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Complex arguments

by ajtgraves
Tags: arguments, complex
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May6-14, 03:27 AM
P: 2
I'm trying to sketch the nyquist plot of
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
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May6-14, 04:52 PM
P: 160
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)##.
May6-14, 05:20 PM
P: 2
Aaah, I'm with you now, that all makes much more sense. This link helped me too:

Thanks very much once more, I really appreciate the help.

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