Why isn't the real part of z just the coefficient R?

[whatisreality]

In the equation 1/z = 1/R +1/(iwL), why isn't the real part just R?

Unless my rearrangement is wrong, z=R+iwl, so Re(z) should definitely be R!

 

[HallsofIvy]

Yes, your rearrangement is wrong. \frac{1}{a}= \frac{1}{b}+ \frac{1}{c} does NOT give "a= b+ c". As a check, suppose b= c= 2. The 1/a= 1/2+ 1/2= 1 so a= 1 But b+ c= 4, not 1.

From \frac{1}{z}= \frac{1}{R}+ \frac{1}{iWL}, multiply on both sides by all denominators, RiWLz, to get RiWL= iWLz+ Rz= (R+ iWL)z. Solve that for z to find the real and imaginary part.

 

[wabbit]

So you're saying 1/(R+iwL)=1/R+1/(iwL) ?

 

[whatisreality]

Yes. I did actually think that was weird, not sure why I didn't check it. Let me try that rearrangement again...

 

[whatisreality]

z = -R²ω²L²/(iωL+R) is my second attempt!Edit: maybe not. L squared too. And that makes life more difficult! What do I do about i on the denominator? Multiply by the conjugate?

 

[HallsofIvy]

Pretty standard: multiply both numerator and denominator by the complex conjugate of the denominator.
\frac{a+ bi}{c+ di}= \frac{a+ bi}{c+ di}\frac{c- di}{c- di}= \frac{ac+ bd+ (bc- ad)i}{c^2+ d^2}=\frac{ac+ bd}{c^2+ d^2}+ \frac{bc- ad}{c^2+ d^2} i

Here you want to multiply numerator and denominator by R- i\omega L.

 

[whatisreality]

z = RiwL/(R+iwL)
Realise the denominator:
(iR²wL+Rw²L²)/(R²+w²L²)

 

[wabbit]

You can do better than that (your first line is OK though).
Try this as a warm up : forget the exercise, just compute \frac{1}{a+ib}. Then go back to z.