102 3. CONTROLLER DESIGNFOR DC-DC CONVERTERSUSINGKHARITONOV’S THEOREM
When MOSFET switch is closed, converter differential equations can be written as
P{
L1
D
r
L1
C r
ds
C r
C 2
L
1
i
L1
r
ds
C r
C 2
L
1
i
L2
v
C 2
L
1
C
v
in
L
1
C
r
C 2
L
1
i
o
P{
L2
D
r
ds
C r
C 2
L
2
i
L1
r
L2
C r
ds
C r
C1
C r
C 2
L
2
i
L2
C
v
C1
L
2
v
C 2
L
2
C
r
C 2
L
2
i
o
Pv
C1
D
i
L2
C
1
Pv
C 2
D
i
L1
C
2
C
i
L2
C
2
i
o
C
2
:
(3.11)
When MOSFET M is opened, diode D1 is forward biased. Figure 3.18 shows the equiv-
alent circuit for this case.
Vin
VD
io
+
+
C2
C1
rC1
rL1
L1
L2
r2
rD
rC2
–
+
–
+
Figure 3.18: Equivalent circuit for opened MOSFET.
3.4. ROBUST CONTROLLER DESIGN FOR SUPER BUCK CONVERTER 103
In this case converter dynamical equations can be written as:
P{
L1
D
r
L1
C r
C1
C r
d
C r
C 2
L
1
i
L1
r
D
C r
C 2
L
1
i
L2
v
C1
L
1
v
C 2
L
1
C
v
in
L
1
C
r
C 2
L
1
i
o
V
D
L
1
P{
L2
D
r
D
C r
C 2
L
2
i
L1
r
L2
C r
D
C r
C 2
L
2
i
L2
v
C 2
L
2
C
r
C 2
L
2
i
o
V
D
L
2
Pv
C1
D
i
L1
C
1
Pv
C 2
D
i
L1
C
2
C
i
L2
C
2
i
o
C
2
:
(3.12)
Assume a super buck converter with the following parameter values: V
in
D 30 V, L1 D
500 H, r
L1
D 0:1 , L2 D 500 H, r
L2
D 0:1 , C1 D 470 F, r
C1
D 0:05 , C 2 D 470 F,
r
C 2
D 0:05 , 5 < R < 25 , D D 0:66, and Fsw D 50 KHz.
D and Fsw show the duty ratio and the switching frequency of the converter, respectively.
Figure 3.19 shows the frequency response of control-to-output transfer function when load
resistor changes from 5 toward 25 with 0:2 steps. For the aforementioned load range the
converter operates in the CCM.
e converter dynamical equation (control to output transfer function) can be written as:
Qv
o
.s/
Q
d .s/
D
n
3
s
3
C n
2
s
2
C n
1
s
1
C n
0
s
4
C d
3
s
3
C d
2
s
2
C d
1
s
1
C d
0
: (3.13)
e denominator is a fourth-order polynomial since there are four energy storage elements
(two capacitors and two inductors) in the circuit.
e following MATLAB
®
code is used to extract the minimum and maximum of transfer
function coefficients. Minimum and maximum of coefficients is shown in Table 3.7.
clc
clear all;
%R=5..20
R_min =5;
R_step =.2;
R_max =25;
NUM=zeros (1,5);
DEN=zeros (1,5);
104 3. CONTROLLER DESIGNFOR DC-DC CONVERTERSUSINGKHARITONOV’S THEOREM
N=(R_max -R_min )/ R_step;
n=0;
for R= R_min:R_step: R_max
n=n+1;
disp(' percentage of work done:')
disp(n/N *100)
VIN =30;
rin =.1;
L1 =500e-6;
rL1 =.1;
L2 =500e-6;
rL2 =.1;
C1 =470e-6;
rC1 =.05;
C2 =470e-6;
rC2 =.05;
rD =.01;
VD =.7;
rds =.1;
D=.66;
%Symbolic equations
syms iL1 iL2 vC1 vC2 vin vD d
%CLOSED MOSFET EQUATIONS
y=(iL1+iL2 -(vC2/R))/(1+( rC2/R));
x=(rC1+rL2)*iL2 -vC1+rds *(iL1+iL2)+rC2*y+vC2;
M1=(vin -x-vC1 +(rL2+rC1)*iL2 -rL1*iL1)/L1;
M2=(-( rC1+rL2 )*iL2+vC1 -rds *(iL1+iL2)-rC2 *y-vC2)/L2;
3.4. ROBUST CONTROLLER DESIGN FOR SUPER BUCK CONVERTER 105
M3=(-iL2 )/C1;
M4=(y)/C2;
%OPENED MOSFET EQUATIONS
z=vD+rD*( iL1+iL2 )+rC2*y+vC2+rL2*iL2;
M5=(vin -( rL1+rC1 )*iL1 -vC1+rL2*iL2 -z)/L1;
M6=(-vD-rD *(iL1+iL2)-rC2*y-vC2 -rL2*iL2 )/L2;
M7=(iL1)/C1;
M8=(y)/C2;
%AVERAGING
MA1= simplify(d*M1+(1 -d)*M5);
MA2= simplify(d*M2+(1 -d)*M6);
MA3= simplify(d*M3+(1 -d)*M7);
MA4= simplify(d*M4+(1 -d)*M8);
%DC OPERATING POINT CALCULATION
MA_DC_1=subs (MA1 ,[vin vD d],[ VIN VD D]);
MA_DC_2=subs (MA2 ,[vin vD d],[ VIN VD D]);
MA_DC_3=subs (MA3 ,[vin vD d],[ VIN VD D]);
MA_DC_4=subs (MA4 ,[vin vD d],[ VIN VD D]);
DC_SOL=
solve(MA_DC_1 ==0, MA_DC_2==0, MA_DC_3 ==0, MA_DC_4 ==0,
'iL1 ','iL2 ','vC1 ','vC2 ');
IL1=eval(DC_SOL .iL1);
IL2=eval(DC_SOL .iL2);
VC1=eval(DC_SOL .vC1);
VC2=eval(DC_SOL .vC2);
%LINEARIZATION
%x=[iL1;iL2;vC1;vC2 ]
%u=[vin;d] where d=duty and vin=input voltage changes
A11=subs(simplify(diff(MA1 ,iL1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A12=subs(simplify(diff(MA1 ,iL2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A13=subs(simplify(diff(MA1 ,vC1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
106 3. CONTROLLER DESIGNFOR DC-DC CONVERTERSUSINGKHARITONOV’S THEOREM
[IL1 IL2 VC1 VC2 D VD VIN]);
A14=subs(simplify(diff(MA1 ,vC2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A21=subs(simplify(diff(MA2 ,iL1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A22=subs(simplify(diff(MA2 ,iL2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A23=subs(simplify(diff(MA2 ,vC1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A24=subs(simplify(diff(MA2 ,vC2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A31=subs(simplify(diff(MA3 ,iL1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A32=subs(simplify(diff(MA3 ,iL2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A33=subs(simplify(diff(MA3 ,vC1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A34=subs(simplify(diff(MA3 ,vC2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A41=subs(simplify(diff(MA4 ,iL1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A42=subs(simplify(diff(MA4 ,iL2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A43=subs(simplify(diff(MA4 ,vC1)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A44=subs(simplify(diff(MA4 ,vC2)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
A=eval ([A11 A12 A13 A14;
A21 A22 A23 A24;
A31 A32 A33 A34;
A41 A42 A43 A44
]);
B11=subs(simplify(diff(MA1 ,vin)) ,[iL1 iL2 vC1 vC2 d vD vin ],
[IL1 IL2 VC1 VC2 D VD VIN]);
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