What is the magnitude of this transfer function?

Question

How can I calculate the magnitude of $G(s)$ ?

$$G(s) = \frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$$

Thank you in advance.

Answer 1

In general, for a complex function $G(s)$, its magnitude can be computed via $|G(s)| = \sqrt{G(s)G^*(s)}$, where $G^*(s)$ denotes complex conjugation.

In your case, $G(s)$ is rational and you can use the fact that $\left|\frac ab\right| = \frac{|a|}{|b|}$, i.e., form magnitude of enumerator and denominator separately.

Moreover, in these types of questions one is often interested in the magnitude of G(s) in the imaginary line, i.e., for $s=\jmath \omega$. This significantly simplifies things. The denominator is then $(\jmath \omega)^2 + 2\jmath\zeta\omega_n\omega+\omega_n^2 = \omega_n^2-\omega^2 + 2\jmath\zeta\omega_n\omega$. This is already in the form $z = \Re\{z\}+\jmath \Im\{z\}$, which allows you to use the rule $|z| = \sqrt{\Re\{z\}^2+\Im\{z\}^2}$.

Connecting the dots, we then get $$|G(\jmath \omega)| = \frac{\omega_n^2}{\sqrt{(\omega_n^2-\omega^2)^2 + 4\zeta^2\omega_n^2\omega^2}} $$.

In case you were looking for $|G(s)|$ outside the imaginary line I guess you can extend what I wrote easily by yourself.