**Editor’s note** : My good friend and long-time colleague, Michael Steffes, expands upon my introduction article, Rauch filter basics, and adds some very useful and practical insights into the Rauch filter for designers.

Going from the simple Multiple FeedBack (MFB or Rauch) low pass active filter transfer function assuming an ideal Voltage Feedback Amplifier (VFA) to one including more of the op amp terms can be done in several ways. An earlier (2006) pass at this^{1} reported a 3^{rd} order LP transfer function where the cubic coefficients were in fact slightly in error – those will be corrected here. A recent discussion^{2} cited a 2005 source^{3} where the amplifier model used did not reflect the full op amp. That source developed 3^{rd} order coefficients assuming the external filter feedback elements were connected into what op amp design teams call the compensation node. This high impedance node sets the open loop DC gain and dominant pole frequency but actually sits behind an output stage that drives the external world with a lower impedance source buffering the compensation node.

Including the output stage in the MFB analysis significantly simplifies the results and moves closer to real implementations. The corrected coefficients for the cubic developed in Reference 1 will be shown here then used in an example design with a very slow op amp to predict the shifted pole locations due to the op amp Gain BandWidth Product (GBW or GBP). Continuing to simulation with that same GBW in a full VFA op amp model will then show how well the cubic prediction is working.

**Improving the MFB performance equations:**

Starting from the ideal op amp equations for the MFB design^{4} , adding in a simple single pole op amp open loop model will modify the transfer function to a 3^{rd} order low pass. Repeating the steps from Reference 1, section 7, for the circuit and RC numbering of Figure 1 will give a 3^{rd} order low pass transfer function with no zeroes.

Correcting the errors in Reference 1, first model the op amp as a single pole model with a DC gain of A_{ol} and dominant pole frequency of ω _{a} (not ω _{o} which is the target filter characteristic frequency) – working in radians throughout. The GBW product for the op amp will be A_{ol} ω _{a} /2 π . For each term the older form (ref. 1) and corrected term will be shown together.

Developing the transfer function with this op amp model for the circuit of Figure 1 gives this form (Reference 1):

As Reference 1 notes, it is possible (and often very useful) to include a capacitance to ground on the inverting summing junction. This tuning C_{t} is either the op amp parasitic and/or an intentional external capacitor added for phase margin tuning. The transfer function stays 3^{rd} order including C_{t} and the coefficient equations become a little more complicated. The simpler forms without C_{t} will be shown here. Getting the cubic into monic form with B3 = 1, gives these corrected expressions:

While certainly somewhat involved, it is an easy matter at this point to build a spreadsheet to calculate each of the coefficient values from the RC solutions and op amp being considered and generate the calculated pole locations from a cubic solver. That will be done here with an intentionally very slow device showing the large shift in resulting closed loop filter shape predicted by the cubic solution and the corrected coefficients. What this more accurate solution is still neglecting is the op amp parasitic capacitance at the inverting input, the open loop output impedance of the op amp, and any higher order poles in the open loop response.

**Testing the MFB performance equations including the op amp’s GBW**

Going to the example designs shown in Reference 4 and choosing an op amp closer to the “required” minimum (3.9Mhz) the ADI tool (Reference 5) reports (assuming a GBW adjust routine will be applied) will shift the final poles quite a bit. The design targets from Reference 4 are here:

- DC gain of -10V/V (20dB)
- Small signal response peaking 1dB
- F
_{-3dB}= 100kHz

These give us targets for the active filter with a gain of -10V/V and

4. F_{o} = 80.62kHz

5. Q = 0.957

To test the cubic solutions with an actual op amp simulation, use the 5.4MHz GBW OPA377 (Reference 6). That 5.4Mhz comes from a TINA (Reference 7) test for Aol frequency response and is slightly lower than the stated 5.5Mhz for the device.

Several RC solutions were shown in Reference 4 where here the lowest noise gain peaking out of the Intersil tool (Reference 8) will be used in the cubic coefficient equation shown above. Those RC values are shown in Table 1.

**Table 1**

**MFB RC values from the Intersil solution**

The OPA377 (Reference 6) op amp terms to use in the cubic coefficients are shown in Table 2.

**Table 2**

**Required single pole op amp open loop model terms**

The resulting coefficient values for the cubic pole expression are shown in Table 3.

**Table 3**

**Cubic coefficient evaluations**

The cubic solver will take these and return a real pole and complex pole pairs reported in real and imaginary terms in radians. Those need to be adjusted to Hz and then into F_{o} and Q format. Table 4 shows those results where a much lower F_{o} and Q than target result from the 5.4Mhz GBW op amp being used.

**Table 4**

**Cubic solution for the F _{o} and Q.**

Continuing on into a TINA simulation using the OPA377 model, gives this resulting frequency response shape in Figure 2:

**Figure 2**

**MFB simulation circuit using the OPA377 model and the Intersil RC values. **

And then extracting the F_{o} and Q from the simulated gain response gives this result including the full op amp model in Table 5.

**Table 5**

**Extracting the F _{o} and Q from the simulated response shape. **

The cubic solution predicted the closed loop 2^{nd} order pole F_{o} = 73.1kHz whereas the full sim showed 73.5kHz. It also predicted the Q =0.919 whereas the full simulation showed 0.914. Clearly the corrected coefficients shown here are doing a very good job of including the op amp gain bandwidth product and that captures most of the pole shift going to the full op amp model in this example. Also, the resulting fit error to target is very large suggesting a GBW adjustment routine should be applied at this point if such a slow device is applied to the design. The ADI tool includes that feature as an option while at this time it appears none of the other online active filter tools implement a GBW adjusted design flow for the Rauch low pass active filter.

**References for MFB LP Cubic development. **

- TI Application note “Design Methodology for MFB Filters in ADC Interface Applications”, Michael Steffes, Feb. 2006.
- Planet Analog, Jan28, 2018, Steve Taranovich, Rauch filter basics
- University presentation, June 2005 “Analog Filter for Telecommunications”, A. Bashirotto,
- EDN article, “Testing op amp tools for the active filter design accuracy and dynamic range”, Michael Steffes, Dec. 2017,
- Entry page for the Analog Devices online Active Filter Wizard. Login required.
- TI, “Low-Cost, Low Noise, 5.5MHz, CMOS Operational Amplifier”
- TINA simulator available from DesignSoft for <$350 for the Basic Plus edition. Includes a wide range of vendor op amps and is the standard platform for TI op amp models.
- Entry page for the Intersil online op amp design tools. Login required.

## 0 comments on “Include the op amp gain bandwidth product in the Rauch low-pass active filter performance equations”