### Author Topic: Multiple Feedback Bandpass filter: transfer function  (Read 8472 times)

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#### ch.snyers

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##### Multiple Feedback Bandpass filter: transfer function
« on: March 28, 2012, 02:24:24 am »
Hi !

I'm a student in electrical engineering in Belgium and I use a multiple feedback bandpass filter in one of my projects.

I tried to find out the transfer function of this filter and this is what I found:

With this expression I made a Bode diagram:

Is that normal that the frequencies other than 1kHz are still amplified (not as much but still) ? Is my transfer function right ?

Thanks !

#### alm

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #1 on: March 28, 2012, 06:58:38 am »
I don't feel like double checking your math, but it sounds quite plausible to me that frequencies other than 1 kHz are still amplified. A filter only has so much roll off, and the gain at 1 kHz is +80 dB. At first glance, I would expect a 40 dB/decade (12 dB/octave) roll-off in the stop band, however, since this is a second order filter.

#### [email protected]

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #2 on: March 28, 2012, 07:00:34 am »
Without doing a single calculation it is obvious your transfer function is wrong. The units don't add up. Vs/Ve should be a unitless factor, but your Vs/Ve is in Ohms.
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#### alm

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #3 on: March 28, 2012, 07:19:14 am »
Good catch, I missed that.

#### ejeffrey

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #4 on: March 28, 2012, 08:16:44 am »
In any case, once you figure out the units the plot of your transfer function looks about right for this sort of filter.  You can increase the Q which will make the filter 'sharper' around the center frequency, but it becomes more sensitive to both component tolerances and the limited gain-bandwidth product of the opamp.

#### amspire

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #5 on: March 28, 2012, 02:30:33 pm »
I don't think your transfer function is quite right. In fact it doesn't look like a particularly useful bandpass circuit to me at all. Are you sure that is the circuit you want? Without any positive feedback like a Sallen-Key filter circuit, or sufficient phase shift to get positive feedback, it will be a pretty lame filter. The 80dB gain cannot happen - I suspect it has to have a maximum gain less then unity.

For analysis, the presence of the input voltage divider R1 and R3 can be replaced with a single resistor and a lower Vin, thanks to Thevenin. Including the R1-R3 divider will add a lot of unnecessary complication to your equation, and you can always add the divider back in at the end and the only thing it is actually doing is reducing the gain of the filter ( as long as Vin is fed from a voltage source like an opamp output).

I could derive the transfer function if you like, but I don't think this is the circuit you actually want.

Richard.

#### slateraptor

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #6 on: March 28, 2012, 06:52:46 pm »
In your diagram, if we label the upper capacitor C1 and the lower capacitor C2, then the transfer f'n in Laplace domain is

$\frac{V_{out}}{V_{in}}=-[\frac{R_3}{R_1+R_3}][\frac{sR_2C_2}{s^2(R_1||R_3)C_1R_2C_2+s(R_1||R_3)(C_1+C_2)+1}]$

Note the voltage divider contribution to gain and 2nd-order bandpass form, which is expected in this multiple feedback topology.

#### slateraptor

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #7 on: March 28, 2012, 07:51:33 pm »
Without any positive feedback like a Sallen-Key filter circuit, or sufficient phase shift to get positive feedback, it will be a pretty lame filter.

What would you recommend as a reasonable 2nd-order sol'n using an opamp with mediocre GBW spec?

#### ch.snyers

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #8 on: March 28, 2012, 10:20:17 pm »
First thanks for all the answers!

I don't think your transfer function is quite right. In fact it doesn't look like a particularly useful bandpass circuit to me at all. Are you sure that is the circuit you want? Without any positive feedback like a Sallen-Key filter circuit, or sufficient phase shift to get positive feedback, it will be a pretty lame filter. The 80dB gain cannot happen - I suspect it has to have a maximum gain less then unity.

For analysis, the presence of the input voltage divider R1 and R3 can be replaced with a single resistor and a lower Vin, thanks to Thevenin. Including the R1-R3 divider will add a lot of unnecessary complication to your equation, and you can always add the divider back in at the end and the only thing it is actually doing is reducing the gain of the filter ( as long as Vin is fed from a voltage source like an opamp output).
Well this filter is from http://www.ti.com/lit/ml/sloa088/sloa088.pdf (p. 31) so I assumed it was a good design. What would you use instead ?

In your diagram, if we label the upper capacitor C1 and the lower capacitor C2, then the transfer f'n in Laplace domain is

$\frac{V_{out}}{V_{in}}=-[\frac{R_3}{R_1+R_3}][\frac{sR_2C_2}{s^2(R_1||R_3)C_1R_2C_2+s(R_1||R_3)(C_1+C_2)+1}]$

Note the voltage divider contribution to gain and 2nd-order bandpass form, which is expected in this multiple feedback topology.
Thanks, this makes more sense now. Actually I was pretty close to the right answer, I just forgot a 1/R1 factor (assuming C1=C2).

#### amspire

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##### Re: Multiple Feedback Bandpass filter: transfer function
« Reply #9 on: March 29, 2012, 01:18:15 am »
Looks like I got things wrong. I assumed that the current phase between the two caps was always 90 degrees, but that is wrong.

So the multi-feedback bandpass can give decent Q's of up to 20.

I think the Sallen-Key circuit puts much less demands on the opamp gain then the your filter will. Running a LM324 at unity gain, you can still get a very decent notch at 100Khz. I think the multi-feedback bandpass is running out above about 1KHz.

http://en.wikipedia.org/wiki/Sallen%E2%80%93Key_topology

Good choices are C1 = C2 and R2 = 2R1. The smaller Rf, the higher the Q. Connecting the minus input to the opamp output will give a bit less then a 0dB gain.

A nice 3 opamp circuit is this one http://electronicdesign.com/files/29/10684/10684_01.pdf  in that the Q is adjustable, and it has constant bandpass gain. And again it does not depend much on opamp gain.

The twin-T circuit is nice in that the main work is done by passive components, but the big negative is you have to look carefully at the effects of component tolerance.

Richard.

Smf