# Nyquist Plot Intersection with Real Axes

• nebbione
In summary: You could start by finding F(s) on the real axis and sketching a curve there. You can then use the arctan equation to find the phase shift for s=jω.
nebbione
Hi everyone, I'm real confused and stucked about a point in applying Nyquist stability criterion... now i'll explain why.

I know that it's needed to know how many times I'm wrapping the nyquist critical point (-1;0) with my plot, and I'm enough good to draw by hand a nyquist plot, but the problem is to determine analitically the intersection value of the plot in which I'm interssecting the real axes, I've heard that i have to substitute 'jω' in my transfer function and after i have to divide the real part from the imaginary part of my transfer function and then find out the ω where I'm intersecting the real axes putting the real part of the transfer function equal to zero. Mathematically speaking : RealPart(F(jω))=0 where F is my transfer function.
Once I've found the ω i substitute that ω in my transfer function to find the exact point where I'm intersecting the real axes.
My first question is : Is there a quicker way to compute this point without using software, since I need it for a test ?

I've heard about the arctan way of computing the phase, but the problem is that if I put an equation (for example) like this :
arctan(w/3)+arctan(w/5)-3arctan(w/10)=-180

it's impossible for me to solve it...i don't know how to handle these arctan equations, my second question is : Is there a way to compute it ? And if not, which is the way to solve this problem ?

nebbione said:
My first question is : Is there a quicker way to compute this point without using software, since I need it for a test ?
You could use Im(F(jω)) (edit: You meant Im(F(jω)) = 0 ?) and just punch in likely values for ω (on a calculator) until you find the sign change. That should put you close enough to the intersection with the real axis for a quick sketch.

nebbione said:
... my second question is : Is there a way to compute it ?
By hand? It's easy for some systems, hard for others. I had a very basic implementation of Newton's method on my calculator to solve equations like that numerically. Came in handy now and then.

Last edited:
Can you make me an example ?
For example if i have a transfer function like : F(s)= 10*(s+5)/(s-1)^3 how should i proceed ?

## 1. What is a Nyquist Plot?

A Nyquist Plot is a graphical representation of the frequency response of a system. It shows the relationship between the frequency of a signal and its gain and phase shift. It is commonly used in control systems and signal processing to analyze the stability and performance of a system.

## 2. How is a Nyquist Plot constructed?

A Nyquist Plot is constructed by plotting the complex values of the transfer function of a system on a plane. The real part of the transfer function is plotted on the x-axis, while the imaginary part is plotted on the y-axis. The plot is then traced in a counter-clockwise direction as the frequency increases from 0 to infinity.

## 3. What is the significance of the intersection of the Nyquist Plot with the real axes?

The intersection of the Nyquist Plot with the real axes represents the critical frequency, also known as the gain crossover frequency. This is the frequency at which the gain of the system is equal to 1, and the phase shift is 0 degrees. It is an important point in the plot as it determines the stability and performance of the system.

## 4. How do I interpret the Nyquist Plot intersection with the real axes?

If the Nyquist Plot intersects the real axes at a point above the origin, it indicates that the system is stable. However, if the intersection is below the origin, it suggests that the system is unstable. The distance of the intersection point from the origin also gives an indication of the stability margin of the system.

## 5. What factors can affect the Nyquist Plot intersection with the real axes?

The Nyquist Plot intersection with the real axes can be affected by the system's transfer function, controller design, and external disturbances. A change in any of these factors can alter the critical frequency and, therefore, the intersection point. It is essential to carefully design and analyze the system to ensure a stable and desirable intersection point.

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