A single Op Amp Rauch Biquadratic cell1
This is a multiple-feedback (MFB) filter architecture in which there is less sensitivity to component variations than many other filter architectures.
In this architecture, the inverting configuration makes the noise gain +1 higher. Therefore, the bandwidth of the amplifier is approximately equal to the Gain Bandwidth (GBW) product divided by the noise gain.
Parasitic impedance (ZpX ) at the input junction Tee where Z1 , Z2 , and Z1’ intersect will be in parallel with Z1’ and will change Z1’ to Z1’ ||ZpX and a frequency response change may occur (Image courtesy of Reference 1)
Let’s look at a lowpass frequency response1
A lowpass frequency response example of the Rausch Biquadratic Cell is shown where Z1 = R1 , Z2 = R2 , Z3 = R3 , Z1’ = (1/s)C1 , and Z2’ = (1/s)C2 (Image courtesy of Reference 1)
So, in the passband we have:
In the case of an in-band maximally flat frequency response where Q = (√ 2)/2:
R1 = R3 = 2 R2 = 2R
C1 = 4C2 = 4C
f o = 1/(4 π √ 2 x R x C)
Again here, the transfer function will be sensitive to parasitic capacitance across C1
Now let’s take a look at the noise and linearity of this Rauch Biquadratic Cell:
Linearity is good for the closed loop architecture
For an out-of-band signal, an R1 C1 pre-filter, will increase the our-of-band linearity
Effects in a real op amp situation
In this real op amp, the transfer function has one zero and three poles in which the zero as far from fp if Ao >> 1. The extra pole is around the unity gain of the op amp.
See the following graph of Gain vs. log of frequency, where K = R x gm :
The two other poles are shifted with respect to the location which was originally designed:
Please see an excellent series of articles on the accuracy and dynamic range accuracy of op amp tools by Michael Steffes on EDN here: [Testing op amp tools for their active filter design accuracy and dynamic range] and [Active filter design tools shootout]
1 Analog Filters for Telecommunications, Andrea Baschirotto, June 2005