1.4. DYNAMICS OF A BUCK CONVERTER WITHOUT UNCERTAINTY 11
%
%where ,
% x=[iL vC]'
% u=[io vg d]'
%since we used the variables D for steady state duty ratio
%and C to show the capacitors values we use AA , BB, CC
%and DD instead of A, B, C and D.
%Calculating the matrix A
A11=subs(simplify(diff( averaged_diL_dt ,iL)),[iL vC d io],
[IL VC D IO]);
A12=subs(simplify(diff( averaged_diL_dt ,vC)),[iL vC d io],
[IL VC D IO]);
A21=subs(simplify(diff( averaged_dvC_dt ,iL)),[iL vC d io],
[IL VC D IO]);
A22=subs(simplify(diff( averaged_dvC_dt ,vC)),[iL vC d io],
[IL VC D IO]);
AA=eval ([A11 A12;
A21 A22 ]);
%Calculating the matrix B
B11=subs(simplify(diff( averaged_diL_dt ,io)),[iL vC d vD io vg],
[IL VC D VD IO VG]);
B12=subs(simplify(diff( averaged_diL_dt ,vg)),[iL vC d vD io vg],
[IL VC D VD IO VG]);
B13=subs(simplify(diff( averaged_diL_dt ,d)) ,[iL vC d vD io vg],
[IL VC D VD IO VG]);
B21=subs(simplify(diff( averaged_dvC_dt ,io)),[iL vC d vD io vg],
[IL VC D VD IO VG]);
B22=subs(simplify(diff( averaged_dvC_dt ,vg)),[iL vC d vD io vg],
[IL VC D VD IO VG]);
B23=subs(simplify(diff( averaged_dvC_dt ,d)) ,[iL vC d vD io vg],
[IL VC D VD IO VG]);
BB=eval ([B11 B12 B13 ;
B21 B22 B23]);
12 1. EXTRACTION OF UNCERTAIN MODEL OF DC-DC CONVERTERS
%Calculating the matrix C
C11=subs(simplify(diff(averaged_vo ,iL)) ,[iL vC d io],
[IL VC D IO]);
C12=subs(simplify(diff(averaged_vo ,vC)) ,[iL vC d io],
[IL VC D IO]);
CC=eval ([C11 C12]);
D11=subs(simplify(diff(averaged_vo ,io)) ,[iL vC d vD io vg],
[IL VC D VD IO VG]);
D12=subs(simplify(diff(averaged_vo ,vg)) ,[iL vC d vD io vg],
[IL VC D VD IO VG]);
D13=subs(simplify(diff(averaged_vo ,d)) ,[iL vC d vD io vg],
[IL VC VD IO VG]);
%Calculating the matrix D
DD=eval ([D11 D12 D13 ]);
%Producing the State Space Model and obtaining the small
%signal transfer functions
sys=ss(AA,BB ,CC,DD);
sys.inputname={'io ';'vg ';'d'};
sys.outputname ={'vo '};
vo_io=tf(sys (1,1)); %Output impedance transfer function
%vo(s)/io(s)
vo_vg=tf(sys (1,2)); %vo(s)/vg(s)
vo_d=tf(sys(1 ,3)); %Control -to -output(vo(s)/d(s))
%drawing the Bode diagrams
figure (1)
bode(vo_io),grid minor , title ('vo(s)/io(s)')
figure (2)
bode(vo_vg),grid minor , title ('vo(s)/vg(s)')
figure (3)
bode(vo_d),grid minor ,title ('vo(s)/d(s)')
1.4. DYNAMICS OF A BUCK CONVERTER WITHOUT UNCERTAINTY 13
50
0
-50
-100
0
-45
-90
-135
-180
10
2
10
3
10
4
10
5
10
6
10
7
From: ? To: ?̃
?
?̃
?
(?)/?(?)
Frequency (rad/s)
Magnitude (dB)
Phase (deg)
Figure 1.5:
Qv
o
.s/
Q
d .s/
D 6257:7
sC210
5
s
2
C1203sC2:52310
7
Bode diagram.
14 1. EXTRACTION OF UNCERTAIN MODEL OF DC-DC CONVERTERS
20
10
0
-10
-20
-30
270
225
180
135
90
From:
To:
()/
()
Magnitude (dB)
Phase (deg)
Frequency (rad/s)
10
2
10
1
10
3
10
4
10
5
10
6
10
7
Figure 1.6:
Qv
o
.s/
Q
i
o
.s/
D 0:0499
.sC210
5
/.sC580/
s
2
C1203sC2:52310
7
Bode diagram.
1.4. DYNAMICS OF A BUCK CONVERTER WITHOUT UNCERTAINTY 15
50
0
-50
-100
-150
0
-45
-90
-135
-180
10
2
10
3
10
4
10
5
10
6
10
7
From: ?̃
?
To: ?̃
?
?̃
?
(?)/?̃
?
(?)
Magnitude (dB)
Phase (deg)
Frequency (rad/s)
Figure 1.7:
Qv
o
.s/
Qv
g
.s/
D 49:875
.sC210
5
/
s
2
C1203sC2:52310
7
Bode diagram.
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