Calculating magnitude of a complex fraction

Question

I am having a hard time determining the magnitude of a complex fraction.

$$G(j\omega) = \frac{\omega^{2}_{n}}{j\omega(j\omega + 2\zeta \omega_{n})}$$

I understand that $$|G(j\omega)| = \sqrt{Re\{G(j\omega)\}^{2}+Im\{ G(j\omega)\}^{2}}$$

How would I break $G(j\omega)$ up into real and imaginary parts

Update:

$ \omega_{n}, \zeta$ are pure values

$j$ is an imaginary number

$\omega$ is the independent variable

Answer 1

Assuming that $\omega, \omega_n,\zeta \in \mathbb{R}$, and using that $\,|z|^2 = z \cdot \bar z\,$:

$$ \big|G(j\omega)\big|^2 = \frac{\omega^{2}_{n}}{j\omega(j\omega + 2\zeta \omega_{n})} \cdot \frac{\omega^{2}_{n}}{-j\omega(-j\omega + 2\zeta \omega_{n})} = \frac{\omega_n^{\,4}}{\omega^2 \left(4 \zeta^2\omega_n^{\,2}+\omega^2\right)} $$

Answer 2

Are you required to approach the problem by finding the real and imaginary parts? It's probably easier to use the following properties of complex magnitude:$$ |ab| = |a| \cdot |b|$$ and $$\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}$$

Answer 3

Use conjugation $$G^*(j\omega) =G(-j\omega)$$ ( assuming everything except $j$ is real )

Then $$ Re(G(j\omega))= \frac 12(G(j\omega)+G(-j\omega)) \\ Im(G(j\omega))= \frac 1{2j}(G(j\omega)-G(-j\omega)) $$