I've been trying to figure this problem out for a while now. I've been given a transfer function $$H(s) = \frac{s(s+100)}{(s+2)(s+20)}.$$ I'm supposed to calculate the bode magnitude and frequency for it and then plot them. When I had done out the calculation I ended up getting $$|H(j\omega)| = \frac{\sqrt{\omega^4 – 10000\omega^2}}{\sqrt{\omega^4 – 484\omega^2 + 1600}}.$$
However this doesn't seem to be correct. I keep attempting my calculations but none of them seem to be correct. I'm also not sure how I can go about to plot it. Could anyone please help me out with this? Thanks!
Best Answer
The magnitude of $H(j \omega)$ is
$$|H(j \omega)| = \frac{\sqrt{\omega^4 + 10000\, \omega^2}}{\sqrt{\omega^4+404\, \omega^2 + 1600}}$$
The reason for this is that $|a + i b| = \sqrt{a^2+b^2}$ for $a,b \in \mathbb{R}$.
The phase of $H(j \omega)$ may be computed by taking phases of individual terms:
$$\arg{H(j \omega)} = \frac{\pi}{2} + \arctan{\frac{\omega}{100}} - \arctan{\frac{\omega}{2}}- \arctan{\frac{\omega}{20}}$$