Frequency response function (polar diagram)

[Sci-Fry]

This isn't really a homework question but I felt this is the most relevant section for this. I therefore apologize for not following the standard post template.

I was going through my electrical engineering notes on frequency response functions. It was explaining how to plot frequency response functions on polar diagrams and had the following two steps, which I didn't quite understand. It got to this:

$$G(j\omega) = x + jy = \frac{1}{R}\frac{1-j\omega T}{1+(\omega T)^2}$$

It is easy to write:

$$x=\frac{1}{R}\frac{1}{1+(\omega T)^2} ; y=\frac{1}{R}\frac{-\omega T}{1+(\omega T)^2}$$

That's fine, but I didn't understand how they jumped to the next step:

From which it is straightforward to eliminate \omegaT to give

$$(x-\frac{1}{2R})^2 + y^2 = (\frac{1}{4R})^2$$

The point of writing it in this form is that the frequency response can now be plotted as a semi-circle on the Argand diagram. However, I don't understand how they did this step so easily. Is there something I'm not spotting? What is the method from getting from the previous step to this one?

Many thanks,
Sci-Fry

 

[tiny-tim]

Hi Sci-Fry!

(have an omega: ω )

square y, and multiply by R²(1 + (ωT)²) …

that gives you (ωT)²/(1 + (ωT)²), which is 1 - … ?

 

[Sci-Fry]

Hi tiny-tim, thanks for the response (and the omega :P)!

Hmmm... I'm probably being incredibly daft, but I see what you're doing...

[note: couldn't get some of the latex to work so I'm using mimtex off another server]

Is that what you were getting at? I'm still not sure what exactly to do after that. Playing around with the numbers gets me to the wrong answer:

That gets rid of the omega, but it's not what I'm looking for. I know I'm being stupid, just wish I knew where...

 

[tiny-tim]

hmm … now i look at it, my "1 -" is missing the point

square (x - 1/2R), and multiply by R²(1 + (ωT)²)

 

[Sci-Fry]

Oh that makes sense now, got it. Thanks.

Still difficult to spot though. I don't get how the lecturer can call that a straightforward elimination :P