Finding Im(z) by putting two terms under a common denominator

Question

Sorry if the title is unclear.

I have the following equation for impedance.

$$z=\frac{RiwL}{R+iwL}+\frac{R}{1+iwRC}$$

And I need to find Im(z).

Can I multiply each by its conjugate, and then put the two terms under the same denominator to add them? Or do I need to put the two terms under the same common denominator first, add and then multiply by the conjugate?

Answer 1

multiply the first denominator by $$R-iwL$$ and the second one by $$1-iwRC$$ and you will get $$\frac{L^2Rw^2}{R^2+w^2L^2}+\frac{R}{1+w^2R^2C^2}+i\left(\frac{LR^2w}{R^2+w^2L^2}-\frac{CR^2w}{1+w^2R^2C^2}\right)$$ and you can solve your problem

Answer 2

If I understand it right, what you are asking is whether $(a+b)\overline{(a+b)}=a\overline{a}+b\overline{b}$, which is false. Hence you have to put the two terms together first, or to use that $(a+b)\overline{(a+b)}=a\overline{a}+b\overline{b}+a\overline{b}+b\overline{a}$. But why do you need to multiply by the conjugate? The formula for the imaginary part is

$\text{Im}(z)=\frac{z-\overline{z}}{2i}$