Nyquist Plot Intersection with Real Axes

[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 ?

 

[milesyoung]

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.

... 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.

 

[nebbione]

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 ?