# How do you determine the approximate frequency where the magnitude starts decreasing -40dB/dec for complex pole?

I am stuck at drawing the approximate Bode plot for the complex pole transfer function as below.

Please see the question and problem in the picture.

How do you determine the approximate frequency where the magnitude starts decreasing -40dB/dec as the figure?

PS:

Sorry I made some silly mistake. Actually I want to say that these poles are in the LHS of the complex plane but I got it backward. Please assume that they are in the LHP now and the system is stable.

• Have you tried expanding the polynomial in the deniminator? Who knows what you might get? – a concerned citizen Nov 17 '16 at 7:51
• The poles are in the right half plane, so unstable – Chu Nov 17 '16 at 8:16
• @Chu: I don't care if it is stable or not, just how to calculate it. – anhnha Nov 17 '16 at 8:17
• If it's not stable there is no frequency response – Chu Nov 17 '16 at 8:18
• @ a concerned citizen: I tried that. I can plot exactly the curve. However, what I want to know is the approximation such as what we did for the real poles. – anhnha Nov 17 '16 at 8:18

This might help a bit when the damping is low enough to get a resonant peak: -

It's based around a low pass filter whose transfer function is this: -

This answer might also help and this too.

• Look nice and impressive. However, I knew how to plot exactly like this. What I want to calculate is how to calculate the break frequency in case the approximation plot like this one below: niig-gateguru.in/photoalbum/preview/niig-gateguru/my1/5546806/… – anhnha Nov 17 '16 at 17:52
• We'll, that plot only shows the point where the graph turns from a flat line to a falling line. It gives no clue about zeta hence what you see is all you can say I.e. the break is where it is shown and this can be presumed to be the 3 dB point. – Andy aka Nov 17 '16 at 19:06
• Your starting point in the question was from the math and now you seem to want to understand it from a graph as starting point. I'm trying to help (because I think I can) but I'm getting confused about your real question along the way. – Andy aka Nov 17 '16 at 19:41
• Thanks. On what basis the graph of the link is drawn and can be assumed to be 3dB point? – anhnha Nov 18 '16 at 15:59
• At $\omega_n$ (by definition). – Andy aka Nov 18 '16 at 16:16

To get the frequency where the magnitude starts to decrease follow these steps. Multiply the poles to get a second order polynomial. Divide the polynomial by the constant term to to get a polynomial with S^0 coefficient is 1. Now the frequency where the magnitude starts to decrease the square root of the S^2 coefficient. How much peaking there is can be determined by using this frequency and the S^1 coefficient to find the damping damping ratio.