# Maginutude and Phase Angle for Nyquist Plots

• EugP
In summary, the conversation discusses finding the magnitude and phase angle for a Nyquist Plot of a transfer function. The magnitude is found using a specific equation and the phase angle involves the use of arctangent. There is also a 180-degree shift in the last term of the phase angle equation, which is explained by considering the directed vector of (jw-3) from the origin.
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.

Last edited:
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).

Last edited:

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.

## 1. What is the significance of magnitude in Nyquist plots?

The magnitude in Nyquist plots represents the gain or amplification of a system at a particular frequency. It is represented on the vertical axis of the plot and is measured in decibels (dB). The higher the magnitude, the greater the amplification of the system at that frequency.

## 2. How is phase angle represented in Nyquist plots?

The phase angle in Nyquist plots represents the phase shift or delay of a system at a particular frequency. It is represented on the horizontal axis of the plot and is measured in degrees. The phase angle can be positive or negative, indicating whether the system is leading or lagging in its response at that frequency.

## 3. How do you interpret Nyquist plots?

Nyquist plots can be interpreted by looking at the shape and location of the plot. The shape of the plot can indicate stability or instability of the system. A circular shape indicates a stable system, while a non-circular shape can indicate instability. The location of the plot on the real and imaginary axes can also provide information about the frequency response and characteristics of the system.

## 4. How are magnitude and phase angle related in Nyquist plots?

Magnitude and phase angle are related in Nyquist plots through the frequency response of the system. As the frequency increases, the magnitude may decrease and the phase angle may change. This relationship can provide insight into the behavior of the system at different frequencies.

## 5. What are some common applications of Nyquist plots?

Nyquist plots are commonly used in control systems, circuit analysis, and signal processing. They can be used to analyze the frequency response and stability of systems, as well as design and optimize control systems. They are also useful in understanding the behavior of electronic circuits and filters.

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