# Manipulate analog filter gain to align op amps

-December 16, 2015

Every great active filter has a carefully selected operational amplifier to create an optimum solution. Understanding the relationship between the filter’s quality factor (Q), circuit gain (G), and amplifier gain-bandwidth-product (GBWP) will help implement the best design, where the amplifier’s bandwidths are carefully balanced.

In this article we use a single-supply, lowpass, fourth-order, Chebyshev filter with a 0.5 dB ripple allowed in the passband as our example circuit. The overall corner frequency of this filter is 5 kHz and the gain is 20 V/V (Figure 1).

Figure 1 5 kHz, fourth-order, lowpass, Chebyshev filter in a gain of 20 V/V

This formula defines the appropriate amplifier for each stage (equation 1):

min OpAmp BW => 100 * Gain * Q * (corner frequency)    (1)

In this formula, min OpAmp BW is the minimum allowed amplifier’s bandwidth to ensure the amplifier does not interfere with the filter’s response. Q is the filter’s quality factor. The unit-less Q value is associated with the damping rate (ζ) with equation 2:

Q = 1 / (2*ζ)    (2)

The design strategy for this circuit is to attempt to match the required bandwidth of both amplifiers. But before we get involved in the final design, let us take some time to become familiar with the Chebyshev lowpass filter.

Lowpass filter

Figure 2 shows the frequency response of a lowpass filter.

Figure 2 A 1 kHz lowpass, second-order filter in a gain versus frequency plot

At DC the filter’s gain is zero dB. The passband region encompasses an area where the filter gain remains at zero dB. Just before 1 kHz the signal starts to attenuate. At 1 kHz a signal is down by 3 dB below the zero dB level. Beyond 1 kHz, and in the stopband region, the signal starts to attenuate at a rate of 40 dB per decade. With a –40 dB per decade attenuation rate, this is a second-order lowpass filter.

The corner frequency definition of the Chebyshev filters differ for the other approximation types, such as the Butterworth, Bessel, linear phase, traditional Gaussian, and so on. Figure 3 shows the characteristics of the lowpass Chebyshev filter.

Figure 3 Lowpass Chebyshev AC filter responses.

For the Chebyshev lowpass filter, the two graphs in Figure 3 define the responses of even and odd order filters. Note the direction of the ripple signal (within the magnitude of Rp). For even order filters, the ripple rides on top of the DC gain of the filter. For odd order filters, the ripple rides underneath. In both graphs, there are also two definitions of the cut-off frequency. These definitions include passband ripple frequency (fp) and the –3 dB frequency (f–3 dB). In our example, we will use the Chebyshev –3 dB frequency.

Generating a fourth-order Chebyshev filter

The circuit for this example has the following specifications:
• Single supply = 0V to 5V
• Lowpass
• fourth order
• Chebyshev filter
• Rp = 0.5 dB
• Corner frequency = 5 kHz
• DC gain = 20 V/V

These specifications can generate several filters, depending on how you handle the gain in both stages. For instance, if the gain of both stages is equal to the square root of 20 or 4.472 V/V, Figure 4 shows the min OpAmp BW that occurs.

Figure 4 Passband gain in stage 1 and stage 2 are matched to 4.472 V/V to achieve 20 V/V for the entire circuit.

In Figure 4, the stage 1 and stage 2 min Op Amp GBW is equal to 941.026 kHz and 6.7817 MHz, inclusively. Appropriate amplifiers for both stages are:

• Stage 1 OPA170 1.2 MHz, 36V, single-supply, SOT553, low-power operational amplifier
• Stage 2 OPA192 Precision, rail-to-rail input/output, low offset voltage, low input bias current op amp with e-trim

If the gain stage 1 and stage 2 is equal to the 2 V/V and 10 V/V, inclusive, Figure 5 shows the min OpAmp BW that occurs.

Figure 5 Passband gain in stage 1 and stage 2 are 2 V/V and 10 V/V to achieve 20 V/V for the entire circuit.

In Figure 5, the stage 1 and stage 2 min Op Amp GBW is equal to 420.886 kHz and 15.164 MHz, inclusively. Appropriate amplifiers for both stages are:

• Stage 1 OPA170 1.2 MHz, 36V, single-supply, SOT553, low-power operational amplifier
• Stage 2  OPA350 38 MHz high-speed, single-supply, rail-to-rail operational amplifier

If the gain stage1 and stage 2 is equal to the 10 V/V and 2 V/V, inclusive, Figure 6 shows the min OpAmp BW that occurs.

Figure 6  Passband gain in stage 1 and stage 2 are 10 V/V and 2 V/V to achieve 20 V/V for the entire circuit.

In Figure 6, the stage 1 and stage 2 min Op Amp GBW are equal to 2.104 MHz and 3.032 MHz, inclusively. The appropriate amplifier for both stages is:

Stage 1   OPA2131 dual 4 MHz FET-input operational amplifier
Stage 2   OPA2131 dual 4 MHz FET-input operational amplifier

It looks like we have found a happy medium in that the OPA131 services both stages.

Conclusion

When you try to optimize the selection of your amplifiers in active analog filters, you will see success, especially if you implement a gain into a multi-stage system. We adjusted the gains of the stages and successfully saw changes in amplifier bandwidth requirements.

The task of generating an accurate lowpass filter is achievable, if you are willing to do a little research. There are many software applications that generate active analog filters, but few will go so far as to recommend appropriate amplifiers for the finished circuit. I used the Texas Instruments’ Filter Designer software to write this article.

References

Also see: