Compute real and imaginary part of parametrized complex rational expression

Question

I would like to compute the real and imaginary part of a complex valued rational function $Z_b=\frac{1}{\frac{1}{\frac{1}{C_1 \text{i$\omega $}}+L \text{i$\omega $}+R}+C_2 \text{i$\omega $}}$ that depends on parameters $R,C_1,C_2,L,\omega$ which are all real.

$Z_b$ is defined in the notebook code below:

Subscript[Z, a] := s*L + 1/(s*Subscript[C, 1]) + R
Zb := 1/(1/Subscript[Z, a] + s*Subscript[C, 2])
Subscript[Z, b] := Zb /. s -> I\[Omega]
$Assumptions = 
 L > 0 && R > 0 && Subscript[C, 1] > 0 && 
  Subscript[C, 2] > 0 && \[Omega] > 0
ComplexExpand[Subscript[Z, b]]

Now I am not completely sure which functions I could use for this. I tried using ComplexExpand but that did not seem to give me the desired result and instead simply gave me back the definition of $Z_b$.

Answer 1

You get a much simple form and better overview applying Together some times.

$Assumptions = 
L > 0 && R > 0 && Subscript[C, 1] > 0 && 
Subscript[C, 2] > 0 && \[Omega] > 0;

Subscript[Z, a] = s*L + 1/(s*Subscript[C, 1]) + R // Together;

Zb = 1/(1/Subscript[Z, a] + s*Subscript[C, 2]) // Together;

Subscript[Z, b] = Zb /. s -> I \[Omega];

ce = ComplexExpand[Through[{Re, Im}[Subscript[Z, b]]], 
     TargetFunctions -> {Re, Im}] // Simplify

Answer 2

I was able to resolve this issue by adjusting /. s -> I\[Omega] to /. s -> I*\[Omega] and simply calling ComplexExpand.

Subscript[Z, a] := s*L + 1/(s*Subscript[C, 1]) + R
Zb := 1/(1/Subscript[Z, a] + s*Subscript[C, 2])
Subscript[Z, b] := Zb /. s -> I*\[Omega]
ComplexExpand[Subscript[Z, b]]

I then manually copied the real and imaginary part to a new cell where I called Simplify on the expression.