2.3. BUCK-BOOST CONVERTER 53
ere are many efforts presented in the literature to achieve satisfactory output impedance
of PWM DC-DC converters, especially buck type. e methods can be categorized into two
groups:
• sophisticated design of control loops in the converter [Chen et al., 2007, Lee et al., 2008,
Xiao et al., 2008, Qahouq and Arikatla, 2011] and
• modifications of the basic structure of the power stage [Singh and Khambadkone, 2008].
e starting point of the first method is the precise description of the converter, in par-
ticular the use of accurate formulas for open-loop output impedance. e given program can do
this step.
2.3 BUCK-BOOST CONVERTER
Schematic of the buck-boost converter is shown in Fig. 2.12.
–
+
Vg
rg
i
?
L
D
C
+
rL rC
R
Figure 2.12: Schematic of PWM buck-boost converter.
When the MOSFET is closed, the diode is reverse biased and Fig. 2.13 is an equivalent
circuit. According to Fig. 2.13, the circuit differential equations can be written as:
di
L
.t/
dt
D
1
L
r
g
C r
ds
C r
L
i
L
C v
g
(2.15)
dv
C
.t/
dt
D
1
C
R
R C r
C
i
O
1
R C r
C
v
C
(2.16)
i
g
D i
L
(2.17)
v
o
D
R
R C r
C
v
C
C
R r
C
R C r
C
i
O
: (2.18)
54 2. ON THE EXTRACTION OF INPUT AND OUTPUT IMPEDANCE OF PWM DC-DC
–
+
Vg
rg
i
?
L
rL
C
+
rds
io
R
rC
Z
in
(s)
Z
0
(s)
Figure 2.13: Equivalent circuit of buck-boost converter with closed MOSFET.
–
+
Vg
rg
i
?
rD
VD
C
+
io
R
rCrL
Z
in
(s)
Z
0
(s)
+
–
L
Figure 2.14: Equivalent circuit of buck-boost converter with open MOSFET.
When the MOSFET switch is opened, the diode becomes forward-biased. Figure 2.14
shows the equivalent circuit for this case. According to Fig. 2.14, the circuit differential equations
can be written as:
d i
L
.t/
dt
D
1
L
r
D
C r
L
C
Rr
C
R C r
C
i
L
R
R C r
C
v
C
R r
C
R C r
C
i
O
v
D
(2.19)
dv
C
.t/
dt
D
1
C
R
R C r
C
i
L
1
R C r
C
v
C
C
R
R C r
C
i
O
(2.20)
2.3. BUCK-BOOST CONVERTER 55
i
g
D 0 (2.21)
v
o
D
R r
C
R C r
C
i
L
C
R
R C r
C
v
C
C
R r
C
R C r
C
i
O
C V
D
: (2.22)
Consider a buck-boost converter with component values as shown in Table 2.2. e fol-
lowing program extracts the small signal transfer function of the assumed converter.
Table 2.2: e buck-boost converter parameters (see Fig. 2.12)
Nominal Value
Output voltage, Vo -16 V
Duty ratio, D 0.4
Input DC source voltage, Vg 24 V
Input DC source internal resistance, rg 0.1 Ω
MOSFET drain-source resistance, rds 40 mΩ
Capacitor, C 80 µF
Capacitor Equivaluent Series Resistance (ESR), rC 0.05 Ω
Inductor, L 20 µH
Inductor ESR, rL 10 mΩ
Diode voltage drop, vD 0.7 V
Diode forward resistance, rD 10 mΩ
Load resistor, R 5 Ω
Switching Frequency, Fsw 100 KHz
%This program calculates the input and output
%impedance of the Buck -Boost converter .
clc
clear all
syms vg rg d rL L rC C R vC iL rds rD vD io
%Converter Dynamical equations
%M1: diL/dt for closed MOSFET.
%M2: dvC/dt for closed MOSFET.
56 2. ON THE EXTRACTION OF INPUT AND OUTPUT IMPEDANCE OF PWM DC-DC
%M3: current of input DC source for closed MOSFET.
%M4: output voltage of converter for closed MOSFET.
%M5: diL/dt for open MOSFET .
%M6: dvC/dt for open MOSFET .
%M7: current of input DC source for open MOSFET .
%M8: output voltage of converter for open MOSFET .
M1=(-(rg+rds+rL)*iL+vg)/L;
M2=(R/(R+rC)*io -vC/(R+rC))/C;
M3=iL;
M4=R*rC/(R+rC)*io+R/(R+rC)*vC;
M5=(-(rL+rD+rC*R/(R+rC))*iL -R/(R+rC)*vC -R*rC/(R+rC)*io -vD)/L;
M6=(R/(R+rC)*iL -1/(R+rC)*vC+R/(R+rC)*io)/C;
M7 =0;
M8=rC*R/(rC+R)*iL+R/(R+rC)*vC+R*rC/(R+rC)*io+vD;
%Averaged Equations
diL_dt_ave =simplify(M1*d+M5*(1-d));
dvC_dt_ave =simplify(M2*d+M6*(1-d));
ig_ave=simplify(M3*d+M7*(1-d));
vo_ave=simplify(M4*d+M8*(1-d));
%DC Operating Point
DC=solve (diL_dt_ave ==0,dvC_dt_ave ==0,'iL','vC');
IL=DC.iL;
VC=DC.vC;
%Linearization
A11=simplify(subs(diff (diL_dt_ave ,iL),[iL vC io],[IL VC 0]));
A12=simplify(subs(diff (diL_dt_ave ,vC),[iL vC io],[IL VC 0]));
A21=simplify(subs(diff (dvC_dt_ave ,iL),[iL vC io],[IL VC 0]));
A22=simplify(subs(diff (dvC_dt_ave ,vC),[iL vC io],[IL VC 0]));
AA=[A11 A12;A21 A22 ];
B11=simplify(subs(diff (diL_dt_ave ,io),[iL vC io],[IL VC 0]));
B12=simplify(subs(diff (diL_dt_ave ,vg),[iL vC io],[IL VC 0]));
B13=simplify(subs(diff (diL_dt_ave ,d),[iL vC io ],[IL VC 0]));
2.3. BUCK-BOOST CONVERTER 57
B21=simplify(subs(diff (dvC_dt_ave ,io),[iL vC io],[IL VC 0]));
B22=simplify(subs(diff (dvC_dt_ave ,vg),[iL vC io],[IL VC 0]));
B23=simplify(subs(diff (dvC_dt_ave ,d),[iL vC io ],[IL VC 0]));
BB=[B11 B12 B13;B21 B22 B23 ];
C11=simplify(subs(diff (ig_ave ,iL) ,[iL vC io],[IL VC 0]));
C12=simplify(subs(diff (ig_ave ,vC) ,[iL vC io],[IL VC 0]));
C21=simplify(subs(diff (vo_ave ,iL) ,[iL vC io],[IL VC 0]));
C22=simplify(subs(diff (vo_ave ,vC) ,[iL vC io],[IL VC 0]));
CC=[C11 C12; C21 C22];
D11=simplify(subs(diff (ig_ave ,io) ,[iL vC io],[IL VC 0 ]));
D12=simplify(subs(diff (ig_ave ,vg) ,[iL vC io],[IL VC 0]));
D13=simplify(subs(diff (ig_ave ,d) ,[iL vC io],[IL VC 0]));
D21=simplify(subs(diff (vo_ave ,io) ,[iL vC io],[IL VC 0 ]));
D22=simplify(subs(diff (vo_ave ,vg) ,[iL vC io],[IL VC 0]));
D23=simplify(subs(diff (vo_ave ,d) ,[iL vC io],[IL VC 0]));
DD=[D11 D12 D13;D21 D22 D23 ];
%Components Values
%Variables have underline are used to
%store the numeric values of components
%Variables without underline are symbolic variables.
%for example:
%L: symbolic vvariable shows the inductor inductance
%L_: numeric variable shows the inductor inductance value.
L_ =20e -6;
rL_ =.01;
C_ =80e -6;
rC_ =.05;
rds_ =.04;
rD_ =.01;
VD_ =.7;
D_ =.4;
VG_ =24;
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