Finding transfer function from Bode plot

Question

While I know how to find the transfer function from the response graph, I don't know how to find it from Bode plot.

What I know about the system:

1. The system is a second order system
2. The system has no zeros
3. According to the Bode plot, I estimate that there is a double pole at w=2.5

therefore, I need to find Wn, zeta and K.

This is the graph:

You can ignore the noise in the end of the phase graph.

Answer 1

For a 2nd-order low-pass filter you can use this to calculate what the damping ratio, (\$\zeta\$) is: -

Image from my basic website.

Look at the peaking frequency in your spectrum and calculate the damping ratio from that. It looks like "k" the gain coefficient is unity (because of the low frequency gain aligning with 0 dB on your picture) so this needn't play a part anymore.

Once you have the damping ratio you can calculate what \$\omega_n\$ (also referred to as \$\omega_0\$) is.

It looks like you have about 8 dB of peaking. That's a ratio of 2.512. So solve the peaking amplitude formula to find \$\zeta\$. I estimate \$\zeta\$ to be about 0.2050.

I estimate the peak position in the spectrum to be at about 0.957053 of the natural resonant frequency, \$\omega_n\$.

Of course you could work a little bit on the graph to find the same answers: -