PKœv‰LñB–Hmimetypetext/x-wxmathmlPKœv‰L‡öżo÷o÷ content.xml Boost Converter, esr state 1 matrices A1: matrix([0, 0], [0, 0]); (%o1) 0000 B1: matrix([1/L, 0], [0, -1/c]); (%o2) 1L00−1c C1: matrix([0, 1]); (%o3) 01 D1: matrix([0, -R]); (%o4) 0−R state 2 matrices A2: matrix([-R/L, -1/L], [1/c, 0]); (%o5) −RL−1L1c0 B2: matrix([1/L, R/L], [0, -1/c]); (%o6) 1LRL0−1c C2: matrix([R, 1]); (%o7) R1 D2: matrix([0, -R]); (%o8) 0−R state variables, dc X0: matrix([Il], [Vc]); (%o9) IlVc input variables, dc U0: matrix([Vin], [Iout]); (%o10) VinIout basic computation; keep it as it is A: D0 * A1 + (1 - D0) * A2; (%o11) −

1−D0

*R
L
−1−D0L
1−D0c0
A: ratsimp(A); (%o12)

D0−1

*R
L
D0−1L
−D0−1c0
B: D0 * B1 + (1 - D0) * B2; (%o13) D0L+1−D0L

1−D0

*R
L
0−D0c−1−D0c
B: ratsimp(B); (%o14) 1L−

D0−1

*R
L
0−1c
C: D0 * C1 + (1 - D0) * C2; (%o15)

1−D0

*R
1
C: ratsimp(C); (%o16)

1−D0

*R
1
D: D0 * D1 + (1 - D0) * D2; (%o17) 0−D0*R−

1−D0

*R
D: ratsimp(D); (%o18) 0−R dc computation, still without d hat, keep it as it is iA: invert(A); (%o19) 0−cD0−1LD0−1c*RD0−1 iA: ratsimp(iA); (%o20) 0−cD0−1LD0−1c*RD0−1 x0: -iA . B . U0; (%o21) −IoutD0−1Iout*RD0−1−L*

VinL−Iout*

D0−1

*R
L

D0−1
x0: ratsimp(x0); (%o22) −IoutD0−1Iout*D0*R−VinD0−1 y0: (D - C . iA . B) . U0; (%o23) Iout*

RD0−1−

1−D0

*RD0−1

−VinD0−1
y0: ratsimp(y0); (%o24) Iout*D0*R−VinD0−1 ac computation, keep it as it is E: (A1 - A2) . X0 + (B1 - B2) . U0; (%o25) −Iout*RL+Il*RL+VcL−Ilc E: ratsimp(E); (%o26) −

Iout−Il

*R−Vc
L
−Ilc
F: (C1 - C2) . X0 + (D1 - D2) . U0; (%o27) −Il*R F: matrix([ratsimp(F)]); (%o28) −Il*R merging E and F; keep it as it is B: addcol(B, E); (%o29) 1L−

D0−1

*R
L
−

Iout−Il

*R−Vc
L
0−1c−Ilc
D: addcol(D, F); (%o30) 0−R−Il*R computing transfer functions; keep it as it is S0: s * diagmatrix(2, 1) - A; (%o31) s−

D0−1

*R
L
−D0−1L
D0−1cs
S: invert(S0); (%o32) ss*

s−

D0−1

*R
L

+

D0−1

2
c*L
D0−1L*

s*

s−

D0−1

*R
L

+

D0−1

2
c*L

−D0−1c*

s*

s−

D0−1

*R
L

+

D0−1

2
c*L

s−

D0−1

*R
L
s*

s−

D0−1

*R
L

+

D0−1

2
c*L
S: ratsimp(S); (%o33) −c*s*L

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−c*D0−c

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1

D0−1

*L

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1

c*D0−c

*R−c*s*L

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
S: facsum(S, s); (%o34) c*s*L−c*s*

D0−1

*R+c*s2*L+

D0−1

2
c*

D0−1

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−

D0−1

*L
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
c*s*L−c*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
tox: S . B; (%o35) c*s−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−D0−1−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−c*s*

Iout−Il

*R−Vc

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−Il*

D0−1

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−D0−1−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

2
*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−c*s*L−c*

D0−1

*R
c*

−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*

Iout−Il

*R−Vc

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−Il*

c*s*L−c*

D0−1

*R

c*

−c*s*

D0−1

*R+c*s2*L+

D0−1

2

tox: ratsimp(tox); (%o36) −c*s

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−1+D0+

c*s*D0−c*s

*R

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−Il−c*s*Vc+Il*D0+

c*Iout−c*Il

*s*R

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
D0−1

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−

D02−D0

*R−s*L

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−Vc−Vc*D0−Il*s*L+

Iout*D0−Iout

*R

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
tox: facsum(tox, s); (%o37) c*s−c*s*

D0−1

*R+c*s2*L+

D0−1

2
1−D0−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−Il*

D0−1

−c*s*

−Vc−Il*R+Iout*R

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
1−D0−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*D0*R−s*L
−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*

Iout*R−Vc

−Il*s*L
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
tox: ev(tox, Il = x0[1, 1], Vc = x0[2, 1]); (%o38) c*s−c*s*

D0−1

*R+c*s2*L+

D0−1

2
1−D0−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
Iout−c*s*

Iout*R+Iout*RD0−1−Iout*D0*R−VinD0−1

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
1−D0−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*D0*R−s*L
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
Iout*s*LD0−1+

D0−1

*

Iout*R−Iout*D0*R−VinD0−1

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
tox: ratsimp(tox); (%o39) −c*s

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−1+D0+

c*s*D0−c*s

*R

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−−Iout−c*s*Vin+Iout*D0

c*s−2*c*s*D0+c*s*D02

*R+

c*s2−c*s2*D0

*L−D03+3*D02−3*D0+1
D0−1

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−

D02−D0

*R−s*L

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
Vin−Vin*D0−Iout*s*L+

Iout*D0−Iout

*R

c*s−2*c*s*D0+c*s*D02

*R+

c*s2−c*s2*D0

*L−D03+3*D02−3*D0+1
tox: facsum(tox, s); (%o40) c*s−c*s*

D0−1

*R+c*s2*L+

D0−1

2
1−D0−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
Iout*

D0−1

−c*s*Vin
−c*s*

D0−1

2
*R+c*s2*

D0−1

*L+

D0−1

3
1−D0−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*D0*R−s*L
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
Iout*s*L−

D0−1

*

Iout*R−Vin

−c*s*

D0−1

2
*R+c*s2*

D0−1

*L+

D0−1

3
toy: C . S . B + D; (%o41) c*s*

1−D0

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−D0−1−c*s*

D0−1

*R+c*s2*L+

D0−1

2

1−D0

*R*

−D0−1−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2

−c*s*L−c*

D0−1

*R
c*

−c*s*

D0−1

*R+c*s2*L+

D0−1

2

+

D0−1

2
*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−R

1−D0

*R*

−Il*

D0−1

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−c*s*

Iout−Il

*R−Vc

−c*s*

D0−1

*R+c*s2*L+

D0−1

2

−Il*

c*s*L−c*

D0−1

*R

c*

−c*s*

D0−1

*R+c*s2*L+

D0−1

2

+

D0−1

*

Iout−Il

*R−Vc

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−Il*R
toy: ratsimp(toy); (%o42) −1+D0+

c*s*D0−c*s

*R

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−−s*L+

−c*s2*L+D02−D0

*R+

c*s*D02−c*s*D0

*R2

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−Vc−Vc*D0−Il*s*L+

−c*Il*s2*L+

Iout−c*s*Vc

*D0+c*s*Vc−Iout

*R+

c*Iout*s*D0−c*Iout*s

*R2

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
toy: facsum(toy, s); (%o43) 1−D0−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*D0*R−c*s2*L*R+s*

c*D02*R2−c*D0*R2−L

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
−c*Il*s2*L*R+

D0−1

*

Iout*R−Vc

+s*

c*Iout*D0*R2−c*Iout*R2−c*Vc*D0*R+c*Vc*R−Il*L

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
toy: ev(toy, Il = x0[1, 1], Vc = x0[2, 1]); (%o44) 1−D0−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*D0*R−c*s2*L*R+s*

c*D02*R2−c*D0*R2−L

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
c*Iout*s2*L*RD0−1+

D0−1

*

Iout*R−Iout*D0*R−VinD0−1

+s*

c*Iout*D0*R2−c*Iout*R2−c*D0*R*

Iout*D0*R−Vin

D0−1
+c*R*

Iout*D0*R−Vin

D0−1
+Iout*LD0−1

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
toy: ratsimp(toy); (%o45) −1+D0+

c*s*D0−c*s

*R

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
−−s*L+

−c*s2*L+D02−D0

*R+

c*s*D02−c*s*D0

*R2

c*s*D0−c*s

*R−c*s2*L−D02+2*D0−1
Vin−Vin*D0−Iout*s*L+

−c*Iout*s2*L+

Iout−c*s*Vin

*D0+c*s*Vin−Iout

*R+

c*Iout*s*D0−c*Iout*s

*R2

c*s−2*c*s*D0+c*s*D02

*R+

c*s2−c*s2*D0

*L−D03+3*D02−3*D0+1
toy: facsum(toy, s); (%o46) 1−D0−c*s*

D0−1

*R
−c*s*

D0−1

*R+c*s2*L+

D0−1

2

D0−1

*D0*R−c*s2*L*R+s*

c*D02*R2−c*D0*R2−L

−c*s*

D0−1

*R+c*s2*L+

D0−1

2
c*Iout*s2*L*R−

D0−1

*

Iout*R−Vin

−s*

−Iout*L+c*Vin*R−c*Vin*D0*R−c*Iout*R2+c*Iout*D0*R2

−c*s*

D0−1

2
*R+c*s2*

D0−1

*L+

D0−1

3
substituting values tox0: ev(tox, R = 0); (%o47) c*sc*s2*L+

D0−1

2
1−D0c*s2*L+

D0−1

2
Iout*

D0−1

−c*s*Vin
c*s2*

D0−1

*L+

D0−1

3
1−D0c*s2*L+

D0−1

2
−s*Lc*s2*L+

D0−1

2
Vin*

D0−1

+Iout*s*L
c*s2*

D0−1

*L+

D0−1

3
tox0: facsum(tox0, s); (%o48) c*sc*s2*L+

D0−1

2
1−D0c*s2*L+

D0−1

2
Iout*

D0−1

−c*s*Vin
c*s2*

D0−1

*L+

D0−1

3
1−D0c*s2*L+

D0−1

2
−s*Lc*s2*L+

D0−1

2
Vin*

D0−1

+Iout*s*L
c*s2*

D0−1

*L+

D0−1

3
toy0: ev(toy, R = 0); (%o49) 1−D0c*s2*L+

D0−1

2
−s*Lc*s2*L+

D0−1

2
Vin*

D0−1

+Iout*s*L
c*s2*

D0−1

*L+

D0−1

3
toy0: facsum(toy0, s); (%o50) 1−D0c*s2*L+

D0−1

2
−s*Lc*s2*L+

D0−1

2
Vin*

D0−1

+Iout*s*L
c*s2*

D0−1

*L+

D0−1

3
tox1: ev(tox, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0); (%o51) 5.0*10−4*s2.5*10−8*s2+0.250.52.5*10−8*s2+0.25−0.25−0.005*s−1.25*10−8*s2−0.1250.52.5*10−8*s2+0.25−5.0*10−5*s2.5*10−8*s2+0.252.5*10−5*s−5.0−1.25*10−8*s2−0.125 tox2: ev(tox, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0.1); (%o52) 5.0*10−4*s2.5*10−8*s2+2.5*10−5*s+0.250.5+2.5*10−5*s2.5*10−8*s2+2.5*10−5*s+0.25−0.25−0.005*s−1.25*10−8*s2−1.25*10−5*s−0.1250.52.5*10−8*s2+2.5*10−5*s+0.25−0.025−5.0*10−5*s2.5*10−8*s2+2.5*10−5*s+0.252.5*10−5*s−4.975−1.25*10−8*s2−1.25*10−5*s−0.125 toy1: ev(toy, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0); (%o53) 0.52.5*10−8*s2+0.25−5.0*10−5*s2.5*10−8*s2+0.252.5*10−5*s−5.0−1.25*10−8*s2−0.125 factor(toy1); rat: replaced 0.5 by 1/2 = 0.5rat: replaced 0.25 by 1/4 = 0.25rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8rat: replaced -5.0E-5 by -1/20000 = -5.0E-5rat: replaced 0.25 by 1/4 = 0.25rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8rat: replaced -5.0 by -5/1 = -5.0rat: replaced 2.5E-5 by 1/40000 = 2.5E-5rat: replaced -0.125 by -1/8 = -0.125rat: replaced -1.25E-8 by -1/80000000 = -1.25E-8(%o54) 20000000s2+10000000−2000*ss2+10000000−2000*

s−200000

s2+10000000
toy2: ev(toy, D0 = 0.5, Vin = 10, Iout = 0.5, L = 50e-6, c = 0.5e-3, R = 0.1); (%o55) 0.5+2.5*10−5*s2.5*10−8*s2+2.5*10−5*s+0.25−0.025−5.125*10−5*s−2.5*10−9*s22.5*10−8*s2+2.5*10−5*s+0.25−4.975−2.2375*10−4*s+1.25*10−9*s2−1.25*10−8*s2−1.25*10−5*s−0.125 factor(toy2); rat: replaced 0.5 by 1/2 = 0.5rat: replaced 2.5E-5 by 1/40000 = 2.5E-5rat: replaced 0.25 by 1/4 = 0.25rat: replaced 2.5E-5 by 1/40000 = 2.5E-5rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8rat: replaced -0.025 by -1/40 = -0.025rat: replaced -5.125E-5 by -41/800000 = -5.125E-5rat: replaced -2.5E-9 by -1/400000000 = -2.5E-9rat: replaced 0.25 by 1/4 = 0.25rat: replaced 2.5E-5 by 1/40000 = 2.5E-5rat: replaced 2.5E-8 by 1/40000000 = 2.5E-8rat: replaced -0.125 by -1/8 = -0.125rat: replaced -1.25E-5 by -1/80000 = -1.25E-5rat: replaced -1.25E-8 by -1/80000000 = -1.25E-8rat: replaced -4.975 by -199/40 = -4.975rat: replaced -2.2375E-4 by -179/800000 = -2.2375E-4rat: replaced 1.25E-9 by 1/800000000 = 1.25E-9(%o56) 1000*

20000+s

s2+1000*s+10000000
−

500+s

*

20000+s

10*

s2+1000*s+10000000

−

s−199000

*

20000+s

10*

s2+1000*s+10000000

PKœv‰LñB–HmimetypePKœv‰L‡öżo÷o÷ 5content.xmlPKoÍ÷