Natural frequencies of a transfer function

I have been told in the university that the natural frequencies (also called $\textit{eigenfrequencies}$) are the poles of the transfer function, however, Matlab compute them as the modulus of the pole (information taken from here).

So the real question is, are the natural frequencies complex numbers or real numbers measured in rad/s?

• it might depend on whether it's the $s$-plane or the $z$-plane. i haven't heard it called "natural frequency", but usually i hear it referred to as "resonant frequency". even so, the frequency of ringing is not exactly the same as that but gets closer as the "resonance" or the $Q$ of a resonant system gets large. if that's the case, the resonant frequency and the ringing frequency is about the same. – robert bristow-johnson May 3 '18 at 23:42

The natural frequency is a real number, and Matlab computes it correctly by taking the magnitude of the (complex-valued) pole.

As an example, the denominator of the transfer function of a second-order continuous-time system is given by

$$D(s)=s^2+2\omega_n\zeta s+\omega_n^2\tag{1}$$

where $\zeta$ is the damping ratio, and $\omega_n$ is the natural frequency. The poles are given by the roots of $(1)$. Assuming an underdamped system ($\zeta<1)$, the poles are given by

$$s_{\infty}=-\omega_n\left(\zeta\pm j\sqrt{1-\zeta^2}\right)\tag{2}$$

from which we get

$$|s_{\infty}|^2=\omega_n^2\left[\zeta^2+(1-\zeta^2)\right]=\omega_n^2\tag{3}$$

So the natural frequency $\omega_n$ is given by the magnitude of the pole $s_{\infty}$.

Note that the natural frequency $\omega_n$ is not the actual oscillation frequency unless there is no damping (i.e., $\zeta=0$).