# Standard parameters of a second order transfer function with some zeros

We already know that we can derive the parameters $\omega_{n}$ and $\zeta$ from a second order system which adopts the canonical form:

$H(s) = K\frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$

How can I find the damping ratio and the natural frequency of a second-order system with one or two zeros:

$H(s) = \frac{K_1s^2 + K_2s + K_3}{s^2 + K_4s + K_5}$

$H(s) = \frac{K_1s + K_2}{s^2 + K_3s + K_4}$

• The numerator s-terms differentiate the classical 2nd order transient responses, so ROT performance parameters are not readily extracted.
– Chu
Feb 26 '17 at 23:45
• Natural frequency and damping factor are related to the denominator, its expression doesn't change in the cases of low, band or high pass systems. Nov 11 '17 at 15:50

For a second-order system featuring a double zero, the transfer function could look that way:

$$H(s)= H_0\frac{1 + a_1s + a_2s^2 }{ 1 + b_1s + b_2s^2} = H_0 \frac {N(s)}{D(s)}$$

you can rearrange both $N(s)$ and $D(s)$ under the form

$N(s) = 1 + (s/\omega_{0n}*Q_n) + (s/\omega_{0n})^2$ and $D(s) = 1 + (s/\omega_{0}*Q) + (s/\omega_{0})^2$

with $\omega_{0n} = 1/\sqrt{a_2}$ and $Q_n = \sqrt{a_2}/a_1$ with $\omega_{0} = 1/\sqrt{b_2}$ and $Q = \sqrt{b_2}/b_1$

In your first expression, factor $K_3$ on top and $K_5$ in $D(s)$, you have

$$H(s) = \frac{K_3}{K_5}\frac{(1 + \frac{sK_2}{K_3} + \frac{s^2K_1}{K_3}) }{ 1 + \frac{sK_4}{K_5} + \frac{s^2}{K_5}}$$ then apply what I described above.

Same for the second expression:

$$H(s)= \frac{K_2}{K_4} \frac{(1 + sK_1/K_2) }{1 + \frac{sK_3}{K_4} + \frac{s^2}{K_4}}$$ then apply what I described above.

If you want to learn more about these techniques, have a look at the APEC presentation available here: