Maginutude and Phase Angle for Nyquist Plots

[EugP]

Magnitude and Phase Angle for Nyquist Plots

Homework Statement

The assignment is to draw a Nyquist Plot of a certain transfer function. The problem is that I can't figure out how they got the angle for the plot.

$G(s) = \frac{k(s+2)}{(s+1)(s-3)}$

Homework Equations

$|G(j\omega)| \angle G(j\omega)$

The Attempt at a Solution

$G(j\omega) = \frac{k(j\omega+2)}{(j\omega+1)(j\omega-3)}$

From that I know that the magnitude is found like this:
$|G(j\omega)| = \frac{k\sqrt{(\omega^2+4)}}{\sqrt{(\omega^2+1)}\sqrt{(\omega^2+9)}}$

Now, the solution says that the phase angle is:

$\angle G(j\omega) = \angle \tan ^{-1} (\frac{\omega}{2})-\tan ^{-1} (\frac{\omega}{1})-(180 - \tan ^{-1} (\frac{\omega}{3}))$

What I don't understand is why there is a $180^o$ shift in the last term.
Any help would be greatly appreciated.

 

[AqA]

If you consider (jw-3) alone, its y coordinate is 'w' and its x coordinate is '-3'.
Thus (jw-3) as a directed vector from the origin makes an angle of 180-tan-1(w/3) with the positive x axis.
The angle with the positive x-axis is to be considered, not just tan-1(y/x).

 

[Mene]

The phase angle in a Nyquist Plot is typically measured in degrees or radians and represents the difference in phase between the input and output signals of a system. In this case, the phase angle is being calculated using the arctangent function, which takes the ratio of the imaginary and real components of the transfer function. This can provide useful information about the stability and behavior of a system.

The 180 degree shift in the last term of the phase angle calculation is likely due to the fact that the denominator of the transfer function, (s-3), has a negative sign. This negative sign causes a 180 degree phase shift in the output signal. Therefore, in order to accurately represent the phase angle in the Nyquist Plot, this 180 degree shift must be taken into account.

In general, it is important to carefully consider all components and signs in a transfer function when calculating the magnitude and phase angle for a Nyquist Plot. Any discrepancies may result in incorrect or misleading information about the behavior of the system.