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Rauch filter basics

January 28, 2018
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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.

Figure 1

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)

 

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

Figure 2

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)

 

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:

We have:

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:

Noise Performance

Linearity Performance

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:

A low value of K = R x gm gives a shift in the poles with respect to the designed target. A K > 40 is suggested with an op amp that has an fT > 20 x f0.

A low value of K = R x gm gives a shift in the poles with respect to the designed target. A K > 40 is suggested with an op amp that has an fT > 20 x f0 .

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]

References

1 Analog Filters for Telecommunications, Andrea Baschirotto, June 2005

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