Roots of Symbolic Transfer Function

I am trying to do some symbolic math. I can get a solution for the desired equation, Htf, ( S.Htf), but it is in a different form than I would like.

syms s gm1 gmp ro1 rop R2 R1 Cgs Cout Zcgs Zcgd Zcout Z1 Z2 Vinp Vinm Vout1 Vd Vout2 Htf

eqn1 = Vout1 == (-gm1*Vinp + gm1*Vinm)*Z1;
eqn2 = Z1 == 1/(1/ro1 + 1/Zcgs);
eqn3 = Vd == (-gmp*Vout1)*Z2;
%eqn4 = Iout1 == (Vout1 - Vd)/Zcgd;
eqn5 = Z2 == 1/(1/rop + 1/(R2 + R1) + 1/Zcout);
eqn6 = Vinm == Vd*(R1/(R1 + R2));
eqn7 = Vout2 == Vd*(Zcout/(Zcout));
eqn8 = Zcgs == 1/(s*Cgs);
%eqn9 = Zcgd == 1/(s*Cgd);
eqn10 = Zcout == 1/(s*Cout);
eqn11 = Htf == Vout2/Vinp;

%S = solve(eqn1,eqn2,eqn3,eqn5,eqn6,eqn7,eqn8,eqn10,eqn11,Zcgs,Zcgd,Zcout,Z1,Z2,Vinp,Vinm,Vout1,Vd,Vout2,Iout1,Htf,ReturnConditions=true);
S = solve(eqn1,eqn2,eqn3,eqn5,eqn6,eqn7,eqn8,eqn10,eqn11,Zcgs,Zcgd,Zcout,Z1,Z2,Vinp,Vinm,Vout1,Vd,Vout2,Htf);

S.Htf looks like:

S.Htf = (R1*gm1*gmp*ro1*rop + R2*gm1*gmp*ro1*rop)/(R1 + R2 + rop + Cgs*R1*ro1*s + Cgs*R2*ro1*s + Cout*R1*rop*s + Cout*R2*rop*s + Cgs*ro1*rop*s + R1*gm1*gmp*ro1*rop + Cgs*Cout*R1*ro1*rop*s^2 + Cgs*Cout*R2*ro1*rop*s^2)

I would like to get this in a symbolic form with factored roots in the variable 's', so that the equation looks like:

S.Htf = A/( (B + s)*(C+s) )

Is there a good way to accomplish this? I am struggling to figure it out.

Answer 1

[Walter Roberson]

syms s gm1 gmp ro1 rop R2 R1 Cgs Cout Zcgs Zcgd Zcout Z1 Z2 Vinp Vinm Vout1 Vd Vout2 Htf

eqn1 = Vout1 == (-gm1*Vinp + gm1*Vinm)*Z1;
eqn2 = Z1 == 1/(1/ro1 + 1/Zcgs);
eqn3 = Vd == (-gmp*Vout1)*Z2;
%eqn4 = Iout1 == (Vout1 - Vd)/Zcgd;
eqn5 = Z2 == 1/(1/rop + 1/(R2 + R1) + 1/Zcout);
eqn6 = Vinm == Vd*(R1/(R1 + R2));
eqn7 = Vout2 == Vd*(Zcout/(Zcout));
eqn8 = Zcgs == 1/(s*Cgs);
%eqn9 = Zcgd == 1/(s*Cgd);
eqn10 = Zcout == 1/(s*Cout);
eqn11 = Htf == Vout2/Vinp;

%S = solve(eqn1,eqn2,eqn3,eqn5,eqn6,eqn7,eqn8,eqn10,eqn11,Zcgs,Zcgd,Zcout,Z1,Z2,Vinp,Vinm,Vout1,Vd,Vout2,Iout1,Htf,ReturnConditions=true);
S = solve(eqn1,eqn2,eqn3,eqn5,eqn6,eqn7,eqn8,eqn10,eqn11,Zcgs,Zcgd,Zcout,Z1,Z2,Vinp,Vinm,Vout1,Vd,Vout2,Htf);

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

[N,D] = numden(S.Htf);
R = solve(D, s);

HTF = N / ((R(1) - s) * (R(2) - s))
pretty(HTF)

                                                           R1 gm1 gmp ro1 rop + R2 gm1 gmp ro1 rop
-------------------------------------------------------------------------------------------------------------------------------------------------------------
/     Cgs R1 ro1 - #1 + Cgs R2 ro1 + Cout R1 rop + Cout R2 rop + Cgs ro1 rop \ /     #1 + Cgs R1 ro1 + Cgs R2 ro1 + Cout R1 rop + Cout R2 rop + Cgs ro1 rop \
| s + ---------------------------------------------------------------------- | | s + ---------------------------------------------------------------------- |
\                                       #2                                   / \                                       #2                                   /

where

                 2   2    2        2          2        2       2          2   2    2        2       2          2    2    2                      2    2    2
   #1 == sqrt(Cgs  R1  ro1  + 2 Cgs  R1 R2 ro1  + 2 Cgs  R1 ro1  rop + Cgs  R2  ro1  + 2 Cgs  R2 ro1  rop + Cgs  ro1  rop  - gm1 gmp Cgs Cout R1  ro1  rop  4

                     2                                     2    2                                                     2                2                                2
      - 2 Cgs Cout R1  ro1 rop - gm1 gmp Cgs Cout R1 R2 ro1  rop  4 - 4 Cgs Cout R1 R2 ro1 rop - 2 Cgs Cout R1 ro1 rop  - 2 Cgs Cout R2  ro1 rop - 2 Cgs Cout R2 ro1 rop

            2   2    2         2          2       2   2    2
      + Cout  R1  rop  + 2 Cout  R1 R2 rop  + Cout  R2  rop )

   #2 == (Cgs Cout R1 ro1 rop + Cgs Cout R2 ro1 rop) 2