# Active Filter Circuit

#### bp21

Joined Aug 31, 2011
4
All,

I am having trouble (I think) coming up with the proper transfer function for the following circuit:

http://imgur.com/mvWga

I do a KCL equation at node A and also at the negative terminal of the op amp.

I come up with the following transfer function...

Vo/Vi = (-s^2*R1R2C1C2)/(s*(R1C1+R1C2+R2C2)+1)

When I model the circuit in Spice, it looks like an RLC circuit, however I can't seem to get a transfer function in the proper form (with an obvious damping factor and resonant frequency).

Is my transfer function wrong?

I hope this wasn't too confusing. Help!

#### t_n_k

Joined Mar 6, 2009
5,455

#### Audioguru

Joined Dec 20, 2007
11,248
The circuit is a second-order Bessel Sallen and Key lowpass filter.
Its capacitor values are almost as low as the input capacitance of the opamp and stray capacitance so its cutoff frequency and transfer function will be different from what you want.

#### bp21

Joined Aug 31, 2011
4
Thanks guys, that helps a lot. I hadn't accounted for the stray capacitance and was assuming an ideal op amp. I've got some more studying to do. I appreciate the help!

#### The Electrician

Joined Oct 9, 2007
2,970
The circuit is a second-order Bessel Sallen and Key lowpass filter.
It isn't a lowpass and it's not a Sallen-Key.

#### bp21

Joined Aug 31, 2011
4
After looking at it a bit more I am in agreement with The Electrician.

Here's what I don't understand:
The response with ideal components in LTspice is showing me a narrow bandpass type of response with the center frequency between 5-10Mhz, but if my above transfer function is correct, I am seeing only one pole (at s = -1/(R1C1+R1C2+R2C2)) and two identical zeros (at s = 0). Two zeros and one pole shouldn't be giving me a bandpass response should it?

#### The Electrician

Joined Oct 9, 2007
2,970
Vo/Vi = (-s^2*R1R2C1C2)/(s*(R1C1+R1C2+R2C2)+1)

When I model the circuit in Spice, it looks like an RLC circuit, however I can't seem to get a transfer function in the proper form (with an obvious damping factor and resonant frequency).

Is my transfer function wrong?

I hope this wasn't too confusing. Help!
You won't be able to get a proper denominator with the topology you have shown. Your topology is not one of the standard topologies.

You could convert it to a Sallen-Key by grounding the right end of R2, and exchanging the + and - inputs of the opamp, connecting the - input to the output: http://en.wikipedia.org/wiki/Sallen–Key_topology.

Or, you could add another capacitor and rearrange things a little and make it into a multiple feedback filter: vhttp://sim.okawa-denshi.jp/en/OPtazyuHikeisan.htm

#### Audioguru

Joined Dec 20, 2007
11,248
Sorry, it is not a lowpass filter, it is a highpass filter.
Also it is an inverting filter circuit. A Sallen and Key filter is non-inverting.

Most opamps do not work at 5MHz to 10MHz radio frequencies.

#### The Electrician

Joined Oct 9, 2007
2,970
After looking at it a bit more I am in agreement with The Electrician.

Here's what I don't understand:
The response with ideal components in LTspice is showing me a narrow bandpass type of response with the center frequency between 5-10Mhz, but if my above transfer function is correct, I am seeing only one pole (at s = -1/(R1C1+R1C2+R2C2)) and two identical zeros (at s = 0). Two zeros and one pole shouldn't be giving me a bandpass response should it?
In the attachment, I show the frequency response of your circuit with an ideal opamp and no parasitic capacitances in the wiring.

Now, imagine that the opamp is not ideal, but rolls off at some frequency around 10 MHz. Combine that rolloff with the ideal response of your circuit and you will have an overall bandpass-like response.

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