summary

　　A common approach to gain sequencing for multistage lowpass active filters is to place most or all of the gain in the first stage. This is the correct approach if the only concern is to reduce low-frequency input-referred noise. However, several other considerations may cause you to change this gain sequencing to achieve a better implementation. These considerations include: the effects of noise peaking within the characteristic frequency range of each stage, overshoot of high-Q, high-gain stages leading to limited slew range and/or clipping, and the amplifier bandwidth required for reliable implementation. This article describes the above situations, finds appropriate implementations for them, and explains the effects of these solutions.

　　Design Considerations for Multistage Low-Pass Active Filters

　　Every multi-stage active filter designer has had to wrestle with the trade-offs between ordering the Q values ​​of the stages and how much gain to allocate to each stage. If the total gain of the filter is to be greater than 1, the simplest design approach is to place most or all of the gain in the first stage. This approach, when properly analyzed, will result in the lowest input referred noise (when the noise frequency is well below the filter characteristic frequency). In addition, for a standard multi-pole design, a range of Q values ​​are placed from low to high. Where to place the highest Q stage is a very important consideration and is actually critical to the success of the implementation. These highest Q stages will see the highest output noise peaking and are most likely to have step response overshoots that will cause slew range limitation and/or clipping. Many design tools place this stage first, which conflicts with the goal of placing most of the gain in the first stage. Some design tools place most of the gain in the last stage, resulting in much more noise peaking than necessary, increasing the overall noise of the filter output. Some design tools take a middle ground approach and place the highest Q stage in the middle (for stages above 4), which seems to work well for some applications.

　　When these filters are actually constructed using real components with their own performance limitations, these considerations are no longer theoretical. Using a recently developed online design tool (Reference 1), it is possible to develop multiple examples that achieve the same target frequency response. Their step response, noise, and required amplifier performance margin can vary greatly depending on the gain and Q ordering.

　　Of course, gain ordering is only a concern if the overall gain in the low-frequency passband of the design is greater than 1. Although the issue of gain and Q ordering also applies to multi-stage feedback (MFB) or infinite gain topologies, the Sallen Key filter (SKF) will be used here to illustrate the issues and results. It has been shown that the gain achieved by a particular SKF stage is limited (Reference 2). This only occurs when the RC solution is subject to some other constraints. It is generally assumed that an equal capacitance design is required, which in practice will limit the minimum gain that can be achieved per stage. However, for board-level implementations, the equal capacitance assumption is artificial and may be more useful for design flows that target integration. The design here is not limited to equal resistance or capacitance, allowing the design to achieve any desired gain in the passband. However, it is important to note that as the gain increases, this will increase the sensitivity of the filter to component variations and gain variations. The increase in gain of a stage requires that the RC components used to set the filter and gain in that stage have smaller RC tolerances. Of course, such components are now available.

　　The design flow in Reference 1 tends to increase the resistors so that the noise they contribute is negligible compared to the intrinsic noise of the op amp. The 1/R2C2 pole is also set to reduce the internal noise gain peaking of the filter stage (Figure 1).

Figure 1. Basic SKF second-order low-pass filter

　　Considerations for required op amp bandwidth in each stage of gain allocation

　　To successfully implement the design shown in Figure 1, the minimum amplifier closed-loop bandwidth must be estimated. Typically, requiring an amplifier bandwidth multiplier of 100 to 200 times the target Fo is easier to achieve. More complex designs calculate the target bandwidth based on the gain and target Q of the stage, resulting in an ideal sensitivity of the amplifier bandwidth as Fo and Q are varied.

　　An example of bandwidth multiplier calculations as a function of Q (at a given gain) is shown in Figure 2. This plot shows the ratio of amplifier bandwidth to Fo for the Figure 1 circuit, with the goal of providing just enough amplifier bandwidth to allow for a high-precision filter implementation, so that Fo does not vary more than 2% for a 15% change in bandwidth (Reference 3). It is certainly possible to use an amplifier with a higher bandwidth margin than this design, but the goal of this design is to lower the bar for a successful design.

Figure 2. Required op amp bandwidth vs. gain and Q.

　　Note that this figure places an emphasis on bandwidth. This allows the SKF topology to be implemented using either current feedback amplifier (CFA) or voltage feedback amplifier (VFA) devices. CFA devices are particularly well suited for high gain because they maintain a constant closed-loop bandwidth over a range of gains. This is particularly evident in the figure (derived from the algorithm in Reference 1). For example, at a gain of 2 and a Q of 0.5, only a bandwidth margin of 7 is required. With a bandwidth margin of 25, a gain of 10 and a Q of 4.5 can be achieved. These relatively modest design margins allow a wider variety of physical devices to be used to achieve a particular filter criteria, but some algorithm that can scale components to support that bandwidth is required to achieve the filter design criteria. Design flows that use ideal equations to calculate RC values ​​require a larger margin in the amplifier bandwidth.

　　If a VFA device is used, the following correction is required: multiply each data point by the gain to obtain the required ratio of the gain-bandwidth product to Fo. As shown in Figure 3, all the curves are shifted up and expanded.


Figure 3. Required bandwidth-gain product and Q parameters for a given gain.

　　Here we can start to look at some of the extreme multipliers needed to achieve higher Q and higher amplifier stage gains for a simple second-order SKF built with VFA devices. For example, if the gain is 10 and the Q is 1, this curve shows that we need an amplifier with a gain bandwidth product of at least 215xFo. At Fo of 1KHz, this requirement is not difficult to achieve. But if Fo is greater than 1MHz, it will be more difficult. This is why higher speed SKFs with stage gains tend to use CFA op amps.

　　Designing a practical, discrete op-amp multistage active filter requires that each stage have only the bandwidth required to meet the needs of that stage. In general, excess bandwidth margin comes at a cost, either in terms of increased power consumption or increased purchase price. In addition, multistage filters can be implemented using multichannel devices made from the same basic amplifier model, provided that the bandwidth and slew rate requirements of each stage can be kept roughly equivalent. Ideally, the exact same bandwidth requirements can be obtained by drawing a horizontal line on Figure 2 or Figure 3, and then using the intersection of the horizontal line with the parameter curves to set the gain and Q of each stage. However, the more intuitive interpretation derived from this is sufficient for our needs. These curves clearly show that as the Q value or Fo of a stage increases, the gain allocated to that stage should decrease. Except for the Butterworth filter, where the Fo value is equal at each stage, the Fo of other filters above third order will change at each stage. Most typical low-pass filters have a shape where Fo increases with Q value. This affects the required amplifier bandwidth more than the Q-dependency shown in Figures 2 and 3, but this effect is highly dependent on the shape of the specific filter chosen.

 

      Below is an example of this effect. Consider a sixth order, 0.5 degree equal ripple phase low pass filter. This filter has an Fcutoff of 200 kHz and an overall gain of 10. We first use the step-by-step gain and step-by-step Q method, then the reverse method with step-by-step gain, and then record the estimated minimum amplifier bandwidth and gain bandwidth product in the table for each method. The latter is important to determine whether only VFAs can be used for these stages. This filter shape has a very good low overshoot step response at the output, but there will be some ringing and overshoot at each stage within the filter.

　　The design shown in Figure 4 is divided into three levels according to the idea of ​​reference 1. The level with the highest Q value is placed in the first level, with the lowest gain; the middle level has a slightly lower Q value and a slightly higher gain than the first level; the last level has the lowest Q value and the highest gain. It can be seen that Fo decreases from left to right and the gain increases gradually.

Figure 4. This gain and Q distribution more closely matches the bandwidth and slew rate requirements.

　　Alternatively, we can design in a more common way, setting the gain of the first stage to the maximum and decreasing it step by step. Generally speaking, this provides lower input referred noise. This is feasible for low frequencies, but it will not have as much impact on the overall output noise as expected. Figure 5 shows the same filter standard, with the three-stage gain ranked 5, 2, and 1 respectively, achieving the same overall filter shape. At the same time, the Q value and Fo ranking also show a trend from high to low from input to output.

Figure 5. Gain and Q assignments using a more common design flow.

　　Using the algorithm that generated the curve above, we can multiply the estimated minimum amplifier bandwidth by the gain of that stage to get the required gain bandwidth product for each stage and tabulate it. Although this algorithm is only mentioned in Reference 1, most design tools have similar calculations for the required amplifier bandwidth from Fo, Q, and gain, so the results are similar. We can also calculate the peak slew rate required for each stage. The specific calculations are described later. Our goal here is to achieve a final output swing of 4Vpp, and we can estimate the required peak slew rate by combining the nominal swing of each stage output and the increased dV/dT peak caused by the overshoot step response of the higher Q value stage.

Figure 6. Comparison of required gain-bandwidth product and slew rate.

　　The bandwidth requirements for each approach show noteworthy differences. It is clear that the bandwidth-gain product required for the second approach varies greatly (35:1, while the bandwidth-gain product for the first approach is essentially constant). In addition, the design flow presented in Reference 1 requires the use of faster devices in the first two stages (Figure 5) compared to the first design, because higher slew rates are now required in the first two stages (GBP is 61MHz for the ISL28191 and 7.7MHz for the ISL28114). The slew rate of the last stage remains the same, but the first two stages now require higher slew rates. The final output always achieves the same slew rate for a fixed target output step size and overall filter shape, but the first design requires lower peak slew rates in the first two stages, which is more desirable when the same amplifier is used in each stage.

　　While it may be possible to find some "optimal" algorithm to achieve the exact gain allocation to achieve a fixed gain-bandwidth criterion, the approximate method of allocating gain described in this article can easily be used to group the required amplifier bandwidth. For a variety of reasons, in the case of more than one filter stage, the gain of the high-Q stage should be kept low, and if a more consistent, moderate amplifier bandwidth criterion is required, more of the total gain should be allocated to the lower Q stages. This will also result in lower sensitivity of Q to Ko changes.

　　But what if the Q values ​​in Design 1 were ordered from high to low from input to output? This does seem to be more consistent in maintaining the required slew rate of the stages, and it may also show advantages in terms of clipping and overall noise.

　　Calculation of the peak slew rate required for the second-order step response

　　Each stage of a multistage active filter produces an output voltage transition, which requires limiting the maximum dV/dT. In this example dominated by the step response, if the dV/dT exceeds the rated slew rate of the selected op amp, the step response will usually deviate significantly from the expected value. Therefore, in a careful gain and Q ranking approach, we need to consider the implicit requirements of the internal and output stages for the peak dV/dt. Frequency-domain dominated applications will have implicit slew rate requirements for each stage output. In this case, it makes more sense to use the SFDR of the overall response as a criterion, but this discussion is beyond the scope of this article, but similar conclusions can be drawn to the step response dominated analysis used in this example.

　　Although the actual waveforms inside a multi-stage active filter can be very complex in the time domain, a reasonably conservative approach can be taken to analyze each stage as if it were excited by an ideal input step. Any real filter will produce inter-stage inputs with slower edge rates than the ideal step, giving us some design margin and allowing the use of simpler equations.

　　If we assume that any second-order low-pass filter stage will produce the target swing of Vopp when excited by an ideal input step, we can find the peak dV/dt by analyzing the output time waveform. Interestingly, the result is hard to find in the literature, but from Reference 4 (Equation 55), we can get a simple approximation, as shown in Equation 1.

　　Equation 1

　　If we know the ideal full step size at the output and the F-3dB of the second stage, this approximate equation can help us easily find the dV/dt peak. We can get F-3dB from the target filter pole by the following formula

　　Equation 2

　　Based on the examples given for Design 1 and Design 2, the F-3db values ​​for each stage are calculated as shown in Table 1. (Remember that we kept the Q and Fo ordering the same, we simply assigned different gains to make the step amplitudes inside the filter different)

Table 1.

　　Now let’s look at the highest Q stage, which will typically also have the highest Fo, or the largest bandwidth. If we can extrapolate a fixed maximum output Vpp criterion, we can get the results of Equation 1 to fall into a smaller range. The simplest approach is to reduce the required Vstep as the F-3db bandwidth increases. Design 1 actually does this by using a lower gain at the input stage. Design 2 shows the slew rate results (shown in Figure 6) without reducing the required Vstep for the higher Q stage (Figure 6). The slew rates for Designs 1 and 2 shown in Figure 6 are based on a target final output swing of 4Vpp. The slew rate at that output is calculated using Equation 1 (using a 2x multiplier for extra margin), then the step size from the previous stage into the last stage is calculated by dividing the stage nominal swing by the stage nominal gain, and then using that swing and the stage’s F-3db to calculate the required slew rate, and so on back to the first stage.

　　Each higher Q stage produces a significant overshoot in the step response. Design 1 avoids clipping by having lower swings in the stages with higher overshoot and increasing the step swing from input to output. The step response of Design 1 is shown in Figure 7 from a simulation run (Reference 1). This plot shows the input and output voltages through each stage of the filter. Note the gradual increase in swing and very low overshoot at the final output stage. This is typical of this type of filter configuration. It produces a +/-2V bipolar swing on a +/-2.5V bipolar supply.

Figure 7. Step response simulation of Design 1

　　In contrast, the step response of Design 2 clips between stages, resulting in a very undesirable simulated response. This is because the large gain of the first stage causes large swings early in the design. As shown in Figure 8, the macromodel correctly predicts clipping at the output of the second stage in the simulation, while the final stage cleanly filters it out.

Figure 8. Design 2's step response shows clipping.

　　This is significantly worse than Design 1. Although this is an extreme example, it does illustrate the importance of using the final stage gain to reduce the risk of interstage clipping.

 

        Consideration of the noise gain peak in Q value sorting of multi-stage filters

　　Anyone who has ever measured the output noise spectrum of an SKF filter will be somewhat surprised to see how high the noise peaks are. One of the most unnoticed characteristics of SKF filters is their high peak noise gain which can be mitigated to some extent by careful choice of resistor values.

　　The "noise gain" of an op amp circuit is the inverse of the ratio of the output voltage divider to the differential input voltage. This also gives the frequency response of the gain from the op amp's own input noise voltage to the output. It is also the ratio of the op amp's open loop gain to this noise gain, which is the loop gain at the passband frequency inside the SKF filter. This is a very useful factor, as it shows that the greater the loop gain, the lower the harmonic distortion. Therefore, the noise gain peak of the SKF should be understood and minimized for a number of reasons.

　　Equation 3 shows the basic form of the noise gain Laplace transfer function for the second-order low-pass SKF filter of Figure 1. The numerator is a quadratic polynomial with real zeros spread over a wide range (one less than ω0 and one greater than ω0), and the poles in the denominator are the poles required for the filter. This denominator expression is actually the pole equation for the passive 2 R and 2 C circuit before the gain element is introduced (removing the amplifier from the circuit in Figure 1 and grounding C2 gives the transfer function from input to output above C2, and the poles of this expression are the denominator of the SKF noise gain expression).

　　Equation 3 Noise Gain 

　　The denominator equation agrees with the desired filter response. If we substitute the Ko/R2C2 term in the linear coefficient in the denominator and rewrite the equation in terms of the filter specification, we see that we actually do not have much margin in the noise gain response.

　　Equation 4 Noise gain of filter index 

　　With the exception of the 1/R2C2 pole, every term in this equation is determined by the desired filter shape, DC gain, and Ko. This phenomenon can be used to reduce noise gain peaking to some extent, but it is still primarily controlled by the desired filter Q. In simple terms, it is best to keep the R1/R2 ratio of the SKF filter between approximately 0.15 and 0.7. It is mathematically correct to keep this ratio as close to zero as possible to reduce noise gain peaking, but R1 of 0 has other problems.

　　So the noise gain starts at Ko and ends at Ko over frequency. There is a large peak near ω0 due to the desired filter shape reflected by the pole equation (Q > .707 will form a peak in the desired frequency response) and the peak caused by the zero formed below ω0. An obvious question is whether the noise gain peaking inside the op amp loop will affect stability. It will affect stability when the peak is within the amplifier open loop response. However, even for the very low bandwidth margin conditions shown in Figures 1 and 2, the noise gain curve will intersect the open loop gain much higher than the peak shown below. But this issue illustrates the importance of not using too large an amplifier bandwidth and Fo multiplier. In theory, almost any amplifier bandwidth can be used and, by iteration, the resistor and capacitor values ​​required for the desired filter shape can be obtained. However, making the ratio of amplifier bandwidth to Fo too large will cause local loop stability problems.

　　Figure 9 shows an example of the noise gain magnitude of the first stage of Design 1. It uses the design algorithm of Reference 1, first controlling the values ​​of R1+R2 to make the noise added to the op amp itself almost negligible, and then setting the R1/R2 ratio within the range recommended above.

Figure 9. Noise gain magnitude of the first stage of Design 1.

　　Here the initial gain is 1.5V/V and the final noise gain is 3.56dB. At around the Fo frequency of 414KHz we find a dramatic peak with a gain increase of nearly 11.4dB, a whopping 15dB, or 3.7 times the required gain of the filter. The filter shape required for this stage only shows a peak from 3.56dB DC gain to a maximum gain of 8.44dB, far less than the noise gain peak hidden in this response. If this higher Q stage could provide more total filter gain, the entire curve would shift up. This will be demonstrated later.

　　It would be prudent to place a lower Q, lower Fo stage after this very high noise peak. This would filter these peaks and result in lower overall output noise.

　　Figure 10 shows the output noise of each stage of the filter in Design 1. It includes all the parameters of the design generated from Reference 1, the noise voltage, noise current, and resistor noise. The output of the first two stages has obvious noise peaks, but the final output noise peak is almost negligible because the Q value of the last stage is small. The noise peak here is 1.76µV/√Hz at 190KHz.

Figure 10. Output noise plot for Design 1.

　　Let's go back to Design 2, which uses the ISL28113 to complete the design. According to the noise graph below (the design will be dominated by the op amp noise voltage after setting the resistor, so this graph mainly reflects the voltage noise effect of the op amp) for a fair comparison of the output noise. The graph shows that there are obvious noise peaks in the first two stages, but due to the relatively low Fo and Q values ​​of the last stage, it has a good filtering effect. The graph shows that the noise peak at 200KHz is 1.49µV/√Hz, which is slightly lower than Design 1.

Figure 11. Output noise of Design 2.

　　So, in this case, putting more gain in the first stage does reduce the overall output noise slightly. But it is very important to place the lowest Q stage last. A design with the highest Q stage last will result in the highest overall noise. Even if the DC gain of that stage is only 1, its higher frequency peaking will result in an undesirable overall noise result.

　　If interested, the output noise Vpp of the two designs can also be calculated. The calculation method is to add the output noise power in the frequency range to get (Vrms)2, then take the square root and multiply the square root by 6 to get the approximate noise Vpp. Figure 12 shows the estimated measured output noise Vpp if the two filters are followed by a rectangular noise filter with a cutoff frequency on the x-axis.

Figure 12. Comparison of the combined output noise voltage of Design 1 and Design 2

　　For example, if we have a 200KHz sixth-order design followed by a 500KHz rectangular filter, we can measure about 5mVPP on Design 2 and 6mVPP on Design 1. Overall, this difference does not affect the noise advantages mentioned above for Design 1. So in this

　　case, putting more gain in the first stage can provide some noise advantages. But the idea here is to recommend putting the medium Q and medium gain stage first, followed by the highest Q and lower gain stage, and finally the lowest Q and higher gain stage. The automatic design algorithm provided in Reference 1, sorting from high to low Q and low to high gain, can achieve excellent characteristics in amplifier bandwidth, slew rate consistency, step response overshoot, and clipping performance, but in some cases there is a moderate increase in output noise.

　　in conclusion:

　　If one of the goals of a multistage lowpass active filter is to reduce the design margin requirements on the op amp while using the same amplifier in each stage, then it makes sense to assign lower gain to the higher Q stages. To avoid noise peaking in the final output, it is best to place the lowest Q stage last. To limit interstage clipping, this low Q stage should have a certain gain (more than 2 if possible) when the total filter gain is greater than 1. Increasing the gain of the first stage can slightly reduce the output noise, while placing the highest Q stage with low gain in the middle can slightly improve the noise performance.