PK¶v‰LñB–Hmimetypetext/x-wxmathmlPK¶v‰LÀÓQْْ content.xml Buck Converter, esr state 1 matrices A1: matrix([-R/L, -1/L], [1/c, 0]); (%o1) −RL−1L1c0 B1: matrix([1/L, R/L], [0, -1/c]); (%o2) 1LRL0−1c C1: matrix([R, 1]); (%o3) R1 D1: matrix([0, -R]); (%o4) 0−R state 2 matrices A2: matrix([-R/L, -1/L], [1/c, 0]); (%o5) −RL−1L1c0 B2: matrix([0, R/L], [0, -1/c]); (%o6) 0RL0−1c 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) −D0*RL−

1−D0

*R
L
−D0L−1−D0L
D0c+1−D0c0
A: ratsimp(A); (%o12) −RL−1L1c0 B: D0 * B1 + (1 - D0) * B2; (%o13) D0LD0*RL+

1−D0

*R
L
0−D0c−1−D0c
B: ratsimp(B); (%o14) D0LRL0−1c C: D0 * C1 + (1 - D0) * C2; (%o15) D0*R+

1−D0

*R
1
C: ratsimp(C); (%o16) R1 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) 0c−L−c*R iA: ratsimp(iA); (%o20) 0c−L−c*R x0: -iA . B . U0; (%o21) IoutL*

Vin*D0L+Iout*RL

−Iout*R
x0: ratsimp(x0); (%o22) IoutVin*D0 y0: (D - C . iA . B) . U0; (%o23) Vin*D0 y0: ratsimp(y0); (%o24) Vin*D0 ac computation, keep it as it is E: (A1 - A2) . X0 + (B1 - B2) . U0; (%o25) VinL0 E: ratsimp(E); (%o26) VinL0 F: (C1 - C2) . X0 + (D1 - D2) . U0; (%o27) 0 F: matrix([ratsimp(F)]); (%o28) 0 merging E and F; keep it as it is B: addcol(B, E); (%o29) D0LRLVinL0−1c0 D: addcol(D, F); (%o30) 0−R0 computing transfer functions; keep it as it is S0: s * diagmatrix(2, 1) - A; (%o31) RL+s1L−1cs S: invert(S0); (%o32) ss*

s+RL

+1c*L
−1L*

s*

s+RL

+1c*L

1c*

s*

s+RL

+1c*L

s+RLs*

s+RL

+1c*L
S: ratsimp(S); (%o33) c*s*Lc*s*R+c*s2*L+1−cc*s*R+c*s2*L+1Lc*s*R+c*s2*L+1c*s*L+c*Rc*s*R+c*s2*L+1 S: facsum(S, s); (%o34) c*s*Lc*s*R+c*s2*L+1−cc*s*R+c*s2*L+1Lc*s*R+c*s2*L+1c*s*L+c*Rc*s*R+c*s2*L+1 tox: S . B; (%o35) c*s*D0c*s*R+c*s2*L+1c*s*Rc*s*R+c*s2*L+1+1c*s*R+c*s2*L+1c*s*Vinc*s*R+c*s2*L+1D0c*s*R+c*s2*L+1Rc*s*R+c*s2*L+1−c*s*L+c*Rc*

c*s*R+c*s2*L+1

Vinc*s*R+c*s2*L+1
tox: ratsimp(tox); (%o36) c*s*D0c*s*R+c*s2*L+11+c*s*Rc*s*R+c*s2*L+1c*s*Vinc*s*R+c*s2*L+1D0c*s*R+c*s2*L+1−s*Lc*s*R+c*s2*L+1Vinc*s*R+c*s2*L+1 tox: facsum(tox, s); (%o37) c*s*D0c*s*R+c*s2*L+11+c*s*Rc*s*R+c*s2*L+1c*s*Vinc*s*R+c*s2*L+1D0c*s*R+c*s2*L+1−s*Lc*s*R+c*s2*L+1Vinc*s*R+c*s2*L+1 tox: ev(tox, Il = x0[1, 1], Vc = x0[2, 1]); (%o38) c*s*D0c*s*R+c*s2*L+11+c*s*Rc*s*R+c*s2*L+1c*s*Vinc*s*R+c*s2*L+1D0c*s*R+c*s2*L+1−s*Lc*s*R+c*s2*L+1Vinc*s*R+c*s2*L+1 tox: ratsimp(tox); (%o39) c*s*D0c*s*R+c*s2*L+11+c*s*Rc*s*R+c*s2*L+1c*s*Vinc*s*R+c*s2*L+1D0c*s*R+c*s2*L+1−s*Lc*s*R+c*s2*L+1Vinc*s*R+c*s2*L+1 tox: facsum(tox, s); (%o40) c*s*D0c*s*R+c*s2*L+11+c*s*Rc*s*R+c*s2*L+1c*s*Vinc*s*R+c*s2*L+1D0c*s*R+c*s2*L+1−s*Lc*s*R+c*s2*L+1Vinc*s*R+c*s2*L+1 toy: C . S . B + D; (%o41) c*s*D0*Rc*s*R+c*s2*L+1+D0c*s*R+c*s2*L+1R*

1c*s*R+c*s2*L+1+c*s*Rc*s*R+c*s2*L+1

−c*s*L+c*Rc*

c*s*R+c*s2*L+1

+Rc*s*R+c*s2*L+1−R
c*s*Vin*Rc*s*R+c*s2*L+1+Vinc*s*R+c*s2*L+1
toy: ratsimp(toy); (%o42) D0+c*s*D0*Rc*s*R+c*s2*L+1−s*L+c*s2*L*Rc*s*R+c*s2*L+1Vin+c*s*Vin*Rc*s*R+c*s2*L+1 toy: facsum(toy, s); (%o43) D0+c*s*D0*Rc*s*R+c*s2*L+1−s*L−c*s2*L*Rc*s*R+c*s2*L+1Vin+c*s*Vin*Rc*s*R+c*s2*L+1 toy: ev(toy, Il = x0[1, 1], Vc = x0[2, 1]); (%o44) D0+c*s*D0*Rc*s*R+c*s2*L+1−s*L−c*s2*L*Rc*s*R+c*s2*L+1Vin+c*s*Vin*Rc*s*R+c*s2*L+1 toy: ratsimp(toy); (%o45) D0+c*s*D0*Rc*s*R+c*s2*L+1−s*L+c*s2*L*Rc*s*R+c*s2*L+1Vin+c*s*Vin*Rc*s*R+c*s2*L+1 toy: facsum(toy, s); (%o46) D0+c*s*D0*Rc*s*R+c*s2*L+1−s*L−c*s2*L*Rc*s*R+c*s2*L+1Vin+c*s*Vin*Rc*s*R+c*s2*L+1 substituting values tox0: ev(tox, R = 0); (%o47) c*s*D0c*s2*L+11c*s2*L+1c*s*Vinc*s2*L+1D0c*s2*L+1−s*Lc*s2*L+1Vinc*s2*L+1 tox0: ratsimp(tox0); (%o48) c*s*D0c*s2*L+11c*s2*L+1c*s*Vinc*s2*L+1D0c*s2*L+1−s*Lc*s2*L+1Vinc*s2*L+1 toy0: subst(0, R, toy); (%o49) D0c*s2*L+1−s*Lc*s2*L+1Vinc*s2*L+1 toy0: ratsimp(toy0); (%o50) D0c*s2*L+1−s*Lc*s2*L+1Vinc*s2*L+1 tox1: ev(tox, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0); (%o51) 2.5*10−4*s2.5*10−8*s2+112.5*10−8*s2+10.01*s2.5*10−8*s2+10.52.5*10−8*s2+1−5.0*10−5*s2.5*10−8*s2+1202.5*10−8*s2+1 tox2: ev(tox, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0.1); (%o52) 2.5*10−4*s2.5*10−8*s2+5.0*10−5*s+11+5.0*10−5*s2.5*10−8*s2+5.0*10−5*s+10.01*s2.5*10−8*s2+5.0*10−5*s+10.52.5*10−8*s2+5.0*10−5*s+1−5.0*10−5*s2.5*10−8*s2+5.0*10−5*s+1202.5*10−8*s2+5.0*10−5*s+1 toy1: ev(toy, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0); (%o53) 0.52.5*10−8*s2+1−5.0*10−5*s2.5*10−8*s2+1202.5*10−8*s2+1 toy2: ev(toy, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0.1); (%o54) 0.5+2.5*10−5*s2.5*10−8*s2+5.0*10−5*s+1−5.0*10−5*s−2.5*10−9*s22.5*10−8*s2+5.0*10−5*s+120+0.001*s2.5*10−8*s2+5.0*10−5*s+1
PK¶v‰LñB–HmimetypePK¶v‰LÀÓQْْ 5content.xmlPKo7“