I'm trying to sketch the nyquist plot of $$\frac{j\omega-1}{j\omega+1}$$ but can't seem to calculate the argument correctly. I think it should be $$\arctan(-\omega) - \arctan(\omega) = -2\arctan(\omega)$$ but this doesn't give the correct nyquist plot behaviour for $\omega \to 0$ and $\omega \to \infty$ - surely $-2\arctan(\omega)$ implies that $\lim_{x\to 0} = 0^\circ$ and $\lim_{x\to \infty} = -180^\circ$? Wolfram Alpha disagrees but I can't see where I'm going wrong. Am I making a glaring error somewhere? Any help would be greatly appreciated. Thanks very much
You're an electrical engineer, right? So j means the square root of -1? Assuming it does, I think you are off by 180 degrees. The formula ##\mathrm{arg}(x + yj) = \arctan(y/x)## is valid when ##x > 0##, but if the real part of ##x + yj## is negative, as it is in the numerator, you need to adjust for the fact that arctan only returns angles in ##(-\pi/2, \pi/2)## by using ##\mathrm{arg}(x + yj) = \pi + \arctan(y/x)##.
Aaah, I'm with you now, that all makes much more sense. This link helped me too: http://en.wikipedia.org/wiki/Complex_number#Absolute_value_and_argument Thanks very much once more, I really appreciate the help.