For example in this 4th order transfer function how the damping ratio would be calculated? enter image description here

in fact I` m encountered with this problem: If I didnt realize the concept of problem plz guide me:

problem: calculate gain magnitude (k) if damper ratio of closed loop system is 0.7 (it means zita=0.7):


open loop transfer function:

enter image description here

the expansion of open loop system fraction:

enter image description here

and now how much is zita? quadratic faraction isn t standard form because there s a 's' in numerator, am I right?

how much is k due to the zita has a magnitude of 0.7? enter image description here

  • 4
    \$\begingroup\$ How the damping ratio is defined for 4'th order system? \$\endgroup\$
    – Eugene Sh.
    Commented Jan 17, 2018 at 17:34
  • \$\begingroup\$ This transfer function does not exhibit any peak in gain. take the derivative and see why. \$\endgroup\$ Commented Jan 17, 2018 at 18:42
  • \$\begingroup\$ In this case, it probably means the damping coefficient of the bracketed 2nd order term, since this is under-damped. But strictly it's undefined for a TF higher than 2nd order. \$\endgroup\$
    – Chu
    Commented Jan 17, 2018 at 18:55

2 Answers 2


The damping ratio is a parameter, usually denoted by ζ (zeta),1 that characterizes the frequency response of a second order ordinary differential equation.

The quote above is taken from Wikipedia: Damping ratio. In other words it relates to a 2nd order transfer function and not a 4th order system. Having said that, if it is possible to reduce the denominator to two multiplying equations each of the form: -

\$s^2 +2s\zeta \omega_n + \omega_n^2\$ (where \$\zeta\$ is damping ratio and \$\omega_n\$ is natural resonant frequency)

Then it would have some meaning.

  • \$\begingroup\$ I didnt find out my response \$\endgroup\$
    – muhamad
    Commented Jan 17, 2018 at 17:47
  • \$\begingroup\$ @somebody If you haven't yet got it, the concept "damping factor" only applies to quadratics. Suppose you have a quadratic: the names of each constant are: quadratic coefficient, linear coefficient and constant coefficient, in that order. The linear coefficient has a certain meaning. Now, suppose someone gave you a 4th order equation. The linear coefficient would NOT have the same meaning, anymore. The whole idea is in the trash, so to speak. There is no understood and well-defined meaning for "damping ratio" on a 4th order system. So you need to define your meaning. \$\endgroup\$
    – jonk
    Commented Jan 17, 2018 at 17:55
  • 1
    \$\begingroup\$ @somebody The idea of a damping factor in 2nd order is as an index of its tendency towards oscillation. If you can work out an index that achieves this goal for 4th order systems, then I think you'd already have your answer anyway. But feel free to discuss it, as a comprehensive study of the set of all possible 4th order filters in this context would actually be interesting to read (to me, anyway.) \$\endgroup\$
    – jonk
    Commented Jan 17, 2018 at 18:02
  • \$\begingroup\$ @jonk I would find it interesting too. \$\endgroup\$
    – Andy aka
    Commented Jan 17, 2018 at 18:06

In this case, it probably means the \$\small \zeta\$ of the bracketed 2nd order term, since this is under-damped. But strictly it's undefined for a TF higher than 2nd order.

Also, this appears to be an open loop TF, so the question may be looking for the 'equivalent' 2nd order closed loop TF where the \$\small \zeta\$ value is given by the ROT: \$\small\zeta\approx 0.01\times PM\$, and \$\small PM\$ is the phase margin in degrees.

  • \$\begingroup\$ how is the manner of calculating its phase margine? \$\endgroup\$
    – muhamad
    Commented Jan 17, 2018 at 19:37
  • \$\begingroup\$ From the open loop Bode plot, it's (180 - phase angle) at the frequency where the gain is 0dB \$\endgroup\$
    – Chu
    Commented Jan 18, 2018 at 7:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.