Deriving expressions from transfer function of state variable filter?

Question

I have been trying to design a state variable filter based on a block diagram provided. The transfer function which I have is of the form:

$$\frac{K_1K_2}{s^2 + K_1s + K_1K_2}$$

I need to obtain and derive expressions for the natural frequency and quality factor of the state variable filter in terms of \$K_1\$ and \$K_2\$. In order to do so I need to compare my T.F with a 2nd order normalized T.F for state variable filters. Below is the T.F I am comparing with from this website.

$$\frac{V_{\text{out}}}{V_{\text{in}}} = \frac{A_o\left(\frac{f}{f_o}\right)}{\left[1 + 2\zeta\frac{f}{f_o} + \left(\frac{f}{f_o}\right)^2\right]}$$

I am getting confused since the 2nd order normalized T.F does not have an \$s^2\$ term.

For the natural frequency I have got:

$$\omega_n\text{ or }\frac{f}{f_o} = \sqrt{K_1K_2}$$

and for the quality factor:

$$2\zeta\omega_n = K_1s$$ $$\zeta = \frac{K_1s}{2\sqrt{K_1K_2}}$$

My question is: Am I comparing to the right transfer function since it does not have s squared terms? If so have I proceeded correctly? If not, are there any tips on what I could do to derive the above expressions correctly?

Answer 1

The transfer function of a second-order low pass filter is given by (reference):

$$H(s)=\frac{\omega_0^2}{s^2+2\zeta\omega_0 s+\omega_0^2}\tag{1}$$

where \$\omega_0\$ is the resonant frequency (in radians), and \$\zeta\$ is the damping ratio.

Comparing \$(1)\$ to your transfer function you get

$$\omega_0=\sqrt{K_1K_2}$$

and

$$\zeta=\frac12\sqrt{\frac{K_1}{K_2}}$$