ripple. Because dc–dc converters have so much application in dc devices, so they should have minimum value of output ripple. As a result, it can be concluded that studying of output voltage ripple is one of important subjects in dc–dc converters. In this section, by using the proposed mathematical model, the output voltage ripple can be calculated. Considering Fig. 9, the value of output voltage ripple is equal with:

DV o ¼ v o1;ss  v o0;ss

v o1;ss ¼ RL

D − Vc

Vi

iL

+

L

S

− C R vo +

RL

Vi

iL

ic io

D −

−

Vc

C R vo + ic io

+

L

(a)

(b)

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

vL

T t2

t1 Vi

2T

is expressed as



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

DV o;ss ¼

eaTþct1 sin xt2 xC

  b1 þ z  eaTþct1 cos xt 2 þ xc sin xt2 b2 1  2eaT cos uT þ e2aT

 ðeðacÞt1  1Þ

ð64Þ

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

80 lossless condition

iL1,ss

loss condition

iL 0,ss

60

Io

t3

t1

t1

t3

t Vo [V ]

ic iL1, ss − I o iL 0 , ss − I o −Io Vo vo 0,ss

vo1,ss

  z  eaTþct1 cos xt2 þ xc sin xt 2 b2 ðacÞt1 e 1  2eaT cos uT þ e2aT ð63Þ

eaTþct1 sin xt 2 b1 þ xC

t

−Vc iL

ð62Þ

Considering (26) and (50), the value of follows:

0

S

ð61Þ

x

t 20

0

Vo vo1,ss

t Fig. 9. Waveforms converter in CCM.

40

0

0.2

0.4

0.6

0.8

1

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

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

LC ¼

RT ½4ð1  D þ rÞ2  4rD 8

DIL;ss ¼ iL1;ss  iL0;ss

ð65Þ

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

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:

r¼

RL ð1  DÞ R

iL1;ss

 eðacÞt1 þ

For theoretical analysis, the converter parameters are considered as follows:

(

f ¼ 1 kHz; RL ¼ 0:5 X

DIL;ss ¼

For theoretical analysis, the following values are selected considering the above ranges:

In this section, the power loss and efficiency analysis of the buck–boost dc–dc converter is presented considering non-ideal components for the converter. Fig. 13a and b shows the equivalent circuit of the converter in on and off-state of the switch, respectively. In the figure, RS, RD, RL, and RC are the resistance of switch in its on-state, resistance of the diode, resistance of the inductor and capacitor, respectively. For the loss calculations, ripple of the inductor current is neglected [24]. According to Fig. 13, the switch current for its on and off-state can be expressed as follows:

(a)

[%]

5

Δ Vo ,ss

[%] 0

10

Vo ,ss

Δ Vo ,ss

[%] Vo ,ss

D

20 60 40 100 80 R [Ω]

0 1 0.5

D

ð69Þ

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

3.2.3. Calculation of inductor current ripple in steady state One of the other effective parameters on the operation of dc–dc converters is the inductor current ripple. Considering Fig. 9, it is observed that the value of the inductor current ripple in steady state is obtained by:

0.5

)   aTct1 sin xt 2 1  eaTct1 cos xt2  xc sin xt 2 b1  e b2 xL 1  2eaT cos uT þ e2aT

Fig. 12a shows the variations of inductor current ripple versus the load resistance and duty cycle for specific values of input voltage and inductance. Considering this figure, it is clear that by increasing the value of load resistance, the value of inductor current ripple increases. Also the value of inductor current ripple increases by increasing the value of duty cycle, but this increment will continue until a specific value of duty cycle and after it by increasing the value of duty cycle, the value of inductor current ripple decreases. Fig. 12b shows the variations of inductor current ripple versus inductor current and duty cycle for specific values of load resistance, input voltage, and capacitance. Considering this figure, it is observed that by increasing the value of inductance, the value of inductor current ripple decreases. Fig. 12c shows the variations of inductor current ripple versus the values of capacitance and duty cycle for specific values of load resistance and input voltage. Considering this figure, it is clear that the value of inductor current ripple is independent of the value of capacitance. As shown in Fig. 12c, by increasing the duty cycle, the value of inductor current ripple increases. This increment will continue until a specific value of duty cycle and after a specific value by increasing the duty cycle, the value of inductor current ripple decreases.

Considering the selected values by using (65), the value of the critical 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 versus load resistance and duty cycle for specified values of input voltage and inductance. Considering this figure, it is observed that the value of output voltage ripple has a reverse relation with load resistance. As a result, by increasing the value of load resistance, the value of output voltage ripple decreases. In addition, it is clear that by increasing the value of duty cycle, the value of output voltage ripple increases. Fig. 11b shows the variations of output voltage ripple versus inductance and duty cycle for specified values of input voltage and load resistance. Considering this figure, it is observed that, the value of inductance does not affect on the value of output voltage ripple. Also Fig. 11b shows that by increasing the duty cycle, the value of output voltage ripple increases. Fig. 11c shows the variations of output voltage ripple versus the capacitance and duty cycle for specified values of Vi, R, and L. Considering this figure, it is observed that by increasing the value of capacitance, the value of output voltage ripple decreases. In addition, by increasing the value of duty cycle, the value of output voltage ripple increases. It is important to note that for higher values of capacitance, the effect of duty cycle on the value of output voltage ripple is less. In modeling the converter, the value of equivalent series resistance (ESR) of the output capacitor has been neglected. The value of ESR has negligible effect on the different parameters of converter such as output voltage ripple. This has been proven with more details in [22].

0 1

ð68Þ

  Vi  eðacÞt1  1 þ ½1  eðacÞt1  RL

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

10

Vi ½1  eðacÞt1  RL

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

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

ΔVo ,ss

(

)   aTct1 sin xt2 1  eaTct1 cos xt2  xc sin xt2 b1  e b2 xL ¼ 1  2eaT cos uT þ e2aT

ð66Þ

20

ð67Þ

0

25

(b)

15

5

L [mH ]

0

Vo ,ss

956

20 10 0 1 0.5

D

0

2

1.2

0.4 0

C [mF ]

(c)

Fig. 11. Variations of output voltage ripple versus; (a) duty cycle and load resistance; (b) duty cycle and inductance and (c) duty cycle and capacitance.

957

1 0 1 0.5

D

0

20 60 40 100 80 R [Ω ]

Δ I L , ss [ A ]

2

2

Δ I L , ss [ A ]

Δ I L , ss [ A ]

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

1 0 1 0.5 0

D

(a)

25

0

5

15

2 1 0 1 0.5

D

L [mH ]

0

(b)

2

1.2

0.4 0

C [mF ]

(c)

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.

iS Vi

RS VS iL

RL L

D

RC

C

−

R vo ic io

S

RL

i D RD V D

Vi

+

L

(a)

iL

ic

RC

−

R vo

C i o

+

(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 ¼

IL

for 0 6 t < DT

0

for DT 6 t < 1

ð70Þ

In (70), IL is the average value of the inductor current which is obtained by substituting (49) and (50) in (27) and averaging (27). The loss of the power switch can be written as follows: 2

PS ¼ RS iS þ V S iS

ð71Þ

Substituting (70) in (71), the power loss of the switch is obtained as:

PS ¼ RS I2L þ V S IL 

0

for 0 6 t < DT

IL

for DT 6 t < 1

2

ð74Þ

ð75Þ

ð76Þ

The capacitor current can be written as:

iC ¼



Io

for 0 6 t < DT

I L  Io

for DT 6 t < 1

ð77Þ

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

Io ¼

Vo R

ð80Þ

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

Po Po þ PLoss

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 components. 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 versus duty cycle for different values of the load resistance. The figure indicates that the efficiency is minimum for the lowest and highest values of the duty cycle. Also, for a specific value of duty cycle, the efficiency of the converter improves as the load resistance increases. The variation of the converter efficiency versus duty cycle for different values of the inductor resistance is indicated in Fig. 14b. As the figure shows, the converter efficiency decreases as the inductor resistance increases. Moreover, the effect of the inductor resistance on the efficiency is not considerable for lower values of the duty cycle. Fig. 14c shows the variations of the converter efficiency versus the duty cycle for various values of the capacitor equivalent resistance. For a specific value of RC, the value of efficiency decreases for lowest and highest values of the duty cycle. Also, for a specific value of duty cycle the value of efficiency is inversely related with the equivalent resistance of the capacitor. Fig. 14d and e shows the variations of the efficiency versus duty cycle for different values of the switch resistance and diode resistance, respectively. As the figures indicate, for a specific value of duty cycle, the converter efficiency decreases as the value of the switch and diode resistances increase.

The power loss of the inductor resistance can be written as follows:

PRL ¼ RL I2L

PLoss ¼ PS þ PRD þ PRC þ PRL

ð73Þ

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

PD ¼ RD I2L þ V D IL

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

g¼

The power loss of the diode is as follows:

PD ¼ RD iD þ V D iD

ð79Þ

ð72Þ

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

iD ¼

PRC ¼ RC I2o þ RC ðIL  Io Þ2

ð78Þ

where Vo is the average output voltage which can be achieved using (61). The power loss of the equivalent resistance of the capacitor is sum of the corresponding losses in the both on and off-state of the switch. This power loss can be written as follows:

ð81Þ

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

1

1

0.8

0.8

0.6

0.6

η [ pu ]

η [ pu ]

958

0.4

0.4

RL = 0.05 [Ω]

R = 15 Ω R = 60 Ω

0.2

0.2

RL = 0.1[Ω ] R L = 0 .2 [ Ω ]

R = 100 Ω 0

0.2

0.4

0.6

0.8

0

1

0

(a)

(b) 1

0.8

0.8

0.6

0.6

0.4

1

0.6

0.8

1

RS = 0.11Ω

0.2

RS = 0.2 Ω

RC = 0.2 Ω 0.2

0.8

RS = 7 mΩ

RC = 0.1 Ω 0

0.6

0.4

RC = 5 mΩ

0

0.4

D

1

0.2

0.2

D

η [ pu ]

η [ pu ]

0

0.4

0.6

0.8

0

1

0

0.2

0.4

D

D

(c)

(d) 1

η [ pu ]

0.8 0.6

0.4

RD = 10m Ω 0.2

RD = 0.09 Ω RD = 0.2 Ω

0 0

0.2

0.4

0.6

0.8

1

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 of the capacitor, (d) power switch resistance, and (e) diode resistance.

30

5 4

vo [V ]

iL [ A]

20 3 2

10 Proposed Model

Proposed Model

1

Average Model

Average Model 0

0

0.01

0.02

0.03

0.04

0

0

0.01

0.02

Time [sec]

Time [sec]

(a)

(b)

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

0.03

0.04

959

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

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

5. Comparison of the proposed method with conventional method

vo [V ]

i L [ A]

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

3.0

3.0

2.0

2.0

1.0

1.0

0.0

0.0

25.0 24.5 24.0 23.5 23.0 22.5 22.0 21.5 21.0

25.00 24.50 24.00 23.50 23.00 22.50 22.00 21.50 21.00

0

0.001

0.002

0.003

vo [V ]

i L [ A]

Time [sec]

i L [A ]

0.1000

3.0

2.0

2.0

1.0

1.0

0.0

0.0 25.00 24.50 24.00 23.50 23.00 22.50 22.00 21.50 21.00 0

0.001

0.002

0.003

0.004

2.0

1.0

1.0

0.0

0.0 25.00 24.50 24.00 23.50 23.00 22.50 22.00 21.50 21.00 0.002

Time [sec]

0.003

0.004

(c)

0.1040

Vo[V]

0.1010

(b)

2.0

0.001

0.1030

0.1020

0.1030

0.1040

Time [sec]

3.0

0

0.1020

IL[A]

0.1000

3.0

25.0 24.5 24.0 23.5 23.0 22.5 22.0 21.5 21.0

0.1010

Time [sec]

3.0

25.0 24.5 24.0 23.5 23.0 22.5 22.0 21.5 21.0

Vo[V]

(a)

Time [sec]

vo [ V ]

0.004

IL[A]

IL[A]

Vo[V]

0.1000

0.1010

0.1020

0.1030

0.1040

Time [sec]

Fig. 16. Waveforms of inductor current and output voltage; left column: experimental results; right column: simulation results using PSCAD; (a) R = 30 X; (b) R = 70 X; (c) L = 6 mH; (d) L = 15 mH; (e) C = 0.4 mF; and (f) C = 1.2 mF.

960

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

3.0

2.0

2.0

1.0

1.0

0.0

0.0

vo [V ]

i L [A ]

3.0

25.0 24.5 24.0 23.5 23.0 22.5 22.0 21.5 21.0

25.00 24.50 24.00 23.50 23.00 22.50 22.00 21.50 21.00

0

0.001

0.002

0.003

Time [sec]

(d)

2.0

1.0

1.0

0.0

0.0

vo [V ]

i L [A ]

2.0

25.00 24.50 24.00 23.50 23.00 22.50 22.00 21.50 21.00

0.001

0.002

0.1010

0.003

0.1020

0.1030

0.1040

0.1030

0.1040

0.1030

0.1040

Time [sec]

3.0

0

Vo[V]

0.1000

0.004

3.0

25.0 24.5 24.0 23.5 23.0 22.5 22.0 21.5 21.0

IL[A]

IL[A]

Vo[V]

0.1000

0.004

0.1010

Time [sec]

0.1020

Time [sec]

(e) 3.0

2.0

2.0

1.0

1.0

0.0

0.0

vo [V ]

i L [A ]

3.0

25.0 24.5 24.0 23.5 23.0 22.5 22.0 21.5 21.0

25.00 24.50 24.00 23.50 23.00 22.50 22.00 21.50 21.00 0

0.001

0.002

0.003

Time [sec]

0.004

(f)

IL[A]

Vo[V]

0.1000

0.1010

0.1020

Time [sec]

Fig. 16. (continued)

In this method, for modeling the converter, two approximations are used. In the first order approximation, the state variables and switching function contain three terms of Fourier expansion and other 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 complicated mathematical methods such as numerical solution methods should be used to solve them, In the proposed method in

961

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

this paper, the obtained differential equations for the output voltage and inductor current are order two which solving these equations is easier by using the Laplace transform. One of the other advantages of the proposed method in comparison with the presented method in [17] is the ability to analyze the steady state parameters of converter such as output voltage and inductor current ripples. Although the proposed method has complex equations, it has the ability to analyze the converter in each two transient and steady states. Fig. 15 shows the step responses of output voltage and inductor current 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 figure, the step response of proposed method is compared with the obtained results of presented method in [17]. Considering this figure, it is observe that by the proposed method have better accuracy than the average state space method. 6. Experimental and simulation results To prove the validity of presented theoretical subjects, the experimental 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 prototype are MJ13005 and BUR460, respectively. In order to generate the gate commands of the power switch, the 89C52 type microcontroller by ATMEL Company has been used. The dc supply existing in the laboratory has been used as the dc voltage source. Tektronix TDS 2024B four channel digital storage oscilloscope has been used for measurements in laboratory.

IL[A]

5.0

Fig. 16a and b shows the simulation and experimental results of inductor current and output voltage for load resistance R = 30 X and R = 70 X in the steady state, respectively. Considering the figures, it is observed that by increasing the value of load resistance, the value of inductor current ripple increases and the value of output voltage ripple decreases. According to Fig. 16a, the average value of output voltage is equal with 22.867 V By applying the above values in (61), the value of Vo is equal with 22.983 V. Also in Fig. 16a, the initial values of output voltage and inductor current in each switching interval are equal with: iL1;ss ¼ 2:593 A; iL0;ss ¼ 1:219 A;

v o1;ss ¼ 21:953 V; v o0;ss ¼ 23:781 V

Considering (49) and (50), the initial values of inductor current and output voltage in steady state are equal with 1.222 A and 23.830 V, respectively. Also, considering (23) and (26), the initial values of inductor current and output voltage during off time interval of switch are equal with 2.597 A and 21.998 V, respectively. As it is observed, the results of the theoretical analysis prove the validity of the simulation and experimental results and the value of error is about 0.5%. For L = 6 mH and L = 15 mH, the waveforms of inductor current and output voltage are shown in Fig. 16c and d, respectively. According to the figure, it is observed that by increasing the value of inductance, the value of the output voltage ripple does not vary but the value of inductor current ripple decreases by increasing the value of inductance. In Fig. 16c, the values of output voltage and inductor current ripples are 1.404 V and 1.625 A, respectively. Considering (64) and (69), the values of output voltage and inductor current ripples are equal with:

Vo[V]

30.0 25.0

4.0

20.0

3.0

15.0 2.0

10.0

1.0

5.0

0.0

0.0

0.000 0.010 0.020 0.030 0.040 0.050 0.060

0.000 0.010 0.020 0.030 0.040 0.050 0.060

Time [sec]

Time [sec]

(a)

(b)

Fig. 17. Step response based on simulation; (a) inductor current and (b) output voltage.

30

5

Experimental

Theoretical 4

vo [V ]

iL [A]

20 3 2 1 0

Theoretical 10

Experimental 0

0.01

0.02

0.03

0.04

0.05

0.06

0

0

0.01

0.02

0.03

0.04

Time [sec]

Time [sec]

(a)

(b)

0.05

0.06

Fig. 18. Experimental and theoretical results of step response; (a) inductor current and (b) output voltage.

962

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

DV o ¼ 1:406V;

DIL ¼ 1:626A

uðmTÞ ¼

As it is observed, the experimental and simulation results prove the validity of results of theoretical analysis. Fig. 16e and f shows the waveforms of inductor current and output voltage for C = 0.4 mF and C = 1.2 mF, respectively. According to figure, it is observed that, by increasing the capacitance, the value of output voltage ripple decreases, but the value of inductor current ripple does not vary. Fig. 17a and b shows the step response of inductor current and output voltage based on simulation. In order to simulation, the selected values for plotting Fig. 3 have been used. Fig. 18 shows the step response of output voltage and inductor current in two experimental and theoretical cases. In this case, the selected values for plotting Fig. 3 are used. As shown in Figs. 3, 17, and 18, there is good agreement between simulation, theoretical, and experimental results.



In this paper, a mathematical model was proposed for modeling the buck–boost dc–dc converter in CCM. The proposed method is based on Laplace and Z transforms. The Laplace transform is used to determine the equations of inductor current and output voltage. The Z-transform is applied as a tool to obtain the initial values of inductor current and output voltage. Also it was shown, the Ztransform can be used for analyzing the responses of transient and steady states. By using the proposed method, the effect of converter elements was investigated on time response of converter in transient and steady states. It was shown that the buck–boost dc– dc converter in CCM has slow transient response for higher values of inductance and capacitance. In addition, the effect of converter elements on output voltage and inductor current ripples was investigated. It was shown that the value of output voltage ripple is independent of the value of inductance, but by increasing the values of load resistance and capacitance, the value of output voltage ripple decreases. In addition the inductor current ripple has a direct relation with the load resistance. In CCM, the inductor current ripple is independent of the value of capacitance and has an indirect relation with the value of inductance. The proposed modeling method can be used as a tool for analyzing the operation of converter by high variations of parameters. The theoretical results were proved by experimental and simulation results.

0

uðmT  t 1 Þ ¼

uðmT  TÞ ¼

ðA5Þ

mT < 0 for m < 0 (

1 mT  t 1 P 0 for m P tT1 0



ðA6Þ

mT  t 1 < 0 for m < tT1

1 mT  T P 0 for m P 1 0

ðA7Þ

mT  T < 0 for m < 1

In (A6), tT1 is the duty cycle of converter (D) in CCM and it is defined as follows:

D¼

t1 T

ðA8Þ

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

 f1 ðmÞ ¼

7. Conclusion

1 mT P 0 for m P 0

f2 ðmÞ ¼



1 06m
D6m<1

0

06m
ðA9Þ

ðA10Þ

1 D6m<1

Appendix B

" X¼

  z  eaTþct1 cos xt 2 þ xc sin xt 2 e

aTþct 1

sin xt2

xC

#

eaTct1 sin xt2 xL



z  eaTct1 cos xt 2  xc sin xt 2



ðB1Þ

 i V i h at2 c ðe  ect1 aT Þ cos xt2 þ sin xt 2 x RL   V i 1 at2 ct1 aT b2 ¼ e Þ sin xt2 ðe RL xC eat2 a0 ¼ 2 ða þ x2 ÞT 2

ðB3Þ

a1 ¼ aT sin xt 2  xT cos xt2

ðB5Þ

a2 ¼ aT cos xt 2 þ xT sin xt2 1 a3 ¼ 2 ða þ x2 ÞT

ðB6Þ

b1 ¼

ðB2Þ

ðB4Þ

ðB7Þ

Appendix A References Considering (3)–(5), the functions f1(t) and f2(t) can be expressed as follows, respectively:

f1 ðtÞ ¼

1 X

½uðnT þ mT  nTÞ  uðnT þ mT  t 1  nTÞ

n¼0

¼

1 X ½uðmTÞ  uðmT  t 1 Þ ¼ nf1 ðmÞ

ðA1Þ

n¼0

f2 ðtÞ ¼

1 X ½uðnT þ mT  t1  nTÞ  uðnT þ mT  T  nTÞ n¼0

1 X ¼ ½uðmT  t1 Þ  uðmT  TÞ ¼ nf2 ðmÞ

ðA2Þ

n¼0

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

f1 ðmÞ ¼ uðmTÞ  uðmT  t1 Þ

ðA3Þ

f2 ðmÞ ¼ uðmT  t 1 Þ  uðmT  TÞ

ðA4Þ

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

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