# MFB Filter Gain Formula

1. Aug 20, 2013

### Deepsatchel

I am attempting to come up with a systematic way to design Multiple Feedback bandpass filters with certain specifications.

• Frequency is some given number
• Bandwidth varies with Frequency by
f0*2^13/12
• The Q should therefore be .4719
• The gain should be 1

I am at an understanding of how to manipulate Frequency and Q (BW follows), but I am having trouble finding information on how to manipulate gain. I am using the following topology:

http://sound.westhost.com/articles/af-f6.gif

With that background, which may prove irrelevant to my actual question, how is the gain for this MFB calculated? Rod Elliott, the author of the above picture, claims that that happens to be a unity gain filter, but does not explain why this is so.

I read a TI note that gave a gain formula for an MFB, but it was a slightly different design. The formula was a simple resistor ratio. Can anyone figure out the gain for this one?

I may have more questions after I sleep on it. Thanks in advance everyone.

Last edited: Aug 20, 2013
2. Aug 20, 2013

### ralph638s

Your schematic has two C2's and both are 10nF. There are also two R2's, but one is 40k and the other is 80k. I'm thinking that there is a mistake, that they should both have the same value, just like the two caps. In that case, the gain at the center frequency will just be R2/R2 = 1.

3. Aug 21, 2013

### Deepsatchel

You are right, there is something strange about that schematic. I think the resistor to ground should be labelled an R3, as the ubiquitous formula for MFB frequency involves 3 resistor values.

Here is another schematic that shows correct lettering.

Rod Elliott, god bless his soul, also provides an MFB component calculator available for download. You put in the components and it tells you Frequency, Gain, and Q, or vice versa. So what I've done is reverse-engineered some of the results from that calculator as it relates to Gain. R1 and R3 (referencing the above) are the only two that affect gain.

Looking at the results, it quickly became clear that the ratio is Gain = R3/(2*R1)

Does that seem correct? It is possible that the calculator is wrong, I'm hoping to be able to analyze this thing for myself and be able to infer the gain.

4. Aug 21, 2013

### Deepsatchel

Sorry for the double post, but I think I see it. When the input signal is at F0, the capacitors "disappear," and you are left with something more like this: