Butterworth Filter Question...?

Question

I have obtained the transfer function for an 8th order butterworth low pass filter with a cut off at 2Hz. What is the next step from here to start selecting component values for a passive ladder filter?

Transfer function:

5.96e008

--------------------------------------...

s^8 + 64.08 s^7 + 2053 s^6 + 4.267e004 s^5 + 6.271e005 s^4 + 6.665e006 s^3 + 5.01e007 s^2

+ 2.443e008 s + 5.957e008

The ladder filter has this equation, where w is angular frequency :

Vout/Vin = 1 / {

w^8 (C2C4C6C8L1L3L5L7)

-W^6 (C2C6C8L1L5L7 + C4C6C8L3L5L7 + C2C4C6L1L3L5 + C2C4C8L1L3L7)

+W^4 (C2C6L1L5 + C2C8L1L7 + C4C8L3L5 + C4C8L3L7 + C2C4L1L3 + C6C8L5L7)

-W^2 (C2L1 + C4L3 + C6L5 + C8L7)

+1

}

What do I do next?

Answer 1

I'm going to assume you have adopted this approach because you want to split the output of some massive audio amplifier into two (or more) parts and the 2Hz filter is to 'block' the feed to the sub-woofer (to avoid burn-out perhaps ?)

I wish you luck - the high power components you are going to need come in limited range of values, so you are going to have to adapt the design to compensate for any non-optimum values you are forced to use.

I suggest you first calculate and select the 'nearest fit' capacitors and then recalculate using the actual capacitor values you have been forced into (since you can wind your own inductors to the the exact required recalculated values).

PS since real world components have some +/- tolerance (and caps are inductive, and inductors capacitive) anything over 2 or 3 orders is likley to be ineffective (i.e. 'lost in the noise').

Answer 2

This is actually quite a difficult problem, going from a transfer function to an LC ladder filter. It's certainly not trivial for an 8th-order filter.

If I had to do it (but I'd avoid LC filters like the plague at low frequencies!), my first port of call would be a standard reference text which tabulates values for LC ladder filters. Since the Butterworth is a very standard filter characteristic, it's bound to be in there. The book will typically give you a filter design with a cutoff angular frequency of omega = 1 rad/s and source and load impedances of 1 ohm. You then need to scale it for frequency, and then rescale it for impedance.

A more practical 2Hz filter would use op amps. You could do it with four second-order sections, but there are probably fancier ways requiring fewer op amps. Anyway, you'll need 8 capacitors and a bunch of resistors.

These days filter design is done using computers, numerical simulation and optimisation routines, which takes all the hassle out of it. It's also easy to investigate the effect of component tolerances.