I need another pair of eyes.

The cutoff frequency for a 2md order multiple feedback filter is 1/(2*pi*sqrt(C1*C2*R2*R3)) which agrees with a TI application note on page 7 save for my swapped positions of R1 and R2.
However, if all values of caps and resistors are equal to 1 then the cut off frequency should be 159mHz. The calculations agree and this webtool agrees; however, Spice does not and the transfer function from the very same webtool put in MATLAB agrees with Spice.

Seriously...what the <snip> am I missing?

 

Where did you find an opamp that works at a frequency as high as 1.6GHz?
Most opamps do not go higher than only 100kHz.

 

I need another pair of eyes.

The cutoff frequency for a 2md order multiple feedback filter is 1/(2*pi*sqrt(C1*C2*R2*R3)) which agrees with a TI application note on page 7 save for my swapped positions of R1 and R2.
However, if all values of caps and resistors are equal to 1 then the cut off frequency should be 159mHz. The calculations agree and this webtool agrees; however, Spice does not and the transfer function from the very same webtool put in MATLAB agrees with Spice.

Seriously...what the <snip> am I missing?

My guess is that you need to use more realistic values of components. Most opamps don't work well with 1 ohm resistors and 1 F capacitors. The required output currents are much too high. If spice and webtool are using realistic opamp models, that is most likely your problem.

 

159mHz or 0.159Hz. I do not know where you got 1.6GHz from. As an aside, modern op-amp's UGBW go way above 100 kHz. Anyway...

The the spice model is ideal so that should not be a problem, but I understand and aware of your concern about realistic components.

The transfer function has no bearing on a real world op-amp; it's just an equation.

I crunched some more numbers, and it seems like at very low frequencies and (mHz range) and very high (100's of kHz) range there is a % error between the equation in the appnote/webtool and what MATLAB/Spice spit out.

Makes no sense!

It must be something so painfully obvious, and I need the Captain to tell me.

 

Looking at your plot, it appears you are using the 3 dB point to define the cutoff frequency. Don't you want the 90 degree point for a second order filter?

The 90 degree phase shift point will mark the pole locations. You can still define an effective cutoff point at 3 dB, but you need to be clear about what you are defining.

If you use the 90 degree point, do you get agreement?

Note: If you don't have the phase response handy, this example has -9.54 dB attenuation at the pole locations.

 

The GBW of an opamp is the frequency where its gain is only 1 like a piece of wire so it is useless as a filter even if the frequency is 1/10th the GBW.

 

What steveb says, from a different angle. The natural resonant frequency is indeed 159 mHz (which is 1/2π, given all the passives are unity), but the Q is 0.33, and this degree of overdamping will shift the -3dB point away from the resonant frequency.

I'm not sure that this is the Q that you want, it's just a product of setting all the passives to unity. This isn't a technique that works well with active filters.

Here's a paper on sensitivity that's got the equations for ω and Q, and page 19 of this fine TI tome has the equations for working out the passive values given the passband gain, Q and rolloff frequency. At 159 mHz the trick is to find a decent sized and decent priced capacitor that still has a decent dielectric, as you need to be in circa-10 μF territory to make the resistor values sensible.

 

What steveb says, from a different angle. The natural resonant frequency is indeed 159 mHz (which is 1/2π, given all the passives are unity), but the Q is 0.33, and this degree of overdamping will shift the -3dB point away from the resonant frequency.

Yep, that is what it was. It came to me while driving to work in the morning. I think I feel asleep trying to figure it out.

The reason why I am starting with 1Ω and 1uF is trying to implement the scaling technique in Don Lancaster's Active Filter Cookbook. Nice book for filters with little math.

 

Lancaster's fine tome is a handy cookbook, but much is omitted, partly for reasons of space, partly so as not to scare off potential readers. Sadly the maths is necessary for a decent understanding of filters. Looking at the equation for Q (from the previous sensitivity link) shows that it will always be 0.33 if all the component values are scaled from unity.

It's horses for courses with filter design, made more difficult by the fact the information is spread about. I don't know of any one filter book that contains everything you'd need to know. Talk of tolerance sensitivity, stability margins, op-amp gain-bandwidth requirements, passband gain limits and DC biasing are usually conveniently ignored.

If you want a bit more depth, ME van Valkenburg's filter book is both detailed and practical.

 

I use only Butterworth filters because they have a flat response up to the cutoff frequency which has a sharp corner then a calculated roll-off rate.
I never looked but I think a Butterworth filter has a Q of 0.707.