Designing MFB Bandpass Filters: Frequency, Bandwidth, and Gain Formula Explained

[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.

 

[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.

 

[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.

 

[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:

http://upload.wikimedia.org/wikipedia/commons/4/41/Op-Amp_Inverting_Amplifier.svg

Could anyone shed some light on what effect the R2 from above might have?

 

[alphy]

I would like to commend you for your efforts in designing a Multiple Feedback bandpass filter and seeking a systematic approach to it. Let me try to provide some insights and answers to your questions.

Firstly, let's start with the gain formula for the MFB bandpass filter. The gain for this topology can be calculated using the following formula:

Gain = -R4/R3

Where R4 is the feedback resistor and R3 is the input resistor. In the topology you have shared, the resistor R4 is connected between the output and the inverting input of the op-amp, while R3 is connected between the non-inverting input and the ground. Therefore, the gain for this topology would be 1, as both R4 and R3 have the same value of 1kΩ.

Now, let's move on to the specifications you have mentioned - frequency, bandwidth, and Q. Frequency is a given number, which means it is the center frequency of the bandpass filter. The formula for determining the center frequency for an MFB bandpass filter is:

f0 = 1/(2πRC)

Where f0 is the center frequency, R is the input resistor, and C is the input capacitor. In your case, the input resistor R3 is 1kΩ and the input capacitor C1 is 0.1μF, which gives a center frequency of approximately 1591 Hz.

Next, the bandwidth varies with frequency, and the formula for this is:

BW = f0 * 2^(1/12)

Where BW is the bandwidth and f0 is the center frequency. In your case, the bandwidth would be approximately 1673 Hz.

Finally, the Q value for an MFB bandpass filter is calculated using the following formula:

Q = f0/BW

In your case, the Q value would be approximately 0.95, which is close to the value of 0.4719 that you have mentioned.

I hope this helps in understanding the calculations and specifications for designing an MFB bandpass filter. It is important to note that these formulas are based on ideal conditions and may vary in practical applications due to component tolerances and other factors. It is always recommended to simulate and test the filter to achieve the desired specifications.