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.

Lengthy phase margin calculation

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)?

  • \$\begingroup\$ Do you have a link to the assertion? \$\endgroup\$
    – Andy aka
    Commented Jun 22, 2016 at 11:06
  • \$\begingroup\$ It's a design ROT for systems that are, ostensibly, 2nd order, and should be treated as such. \$\endgroup\$
    – Chu
    Commented Jun 22, 2016 at 13:59

2 Answers 2


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

enter image description here


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).


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