# Does movement along the imaginary axis of a root locus change the natural frequency?

I have two root locus plots (x-real, y-imaginary) of the same oscillating mode: (1) for a base case system and (2) a system with added feedback. They are so similar, almost identical that I am having a hard time interpreting it but the author comments that "the frequency of the mode has been hardly changed". So he concludes it has not been changed significantly but how can he tell? Is it because the root locus has not moved along the imaginary axis?

On a root locus, zeroes and poles are plotted. What could be changed about these zeroes and poles without changing the frequency of the mode?

• What's the open-loop TF? – Chu Nov 23 '15 at 0:55
• I don't know unfortunately. It's from a textbook. I think it is not legal for me to upload the two figures here? – Martin Bakardjiev Nov 29 '15 at 14:25

## 1 Answer

If a system has a pole at $s = a + ib$, then it has a response that looks like $$r(t) = e^{(a + ib)t} = e^{at}e^{ibt}$$ If these poles have a complex part (ie: $b \neq 0$), then the pair of poles create a real response that looks like $$r(t) = e^{at} \sin(bt) \text{ or } e^{at} \cos(bt)$$

To answer your question:

• Moving the poles along the real axis doesn't change the response frequency. Instead, it changes how quickly the response decays.
• Moving the poles along the imaginary axis changes how fast the response oscillates.

So, like the author suggests, if the imaginary part of the roots doesn't change, then the frequency doesn't change either.

• For a complex root, $\omega_n$ is the length of the vector from the origin to the root, so moving the real part of the root does potentially influence $\omega_n$. By how much depends on the $\zeta$ value. – Chu Nov 23 '15 at 16:39
• Excellent answer. So, to be most general both real and imaginary parts could possibly change the natural frequency in certain cases. Now I understand very well why when the root locus is circular around the origin - the natural frequency does not change. Am I right or did I get confused? – Martin Bakardjiev Nov 23 '15 at 18:23