**Using a Decompensated VFA in a Multiple FeedBack (MFB) Active Filter **

Another very common VFA application would be in an MFB (or Rauch) active filter. Legacy literature constrains these solutions to unity gain stable VFA devices since the direct feedback capacitor required by the topology shapes the NG to unity gain at higher frequencies. However, a direct extension of the inverting compensation technique described above will allow decompensated devices to be applied to MFB solutions (Reference 8). Expanding the solution universe to decompensated devices will allow much higher slew rate devices giving higher full power bandwidth and lower harmonic distortion. Use the 2.8GHz Gain Bandwidth Product (GBP) OPA818 to implement a 2^{nd} order MFB design delivering:

- Inverting gain of -5V/V
- 5MHz F
_{-3dB}
- Butterworth response (Q=0.707)

The RC solution of Figure 6 is using a reduced noise and NG peaking flow (Reference 8) to get the filter RC values. The required added step is to target a higher frequency noise gain using the filter feedback C_{2} and an added C_{t} to ground on the inverting summing junction. Targeting a capacitor divider NG of 10V/V for this minimum stable gain of 7V/V device gave the added 120pF to ground on the op amp’s inverting node in Figure 6. This C_{t} capacitor can be included in the 3^{rd} order transfer function coefficient polynomials for a GBP adjusted RC solution without increasing beyond 3^{rd} order (Reference 9). Adding C_{t} will effectively reduce the available GBP by that higher frequency NG value - in this case reducing the available GBP to 280MHz for the MFB RC solutions. Some legacy MFB solution flows show a very high required GBP for this design point (1.77GHz from the TI active filter designer, Reference 10). However, accounting for the equivalent GBP in this design gave the reasonably accurate solution of Figure 6. These RC values are both adjusted for GBP (including the effect of C_{t}) and selected for the best fit E96 resistor and E24 capacitor values.

**Figure 6**

Another important indicator of both active filter design margin and stability come from the LG simulation of Figure 7. This is showing a very good 31dB of LG at the filter F_{o} = 5MHz. Then, at the LG=0dB frequency of 235MHz we also see a very good 55degrees phase margin. Removing that C_{t} capacitor moves the LG=0dB crossover out to 956MHz with only 15degrees phase margin.

**Figure 7**

This application of a decompensated VFA is of course peaking the output noise to get stability. Figure 8 shows the output spot noise with and without C_{t} in the design. The stable design will have an increased integrated noise to get that stability. The first peak in the output noise is the filter noise gain peak while the 2^{nd} is the LG=0dB crossover peak. A post RC filter can be used to reduce this higher integrated noise. Removing C_{t} definitely reduces the output spot noise but that sharp peak around 1GHz is typical of very low phase margin designs.

**Figure 8**

Essentially extending the inverting compensation approach to this active filter design easily allows decompensated VFAs to be applied to these MFB filter requirements. This can vastly extend the available full power bandwidth using the intrinsically higher slew rates available from decompensated devices. If possible, the full design flow should consider a simple post RC filter to reduce the more peaked output spot noise intrinsic to this approach. It does not appear at this time that any of the vendor online MFB design tools will apply a decompensated VFA to solutions. To force a solution, use a unity gain stable device, to be later replaced by a decompensated device with this added C_{t} targeting a high frequency noise gain to be developed later, and divide that into the decompensated GBP to get the equivalent unity gain device GBP to target. Execute a design with a device near that target using a unity gain stable device, then replace it with the decompensated device with the added C_{t} element.

**Applying Decompensated VFAs to Photodiode Transimpedance Applications**

Probably the most ubiquitous application of decompensated VFAs comes in the transimpedance application. The simplest form of this is just the detector capacitance, a feedback resistor that sets the gain, and (most importantly) a feedback capacitor that sets the Q of the closed loop response. The Bode plot for the loop gain is very similar to Figure 3 except the DC NG is 0dB (see Figure 2, Reference 11). One common assumption is that the feedback C_{f} will eventually be chosen to deliver a closed loop maximally flat Butterworth response. With that assumption, Equation 7 (Equation 12, Reference 11) gives the required GBP dependent on the other terms in the design. Increasing the desired gain (R_{f}), bandwidth, and source C_{s} all act to increase the required GBP in the VFA to deliver a closed loop 2^{nd} order maximally flat Butterworth response shape.

The next key is to decide if a bipolar input or JFET input device is preferable. That normally becomes an input referred equivalent spot current noise question. The simplified expression in Equation 8 (Equation 13, Reference 11) applies when there is a postfilter at a frequency less than the feedback pole frequency. Making that assumption allows a simpler integration of the rising portion of the spot output noise due to the noise gain zero formed by the feedback resistor and source capacitance (C_{s}). Equation 8 is delivering the equivalent flat input current spot noise that will integrate to the same total output noise power as the highly peaked actual output noise response.

What this means in practice is the target F_{-3dB} in the transimpedance stage should be slightly higher than desired. This is a good practice in any case to account for GBP variation in the op amp stage where an extrinsic postfilter will deliver a lower tolerance operating bandwidth and constrain the integrated noise.

The F_{NPBW} is the Noise Power BandWidth (NPBW) in Hertz of the postfilter which needs to be less than the feedback pole frequency for Equation 8 to be accurate. It is often difficult to predict which decompensated VFA will give the lowest input referred current noise as different combinations of terms in Equation 8 can be dominant depending on source capacitance, the NPBW, the gain, and the op amp noise terms. This often it comes down to trying different physical op amp terms in Equation 8. Normally, for transimpedance gains > 100kΩ, a JFET or CMOS input device will be preferable.

Equation 7 can be used in a couple of ways to set the limits for a transimpedance design. Since increasing the feedback R_{f} uniformly reduces the input referred current noise as shown by Equation 8, one common approach is to solve Equation 7 for the maximum R_{f} to satisfy the target F_{-3dB} with the source capacitance and available GBP in a candidate device as shown in Equation 9.

Once we have this maximum gain to meet the intended bandwidth target, the last step is to solve for the required feedback C_{f} to hit that. Equation 10 shows an approximate solution that works very well where the Q relates to the desired or allowed frequency response peaking in the transimpedance stage.

Continuing with the OPA818 2.8GHz GBP JFET input device, target a final F_{-3dB} using a 2^{nd} order passive Butterworth postfilter to 10MHz while targeting the transimpedance stage at 14MHz. Assume a 20pF diode capacitance and add the internal C_{cm} = 1.9pf + C_{diff} = 0.5pF. Putting these numbers into Equation 9 yields:

R_{fmax} = 102kΩ

And then solving for the required C_{f} to hit a Q=0.707 using Equation 10 yields:

C_{f} = 0.16pF

Putting this into a TINA (Reference 12) simulation, along with an 11MHz 2^{nd} order Butterworth RLC filter with only -1dB insertion loss (Reference 13), gives the design of Figure 9. Here, the response to the OPA818 output pin matches the 14MHz Butterworth target while combined with the 11MHz passive postfilter gives a nearly perfect 10MHz F_{-3dB} to the output.

**Figure 9**

The input referred current noise is a low 0.4pA/√Hz through most of the span rising to 3.3pA/√Hz at 10MHz due to increasing NG for the input voltage noise and the filter rolloff. Evaluating Equation 8 using a NPBW = 11MHz*1.11 = 12.2MHz (the F_{-3dB} to NPBW adjustment for a 2^{nd} order Butterworth filter) gives an i_{eq} = 2.2pA/√Hz

**Figure 10**

This design flow was simplified by assuming a closed loop Butterworth design target. Testing the LG phase margin in Figure 11 showed very nearly an exact (65.5deg) Butterworth phase margin.

**Figure 11**

The nearly 180deg phase shift around the loop at lower frequencies is normal for transimpedance design as the NG zero (70kHz in this example) starts rising at lower frequencies. That loop phase is pulled back up to 65.2deg phase margin at LG = 0dB crossover by the feedback pole. The high frequency noise gain is approximately 1+(22.4pf/0.16pF) = 235V/V. Such a high noise gain at higher frequencies shows why the higher order poles of a decompensated VFA can normally be ignored and only the true single pole GBP is required for the design.