\( \DeclareMathOperator{\abs}{abs} \)

Buck Converter, esr

state 1 matrices

(%i1) A1: matrix([-R/L, -1/L], [1/c, 0]);
\[\mathrm{\tt (\%o1) }\quad \begin{pmatrix}-\frac{R}{L} & -\frac{1}{L}\cr \frac{1}{c} & 0\end{pmatrix}\]
(%i2) B1: matrix([1/L, R/L], [0, -1/c]);
\[\mathrm{\tt (\%o2) }\quad \begin{pmatrix}\frac{1}{L} & \frac{R}{L}\cr 0 & -\frac{1}{c}\end{pmatrix}\]
(%i3) C1: matrix([R, 1]);
\[\mathrm{\tt (\%o3) }\quad \begin{pmatrix}R & 1\end{pmatrix}\]
(%i4) D1: matrix([0, -R]);
\[\mathrm{\tt (\%o4) }\quad \begin{pmatrix}0 & -R\end{pmatrix}\]

state 2 matrices

(%i5) A2: matrix([-R/L, -1/L], [1/c, 0]);
\[\mathrm{\tt (\%o5) }\quad \begin{pmatrix}-\frac{R}{L} & -\frac{1}{L}\cr \frac{1}{c} & 0\end{pmatrix}\]
(%i6) B2: matrix([0, R/L], [0, -1/c]);
\[\mathrm{\tt (\%o6) }\quad \begin{pmatrix}0 & \frac{R}{L}\cr 0 & -\frac{1}{c}\end{pmatrix}\]
(%i7) C2: matrix([R, 1]);
\[\mathrm{\tt (\%o7) }\quad \begin{pmatrix}R & 1\end{pmatrix}\]
(%i8) D2: matrix([0, -R]);
\[\mathrm{\tt (\%o8) }\quad \begin{pmatrix}0 & -R\end{pmatrix}\]

state variables, dc

(%i9) X0: matrix([Il], [Vc]);
\[\mathrm{\tt (\%o9) }\quad \begin{pmatrix}\mathit{Il}\cr \mathit{Vc}\end{pmatrix}\]

input variables, dc

(%i10) U0: matrix([Vin], [Iout]);
\[\mathrm{\tt (\%o10) }\quad \begin{pmatrix}\mathit{Vin}\cr \mathit{Iout}\end{pmatrix}\]

basic computation; keep it as it is

(%i11) A: D0 * A1 + (1 - D0) * A2;
\[\mathrm{\tt (\%o11) }\quad \begin{pmatrix}-\frac{\mathit{D0}\cdot R}{L}-\frac{\left( 1-\mathit{D0}\right) \cdot R}{L} & -\frac{\mathit{D0}}{L}-\frac{1-\mathit{D0}}{L}\cr \frac{\mathit{D0}}{c}+\frac{1-\mathit{D0}}{c} & 0\end{pmatrix}\]
(%i12) A: ratsimp(A);
\[\mathrm{\tt (\%o12) }\quad \begin{pmatrix}-\frac{R}{L} & -\frac{1}{L}\cr \frac{1}{c} & 0\end{pmatrix}\]
(%i13) B: D0 * B1 + (1 - D0) * B2;
\[\mathrm{\tt (\%o13) }\quad \begin{pmatrix}\frac{\mathit{D0}}{L} & \frac{\mathit{D0}\cdot R}{L}+\frac{\left( 1-\mathit{D0}\right) \cdot R}{L}\cr 0 & -\frac{\mathit{D0}}{c}-\frac{1-\mathit{D0}}{c}\end{pmatrix}\]
(%i14) B: ratsimp(B);
\[\mathrm{\tt (\%o14) }\quad \begin{pmatrix}\frac{\mathit{D0}}{L} & \frac{R}{L}\cr 0 & -\frac{1}{c}\end{pmatrix}\]
(%i15) C: D0 * C1 + (1 - D0) * C2;
\[\mathrm{\tt (\%o15) }\quad \begin{pmatrix}\mathit{D0}\cdot R+\left( 1-\mathit{D0}\right) \cdot R & 1\end{pmatrix}\]
(%i16) C: ratsimp(C);
\[\mathrm{\tt (\%o16) }\quad \begin{pmatrix}R & 1\end{pmatrix}\]
(%i17) D: D0 * D1 + (1 - D0) * D2;
\[\mathrm{\tt (\%o17) }\quad \begin{pmatrix}0 & -\mathit{D0}\cdot R-\left( 1-\mathit{D0}\right) \cdot R\end{pmatrix}\]
(%i18) D: ratsimp(D);
\[\mathrm{\tt (\%o18) }\quad \begin{pmatrix}0 & -R\end{pmatrix}\]

dc computation, still without d hat, keep it as it is

(%i19) iA: invert(A);
\[\mathrm{\tt (\%o19) }\quad \begin{pmatrix}0 & c\cr -L & -c\cdot R\end{pmatrix}\]
(%i20) iA: ratsimp(iA);
\[\mathrm{\tt (\%o20) }\quad \begin{pmatrix}0 & c\cr -L & -c\cdot R\end{pmatrix}\]
(%i21) x0: -iA . B . U0;
\[\mathrm{\tt (\%o21) }\quad \begin{pmatrix}\mathit{Iout}\cr L\cdot \left( \frac{\mathit{Vin}\cdot \mathit{D0}}{L}+\frac{\mathit{Iout}\cdot R}{L}\right) -\mathit{Iout}\cdot R\end{pmatrix}\]
(%i22) x0: ratsimp(x0);
\[\mathrm{\tt (\%o22) }\quad \begin{pmatrix}\mathit{Iout}\cr \mathit{Vin}\cdot \mathit{D0}\end{pmatrix}\]
(%i23) y0: (D - C . iA . B) . U0;
\[\mathrm{\tt (\%o23) }\quad \mathit{Vin}\cdot \mathit{D0}\]
(%i24) y0: ratsimp(y0);
\[\mathrm{\tt (\%o24) }\quad \mathit{Vin}\cdot \mathit{D0}\]

ac computation, keep it as it is

(%i25) E: (A1 - A2) . X0 + (B1 - B2) . U0;
\[\mathrm{\tt (\%o25) }\quad \begin{pmatrix}\frac{\mathit{Vin}}{L}\cr 0\end{pmatrix}\]
(%i26) E: ratsimp(E);
\[\mathrm{\tt (\%o26) }\quad \begin{pmatrix}\frac{\mathit{Vin}}{L}\cr 0\end{pmatrix}\]
(%i27) F: (C1 - C2) . X0 + (D1 - D2) . U0;
\[\mathrm{\tt (\%o27) }\quad 0\]
(%i28) F: matrix([ratsimp(F)]);
\[\mathrm{\tt (\%o28) }\quad \begin{pmatrix}0\end{pmatrix}\]

merging E and F; keep it as it is

(%i29) B: addcol(B, E);
\[\mathrm{\tt (\%o29) }\quad \begin{pmatrix}\frac{\mathit{D0}}{L} & \frac{R}{L} & \frac{\mathit{Vin}}{L}\cr 0 & -\frac{1}{c} & 0\end{pmatrix}\]
(%i30) D: addcol(D, F);
\[\mathrm{\tt (\%o30) }\quad \begin{pmatrix}0 & -R & 0\end{pmatrix}\]

computing transfer functions; keep it as it is

(%i31) S0: s * diagmatrix(2, 1) - A;
\[\mathrm{\tt (\%o31) }\quad \begin{pmatrix}\frac{R}{L}+s & \frac{1}{L}\cr -\frac{1}{c} & s\end{pmatrix}\]
(%i32) S: invert(S0);
\[\mathrm{\tt (\%o32) }\quad \begin{pmatrix}\frac{s}{s\cdot \left( s+\frac{R}{L}\right) +\frac{1}{c\cdot L}} & -\frac{1}{L\cdot \left( s\cdot \left( s+\frac{R}{L}\right) +\frac{1}{c\cdot L}\right) }\cr \frac{1}{c\cdot \left( s\cdot \left( s+\frac{R}{L}\right) +\frac{1}{c\cdot L}\right) } & \frac{s+\frac{R}{L}}{s\cdot \left( s+\frac{R}{L}\right) +\frac{1}{c\cdot L}}\end{pmatrix}\]
(%i33) S: ratsimp(S);
\[\mathrm{\tt (\%o33) }\quad \begin{pmatrix}\frac{c\cdot s\cdot L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{c}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot L+c\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i34) S: facsum(S, s);
\[\mathrm{\tt (\%o34) }\quad \begin{pmatrix}\frac{c\cdot s\cdot L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{c}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot L+c\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i35) tox: S . B;
\[\mathrm{\tt (\%o35) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}+\frac{1}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}-\frac{c\cdot s\cdot L+c\cdot R}{c\cdot \left( c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1\right) } & \frac{\mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i36) tox: ratsimp(tox);
\[\mathrm{\tt (\%o36) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{1+c\cdot s\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i37) tox: facsum(tox, s);
\[\mathrm{\tt (\%o37) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{1+c\cdot s\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i38) tox: ev(tox, Il = x0[1, 1], Vc = x0[2, 1]);
\[\mathrm{\tt (\%o38) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{1+c\cdot s\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i39) tox: ratsimp(tox);
\[\mathrm{\tt (\%o39) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{1+c\cdot s\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i40) tox: facsum(tox, s);
\[\mathrm{\tt (\%o40) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{1+c\cdot s\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i41) toy: C . S . B + D;
\[\mathrm{\tt (\%o41) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}+\frac{\mathit{D0}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & R\cdot \left( \frac{1}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}+\frac{c\cdot s\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\right) -\frac{c\cdot s\cdot L+c\cdot R}{c\cdot \left( c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1\right) }+\frac{R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}-R & \frac{c\cdot s\cdot \mathit{Vin}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}+\frac{\mathit{Vin}}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i42) toy: ratsimp(toy);
\[\mathrm{\tt (\%o42) }\quad \begin{pmatrix}\frac{\mathit{D0}+c\cdot s\cdot \mathit{D0}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L+c\cdot {{s}^{2}}\cdot L\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}+c\cdot s\cdot \mathit{Vin}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i43) toy: facsum(toy, s);
\[\mathrm{\tt (\%o43) }\quad \begin{pmatrix}\frac{\mathit{D0}+c\cdot s\cdot \mathit{D0}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{-s\cdot L-c\cdot {{s}^{2}}\cdot L\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}+c\cdot s\cdot \mathit{Vin}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i44) toy: ev(toy, Il = x0[1, 1], Vc = x0[2, 1]);
\[\mathrm{\tt (\%o44) }\quad \begin{pmatrix}\frac{\mathit{D0}+c\cdot s\cdot \mathit{D0}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{-s\cdot L-c\cdot {{s}^{2}}\cdot L\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}+c\cdot s\cdot \mathit{Vin}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i45) toy: ratsimp(toy);
\[\mathrm{\tt (\%o45) }\quad \begin{pmatrix}\frac{\mathit{D0}+c\cdot s\cdot \mathit{D0}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L+c\cdot {{s}^{2}}\cdot L\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}+c\cdot s\cdot \mathit{Vin}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i46) toy: facsum(toy, s);
\[\mathrm{\tt (\%o46) }\quad \begin{pmatrix}\frac{\mathit{D0}+c\cdot s\cdot \mathit{D0}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{-s\cdot L-c\cdot {{s}^{2}}\cdot L\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}+c\cdot s\cdot \mathit{Vin}\cdot R}{c\cdot s\cdot R+c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]

substituting values

(%i47) tox0: ev(tox, R = 0);
\[\mathrm{\tt (\%o47) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot {{s}^{2}}\cdot L+1} & \frac{1}{c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i48) tox0: ratsimp(tox0);
\[\mathrm{\tt (\%o48) }\quad \begin{pmatrix}\frac{c\cdot s\cdot \mathit{D0}}{c\cdot {{s}^{2}}\cdot L+1} & \frac{1}{c\cdot {{s}^{2}}\cdot L+1} & \frac{c\cdot s\cdot \mathit{Vin}}{c\cdot {{s}^{2}}\cdot L+1}\cr \frac{\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i49) toy0: subst(0, R, toy);
\[\mathrm{\tt (\%o49) }\quad \begin{pmatrix}\frac{\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i50) toy0: ratsimp(toy0);
\[\mathrm{\tt (\%o50) }\quad \begin{pmatrix}\frac{\mathit{D0}}{c\cdot {{s}^{2}}\cdot L+1} & -\frac{s\cdot L}{c\cdot {{s}^{2}}\cdot L+1} & \frac{\mathit{Vin}}{c\cdot {{s}^{2}}\cdot L+1}\end{pmatrix}\]
(%i51) tox1: ev(tox, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0);
\[\mathrm{\tt (\%o51) }\quad \begin{pmatrix}\frac{2.5\cdot {{10}^{-4}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1} & \frac{1}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1} & \frac{0.01\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1}\cr \frac{0.5}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1} & -\frac{5.0\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1} & \frac{20}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1}\end{pmatrix}\]
(%i52) tox2: ev(tox, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0.1);
\[\mathrm{\tt (\%o52) }\quad \begin{pmatrix}\frac{2.5\cdot {{10}^{-4}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1} & \frac{1+5.0\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1} & \frac{0.01\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1}\cr \frac{0.5}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1} & -\frac{5.0\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1} & \frac{20}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1}\end{pmatrix}\]
(%i53) toy1: ev(toy, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0);
\[\mathrm{\tt (\%o53) }\quad \begin{pmatrix}\frac{0.5}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1} & -\frac{5.0\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1} & \frac{20}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+1}\end{pmatrix}\]
(%i54) toy2: ev(toy, D0 = 0.5, Vin = 20, Iout = 1, L = 50e-6, c = 0.5e-3, R = 0.1);
\[\mathrm{\tt (\%o54) }\quad \begin{pmatrix}\frac{0.5+2.5\cdot {{10}^{-5}}\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1} & \frac{-5.0\cdot {{10}^{-5}}\cdot s-2.5\cdot {{10}^{-9}}\cdot {{s}^{2}}}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1} & \frac{20+0.001\cdot s}{2.5\cdot {{10}^{-8}}\cdot {{s}^{2}}+5.0\cdot {{10}^{-5}}\cdot s+1}\end{pmatrix}\]
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