# 2nd order filter transfer function normalization

1. Feb 3, 2010

### bitrex

I'm looking at a guide by Texas Instruments on active filter design. In it are the following equations for a second order lowpass filter, verbatim:

The coefficient form of the denominator: $$s^2 + a_1s + a_0$$

Normalized: $$P(s) = (\frac{s}{\sqrt{a_0}*\omega_c})^2 + \frac{a_1s}{a_0*\omega_c} + 1$$

Substituting $$s = j2\pi f, \omega_c = 2\pi f_c, a_1 = \frac{1}{Q}, \sqrt{a_0} = FSF$$

$$P(f) = -(\frac{f}{FSF*f_c})^2 + \frac{1}{Q}\frac{jf}{FSF*fc} + 1$$

Maybe I'm missing something obvious here, but why is it that it is not FSF^2 in the second term of the last equation, if $$\sqrt{a_0} = FSF$$?

Last edited: Feb 4, 2010
2. Feb 4, 2010

### bitrex

I made an error in the LaTeX of the first equation, I've corrected it.

3. Feb 5, 2010

### bitrex

I think I see the problem, it must be a misprint on their part. The equation should be:

$$P(s) = (\frac{s}{\sqrt{a_0}*\omega_c})^2 + \frac{a_1s}{\sqrt{a_0}*\omega_c} + 1$$

Also remembering that $$\omega_c = \sqrt{a_0}$$ helps.