Is this transfer function impossible with a low-pass Sallen-key filter?

Question

I was given this transfer function and instructed to use a Sallen-key cascaded with a 1st order filter to achieve it.

$$H(s)=\frac{628^3}{(s+628)(s^2+628s+628^2)}$$

The first order transfer function is

$$H(s)=\frac{628}{s+628}$$

So if I choose \$c=10\mu F\$, then \$R=159\Omega\$.

That leave the Sallen-key transfer function to be

$$H(s)=\frac{628^2}{(s^2+628s+628^2)}=\frac{\frac{A}{R^2C^2}}{s^2+\frac{2}{RC}s+\frac{1}{R^2C^2}}$$

So \$A\$ must be equal to \$1\$ and \$\frac{1}{R^2C^2}=628^2\$. The same values for R and C as the first order filter work here.

The problem is that middle term, \$628s=\frac{2}{RC}\$. This contradicts the other values. It works fine if this was a typo and that term was meant to be \$\frac{1}{2}\$ that value. Did I make a mistake here? I've wasted hours deriving and rederiving the transfer function for the filter to no avail.

Answer 1

Start with a 2nd order low pass filter transfer function in its general form: -

$$H(s) = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_ns + \omega_n^2}$$

And then mulitply this with a single order low pass filter transfer function: -

$$\dfrac{\omega_n}{s+\omega_n}$$

Giving you this: -

$$\dfrac{\omega_n}{s+\omega_n}\cdot \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_ns + \omega_n^2}$$

Can you see that this is the same form as your equation and therefore, \$2\zeta\omega_n\$ must equal \$\omega_n\$ or put differently, \$\zeta\$ = 0.5.

The problem you have in your analysis is that you haven't factored-in the effect of zeta (\$\zeta\$) and a sallen key filter can quite often have a zeta below 1 because this can optimize the transfer function. Look up butterworth sallen key filters for example.