# Phase Margin and Damping Ratio Approximation

I have read in many texts that the closed loop system damping factor can be approximated as:

$$\\Phi_m= 100 * \zeta\$$

With $$\\Phi_m\$$ as the phase margin and $$\\zeta\$$ as the damping ratio.
The actual relation between the two is more complicated and I think requires numerical method to solve.

How is the approximation made in a second order control system, and when is it valid to consider the approximation (what's the range of the phase margin)?

• Do you have a link to the assertion? – Andy aka Jun 22 '16 at 11:06
• It's a design ROT for systems that are, ostensibly, 2nd order, and should be treated as such. – Chu Jun 22 '16 at 13:59

You can see when the approximation is good simply by plotting the two curves.

The mentioned approximation for the phase margin (100*damping factor) applies to a second order system only when the damping factor is smaller than 1/SQRT(2)=0.7071 or when the phase margin is smaller than app. 65 deg. (Ref.: R.C. Dorf, Modern Control Systems, 6th edition, Addison-Wesley).