A doubt about symbolic rational expression

Question

I have this expression:

$$PB=\frac{(R3 + R4) R6}{R4 R5 + R4 R6 + C2 R2 R3 R5 S + C2 R2 R4 R5 S + C1 C2 R1 R2 R3 R5 S^2 + C1 C2 R1 R2 R3 R6 S^2}$$

I need to put it in this form:

$$PB=\frac{\frac{(R3 + R4) R6}{C1 C2 R1 R2 R3 (R5 + R6)}}{\frac{R4}{C1 C2 R1 R2 R3} + \frac{(R3 R5 + R4 R5)}{C1 R1 R3 (R5 + R6)}S + S^2}$$

I made this code:

In[]:T=(R3 + R4) R6/(R4 R5 + R4 R6 + C2 R2 R3 R5 S + C2 R2 R4 R5 S + C1 C2 R1 R2 R3 R5 S^2 + C1 C2 R1 R2 R3 R6 S^2);
     Den = Cancel[Collect[Denominator[T] /aux, S]]

And I obtained as output:

$$\frac{R4}{C1 C2 R1 R2 R3} + \frac{(R3 R5 + R4 R5)}{C1 R1 R3 (R5 + R6)}S + S^2$$

Then I made:

In[]:Num = Cancel[Numerator[T]/aux]

And I obtained as output:

$$\frac{(R3 + R4) R6}{C1 C2 R1 R2 R3 (R5 + R6)}$$

But when I made:

PB=Num/Den

The output was:

$$\frac{(R3 + R4) R6}{C1 C2 R1 R2 R3 (R5 + R6)(\frac{R4}{(C1 C2 R1 R2 R3)} + \frac{(R3 R5 + R4 R5) S}{C1 R1 R3 (R5 + R6)} + S^2)}$$

My question is: Is there a way to put the expression in the form I need?

Answer 1

Clear["Global`*"]

T = (R3 + 
     R4) R6/(R4 R5 + R4 R6 + C2 R2 R3 R5 S + C2 R2 R4 R5 S + 
      C1 C2 R1 R2 R3 R5 S^2 + C1 C2 R1 R2 R3 R6 S^2);

eq = 1/T == (b + c*S + S^2)/a;

sol = Solve[CoefficientList[#, S] & /@ eq, {a, b, c}][[1]] /.
   Rule[lhs_, rhs_] :> 
    Rule[lhs, Simplify[Numerator[rhs]]/Simplify[Denominator[rhs]]];

T2 = HoldForm[a]/(b + c*S + S^2) /. sol

T2 // ReleaseHold

EDIT: To keep S separate

(T3 = HoldForm[a]/(b + HoldForm[c]*S + S^2) /. sol) // TraditionalForm