I have second order transfer function $$ G(s) = \frac{1.247}{s^2+9.76s+23.8}$$ G(s) is in the forward path of a unity feedback system.

How do I find the closed loop transfer function and damping constant from this? Now I know that the equation to find the closed loop transfer function is $$TF = \frac{G}{1+G}$$

If I use that here, I get $$TF = \frac{1.247}{s^2+9.76s+25.047}$$ and It is not in the form of a standard second order system which is $$TF = \frac{\omega_n^2}{s^2+2\delta\omega_ns+\omega_n^2}$$

That is, the numerator in TF is not 25.047. So How do I calculate damping ratio in this situation? Can I still take 25.047 as the square of natural frequency and equate that with the coefficient of s to find damping ratio?

  • 1
    \$\begingroup\$ $$TF =K \frac{\omega_n^2}{s^2+2\delta\omega_ns+\omega_n^2}$$ this is standard form \$\endgroup\$
    – user215805
    Apr 5, 2021 at 18:02

1 Answer 1


As user215805 stated, the standard form is $$TF=K\frac{\omega^2_n}{s^2+2\delta\omega_ns+\omega^2_n}$$ And you have $$TF=\frac{1.247}{s^2+9.76s+25.047}$$ This means that in your case $$\omega^2_n=25.047$$ Which you already know, but notice that from the standard form you can deduce that $$K\omega^2_n=1.247$$ And simply $$K=\frac{1.247}{\omega^2_n}=\frac{1.247}{25.047}≈0.05$$ So now you have $$TF=0.05\frac{25.047}{s^2+9.76s+25.047}$$ Similarly you can deduce that $$2\delta\omega_n=9.76$$ And therefore $$\delta=\frac{9.76}{2\omega_n}≈0.975$$ And so you have reached the standard form $$TF=0.05\frac{25.047}{s^2+2(0.975)\sqrt{25.047}s+25.047}$$


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.